■'
-
■
■
- ■
\
■ <• . ::
-
.
V ' '
' ;;?'
**" ■ j.
•
|
1 |
OEJEj |
|
|
■ |
VA:;.
.m -« j-a
|
. |
|
|
;•' |
|
|
■ ^a s |
:
* i
i
.
i
^^—^^■■—■^^^■—^m^^— ii in lining — M^^^^M^^M^— ^I^M
JfniiHHIlillllllllilllilllliilliiiiillililiiiiiiiiiiiiiiiiiniikiiiimiiiiiiiiiiiiiiiiiiiiiiiiiifiiiiiiiiniiniiiiiiiiiiiiiHiiininiiiiiHHiiiiiiiiiHiiiiiiiiHiiiiiii!
PROBLEMS OF RELATIVE GROWTH
HETEROCIfELY AND REGENERATION IN THE COMMON LOBSTER
i. Marine Lobster, Homarus (Gammarus) vulgaris, Normal Left-handed Specimen.
2. The Same Regenerating the Left 'Crusher' Claw after Autotomy.
3. Shed Skin of the Same at the Next Moult.
4. The Same after this Moult, showing Direct Regeneration of Left Crusher.
Note the slenderer propus and absolutely longer dactylus in the smaller (' nipper ') claw.
PROBLEMS
OF RELATIVE GROWTH
BY
JULIAN S. HUXLEY, M.A.
Honorary Lecturer in Experimental Zoology, King's College, London
WITH IO5 ILLUSTRATIONS
LINCOLN MAC VEAGH THE DIAL PRESS NEW YORK • MCMXXXII
PRINTED IN GREAT BRITAIN
TO
D'ARCY WENTWORTH THOMPSON
' The morphologist, when comparing one organism with another, describes the differences between them point by point, and " character " by " character." If he is from time to time constrained to admit the existence of " correlation " between characters (as a hundred years ago Cuvier first showed the way), yet all the while he recognizes this fact of correlation somewhat vaguely, as a phenomenon due to causes which, except in rare instances, he cannot hope to trace ; and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point, and character by character. But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different co-ordinate systems, this fact will of itself constitute a proof that a com- prehensive " law of growth " has pervaded the whole structure in its integrity, and that some more or less simple and recog- nizable system of forces has been at work.' — D'Arcy Thompson (Growth and Form, p. 727).
vn
PREFACE
IN this book I have attempted to give some account of the chief results emerging from a study of the relative growth of parts in animals which I have undertaken during the last ten years. I have tried to correlate my own findings and conclusions with those of other workers in the same and related fields, but am well aware of the many gaps that remain. However, it has not been my main intention to produce an exhaustive survey of the subject, but rather to set forth certain new facts and ideas and some of their chief implications.
There are, I think, four chief points in the book which are more or less new. One is the quantitative formulation of heterogonic growth (Chapters I and II) ; a second is the dis- covery of the widespread existence of growth-gradients, and their quantitative analysis (Chapters III and IV) ; a third is the recognition that growth of logarithmic spiral type as seen in Molluscan shells, etc., operates with the same growth- mechanisms (growth-centres and growth-gradients) as does growth of ordinary type as seen in a Crustacean antenna or a sheep's leg (Chapter V) ; and the fourth is the application of these results to certain evolutionary problems, as set forth in the final chapter.
I owe a great deal to previous work in this field : first and foremost to D'Arcy Thompson's Growth and Form, but also to the books and papers of Champy, Teissier, Schmalhausen , and others too numerous to mention.
I have to thank Professor L. T. Hogben and Dr. R. A. Fisher, F.R.S., for reading the book in typescript and making various useful suggestions ; and I owe a great deal to Pro- fessor H. Levy for helping me with some of the mathematical problems involved. My thanks are also due to Dr. C. F. A. Pantin and Professor Selig Hecht, whose discussions with me of various problems raised in this book have helped greatly in clarifying my ideas. And especially I would like to thank my pupils and co-workers, Mr. E. B. Ford, Mr. F. N. Rat- cliff e, Miss M. Shaw (Mrs. White), Miss M. A. Tazelaar, Mr.
ix
x PROBLEMS OF RELATIVE GROWTH
S. F. Bush, Professor F. W. Kunkel, Mr. J. A. Robertson, Miss I. Dean, Mr. A. S. Edwards and Mr. F. S. Callow, without whose collaboration I should never have been able to collect and analyse the data on which this treatment of the subject is founded. Finally, I must not forget my secretary, Miss P. Coombs, whose aid has been invaluable in preparing the book for the press.
Many of the figures have been drawn for this book. As regards the others, I would like to express my thanks for the willingness of the authors and publishers concerned for allow- ing me to reproduce them. Acknowledgements have been made in the list of illustrations : the citations there made refer to the literature list for fuller details.
JULIAN S. HUXLEY King's College, London December, 1931
ERRATUM
Page 84, Fig. 46, legend. For " dactylus ; ischium " read " dactylus ; carpus ; ischium "—problems of relative growth.
CONTENTS
PAGE
Preface ......... ix
List of Illustrations. ...... xiv
CHAPTER I CONSTANT DIFFERENTIAL GROWTH-RATIOS
§ i. Introductory ........ i
§ 2. Constant Differential Growth-Ratios ... 4
§ 3. Examples of Constant Differential Growth-Ratios. 13
§ 4. Inconstancy of Form and Constancy of Form-change 38
CHAPTER II
THE COEFFICIENT OF CONSTANT GROWTH- PARTITION ; AND SOME SPECIAL CASES
§ 1. The Heterogony of Deer Antlers .... 42
§ 2. The Coefficient of Constant Growth-partition . 49
§ 3. Heterogony in Holometabolous Insects 55 § 4. Heterogony and Polymorphism in Neuter Social
Insects ......... 61
§ 5. Heterogony, Moulting, and Dimorphism ... 68
CHAPTER III GROWTH-CENTRES AND GROWTH-GRADIENTS
§1
§2
§3
§4 §5 §6
§7 §8
Growth-gradients within Single Organs ... 79 Steepness of Growth-gradient within an Organ and
Growth-intensity of the Organ as a whole . . 83 Reversal of the Sign of the Growth-gradient in
Negative Heterogony ...... 87
The Form of Growth-gradients. . . . .90
Growth-gradients in Regions of the Body . . 92 Graded Growth-intensity in the Different Planes
of Space ......... 95
Gradients in Growth-rate of Epidermal Structures 100
Conclusion ......... 102
xi
xii PROBLEMS OF RELATIVE GROWTH
PAGE
CHAPTER IV
GROWTH-GRADIENTS AND THE GENERAL
DISTRIBUTION OF GROWTH-POTENTIAL IN
THE ANIMAL BODY
§ i. General Growth-gradients : D'Arcy Thompson's
Graphic Method . . . . . . .104
§2. General Growth-gradients: Quantitative Analysis iio
§ 3. The Two Phases of Growth . . . . .118
§ 4. Growth-changes Correlated with High Local Growth- intensity ........ 120
§ 5. Some Cases of Teratological and Abnormal Growth 128
§ 6. The Law of Antero-posterior Development and its
Effect upon Growth ...... 132
§ 7. The Mathematical Formulation of Relative Growth
in Embryonic Life ....... 139
§ 8. Conclusion ......... 147
CHAPTER V
GROWTH-CENTRES AND GROWTH-GRADIENTS IN ACCRETIONARY GROWTH
§ 1. The Accretionary Method of Growth . . . 149
§ 2. Logarithmic Spirals as the Result of Growth- gradients . . . . . . . .151
§ 3. Growth-gradients and the Shells of Molluscs . 154
§ 4. Conclusion ......... 163
CHAPTER VI
HETEROGONY, GROWTH-GRADIENTS AND PHYSIOLOGY
§ 1. Normal Proportions as Result of a Partition-equili- brium ......... 165
§ 2. The Initial Determination and Physiological Basis
of Growth-gradients . . . . . .167
§ 3. Other Gradient Theories . . . . .170
§ 4. Heterogony and Hormones . . . . .176
§ 5. Heterochely and Relative Growth-rates. . . 1S9
§ 6. Specific Growth-intensities and their Interaction . 191
CONTENTS
Xlll
§ 7. The Influence of External Conditions
§ 8. Rhythmical Irregularities of Growth-ratio
page 197 203
§ 1.
§2. §3- §4- §5- §6.
§7-
§8.
CHAPTER VII
BEARINGS OF THE STUDY OF RELATIVE GROWTH ON OTHER BRANCHES OF BIOLOGY
Heterogony and Taxonomy : Sub-species and Taxo-
nomic Forms ........ 204
Heterogony in Groups Higher than the Species . 212
Heterogony and Evolution . . . . .216
Heterogony and Comparative Physiology . . 224
Relative Growth and Genetics . . . .229
Relative Growth, Embryology, and Recapitulation 234
General Approach to the Problem of Qualitative
Form-change ........ 240
Conclusion ......... 243
Bibliography ........ 245
Addenda . . . . . . . ... 257
Index of Authors ....... 267
Subject Index ........ 270
Index of Organisms ....... 274
LIST OF ILLUSTRATIONS
FIG. PAGE
Heterochely AND REGENERATION IN THE LOBSTER
Frontispiece (From Przibram, 1931) i. Diagram illustrating some quantitative aspects of
heterogony ........ 5
2. Changes in absolute and percentage weight of the
large claw during growth in fiddler-crabs . 9
3. heterogony of the large claw in 4oi male fiddler-
CRABS ......... IO
(From Huxley, 1927k) ' 4. HETEROGONY OF STEM-WEIGHT AGAINST ROOT-WEIGHT IN
VARIOUS PLANTS . . . . . . .13
(From Pearsall, 1927)
5. HETEROGONY OF PETIOLE LENGTH AGAINST LAMINA DIA-
METER in Nasturtium leaves . . . . -14
(From Pearsall, 1927)
6. Changes in relative abdomen-breadth during the
growth of the shore-crab . . . . .15
7. heterogony of abdomen-breadth in 625 shore-crabs io
(After Huxley and Richards, 1931)
8. HETEROGONY OF THE CHELA IN THE PRAWN PALAEMON MAL-
COMSONI . . . . . . . . 17
g. heterogony of face relative to cranium in sheep- dogs and baboons . . . . . . . 18
10. Baboon skulls of various ages to show change in pro-
portions ......... 19
(From Zuckerman, 1926)
11. HETEROGONY OF TAIL-LENGTH IN THE MOUSE PHENACOMYS 22
12. HETEROGONIC RELATION OF DORSAL AND VENTRAL EYE-
LOBES IN THE BAR-EYED MUTANTS OF DROSOPHILA . 22
(Modified from Hersh, 1928)
13. HETEROGONY OF INTER-OCULAR DISTANCE IN CRABS . 23
(After Teissier, 1931)
14. Change of proportions in shore-crab ... 24
(From unpublished drawings kindly supplied by Dr. G. Teissier)
15. HETEROGONY OF THORACIC GANGLION IN CRABS . . 25
(After Teissier, 1931)
16. HETEROGONY OF WATER-CONTENT IN TENEBRIO LARVAE 26
(After Teissier, 1931)
17. HETEROGONY OF NITROGEN-CONTENT IN TENEBRIO LARVAE 26
(After Teissier, 1931)
18. HETEROGONY OF PHOSPHORUS-CONTENT IN TENEBRIO
LARVAE ......... 28
(After Teissier, 1931)
xiv
LIST OF ILLUSTRATIONS xv
FIG. PAGE
ig. Heterogonic relation of heat of combustion to body- weight in Tenebrio larvae ..... 28
(After Teissier, 1931)
20. Heterogony of water-content in wax-moth larvae
(Galleria) . . . . . . . -3i
(After Teissier, 1931)
21. Heterogony of male and female chelae in two species
of prawns (palaemon) ...... 33
(A, from Tazelaar, 1931)
22. Variation in the relative growth of the female
abdomen in fiddler-crabs ..... 35
(A, from Morgan, 1923 ; B, after Huxley, 1924)
23. Relative growth of male and female abdomen in
spider-crabs (inachus) ...... 36
(From Shaw, 1928)
24. ISOGONIC GROWTH OF PARTS IN THE FISH ORTHOPRISTIS . 37
(From Hecht, 1916)
25. Heterogony of antler-weight in adult Red Deer . 43
(From Huxley, 1931A)
26. Antler-weight against body-weight by age in Red
Deer ......... 44
(From Huxley, 1931A)
27. Negative heterogony of antler-weight in Roe Deer 46
(From Huxley, 1931A)
28. Diagram of antler-growth in Red and Roe Deer . 47
(From Huxley, 1931A.)
29. Body-weight and antler-weight against age, Red
Deer ......... 48
(From Huxley, 1931A)
30. Decrease of growth-coefficient during regenera-
tion, Sphodromantis ...... 50
(Modified from Przibram, 1917)
31. Regulation of size in grafted eyes of Amblystoma . 52
(From Twitty, 1930)
32. Interrupted heterogony in the male chela of spider-
crabs (Inachus) ....... 53
33. Heterogony of the fore-limb in the beetle Euchirus 56
(From Champy, 1924)
34. Heterogony of the ' tail ' appendage of the wting in
the butterfly papilio dardanus . . . -57
(From Champy, 1924)
35. Heterogony of the male mandible in three species
of stag-beetles (Lucanidae) ..... 58
(From Huxley, 1931c)
36. Change of proportions with increased size in neuter
ants (Pheidole) ....... 63
(From Wheeler, 1910)
37. Heterogony of head-size in neuter ants (Anomma
and Camponotus) ....... 64
38. Change of proportions with increased size in neuter
termites (Termopsis) ...... 66
(From Heath, 1927)
39. Dimorphism of female abdomen and male chela in
spider-crabs (Inachus) ...... 69
(From Shaw, 1928)
xvi PROBLEMS OF RELATIVE GROWTH
FIG. PAGE
40. Dimorphism and heterogony of the male forceps in
earwigs (forficula) . . . . . -71
(From Huxley, 1927s)
41. Diagram of the possible origin of forceps-dimorphism
in male earwigs . . ..... 73
(From Huxley, 1931c)
42. Differential effect of adverse conditions on
forceps-length and body-length in male earwigs 75 (From Huxley, 1927s)
43. Dimorphism and heterogony in the 3RD leg of the
mite Analges ........ 77
(From Jucci, 1924)
44. Difference in growth-coefficients of various regions
of the large claw of fiddler-crabs . . . 80
45. Growth-gradients within the male chela of crabs
(uca and maia) ....... 83
46. Changes in proportion of parts during growth of the
chela in prawns (palaemon) ..... 84
(From Dean, unpublished)
47. Growth-gradients in the antennae of Copepods . 86
48. Growth-gradients and evolutionary change in the
feet of Ungulates ...... 88
(From D'Arcy Thompson, 1917)
49. Reversed growth-gradient in the limbs of sheep . 89
(From Huxley, 1931B)
50. Change in form of growth-gradient with change of
growth-rate in the large claw of hermit-crabs (eupagurus) . . . . . . . .91
(After Bush, 1930)
51. Growth-gradients in pereiopods and chelae of prawns
(Palaemon) ........ 92
(From Tazelaar, unpublished)
52. Growth-gradients in the abdomen of crabs (Pinno-
theres AND TELMESSUS) ...... 93
(From Huxley, 1931B)
53. Female pea-crabs (Pinnotheres) of various sizes,
showing changes in proportion of the abdomen during growth ....... 94
(From Atkins, 1926)
54. Relative growth in the three planes of space in the
crusher and nipper claws of lobsters ... 97
55. Relative growth in length and breadth of the chela
in females, males and parasitized males of upogebia 98
56. Gradients in feather-growth in the fowl . . 101
57. Growth-gradients and evolutionary change in fish
(Diodon and Orthagoriscus) ..... 105 (From D'Arcy Thompson, 1917)
58. Cartesian transformations of the carapace of
various crabs . . . . . . .107
(From D'Arcy Thompson, 1917)
59. Reconstructions of evolutionary stages in the avian
pelvis, by the method of Cartesian transformation 108 (From D'Arcy Thompson, 1917)
LIST OF ILLUSTRATIONS xvii
FIG. PAGE
60. Reconstructions of evolutionary stages in the evo-
lution OF THE HORSE SKULL, BY THE METHOD OF CAR- TESIAN TRANSFORMATION ; AND COMPARISON WITH ACTUAL FOSSIL SKULLS . . . . . log
{From D'Arcy Thompson, 1917)
61. Growth-gradients along the body- axis of the hermit
crab eupagurus . . . . . . .112
{From Bush, 1930)
62. Growth-intensities of various parts in male and
female Stag-beetles (Lucanus) . . . .114
63. Growth-gradients of male and female stag-beetles 115
64. Growth-profile of metamorphosing herring . .116
{From Huxley, 1931B)
65. Effect of a region of high growth-intensity on
growth of neighbouring parts in spider-crabs (Maia and Inachus) ...... 122
{A, from Huxley, 1927 a ; B, from Shaw, 1928)
66. Effect of a region of high growth-intensity on
growth of neighbouring parts in prawns (Palae- mon) . . . . . . . . .124
{A, from Shaw, 1928 ; B, from Tazelaar, 1930)
67. Relative growth of parts anterior and posterior to
a region of high growth-intensity in prawns (Palaemon) ........ 125
{From Tazelaar, 1930)
68. Effects of a regenerating limb upon the growth of
neighbouring limbs in Sphodromantis . . . 127
69. Graded growth-effects in two human monsters . 130
{A, from Nanagas, 1925 ; B, from Mead, 1930)
70. Change in relative weight of various organs of the
cat during fetal life . . . . . -134
{After Latimer and Aikman, 1931)
71. Positive heterogony of head-length in whalebone
WHALES ......... 137
72. Changes in relative weight of various organs of the
chick during embryonic life .... i43
{After Schmalhausen, 192713)
73. Growth-rates of various organs of the chick during
embryonic life ...... i44-i45
74. Diagram illustrating the co-operation of two
growth-ratios in determining the form of mol- luscan shells . . . . . . 1 56
75. Origin of the spiral form of the shell in Limacina . 159
{From original by Dr. Lebour)
76. Diagram of the growth-gradients operating to pro-
duce THE TURBINATE SPIRAL SHELL OF MOLLUSCS . l6o
77. Heterogony of normal and regenerating claws in
portunus ........ 166
78. Asymmetry in the thoracic ganglia of the male fid-
dler-crab ........ 168
{From Ratcliffe, unpublished)
79. Chemical and metabolic gradients in Crustacea and
earthworms . . . . . . . .170
{From Perkins, 1929)
xviii PROBLEMS OF RELATIVE GROWTH
FIG. PAGE
So. Disproportionate effect of starvation on a hetero-
GONIC ORGAN (DORSAL CREST IN MALE TRITON) . . l8o
(From Champy, 1924)
81. Disproportion of limbs caused by precocious meta-
morphosis in the frog ...... 182
(Modified from Wells, Huxley and Wells, 'The Science of Life,' London, 1931)
82. Differential effect of thyroidectomy on the growth
of various organs of the rat . . . 1 85
(From Hammett, 1929)
83. Effect of thyroidectomy on the growth of the hypo-
physis IN THE RAT . . . . . . I 87
(From Hammett, 1929)
84. Diagram illustrating Przibram's hypothesis of the
effect of differential growth-rates on chela- reversal in heterochelous crustacea . . . igi
85. Growth of normal and grafted eyes in two species
of Amblystoma ...... 192-193
(From Twitty and Schwind, 1931)
86. Relative growth of eye in two species of Ambly-
stoma ......... 194
(From Twitty and Schwind, 1931)
87. Interaction of parts of the eye in Amblystoma . 196
88. Regulation of growth in eyes grafted on to indi-
viduals of different age ..... 198
(From Twitty, 1930)
89. Changes in relative weight of different parts of the
body caused by under-nourishment in rats . . 202
(From Jackson, 1925)
90. Comparison of absolute size and proportions of
modern and prehistoric scottish red deer . . 206
(After Ritchie, 1920)
91. The female and five different forms of male in the
stag-beetle cyclommatus ..... 209 (After Dudich, 1923)
92. Relative growth of mandible in the five male forms
of Cyclommatus . . . . . . .211
(From Huxley, 1931c)
93. Change of form with change of absolute size in the
mandible of male stag-beetles . . . .212
(From Grijfini, 191 2)
94. Heterogony in groups larger than the species : PHE-
NOMENON OF LAMEERE IN THE GENUS GOLOFA . 213
(After Champy, 1929)
95. Specific variations in the detail of a heterogonic
organ (cephalic horn) in goliath beetles . . 217
(From Champy, 1924)
96. Developmental changes of proportion in wild and
domestic sheep ....... 223
(After Hammond, 1928)
97. Graph showing relation of egg-weight to body-weight
in 432 species of birds ...... 226
(From Huxley, 1927c)
LIST OF ILLUSTRATIONS xix
FIG. PAGE
98. Diagram showing the effect of rate-genes on eye-
pigmentation in Gammarus ..... 228 (From Wells, Huxley and Wells, 'The Science of Life,' London, 1931)
99. Rate of eye-darkening in two genetic strains of
Gammarus ........ 230
(From Ford and Huxley, 192Q)
100. Effect of rate of general growth on rate of eye-
darkening IN A PURE STRAIN OF GAMMARUS . . 230
(From Ford and Huxley, 1920)
101. Multimodal frequency-distribution of body-build
index in man ........ 233
(From Davenport, 1923)
102. Diagram illustrating the effect of rate-genes upon
vestigial organs ....... 236
103. Effect of temperature on rate of eye-darkening in
a pure strain of gammarus ..... 238 (From Ford and Huxley, 1929)
104. Diagram illustrating the relation of mutations in
rate-genes to recapitulation and paedomorphosis 24o
PROBLEMS OF RELATIVE
GROWTH
CHAPTER I CONSTANT DIFFERENTIAL GROWTH-RATIOS
§ i. Introductory
THE problem of differential growth is a fundamental one for biology, since, as D'Arcy Thompson especially has stressed (1917), all organic forms, save the simplest such as the spherical or the amoeboid, are the result of dif- ferential growth, — whether general growth which is quantita- tively different in the three planes of space, or growth localized at certain circumscribed spots. But the subject has received little consideration. D'Arcy Thompson's own treatment, though exhaustive on certain points (e.g. the logarithmic spiral), profoundly original and important in others (e.g. his use of Cartesian transformations to illuminate the evolution of one form from another), and interesting throughout, is admittedly incomplete. Certain large bodies of data, such as those included in Donaldson's The Rat (1924) and in various treatises on physical anthropology, e.g. R. Martin (1928 ) exist on differential growth in mammals, but have so far not been analysed save by the use of purely empirical formulae ; Champy (1924) has written a very stimulating book on differential growth of such extreme type as to warrant the term ' dys- harmonic ', and has later given further examples (1929) ; Przibram has recently (1930) collated some of his interesting results and ideas. But, apart from this, little that is con- nected or general has been written on the subject ; and even the individual papers dealing with the topic are few and on the whole disconnected.
