228
LATITUDE AND RELATIVE GROWTH IN THE
RAZOR CLAM, SILIQUA PATULA1
BY
F. W. WEYMOUTH, H. C. M C M I L L I N
AND
WILLIS H. RICH.
(From the Department of Physiology, Stanford University.)
(Received 25th September, 1930.)
(With Ten Text-figures.)
INTRODUCTION.
has long attracted the attention of biologists, but it is difficult to find any
quantitative treatment of the subject before the middle of the last century, and the
40 years dating from the publication of Minot's first paper in 1891 include practically
all the notable studies of animal growth.
Although the regularity of the observed course of growth has repeatedly tempted
workers to a mathematical expression, there has resulted no agreement as to the
most satisfactory type of formula nor even of the proper method of attack upon this
attractive problem.
It was therefore with great interest that following work on the growth of the Pismo
clam by the senior author (Weymouth, 1923 b), the writers undertook for the United
States Bureau of Fisheries an investigation of the Pacific razor clam (Siliqua patula),
a species of economic importance distributed over more than 2600 miles of the
Pacific Coast. This presented an unusual opportunity for the study of an invertebrate which we believed to exhibit a simplified type of growth in contrast to mammals
which have formed the bases of most growth studies. Since 1923 field work for the
Bureau of Fisheries has been carried out in various localities between Pismo,
California, and Chignik Bay, Alaska, and an immense amount of material has been
collected and measured. Analysis of these data has now reached a point where some
of the findings and their relation to current views of growth may advantageously be
placed on record. Before discussing growth, it will be desirable to present some
facts concerning these data which, we believe, make them uniquely valuable for the
intended purpose. The following points will be treated in order: (a) the homogeneity
of the material, (b) sexual differences, (c) the method of age determination, (d) the
dimensions selected for measurement and the technique of measuring, (e) the
measure of variability used, and (/) the age attained as shown by mortality records.
GROWTH
1
Published with the permission of the United States Commissioner of Fisheries.
Latitude and Relative Growth in the Razor Clam
229
Since we purpose to compare growth in different localities, it is of prime importance that the material compared be homogeneous. In a paper to be published
elsewhere we discuss at length the systematic relation of the species of the genus
Siliqua found on the west coast of North America. On the basis of a critical
examination of a great amount of material, we are convinced that all individuals
here considered belong to one species {Siliqua patula). This species has been stable
since recent geological times and the growth in various localities is directly comparable. We have, thus, available records of animals genetically homogeneous
which have grown under normal conditions for thousands of generations. None of
the laboratory animals have histories promising such stability of type.
In some mollusca there is a sufficient sexual dimorphism to permit the recognition of the sex of an individual by the shell (Chamberlain, 1927), but in the present
case the sex can only be determined by examination of the gonad. Do the sexes
grow at the same rate and can sex be disregarded in assembling the data ? Samples
of 150 specimens of each sex from Swickshak beach and from Cordova, Alaska
failed to reveal differences and a larger number from Copalis Beach gave similar
results. A sample of 115 males and 113 females from Hallo Bay, Alaska, however,
showed a difference in the growth-rate between sexes, but other facts make the
findings inconclusive. We shall discuss these differences later.
We may safely conclude that where the numbers are adequate and reasonably
divided between the sexes, as is true of all but the oldest age groups for each locality,
no distortion can result from the combination of the data for two sexes. In extreme
age the small number available is a source of variability exceeding that related to
sex and the averages for the oldest age groups should, of course, have little
weight.
In order to obtain the time element for growth data it is necessary to determine
the age of the specimen at hand. The method here used, which has been presented
elsewhere in detail (Weymouth, 19236; Weymouth, McMillin and Holmes, 1925),
rests on the now well-established fact that the shell shows in its structure evidence
of the slow winter growth in the form of rings which outline the margin of the shell
at the times in question. As a result it is possible to determine not only the age of
the particular specimen, but to obtain from this specimen the length and width of
the shell at each winter during the life of the individual.
Length has been selected as the basis of this growth study because, in the present
case, we consider it the best measure of size. The weight of a clam varies widely due
to two causes. The mantle cavity and the sinuses of the foot hold a variable amount
of water, which may or may not be lost at the time of digging. The sexual products
of the clam compose 10 to 30 per cent, of the total weight, and a further error in
weight is introduced by seasonal fluctuations in the size of the gonad. In advanced
age a smaller proportion of body weight is taken up by sexual products and the clam
gains little in weight although the shell continues to grow. For these reasons we
consider length a better measure of growth than weight. Calipers with which
accurate measurements of length may rapidly be made are easily used in the field
where equally accurate scales cannot be carried.
16-2
230
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
By measuring each ring, a growth-curve of the individual is, in effect, obtained.
Therefore, the norms of growth which we present are based, as in the case of
laboratory reared mammals, upon the average values of a large series of complete
records of individuals throughout life. The length measurements will thus obviously
present a higher correlation than if derived each from a separate individual, and the
growth-curves are correspondingly smoother.
The number of measurements each representing the length of a clam at a known
age is such as to inspire confidence in the statistical results. We have used 13,797
lengths in computing the growth-curves here presented, and our data include many
more measurements which have been valuable for comparison.
The size of clams at any age from one locality has been derived from a large
number of measurements. In each distribution of sizes at a given age, the median
has been used as a measure of central tendency instead of the arithmetic mean. By
locating the 10th and 90th percentiles of each distribution, relatively stable points
are located, and the difference between them (D) (Kelley, 1921) gives a reliable
measure of variability. The relation of variability to the rate of growth will be
discussed later.
