AMER. ZOOL., 14:633-645 (1974).
The Basal Inhibition in Hydra may be Mediated by a
Diffusing Substance
HARRY K. MACWILLIAMS AND FOTIS C. KAFATOS
The Biological Laboratories, Harvard University,
Cambridge, Massachusetts 02138
bt NUI'MS. The decay, following basal disk removal, of the disk-mediated inhibition ot
disk formation in hydra is as slow as would be expected, were the inhibition mediated
by a diffusing substance.
When a hydra's size is changed, either by adding or by removing mass, the inhibition
intensity changes in the manner expected if the vehicle of the inhibition is a substance
diffusing from the disk and destroyed by all the cells of the body.
INTRODUCTION
There has been considerable interest
recently in the idea that some of the signals which control morphogenesis may be
diffusing substances. This idea seems particularly plausible in the case of the basal
disk-mediated inhibition of basal disk formation in hydra (MacWilliams and Kafatos, 1968); an inhibitory signal with a continuously variable intensity is involved
(MacWilliams et al., 1970); this intensity
decreases smoothly with increasing distance
from the disk (Shostak, 1972; MacWilliams
and Kafatos, unpublished). In this paper
we report on two further properties of this
inhibition; its decay time after its source is
removed and its response to changes in animal size. In both cases our results support
a diffusion mechanism.
DECAY KINETICS OF THE INHIBITION
One can construct hydra which may
form basal disks in abnormal positions
either by cutting a hydra in two and reassembling the pieces, or by transplanting a
small tissue fragment to the side of an intact animal. In such experiments, if the animals' original basal disks are removed, the
Present address of Harry K. MacWilliams: MaxPlanck-Institut fur Virusforschung, Molekularbiologische Abteilung, 74 Tubingen, West Germany.
frequency of disk formation at the abnormal site increases. This observation is the
source of the idea that hydra's basal disk
inhibits disk formation, and of the idea
that the intensity of inhibition decreases
following disk excision. The time course
of the inhibition change following disk
removal is of interest: a rapid decay (e.g.,
a half-time of seconds) would not be reasonably reconcilable with the idea that the
inhibition is transmitted by a substance
diffusing from the disk (Crick, 1970). We
have studied the kinetics of basal inhibition decay by the simple procedure of
measuring the apparent inhibition at various times before and after the basal disk
is removed. To interpret these measurements, it is first necessary to consider the
relationship between actual and apparent
inhibition intensity changes. We will derive a model which predicts that if the
actual inhibition decreases rapidly following disk removal, all of the decrease will
appear to occur before excision, while if
the decrease is slow, the apparent inhibition will continue to decrease for some
time afterwards. Experimental data will
then be presented which support the slowdecrease hypothesis. Finally, we will show
that if we assume that the inhibition decay
is exponential, our model and data permit
the calculation of a minimum estimate of
the half-time which is in accord with expectations if the inhibition is diffusionmediated.
633
634
HARRY K. MACWILLIAMS AND FOTIS C. KAFATOS
Apparent and actual inhibition
intensity changes
The intensity of the basal inhibition in
hydra can be measured by transplanting
a small piece of tissue to the site at which
the measurement is required. An appropriately selected transplant will sometimes
form a basal disk and sometimes not. The
frequency of disk formation, observed
when the experiment is repeated a number of times, appears to be simply related
to the difference between the prevailing
inhibition intensity and the transplant's
"basal threshold"—the level of inhibition
which is, on the average, just sufficient to
prevent disk formation (MacWilliams and
Kafatos, unpublished; see also MacWilliams et al., 1970, in which the same
relationship appears in a slightly different
experiment). One can thus obtain measurements of the inhibition intensity, relative to the threshold of the transplant,
from measurements of the disk formation
frequency.
It is not clear how one should interpret
the disk formation frequency if the inhibition intensity is changing when transplants are emplaced. Is it a measure of the
inhibition intensity at the moment of
transplantation, of the average over some
defined period, or of the minimum or
maximum during some period? Since relevant experimental evidence is not obtainable (no independent method for measuring inhibition intensity now exists), we
have approached this problem by model
building. Our solution is a straightforward
consequence of the following three
assumptions:
1) In transplants exposed to subthreshold
inhibition the threshold increases.