Since 1920 I have been studying certain phases of the problem : the purpose of the present review is to bring to- gether the various aspects which have presented themselves,
2 PROBLEMS OF RELATIVE GROWTH
to demonstrate the existence of certain broad empirical laws which appear to govern most cases of differential growth so far studied, to discuss their bearing on other branches of biology, and to point the way to further attack on the subject by those trained in other methods.
The first step, it appeared to me, was to study a number of clear-cut cases of differential growth and to see whether they were capable of quantitative expression. My own mathe- matics are regrettably deficient, but I was able (see Section 2) to obtain a simple formula which appears to be at any rate a first approximation to a general law for differential growth. Among many morphologists and systematists there appears still to linger a distrust of the application of even such element- ary mathematics to biological problems. The usual criticism is that the formulae arrived at may have a certain convenience, but can tell us nothing new, and nothing worth knowing of the biology of the phenomenon. This appears to me to be very ill-founded. In the first place, to have a quantitative expression in place of a vague idea of a general tendency is not merely a mild convenience. It may even be a very great convenience, and it may even be indispensable in making certain systematic and biological deductions. But further, it may suggest important ideas as to the underlying processes involved ; and this is precisely what the quantitative analysis of relative growth is doing. As will be seen in this and the subsequent chapters, there are certain hypotheses which square with the formula, others which do not : without the quanti- tative expression, we should be largely theorizing in the air. I would not trouble to spend my time on this point if it had not been urged on several occasions in my hearing ; other- wise, one would expect that the interaction of quantitative theory with observation and experiment devoted to testing the theory, so fruitful not only in other sciences but in genetics within the field of biology, would automatically be welcomed.
Furthermore, the establishment of one quantitative rule leads on to the discovery of others. Chapters I and II will be devoted to showing that, when we consider the growth of whole organs relative to the rest of the body, the results can be understood if we postulate that the ratio between the intensity (or relative rate) of growth of the organ and that of the body remains constant over long periods of the animal's life. To borrow a term from another branch of science, there
INTRODUCTORY 3
is a constant partition-coefficient of growth-intensity between organ and body. It was next found that in many organs, especially those growing at markedly different rates from the body as a whole, growth-intensity was not distributed uni- formly, but in a more or less regular pattern. This led on to the notion, already suggested on different grounds by D'Arcy Thompson, that the growth-intensity of the body as a whole (or, if you prefer it, the relative growth-rates of its various parts) is distributed according to an orderly system of ' growth-gradients '. These conclusions will be discussed in Chapters III to V.
The physiological mechanism underlying these general rules still remains very obscure, in the absence of experiment specifically directed to the point : but there are some inter- esting hints and possibilities, and these will be discussed in Chapter VI.
Finally, the facts derived from the study of relative growth have a number of important bearings upon other branches of biology ; and the concluding chapter will be devoted to these. I hope to convince the systematist that by a knowledge of the laws of relative growth we are put in possession of new criteria bearing on the validity of species, sub-species, and ' forms ' ; the nature of certain dwarf forms ; and the import- ance (or the reverse) of size-differences in general for system- atics. In regard to that special branch of systematics usually called physical anthropology, it will be found that these laws have a bearing on the important question as to whether true evolutionary change has taken place in civilized populations during historical time. As regards evolution, it will be found that the subject throws light upon the question of adaptation, on the general theory of orthogenesis, and on the selection problem. Furthermore, the existence of growth-gradients, as D'Arcy Thompson has already pointed out, makes it much easier for us to understand how certain types of evolutionary transformation can have been brought about, since a single genetic change affecting a growth-gradient will automatically express itself in a changed relation in the size of a large number of organs or regions.
Then comparative physiologists will find it necessary to know precisely how to discount the effects of differences in total absolute size when they wish to estimate the compara- tive development of an organ in a series of related species or groups ; and will further find interesting hints as to the
4 PROBLEMS OF RELATIVE GROWTH
nature of factors which tend to limit the size of an organ at high absolute sizes.
Nor can genetics be left out. A constant partition of growth-intensity between different regions implies constant differences in their rates of growth. Thus any genes control- ling relative size of parts will have to exert their action by influencing the rates of processes, and so fall into line with the numerous other rate-factors whose importance has been summarized by Goldschmidt (1927) and by Ford and Huxley (1929). The fact, however, that the ratios between growth- rates, and not their absolute values, are the determining factors introduces certain complications, whose discussion will be found to have an interesting bearing upon the analysis of other genetic ' characters '.
Finally, the ancient problem of embryological recapitulation will be found to be illuminated from a new angle ; and many undoubted cases of recapitulation will be found to owe their origin not to any mysterious phyletic law, but to embryological convenience, adjusting evolutionary changes in the size of an organ to the general rules of relative growth during individual development.
This brief introductory sketch will, I hope, have shown some of the chief points of interest in the study of relative growth. We must now come to grips with the subject, and for the reasons above stated propose to do so by con- sidering what at first sight seems a rather arid point — the quantitative expression of the relation between the body as a whole and an organ whose proportionate size changes during life.
§ 2. Constant Differential Growth-ratios
Champy (1. c.) and others have pointed out that certain organs increase in relative size with the absolute size of the body which bears them ; but so far as I am aware, I (Huxley, 1924B) was the first to demonstrate the simple and significant relation between the magnitudes of the two variables. In typical cases, if x be the magnitude of the animal (as measured by some standard linear measurement, or by its weight minus the weight of the organ) and y be the magnitude of the dif- ferentially-growing organ, then the relation between them is y = oxk, where b and k are constants.1 The constant b is
1 This can also be written log y = log b + k log x, which means that any magnitudes obeying this formula will fall along straight lines if plotted on a double logarithmic grid.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 5
here of no particular biological significance, since it merely denotes the value of y when x = 1 — i.e. the fraction of x which y occupies when x equals unity. We may call it the
3000
I
^2000
I
•is
"a 1000
500
100
|
k=20 |
1 ' ■ ■ 1 |
|
|
b = |
||
|
- OO 003 |
||
|
h = |
||
|
f® /•/ x I0'5 |
||
|
® /Ox/0"5 |
||
|
/ |
||
|
k -27 < |
®09*I0'S d |
|
|
O0-002 |
||
|
b - /0~5x/-7+\ |
1 1 1 1 1 1 f |
|
|
t . \H+tr |
||
|
- O 0-001 |
b - /0 x i/o 9 / t / |
|
|
■ O00005 |
/ 05 + S |
|
|
/k = l3 (8/0 )x/0~5= h S [(DOS ) 0 |
||
|
< |
||
|
o-oooi s eJOdbOOl |
J 000 5000 10,000 15,000
size of body
-3000
2000
1000
Fig. 1. — Diagram to show the quantitative effect of varying the constants
in the simple heterogony formula, y = bxk, assuming that the origin of growth
in x and y begins at the same time.
The dotted line gives the growth of the organ (y) when k = 2-0 and b = -ooooi. The points to the left show values of y for different values of b when the rest-of-body is of size 1000. Those to the right show the effects, at body-size io,ooo, of varying both k and b.
6 PROBLEMS OF RELATIVE GROWTH
fractional coefficient. But the value of k has an important meaning.
It implies that, for the range over which the formula holds the ratio of the relative growth-rate of the organ to the relative growth-rate of the body remains constant, the ratio itself being denoted by the value of k. By relative growth-rate is meant the rate of growth per unit weight, i.e. the actual abso- lute growth-rate at any instant divided by the actual size at that instant.
This is at once seen by plotting the logarithm of y against
the logarithm of x. In unit time the increase in the logarithm
of y is k times the increase in the logarithm of x, which may
be written :
d , ■, d ,
g.logy-Jglog*
dy , ,dx ,
ily - kTtlx
This formula, on which I have had the advantage of con- sulting Professor Levy, of the Imperial College of Science, can be deduced on the basis of very simple assumptions about growth in general. One essential fact about growth is that it is a process of self-multiplication of living substance — i.e. that the rate of growth of an organism growing equally in all its parts is at any moment proportional to the size of the organism. A second fundamental fact about growth is that the rate of self -multiplication slows down with increasing age (size) ; a third is that it is much affected by the external environment, e.g. by temperature and nutrition. The two latter considerations affect all parts of the body equally, so that we may suppose that the growth-rate of any particular organ is proportional simultaneously (a) to a specific constant characteristic of the organ in question, (b) to the size of the organ at any instant, and (c) to a general factor dependent on age and environment which is the same for all parts of the body.
If y stand for the size of the organ, and x for that of the rest of the body, we shall then have
— = oaG and -- = #yG, dt at
where a and /? are the specific constants for the rest of the body and for the organ in question, and G measures the
CONSTANT DIFFERENTIAL GROWTH-RATIOS 7
general conditions of growth as affected by age and environ- ment,1 then — = — •
ax ax
Thus log y = - log x + log b, where b is a constant : i.e.
y = bxp/a.
And fi/a, which can also be written k, is a constant, and is also the ratio of the specific components of the growth-rates of y and x respectively.2
I am, of course, aware that the existence of growth-cycles and other facts make it impossible to suppose that the ex- pression for change of growth can be so simple as here set forth. We must suppose that each cycle may have its own general and specific components of the growth-rate — i.e. that a, /5 and y may change comparatively abruptly during the life-cycle, and also it is quite possible that other inherent alterations, such as the gradual increase of viscosity of proto- plasm with age (Ruzicka, 1921), will cause gradual and pro- gressive diminution of the specific constants which would account for the various distortions of the S-shaped curve of growth from the form expected on the simplest assumptions. But I am convinced that some such general method of en- visaging growth is sound ; and it is interesting to find our empirical formula for constant differential growth-ratios deducible from it. (See also Schmalhausen, 1927B, 1930.)
Exactly the same formula would apply to two sums of money put out at different rates of compound interest, pro- vided that they were not accumulating discontinuously by quarterly or annual interest payments, as in financial fact, but continuously, as in the Compound Interest Law, and as in biological growth, k would here denote the ratio of the
1 One might expect, from certain experimental data, that the factor G would be a simple function of the defect of the size of the organism at any given time from its final size ; but this would not interfere with the validity of our more general formula.
2 It may well be that the ' general factor ' is not capable of such a simple formulation. But provided that such a general factor does exist — i.e. that the growth both of organ and of rest-of-body is related to some general law of growth affecting the organism as a whole, the deduction of constant differential growth-ratios remains valid. And that such a relation does exist is shown by the work of Przibram, Harrison and others discussed in Chapter VI.
8 PROBLEMS OF RELATIVE GROWTH
two rates of interest. (In our biological parallels, we know nothing of the actual rates of growth, for since the organ and the body have both existed for the same length of time when we measure them, the time-factor cancels out, in point of fact.)1 The actual rates, unlike those for the two sums of money, will obviously be altering continuously; they will be high in youth, low in age ; increased by high temperature, decreased by low ; and so forth. What concerns us is that if our formula holds, the ratio of the relative rates of growth remain constant.
In such cases, therefore, we have a constant differential growth-ratio , denoted by the value of k. If we prefer to con- centrate upon the growth of the organ relative to the growth of the body considered as a standard, then we may speak of k as denoting the growth-coefficient of the organ. An organ which is thus growing at a different rate from the body as a whole may be called heterogonic, to use the convenient term coined by Pezard (1918). If it is growing at the same rate as the body it must be styled isogonic ; as will be apparent, isogony is merely a special case of heterogony, as the circle is a special case of the ellipse.
It is clear that comparatively small variations in the value of k will have large results provided that growth continues over a considerable range of size. An attempt to show this graphically has been made in Fig. 1.
The best worked-out example of this law so far concerns the large chela of male fiddler-crabs, Uca pugnax (Huxley, 1927A). This obeys the law of constant growth-ratio from crabs of only about 60 milligrams total weight to the largest found, weighing sixty times as much ; the value of k, however, changes quite abruptly at about i-i g. total weight, a point which probably denotes the onset of sexual maturity, decreasing here to less than 80 per cent, of its value for the earlier growth- phase. (It is a noteworthy and unexpected fact that the growth-coefficient of this secondary sexual character should be reduced instead of increased when the gonad begins to function.) See Figs. 2 and 3.
In our examples of the fiddler-crab, the weights of chela and rest-of-body behave, over the earlier and longer growth-phase, like £2 and £100 put out at 8 per cent, and 5 per cent, (continuous) compound interest respectively 2, and a calculation on this basis
1 For cases where an organ is laid down considerably later than the body as a whole, see Chapter IV.
2 Strictly speaking, of course, 8-i and 5 per cent, (see p. 10).
CONSTANT DIFFERENTIAL GROWTH-RATIOS 9
will reproduce the actual figures for weight.1 But we can be perfectly sure that the actual growth-rate of the crab and of its claw slows off with age, that it differs in summer and winter, and is further subjected to all kinds of irregular fluc- tuations due to temperature, food and other factors. The actual rates may be as 8,000 : 5,000 in early life, as 160 : 100 later, as 4 : 2-5 in maturity, and as 0-08 : 0-05 in extreme old age ; yet so long as the ratio 8 : 5 is preserved, claw-size will always be the same function of body-size — a body of given
0 100 ZOO 300 400 600 BOO 1000 IZSO H00 1750 Z000 ZZSO
weight of rest of body.mg
Fig. 2. — Increase of absolute and relative chela-weight in the large chela of the male fiddler-crab, Uca pugnax.
(Constructed from the data of Huxley, 1927A.)
weight will have attached to it a claw whose weight would be the same whether the body had taken three weeks or three years to reach its present size. 2 The differential growth-ratio re- mains the biologically and morphogenetically important factor. The expression^ = bxk can be written log y = log b + k log x.
1 y0 = 2, x0 = 100. At time t, y, = y0eom, xt = A'oe005(. After ten years, y10 = 2eos = ^4-45, and x10 = iooe0'5' = ^164-92. After twenty years, y20 = ze16 = ^9-9i. and x20 = iooe10 = £271-9, and so on. Double logarithmic plotting of these figures gives a straight line.
2 In all probability this is only true as an approximation. It is a priori unlikely that there is no differential effect of environmental agencies on the growth-rates of body and chela respectively.
10
PROBLEMS OF RELATIVE GROWTH
In other words, if the logarithms of the magnitudes are plotted, we should expect a straight line, from the slope of which the value of k can be read off ; (if a be the angle the line makes with the x axis, then tan a = k). Fig. 3 shows the excellent
|
1.5 |
2.0 |
2.5 3.0 |
|
I 3.0 |
1 |
1 1 Q |
|
rT |
||
|
2.0 |
Oj |
|
|
0 |
||
|
1.0 |
yO |
J> Upugnflxb" |
|
M 1 |
1 |
3.5
I
H3.0
2.0
1P
17 2.0
0.5
3P 3.4
Fig. 3. — Increase of the logarithm of absolute chela-weight with the logarithm of body-weight in male fiddler-crabs.
approximation of the actual points to a straight line when so plotted. In this particular case, the constants are as follows — first phase (to total weight i-i g. or just over ; rest-of-body weight about 075 g.) : b = 0-0073, k = 1-62 ; second phase (from this point to maximum size, in this sample maximum
CONSTANT DIFFERENTIAL GROWTH-RATIOS u
total weight rather over 3-5 g.) : b = 0-083, k = I#255- As purely graphic methods, especially with logarithmic plotting, are not sufficient to establish the accuracy of an empirical formula of this sort (see, e.g., Gray, 1929), I have calculated the values to be expected from the formula. As will be seen from Table I, the deviations from expectation are slight — only in four cases over 5 per cent., and these all in the first phase, where errors in weighing are liable to be relatively greater ; the mean deviation for the second phase is only + 0-19 per cent., for the second phase it is + 0-35 per cent., and would be smaller but for the large deviation of the last class, which consists of only a few individuals. Further, there is no trend of the deviations from predominantly positive to predominantly negative or vice versa. We may thus take the formula as a rather surprisingly close approximation to reality. It is possible that the delimitation of the beginning of the second phase after the 14th instead of after the 15th class would have improved matters ; and also that small alterations in the values of k would have given an even better fit,1 but I am only concerned to show that the data conform to this type of mathematical expression, not to obtain accuracy in an extra decimal place in the formula itself.
We are accordingly justified in saying that the large chela of the male Uca grows in close approximation to the formula of constant differential growth-ratio, namely : y — bxh.
In passing, it is worth noting that the logarithmic method of plotting brings into true relief an important point which is entirely obscured by the usual method of plotting on the absolute scale — namely the fact that growth is concerned essentially with the multiplication of living substance. On the logarithmic scale, equal spaces on the graph denote equal amounts of multiplication, whereas on the ordinary absolute scale they denote equal additions. From the point of view of growth, the increase of weight of our fiddler-crabs from 5 mg. to 25 mg. is equivalent to that from 1 g. to 5 g. ; but on the absolute scale the former interval cannot even be repre- sented on the same graph as the latter. Thus when I speak of a fraction of the growth-period, I shall invariably be think- ing in terms of multiplicative growth, in which an «-fold
1 The last two columns of the table give the expectation if 0-0074 be substituted for 0-0073 as the value of b, and show that this gives a greater deviation, but one of opposite sign.
12
PROBLEMS OF RELATIVE GROWTH
TABLE I
Uca pugnax (401 Specimens) Growth-ratio of Large Chela {y)
and Rest of Body (x)
(a) First phase: to total weight (x + y) = i-i g. y = 0-0073 *162 (mg-)'
|
# = mean |
||||||
|
weight cf |
y = mean |
Per cent. |
y calculated |
Per cent. |
||
|
rest of |
weight of |
y |
deviation |
on formula |
deviation |
|
|
body after |
large chela |
calculated |
actual from |
y = |
actual from |
|
|
removal of large chela |
(actual) |
calculated |
o-oo74sr1'62 |
calculated. |
||
|
mg. |
mg. |
mg. |
mg. |
|||
|
I |
57'6 |
5'3 |
5-16 |
+ 2-7 |
5-24 |
+ I-I |
|
2 |
80-3 |
9-0 |
8-89 |
+ 1-2 |
9-02 |
— 0-2 |
|
3 |
109-2 |
13-7 |
14-59 |
- 6-i |
14-79 |
- 7-4 |
|
4 |
156-1 |
25-1 |
25-88 |
- 3-o |
26-24 |
- 4-0 |
|
5 |
199-7 |
38-3 |
38-90 |
- i-5 |
39-45 |
— 2-9 |
|
6 |
238-3 |
52-5 |
51-76 |
+ 1-4 |
52-48 |
4- 0-4 |
|
7 |
270-0 |
59-o |
63-53 |
- 7-i |
64-42 |
-8-4 |
|
8 |
300-2 |
78-1 |
75-34 |
+ 3-7 |
76-38 |
+ 2-3 |
|
9 |
355-2 |
104-5 |
98-63 |
+ 5-9 |
ioo-o |
+ 4-5 |
|
10 |
420-1 |
i35-o |
129-4 |
+ 4-3 |
131-2 |
4" 2-9 |
|
11 |
470-1 |
164-9 |
155-2 |
+ 6-2 |
157-4 |
+ 4-8 |
|
12 |
535-7 |
195-6 |
191-9 |
+ i-9 |
194-5 |
4- o-6 |
|
13 |
617-9 |
243-0 |
242-7 |
+ o-i |
246-0 |
— 1-2 |
|
14 |
68o-6 |
271-6 |
283-8 |
- 4-3 |
287-7 |
-5-6 |
|
15 |
743-3 |
319-2 |
327-3 |
- 2-5 |
331-9 |
-3-8 |
Algebraic sum of deviations 4- 2-9 Mean deviation 4-0-19
— 16-9
- i-i3
|
(6) |
Second phase : from total weight 1-2 g. |
onwards. |
y = 0. |
|||
|
y calculated |
||||||
|
y |
Per cent. |
on formula |
||||
|
X |
y |
calculated |
deviation |
y = 0-084 S1'256' |
Per cent, deviation |
|
|
16 |
872-4 |
417-6 |
406-8 |
4- 2-6 |
411-7 |
4- i-4 |
|
17 |
983-1 |
460-8 |
472-4 |
- 2-5 |
478-0 |
-3-6 |
|
18 |
1,079-9 |
537-o |
531-9 |
+ I-I |
538-3 |
— 0-2 |
|
19 |
^^SS |
593-8 |
585-6 |
4- i-4 |
592-7 |
4- 0-2 |
|
20 |
1, 211*7 |
616-8 |
628-7 |
- 1-9 |
636-2 |
- 3'i |
|
21 |
1,291-3 |
670-0 |
665-6 |
+ 0-7 |
673-6 |
- 0-5 |
|
22 |
1,363-2 |
699-3 |
720-6 |
- 3-o |
729-3 |
- 4-i |
|
23 |
1,449-1 |
777-8 |
769-1 |
4- 1-7 |
778-4 |
— o-i |
|
24 |
1,807-9 |
1,009-1 |
1,015-0 |
- o-6 |
1,028-0 |
- i-8 |
|
25 |
2,235-0 |
1,380-0 |
1,327-0 |
+ 4-° |
1,344'° |
4- 2-6 |
|
Algebrai< |
; sum of deviations |
+ 3-5 |
- 9-2 |
|||
|
Mean deviation |
+ o-35 |
— 0-92 |
In Huxley, 1927A (p. 152), this was stated as 1-33 owing to an error.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 13
increase from one absolute size is regarded as equivalent to an «-fold increase from another absolute size.
