Parameters for normal growth
In our laboratory colony, larvae vary in their growth rate and in the peak size they attain before purging their gut. They purge on days 17 to 20 after hatching. The growth trajectory of a single representative larva of Manduca is shown in Figure . This individual purged its gut on the 18th day after hatching from the egg. Growth stops periodically as the larva molts from one instar to the next, and after day 18 the mass declines as the larva purges its gut contents, enters the wandering stage, and prepares for pupation. Overall growth during the feeding period is approximately exponential (Figure ), as is typical for insects. Within each instar, growth is also approximately exponential, but it is clear that the exponent declines progressively from instar to instar. Regression of the exponent value on instar number (Figure inset) shows that the growth exponent declines approximately linearly from instar to instar. In any given instar the value exponent is given by: exponent = 1.01 - 0.098*instar. Thus, in the fifth instar the growth is described approximately by mass = W0*e0.52*t, where t is the time in days from the beginning of the fifth instar and W0 is the initial mass of the instar.
Figure 3 Growth trajectory of a typical larva. Growth occurs during five larval instars, separated by brief periods of molting during which no growth occurs. (a) Linear plot: the peak on day 18 is the maximum size the larva attained and marks the time at which (more ...)
Exponential size increase and the critical weight
Although the growth exponent decreases gradually from instar to instar, the size increment from instar to instar is constant. That is, the final mass of each is a constant multiple of the final mass of the previous instar. As a consequence, the mass of a larva at each larval molt increases exponentially from instar to instar (Figure ). The only exception to this rule is the final mass of the last larval instar, which is substantially larger than expected. Figure illustrates that the predicted final mass of the fifth instar should be about 5.4 g, on the basis of a projection from an exponential regression on the masses of earlier instars. Thus, if Manduca had six or more larval instars, we would expect the fifth instar to molt to the sixth at a mass of about 5.4 g. The actual final mass of the fifth instar is about 11.5 g, about twice the expected mass.
Figure 4 Final sizes of the five larval instars of Manduca sexta. The size of the first four instars increase exponentially, but the final size of the fifth instar is about twice (10.75 g) what would be expected (5.39 g) from the regression on the earlier instars. (more ...)
Interestingly, the predicted mass at a presumptive fifth to sixth instar molt (5.4 g) is very close to the critical weight (approximately 5.3 g) for the strain in which these measurements were made. We have also measured the critical weights and growth coefficients for several other genetic strains of Manduca that differ substantially in growth rate and body size (Figure ). We found that there is an almost perfect linear relationship between the critical weight and the expected final mass of the fifth instar predicted by the size increments of earlier instars.
Figure 5 Relationship between the empirically measured critical weight of a fifth-instar larva and the predicted weight at which a fifth instar would have molted to a sixth larval instar. The predicted weight is based on the projected terminal weight of the last (more ...)
This finding suggests that the physiological changes initiated at the critical weight are somehow related to those that accompany a normal larval-larval molt. Moreover, this finding shows that the critical weight is a simple multiple of the mass of the larva at the beginning of the final larval instar.
It is therefore possible to derive an equation that relates the critical weight to the size increment and initial conditions. In general, the final size of each of the first four instars is a function of the size increment, and is given by the equation
final mass of instar = W1*eD*instar, (1)
where W1 is the mass of the hatchling larva and D is the size increment for the first four instars (= 1.66 in Figure ). The critical weight (CW) in the fifth instar can also be estimated from the initial weight of the fifth larval instar (W5) as follows:
CW = W5*eD. (2)
In the several genetic strains of Manduca we have examined, W5 varies from 0.85 g to 2.25 g, and D varies from 1.45 g to 1.85 g. Genetic and environmental variation in the values of W5 and D will have profound effects on the value of the critical weight and, by extension, the final weight of the larva.
Growth of the fifth larval instar
A mean growth curve for a cohort of fifth-instar larvae from a laboratory colony of Manduca from ecdysis to the time of the gut purge is shown in Figure . Clearly, the overall growth during the fifth instar is not exponential but resembles a rather flat sigmoid. The slowdown and cessation of growth at the end of the instar are due to the secretion of ecdysteroids, which cause the larval to stop feeding and enter the wandering stage in preparation for pupation. The low growth rate at the beginning of the instar reflects the time necessary for the biochemical and physiological processes of molting to cease and those for feeding and growth to reactivate.