No clams over 5 years old have been found at Pismo, California. The Washington
beds produce clams up to 9 years of age, while the commercial catch of Alaska
contains a large number of 13-year-old clams, and ages up to 19 years have been
recorded. The maximum age is, of course, an unsatisfactory measure of life span at
each bed, and mortality data in the form of survival curves are necessary for valid
comparisons.
Table I. Percentage surviving annually of clams from each bed.
Year
Pismo
Crescent
City
Channel
i
2
IOO-O
ioo-o
IOOO
920
980
97-3
3
4
537
936
911
105
1-7
88-i
79'3
62-8
5
6
7
8
603
119
9
IO
II
12
13
14
15
16
18
19
—
—
—
—
—
3°'4
Sink
Copalis Massett
IOOO
973
964
ioo-o
IOOO
98-5
987
913
88-6
76-5
836
95-8
53-8
37-1
17-2
3-8
711
—
—
—
—
—
296
2-6
i-8
—
—
—
—
—
—
—
—
20-9
4'4
o-6
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
9-2
I-I
—
—
Controller
Bay
IOOO
990
990
990
Karl
Bar
Swickshak
Hallo
Bay
IOOO
IOOO
ioo-o
98-3
98-3
96-2
96-2
95-8
95-3
887
33°
978
89-1
71-8
48-2
12-2
159
240
7'4
14-6
9-7
5-8
976
976
896
84-3
80-4
769
72-0
62-5
586
44-8
27-2
3-2
1-3
i'3
95
i-8
0-4
93-4
718
547
477
2O
6-i
37
1-2
—
970
96-5
94-4
87-3
860
54-3
i-3
807
76-9
722
580
467
32-3
218
114
—
30
—
09
'•3
—
Those surviving the first winter are taken as 100 per cent.
Latitude and Relative Growth in the Razor Clam
231
This need we were forced to supply from age frequency data from the different
beds. We have assumed, as did Lea for the herring (1924), that the frequency of
the older age groups represents a practical survival curve for that locality. The earlier
portions of these curves have been supplied by scaling all series of data to correspond
with those containing the most adequate representation of the young. Only clams
having passed through one winter are considered in these curves, although some
data for the mortality of smaller clams have already been published (McMillin
1924; Weymouth, McMillin and Holmes, 1925). (See Table I.) Although sources
of error remain, the survival curves thus obtained, when plotted with those of man
and of Drosophila (Pearl, 1928) for comparison, appear of the usual type and regular
in course. As the number of survivors falls off, the curves become irregular; 5 per
cent, proved to be the lowest value that could be accurately read for all localities.
The ages corresponding to this value were read off the graphs for all the beds and
have been considered the maximum ages. The life span used is, therefore, that age
reached by 5 per cent, of all the clams surviving the first winter.
GROWTH.
The growth data on the razor clam will now be presented concisely both from
the absolute and from the relative aspect. We will then consider more in detail the
relative growth rate and a method of developing a mathematical expression for the
course of growth considered from the relative aspect. (See Table II.)
Table II. Lengths!
Age* Pismo
1
2
i-73
9-O7
3
4
5
6
7
11-63
I2-IO
1268
8
9
10
11
12
13
14
15
16
17
18
'9
—
—
—
—
—
—
—
—
—
—
—
—
—
—
Crescent
City
Channel
Sink
1-19
6-75
io-35
1-09
559
9-10
10-54
11 08
11-50
11-89 M.
11-89 M.
I-2I
8-35
10-47
11-18
11-78
11-96
12-70 M.
13-40 M.
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
II-8I
12-58
13-03
13-32
13-84 M.
13-51 M.
—
—
—
—
—
—
—
—
—
—
Copalis
Massett
2-04 0 7 0
8-6i
5-35
10-87
9-35
12-04 10-97
1281 11-78
1340 12-58
13-84 13-27
14-19 I3-58
14-50 1369
—
14-61 M.
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
Controller
Bay
0-34
2-12
4-18
6-52
8-45
971
10-51
11-25
11-90
12-6013-00 M.
13-28 M.
13-51 M.
—
—
—
—
—
—
Karl Bar
0-38
2-73
6-4i
9-28
11-49
12-74
13-70
14-19
14-63
14-94
15-25
15-61
0-38
2"43
5"49
8-57
10-92
12-66
13-78
14-52
15-03
15 43
15-63
15-95
16-05
15-90
15-95
16-15
16-25
1640
Swickshak
16-12
M.
M.
M.
M.
15-96 M.
1672 M.
—
M. = Mean (used when number cases too small to determine median).
• Expressed as ring number; for true age subtract one-half year.
—
—
—
Hallo
Bay
o-34
2-25
m
10-96
12-37
13-17
1365
14-06
14-44
1475
15-08
I5-38
15-50
15-80
15-61 M.
1574 M.
16-31 M.
16-74 M.
232
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
We have plotted the regression of length on age which gives the common sigmoid
curve found in most growth studies (Figs, i and 2). The regression of age on length
(Weymouth, McMillin and Rich, 1925) which does not differ significantly in the
early stages cannot be substituted for length on age after the median duration of life.
The length of the first ring varies roughly with the latitude of the beds from which
the clams are taken, the size decreasing from south to north. From Massett, B.C.
southward, the young clams may be found on the beach in late summer and autumn;
the Alaskan clams, on the other hand, pass through metamorphosis in late autumn
and are extremely small even during the following summer. From this it would
appear that the two factors that most influence the growth of the young clam are the
temperature and the length of the first growing season.
e PISMO
T CRLSCENT CITY
iwssrrr
• CONTROLLER BAY
• KARL5 BAR
"•HALLO BAY
10
12
CALIFORNIA
CALIFORNIA
BRITISH COLUMBIA
ALASKA
ALASKA
ALASKA
ALASKA
18
19
Ring number
Fig. 1. Growth-curves of clams from seven localities. To determine actual age, subtract
one-half year from ring number.