The same total increase could be subdivided into fractions of equal absolute size ; but this method of subdivision in terms of additive growth has not the same biological value, and will not be adopted. Thus for an increase from 1 g. to 256 g., equal fractions of the growth-period are best repre- sented by the equal multiplicative increases from 1 to 4, 4 to 16, 16 to 64 and 64 to 256 g. ; and not by the equal additive increases to 64, 128, 192 and 256 g.
§ 3. Examples of Constant Differential Growth-ratios
The expression y = bxu I shall refer to as the simple hetero- gony formula. This formula, or an approximation to it, has been found to hold for a number of other organs, e.g. the
Fig. 4. — Increase of the logarithm of stem-weight against the logarithm of
root-weight in various plants.
14
PROBLEMS OF RELATIVE GROWTH
abdomen of some female crabs (Shaw, 1928 ; Sasaki, 1928) ; the chelae of many male and some female Decapoda (Huxley, 1927 ; Shaw, 1928 ; Tazelaar, 1930 ; Tucker, 1930 *) ; other appendages of various Crustacea ; the trunk of Planarians as
Fig. 5. — Increase of the logarithm of petiole-length against the logarithm of lamina diameter in nasturtium leaves, Tropaeolum.
against the head (Abeloos, 1928) ; the face as against the cranium of dog and baboon (Huxley, 1927, analysing Becher, 1923 ; Huxley, unpublished, analysing Zuckerman, 1926) ; the shoot as against the root of certain plants (Pearsall, 1927) ; the size (facet-number) of the two lobes of the eye in the
1 Tucker states that his data indicate a linear relation between male chela length and breadth, and carapace length. However, his graphs and his percentage measurements indicate that this does not give an accurate fit, whereas logarithmic plotting (Fig. 55) gives an excellent approximation to a combination of two straight-line curves, as in male Uca.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 15
bar-eye mutant of Drosophila melanogaster (Hersh, 1928) ; the linear dimensions of certain Molluscs (Nomura, 1928 ;
30" "To
/o
40-
30-
20-
10
-— — ---»t-.
10
20
30
40 50 60
mm
Fig. 6. — Increase in relative width of abdomen with increase of carapace length in the shore-crab, Carcinus maenas : above, female ; below, male.
The ordinates represent °_?5 — \ O/ • f he abscissae represent carapace length in mm.
carapace length /<J '
Nomura and Sasaki, 1928) ; the tail of the mouse Phenacomys (Taylor, 1915) ; the dimensions of the casques of Hornbills (Banks, unpublished) ; the weights of various organs of the
i6
PROBLEMS OF RELATIVE GROWTH
rat (analysis of the data of Donaldson, Hatai, Jackson, etc., — see Donaldson, 1. c.) ; the length of the head of whalebone whales during post-natal life (Mackintosh and Wheeler, 1929) ; the lamina diameter and petiole length in Tropaeolum leaves
6 8 10 20
carapace length, mm.
30 40 50
Fig. 7.
-Increase of width of abdomen with increase of carapace length in the shore crab, Carcinus maenas : logarithmic plotting.
+■ , unsexables ; 0, females ; H males. The growth coefficient for unsexables and young females is 1-26, for older females 1-42, that for young males 107, for older males 0-94.
(Pearsall, 1927), and even for the amounts of various chemical substances in the growing larvae of the wax-moth Galleria and the meal-worm beetle Tenebrio (Teissier, 1929, 1931), as well as for organs which, physiologically speaking, represent special cases, as the antlers of deer which are shed every year
CONSTANT DIFFERENTIAL GROWTH-RATIOS 17
PaJaemon malcomsoni
2 6
2 4
2 2
2 0
(Huxley, 1926, 1927, 193 1), and the horns, mandibles, etc., of various holometabolous insects (Huxley, 1927 — see Chapter II). It is probable that any other changes of proportion which have not yet been analysed from this point of view will turn out to obey the same law, e.g. the progressive increase of rela- tive tail-length in the salamander Eurycea (Wilder, 1924), or that of wing-rudiments in dragonflies (Balfour-Browne, 1909). See also p. 258 (Daphnia), p. 263 (rat).
Most of the organs thus far cited are positively heterogonic, increasing in relative size with growth ; others, however, de- crease in relative size with increase in absolute size, and are accordingly negatively hetero- gonic. Such of these cases as have been analysed also appear to obey the rule of constant differential growth-ratios : e.g. nucleus in oocytes of Hydractina (Teissier, 1927) ; brain in vari- ous mammals (Lapicque, 1907 ; Dubois, 1914, 1918) ; the number of nerve-fibres and /or neurones in mammals (Lapicque and Giroud, 1923 ; Dubois, 1918) ; heart in many vertebrates (Klatt,
1919 ; Clark, 1927, for references) ; limbs in post-natal sheep (Hammond, 1927, 1929) ; certain limbs in Hermit-crabs (Bush, I93°) '> pereiopods in the racing-crab Ocypoda (Cott, 1929) ; legs in Orthoptera (Przibram, 1930) ; Gammarus eyes (p. 260).
Some of the facts are graphically illustrated in Figs. 4, 5 (plant organs) ; 6, 7, 22, 23 (crab abdomen) ; 8, 21, 32, 55, 77 (Crustacean chelae) ; 9, 10 (mammalian cranium) ; 11 (mouse tail-length) ; 12 (facet-number, Drosophila eye) ; 13-15 (various organs of crabs) ; 16-20 (chemical substances in insect- larvae) ; 71 (head-length, whales) ; 25, 27 (deer antlers) ; 33-35, 40, 91, 92 (organs of holometabolous insects).
We may give some tables and figures in support of these statements.
log body length, mm.
Fig. 8. — Relative growth of the chela in the prawn, Palaemon mal- comsoni : logarithmic plotting.
(From the data of Kemp, and Henderson and Mathai.)
i8
PROBLEMS OF RELATIVE GROWTH
Sheep Dog (Fig. 9)
Data from Becher (1923) Analysed in Huxley, 1927
x 42-0
05-3 74-5 85-5 99-3
II2-6 I20-O
y 22 -o
48-3 58-0
73-5 89-1
102-0 II2-0
x
y k
30-
cranial region (mm.).
facial region (mm.).
1-5 (except for highest values of x).
|
160 |
— 1 \ 1 — |
- |
|
|
130 |
- |
I/K |
|
|
100 |
• |
r |
|
|
- |
■ |
||
|
5? «5! |
■ |
/ 1 / 1 |
- |
|
1 ™ |
/ X |
7 ' / 1 / 1 / 1 1 1 (6) |
■ |
|
20 |
1 |
— 1 1 1 |
1 |
40 60 80 100
cranium length, mm.
120
Fig. 9. — Growth of the facial region relative to the cranial region in the skulls of sheep-dogs (x) and baboons (©). k for sheep-dog about 1-49 ; for baboon, points 2-5, about 4-25. [Constructed from the data of Becher, 1923, and Zuckerman, 1926.)
CONSTANT DIFFERENTIAL GROWTH-RATIOS 19
Values of Cranium-length and Face-length in the Baboon Papio porcarius at Different Sizes (from Zuckerman, 1926) (Figs. 9, 10)
No. of Cases
1
4 7 3 6
4
The growth-coefficient of face-length on cranium-length for Classes 2-5 is about 4-25, a very high figure. The curve shows irregularities at both ends.
|
Mean Face- |
|
|
Mean Cranium- |
length (Naso |
|
length |
prosthion) |
|
mm. |
mm. |
|
• 78-5 |
31-0 |
|
100-25 |
64-6 |
|
108-9 |
94-8 |
|
• "47 |
131-0 |
|
. 118-25 |
140-8 |
|
122-0 |
M4-25 |
Fig. 10. — Baboon skulls of various sizes, to show the increase in relative size
of facial region with absolute size of skull.
1, new-born; 2, juvenile (with milk dentition); 3, adult female; 4, adult male.
20
PROBLEMS OF RELATIVE GROWTH
TABLE Ia Abdomen-breadth against Carapace-length in Carcinus
MAENAS FROM PLYMOUTH (FigS. 6, 7)
(Data of Huxley and Richards, 1931)
|
Mean |
Mean |
|
|
V |
car.-l. |
abd.-br. |
|
(mm.) |
(mm.) |
|
|
(a) Unsexable (74 specimens) |
||
|
13 |
3-09 |
0-578 |
|
22 |
3-80 |
0-680 |
|
20 |
4-22 |
0-823 |
|
11 |
476 |
0979 |
|
8 |
5-19 |
1-019 |
|
(b) Fern |
ales (281 specimens) |
|
|
12 |
5-56 |
1-16 |
|
16 |
6-52 |
i-45 |
|
14 |
7-41 |
1-67 |
|
12 |
9-32 |
2-30 |
|
15 |
10-37 |
2-80 |
|
16 |
n-35 |
3-n |
|
17 |
12-33 |
3-37 |
|
23 |
13-29 |
3-81 |
|
19 |
14-35 |
4-06 |
|
12 |
I5-3I |
429 |
|
19 |
16-35 |
4-82 |
|
15 |
17-36 |
5-15 |
|
15 |
18-16 |
5-48 |
|
7 |
19-34 |
5-93 |
|
6 |
20-33 |
6-72 |
|
10 |
21-51 |
6-82 |
|
12 |
22-40 |
7-54 |
|
8 |
23-34 |
7-76 |
|
6 |
24-33 |
8-33 |
|
5 |
25-52 |
9-16 |
|
10 |
26-45 |
9-59 |
|
6 |
27-65 |
9-79 |
|
2 |
28-45 |
10-40 |
|
3 |
29-30 |
1 1 03 |
|
4 |
3o-55 |
11-62 |
|
4 |
31-49 |
12-46 |
|
3 |
32-37 |
12-75 |
|
5 |
33-32 |
12-70 |
|
4 |
34-33 |
I3-56 |
|
1 |
43-5o |
17-50 |
|
2 |
45-3o |
19-20 |
|
1 |
46-30 |
19-90 |
|
1 |
50-20 |
23-70 |
CONSTANT DIFFERENTIAL GROWTH-RATIOS 21
Values of Constant Differential Growth-ratios for Shoot- weight against Root-weight in Various Plants (from Pear- sail, 1927) (Fig. 4)
|
Plant |
Value of k |
|
Daucus carota (carrot) |
°'55 |
|
Brassica rapa (turnip) |
0-65 |
|
Gossypium roseum (cotton) |
090 |
|
Impatiens sp. .... |
i-oo |
|
Pisum sativum (pea). . |
0-90-1-15 (3 expts.). |
|
,, ,, (etiolated) . |
1-75-2-65 |
|
Triticum vulgare |
1-05 |
|
Hordeum distichum (low N) |
1-20 |
|
(highN) |
i-55 |
|
Linum usitatissimum |
1-30 |
Values of Cell-diameter and Nuclear Diameter in the Oocytes of Hydractinia Echinata, in /i. (From Teissier, 1927)
k = 0-69 ; b — 1-5
|
x = oocyte diameter y = nuclear Deviations from cal- culated values |
6-8 5-6 o-o |
io-o 7-3 o-o |
13-6 9-0 — o-i |
17-5 1 1*4 +o-i |
22-2 12-5 — 0-2 |
25-3 14-4 + 0-5 |
34'° 16-9 — 0-2 |
|
x = oocyte diameter y = nuclear ,, Deviations from cal- culated values . |
43-o 22-5 + 2-4 |
53-5 24-8 + i-5 |
70-0 29-6 + i-5 |
•88-o 3o-3 -2-6 |
n8-o 36-5 -3-8 |
136-0 42-7 -2-5 |
168-0 52-0 -f-O'2 |
Mean Values of Pre-ocular Length and Total Length in Planaria gonocephala, calculated from Abeloos (1928)
k — 0-63 approx.
Total length . 1-5 3-0 50 18-0 mm.
Pre-ocular length . 0-231 0-43 0-50 1-125 mm.
Mean Values for Facet-Number in the Two Lobes of Bar-eyed Drosophila at Different Temperatures (from Hersh, 1928) (Fig. 12)
|
Temperature |
32-0° |
29-5° |
27-5° |
25-5° |
21-5° |
18-0° |
15-0° |
|
|
No. facets dorsal lobe No. facets ventral lobe |
17-63 12-73 |
22-15 18-86 |
23-30 19-24 |
28-63 19-11 |
43-35 36-24 |
62-08 57-62 |
66-04 67-28 |
I Homozy- rgous bar J ?? |
|
No. facets dorsal lobe No. facets ventral lobe |
81-22 41-93 |
152-15 9020 |
150-29 103-66 |
193-54 114-29 |
196-21 151-51 |
222-45 I59-OI |
256-12 185-89 |
\ Heterozy- 1 gous (Bar [ X normal) ) 99 |
In both cases k for ventral lobe on dorsal lobe is about 1-5, but the value of b is considerably higher for the homozygotes.
Fig. i i . — Tail-length against total length
during growth in the mouse, Phenacomys
longicaudus ; logarithmic plotting.
k = about i -41.
(Recalculated from the data of Taylor, 1915, p. 129.)
100 130 160 ZOO
total length, mm.
15
20
30 40 50 60 70 8090100
dorsal facet number
150 ZOO
300
Fig. 12. — Relation of facet-number in dorsal and ventral lobes of mutant female fruit-flies of the bar-eye series ; logarithmic plotting. With decreasing temperature, the total number of facets in the eye increases ; but the number in the ventral lobe of the eye increases heterogonically relative to the number
in the dorsal lobe.
The curve on the right denotes heterozygotes between bar and wild-type (full eye). In the curve on the left, x denotes homozygous ultra-bar and heterozygotes between ultra-bar and bar ; o, heterozygotes between ultra-bar and wild-type ;• , homozygous bar.
k for all is close to 1-5 ; 6 is lowered by admixture of the wild-type gene, and is at its maximum in pure ultra-bar (i.e. rises with decreasing absolute size of the eye). It is clear that facet-formation must begin ontogenetically in the dorsal lobe.
22
CONSTANT DIFFERENTIAL GROWTH-RATIOS 23
Dimensions (in inches) of Parts of the Bill and Casque of the Hornbill Antheracoceros malaganus (from E. Banks, unpublished)
|
$$ |
<? |
|||||||
|
Length of gape along |
||||||||
|
curve .... |
3-5 |
3-8 |
4-25 |
4'25 |
5-o |
5*3 |
5-6 |
5-8 |
|
Length of casque |
||||||||
|
(straight) . |
* |
* |
2-6 |
3-2 |
3-55 |
4.6 |
5-2 |
6*7 |
|
Height of casque |
||||||||
|
above bill . |
o-5 |
O-Q |
I-O |
0-9 |
1-4 |
i-6 |
i-5 |
1-85 |
|
Height of bill proper |
1-2 |
1*2 |
i-5 |
1-25 |
i-5 |
i-75 |
1-6 |
2-05 |
* Indistinguishable from bill.
Both in length and height the casque is highly heterogonic relative to the length and /or the height of the bill proper.
■ 50
/
100
- 20
20 1
50
100
200 mm,
1 ' '
Fig. 13. — Increase of interocular distance with carapace width in the crabs Cancer (i) and Eriphia (3), and of carapace length with carapace width in
Cancer (2) ; logarithmic plotting. k for interocular distance, in Cancer 0-70, in Eriphia 0-76.
3 mm.
10mm.
10
20mm
100
e
Fig. 14. — Change of proportions in the crab, Carcinus maenas.
A— D, outline of carapace and of thoracic ganglion in four specimens of different absolute size, to show negative heterogony of interocular breadth {k = 0-85), and of thoracic ganglion (k= 06). The carapace lengths have been made the same for all: actually they were 3-1, 109, 30-0 and 71-0 mm. respectively.
E — H, ommatidia of the same four specimens, all drawn to the same absolute scale. k for ommatidial diameter = 0-32.
24
CONSTANT DIFFERENTIAL GROWTH-RATIOS 25
mm 5
.0
I'
I'
10 20
carapace breadth
50 mm.
Fig. 15. — Diameter of the thoracic ganglion against carapace breadth in crabs : (1) Pachygrapsus ; (2) Carcinus ; logarithmic plotting.
k in both cases about o-6. (After Teissier, 193 1 : in the original paper, the Carcinus curve is erroneously ascribed to Pachygrapsus and vice versa.)
Since the above was written, the important paper of Teissier (1931) has provided numerous fresh examples. These we may summarize in tabular form (and see Figs. 13-20).
|
Animal |
y = organ |
x = standard to which organ is compared |
Value of k |
|
Mealworm, Tenebrio mo- |
weight of moult- |
total fresh weight |
o-8 |
|
litor larva |
ed skin or cara- pace |
||
|
Water-beetle, Dytiscus |
, , , , |
>) > t |
o-8 |
|
marginalis, larva |
|||
|
Water-boatman, Noto- |
11 11 |
)> it |
10 |
|
necta glauca |
|||
|
Shore-crab, Carcinus |
11 11 |
j » ,, |
10 |
|
maenas |
|||
|
Crab, Pachygrapsus raar- |
11 11 |
11 > 1 |
10 |
|
moratus |
|||
|
Stagbeetle, Lucanus cer- |
weight of desic- |
weight of desic- |
2-0 |
|
vus $ |
cated mandibles |
cated elytra |
|
|
>i 11 11 |
weight of desic- |
» > M |
i-5 |
|
cated head |
|||
|
11 11 11 |
weight of desic- cated legs |
,. |
10 |
|
1. |
mean diameter |
length of elytron |
10 |
|
eye |
|||
|
11 11 11 |
maximum breadth head |
,. |
20 |
mgr. 100
50
•w 20
I
i
C. 10
•§
* 5
A
?
12 5 10 20 50 100 200 mgr.
body - weight
Fig. i6. — Water-content against body-weight in the larval mealworm, Tenebrio.
Solid line, fresh weight ; dotted line, dry weight ; logarithmic plotting.
k (fresh weight), 0-975 ; (dry weight), 0-92.
mar
Tig,
Ho
0,5
§
0,2
,6
12 5 10 20 50 100 mgr.
body - wetight
Fig. 17. — Total nitrogen against body-weight (solid line, fresh weight ; dotted
line, dry weight) in the larval mealworm, Tenebrio ; logarithmic plotting.
k for fresh weight, 0965 ; for dry weight, o-9r.
26
CONSTANT DIFFERENTIAL GROWTH-RATIOS 27
|
Animal |
y = organ |
x = standard to which organ is compared |
Value of k |
|
Stick insect, Dixippus |
length prothorax |
total length |
I-O |
|
morosus |
|||
|
it it tt |
length head |
it ti |
0-71 |
|
tt 11 11 |
breadth head |
> » 1 1 |
0-71 |
|
it 11 11 |
diameter eye breadth head |
11 11 |
0-48 |
|
Mealworm, Tenebrio |
ty total weight |
o-95 |
|
|
molitor, larva |
|||
|
Crab, Carcinus maenas |
interocular breadth |
carapace breath |
0-85 |
|
}* tt a |
carapace length |
11 11 |
about i-o |
|
Pachygrapsus mar- |
carapace length |
It 11 |
about i-o |
|
moratus |
|||
|
ts II II |
interocular breadth |
11 11 |
0-85 |
|
Eriphia spinifrons |
> 1 it |
11 |
0-76 |
|
Cancer pagurus . |
11 11 |
11 |
0-70 |
|
Water-boatman, Noto- |
diameter of a |
total length ^ |
0-46 |
|
necta glauca |
single ommati- dium |
||
|
Cockroach, Blatta ori- |
11 11 |
11 a |
0-36 |
|
entalis |
|||
|
Stick insect, Dixippus |
11 > > |
11 a |
0-37 |
|
morosus |
|||
|
Crab, Carcinus maenas |
11 11 |
carapace breadth |
0-32 |
|
,, Pachygrapsus mar- |
11 11 |
a a |
0-47 |
|
moratus |
|||
|
Crayfish, Potamobius as- |
11 it |
length |
0-40 |
|
tacus |
|||
|
Crab, Carcinus maenas |
diameter thoracic ganglion |
carapace breadth |
about o-6 |
|
Pachygrapsus mar- |
11 ti |
tt n |
o-6 |
|
moratus |
|||
|
Insect larva, Chaoborus |
diameter cerebral |
body length |
o-6 |
|
crystallensis |
ganglion |
||
|
a a a |
mean diameter abdominal ganglia |
tt a |
o-6 |
|
Stick insect, Dixippus |
diameter cerebral |
total length |
o-6 |
|
morosus |
ganglion |
||
|
>» a a |
mean diameter thoracic ganglia |
a a |
„ o-6 |
|
a a a |
mean diameter abdominal ganglia |
a a |
o-6 |
|
Mealworm, Tenebrio mo- |
mean diameter |
total length |
o-6 |
|
litor larva |
abdominal ganglia |
||
|
Water-boatman, Noto- |
diameter 2nd |
11 11 |
„ o-6 |
|
necta glauca |
thor. ganglion |
||
|
Insect larva, Chaoborus |
mean diameter |
body length |
0-4 |
|
crystallensis |
nucleus, nerve- cells |
mgr. 0,5
0,2
|
0,1 |
|
|
-tr |
|
|
^ |
|
|
o |
|
|
<: ^ |
0,05 |
|
J? |
|
|
-to 3 |
|
|
0,02 |
0,01
12 5 10 20 50 100 200 mgr.
body - weight
Fig. i 8. — Total phosphorus against body- weight in the larval mealworm,
Tenebrio ; logarithmic plotting.
k for fresh weight, 1-03, later 1-08 ; for dry weight, 0-975, later 1-02.
cod.