2) The basal threshold in transplants exposed to suprathreshold inhibition relaxes towards locally prevailing values
(in particular, it decreases in the experiment to be reported here).
3) An irreversible committment is made
to form a disk if the threshold is allowed to increase without bounds.
Basal threshold increase (assumption 1)
disk disk
no disk
disk
time
FIG. 1. Apparent inhibition ( x — X ) when the
actual inhibition (O
O) undergoes a step increase. The apparent inhibition anticipates the
change. Dotted lines (
) indicate threshold
trajectories of possible transplants.
has been demonstrated (MacWilliams et
al., 1970). Apical threshold decrease (analogous to assumption 2) was found to occur
by Webster (1966) in a study of apex formation by transplants. There is as yet no
evidence for the third assumption. In the
discussion that follows, we will assume
(for the sake of simplicity) that threshold
increase and decrease are linear with time.
We also require that the threshold at any
time may be determined unambiguously
from the threshold at a time point which
is just subsequent. No assumptions as to
the absolute or relative rates of threshold
change are implicit in our arguments.
First consider the case in which the true
value of inhibition undergoes a step increase (solid line, Fig. 1). Predicted threshold trajectories (dotted lines) are shown
for transplants made at several times with
several initial thresholds. Any transplant
with a threshold at any time above the
dashed line will form a disk while all
disk
'"•*no disk
time
FIG. 2. Apparent inhibition (X — X) when the
actual inhibition (O
O) increases slowly. The
apparent and actual curves coincide. The dotted
lines (
) indicate threshold trajectories of
possible transplants.
635
DIFFUSION IN HYDRA
ibil ion
disk
.-r
•
•fc
•
/
10 disk
time
FIG. 3. Apparent inhibition (x — X) when the
actual inhibition (O
O) undergoes an asymptotic increase. Dotted lines (
) indicate
threshold trajectories of possible transplants.
transplants with thresholds below the line
will not. The dashed line, therefore, gives
the apparent inhibition curve. Here we
see that the apparent inhibition rises, in
anticipation of the increase in actual inhibition, at a rate equal to the rate of
threshold increase.
Next examine a case in which the inhibition rises at a rate slower than the
rate of threshold increase (Fig. 2). Here at
each point a transplant with threshold
higher than the actual inhibition is expected to form a disk, while one with
lower value will be inhibited. Thus, the
observable and actual curves coincide.
Reasoning similarly from the predicted
threshold trajectories of possible transplants, one can find the apparent inhibition curve in any situation in which the
actual inhibition rises. Beginning at a
point after the increase is complete, one
follows the true curve backwards, so long
as its slope is less than the rate of threshold
increase. One extrapolates backwards at
the rate of threshold increase whenever
the slope exceeds this value. The application of these rules to a case in which the
inhibition increases to an asymptote is
illustrated in Figure 3.
In cases of true inhibition decrease, the
course of apparent inhibition change may
be determined in the same way, mutatis
mutandis: one follows the curve backwards until a point is reached where its
\
disk
a
"^
1y
time
"A no disk
disk
a
J
inhit
c
o
\
'•^ no disk
c
disk
\
v!
c
*-T
' A no disk
«
time
FIG. 4. Apparent inhibition (X---X) when the
actual inhibition (O
O) decreases. The dotted
) indicate threshold trajectories of
lines (
possible transplants.
time of
disk excision
time
FIG. 5. Apparent (- — ) and actual (
) inhibition if the actual inhibition disappears instantaneously following disk removal and reappears
full-blown.
HARRY K. MACWILLIAMS AND FOTIS C. KAFATOS
636
slope exceeds the rate of threshold decrease, then extrapolates backwards at this
slope until the actual inhibition curve is
once again encountered. Several examples
are shown in Figure 4.