Figure 6 Growth of a fifth-instar larva with a critical weight of 5.2 g. The vertical dotted line is drawn through the time point at which the critical weight is passed. The growth trajectory before this time is concave upward and the trajectory after this time (more ...)
In order to derive an equation that describes growth during the fifth instar it is useful to know what the trajectory would look like in the absence of the influence of ecdysteroids, which can be thought of as prematurely terminating the growth phase. In the last larval instar, ecdysteroid secretion is inhibited by JH, and when larvae are treated with JH they continue to grow well beyond their normal final size [25
]. It is therefore possible to deduce the shape of the uninterrupted growth trajectory by inhibiting the secretion of ecdysone with exogenous JH. When a topical application of 50 μg methoprene (a stable JH analog) is given on days 1 and 2 of the fifth instar, ecdysone secretion is inhibited and the larva continues to grow for at least a week beyond the time that growth would normally have stopped. The growth trajectory of JH-treated larvae is shown in Figure .
Figure 7 Growth trajectories of normal and JH-treated larvae of Manduca. JH-treated larvae (open circles) received a topical application of 50 μg methoprene (a stable JH analog) when they reached a weight of 3 g and again when they reached a weight of (more ...)
The overall growth curve of methoprene-treated larvae shows a gradually increasing growth rate until they reach a mass approximately equal to the critical weight (5.3 g for this strain of Manduca), followed by a decreasing growth rate above that weight. We assume that the integument poses an increasing resistance to growth as the larva increases in size, and that this accounts for the decreasing growth rate as the larva gets bigger. It is likely that growth in these larvae finally stopped at the maximal size allowed by the stretch of the epicuticle.
The critical weight
The critical weight has an important role in controlling the final size of the larva. In the last larval instar, the critical weight marks the initiation of a dramatic change in physiology. After reaching the critical weight, the level of JHE in the hemolymph rises abruptly [21
] and the JH titer gradually drops to zero. Once JH has disappeared, the secretion of PTTH and ecdysone are disinhibited. When ecdysone is secreted the larva stops feeding and growth stops.
The mechanism by which a larva assesses its critical weight is unknown at present, but the data presented above show that it corresponds to the weight at which the fifth instar larva would have molted to the next larval instar, had it not been the final larval instar (see Figure ). In addition, we have found that there is a simple linear relationship between the critical weight and the initial weight of the fifth instar larva across a broad range of body sizes and genetic backgrounds (Figure ). The critical weights used to construct Figure were determined using the method outlined in Figure , and show that the critical weight is approximately 5.3 times the initial weight of the instar, minus 0.8 g. This is in close accord with the interpretation of Figure . The critical weight thus has a simple linear relation to the initial weight of the final instar larva, and variation in the initial weight accounts for 95% of the variation in the critical weight (Figure ).
Figure 8 Relationship between the mass of the larva at the beginning of the last larval instar and the critical weight (CW), at which the decision to initiate the endocrine events that lead to metamorphosis is made. Each point is from a different genetic strain (more ...)
There are various mechanisms that could have this property. What would be required is a measure or process that changes with the mass of the larva and for which the larva can measure the ratio between the current state and the state at the beginning of the instar. Stretch reception in which the length of the stretch receptor is set at the beginning of the instar provides a plausible mechanism [28
], as does the prothoracic gland size measure described by [30
A mathematical description of growth and size determination
Growth before the critical weight
A model for growth and size determination must accurately replicate both the normal growth trajectory of a larva and the normal duration of the growth period. In other words, the model must account for both the growth trajectory and the decision point at which growth stops.