Following the formation of the first ring, growth is rapid. In the southern beds
over two-thirds of the total length is reached during the second summer. In the
north, growth is slower and a comparable increase requires a period of over 4 years.
In our early studies on this species, growth-curves for the Washington beds were
presented which did not indicate an inflection. In the slower growing Alaskan forms,
however, an inflection is always present and this has led us to look more carefully
for it in the southern forms. Ultimately a method of fitting to be discussed later
convinced us that an inflection is present in all, and the curves of absolute growth
have been so drawn. On each curve the inflection falls at a different age which
varies from 13 to 33 per cent, of the total life span. Following the inflection in all
cases the growth-curve rises less rapidly until a final characteristic size is reached
ranging from 12 to over 16 cm. As will be pointed out later, there is a tendency for
the slower growing clams to reach the greater size.
To show the absolute rate of growth, we may plot the increments of length for
the successive years against the ages. The maximum of this curve corresponds, of
course, to the inflection of the length-curve. Since the only time interval available
Latitude and Relative Growth in the Razor Clam
233
is i year, the form of the derived curve is clearer in the clams reaching the greater
ages. Subsequent to the inflection of the length-curve the annual increments decrease
regularly and, for a time, closely approximate a descending geometric series so that
the logarithm of the increment plotted on age gives a straight line. In extreme old
age, however, the increments deviate from this linear relationship exceeding the
expected values; that is, the annual growth is greater in old age than the relations
found in middle life would lead one to expect. In cases where the inflection falls
15
H
t
14
!^<—
e
f/
12
4i
t
i
—
\
^
—
_
10
/
t
A
I
:
•—
—
& COPALIS
® SINK
• CHANNEL
1
2
3
4
5
6
7
8
Age (years)
Fig. 2. Growth-curves of clams from Washington Coast. The " Sink" and " Channel"
are atypical habitats near Copalis.
early and is not conspicuous, it is possible to fit the greater portion of the growth
curve by several formulae based on the just mentioned approximation of the annual
increments to a decreasing geometrical series.
For example, Putter (19206), proposed the following expression for growth
(i),
in which A = length at time t, L= final length, a and c = constants and e = base of
natural logarithms. In 1923 the senior author, in ignorance of the above work, fitted
the growth curve of the Pismo clam to a formula of similar significance
y=a-bc*
(2),
F
234
- W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
<
(
'
89
HALLO B
t
3
J
3
//
2
4
6
8
10
12
14
18
16
19
Age (years)
Fig. 3. Growth-curves of male and female clams from Hallo Bay, Alaska. All of the
specimens were collected on the same day.
s
t
\
©
V
B
Pisno
Cf/nctNT CITY
MASserr
H
HALLO BAY
\
3
\
"S
\
\
I
M
M
B
O
/
2
J
1 *
HJ
fS
0
1
S
H
..
i
6
Age (years)
Fig. 4. The relative growth-rates (
——J at each age. Relative growth-rate x ioo=percental growth-rate.
7
Latitude and Relative Growth in the Razor Clam
235
in which y = length at time x, a —finallength, c = a constant less than unity (the
ratio of any annual increment to the preceding increment) and b = a constant.
Brody in 1923 proposed the formula of a monomolecular reaction
L=A(i-e'u)
or in the form later applied to weight
(3),
W=A-Be~u
(4),
10
/
§
a
t
/
HALLO
/
BAY
C LENGTH
D & LENGTH
p
•
>
—^
1
Z
3
4
5
6
Ring number
1
7
e
IO
Fig. 5. Growth-curve of all clams from Hallo Bay with first differential.
in which L = length at time t or W = weight at time t, A = final length or weight,
k and B = constants and e = base of natural logarithms.
Since these three formulae give curves without inflections and too low a value,
in the case of the clam, for the greatest ages we have considered them unsatisfactory,
graduating, as they do, the central portion only of the growth curve.
Neither can the formula of Robertson be accepted as giving a satisfactory representation of the absolute growth curve. In its early form (1908) the fit is admittedly
poor, in its present form (1926), while the graduation of the growth curve of the
white mouse is very close, the presence of 11 constants in three concurrent formulae
rob the expression of any possible biological significance.
236
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
Relative growth may be defined as the increase per unit mass (or length) per
unit of time and may be represented for the clam by y-j- as contrasted with -5-, the
absolute growth-rate. The course of relative growth may be presented by a ratio
diagram in which the logarithms of the lengths are plotted against the arithmetic
values of time. This method, used by economists to show proportional changes in
prices, volume of business, and the like, presents equal percental changes as equal
slopes of the curves. The differential of this curve would give the relative growthrate, since
_ ^
dXogL
(s).
dt ~ Ldt
\
+J.0
. '
\
-
•2.0
*I.O
/
f
—
0
•
/
-1.0
1
HALLO
B*r
LENGTH
U•>G, LeNG TH
A Los,
-2.0
BA
/
. .1
.05
-3.O
I
•4.0
O
•
.03
.OZ
/
Z
3
4
5
G
7
G
9
I
O
Ring number
Fig. 6. Ratio diagram of growth of all clams from Hallo Bay with first differential.