It is worthy of note that rapid inhibition
changes—increases faster than the rate of
threshold increase, and decreases faster
than threshold decrease—cannot, according
to the model, be directly observed: they
will seem to occur, when measured by the
transplantation technique, only as fast as
the corresponding rate of threshold change.
This severely limits the information obtainable about the true course of inhibition
decay. Consider, however, that the inhibition decreases and then increases, as expected when the basal disk is excised and
subsequently regenerates. If the inhibition
disappears instantaneously, the observable
inhibition curve will have the shape of a
V or a flat-bottomed V (Fig. 5) which
attains its minimum at the time of disk
excision. If the inhibition decreases slowly
(Fig. 6) a V-shaped or approximately
V-shaped curve is also expected, but the
minimum will occur afier disk removal.
donor
transplantation
n
,\
I
transplantation
disk excised
2 hours
,\
disk excised
1 hour
disk excised
transplantation
no delay
disk excised
transplantation
I
I
1 hour
[
2 hours
disk excised
transplantation
disk excised
transplantation
i
3 hours
*t
transplantation
[disk never excised]
FIG. 7. Design of the inhibition decay experiment.
The third-from-apicalmost eighth of a donor hydra
was transplanted to a wound made in a second
animal one-fourth of the body length from its apex.
The host's basal disk was removed 2 or 1 hr after
the transplantation, simultaneously with the transplantation, 1, 2, or 3 hr before transplantation, or
not at all.
Experiment and data reduction
time of
disk excision
time
FIG. 6. Apparent (
) and actual (
) inhibition if the actual inhibition decreases and increases slowly following disk removal.
An experiment to measure the inhibition
before and after disk excision (Fig. 7) gave
the results shown in Table 1. Transplant
donors were algae-free H. viridis and hosts
were normally pigmented animals. Methods of grafting and conditions of culture
were as previously described (MacWilliams
and Kafatos, 1968; MacWilliams et al.,
1970).
637
DIFFUSION IN HYDRA
TABLE 1. Results of the inhibition decay experiment.
Transplantation time
relative to time of
disk removal (hours)
Disks/total transplants
Percent inhibition
± binomial uncertainty
Inhibition intensity
± uncertainty (la)
-2
-1
0
1
25/80
39/80
41/80
58/80
67 ± 5
.44 ± .14
2
40/80
49 ± 6
51+6
28 ± 5
50 + 6
.03 ± .15 -.03 ± .15 -.58 ± .16 .00 ± .15
Since the transplants, and thus the
transplant thresholds, were the same at
each time point, the observed disk formation frequency changes are due to changes
in effective inhibition intensity. Values for
the inhibition intensities in standard deviation units (MacWilliams et al., 1970)
were found by determining for each inhibition frequency / the standard deviation
value I for which the integral of the normal curve from minus infinity to I equals
/. These inhibition values are given in
Table 1 and plotted versus transplantation
time in Figure 8.
Since the experimental curve is approximately V-shaped, the V-shaped curve which
best fits the data was found. A brute-force
approach was adopted in which a grid o£
FIG. 8. Inhibition intensity versus time of transplantation (relative to the time of host disk removal) observed in the inhibition decay experiment. Bars indicate one standard deviation.
23/80
disk not
removed
20/80
71 + 5
.55 + .16
75 + 5
.67 + .17
3
points, separated by 0.1 <x increments in
inhibition intensity and 0.1 hr increments
in time, was established over the entire
area of Figure 8 and the acceptability of
each point as the vertex of a V-shaped
curve fitting the experimental data quantitated. For each grid point, values for the
slopes of the two arms of the V were
found which optimized the fit of the V to
the experimental data according to a
weighted least-squares formula (appendix) . The weighting procedure used was
to divide the deviation between each experimental value and the corresponding
predicted value by the uncertainty (1
standard deviation) of the experimental
value, before squaring and summing. This
has the result that if the predicted values
are equal to the true apparent inhibition
intensities (i.e., the correct V-shaped curve
has been found) then each of the weighted
deviations between curve and data will
have an expected mean of zero and standard deviation of 1. The sum of n such
variables squared has a chi-squared distribution with n degrees of freedom. This
means that if the sum of squared deviations between the data and the optimized
V-shaped curve at any grid point is greater
than can reasonably be expected of a
variable with the appropriate chi-square
distribution (four degrees of freedom; two
must be subtracted from the original six to
compensate for the two variables optimized) , then the hypothesis that the point
in question is the true inhibition minimum
can be rejected. Conversely, if the sum of
squares is a reasonable value for the chisquared variable, the hypothesis must be
regarded as acceptable.