500
HO
^ 200
I
o 100
1
50
5 10 20 50 100 mgr.
body - weight
FlG. 19. — Heat of combustion against body-weight in the larval mealworm,
Tenebrio ; logarithmic plotting.
k for fresh weight, 1-07 ; for dry weight, 1-02. 28
CONSTANT DIFFERENTIAL GROWTH-RATIOS 29
Animal
Mealworm, Tenebrio molitor larva
Wax-moth, Galleria mell onella, larva
y = organ
x = standard to which organ is compared
total fat
fat
phos- phos-
growth
1-07 i-o6 1-04
i-oi 1 00 098
103 0-975
(i-o8 1-02)
|
I 03 |
o-975 |
|
I-OO |
o-945 |
|
o-975 |
0-92 |
|
0-965 |
o-gi |
|
0-87 |
0-82 |
|
0-83 |
o-8o |
|
0-82 |
078 |
|
o-8i |
0-76 |
|
1 07 |
1 02 |
about o-8
about 0-9
total carbon
dry weight
dry weight, removed
total phosphorus, until late in lar- val life
(total phosphorus, end of larval life)
protein nitrogen
fresh weight
total water
total nitrogen
extractives
lipidic phorus
nucleic phorus
ash
total heat of com- bustion
Oxygen consump- tion at rest and fasting
Oxygen consump- tion under nor- mal conditions, but on a ration permitting only slow (flour)
ditto, but fed a ration permit- ting normal growth
total water, early phase*
total water, late phase
dry weight, early phase
dry weight, late phase
total heat of com- bustion, early phase
late phase ' refer to the fact that after a period of regular differential growth lasting for most of the larval period, there occurs a short phase of irregularity, denoting rapid change in character of metabolism, followed by a second regular period resembling the first but with different quantitative relations.
Value of k
(«) (b)
fresh dry weight weight
(a) i-i3
(b) i-o6
about 0-95
10
i-o
0-96 0-91
i-o
i-oS
i-i3
i-o
i-o
I-OQ
Early phase ' and
30
PROBLEMS OF RELATIVE GROWTH
|
Animal |
y = organ |
x = standard to which organ is compared |
Value of k |
|
(a) (b) |
(«) (b) |
||
|
Wax-moth, Galleria mell- |
total heat of com- |
fresh dry |
I-l6 i-oq |
|
onella, larva |
bustion, late phase total phosphor- \ |
weight weight |
|
|
us, early phase 1 total phosphor- j |
total water |
O-Q |
|
|
us, late phase / |
|||
|
total fat, early ~\ phase [ total fat, late 1 phase / |
|||
|
dry weight |
1-32 |
||
|
dry weight, fat\ |
|||
|
removed, early |
|||
|
phase dry weight, fat [' |
»» |
0-82 |
|
|
removed, late |
|||
|
phase / |
Now it is to be observed that a constant differential growth- ratio, during some at least of the period of growth, is not merely an empirical rule found over a large range of organs and groups, but is what one would expect on a priori grounds. For it is the biologically simplest method we can conceive of obtaining the enlargement (or diminution) of an organ.
If an organ is to begin its career small and end it large, the obvious method is to make it grow at a higher rate than the rest of the body ; the difference once initiated (by what- ever physiological means) there is no a priori reason why it should not be maintained at approximately the same relative level throughout, since changes which affect the growth of the body will be expected to have, within narrow limits, a pro- portionate effect on the growth of the organ (and see p. 6).
We shall later consider certain special cases where growth of organ and body appear not to be equally affected in cer- tain circumstances (pp. 200, 259) ; others which show that the formula for constant differential growth-ratio can only be an approximation when we are dealing with an organ as a whole (p. 81) ; and still further facts which rule out some of the early stages of an organ's development from the operation of the law (p. 139, seq.). However, both empirical facts and theoretical considerations warrant us in regarding a constant differential growth-ratio, operating over a longer or shorter period of time, as the primary law of the relative growth of parts, once they have reached the stage of full histological
CONSTANT DIFFERENTIAL GROWTH-RATIOS 31
differentiation. It may (like Boyle's law) prove only to be an approximation, and to be capable of modification in cer- tain circumstances ; yet (again like Boyle's law) it may remain fundamental.
We may now proceed to consider a little more in detail some of the cases. In the first place, we can utilize our formula to deduce the moment of onset, in male Uca pugnax, of the large chela's heterogony. To do this we must first know the weight of the small claw. This in males is identical in form
mqr. 100
50
I
• 10
k 5
1 2 5 10 20 50 100 200 mgr.
body - weight
Fig. 20. — Increase of water-content with total weight in the larva of the
wax-moth, Galleria ; logarithmic plotting. k for fresh weight, early phase i-o, late phase i-o ; for dry weight, early phase 0-96, late phase 0-91.
with both claws of females, and does not change its relative weight with increasing body-size : at all stages it weighs, as does a single female chela, almost precisely 0-02 of the rest- of-body weight. If we make the assumption that our formula for the first phase holds from the first moment of increase of the large chela, we have simply to extrapolate from our formula and find the point on the curve at which rest-of-body weight is fifty times chela weight. This is found to be close to 5 mg. body- weight, when the chela should weigh o-i mg.
32 PROBLEMS OF RELATIVE GROWTH
We can now proceed to check this deduction. Morgan (1923A, 1924) has found that the very youngest post-larval U. pugnax he could obtain have both claws alike, of female or small type, in both sexes. In this stage, the chelae are autotomized very readily ; and only when one is thus thrown off does the other proceed to transform into a large chela. After this has once happened, the fates of the two chelae are irreversibly determined, though initially either may become a large chela through the accident of the other's autotomy. The moment of determination of the large chela appears normally to take place very early, during the first or second instar of post-larval life.
Accordingly, I collected and weighed a number of the smallest fiddler-crabs to be found on the beach, in which the sexes could not be determined by casual inspection of either chela-size or abdomen-shape. Their mean weight was about 6-7 mg. — an excellent approximation to the 5 mg. prophesied on theoretical considerations.
Then again, we can compare relative growth in different species of the same genus. Uca minax is much scarcer near Wood's Hole, and the comparatively few specimens available were all of a size to be in the second phase of U. pugnax. However, they yielded one or two interesting results. The double logarithmic plot of chela against rest-of-body clearly approximated to a straight line ; but owing to the smaller number available it was impossible to determine the growth- coefficient of the chela so accurately. It was, however, cer- tainly between 1-58 and 1-66 — in other words, almost exactly the same as that of U. pugnax for the first phase. Either U. minax has no change in the growth-coefficient of the chela at or near maturity, or at all periods its chelar growth-coeffi- cient is higher than in pugnax. U. minax also differs from its relative in the greater size which it attains ; the biggest specimens found weighed 17-8 g. as against 3-6 g. for U. pugnax. Correlated with this, as was to be expected, was the greater rela- tive weight of the large chela to be found in minax. This in one specimen amounted to no less than 77 per cent, of rest-of-body weight, as against a maximum of 65 per cent, in U. pugnax.
It is clear that the large chelae of big specimens of U. minax must be getting close to their maximum limit of relative size. A claw as big as the rest of the body would not be very prac- ticable, and these are already over three-quarters this relative size. In U. pugnax, where our figures are more accurate,
CONSTANT DIFFERENTIAL GROWTH-RATIOS 33
the large chela has attained half the rest-of-body size when the total weight is about 1-65 g. If the animal could grow to 24 g. (less than 30 per cent, bigger than the biggest U. minax) , its large chela would be the same weight as all the rest of it together. Twenty-four grams is a very small weight for many crabs, in- cluding forms of sim- ilar semi-terrestrial and burrowing habits to Uca, such as Ocy- poda, yet no species of fiddler-crab has grown to a size much over that of U. minax. It may thus be plausibly sug- gested that the exist- ence of this continu- ously high growth- ratio in the large claws of the male fiddler-crabs may have acted as a limiting factor in their size-evolution, any possible advan- tage to be obtained by increase in size not countervailing to cause selection to alter the chela's growth-mechanism . Analysis of the 3
3 4 5 6 7 8
carapace length, cm. A
zzo
2-00
/■BO
1-60
/■40
I 20
100
log carapace length, mm.
120
B
1-60
Fig. 21. — Relative growth of male and female
chelae. (A) in Palaemon carcinus, (B) Palaemon
bengalensis ; logarithmic plotting.
(From data of Kemp, I9i3-X5-)
34 PROBLEMS OF RELATIVE GROWTH
relative growth of homologous organs next shows that the same organ may behave very differently, as regards its growth-behaviour, in different forms. For instance, the chela of male Uca pugnax shows heterogony on one side of the body only, but shows it throughout all but the first instar of post-larval life, with a decrease in its growth-coefficient apparently at the time of sexual maturity. In the spider- crab Maia squinado (Huxley, 1927 ; and unpublished), both chelae are heterogonic, but heterogony does not set in until quite late in life, presumably at sexual maturity, and then continues till death. The same appears to be the case with the large prawn Palaemon carcinus (Tazelaar, 1930), though here the chelipeds are the second and not the first pereiopods ; and with the spider-crabs of the genus Inachus. But in the latter the male chelae revert more or less completely to the female type in the non-breeding season ; this reversion is much less marked in I. dorsettensis (Shaw, 1928) than in 7. mauritanicus (Smith, 1906A). In various other crabs, and in lobsters, crayfish and pistol-crabs (Alpheus), both chelae are heterogonic, but with different growth-coefficients, leading to the condition of heterochely. Furthermore, the sex-difference as regards the growth-coefficient of the chela may vary, some forms having equal positive heterogony in both sexes, others showing positive heterogony in both sexes, but with a lower growth-coefficient in the female, and still others showing male heterogony but female isogony. This variability is particu- larly well shown in prawns (Palaemonidae) ; in these, further, the large chela is the second, not the first pereiopod. Finally in Gammarus chevreuxi, Kunkel and Robertson (1928) have shown that the marked heterogony of the male gnathopod begins shortly before sexual maturity and ends shortly after, its growth being roughly isogonic for the much longer previous and subsequent periods. (Cf. also birds' wings, p. 263.)
A similar state of affairs is seen in regard to the abdomen of female Brachyura. This must always be heterogonic for some part of its development, since it is always broad in the adult, always narrow and of male type in the young juvenile. A state of affairs similar to that of the large male chela of Uca is found in the female abdomen of Carcinus maenas — it is hetero- gonic (with a late increase in intensity) from the earliest stages until the end of life. The details for both sexes at all ages are shown and described in Fig. 7. In female Uca, on the other hand, while heterogony is initiated at the beginning of post-
B
Fig. 22. — Variation in relative growth in the female abdomen of fiddler-crabs
(Uca pugnax).
(A) Left, medium-sized crab (carapace breadth, 10 mm. ; abdomen, 6 mm., broad and reaching the bases of the legs). Right, medium-sized crab (carapace breadth, n mm. ; abdomen, 5J mm., broad, not reaching the bases of the legs).
(B) Means (A — A) and extremes (B — B and C — C) of relative abdomen breadth in female crabs between 4 and 14 mm. carapace breadth. There is a clear bimodality of mean abdomen size, with marked heterogony between 8 and 11 mm. carapace breadth.
35
36
PROBLEMS OF RELATIVE GROWTH
larval life, a state of equilibrium (adult female pro- portions) is event- ually attained, when the lateral margins of the abdomen have reached the bases of the legs (Mor- gan, 1924 ; Huxley, 1924A) (Fig. 22).
(In passing, it may be noted that Morgan was led to postulate female intersexuality^ in this species on rinding certain ap- parently mature females with ab- domens propor- tionately narrower than the full female type. Hux- ley, however (I.e.), was able to show that these were merely the ex- treme minus vari- ants for the normal variation curve of female abdomen - growth, and that the sup- position of inter- sexuality was un- called for — an interesting appli- cation of the study of heterogony.)
That the equili- brium-position is
Carapace
Carapwce
Relative abdomen breadth, %
Fig. 23. — Relative growth in the male and female abdomen in the spider-crab, Inachus dorsettensis.
(A) Left, mature female abdomen ; right, mature male abdomen. Abdominal breadth was taken on the 6th segment (n — o ; v — w).
(B) Change of means and extremes of relative abdomen breadth
with increase of carapace length in males and females.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 37
not in any way automatically, and still less mechanically, brought about when the sides of the abdomen reach the legs is shown by Pinnotheres, the pea-crab (Atkins, 1926)
8 10 12 14- 16 IS 20 22 ZA-
LtriGTri -Cns,
Fig. 24. — Relative growth in the teleost fish, Orthopristis.
The abscissae represent total length. The ordinates represent (above) the length of the head, trunk (body), and tail ; the division between head and trunk is taken at the hind end of the opercular bone, between trunk and tail immediately above, or below, the end of the hindmost median fin ; (below) maximum depth and width of body.
38 PROBLEMS OF RELATIVE GROWTH
in the adult female of which the abdominal margins far over- lap the leg-bases.
Then we have the quite different case of Inachus (Fig. 23) which resembles that of the male gnathopod in Gammarus, only here all the marked heterogony appears to be achieved in a single moult-period. And finally we have the fact first discovered by Geoffrey Smith (1906A), that parasitization of Inachus with Sacculina, while reducing or abolishing the heterogony of the male chela, actually increases the growth- ratio of the female abdomen, to remind us that the differential growth-ratios are only constant in certain conditions.
Similar relations would doubtless be found for other organs, but these are the best analysed. They show that the dif- ferential growth-coefficient of an organ, though it may remain constant throughout post-larval or post-embryonic life, may equally well be confined to the beginning, the end, or the middle (and here sometimes to a very small period) of the life-history, or may change its value slightly but definitely. Since, however, it is justifiable to regard isogony as a special case of heterogony, with growth-coefficiency unity, it remains true that in all cases the growth-coefficients of parts or organs remain constant over definite periods, and that these periods are in the great majority of cases few in number and long in time. (See Hecht, 1916, on the proportions of fishes for a case of long-continued isogony in all measured dimensions : Fig. 24.) 1
§ 4. Inconstancy of Form and Constancy of
Form-change
In concluding this chapter, it may be pointed out that the constancy of growth-ratio over considerable periods of the life-history in spite of environmental fluctuation, is of very considerable importance for analytical morphology. Where- ever it obtains, it implies that the form of an animal, as given by the proportions of its parts, depends (naturally within the
1 Even here, the isogony is not permanent. Up to a length of 30 cm. — i.e. about a year old — Kearney (reference in Robbins, Brody et al., 1928, p. 123) shows that there is heterogony. Hecht further points out that so far as known, all vertebrates with determinate growth change their proportions continuously up to the adult phase. It is only in forms with indeterminate growth like fishes that there exists a long-continued period with no change in external proportions (though even here the relative size of the viscera changes). The work of Keys (p. 259), however, indicates that Hecht's conclusions are riot strictly accurate. Compare also Olmsted and Baumberger (p. 261).
INCONSTANCY OF FORM 39
limits of normal variation, of which more later) solely upon its absolute size, not upon the length of time it has taken to reach that size, nor upon changes in any other external variable. The modifications of and exceptions to this statement we shall consider later ; here we can accept it as our first general rule.
As immediate corollary of this we have the fact that only animals in which all organs are growing at the same rates will preserve their form unchanged with increase of size ; and this is as much as to say that no animal will keep its form identical while increasing in size, for it appears highly improb- able that any animal will be found in which some organs do not grow at a different rate from the body as a whole. And even if for the moment we stick to external form, and further if we only consider quite large differences in growth-activity, we shall find many animals in which, as in the male fiddler- crab, the only constancy of form is the constancy of its mode of change. This is less obvious and in some ways less important in the higher animals (notably almost all mammals and birds among vertebrates and almost all insects and spiders among arthropods), in which growth ceases at a definite size, and there supervenes an adult stage of constant size and often of long duration. For here we can often afford to consider only the adult forms, in which the proportions of form have been fixed by the cessation of growth. It is the limitation of form at a fixed absolute size which confers this convenience upon the systematist and the morphologist.
Even here, however, as we shall see in detail later, the rule has many applications. To take the most obvious case, the absolute size at which growth ceases may be altered by treat- ment such as feeding ; in such case, the permanently stunted individual will approximate in proportions to a normal juvenile stage, the well-fed, abnormally large specimen will have pro- portions not met with at all among the normal population of adults.1 It is, in other words, a mere biological accident that adult proportions, even in species with limited growth, are relatively fixed ; and to neglect the fundamental fact of change
1 As a matter of fact, the degree of development (including growth) of a higher vertebrate appears to be simultaneously dependent upon at least two variables, size and age. This is well shown by Appleton (1925, see his chart 3) as regards the degree of ossification in new- born rabbits ; and by Jackson (1925) for young rodents stunted by underfeeding. Jackson's results are discussed further in Chapter VI. See also the work of Adolph (p. 258).
40 PROBLEMS OF RELATIVE GROWTH
of proportions with absolute size, and to proceed as if certain arithmetic (percentage) proportions were immutable ' charac- ters ' of the species, may lead to serious error.
But we must remember that the limitation of growth and the consequent establishment of a small range of stable adult size is a late and specialized feature in evolution. The majority of animals show unlimited growth : they continue growing, though usually at a constantly diminishing rate, until they die, or in asexually-reproducing forms, until they divide. A lobster or a plaice may increase its linear dimensions several fold after the attainment of sexual maturity. In such types, there is no fixed or adult form ; the change of proportions continues unabated throughout life, and may be as obvious during post-maturity as during pre-maturity. An excellent example of this is provided by the detailed studies of Mrs. Sexton (1924) on the successive instars of Gammarus chevreuxi, supplemented by the work of Kunkel and Robertson (1928) on the same species.1
Even among mammals a change of proportions may con- tinue throughout life. In the voles (Microtinae) Hinton (1926, Chap. II, 8-14, Pis. Ill, IV, IX) finds that slow growth occurs long after the adult state has been arrived at, the epiphyses of the long bones never uniting. This continuous growth is accompanied by continuous change of proportions. With increasing size of the skull, for instance, the rostrum becomes relatively narrower and slightly longer, the interorbital region narrower and the molars relatively smaller. Unfortunately the measurements given do not permit of any accurate state- ment as to the changes involved, or as to the distribution of growth-potential in different regions. Here is an interesting field for the student of relative growth. It would be par- ticularly interesting to discover whether the relative growth- rates of tail and parts of skull, limbs, etc., remained the same after the attainment of sexual maturity as they did before. It would be easier to investigate this on the limb-segments than on the skull, which undergoes complex distortions and curvatures. It would also, of course, be necessary to keep
1 Sexton states that sexual maturity occurs at the seventh instar, that proportions continue to change for two further instars in the male, one in the female, but after this no further proportion-changes occur (though the males at least may increase about 40 per cent, in length). That this statement is not accurate is shown by Kunkel and Robertson, whose graphs demonstrate a change in the proportions of several organs up to the largest sizes found.
INCONSTANCY OF FORM 41
the animals under standard conditions, since the work of Sumner and of Przibram has shown that increased temperature causes an increased relative size of appendages in rodents.
It is true the change will not usually be of the same extent after sexual maturity, for although the changes in absolute size may be greater between maturity and death than in the period from the post-embryonic or post-larval phase to maturity, yet the fraction of total growth which takes place after maturity is always a good deal less, if measured by the true criterion, namely the amount of multiplication of initial size. For the fiddler-crab, for instance, the pre-maturity multiplicative increase in weight is about 250-fold, the post- maturity increase about three- to four-fold, though the absolute (additive) increases are roughly as 1 to 2-5. None the less, the post-mature alterations may be very considerable. In the male fiddler-crab, after his attainment of sexual maturity, the proportion of the weight of the large chela increases from 43 per cent, of rest-of-body weight to nearly 62 per cent. — an increase of nearly 45 per cent, in relative size.
No two male Uca pugnax have the same proportions unless they happen to be of the same absolute size : any diagnosis made on the basis of percentage measurements of chelae (and also, though much less markedly so, for other organs such as the pereiopods) would be valueless. But in spite of the fact that the form of the animal is continually changing, it does so in an orderly way ; and though percentage values for the limbs have no diagnostic significance, the constants in the growth-ratio formula are true specific characters. In a word, the systematist needs algebra as well as arithmetic in making any diagnoses based upon the size of parts of the body.
Note. — S. A. Allen (1894), Amer. Mus. Nat. Hist. Bull., 6, 233, also finds a progressive change of proportions in a rodent (see p. 40). Neotoma shows a steady increase of dolichopy and dohchocephaly with increase of absolute size.
CHAPTER II
THE COEFFICIENT OF CONSTANT GROWTH- PARTITION ; AND SOME SPECIAL CASES
§ i. The Heterogony of Deer Antlers
THERE are certain special cases so important to a study of relative growth that they deserve a chapter to themselves.
The first is that of the antlers of deer. As is well known, these are shed each year, and replaced the year after by a totally new growth. Usually, each new growth is larger than the preceding growths, but in old age, illness, or other especially unfavourable conditions, the weight (and number of ' points ') may decrease. An analysis of the normal growth of the antlers of a number of individual red deer (Cervus elaphus) and of the factors affecting that growth, is given in Huxley (1926).