Due to the regularity of the data and the method of treatment to be described, it is
possible closely to approximate - —^— by —
that is, by the annual increments
of the logarithms of the lengths.
The ratio diagram of the growth of the razor clam rises steeply at first and
gradually becomes less steep throughout its entire length. In the period for which
we have records, that is, after the first winter, there is no inflection. In other words,
the relative growth-rate (the first differential of the ratio diagram) is greatest at first
and falls constantly; or, still differently expressed, there is after the end of larval life
a decreasing negative acceleration (or retardation) of growth per unit length.
Latitude and Relative Growth in the Razor Clam
237
These relations are shown in Fig. 6 which is a ratio diagram of the lengths at
each age for clams from Hallo Bay. The logarithms of the lengths (expressed as
natural logarithms for ease in calculation) have been plotted against the age in years.
It will be noted that the growth-curve so drawn rises smoothly without sign of an
inflection. The first differential of this growth-curve, the relative growth-rate, falls
regularly without a maximum such as would result from an inflection in the corresponding integral curve. Its shape suggests the logarithmic-exponential relation and
it is found that if the logarithms of the relative growth-rates are plotted on time that
they give, in many cases, a close approach to a straight line. It is interesting to note
that Thompson (1917), in criticising Minot, says that the use of the more complex
HALLO BAY
© L09.
•
•So
6
SLOP?
CALCULATED MOM LCNGTH
CALCULATED FROM {±LoeL
e
Ring number
Fig. 7. Ratio diagram of growth of clam from Hallo Bay and the logarithm of the first
differential plotted on the same time scale.
relative differential I w , instead of —=- ] would be justified only if the rate curve
so derived should give a straight line or other simplified expression. In the present
case this is exactly what occurs. This treatment is indicated in Fig. 7. The upper
curve is a ratio diagram of length (loge L) similar to that of Fig. 6, but extended
to the fourteenth winter. The lower curve is the plot of the logarithm of the
relative growth-rate. Each short horizontal line extends over one unit of time
and the height above the base line is equal to the logarithm of the relative growthrate, loge (A loge L). The area of each rectangle under these horizontal lines, therefore, is equal to the respective growth-rate during that unit of time. The area under
the first derived curve should equal that of these rectangles and it will be seen that
the first eight are so placed that a straight line accomplishes this very accurately.
The correctness of this was checked by calculating the point slope at the middle of
238
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
each group of three lengths1. It will be seen that the agreement of the two entirely
independent methods is very close. It is to be noticed, however, that after the eighth
winter the slope is distinctly less and toward the end, due to inadequate data, the
curve is irregular. It is true of all curves that, even before they become irregular
because of small numbers, the curves deviate from the earlier trend, indicating that
the growth is maintained at a higher rate than the first part of the curve would lead
one to predict.
Since the relative growth-rate decreases in a logarithmic-exponential fashion,
the length-age relation may therefore be represented by a Gompertz curve of the form
(6)2.
L = Be-™-kt
That is, growth is an exponential function of time, but the exponent is a changing
one, decreasing with time in an exponential fashion. The Gompertz curve was first
applied to the graduation of mortality data and has since proved applicable to a
variety of phenomena. As can be seen from Table III, the fit is excellent. The
formula explains on simple mathematical grounds the presence of an inflection
which is without biological significance. Our knowledge of the complex phenomena
of animal growth is as yet insufficient to permit the construction of a rational formula
to represent its entire course. We feel, however, that the increase in mass of a
metazoan must represent basically an exponential relation such as suggested by the
present formula. This formula may also be derived from the fact so long insisted on
by Minot and since confirmed by others that the relative growth-rate declines through
most of the life qycle; it is only necessary to postulate that this decline is at a constant percental rate. Even though the decline in some cases should be found to be
less simple in course, the same biological situation would still obtain.
Since we have comparable lengths here presented, beginning with an age of
approximately one half-year and including only the winters of the clam's life, we
should not, strictly speaking, either interpolate or extrapolate. Intermediate points
on the curve do not represent the actual growth, since this shows an annual cycle.
The larval growth during the first half-year may well differ from that of the adult
and must show a portion of an annual cycle. Extrapolation into this period checked
1
We wish to acknowledge our indebtedness to Harold Hotelling of Stanford University for
assistance in applying to this problem a method developed by him (1927), and for other advice on
mathematical questions in the present paper.
2
Relative growth rate = £%- =
log
dt '
dlogL
dt
a-kt;
Ae~u
here A =
logl =
= b-
<?•
1-6
kt
ce- , where c
L = Be-ce~ki,
where B
=
~k '•
-A
Latitude and Relative Growth in the Razor Clam
239
against the size of the egg and the young stages makes it clear that the relative growthrate must be falling during most of this time. There is good reason, however, for
believing that immediately following fertilisation there is a "lag period" during
which the growth is less than in subsequent stages. In the absence of all data on
this period, we cannot make a more definite statement regarding the clam1.
Table III. Growth of Hallo Bay clams.
Year
i
2
3
4
5
6
7
8
9
IO
II
12
13
14
15
16
17
18
19
A,
(cm.)
(cm.)
O-348
2-252
5-422
8600
10962
12-405
13-175
13-670
14093
14-444
14-767
15-077
15-375
15-500
0-348
2-140
5-4I5
8590
11-096
12-562
13-385
13618
14-036
14-416
14-760
15-070
15350
15-601
15826
16027
16-207
15733
15610
1574°
16-310
16-740
16367
16-510
(cm.)'