All reasonable (P > .05) positions for
the V's vertex are enclosed in the ellipsoid
G38
HARRY K. MAC WILLIAMS AND FOTIS C. KAFATOS
ment, but only after a healing delay. The
duration of this is unkown, although it has
been found that the establishment of tight
junctions between host and graft cells requires several hours (L. Wolpert, personal
communication). To correct for this we
could consider the effective time of transplantation to be later than the actual
moment of surgery. This would move all
points of the observed inhibition curve to
the right with respect to the vertical axis
in Figures 8 and 9.
The rapid-decay hypothesis rejected
According to the hypothesis that the inhibition decays very rapidly, the apparent
inhibition should not decrease after the
FIG. 9. The V-shaped curve which best fits the
data of the inhibition decay experiment, and the
region (dashed ellipsoid) within which lie the
vertices of all V-shaped curves which are statistically acceptable (P > .05). Bars indicate one
standard deviation from the observed inhibition
intensities.
c
o
N
Ml
,'J
time
M2
shown in Figure 9. The central cross marks
the best position (P > .80) and the V
drawn is the best-fit V at this point. The
best-fit rates of threshold decrease and increase are the slopes of the arms of the V.
Within this ellijjsoid, these values varied
from —.13 to —.34 a/hr (threshold decrease) and from .24 to 1.14 o-/hr (threshold increase).
c
o
time
Possible systematic errors in locating the
inhibition minimum
If the apparent inhibition curve were
not V-shaped, the minimum of a fitted
V-shaped curve could be systematically displaced from the true apparent inhibition
minimum (Fig. 10). We believe that a
M2
time
serious error of this sort is unlikely in our FIG. 10. True inhibition curves ( ) with flat
case, since a measurement of the inhibition or gently sloping bottoms, showing the possible
at a time point near the fitted minimum displacement of the apparent inhibition minimum
gives a value in good agreement with the (Ml) from the position estimated by fitting Vshaped curves (M2). The dashed lines (
) are
value predicted according to the V.
the linear portions of the apparent inhibition
Transplants probably do not become curves; dotted lines (
) are extrapolations of
fused to the host immediately on emplace- these to complete the Vs.
639
DIFFUSION IN HYDRA
y = k, + /t.,e"fcj'
. time of
disk excision
where t is time and k,, U, and l<j are constants, specifying respectively (i) the value
asymptotically approached, (ii) the difference between initial value and klt and (iii)
the rate of decay. The half-time of an exponential decay (the time required for
one-half of the final hange) is related to
the value of k3 by:
T.5O = .69/*,
FIG. 11. Two true inhibition curves (
) which
would give rise to the same apparent inhibition
Thus, the value of T.5O increases with a
decrease in ks.
curve
In our case, we can write for the inhibition intensity 1:
(-•--)•
time of disk excision. However, the observed decrease in inhibition frequency
between 0 and 1 hr is significant (^2 = 8.1,
P < .005). No V-shaped curve with a vertex occurring at or before the time of disk
removal fits the data acceptably (P <
.005). These conclusions are only strengthened if the data points are displaced to the
right to correct for a healing delay. Therefore, the rapid-decay hypothesis must be
rejected.
The half-time of inhibition decay
Figure 11 shows that a given V-shaped
apparent inhibition curve is consistent
with two quite different true inhibition
curves. These curves, furthermore, are not
the only possibilities, but only representatives of a continuous range. It therefore does not seem practical to determine
much about the course of true inhibition
decay from an observed V-shaped inhibition curve. However, it is not difficult to
show (assume the opposite) that all of the
true curves consistent with a given
V-shaped observable curve have two characteristics in common: (i) all pass through
the vertex of the V; and (ii) at this point
they are steeper than, or have the same
slope as, the left arm of the V.