Further, casual inspection is sufficient to indicate that relative antler-weight increases with absolute body-weight, as is stressed by Champy (1. c). To obtain quantitative data, however, was not easy. After much search, I hit on the papers of Dombrowski (1 889-1 892) published many years ago in an obscure periodical — the only papers to my knowledge to contain the body-weights and antler-weights of large numbers of Red deer and Roe deer. These data, supplemented by those of Rorig (1901), by scattered cases in the literature, and by information privately supplied to me by sportsmen, have now been analysed by me (Huxley, 1927, and 1931). It appears quite definitely that although there may be much individual variation even in one locality, and though extraneous agencies such as the amount of lime in the soil affect relative antler-weight considerably, yet when the mean of considerable
42
|
¥-&- |
■G |
||||
|
p |
|||||
|
f\*l |
|||||
|
V |
|||||
|
/ Gi |
|||||
|
cpw |
|||||
70 I0O
Kg Body wt
\5Q
20 0 250
Fig. 25. — Antler-weight against body-weight in 527 adult red deer (Cervus elaphus) ; logarithmic plotting. See Table II.
k, except for the last two points, is close to i-6.
43
44
PROBLEMS OF RELATIVE GROWTH
numbers is taken, the results approximate to the formula for
constant differential growth-ratio.1
This applies to adult ani- mals. When antler-growth is taken by age for single individuals, it will be seen that the differential growth- ratio of the antler-weight is not constant, but declines steadily with age, being first about 3-0, and declining to close to i-o. (When regres- sion of body-weight occurs, it appears certain that regres- sion in antler-weight accom- panies it, though it cannot yet be stated whether the regression is heterogonic.)
It is doubtless affected also by numerous subsidiary fac- tors such as abundance of food and specific dietary
1 A discrepancy occurs as re- gards the antlers of those beasts with highest body-weight ; the weights of these when expressed as relative (percentage) weights, fall below those of the body-size class next below. This appears to be merely a classificatory phenomenon. There being con- siderable individual variation as to what we may call the par- tition-coefficient of material be- tween antlers and body, those animals with the very largest body- weights are likely to repre- sent extreme variants in the direction of heavy body but light antlers. Further, and poss- ibly more important, since body- weight is extremely variable owing to fluctuations in amount of fat, most very heavy beasts are likely to owe their exceptional weight to exceptional nutritive con- ditions ; and therefore their relative antler-weight will go down rela- tively to this excess of fat, which is presumably without immediate significance in determining antler-size.
|
/ |
|||||
|
/ |
|||||
|
/ |
|||||
|
/ |
|||||
|
n |
|||||
|
/ |
|||||
|
5 |
// |
||||
|
tj |
|||||
|
/ f |
|||||
|
4 |
/ / i t / |
/© |
|||
|
3 |
/ |
/ J |
|||
|
So' |
/ |
||||
|
i 2 |
|||||
|
c |
t i |
||||
|
■s |
i |
||||
|
So |
/ / |
||||
|
1 |
/ / t |
||||
|
0-5 |
r |
60
80
ICO 130
Kg. Body wt.
170
Fig. 26. — Red deer, antler-weight
against body-weight at various ages ;
logarithmic plotting.
The antler- and body-weights for the first 7 years of life (in stags from Warnham Park, Sus- sex) are plotted. The curve (solid line) bends over and approximates to the straight line curve (dotted line) for an tier- weight against body- weight in adults (see Fig. 25).
k begins with a value of 3-0 or over, and declines to i-6 or under.
HETEROGONY OF DEER ANTLERS
45
TABLE II
Body-weight, Antler-weight, Point-number and Relative Antler- weight of 527 Red Deer shot in various parts of Europe (392 collected by Dombrowski ; 10 by Baillie Grohman ; 125 by Huxley. Analysed by Huxley, 1931A)
Arranged by body-weight classes, all of 20-kg. interval (except the last class, of 40-kg. interval) . Note that for Classes 2 to 8 (comprising over 90 per cent, of the animals) the relative antler-weight rises steadily with increasing body-weight, k for antler-weight (except for the last two classes) is about i-6 ; b = -00162.
|
kg. Class body- weight |
No. of specimens |
Mean body- weight kg. |
Mean antler- weight kg. |
Mean point- number |
Relative antler- weight per cent. of body |
Relative pt. no. • pt. no. n Vbody-wtJ |
|
I 60- 80 |
19 |
74-4 |
1-64 |
7-50 |
2-20 |
o-ioi |
|
2 80-100 |
119 |
93-4 |
2-03 |
820 |
2-17 |
0-088 |
|
3 100-120 |
106 |
110-4 |
3-16 |
9-81 |
2-86 |
0-089 |
|
4 120-140 |
113 |
130-6 |
3-96 |
11-64 |
303 |
0-089 |
|
5 I40-160 |
65 |
1489 |
478 |
1 1 74 |
3-21 |
0-079 |
|
6 160-180 |
29 |
1707 |
6-21 |
13-10 |
364 |
0-077 |
|
7 180-200 |
33 |
191-1 |
7-28 |
14-77 |
3-8i |
0-077 |
|
8 200-220 |
18 |
2II-8 |
8-91 |
I5-4I |
4-21 |
0-073 |
|
9 220-240 |
H |
231-7 |
879 |
13-62 |
379 |
0059 |
|
IO 240-280 |
11 |
259-1 |
863 |
1378 |
3*33 |
0-053 |
TABLE III
Body- weight, Antler-weight, Point-number and Relative Antler- weight of 405 Roe Deer shot in various parts of Europe (data from Dombrowski ; analysed by Huxley, 1931A)
k for antler- weight = about 0-57 ; b = -0455.
|
Class, Body-weight |
No. of specimens |
Mean body- weight kg. |
Mean antler- weight g- |
Mean point- number |
Relative antler- weight per cent, of body |
|
1-5 (13-18 kg.) 6-12 (19-25 kg.) 13-15 (26-39 kg.) |
127 254 24 |
16-6 20-9 28-3 |
225-7 257-0 306-5 |
5-92 6-05 6-21 |
1-36 1-23 1-08 |
46
PROBLEMS OF RELATIVE GROWTH
factors. The reason that the weights for adults fall upon the line corresponding to a constant growth-ratio with k = about i-6 appears simply to be that during the decline of the antler's growth-ratio, a rather narrow range of values for the growth- coefficient is attained during adult life (see Fig. 26), the great majority falling between say i-8 and 1-4.
o-4
0-3
SJ
0-2
1 >
/ /
/
f / ^ / S^
js£
s /
_^£L y
^^ /
/ / / /
/
/
. L
30
40
Fig. 27.
15 20
Kg Body- wi
-Relative size of antlers in adult Roe-deer (Capreolus caprea).
Solid line, antler-weight against body-weight in 405 Roe-deer ; k = 0-57. Dotted line, prolongation of corresponding curve for adult Red Deer (see Fig. 25). Logarithmic plotting. See Table III.
Corresponding data for the Roe deer (Capreolus capreolus) gave what at first sight appeared a quite paradoxical result — namely a decrease of relative antler-weight with increase of absolute body- weight among adult males (Fig. 27). There is thus negative heterogony of the antlers, with a growth-co- efficient of about 0-57. Reflection suggests the probable ex- planation. There is no reason why the decline in the antler's growth-coefficient with age should not in another species proceed much faster than in the Red deer, and reach a stage
HETEROGONY OF DEER ANTLERS
47
where by the attainment of maturity it was normally below i*o. This purely quantitative difference in the rate of change with age would suffice to explain the apparently contradictory results (Fig. 28). As to the biological causes underlying this
Kg. Body wt
o-i
hO-08 0-06
1-0O4
8 10 10 30 40 60 80 100 150 200
Fig. 28. — Diagram to compare probable method of antler-growth in Red and Roe-deer ; logarithmic plotting.
X — X, the curve for adult red deer (see Fig. 25). A — A and B — B, probable curves for individual antler-growth with age in a small and a large specimen respectively. Y — Y, the curve for adult roe-door (see Fig. 27). C — C, probable curve for individual antler-growth in a typical roe-deer specimen, rising at first more rapidly, but then sinking much lower than the corresponding curve for red deer.
48
PROBLEMS OF RELATIVE GROWTH
quantitative difference we can only speculate : it would seem probable that the Red deer type of slow decrease, with positive heterogony throughout, is the normal course of events in Cervidae, but that it was for some reason biologically desirable for the Roe deer to have small antlers.
12 13 14
Fig. 29.
Solid line, body-weight against age in 212 male red-deer from Wamham Park. Dotted line, antler- weight against age in a smaller and selected group of stags from the same locality. The thin continuous lines below the curve for antler-weight represent diagrammatically the actual growth and shedding of the antlers year by year. The fact that the antler curve inflects later than that for body-weight is probably due to the antlers being from a selected group of beasts, of size above the average.
After this digression, we will return to the general problem involved in the growth-ratio of the antlers. We have seen that the apparent constancy of their growth-ratio, obtained by plotting antler-weight against body-weight in adults, is shown to be a particular consequence of the steady decline of
THE COEFFICIENT OF GROWTH-PARTITION 49
individual growth-ratio with age. But even this does not exhaust the complexity of the phenomenon. The curve for age-change of growth-ratio is obtained by plotting the weights of fully-formed antlers of known age against body-weight for the same age. It will be at once clear that the actual growth- ratio of the antlers can never be the same as that thus obtained, but must always be higher (Fig. 29). For the points on the curve are those which would be obtained if the antler grew with a constant differential growth-coefficient from its incep- tion ; whereas actually, it has to begin its growth anew each year from zero.
Now this is of considerable importance, since it indicates that it is not necessarily the actual rate of growth which is regulated in accordance with our formula, but the limitation of the total amount of growth achieved. What our results tell us is that at any given body-size the total amount of material which can be incorporated in the organ is proportional to the body-size raised to a power (the exact value of the power also varying with age). The mechanism of this relation is at present obscure. We do not know whether the total bulk of material in the body imposes the relation directly, which is unlikely ; whether some substance is formed in the body in this particular quantitative relation, as an exponential function of body-weight, and the final size of the organ is then directly proportional to the amount of this substance ; or whether there be after all a true constant differential growth- ratio between organ and body, determined by some peculiarity of the organ, but that this growth-ratio represents a limiting value, higher values being possible and indeed necessary whenever the relative size of the organ is below its limiting amount. The last supposition is perhaps the most probable, on the close analogy with regeneration (see below) , but experi- ment alone can decide the point.
§ 2. The Coefficient of Constant Growth-partition
In any case, to speak simply of growth-coefficients in such a case is misleading ; yet we require a term for the exponent of body-size according to which relative organ-size is limited. Two terms are possible — either coefficient of growth-limitation, or else growth-partition coefficient ; I shall adopt the latter.1
1 Since writing this passage, I find that Robb (1929) had previously suggested the same idea of growth-partition, which has later been adopted by Twitty and Schwind (193 1).
4
50
PROBLEMS OF RELATIVE GROWTH
In general, it would appear that the existence of a growth- partition coefficient is the most fundamental fact in consider- ing relative growth of parts, and that when true constant growth-coefficients or constant differential growth-ratios are found, they represent special limiting cases of this more general conception.
Let us now consider three further examples which support this conclusion. I have mentioned regeneration. I shall deal with this more fully in a later chapter. Here it suffices to recall the fact that in an animal capable of full regeneration, any organ, heterogonic or not, will, after amputation, be restored in favourable conditions to its normal proportionate
II III IV V VI VII VIII IX x XI XII
50 40 3 0 - 20- 10
126
Fig. 30. — Decrease of growth-coefficient during regeneration in the legs of
Sphodromantis bioculata.
The abscissae represent moult-stages. The ordinates are growth-quotients : i.e. the ratio of the length of the leg at a given moult to its length at the preceding moult. The dotted line represents
the mean growth-quotient (1-26 = V 2) for normally-growing limbs. The solid line is the curve for a middle-leg amputated before the Illrd moult (mean of 3 specimens).
size. In other words, during the process of regeneration its growth-ratio will be much higher than normal, and will gradu- ally sink until it reaches the normal value, at which it will then continue. This emphasizes the generally accepted idea that regeneration is simply a special case of growth, and furthermore makes it clear that here at least the normal growth- ratio of an organ merely represents a limiting value. Thus, as was suggested above with regard to deer-antlers, relative size of organs appears to be determined in the first instance as an equilibrium between amount of material in the organ and amount of material in the body, the equilibrium being determined according to our general formula y = bxk. If the
THE COEFFICIENT OF GROWTH-PARTITION 51
equilibrium be upset, regulation towards the equilibrium position will occur during later growth. The particular mechanism by which the equilibrium is attained does concern growth-ratio ; the more the organ is below equilibrium-size, the higher will be its growth-ratio.1 (See Fig. 30.)
These conclusions are supported by various lines of evidence. In the first place, in cases of grafting of organs we should expect the organ of a young animal grafted on to an older and larger animal to be accelerated in its growth until it reached a size prescribed by its growth-partition coefficient, and the organ of an older animal grafted on to a younger and smaller animal to be correspondingly retarded in its growth. For the first, we may turn to the results of Wachs (1914). When he inserted the lens of a young Urodele larva into the eye of an older larva from which the lens had been previously removed, the small lens was accelerated in its growth. For the second, as well as the first, we have an example in the work of Twitty (1930). Here cross-trans- plantation was made between larvae of Ambly stoma tigrinum and A. punctatum. The latter species grows much more slowly than the former. Twitty removed the eye of a punctatum larva and replaced it by one of the same size from a tigrinum larva ; owing to the higher growth-rate of tigrinum, the age of the donor was much less than that of the host. Even when the host was fed only minimally, the grafted eye now increased in size much more rapidly than that of the host : in one case it increased 50 per cent, in diameter while the host remained stationary in length. (See p. 197.)
The converse experiment consisted in removing the eye from a tigrinum larva and engrafting in its place an eye of the same size from a considerably older punctatum larva. In this case, the grafted eye made very slow growth. Here the rate of growth could be compared with that made by punctatum eyes grafted into A. tigrinum during the embryonic period.
1 In some cases at least the change in growth-ratio will occur accord- ing to the law enunciated for Sphodromantis by Przibram, 1917 : When Z is the normal final length of the regenerating limb, n the length after amputation, r its length at the beginning of a given period of time t, R the length at the end of the time t, Va the normal coefficient of increase of the limb between one moult and the next,
then Z — n — y = — — ; and the growth-partition coefficient
represents the limiting value of the growth-ratio when equilibrium is established.
52
PROBLEMS OF RELATIVE GROWTH
It can at once be seen (Fig. 31) that the growth of the older eyes slowly approaches the normal growth-curve (see also Figs. 85 to 87, and especially 88).
2.0
mm.
i.TS
ISO
UJ >
UJ
IL O
arm
UJ
i- uJ Z d
1.0
•is
20 30 HO SO 60
BODY LENGTH
70
£0
90 mm
Fig. 31. — Regulation of eye-size in eyes of one species of Amblystoma grafted
on to another.
Dotted lines, curves for two cases when embryonic eyes were grafted to a host of the same stage of development as the donor. Solid lines, curves for two cases when eyes were taken from an older larva and grafted on to a younger host larva ; in this case the eyes hardly grow at all until they reach the correct relative size.
We thus are driven to the conclusion that though the eyes of the two species have different specific growth-intensities, and therefore different coefficients of growth-partition when both are present in the same body, there is for each body-size
THE COEFFICIENT OF GROWTH-PARTITION 53
carapace length, mm 13 16 20
a characteristic eye-size towards the attainment of which the rate of eye-growth is regulated.1
In general, as is well known, the rate of regeneration is higher the more material is removed. This fits in with the ideas here presented and with the concep- tions of Przibram (1. c), but it throws into relief the very real difference between the idea of a constant differential growth- ratio and a constant coefficient of growth-limitation. Let us consider a Planarian worm, in which according to Abeloos (I.e.) the trunk grows heterogonically with reference to the head. During normal growth the rela- tion is one of a constant differ- ential growth-ratio. But dur- ing regeneration, the more is cut off, the more rapidly regen- eration takes place : i.e. the smaller the fraction of the body left, the more rapid is the growth-ratio of the regenerate. What is constant is the final partition-coefficient between head-material and trunk- material ; the normal constant differential growth-ratio is the special case of growth during which the partition-coefficient is always of this limiting value.
Finally, the case of Inachus, investigated by G. Smith (1906A) whose data have been further analysed by me (unpublished) also supports this view-point. At Naples, /. mauritanicus , the species of Inachus studied by Smith, shows three forms — ' low '
1 Further details as to the specific growth-intensities of eyes and other organs when heteroplastically transplanted are recorded in Chapter VI.
Fig. 32.- — Chela breadth against carapace length in the male of the spider-crab, Inachus mauritanicus ; logarithmic plotting. (See Table IIIa.)
The ' low ' males (below about 14 mm. cara- pace length) and the ' high ' males (above about 20 mm.) fall on a single curve with diminishing growth-coefficient (mean value of k over 2-3 ; for ' low ' males, about 2-6; for 'high' males, about 1-4). Between these sizes the chela regresses to a narrow female type, then enlarg- ing again with a very high growth-coefficient.
(Constructed from the data of G. W. Smith, 1906A.)
54
PROBLEMS OF RELATIVE GROWTH
00
s s
53
s
53
oo S
<o
5S
s
<
o
i
o
«
O
w >
I— I
H
<
o
O
ty>
|
p |
CD |
|
|
»— i |
l-l |
|
|
eu |
* |
|
|
tfl |
o CD |
|
|
H |
O |
|
|
X |
||
|
H |
o |
|
|
i— i |
In |
|
|
i— i |
o |
to o |
|
vO |
Ph |
|
|
fa o |
vO~ |
|
|
H |
4) |
|
|
W |
rQ |
|
|
H |
cd |
|
|
Q |
H |
|
|
< |
||
|
W w 1 |
a o |
|
|
<; |
||
|
i-l |
T) |
|
|
W |
<D |
|
|
u |
+-> O P i-i |
|
|
In |
-t-> |
|
|
o |
||
|
H |
o U |
o
vO iO " CM M
<N H £
M M O
ro CO ^oo
O*
M N N n
N ■*" ~
On
On cs co^
o
O ff)m CN CN •
vO
M N •
in vo
CO fON
M <N ^
K. -. ^
M CN •
rO
o o r^
m co • ro
iooo § m ro •
|
tJ- On |
lO -t- |
|
|
M |
Tf |
to ro. |
|
ro |
M |
On |
|
H |
vO |
<N |
o
H t' CM
w N o
H M ■
CM O N ^
O o CM 9
■ CO -
g P i-
eK
■> -i-i Co co o — '
!2 <" 22 o <" u
-i O
a a
5£ S
1
On
o°P
co
m ■ CO
ST*
co
VO 8
<°cO
HI •
to
o> 3 9
to
o "9
lO
53
00 00 CO
53
P U cd _q
I-
O •— " - CU £ ^ 53 CUXi oTt^ CO O
*-•*> s o S -O .o
■5V
a s£
k;-?
p
CD
|
oo to |
co |
|
^^ Co 53 <A |
P in CU £) |
|
S 53 |
H _, |
|
^ *^~ |
■P rt |
|
<-> TJ |
|
|
<» -o |
cd « |
|
^rS |
a-i-1 |
|
^^ |
CO CJ |
|
•V1 O |
*•"< _, |
|
° s |
|
|
»Si .^ |
. rt |
|
O CD |
o
l-(
o
CO
ro
M CO
VO
>o
CO
ro
ro m
O M
lO (N
■^" o>
CO
N vO
|
co" CO |
CO p |
||
|
rt |
CD |
||
|
-*oi |
-• O |
a |
|
|
53 |
0i |
11 |
|
|
« b |
Ph |
||
|
hQ |
s -° |
CO |
|
|
« |
5^ M <U CO |
a |
O |
|
a |
6 |
||
|
o |
u |
14 |
"53
HOLOMETABOLOUS INSECTS 55
males, of small size with relatively small but male-type chelae, ' high ' males of large size, with relatively large male-type chelae, and those of intermediate size, which have extremely small, female-type chelae.
When chela-size is plotted double-logarithmically against body-size, it is found that the means for the ' low ' and ' high ' males fall on two segments of a single simple curve, thus con- firming Smith's view that these two types are merely breeding males in their first and second seasons respectively, and that those with female-type chelae are males in the non-breeding phase, during which the secondary sexual characters of their chelae have regressed to the female or neuter type.
Further, on the double logarithmic plot, the curve for these intermediate males first actually declines, and then mounts very steeply until it meets the prolongation of the straight-line curve for the low males, upon which it bends over and con- tinues as the line for the ' high ' males. In other words, after the regression period, the growth-ratio of the claw is much higher than normal, but becomes normal as soon as the theor- etical equilibrium-size is reached.1 (Table IIIa and Fig. 32). It is also seen that the frequency for chela-breadth is bimodal : this will be discussed in § 5.
§ 3. HETEROGONY IN HOLOMETABOLOUS INSECTS
A somewhat different set of special cases is that provided by holometabolous insects. Many of these possess organs (usually of secondary sexual character, and these usually in the male sex), which increase in relative size with increase of absolute size of body. The most familiar of these are the mandibles of the stag-beetles (Lucanidae) and the ' horns ', cephalic or thoracic or both, of various other beetles such as the Dynastidae ; but Champy (1. c.) has collected numerous other examples, ranging from antennae (e.g. Acanthocinus : Champy, 1924, p. 167) and forelegs, to the ' tail' on the hind wing of Papilios and the swollen segments of the hindlegs in certain Hemiptera, such as Anoplocnemis (Champy, 1924, p. 173). See Figs. 33, 34, 91.