0
O-II2
O-OO7
OOIO
-0-134
- O-I57"
— O-2IO
O-O52
0-057
O-O28
O-OO7
O-OO7
O-O25
— O-IOI
- 0093
0583
0-543
- 0057
0-230
(cm.)
O-OI2
0-048
O-II4
O-I38
0-078
CO49
O-O5O
O-O45
O-O4I
OO45
0-048
0-063
0-056
0-081
0-095
—
—
—
Lo = observed length combined males and females.
Lc = length calculated from formulae
1 to 7 years, inc. L = i4-3O32e-'-26™!3<r0-6"'84'"
8 to 19 years, inc. L = i7-6254<r°-l!')1)9423<r°-m"2'.
PEL = probable error of observed length (in the last four years the number of specimens was too
small to calculate this value).
1
Since this was written two papers have been found which bear on the growth formula here
proposed.
In 1926 Sewall Wright suggested in a book review (Journ. Am. Statistical Assoc. 21, 493-497,
1926) that the relative rate of increase declines at a uniform relative rate. In a personal communication he says: "The idea that the average growth power of the cells ( P= ™rj J falls at a more or less
uniform percentage rate as differentiation proceeds ( 5-3- = — k J, was suggested by reading Minot."
He states that the growth of the guinea pig exhibited such a decline of relative growth-rate,
although the observations were never published. With the statement quoted, including the suggestiveness of Minot's work, we are in hearty accord.
In 1928 Davidson (Univ. of III. Agri. Exp. Station, Bull. 302, p. 196) developed independently
and applied with Wright's assistance the formula
\ogW=A-be-kt,
in which W = weight at time t and A = final weight, to the growth of Jersey cattle. The fit is satisfactory but since none of the cows were under the age of milk production the inflection was not
included in the fitted curve. Since this formula is identical in significance to the one proposed in this
paper and both may be derived from Wright's assumptions above quoted, it is clear that the credit
of first suggesting this expression should go to Wright. The present is, however, the first case in
which this formula has been applied to a growth-curve including an inflection and the excellence of
the fit obtained is a striking justification of the soundness of the underlying assumptions. Dr Davidson
has been kind enough to examine part of this manuscript and is in accord with the general views
expressed regarding the growth formula.
240
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. RICH
That a relative growth-rate declining throughout life in a logarithmic-exponential
curve is not peculiar to the clam, may be seen if we turn to the data for other animals.
We have applied the method here employed to Minot's data on guinea-pigs and
find essentially the same picture as he has presented. The defects of his simple
percentage method for which he has been criticised (Brody, 1927 a, p. 17) and of
which he was clearly aware, are not such as to invalidate the idea of a declining rate.
Neither does the far more extensive data of Wright on the adult guinea-pig and of
Ibsen on the foetus (both from Brody, 1927 b, pp. 142-143) essentially alter the
picture originally drawn by Minot.
The data of Donaldson (1924) on rats gives a smooth weight curve with an
inflection at the age of 80 days. A plot of the logarithms of the relative growth-rates
shows a steady decline which is essentially the same as for the clam, and shows no
a i6 \
JS
to
3
'
>
\
*
\
W LoG.faLoGeW) ON AGE
\
f
i
®
/
WEIGHT
O*V
o
AGC
r
\
^oo
eo
100 izo
Age (after birth)
Fig. 8. Growth-curve of rat (data from Donaldson) with the logarithm of the relative
growth-rates plotted on the same time scale.
change of course at the age of inflection on the absolute curve. However, the
extreme delicacy of the differential method emphasises irregularities that pass unnoticed in the absolute weight curve.
Since, in a consideration of growth from the relative aspect, the inflection does
not appear, the question of its importance arises. Is the inflection as significant as
its universal presence in the absolute growth curve would indicate, or as unimportant
as its complete absence from the ratio diagram would suggest ? Brody has emphasised
the importance of the inflection in the growth curve of domestic animals. He says
(1927 a, p. 33): "The point of inflection, then, seems to have significance in indicating: (1) the time of maximum velocity of growth, that is, the transition from the
increasing to decreasing growth velocity; (2) the age of puberty; (3) the lowest
specific mortality, that is, the beginning of the period of increasing specific mor-
Latitude and Relative Growth in the Razor Clam
241
tality; and (4) finally it represents a point of reference for the determination of
equivalence of age in different animals (and, also, equivalence of age in the growth
of populations). The point of inflection should, therefore, be considered as a rather
important growth constant."
Let us consider these claims in detail:
1. The inflection is the maximum of the first differential of the absolute growthcurve. This and other mathematical properties do not prove its biological significance.
2. The inflection corresponds to the age of puberty in man and the rat. In the
growth-curves of other animals given by Brody, it falls at too early an age for puberty, for example about 2 months in the sheep and 5 months in the cow. In the
clam the inflection occurs much earlier than sexual maturity. It thus appears that
the exceptions are more numerous than the examples.
3. As he states, the time of inflection corresponds in man to the time of lowest
specific mortality. As we have no adequate mortality data aside from man by which
the point of lowest specific mortality may be determined, no generalisation is
possible.
4. It may be used as a "point of reference for the determination of equivalence
of age in different animals," but the choice of this point, however convenient as a
working basis, has no influence on its possible biological significance.
We know of no physiological epoch in the life of the clam corresponding either
to the age or to the size at the inflection. It is not related to final size or to the length
of life, and it falls at one-third to one-half the length of maturity. Therefore we see
no physiological significance to the inflection in the length-age curves of razor clams.
We also see that no important biological event corresponds to the inflection in the
growth-curve of many other animals, and hence no general biological significance
can be claimed for it.