If we assume that the decrease of inhibition following disk removal approximates an exponential decay, these properties are sufficient to allow calculation of a
minimum decay half-time. An exponential
decay has the general form:
I = x + (.67 - x) e~u
where .67 is the initial value (taken from
Fig. 8) , x is the unknown final value, and
k is the ks above.
Since we know that the minimum point
of the observable inhibition curve, whose
coordinates are 1.05 hr and —.52 standard
deviation, lies on the true inhibition curve,
we know that:
I = - .52 = x + (.67 - x) (e-1-05*).
Solving for x, we obtain:
CO
x =
1
fi7^-1.05fc
g-1.05ft
The slope of the exponential decay curve
is given by:
4
kt
= (x - .67) keand at the minimum of the observed inhibition curve:
^
= (x - .67) /{<r1
Substituting for x, we have:
-.52 - .67e-10*
We do not know the actual slope of the
true inhibition curve at this point, but we
know that it is as steep as or steeper than
the descending arm of the fitted V, whose
slope is — .34 a/hr; thus:
640
HARRY K. MACWILLIAMS AND FOTIS C. KAFATOS
each case the best-fit value for the slope of
the left arm of the V computed for the point
in question. The least favorable point,
We calculated explicitly the value of dl/dt found after an exhaustive exploration, was
for a number of values of k. The results on the upper left rim of the ellipsoid
are seen in Figure 12 (0-hr curve). The (coordinates: .55 hr, —.38 standard devirequirement that d//d< be less than — .34 ation). The lower limit for T. at this
5O
implies that k cannot be greater than 1.96, point was .13 hr when instantaneous
healwhich allows us to conclude that the half- ing was assumed, but rose to .85 hr and 4.6
time is not less than .35 hr.
hr when 1- and 2-hr healing delays were
If we correct for the healing time as dis- allowed.
cussed above, the computational change
For purposes of comparison, we may conrequired is merely substitution of a larger sider Munro and Crick's (1971) study of
value for 1.05; 2.05 if 1 hr is allowed for increase to steady state concentrations in
healing, 3.05 if 2 hr are allowed. Curves source-to-sink diffusion models of the type
calculated with these values (Fig. 12, 1- proposed by Crick (1970). The relaxation
and 2-hr curves) give increased minimum at the midpoint was found to be approxivalues of half-time: for 1 hr healing, T 5 0 mately exponential with a half-time of
= 1.4 hr; for 2 hr, 7.7 hr.
.07 L2/D, where L is length (cm) and D
To determine the uncertainty in these diffusion constant (cm2/sec). With a
values clue to the uncertainty of the loca- length of 0.2 cm and Crick's (1970) sugtion of the apparent inhibition minimum, gested value for intracellular D of .8 X
we performed similar calculations at other 10"°, the half-time is 1.7 hr. Within the
points within the ellipsoid shown in Fig- limits of the validity of an exponential
ure 9, using as an upper limit for dl/dt in approximation, our results thus show that
the inhibitor relaxation time is not unreasonably
short for a system based on
\U value of T
0.35
diffusion.
BASAL INHIBITION AND BODY SIZE
o
x
0 hours
1 hours
2 hours
FIG. 12. True value of dT/dt at the apparent inhibition minimum for three values of healing time,
assuming that the actual inhibition value decreases
exponentially. The curves show that our upper
limit (—.34) on dl/dt justifies a lower limit on
the half-time of the inhibition decay. For definition
of k, see the text.
We have also measured the response of
the basal inhibition to increases and decreases in hydra's mass. The original
rationale for these experiments was to establish the plausibility of the idea that the
inhibition is involved in maintaining a
constant proportionality between disk size
and the size of the animal. However, the
results turn out to be relevant to the diffusion question as well. Methods were as
described above.