Analysis shows (Huxley, 1927 and 1931) that the relation between the dimensions of the organ and the body here too
1 It is interesting to find that in I. dorsettensis, studied by Shaw (1928), the regression towards female type in the non-breeding season, though present, appears to be much less marked.
56
HOLOMETABOLOUS INSECTS
57
approximates closely to the formula for a constant differential growth-ratio (Table IV, Fig. 35). Again, however, there can be no growth-ratio in the literal sense in which we have found
Fig. 34. — Heterogony of the ' tail ' in the male of the swallow-tail butterfly,
Papilio dardanus (the heterogony is stated by Champy not to occur in species
in which the ' tail ' occurs in both sexes).
it apply, e.g., to the partition of growth-potential between the large chela of Uca and the rest of the body. There cannot be, for the simple reason that in holometabolous insects the organ, as regards its imaginal characters, is not formed until
58
PROBLEMS OF RELATIVE GROWTH
the pupal instar, to emerge at the final moult in its definite shape and size. And as there are no further moults, it is incapable of further growth or form-change.
We are thus driven to suppose either that all the processes connected with the organ's heterogonic growth are confined to a very brief period, presumably just before and just after
the moult from last larval in star to pupa ; or else that, although the visible growth of the organ is confined to this short period, it depends for its amount on some substance whose chemical ac- cumulation during the larval phase has had a constant differential growth -coefficient relative to body- weight (see e.g. Teiss- ier, 1 93 1, for a confir- mation of this latter possibility).
As we shall see later in considering dimor- phism (p. 68) the for- mer hypothesis is the more probable ; but in any case we have, as in deer's antlers, the fact that the amount of growth attained is proportional to body- size raised to a power, the value of the power
30 40 50607080
Total length, mm.
Fig. 35. — Relative growth of male mandibles in three species of stag-beetles (Lucanidae).
-f , Lucanus lunifer ; X , L. cervus ; 0, Cyclommatus laran- dus. ' Total length ' is true total length for Cyclommatus ; for the others it is represented by (elytron length + mandible length). All the curves inflect at large absolute sizes (see text) ; for the remainder of the curves k is about 1-6 for L. lunifer, 2-3 for L. cervus, and nearly 2-0 for C. tarandus.
being equivalent to that of the constant differential growth-ratio in cases where visible growth is continuous over long periods. Thus the true growth-ratio of the organ during its short growth-period is far more rapid than indicated by the value found for the ' growth- coefficient ' by the method of comparing organ and body at different absolute sizes. We are, in fact, again in the same
HOLOMETABOLOUS INSECTS
59
predicament as with the deer's antlers, the chelae of male Inachus, or any heterogonic organ which is regenerating, and are driven to think of a limiting factor to growth, which we have defined as the growth-partition coefficient.
TABLE IV
Mandibles in Lucanidae
(a) Cyclommatus tarandus
(data from Dudich, 1923 : ana- lysed Huxley, 1927)
2
k
x 20-38 24-01 26-38 27-76 29-65 32-20
33-n 35-oi 36-13 37-32
38-44 39-26
41*34 43-22
45-51 46-32 47-28 48-40 50-04 5i-5o 52-50 54-23 56-01
62-06
66-06
69-00
74-00
178
1-97, b
y
3-88 5'3i 6-33 7-32 8-17
9-73 10-71 11-49 12-08 12-73 14-11 14-70
15-84 17-39 18-83 19-19 19-92 20-79
21-53 22-54
23-25 23-96
25-38
28-49 30-69
32-00
34-50
= just over o-oi
(b) Lticanus cervus
(data from Bateson and Brindley, 1892 : analysed Huxley, 1927)
x
(3i-o)
38-65
40-50
42-55 45 -oo
46-93 49-18
53-6o
2 = 48
y
(6-o)
7-75 9-00
io-oo
II-20
n-86 12-82 14-40
k = about 2-3
Lucanus lunifer (data and analysis, Huxley, 1927 )
|
X |
y |
|
|
(38-6) |
(16-8) |
|
|
42-4 |
19-9 |
|
|
45-6 |
22-4 |
|
|
49-3 |
24-5 |
|
|
5i-9 |
26-0 |
|
|
57-3 |
28-8 |
|
|
2 = |
18 |
|
|
k = |
aboul |
: i-55 |
Figures in brackets ( ) indicate insufficient number of individuals in class.
In L. cervus and L. lunifer, x = 'total' length = (elytron length + mandible length), mm.
In C. tarandus, x = true total length = (body length + mandible length), mm.
y — mandible length, mm.
k = growth-partition coefficient : the values given do not hold at high values of x (see text).
60 PROBLEMS OF RELATIVE GROWTH
There is a further point to consider in regard to heterogony in holometabolous forms. In other organisms — a fiddler-crab, for example — the growing system is an open one, in that it is continuously taking in food as it grows. The beetle or other holometabolous insect, however, during most of the period when the form of its adult organs is being laid down, is a closed system, taking in no further food, but depending on accumulated reserves and on the material derived from the breaking down of larval organs. This has two conse- quences for our problem. In the first place, in other forms the fairest comparison of relative size would seem to be between heterogonic organ and rest-of-body, since the size of the ingestive and digestive systems are functions of the size of the rest of the body, not of total size, as may easily be realized by reference to the fiddler-crab. Here two large specimens, male and female respectively, of equal rest-of-body weight and therefore presumably equal-sized alimentary systems, will be of very different total weight, since in the female either chela will weigh only 2 per cent, of the rest-of-body, while in the male the large chela may weigh up to 70 per cent, or more. But in a stag-beetle, for example, the conditions are quite different. Larvae of both sexes have jaws and guts of the same relative size. A male and a female larva of the same total size will have the same amount of reserve material, but during the pre-pupal and pupal period, say 1 per cent, of this is converted into imaginal female mandible, while perhaps 10 or 15 per cent, has to be converted into imaginal male mandible. Since the amount of reserve material is here the important factor for growth, it is total bulk, and not rest-of- body bulk, which should be here used as the standard against which to plot the bulk of the heterogonic organ (see Huxley, 1931c).
This is a minor methodological point ; but it has further consequences. To continue the example of stag-beetles, we should accordingly expect, in an exceptionally large male specimen where the theoretical relative size of the mandibles would be huge (as long as the rest of the body in some speci- mens of Cyclommatus tarandus : Dudich, 1923 ; see Fig. 91), that during the longer time necessary to lay down this large organ, the limited reserve-supply would come to an end, used up by other competing organs, and therefore that the organ would fall below the theoretical size expected on the formula for a constant coefficient of growth-partition. Thus, owing to the
POLYMORPHISM IN NEUTER INSECTS 61
limitation of raw material due to the system being a closed one, we should expect, from a certain absolute size upwards, the actual values for the size of the heterogonic organ to fall progressively more and more below the theoretically expected value. And this is what we actually find. When organ-size is plotted against total size on a double logarithmic grid, the first part of the curve is a good approximation to a straight line, but the end portion curves over so as to be concave to the x-axis. This is so for all cases so far investigated, includ- ing the mandibles of three species of two genera of stag-beetles, the horn of Xylotrupes, the heads of polymorphic neuter ants, etc.1; Figs. 35, 37, 92.
§ 4. Heterogony and Polymorphism in Neuter Social
Insects
We shall later note some further complications introduced into the situation by the fact of moulting. Here we may refer to the particularly interesting case, just mentioned, of poly- morphic neuter ants. It is well known that in what are apparently the more primitive examples of such polymorphism, there is an unbroken array from smallest to largest neuters, the continuous series being quite arbitrarily divided up into ' worker minimae ', ' worker mediae ', and so on up to ' soldier mediae ' and ' soldier maximae ' ; and some myrmecologists have introduced even more elaborate terms (see Wheeler, 1920). Now these series are invariably characterized by a relative increase of head- and especially mandible-size with an absolute increase of total size. Measurements of the weights of head and rest-of-body in species of two genera (the huge Camponotus gigas, from Borneo ; and a driver ant of the genus Anomma from Africa) show that, over the major portion of the size-range, the formula for constant growth- partition coefficient is nicely adhered to 2 (Huxley, 1927 ; Huxley and Bush unpublished) (Table IVa and Fig. 37).
1 Teissier, 193 1 (p. 97), using weight and not linear measure, shows in his Fig. 20 no curving over of this type for the mandibles of Lucanus cervus. This might mean that my interpretation is wrong, and that mechanical reasons are interfering with great growth in length rather than nutritive reasons with growth in mass. On the other hand, the curvature in my material does not begin until elytron-length 33 mm., and Teissier has hardly any specimens as large as this.
2 The Anomma curve bends over at high sizes, as described for stag- beetle mandibles, etc. This may presumably be accounted for as suggested earlier in this chapter. But the formula is also not obeyed
62 PROBLEMS OF RELATIVE GROWTH
TABLE IVa
Anomma nigricans
Data from Huxley and Bush (unpublished)
Analysed in Huxley (1927)
(1383) (165-3)
176-0 223-7
220-3 300-2
272-7 427-6
324-2 567-3
367-5 661-9
2 = 267
k = about 1-55 (after first 2 points)
x = abdomen-length
y = head-breadth
(in arbitrary units) Figures in brackets ( ) indicate insufficient numbers in class.
Camponotus gigas
Data from E. Banks (unpublished)
Analysed in Huxley (1927)
x y
75-0 9-6
in-5 24-3
241-0 82-0
346-8 142-0
S =357
ft — about i-6 (after first point)
x = total weight, mg.
y = head-weight, mg.
The obvious suggestion is that these series of workers and soldiers do actually represent nothing more than a series of size-forms of a single genetic type, possessing a mechanism for heterogony of the mandible and head. The difference from the other holometabolous cases hitherto considered is that the absolute size-range is much greater, and that the differ- ences in size can only be supposed to be brought about by definite treatment of the larvae by their nurses, the largest types being fed to the limit, the smallest types being deprived of food, and so forced to pupate, while still quite small larvae. That enormous differences in size may be produced by cutting down or cutting off the food supply of insect larvae is estab- lished through experimental work on blowflies (Smirnov and
at very small absolute sizes, where the double-logarithmic curve is distorted in the opposite sense, with concavity upwards : the meaning of this, if not merely statistical, is unknown.
POLYMORPHISM IN NEUTER INSECTS
63
Zhelochovstev, 1926), housefiies (Herms, 1928), Drosophila (Gause, 1931), etc. And the postulated differential treatment of different worker larvae by their nurses is well within the known range of complexity of ant behaviour (see also Emery, 1921).
Fig. 36.-
-Increase of relative size of head with absolute size of body in the neuters of the ant, Pheidole instabilis.
That the effect of size-changes may be differential is also known ; e.g. Eigenbrodt (1930) finds that the increase of total size in Drosophila caused by low temperature is accompanied by a decrease in wing-size, which is somewhat greater for breadth
-zoo
(A) Heterogony of the head in neuter ants; logarithmic plotting. © — ©, Anomma nigricans, head-breadth against abdomen-breadth (in arbitrary units : scale
below and to the right). H h Cam-
ponotus gigas, head-weight against total weight, in mg. (scale above and to the left).
(B) Relative head-size, Anomma nigri- cans. Increase in percentage head-breadth with increase in absolute abdomen-breadth. (In this figure the classes into which the specimens were classified were delimited on the basis of [abdomen-breadth + head- breadth].)
140 ZOO
%
180
170
160
140
130
Anomma nigricans
B
200 250 300
cubdomen breadth
350
Fig. 37,
-Graph showing the increase of size of head with absolute size of body in neuter ants.
64
POLYMORPHISM IN NEUTER INSECTS 65
than for length ; Alpatov (1930) rinds the same phenomenon. Smirnov and Zhelochovstev (I.e.) for the blowfly Calliphora find a differential effect in different regions of the wing when total size is reduced by cutting off larval food-supplies after a given time. On the other hand, Alpatov (1. c.) in flies of the same species reduced in size by depriving larvae of food before they reached full growth, also finds a decrease in relative breadth of the wings, showing that absolute size is not the only factor deter- mining proportions.
When, as in some more specialized forms, the neuter series is not continuous, but there exist only two or a few fairly sharply defined types — e.g. only large and relatively large- headed soldier and small and relatively small-headed worker — we need only suppose a slight specialization of the nurses' behaviour. An adumbration of this is seen in the fact, elicited by unpublished work of Miss Edmonds, of Sydney, that even in forms with a continuous series of neuters, the frequency curve for body-size is definitely multimodal (see also Palen- itschenko, 1927).
It is to be hoped that those familiar with the technique of rearing ants in captivity will attack this problem experi- mentally. If my suggestion be verified, it will materially simplify the evolutionary problems connected with the poly- morphism of ant neuters, for instead of having to postulate a large number of genetically distinct types, we need only assume one single genetic type of neuter, but provided with a heter- ogenic mechanism for head-growth, which will produce all the different head-types as secondary by-products of the animals' absolute size.
In termites, the case is somewhat different.1 Within the soldier caste of primitive termites, we do sometimes meet with phenomena which appear to be quite parallel with what I have discussed for ants — considerable variation in absolute body-size accompanied by a progressive alteration in relative head- and jaw-size. This is well shown in the figures given by Heath (1927) ; see Fig. 38. Heath also (see his p. 402) demon- strates fairly conclusively the interesting fact that the poly- morphism is due to the presence of forms which have gone through different numbers of moults ; a fact which strengthens our hypothesis that there is but one genetic type with a heter-
1 For much of the information concerning termites I am indebted to Professor A. E. Emerson, of Chicago University, to whom I should here like to express my thanks.
5
66
PROBLEMS OF RELATIVE GROWTH
ogonic mechanism as regards head-size. The different soldier forms are discontinuous as regards size and head-proportions, which provides us with a further case of di- or poly-morphism due to a combination of heterogony with moulting (§ 5). Reference may also be made to some of the results given in the important paper by Kalshoven (1930), and in that by Hare (1931). Such a method of arriving at polymorphism of neuters would, of course, be impossible in the holometabolous ants. (See also the work of Light, p. 258.)
There are however other termite cases in which my hypo- thesis would not seem to apply. For in- stance, A canthotermes acanthothorax (Sjostedt, 1925 : his Fig. 21, p. 61) has certainly two qualitatively different types of soldiers, which it would appear necessary to regard as differ- ent genetic types, or at least as produced by qualitative differences in feeding. Sjostedt himself figures no less than five kinds of soldiers, the largest number of types decribed for any species of Termite. Two of these, his Nos. 4 and 5, would appear to be growth- forms of one main type. To inspection, his Nos. 1-3 look as if they were growth-forms of another qualitatively different type ; but Dr. Emerson informs me that he has found that the smaller forms (Sjostedt's Nos. 2 and 3) are both infested with an insect larva which inhabits the head, and disproportionately reduces its size ; these forms are not found in nests from which the parasites are absent. The differential retardation of the head might be due to the existence of a true heterogony- mechanism for the head, which is not nor- mally manifested, but is revealed when the parasite produces general size-reduction of the imago ; or it might be due merely to the fact that the head is the seat of the infestation.1 The difference between worker and soldier would appear
1 In passing, we may note that the effects of such parasites may be very striking ; e.g. in Termes gilvus, Silvester (1926) figures extra- ordinary qualitative changes in shape of head and jaws, as well as a general size-reduction and a highly disproportionate reduction in jaw- size, as result of the presence of a similar parasite in the head. And see, for a discussion of the corresponding problem in ants, Wheeler (1928) and Vandel (1930).
Fig. 38. — Heads of largest and smallest workers in a colony of the termite Ter- mopsis angusticollis, showing heterogony and change of pro- portions with in- crease of absolute size.
POLYMORPHISM IN NEUTER INSECTS 67
to be of another nature. Recent work such as that of Emerson (1926), John (1925), Heath (1927, 1928), etc., makes it highly- probable that in all primitive termites, the forms which do the work of the colony are not a distinct caste, but are the juvenile forms of the soldiers. There is thus a marked heterogony of the head and jaws between the last worker instar and the first soldier instar (we have already seen that there may be more than one soldier instar : Heath, 1927 : see also Hare,
I93I-)
In more specialized forms, however, while some of the
workers are juvenile soldiers, a true worker caste, distinguish- able by large size and different proportions, also exists (Emerson, 1926, etc.) It would be extremely interesting to measure the head and body of these two types throughout their growth.
These results, combined with the fact that in some primitive forms, fertile soldiers are met with (Heath, 1928 ; Imms, 1920), and that soldiers with wing-buds are not unknown, whereas no true workers are known ever to be fertile or to possess wing-buds, indicates that workers have been derived from soldiers by a suppression of their final development into the normal big-jawed type — that, in fact, they are neotenic. This neoteny, however, possibly at first facultative, must at least in higher forms have been fixed as a constant caste-character- istic, either by differential feeding or perhaps more probably by some genetic mechanism (see Thompson, 1917).
This has a bearing on our problem, since the main feature in the evolution of the worker from the soldier would simply be the delay in the onset of the head's heterogony, a delay which, when it exceeded the time to the final instar, would eventually lead to the total absence of soldier characteristics. A less degree of delay would give rise to forms of intermediate soldier-worker type ; these are known to exist in various genera, e.g. Armitermes and related genera.
Thus, although heterogony appears to play its part in the origin of polymorphism both in ants and termites, its role is different in the two groups. In ants, the control of absolute size through the feeding of the larvae appears to be the main method. In termites true heterogonic polymorphism is rare, and when present seems due to irregularity of moult-number ; but neoteny due to postponement of the onset of heterogony also plays a role in the differentiation of castes.
68 PROBLEMS OF RELATIVE GROWTH
§ 5. Heterogony, Moulting, and Dimorphism
Finally, certain quite different special problems arise from the interaction of constant differential growth-ratios with the characteristic Arthropod process of moulting. It will be clear, since growth in a typical Arthropod only occurs between the shedding of one exoskeleton and the hardening of the next, that in the life-history of any single specimen the theoretical growth-curve relating organ-size with body-size will never be realized. Instead, growth of both organ and rest-of-body will take place in a series of jumps, but the points thus arrived at will all lie on the theoretical curve.
Although moulting in Crustacea and some insects tends to take place at each doubling of weight (Przibram, 1930), the large variations encountered, together with the variation in the initial post-larval weight, are often sufficient to obscure any recurrent modality in the sizes at which moulting occurs. In a large population of such species, moulting thus occurs at random at any size, and measurements of a heterogonic organ whose growth-ratio is constant over long periods accord- ingly fall on to a continuous curve when made on such a population.
The state of affairs, however, is entirely different if (a) the organ's high growth-coefficient is confined to one or a few instars and (b) is initiated in a more or less constant phase of an instar — e.g. always immediately after moulting. As the simplest case, let us take one in which the high growth- ratio lasts but one instar, as occurs with the female abdomen of Inachus (Shaw, 1928). When the data are grouped into classes by body-size, and then simply the means of the various size-classes taken, a curve is obtained which merely indicates two periods of approximate isogony, separated by a short phase of intensive heterogony (Figs. 22, 23). But when frequency- curves for relative abdomen-size are prepared for each body- size class, the true state of affairs is revealed (Fig. 39). All female Inachus are then seen to fall into one or other of two sharply non-overlapping groups as regards relative abdomen- size. Those below a certain absolute body-size all have narrow abdomens, those above another higher body-size all have broad abdomens. The curves for the short intervening range of body-size, however, are all bimodal, some individuals having narrow, others broad abdomens, but none being of intermediate type. When the curves for all the classes are summed, we
Inachus ?
22 21 20 19 18 17 16 15 14 13 12 II 10
- T T
T T
T T .
T T
35 40 45 50 55 60 65 70 75 80°/ -40-45-50-55 -60-65-70-75-80-85 /t>
|
30 |
|
- r T T |
||||
|
29 |
||||||
|
28 |
||||||
|
27 |
||||||
|
26 |
- T T |
|||||
|
2 b |
— |
|||||
|
24 |
- |
|||||
|
23 |
- |
|||||
|
22 |
T T T T |
|||||
|
21 |
||||||
|
20 19 |
- |
T , . |
||||
|
18 |
- |
|||||
|
17 |
_ |
T |
T _ |
|||
|
16 |
- |
|||||
|
11 |
T |
T |
||||
|
14 |
- |
|||||
|
13 |
- |
T |
||||
|
12 |
- |
|||||
|
11 |
- |
|||||
|
10 |
- |
|||||
|
9 8 |
3^- |
T |
1 1 1 — — 1 — |
— I 1 1 1 1 1 1 1 1 ' |
0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
Fig. 39. — Dimor- phism due to moult- ing in (A) female abdomen (breadth of 6th segment) and (B) male chela (breadth of propus) in the spider-crab, Inachus dorsetten- sis.
The values are relative, expressed as percentages of carapace length. The frequencies for each value of abdomen and chela are given for various values of carapace length, and also for all specimens taken_together.
70 PROBLEMS OF RELATIVE GROWTH
again of course obtain a bimodal curve for the whole population
(Fig- 39)-
This can only mean that the change from narrow to broad
type is effected during a single instar; and further that it must be initiated at a relatively constant phase of the instar, for otherwise there would be greater variability in the broad- type abdomen than is actually found. Finally, the fact that the bimodal curves are found over a considerable range of body-size indicates that there is not one particular serial number of moult, or one limited body-size, at which the transformation (which presumably is associated with sexual maturity) is initiated. If the high growth-ratio of the abdomen had extended over two instars, we should have had a trimodal instead of a bimodal curve ; but the more instars over which it extended and the greater the variation in the body-size at the initiation of heterogony, the more obscured would the curve's multimodality become, owing to the variation in the amount of growth at each moult, which becomes cumulative with the increase in the number of moults concerned.