The inflection may, however, have another meaning. Robertson, although he
claims no biological significance for the inflection, looks upon it as dividing the
growth-curve into two portions. Brody holds a similar view and has given to the
segments the terms "self-accelerated " and "self-inhibited phase." This would make
it an important quantitative landmark even if devoid of biological meaning.
But upon what does the quantitative significance rest ? Does the rate of growth
increase up to the inflection and thereafter decrease? We think this cannot be said
without introducing an inaccuracy of terminology bound to lead to serious confusion.
Minot has spoken clearly on this point (1891, p. 148):" It is evident that the increase
in weight depends upon two factors, first upon the amount of body substance or,
in other words, of growing material present at a given time; second, upon the
rapidity with which that amount increases itself."
Among recent workers Murray, in his notable studies on physiological ontogeny,
has expressed views with which we are in accord. He says (1925, p. 41): " Curves
of the average weights and of their logarithms have been plotted as functions of age.
The latter, which is comparable to percentage weight increments, shows by its slope
that the greatest relative changes in weight occur in the early days. The percentage
242
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. RICH
increase in mass rather than the actual increments of weight is taken as the rate of
growth, because this aspect is considered biologically more significant. Either
definition is permissible, but from a functional standpoint it would seem that the
object of interest is the growth and divisional rate per unit mass. When the weight
increments are taken as a basis of growth rate, the quantity of tissue taking part in
the reaction is left out of consideration.... From a physiological standpoint this
conception of the rate of growth is misleading and does not allow for a comparison
between the same organism at different ages or of organisms of different species at
the same age."
Schmalhausen, also, in his recent significant studies (1926) has emphasised the
importance of the relative growth rate which he calls "die wahre Wachsrumgeschwindigkeit."
That these views agree with our common idea of rate of growth may be made
clear by an illustration from Brody and Ragsdale (1921). From the table we find that
a new-born Jersey calf and a 2-year-old cow each increased 21 lb. in 1 month. No
one would say, however, that they were growing at the same rate, since in the calf
this gain was added to a birth weight of 55 lb. and in the cow to an initial weight of
716 lb.
To complete the analysis of the inflection, it will be apparent that an animal
growing at a constantly decreasing relative rate, will, if starting at a rate initially very
high, show for a time an increasing absolute rate, each increment being larger than
the preceding. But the falling relative rate will, after a time, more than offset the
increasing body size, and the total gains will slacken and, having passed through a
maximum, finally become progressively less, thus producing an inflection in the
absolute rate. Viewed in this light, the inflection becomes a mere mathematical
consequence of the course of relative growth and not a point which corresponds to
any physiological stage. If a small sum of money be placed on interest at a very
high initial rate and the rate constantly decreases with time, the compounded
principal will give a typical growth-curve with an inflection, and the annual income
will be equivalent to the increments of growth. This shows how an inflection may
come about without being related to a physiological epoch and without biological
significance.
VARIATION.
As already stated, we have used the median as a measure of central tendency and
the measure of variability normally associated with it, the interdecile range D. The
plot of D on age resembles that of absolute growth-rate, the values rising to a
maximum which, however, falls somewhat later than the maximum of the rate. In
other words, absolute variability is highly correlated with the absolute growth-rate.
The interdecile range D expressed as a percentage of the median length may be used
as a measure of relative variability, and is found to change with age in much the
same course as the relative growth-rate, showing an even higher correlation than do
the absolute measures. Therefore, in any homogeneous sample of razor clams, the
greater variability is closely associated with the smaller size and the more rapid
Latitude and Relative Growth in the Razor Clam
243
growth. During periods of rapid growth the individuals of any age group differ
markedly in size, but in the older ages where growth is slow there is a tendency to
reach a common size, as shown by the decline not only of the relative but also of the
absolute variability (Fig. 9).
160
140
1
W
IZO
A
•
*
CctESceNT CITY
STICKS
KARL BtW
100
80
60
1
V
20
0
0
t ~ ^ , > - £ ! •—5— 2
4
6
8
/O
/Z
/4
Age (ring number)
Fig. 9. The percentage variability \°tj\ ) °^ e a c h a g e c ' a s s ^ r o m t^liee
plotted against the age.
DIFFERENCES IN GROWTH AT DIFFERENT LOCALITIES.
We have presented features common to the growth-curves for all localities.
These features we consider the most significant and from them a general description
of the course of growth may be drawn. This we have attempted to do. It now remains to discuss the difference in the growth-curves for the various localities. In
studying these, a table was prepared showing seventeen constants or values which
differed from place to place. These were compared in many cases by scatter
diagrams, and by coefficients of correlation in an attempt to select the significant
characteristics of the growth-curves. Seven of the more important constants are
presented in Table IV.
JEB-Vllliii
1
7
244
F
- W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
Table IV. Table of differences..
Pismo
Latitude ...
...
...
...
...
Distance from Pismo (miles)
5 per cent, survival age* (yr.) ...
5 per cent, survival length (cm.)
Average growth (cm.) ...
Relative growth-rate per cent, per year:
Initial
...
...
...
Latitude ...
...
...
...
Distance from Pismo (miles)
5 per cent, survival age* (yr.) ...
5 per cent, survival length (cm.)
Average growth (cm.)
Relative growth-rate per cent, per year:
Initial
.
..
...
...