Experiment 2: effect of removing
portions of the gastric region
A fragment of peduncle tissue, the
second-from-basalmost eighth of a donor's
body column, was transplanted to a wound
made at the midpoint of the body of a host
from which a portion of the gastric region,
varying from none to three-eighths of the
641
DIFFUSION IN HYDRA
TABLE 2. Effect on the basal inhibition of removing a portion of a hydra's gastric region.
i
Donor
2A
Fraction of animal's
3/8
body length removed
2/18
Disks/total transplants
Significance*—difference P < .001
from control
2B
2C
Control
2/8
1/8
0/8
16/20
P > .80
18/20
P > .80
51/60
* Significance determined by the x 2 test.
host's length, was simultaneously removed.
Removal of three-eighths of the host significantly decreased disk formation by the
transplants, while truncation at more
apical sites was without significant effect
(Table 2).
Three interpretations suggest themselves:
1) The apparent increase in basal inhibition when three-eighths of the animal
are removed may be real, caused by the
decrease in host mass. If this is true, the
failure to observe any effect when smaller
amounts of the host are removed must be
attributed to sampling error. It should be
noted, however, that the sample size, although small enough to conceal a moderate inhibition increase in 2B and 2C, is
sufficient to indicate that the increase is
less than proportional to the amount of
tissue removed. (The highest inhibition
probability consistent [P > .05, by confidence intervals] with the observed four
failures of disk formation in twenty cases
when one-fourth of the animal is removed
is 0.45. Even this is closer to the frequency
of inhibition in the controls [0.15] than it
is to the frequency observed in 2A [0.90];
with respect to the amount of tissue removed, however, this experiment is closer
to 2A [three-eights of the animal removed]
than to the controls). If an explanation in
terms of a change in host mass is accepted,
this failure of proportionality must be
explained.
2) Alternatively, the apparent increase
in basal inhibition may result from apical
regeneration at the cut site. One might
imagine that in the process of apical regeneration, the tissue immediately surrounding the regeneration site is "reorganized," and its potential to differentiate
other than apically is suppressed. One
could then explain the difference between
2A and the other experiments as a consequence of the fact that only in 2A is the
transplant within the suppressed region. If
this explanation is correct, no "inhibition
increase" should be observable, even upon
removal of three-eighths of the host, if
transplants are made to a site at least onefourth body length from the cut surface.
3) Finally, it is possible that the "inhibition increase" upon truncation is in
fact due to the loss of an influence of the
head which stimulates disk formation. This
would be expected, were the inhibition mediated by a substance diffusing from the
disk and destroyed by the apex (Crick,
1970). One could then explain the observed differences among 2A, 2B, and 2C
by supposing that in the second and third
cases the apex is rapidly regenerated, while
in the first case it is not (Kass-Simon and
Potter, 1971) and a "head-absence effect"
642
HARRY K.
MACWILLIAMS AND FOTIS C.
TABLE 3. ISfJect of the apex on disk formation
KAFATOS
nearby.
—— *~~ — ~— ^
....
r
\
\
\
9
\
\
X.
V.
x
c
V.
V.
s
••
'
Donor
Position of
transplanted apex
Disks/total transplants
Significance*—difference
from control
Significance*—difference
from 2A
X
N
<1
/
1
"X.
y
k
Control
3A
3B
Outside
Inside
No apex
15/20
P > .40
17/20
P > .99
34/40
P < .001
P < .001
P < .001
* Significance determined by the x* t e s t -
persists long enough to influence the fate
of the transplant. This hypothesis makes
no prediction about the relation between
inhibition increase and distance to a
wound, but may be tested by directly assaying apices for influence on disk formation.
Experiment 3: apices assayed directly
The effect of the apex on foot formation
was determined by transplanting a peduncle fragment to a midbody site, as
before, and simultaneously transplanting
TABLE 4. Body size and the distance from transplant to apical cut surface varied independently.
Donor
Control
(4)
Distance—transplant to
presumptive apex
(body lengths)
Fraction of host's body
length removed
Disks/total transplants
(experimental)
Disks/total transplants
(control)
Significance*—difference
between experiment
and corresponding
control
2A
2B
2/8
1/8
2/8
3/8
3/8
2/8
8/19
2/18
10/20
16/19
P < .01
* Significance determined by the x* test.