A similar but less-marked bimodality occurs in this species as regards chela-size in males, and is doubtless to be explained in the same way ; see also Table IIIa, p. 54, for a similar phenomenon in /. mauritanicus .
The important point to notice is that the unusual and apparently abnormal fact of dimorphism in one sex has here been brought about by a combination of the two normal and common processes of moulting and heterogony.
It will at once be seen that this type of dimorphism is quite different from that noted by G. Smith (1. c.) which we have just considered in regard to the chelae of males of the same genus. The ' low ' and ' high ' males in this case were simply the groups of small and large body-size in what would have been a case of continuous heterogony resembling that of the chela of male Uca, if it had not been for the intercalation of a non-breeding period in which the claw reverted to female type. This dimorphism has nothing to do with moulting, but is due to an interruption of a long-continued heterogony by a non-breeding phase ; while that of the female abdomen of the same genus is due to the restriction of heterogony to a single moult-period. Both, however, are dimorphisms of developmental origin, and have nothing in common with genetic dimorphisms like those of ' diphasic ' mammals or birds such as arctic fox, certain squirrels, herons, owls, etc.,
butterflies. They also differ
HETEROGONY, MOULTING, AND DIMORPHISM 71
or the genetic polymorphism of the females of various mimetic
from the environmental di- morphisms, the best-known case of which is that of the wet-season and dry-season forms of certain tropical butterflies, where the differ- ence between the two types is elicited by different envir- onmental conditions bring- ing out different expres- sions of the same gene- complex.
O
I I
J_Jj.
■ ■ -- 1 I I
J-..
All
L
a
I I- 1 --T .
,_L
17
16
15
14
m m 8
13
,+
4
12
h
+JL
3-5 4-5
—r-
5-5
6-5 7-5 8
forceps-length, mm
■5 9-5 10-0
10
10 11 12131415l61718Mm
Fig. 40. — Bimodality and hetero-
gony of the male forceps in the
earwig, Fovficula.
Left, absolute frequency of different forceps- length at various body-lengths in a random sample of 445 specimens. Right, logarithmic plot of means of ' high ' forceps {h — h), ' low ' forceps (I — I), and both together (dotted), against body-length in 1,519 specimens, k for all specimens, about i-6.
Can we apply the results found in the abdomen of female spider-crabs to the classical cases of dimorphism in holometa- bolous insects — those of the earwig Forficula and the beetle
72
PROBLEMS OF RELATIVE GROWTH
Xylotrupes (Bateson and Brindley, 1892) ? I think that we
can. Let us begin with the clearer-cut case of Forficula, in
which there is a definite dimorphism of the male forceps, but
no dimorphism of female forceps, or of the body-size of either
sex.1
TABLE V
Measurements of 1,519 Earwigs collected by Djakonov : ana- lysed in Huxley, 1927B, Table I (see Fig. 40)
|
fcuD |
,d" -t-> |
Low type |
High type |
|||||||
|
d |
d |
, |
||||||||
|
in +-» |
</i ^j |
|||||||||
|
.— 1 |
cue |
a c |
||||||||
|
>> |
in en O |
0 3 .0 d d |
6 d "rt |
cxd ££ 0 . |
O !-c 4-1 4J > - |
6 |
0 - |
0 >-i |
0 d.5 |
|
|
£> |
d |
rt |
O |
d ko |
■43-d |
0 |
S 5P |
■BtS |
0 |
|
|
<r. (A 0 |
6 |
H |
rt d |
5 M 05 .s |
H |
-2 b£ 0) d 055 |
In 0) |
|||
|
10 |
5 |
_ |
370 |
5 |
370 |
37'° |
. |
|
. . |
o-o |
|
II |
40 |
|
4-15 |
38 |
404 |
367 |
2 |
6-25 |
56-8 |
5-o |
|
12 |
160 |
|
4-4i |
139 |
4'°5 |
33-8 |
21 |
6-8i |
56-8 |
i3-i |
|
13 |
363 |
|
5-24 |
219 |
4-18 |
32-2 |
I44 |
6-86 |
52-8 |
397 |
|
14 |
481 |
|
5-80 |
216 |
4-26 |
3o-4 |
265 |
7-06 |
50-4 |
55-i |
|
15 |
326 |
|
6-6i |
69 |
4-32 |
28-8 |
257 |
7-23 |
48-2 |
78-8 |
|
16 |
129 |
|
7-28 |
15 |
477 |
29-9 |
II4 |
7-61 |
47-6 |
88-4 |
|
17 |
15 |
|
8-17 |
■ — ■ |
■ ■ |
|
15 |
8-17 |
48-1 |
ioo-o |
|
Total |
1,519 |
I3-87 |
5-80 |
701 |
4-19 |
— |
818 |
7-17 |
— |
53-9 |
Analysis of Djakonov's results (Huxley, 1927B) has shown that when the data are tabulated by body-size, the mean
1 Recently Kuhl (1928) has attempted to show that the dimorphism of the forceps of male earwigs is apparent only, due to unconscious selection of largest and smallest specimens in collecting. This, how- ever, does not account for the monomorphism of male body-size or of female forceps ; and in any case is quite unable to account for the degree of dimorphism found by Bateson and Brindley, Djakonov (1925), etc. If apparent dimorphism is so easily produced by such means, existing collections should demonstrate it for large numbers of species ; instead it is quite exceptional.
It would appear that the absence of, or slight tendency to, bimodality shown in Kuhl's material (his pp. 362-3) is to be accounted for by the almost total absence of ' high ' forms in his material, which again is to be correlated with low body-size. The maximum forceps-lengths in his four samples are 6-5, 6-5, 7-5 and 8-o mm. respectively. As my figures show, the ' high ' mode only occurs at 7-5 to 8-o mm. In Djakonov's material the maximum forceps-length is 10 mm., and Brindley (referred to on my p. 313) records a maximum of 12-25 mm. !
HETEROGONY, MOULTING, AND DIMORPHISM 73
values for forceps-length against body-length give a good approximation to the formula for constant differential growth- ratio between forceps-length and body-length. When, how- ever, frequency curves for forceps-length are plotted for each class separately, a situation is revealed analogous to that for Inachus female abdomen. The smallest specimens show uni- modal curves, their forceps being all of ' low ' type ; the largest, also uni- modal curves, with forceps all of ' high ' type ; and the medium-sized show b i m o d a 1 curves, the num- bers of low forceps diminishing and high forceps in- creasing with in- crease of body- size. It should be noted that low and high types, as with Inachus chela and abdomen, d i ff e r only in size ; fur- ther, that in this case there is slight overlapping of the two modal types (Figs. 40, 42.)
Przibram (1927) has suggested that the bimodality is due to some male earwigs having one or more extra moults ; but his contentions, as they stand, will not explain the facts, for they should give bimodality of body-length as well as of forceps, and they do not take account of the overlap of ' low ' and ' high ' types, between which there is neither qualitative nor quantitative difference, but merely a modal distinction depending on their frequency distribution.
Furthermore, it is difficult to see how more than one extra moult can ever occur without leading to tri- or multi-modality.
|
Y |
' |
||||||
|
/ |
,C |
c |
|||||
|
; L |
|||||||
|
5? |
/ |
||||||
|
"> |
/ / |
||||||
|
O J, |
A / e |
'b |
A |
||||
|
yx |
|||||||
|
L |
M |
||||||
|
L |
M |
||||||
|
L |
/ |
M |
2 n — 1 n n -j- 1
Moult-stages {and Body-size)
Fig.
— Diagram to show the possible origin of
dimorphism of the male forceps in earwigs, Forjicula.
LM, LM . . . sizes of larval forceps at successive moults ; growth isogonic. X, point at which heterogonic growth of forceps is supposed to be initiated. If the change to the imago occurs at the nth moult, the modal level of adult forceps-size is at A — A ('low' males) ; if at moult (n + i), the forceps, after continuing of larval type for one more instar (B — B), would reach the modal size-level, C — C (' high ' males).
74 PROBLEMS OF RELATIVE GROWTH
The hypothesis of an extra moult may however be used to account for the facts, along the following lines. The processes responsible for the heterogony of the male forceps begin operating, we will suppose, late in larval life, and usually at a fairly definite phase of an instar. They cannot, however, be expressed in the form and size of male-type forceps until the imaginal instar is reached. This may be reached either at the next moult after the initiation of the heterogonic process, or only at the second moult. The result is that the heterogonic processes have either been operating for less than one instar, or for a period more than double as long, with resultant bimodality. Variation in the intensity of the heterogony, and in the time after moulting at which it is initiated, will bring about the overlap of high and low ; there is, however, no qualitative difference to be expected between the two forceps-forms (Fig. 41).
The interesting fact discovered by Djakonov (1. c.) should be noted, namely that unfavourable nutritive conditions cause a decrease in mean and modal body-size, both for the population as a whole, and for the ' low ' and ' high ' groups considered separately. As regards forceps-length, however, the main effect is to cause a shift in the distribution of the forceps, there being fewer high-type and more low-type, but without alteration of either modal value (Fig. 42). Furthermore, though in unfavourable conditions there are fewer large indi- viduals in both high and low series, yet for classes of the same mean body-size, whether we consider the population as a whole or the high and low groups separately, the mean forceps- size is actually greater for the animals which have grown up in the unfavourable conditions.
This apparently paradoxical fact may be explained in terms of our previous discussion. The unfavourable conditions reduce the total body-size. But the initial growth-ratio of the heterogonic forceps, starting at a late moult, will always be the same ; it is only its final value which will tend to a limit imposed by the growth-partition coefficient. For most body-sizes therefore we should expect that the absolute size of the forceps would depend only on the time for which it had grown, not on the body-size at which it began to grow, and therefore with stunting of total bulk, a given body-size will show an absolutely as well as a relatively greater forceps- size. It is, further, quite possible that the equilibrium-size and final theoretical growth-ratio is in practice never attained,
HETEROGONY, MOULTING AND DIMORPHISM 75
the imaginal stage with its cessation of growth supervening before this point is reached. (This is perhaps supported by the fact that the double-logarithmic plot for all forceps taken
7
to
20 15 10
& 5
K
I 020
15
10
5
|
X |
|||||||
|
I s / |
|||||||
|
D, |
\ |
\ |
|||||
|
,* |
/c |
||||||
|
— |
/ / / / |
D\ |
\c |
||||
|
- |
D, |
/--% \ |
|||||
|
/ |
/c |
\ |
\c |
||||
|
1 -< |
•** s |
1 1 |
D 1 |
1 |
"-J^ |
-j 1 |
10 11 12 13 14 15 16 17 18
body -length, mm.
•St
5
35 4-0 5-0 60 7-0
forceps-length, mm.
B
80
90 9-5
Fig. 42.— Different types of changes produced by unfavourable conditions (D — D) as against favourable conditions (C — C) on (A) body-length and (B) forceps-length in male earwigs. The modal body-length is reduced ; the frequency of forceps of low or high modal lengths is altered, but the modal
length remain the same.
76 PROBLEMS OF RELATIVE GROWTH
together does not curve over downwards at high body-sizes, as with various other holometabolous forms.) If so, then with earwigs within the known range of body-size, those with stunted body will always have larger forceps, though this should not hold if we could produce individuals of a much larger body-size.
A further peculiarity of the curves is that those for high and low forms taken separately (Fig. 40) both show a pheno- menon the reverse of that found in most holometabolous insects — namely that they are concave upwards at their upper ends ; in addition, they are concave downwards at their lower ends, indicating a rapid growth-coefficient at either end. This may, perhaps, be accounted for if we suppose that growth-ratio is accelerated during the period just before and just after a moult. The high-sloping early portion of the ' low ' curve would be due to the normal initial high growth-ratio of the forceps-rudiment at the first onset of heterogony, and would comprise those individuals which started their forceps- heterogony late in the instar prior to the imago. The high- sloping end portion of the same curve would include those which started their heterogony relatively early in the same instar. But those in which heterogony was initiated towards the middle of the instar would only have one period of accelera- tion at the end of a period of slow growth-ratio, and therefore would show less change of forceps-size for a given increase of body-size. If an extra instar is added, the rapid growth- periods are repeated, with corresponding results on the curve. This is, of course, purely speculative, but may serve as the basis for further work.1
The case of Xylotrupes (Huxley, 1927c) has been less thoroughly analysed. It presents various complications, most notable being a tendency to trimodality over a certain range of body-size.
In Lucanidae (Huxley, 1931c) analysis of Dudich's paper (1923) and other data show several interesting facts. First, that in Cyclommatus the range of male body-length (without mandibles) is much greater than in other stag-beetles, being about as much as the mean body-size, whereas in Lucanus cervus and L. lunifer it is less than half the mean. (The ratio of largest to smallest body-length in Dudich's Cyclommatus specimens is 2-47 ; in Bateson and Brindley's Lucanus cervus,
1 Most of the suggestions here advanced concerning Forficula modify or extend those put forward in my paper on the subject (1. a).
HETEROGONY, MOULTING, AND DIMORPHISM 77
the corresponding ratio for elytron-length is i-6.) Secondly, the range of the male's body-size is much greater than that
Fig. 43. — Heterogony and moulting of the 3rd pair of legs of the mite, Analges
accentor inns.
Above : (1) large male with relatively huge 3rd legs ; (2) small male with 3rd legs of the same relative size as in females ; (3) small male in which the onset of heterogony has apparently taken place earlier, giving slightly enlarged 3rd legs. Below : the 3rd legs of the 3 forms, showing details.
of the female's, though both start at the same minimum size (16 mm.) : the largest males have bodies 40 mm. long, the
78 PROBLEMS OF RELATIVE GROWTH
largest females only 25 mm. As the mandibles of the male are highly heterogonic (ranging up to 88 per cent, of body- length), the disparity in total size is even greater. This differ- ence would appear only to be obtained on the basis of possible extra moults in the male. In this form the frequency curve for all mandibles is again multimodal, with two main and one or more subsidiary modes. But the multimodality is not so well defined when curves for separate body-size classes are plotted. There is also a slight tendency to multimodality or at least irregularity in the male body-length frequency curve. No multimodality is apparent in the Lucanus curves. Thus here again the insect with the presumption of a facultative extra moult is characterized by multimodality of heterogonic organ.
A case of male dimorphism in the Acarine mite Analges accentorinus which would appear to be due to similar causes has been described by Jucci (1924). Here the large adult males are characterized by a great hypertrophy of the third pair of limbs. There further exist small but also adult males whose third limbs are almost identical with those of the female. From the scale drawings given by Jucci, we can say that the size-difference between the two forms is very close to what we would expect (on the supposition that bulk is approxim- ately doubled at each instar) if the larger had had one more instar in its development than the smaller (Fig. 43).
In addition, occasional small males are found with slightly enlarged third limbs : these would be specimens in which the onset of heterogony had taken place slightly earlier than usual. The case is thus very similar to the earwig, save that the normal ' low ' type is more like the female than in Forficula.
Having concluded this survey of special cases, I shall in the next chapter pass to a more detailed analysis of the empirical laws of relative growth in heterogonic organs or regions.
CHAPTER III
GROWTH-CENTRES AND GROWTH- GRADIENTS
§ i. Growth-gradients within Single Organs
SO far we have only dealt with the question of relative growth in whole organs or regions of the body ; and in our first chapter we have found an approximation to a simple mathematical formulation of relative growth, which we have called the law of constant differential growth-ratio. This, as we have further seen, is what we should have expected if we had worked on the problem a priori. It teaches us the striking fact that relative growth-rates of different parts of the body may stay constant over long periods of growth, which clearly is important as a contribution to the problem of form co-ordination and the orderliness of form-change, but it sheds little light upon any aspect of the growth-process itself.
In this chapter, however, we shall deal with certain further empirical laws or rules which will, I think, have to be taken into careful consideration in any future investigation of the biology and physiology of growth. It is of some interest that these rules, which to my mind constitute the most important part of any contribution made by me to the study of relative growth, emerged quite incidentally out of the investigations on differential growth-rates. In studying these latter, I had a perfectly clear-cut aim — to see whether change of propor- tions could be envisaged as the result of any simple laws of relative growth. But of the growth-gradients to be discussed in this chapter, I had no suspicion : their existence thrust itself upon me as a new empirical fact, any explanation of which is for the moment entirely problematical. I say a new empirical fact, for although D'Arcy Thompson (1. c.) had already adumbrated a similar view, for one thing he had not fully generalized it or pursued its consequences to their limit, and for another, it was new to me, as I had not at first grasped
79
8o
PROBLEMS OF RELATIVE GROWTH
ilea pugnax 6 large claw
i>
the full implications of his ideas, which only became clear on re-reading his book after obtaining certain empirical results for myself.
The starting-point of these investigations was afforded by the fact, obvious to simple inspection, that whereas the pro- portions of the separate joints of the female-type chelae in the sexes of Uca do not change appreciably during growth, those of the joints of the large or male-type chela do change,
and very markedly. The most obvious alter- ation is a relative in- crease of the size of the propus with absolute increase of the size of the whole chela. How- ever, when the weights o f different chela- regions were accurately determined, it was found that there existed within the limb what we may call a growth- gradient, the distal re- gion (chela + propus) having the highest rela- tive growth-rate, the central region (carpus) the next highest , and the basal region, nearest the breaking-joint, (merus + part of ischium) the lowest, although its relative growth-rate was still above that of the body. If we put this crudely into graphic form, using growth- coefficients as ordinates and spacing the different regions arbitrarily along the *-axis, we obtain a curve representing the distribution, along the main axis of the limb, of what we may for brevity's sake, without introducing any theoretical ideas, speak of as growth-potential. This curve is inclined to the horizontal, and is therefore the graphic representation of a growth-gradient within the appendage, the inclination of the
carpus
Fig. 44. — Graph to show different relative
growth-rates of different parts of the large
claw of the fiddler-crab, Uca pugnax.
Weights in mg. of dactylus + propus (o, scale on right) and merus + ischius (+, scale on left) against weights of carpus ; logarithmic plotting. The distal region shows positive heterogony (k about 1-05) relative to the inter- mediate region (carpus) ; the proximal region shows nega- tive heterogony {k about 0-9). (See Table VI.)
GROWTH-GRADIENTS 81
curve representing the steepness of the gradient, or in other words the difference in absolute growth-potential between the two ends of the gradient (as measured by growth-coefficients, which for our present purpose afford the only comparable standard for measuring intensity of growth-potential in a number of different regions or forms) in relation to length of the gradient — i.e. the relative length of the appendage. Such a graph, however, is as I say only a crude representation of the true growth-gradient. The most obvious reason for this is the impossibility of assigning fixed points along the abscissa- axis to the several joints, since the very fact of their differential growth is causing their centres (or ends) to shift differentially with increase in absolute size.
And secondly, we have the difficulty that the values we have obtained for the growth-coefficients are merely mean values for large regions of the organ, whereas if the idea of a growth-gradient be really justified, we should expect a pro- gressive change of the growth-coefficient from point to point along the axis, even within the limits of a single joint — a theoretical consideration supported by certain actual evidence in other forms (pp. 98, 261-2).
The full solution of the problem, so as to obtain a quantita- tively accurate picture of the graded change in growth-potential along an organ, will be a matter of considerable difficulty, partly owing to the formal difficulties arising from the con- stant change of the relative size of the parts measured, partly owing to the practical difficulty of finding sufficient distinctive points within the limits of a region such as the segment of a limb, on which to take measurements to determine the detailed form of the gradient empirically and not by mere extrapolation from a few mean values.
Finally, there is still another difficulty. As pointed out to me by Mr. J. B. S. Haldane, if y be the value of the weight (or linear measurement) of the organ as a whole, and if y 1}y2, . . . yn be the corresponding values of its constituent joints or segments, then if y = bxk be a correct expression for the limb as a whole, then yx = b^1, y2 = b2xkt, and so forth cannot be accurate expressions for the separate parts (or vice versa), since the sum of the several expressions for the parts will not exactly fit the expression for the whole. On the other hand, within certain limits of the value of k, the discrepancy will only be slight, and we are justified, from the actual figures obtained, in taking an expression of the 6
82
PROBLEMS OF RELATIVE GROWTH
above general form as giving a close approximation to the truth, and therefore in using the values of the growth-coefficients (k) as obtained from this type of expression as standards of growth-intensity.
UCA PUGNAX c? .
Region
TABLE VI
Mean Weights (mg.) of Three Regions of Large Chela (57 Specimens)
Distal (dactylus + propus) Intermediate (carpus) Proximal (merus + ischium
to breaking- joint)
|
Class 1 |
2 |
3 |
4 |
5 |
|
114 |
179 |
222 |
280 |
344 |
|
17-5 |
257 |
33-9 |
40-6 |
507 |
|
28-8 |
42-2 |
557 |
63-1 |
76-9 |
515 70-8
103
TABLE VII
Growth-coefficients of Different Regions of the Large Male Chelae in Different Crustacea (based on Huxley, 1927 and unpublished, and dean, unpublished, analysis of kemp)
Units of Measurement.
Species
Weight, relative to total : Uca pug- chela weight (distal ! nax to breaking-joint)
Maia squinado
Weight, relative to body-weight
Length, relative to total Palaemon
cheliped length (dis tal to breaking-joint)
rudis
|
Merus + ischium |
Carpus 1 |
|
0-89 |
0-97 |
|
o-8i |
o-93 |
|
I-5I |
172 |
|
ischium |
1-04 |
|
0-83 |
|
|
merus |
|
|
1-02 |
1-19
2-22 2*IO
Investigations are now in progress which have for their aim the clearing up of some of the practical and theoretical diffi- culties in the way of obtaining a quantitatively accurate picture of the growth-gradient within an organ. Until these have been completed, I shall here content myself with estab- lishing the fact that a gradient of some sort exists ; and in the graphic representation of growth-gradients I shall
GROWTH-GRADIENTS & GROWTH-INTENSITY 83
arbitrarily represent the centres of homologous regions as equidistant along the abscissa axis, and shall use the ^-values, as obtained from the formula for constant differential growth- ratios, as reasonable approximations for the values of growth- intensity.