35° " '
o
Crescent Channel
City
41° 45'
46° 58'
860
Sink
Copalis
46° 58'
46° 58'
860
700
860
4-40
12-05
2-74
500
7-90
13-40
1-69
505
25
467
43
459
49
469
23
521
23
Massett
Controller
Bay
Karl
Bar
Swickshak
Hallo
Bay
58° 5'
58° 5'
2405
13-40
16 00
1-19
2425
1565
15-70
1 00
353
85
339
88
53° 20'
1380
955
1370
i-43
414
56
7-00
11-40
1-63
6o°
6o° 27'
2040
11 00
13-00
118
2105
1205
1595
1-32
343
68
353
90
1200
172
885
14-40
163
* Given as ring number. To calculate actual age subtract one-half year.
For several reasons only a partial analysis of these extremely suggestive data can
be now attempted. Although the number of individual measurements is large, the
number of separate localities, ten, is so small as to give a large probable error to
simple correlations and a prohibitive one to partial correlations. Certain of the
localities are known to be definitely atypical habitats; this is true of Controller Bay,
and the "Sink" and the "Channel" near Copalis were deliberately selected as
presenting abnormal conditions. We have on hand data from other localities for
which the constants have not yet been determined, and our plans include the
gathering of additional data. For these reasons we shall merely present the results
of the simple correlations as diagrammed in Fig. 10. This word picture of the
observed relations is given without the analysis which, as we have said, is as yet
inadequate. For the same reason a consideration of the results of other workers is
omitted and only a few of the more valuable articles cited.
An analysis of environmental factors is impossible because no adequate records
exist for the localities with which we are concerned, but as a numerical measure of
the environment we have used in our calculations the distance measured along the
coast north from Pismo, California. For all except the Alaskan beds the latitude
would have served equally well, but here the condition requires a word of explanation. The coast of the Gulf of Alaska trends north and west, reaching the highest
latitude near Cordova, after which it sweeps again to the south and west, so that the
beds on Shelikof Straits (Swickshak and Hallo Bay) lie about z° farther south than
those near Cordova (Controller Bay and Karl Bar). The isotherms follow in general
the sweep of the gulf, but the beds in Shelikof Straits lie north of the mean annual
Latitude and Relative Growth in the Razor Clam
245
0
isotherm of 40 F. which passes approximately through those near Cordova.
Agreeing with this is the fact that we pass out of the range of timber at about Hallo
Bay in the Alaskan Peninsula. For these reasons we have used the geographical
position on the coast, rather than the latitude, as representative of the environment.
We have discussed the method of determining the 5 per cent, survival age and
length which we treat as the maximum age and length. It may be characterised
briefly as the age, or length, at which 5 per cent, of the clams reaching the first
winter are still surviving.
The average growth is the quotient obtained by dividing this maximum length
by the maximum age. Of course, the growth is by no means uniform throughout
life, but this average serves as an approximate measure of the rapidity of growth
which can readily be understood.
Fig. 10. Diagram to show the coefficientsof correlation between constants derived
from growth-curves.
We have already emphasised the significance of the relative growth-rate, and
have found this constant to be one of the most useful in the present comparison.
Because of the nature of the course of growth no single measure of rate is adequate;
for simplicity we have selected the initial growth-rate and that at 2 years.
It will be noted first that the correlations between the five most significant
constants are high for biological data although, due to the small number of cases,
the probable errors are large. Due to this fact the customary coefficients of correlation and, in particular, their probable errors are misleading when compared, and
it is better to use the Z or normalised coefficient of correlation of Fisher (1925).
In Table V are given the coefficients of correlation (r) arranged in descending rank
followed by Z and its probable error.
In Fig. 10 the block containing the two relative growth-rates should be considered as a single feature, a type or course of growth, within which are associated
the highly correlated items of early and late relative growth-rates, always in the
17-2
246 F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
sense that high initial rate is coupled with high negative acceleration and therefore
a low rate in later life.
Table V. Correlations between constants for growth in different localities.
Between
r
z
Geographical position and Age
Initial relative growth-rate and relative growth-rate (2 yr.)
Initial relative growth-rate and Geographical position
Relative growth-rate (2 yr.) and Geographical position
Age and relative growth-rate (2 yr.)
Age and initial relative growth-rate
Age and Length
Length and Geographical position ...
Length and initial relative growth rate
Length and relative growth-rate (2 yr.)
+ 0-949
—0-945
—O-934
+ 0905
+ 0876
—0856
+ 0-808
+ 0731
—0-722
+ 0-713
+ 1-82
—1-78
—1 6 9
50
+ 1I-16
+
—1-28
+ I-I2
+ 00 99 13
—
+ 089
PEZ
±
±
±
±
0275
0275
0-275
0-275
0-275
±
i t 0-275
± 0-275
± 0275
± 0275
± 0275
r = coefficient of correlation.
z = tanh" 1 r = normalised coefficient of correlation.
Geographical position also shows a high, in fact the highest, correlation with
maximum age and a distinctly lower one with maximum length. The most northern
clams live the longest and become the largest. The initial growth-rate shows a high
correlation with age and a lower one with length in the sense that rapid initial
growth is correlated with a short life and a small size. The reverse correlations must,
of course, exist between later growth-rate, age and length. Lastly, between age and
length there is a high positive correlation, the clams reaching the greatest age being
also the largest.