51/60
P < .001
P > .80
Control
(2A, 2B)
643
DIFFUSION IN HYDRA
an apex to the peduncle fragment. The
experiment was also performed with the
pieces emplaced in the reverse orientation.
With the peduncle fragment innermost,
a moderate decrease in foot formation was
obtained; with the peduncle fragment
outermost, the frequency of foot formation
was unaffected by the subjacent apex
(Table 3). The data provide no support
for the hypothesis that the apex stimulates
foot formation, contrary to the prediction
according to Crick's model. The data also
fail to support explaining the result of 2A
as due to a local inhibition of foot formation by the apex; an inhibition was observed in only one of the two experiments
and, when observed, was significantly
(P < .001) less than the effect observed
in 2A. In this connection one might also
mention that MacWilliams and Kafatos'
measurements of the basal inhibition gradient give no indication of an anomaly
near the apex (unpublished).
ous insertion site, and the apicalmost
three-eighths of the host was simultaneously removed. The results, together with
relevant comparisons from experiment 2,
are given in Table 4. A significant decrease in disk formation was observed.
This shows that it is not necessary for the
transplant site to be within one-eighth o£
the body length from the site of truncation for a truncation effect to be observed,
contrary to the prediction of the regenerating-apex-proximity hypothesis. However, it
also appears that the inhibition increase
observed in this experiment is less than
that obtained in experiment 2A, which
suggests that the distance from the truncation site is not without influence.
The statistical reliability of these two
conclusions may be determined by testing
two corresponding null hypotheses: that
the inhibition change between experiment
and control is the same in 4 and 2B (the
distance to the wound is the only important effect); that the change is the same in
4 and 2A (the mass change is the only
important effect). For our tests we can
take advantage of the fortuitous fact that
the frequency of disk formation in experiment 4's control, 16/19, is close to the
same as that found in the controls for 2A
and 2B, 51/60. (The foot formation frequency given by any particular experiment
Experiment 4: host mass and transplant
proximity to the regenerating apex
varied independently
Peduncle fragments, isolated as before,
were transplanted to a site three-eighths of
the animal's length from the base, that is,
one-eighth body length basal to the previ-
TABLE 5. Effect on the basal inhibition of increasing the body size.
\
\
\
C\
N
N
X
•*
Donor
1
1
t
-
y
5A
3/8
Tissue added
(body lengths)
Disks/total transplants 9/17
Significance*—
P < .02
difference from
control
' Significance determined by the ^ test.
c
^^
\
\
\N.
%
V
"v
/
*
/
\
V.
N.
s
/
1
Control
5B
none
1/8
2/19
4/19
P > .80
Donor
644
HARRY K. MACWILLIAMS AND FOTIS C. KAFATOS
body tissue would then decrease the aggregate destruction rate, and, assuming production is unchanged, cause the inhibitor
concentration to rise. If destruction is
greater than zero-order with concentration,
a new steady state, at which destruction
again equals production, would ultimately
be attained at a higher concentration.
The existence of the body size effect,
however, is not strong evidence for a
diffusing-morphogen model of the inhibition, since nervous, electrical, or biochemical-wave inhibition mechanisms are certainly capable of producing such an effect,
and can reasonably be expected to do so
if it is useful in regulating the basal disk
size. It would seem, however, that the inhibition
intensity matters particularly to
Experiment 5: increasing the host ?nass
disk-size regulating mechanisms only near
the disk, where disk differentiation occurs;
An independent confirmation that no obvious advantage accrues from having
changes in host mass influence basal inhi- the increase in inhibition when apical
bition intensity can be obtained by in- parts are removed be, as observed, greatest
creasing the mass of the host. In such an near the site of truncation. According to
experiment, the third-from-basalmost eighth the model assuming inhibitor diffusion
of a donor was transplanted to the mid- from the disk and destruction by body
region of a host while the apicalmost cells, exactly this behavior is expected for
three-eighths of the donor, or the apical- the following reasons:
mosfeighth only, was simultaneously transplanted near the host's basal disk. The 1) The slope of the gradient in the steady
state is governed by the necessity that
large apical fragment produced a signifithe total rate of diffusion across any
cant increase in disk formation (Table 5),
plane be equal to the total rate of dethus a significant decrease in inhibition instruction
apical to that plane. A detensity. With the apex alone, the effect was
crease
in
destruction
brought about by
less and the difference from the control
removal
of
apical
tissue
will cause not
was not significant.