§ 2. Steepness of Growth-gradient within an Organ and Growth-intensity of the Organ as a whole
The fact of a growth-gradient once established for the heterogonic chela of Uca, the next step was to see if similar growth-gradients occurred in other heterogonic organs. This proved to be the case. We will first take other examples from Crustacean appendages. The weights of the separate joints of the chela (five of them distal to the breaking- joint) were taken for both male and female Maia squinado (Huxley,
-10 c //
£ 10
OS
0 §>*«
•a-"" ^
merits + ischium
carpus axis of chela. —
dactylus + propus
Fig. 45. — Growth- gradient in the large chela (x) of the fiddler-crab, Uca,
(©) of the spider-crab, Maia.
The growth-coefficients of the different regions are here taken relative to the total weight of the chela distal to the breaking-point, not, as in Fig. 44, to the carpus-weight.
1927, and unpublished). It was found that while the pro- portionate weight of the joints of the female chela remained approximately constant within the limits of variation at all absolute sizes, — i.e. their growth-coefficients relative to the chela as a whole, like that of the chela relative to rest-of- body, were all = i-o, — those of the male chela during its period of heterogony were all greater than unity and were arranged in a regular growth-gradient. This was double in form, with high point in the propus, a slight fall towards the tip (dactylus), and a more rapid and more prolonged fall towards the body. The high point of the gradient we will call the growth-centre.
84
PROBLEMS OF RELATIVE GROWTH
Presumably the growth-gradient in the large chela of Uca was of the same form, but this was not apparent, owing to the propus and dactylus having been lumped together for purposes of measurement. (See Table VII ; Fig. 45.)
The gradient is steeper in the chela of Maia than in that of Uca. This appears to be due to the fact that though the c? chela of Maia begins its period of heterogony much later
10 20 30 40 50 60 70 BO 90 100
120
140 160 160 200
I
<*.
, 30 ■u
g 28
I <0 26
° 24
.-&■
O
•&■
10 20 30 40 50 60
80 100 120 140
cheliped length, mm.
160
180
200
Fig. 46. — Changes in relative length of different segments of the cheliped of the prawn, Palaemon ritdis, with increase of cheliped length.
Solid lines, males ; dotted lines, females. From above downwards : merus; dactylus; ischium (0), and propus (v->). The ischium decreases markedly, the merus and the carpus are almost constant; the propus increases markedly (growth-centre), the dactylus slightly.
in life than that of Uca, and therefore never attains the same enormous relative size, yet during this period, its growth- coefficient is higher than that of Uca (about 1-85 as against i-6 in Uca's first phase and 1-3 in its second). It would appear natural that the greater is the growth-coefficient of an organ as a whole, the steeper will be the growth-gradient of its parts ; this is confirmed by all the evidence so far collected.
GROWTH-GRADIENTS & GROWTH-INTENSITY 85
The only references I can find to these striking changes in the proportions of heterogonic Crustacean limbs are those of Kemp and his fellow-workers (Kemp, 1913, 1914, 1915 ; Hen- derson and Mathai, 1910). But they only draw attention to the change in percentage length of the joints, and have not proved the existence of constant differential growth-ratios, or propounded the idea of a growth-gradient. The figures for Palaemon rudis have, however, been analysed by Miss I. Dean (unpublished) and in both series show a definite growth- gradient with growth-centre in the propus (Table VII ; Fig. 46).
In general, measurements of crustacean limbs show that wherever there is marked heterogony, there is a comparatively steep growth-gradient within the limb, with well-marked growth-centre near the tip (apparently always in the propus) ; when the heterogony is only slight, the growth-gradient is far less steep, and its centre usually near the middle of the limb : in most cases examined either in the merus or carpus. Thus, although the joints of the small (female-type) chela of male Uca do not, to simple inspection, appear to alter in propor- tionate size, measurement shows that in respect at least of linear dimension, they do so, albeit slightly ; the growth- centre here is in the carpus (Huxley and Callow). (It will later be shown that precisely similar relations hold for those brachyuran abdomens which have been measured.)
Benazzi (1929) has given results on the regeneration of the limbs of the larva of the dragonfly Aesckna grandis from which it can be calculated that for regeneration during two instars, the growth-coefficients (k) of femur, tibia and tarsus, relative to the sum of the three parts, are as follows : femur, 1-12 ; tibia, 1-02 ; tarsus, 072. There is thus a regeneration-gradient in the limb with high point proximally.
Whereas the gradients of most organs appear to be of the simple form above described, there are some of unusual type in which part of the organ has growth-coefficients above unity, the rest below unity. This is the case with the first antennae of certain copepods (Seymour Sewell, 1929) (see Fig. 47). There is a positive growth-centre at the eighth or ninth segment, and a negative growth-centre at the extreme tip. The change from positive to negative heterogony of the segments, relative to total antenna-length, occurs close to the joint between the eighteenth and nine- teenth segments. It is worth noting that when a hinge is developed in the male's grasping antenna, it is formed at this
86
PROBLEMS OF RELATIVE GROWTH
joint. In almost all cases relative growth again falls off steadily on the proximal side of the positive growth-centre.1 The first and sometimes a few more of the basal segments usually exhibit negative heterogony, but occasionally show a very low positive heterogony. The first antenna as a whole shows a slight negative heterogony relative to total length (I.e., p. g).2
We may suggest that it is biologically desirable for the terminal portion of the antenna to decrease, the proximal
10 II 12 13 /■» IS 16 17 IB 19 20 21 22 23 24 2i
antennal segments
Fig. 47. — Change in proportions of segments of first antenna of Copepods during growth. Constructed from the data of Seymour Sewell, 1929. He gives the proportionate sizes of the antenna segments relative to total antenna
length at various sizes.
The graph gives the percentage change in proportionate size between the smallest and largest stages measured (the segments have been grouped as indicated, and the means taken for the groups). In every case the growth-rate of the basal region is low, usually negatively heterogonic ; there is a centre of maximum growth at the 8th or 9th segment, and a centre of minimum growth at the distal end. The transition from positive to negative heterogony occurs at about the 18th or 19th segment. X Nannocalanus minor. + Eucalanus subcrassus.
region to increase in relative size. Since changes in relative size appear to operate by means of growth-gradients, the nega- tive growth-centre in the tip of the antenna will be connected
1 Seymour Sewell's data for Undulina vulgaris indicate that in this species, after a low point of no change in proportions in the fourth segment, the growth-gradient again turns upward as we pass towards the body, which would give a still more complex growth-gradient.
2 The increase in total body-length, however, is due partly to the formation of new segments in the growing zone in the sub-terminal region of the abdomen. As I shall attempt to show in a later chapter, growth during early stages of the process, during which differentia- tion from embryonic tissue is actively proceeding, obeys different laws from those concerned with heterogony of parts which are already differentiated. It would be better to compare the growth-rate of the antenna with some definitely-formed part of the body, e.g. cephalo- thorax-length, in which case it would probably show slight positive heterogony.
GROWTH-GRADIENTS IN NEGATIVE HETEROGON Y 87
by a continuous growth-gradient with the positive centre in the region of the ninth segment. Further, we may safely assume that the copepods, in common with almost all other animals, show a negative heterogony of the head region : accordingly this centre of low growth-intensity will again be connected via a continuous growth-gradient with the positive growth-centre of the antenna. Since the gradient is of the same type in both sexes, the functional differentiation of the terminal region of the male antenna as a clasping organ cannot have any causal significance in determining the low growth- rate of this region. On the other hand, the fact that the heterogony passes from positive to negative at about the eighteenth or nineteenth segment may have had something to do with the fixing of the hinge- joint between the clasping region and the rest of the male antenna at this spot ; such a suggestion must, however, be regarded for the moment as purely speculative.
§ 3. Reversal of the Sign of the Growth-gradient in
Negative Heterogony
Those pereiopods which are used as walking legs appear usually to show slight but distinct positive heterogony, and to have a definite but slight growth-gradient with centre in the merus (Bush, 1930). It is of interest that in the actively- running shore-crab Ocypoda, the young (like the active young of Ungulates) must be provided from the start with relatively large legs if their speed is to be sufficient, so that their pereio- pods show a definite negative heterogony, or decrease in rela- tive size with increase of absolute size : and that here the low point of growth, or ' negative growth-centre ', is also in the merus1 (Cott, I.e.; Huxley, 1931B).
A similar reversal of gradient-sign appears to occur in the individual development of Ungulates. D'Arcy Thompson (1. c.) gives a figure (Fig. 48), of the proportions of the foot in ox,
1 This is from length-measurements kindly supplied in answer to a query of mine by Mr. Cott ; unfortunately, he only had a few speci- mens available for measurement, and the results, while clearly showing the merus as the joint of lowest growth-ratio, are not sufficient to construct a growth-gradient. The indication is that the gradient is complex, first rising above the level for the body-standard (carapace length), then sinking well below it in the merus, then rising again. It would be of great interest to establish this by obtaining statistically adequate data, as this is the only indication so far obtained of a com- plex growth-gradient with two points of inflexion within a single limb.
88
PROBLEMS OF RELATIVE GROWTH
sheep and giraffe which together with inspection of skeletons makes it fairly clear that in the phylogenetic elongation of the giraffe's leg, the growth-centre has lain in the cannon-bone, with a steep gradient distally, a less steep one proximally. Meanwhile actual weight (and length) measurements made by Hammond (1927, 1929) and analysed by Huxley (1931B) on the individual growth of the hind-limbs in sheep, show that in regard to the pelvis and the three segments femur, tibia and cannon-bone (unfortunately the digits were not measured) there is, correlated with the negative heterogony of the whole
limb relative to the body, a reversed growth-gradient with low point dis- tally (Table VIII ; Fig. 49). Though these constitute but two isolated bits of evidence, they indicate, so far as they go, that the growth-mechanisms underlying all heterogony are similar, and that when heterogony is negative, the sign of the gradient is simply reversed.
Hammond (1928, see also 1921) has also shown that the growth- gradients in the limbs and elsewhere affect the muscles as well as the bones, so that the study is of practical as well as theoretical importance. An important point made by Hammond may be given in his own words.
Sheep
Fig. 48. — Comparison of the skeleton of the foot in Ox, Sheep and Giraffe, to show- graded alteration in propor- tions of parts.
To effect the transformation form a typical (e.g. ox) form to that in the giraffe, y — c has been enlarged, c — b has remained nearly constant, b — a has been decreased, and a — o markedly decreased. In addition, the length : width ratio has been increased.
"As the animal grows, it changes its conformation ; at birth the calf or lamb is all head and legs, its body is short and shallow, and the buttocks and loin are comparatively underdeveloped ; but, as it grows, the latter — buttocks, loin, etc. — grow at a faster rate than the head and legs, and so the proportions of the animal change. . . . The extent to which these proportions change determines its con- formation ; those which develop most for their age have the best meat conformation, while those which develop least have the worst . . . Breed improvement for meat, therefore, means pushing a stage fur- ther the natural change of proportions as the animal matures. . . . The adult wild Mouflon ewe is in its proportions but little in advance of the improved Suffolk lamb at birth, although it is much larger.
What this means to the butcher and consumer is that of 100 lbs. live weight of an animal shpaed like the Suffolk lamb four days old the butcher can hang up as carcase in his shop 53 lbs., and the cus-
GROWTH-GRADIENTS IN NEGATIVE HETEROGONY 89
tomer can eat as flesh only 30 lbs. ; on the other hand, when the animal is shaped like that of the adult Suffolk ram, from 100 lbs. live weight 67 lbs. of carcase is obtained, and of this 61 lbs. is flesh which can be eaten — more than double that from of the badly shaped animal."
Phase 1
(Smallest to largest
new-born
Phase 2
(Largest newborn to medium sized
-J*
o
-i
S-j
o txo
08
0-6
C4
12 3*3 12345
Axis of limb : distal — >
Fig. 49. — Reversed growth-gradient in organs showing negative heterogony
(limbs of sheep).
The figures on the abscissa represent : i, limb-girdle ; 2, humerus, or femur ; 3, radius + ulna, or tibio-fibula ; 4, carpals, or tarsals ; 5, metacarpals, or metatarsals, (solid line, forelimb ; dotted line, hind-limb). The ordinates denote growth-coefficients (k), those below i-o signifying negative hetero- gony : they are taken relative to vertebral column weight. Size-phase 1 includes smallest to largest new-born specimens (100 to 256 g. vertebral column weight) ; size-phase 2, largest new-born to half- grown (256 to 690 g. vertebral column weight). The growth-gradient is at first flat, with slight positive heterogony ; then steeply tilted downwards distally, upwards proximally.
TABLE VIII
Relative weights of parts of the skeleton of Suffolk sheep at three different ages (c? Hammond, 1929; $ Hammond, 1927), reduced to proportions of weight of cannon-bone taken as 100. From Huxley, 1931B.
|
</l |
Vertebrae |
|||||||||||||
|
c • ° tao 1 v §1 Us |
3 6 IS In 286 |
U s |
a Is a*— |
a a ~3 0 |
3 cu 3 X |
"3 O, nj 0 |
T3 C a ~5 |
|||||||
|
0 "> u 0 |
'0 « u 0 |
a t- -s a |
"3 |
S3 ■2 •> |
||||||||||
|
u 100 |
212 |
O |
^4 |
O |
H |
J |
||||||||
|
Birth . . |
220 |
IOO |
187 |
150 |
75 |
1097 |
404 |
489 |
330 |
1230 |
722 |
|||
|
6" 5 months |
100 |
352 |
320 |
420 |
IOO |
213 |
241 |
169 |
1 1 79 |
58l |
564 |
633 |
1778 |
1205 |
|
i 4 years . |
100 |
380 |
361 |
619 |
IOO |
276 |
280 |
257 |
1235 |
854 |
775 |
794 |
2423 |
1706 |
|
Ratio 4 vrs.: |
I -00 |
1 -3° |
i- 70 |
2-8i |
— |
i-47 |
I-Q3 |
V43 |
IIS |
2-II |
1 -.58 |
2-34 |
i-97 |
236 |
|
birth |
||||||||||||||
|
Birth |
IOO |
147 |
217 |
142 |
||||||||||
|
9 5 months |
IOO |
245 |
285 |
430 |
||||||||||
|
? 4 years . |
IOO |
272 |
324 |
'Sfc'Q |
||||||||||
|
Ratio 4 vrs.: |
||||||||||||||
|
birth |
I -00 |
1-38 |
1-50 |
4-07 |
go PROBLEMS OF RELATIVE GROWTH
Thus it would appear that one of the chief advances made by man in creating improved breeds of sheep and other meat animals has been simply to steepen growth-gradients which already operate during post-natal development in the wild ancestral forms. Hammond himself (1927) has expressed a similar idea. ' The improver of meat-producing animals has apparently not chosen mutations occurring in isolated points independently, but rather has based his selection on the generalized correlated changes of growth '. (See also Fig. 96, p. 223.)
In consequence of the gradient, there will be much less difference in the size of the metatarsal between a semi-wild and an improved breed than in the size of the femur, This is well brought out by Hammond (1927) in his Fig. 4.
In this connexion, it is well to remember that during em- bryonic life, the limbs of sheep must show a growth-gradient precisely opposite in sign to that of their post-natal period. Lambs are born with relatively long legs, as an adaptation to accompanying their dams almost from birth. To achieve these unusual proportions, the leg must have exhibited posi- tive heterogony during foetal life ; and to allow for the fact of the later centre of negative heterogony in the cannon-bone, this same region must have been the positive growth-centre in the earlier period. The same reasoning applies to Ocypoda, whose young are similarly precocial.
§ 4. The Form of Growth-gradients
Analysis of the data of Kemp and his co-workers on Palae- mon spp. undertaken by Miss I. Dean (unpublished) gives a further interesting result. In these prawns, both male and female have obviously heterogonic chelae, but the male's heterogony is considerably higher. Thus a male and a female of the same absolute size will possess chelae of very different sizes, the female's being considerably the smaller. But if we take a male chela and a female chela of the same absolute size (which will of course be borne by a small male and a large female body) the proportions of the separate joints will be found to be fairly similar. This indicates that whenever marked heterogony, or at any rate heterogony designed to give rise to a large chela, is present, it must operate by essen- tially the same growth-mechanism within the limb (and a mechanism quite different from that in a slightly heterogonic pereiopod), whether the growth-coefficient of the whole limb
THE FORM OF GROWTH-GRADIENTS 91
relative to the body be moderate or high. The growth- gradient of the female is not quite so steep as that of the male, a fact also brought out by Tazelaar on P. carcinus (p. 92) ; but the male and female chela-gradients are much more like each other than they are to the gradients of any of the pereiopods.
Still further proof of the radical difference of the growth- gradients leading to pereiopod and to large chela is afforded by the male hermit-crab Eupagurus (Bush, 1930 ; Bush and Huxley, 1930). Here the right chela during early life is not much enlarged, and its growth-coefficient is no greater than that of the pereiopods ; only later does it begin the marked
1,6
1,5
1,4
1*.
1,0
I TTL C
dJstaJ -
|
-■ 1 |
1 |
1 1 |
— B |
|
- |
- |
||
|
-B^<7 |
- |
||
|
/ |
*^ |
||
|
• |
^ |
||
|
/ |
|||
|
/ |
|||
|
Ax' ■ |
1 |
1 ' |
-xA 1 |
Fig. 50. — Change in form of growth-gradient with increase of growth-rate in large (right) male claw of the hermit-crab, Eupagurus.
i, ischium ; m, merus ; c, carpus ; p. propus ; d, dactylus. A — A, juvenile phase ; the growth- gradient resembles that of a pereiopod. B — B, phase of heterogony of right chela ; the main growth- centre shifts distally.
heterogony which provides its definitive enlargement. And during the earlier period its growth-gradient is similar to that of a pereiopod, with centre in the merus ; while so soon as the final heterogony becomes marked, the main growth-centre shifts to the propus (Fig. 50).
Tazelaar (unpublished) has also collected facts bearing on this subject. In Palaemon carcinus, there is a change in the growth-coefficient of the chela in both sexes at about 4-5 cm. carapace length. In the female, before this, the chela has been growing less rapidly than the neighbouring pereiopods ; after this it exhibits a considerable heterogony. During the first of these phases its growth-gradient is almost flat, like those of the pereiopods, but with a slight growth-centre in
92 PROBLEMS OF RELATIVE GROWTH
the propus. Later it exhibits a marked growth-gradient with centre in the dactylus.
In the male, the chela in the first phase shows definite heterogony, about the same as the female chela in the second phase. During this phase it shows a growth-gradient with
isch. merus carpus prop dact isch. merus carpus prop dact. isch merus carpus prop, dact 1st. pereiopod Cheliped (2nd pereiopod) Cheliped( 2nd. pereiopod)
Fig. 51. — Growth-gradients in the 1st pereiopod, and the chela of the prawn,
Palaemon carcinus.
, male ; , female, (a) 1st pereiopod ; the gradient is flat and close to unity throughout ;
(b) and (c) 2nd pereiopod (chela). (6) 1st phase ; female with slight initiation of growth-centre distally ; male with regular growth-gradient (distal growth-centre), (a) 2nd phase; female with definite growth-gradient but incomplete proximally ; male with very marked growth-gradient (sub- termunal growth-centre).
centre in the dactylus. During the second phase, the male chela shows extremely marked heterogony ; and it now possesses a striking growth-gradient, with centre in the propus (Fig. 51). It would seem as if the steepening of the gradient began near the top, and then gradually extended centripetally (cf. p. 168).
§ 5. Growth-gradients in Regions of the Body
Precisely similar gradients to these found in appendages may be traced in the growth of whole regions of the body.
REGIONAL GROWTH
93
The most clear-cut examples concern the abdomen of crabs, which in all cases are narrow in the male, broadly expanded in the female. A large series of measurements has been made by Sasaki (1928) on both sexes of the Japanese species Tel- messus cheiragonus. Analysis of these data shows that whereas the growth-gradient for breadth in the male abdomen is nearly
2.5
©
id 4)
a
"aS-d 2.0
s — .2 a.
4J
1.3
o u
O
3 4
Segments of abdomen: distal — >
Fig. 52. — Growth-gradients in the abdomen of crabs.
Solid lines, for breadth of abdominal segments: ©, Telmessus cheiragonus, <$ ; X, Telmessus cheiragonus, 9 : +, Pinnotheres pisum, $. Dotted line, for length of abdominal segments in Pinnotheres pisum, ? .
flat, with its growth-centre, if so it may be called, near the centre of the region, in the female it is steeper, with its centre (as in the typical male chela) in the penultimate segment (Fig. 52). These figures may be compared with those cited for the edible crab, Cancer pagurus, by Pearson (1908, p. 21) in two large specimens of the same size but opposite sex (Table IX).
TABLE IX
Cancer pagurus; from data of Pearson, 1908.
|
Carapace-breadth 235 mm. |
9/6" ratic |
per cent. |
||||
|
Abdomen |
1 |
|||||
|
segment |
Length |
Breadth |
Length |
Breadth |
Length |
Breadth |
|
mm. |
mm. |
mm. |
mm . |
|||
|
I |
17 |
22 |
17 |
25 |
IOO |
114 |
|
2 |
8 |
17 |
8 |
22 |
100 |
129 |
|
3 |
7 |
23 |
7 |
30 |
IOO |
130 |
|
4 |
8 |
20 |
8 |
32 |
IOO |
160 |
|
5 |
9 |
17 |
10 |
35 |
|