Expressed more generally, we have the following picture. That type of growth
showing high initial and low later growth-rate leads to a rapid assumption of adult
size (measured as some specified fraction, as 75 or 90 per cent, of the maximum
length) and an early sexual maturity (Weymouth, McMillin and Holmes, 1924,
p. 224). The maximum age appears to be a function of the age at which adult size
is reached or even more closely of the age of reaching sexual maturity. This corresponds to the generally recognised relation in mammals, in which a long infancy is
associated with a long life. The rapid type of growth is also linked with a lesser
maximum length. Since age and length show a high positive correlation, it might
naturally be expected that the clams reaching the greater age will be the larger. This
is, however, not the significant relation. As may be seen from Fig. 1, the curves
showing rapid initial growth finally cross those with the initially slower but more
sustained growth, and thereafter lie below at the same ages, so that even if the life
span were extended they would still be smaller. It is as though the inherited growth
impulse were more wastefully expended in the rapid type of growth and accomplished less total growth metabolism, a relation strongly suggested by the experiments of Northrup (1925) in which the total C0 2 production during the life of
Drosophila at 300 C. was but little more than half that of flies raised at 160 C. What
such a simile means in physiological processes we cannot now say.
That the above relationship is not peculiar to a comparison of different localities
but is a fundamental character of growth, is indicated by its appearance in a com-
Latitude and Relative Growth in the Razor Clam
247
parison of different individuals from the same bed or different sexes from the same
locality. Thus, in the case of Hallo Bay (Fig. 3), we see the males growing more
slowly during early life, gradually equalling, then exceeding the females in length and
finally living the longer. Similar relations may be noted in the growth of other animals.
The following papers contain data in some degree comparable with those given
in this article. Comparisons are in all cases difficult. We have presented extensive
figures on the mortality and the growth under natural conditions, the exact features
of which we do not know. In the majority of the papers cited, growth was under
artificial conditions in which the physicochemical factors, chiefly temperature, were
controlled but often did not cover the entire life cycle. For these reasons our
comments are of the briefest.
Bayliss (1918, pp. 44, 61) points out that, whereas a rise of temperature always
increases the rate of a chemical reaction, it may either raise or lower the final
equilibrium concentration of a particular substance, depending upon whether the
reaction is exothermal or endothermal. In the case of the adsorption of Congo red
by filter paper, the amount adsorbed is decreased, although the rate is increased by
a rise of temperature (Fig. 29). The chemistry of growth is too complex and too
little known to permit analysis in terms of as simple a reaction as adsorption,
although the curves are suggestively similar.
In the population growth of unicellular organisms, particularly bacteria, the
effect of temperature and other factors has been extensively studied. For example,
Graham-Smith (1920, p. 153) found that between 170 and 37° C. the initial rate of
growth for cultures of Staphylococcus aureus under otherwise similar conditions is
more rapid at the higher temperatures, but that the final number attained is far
greater at the lower temperatures. The analogy between the growth of a population
and a metazoan, while suggestive, must not be pushed too far, and in addition the
restricted food condition of the culture differs from that of the clam. Nevertheless
the relations observed merit comparison.
Putter has considered the effect of temperature on the rate of growth and the
length of life. In 1920 (1920 b) he gave a theoretical example of the working of his
growth formula, discussed above, applied to animals living at different temperatures. He inferred that the animal at the higher temperature should show the more
rapid growth, but the smaller final size. He gives no observations beyond the
general fact cited by certain zoologists that species from colder waters are usually
the larger. In this paper and in a second appearing the same year (1920 a) he
considers longevity, and arrives at the conclusion that a higher temperature should
be associated with a shorter life. In neither case are examples given. Although our
results are in accord with the expected relation of growth-rate, size and longevity,
Putter's reasoning is not easily applied to the conditions of clam growth.
The most striking parallel to our results is given by Gray (1928) in a study of the
development of Salmo fario eggs. After reaching an age of 35 days the eggs were
separated and reared at temperatures of 50, io° and 150 C. There resulted marked
differences of rate. Those kept at 150 present a much steeper growth-curve but their
final size at hatching is, however, only about two-thirds of those reared at 50. Gray's
248
F. W. WEYMOUTH, H. C. M C M I L L I N , and W. H. R I C H
explanation is direct and convincing. The yolk represents a limited store of energy;
this is expended in two ways, first in the maintenance of the embryo, second in its
growth. At the higher temperatures the fraction consumed in maintenance is
larger, and therefore the fraction available for growth is less.
The food supply of the clam can, however, hardly be considered as limited in
this manner, and the application of these extremely suggestive findings to our
problem is not clear.
The most extensive study of the mortality of animals is that of Pearl (1928), who
experimented with Drosophila reared under varied conditions. He also raised
cantaloup seeds without any food, except that stored in the cotyledons, and presents
the correlations between the rate of growth, length of life, size and other factors.
He considers as most significant the correlation of — 0-643 i O-O57 between the
rate of growth and the length of life; that is, those showing the most rapid growth
die first. Since this experiment covers but a portion of the life cycle of the cantaloup
in the absence of a food supply, whereas we are considering the entire life of the
clam with a normal food supply, the comparison, although suggestive, is not
complete.
SUMMARY.
1. The present paper is a study of the growth of a clam {Siliqua patula) under
natural conditions and over a wide range of latitude.
2. Various constants derived from the growth data are compared for the
different localities. For this species, over the range considered, growth in the
southern localities as compared with the northern is initially more rapid but less
sustained, leads to a smaller total length and is associated with a shorter life span.
3. Reasons are presented for considering the relative growth-rate as a particularly significant constant leading to more sound biological conclusions than the
use of the absolute growth-rate.
4. On the basis of the relative growth-rate, current mathematical expressions for
the course of growth are discussed and a formula used which emphasises Minot's
conception of a growth-rate constantly declining with age.
This expression
^ = Be-ce~M
in which L = length at time t, e = base of natural logarithms, and B, c and k are
constants, is found to graduate the extensive data in clam growth with significant
accuracy.
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•
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