only an inhibition increase, but also a
decrease in slope along the entire inDISCUSSION: IMPLICATIONS OF THE
hibition gradient. The final inhibition
BODY SIZE EFFECT
increase—the change between old and
new steady state gradients—will therefore be greatest at the site of truncation.
It appears that the intensity of the basal
inhibition in hydra is determined by the 2) The net rate of diffusion of inhibitor
size of the animal as well as by the size of
from the cells at the truncation site will
the basal disk. This property, rather than
become zero immediately upon truncaa "positional" effect (Wolpert, 1969),
tion. In more basal positions, the flow
could be the key to understanding hydra's
of inhibitor down the gradient will
ability to regenerate a basal disk with a
continue at the steady-state rate until
size in correct proportion to the animal's
the slope of the gradient has changed.
body (the "French Flag problem").
The rate of inhibition increase following truncation will thus be greatest in
Our results are consistent with the idea
the cells near the cut.
that an inhibitor diffusing from the disk is
destroyed by all body cells; removal of
Our data may thus be taken to support
varies from day to day; this may explain
the otherwise unanticipated failure to observe a dependence of inhibition frequency
on position in these data.) Making the
assumption that the mean inhibition intensity in all controls is the same, the null
hypotheses can be tested by direct comparison of the experimental results. Comparison of 4 with 2A yields a chi-squared
of 8.08 (P < .005). Comparison of 4 with
2B gives a chi-squared of 5.93 (P < .025).
Thus, both null hypotheses are rejected.
Therefore, it seems that both the mass of
the animal and the proximity of the transplant to the truncation site influence basal
inhibition intensity.
G45
DIFFUSION IN HYDRA
the hypothesis that the vehicle of basal
inhibition is a substance diffusing from
the foot and destroyed by the cells of the
body.
APPENDIX: OPTIMUM VALUES FOR THE
SLOPES OF THE ARMS OF A
V-SHAPED CURVE
Consider that the curve has a minimum
at (X,Y) and we have data points (Xi,)>i)
and weighting factors A'j as described in
the text. Let B, equal the slope of the left
arm. The weighted sum of squared deviations for the left arm of the V is:
^[N^Y
-y,)2]
+ B}(Xi-X)
summed over all data points to the left of
the minimum. To find the optimum value
of Blt we take the partial derivative with
respect to Bt and set it to zero.
0 =
[2i[Ni
{Y + Bi (Xi
~
x)
~ y i ) 2]]
^MNi
(Y* + B* (*i - x)2 + yr +
2YBt (x4 - X) - 2ytB, (x, - X ) - 2Yy<) ]
0 =
[
2Y(xt-X)
0 = zi[Ni(B1(xi-X)
0=B1Zi[Ni(xi-X)]
-2yi(xi-
+
+
Y-yt)]
2,[2V, (x, - X) ]
For the right arm of the V the derivation
is similar and the result identical, except
that the summation is over data points
lying to the right of the minimum.
REFERENCES
Crick, F. 1970. Diffusion in embryogenesis. Nature
225:420-422.
Kass-Simon, G., and M. Potter. 1971. Arrested regeneration in the budding region of hydra as a
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363-378.
MacWilliams, H. K., and F. C. Kafatos. 1968.
Inhibition by the basal disk of basal disk differentiation in hydra. Science 159:1246-1247.
MacWilliams, H. K., F. C. Kafatos, and W. H.
Bossert. 1970. The feedback inhibition of basal
disk regeneration in Hydra has a continuously
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Munro, M., and F. H. C. Crick. 1971. The time
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