Growth is exponential until the critical weight is reached, after which the growth rate declines gradually. Exponential growth is described by
dW/dt = k*W, (3)
where W is the mass in g and k is the growth rate. Equation (3) has a solution:
W(t) = W5*exp(k*t), (4)
where W(t) is the mass at time t, W5 is initial weight at the beginning of the fifth instar, and k is the growth exponent. The growth exponent can be deduced from the size of the larva at a given time by solving equation (4) for k:
k = ln(W(t)/W5)/t. (5)
Because the overall growth curve is a rather flat sigmoid with the inflection point at around the critical weight (on day 3 under our 'standard' conditions), the growth rate between days 2 and 4 of the fifth instar larva is approximately linear. We have found that a close approximation of the growth exponent, k
, can be obtained from the growth rate on day 3 and the initial weight of the instar. This eliminates the need to obtain a long series of weight measurements to determine the value of k
. We begin by establishing the relationship between the growth exponent (k
) and the growth rate (GR
) on day 3 (Figure ). The best fit to this relationship is given by k
) + C
, where C
is a constant that depends on the initial weight of the fifth instar larva, which as shown in Figure is given by C
. Combining these two equations gives:
Figure 9 Derivation of growth exponent from growth rate on day 3. (a) The relationship between growth rate and growth exponent for larvae with the same initial weight. The relationship is best fit by a logarithmic regression where k = 0.2*ln(GR) + C, where C (more ...) k
Equations (3) and (4) thus describe growth until the critical weight is achieved. The time at which the critical weight is reached (tCW) can be obtained by setting the left-hand side of equation (4) to the value of the critical weight and solving for t, which gives:
tcw = ln(CW/W5)/k (7)
Growth after the critical weight
As noted above, the critical weight also marks the point at which the growth rate of the last instar larva changes from exponentially increasing to gradually declining (see Figure ). This implies that the growth exponent must decline as the larva grows past its critical weight. The rate of decline of the growth exponent can be derived empirically from the growth trajectories of JH-treated larvae (Figure ). We found that the rate of this decline is the same in larvae with different growth rates and different maximal sizes, and we assume, therefore, that it is characteristic of the species, rather than of a particular individual or genetic strain. After the critical weight, the growth of a larva is therefore given by
Variation in growth rate constant. Empirical growth data for three different strains: H (filled circles), B (triangles) and D (open circles), shows that all have the same rate of decay of the growth constant.
dW/dt = k*d*W, (8)
where d describes the rate of decline of the exponent k. The analysis in Figure shows that d = 1.43*exp(-0.11*t). Substituting this formula for d into equation (8) and solving the differential equation gives the following expression for growth after the critical weight:
where CW is the critical weight and t is the time in days.
Duration of the growth period
In our strains of Manduca, ecdysis to the fifth instar occurs between 2 hours and 6 hours after the lights are switched on, on a cycle of 16 hours light and 8 hours dark (a 16L:8D photoperiod), and feeding begins within an hour after ecdysis. So, for the purposes of the model, we assume that a larva begins to grow 4 hours after lights-on. We define a day as the interval between lights-off signals, and designate the day on which growth begins as day 0 (zero).
The growth period ends with the secretion of PTTH and ecdysone. During this period growth is partitioned between the pre- and post-critical weight growth. The duration of pre-critical-weight growth is given by Equation (7). The duration of post-critical weight growth is determined by the mechanism that controls PTTH secretion. PTTH secretion can occur only during a well defined photoperiodic gate, and in fact occurs during the first photoperiodic gate after JH disappears from the hemolymph [14
]. The mean time required for these processes differs in different genetic strains and must be determined by means of a critical weight experiment, as outlined in Figure .
The photoperiodic gate for PTTH secretion is between 14 hours and 24 hours after lights-off [11
]. Thus the interval between the closing of a photoperiodic gate and the opening of the next one is about 16 hours. A larva that becomes competent to secrete PTTH just before a gate closes will do so, but if a larva becomes competent to secrete PTTH just after a gate closes it will continue to grow for an additional 16 hours, during which it can add an additional 1–2 g of weight (depending on its growth rate).
Parameters of the model
The overall model thus consists of Equations (3) and (8) and requires two parameters that relate to size and growth: the initial mass of the fifth larval instar, W5
(or the critical weight, CW
, which is a simple linear function of W5
), and the growth rate on day 3, GR
(or the growth exponent, k
, which is related to GR
as shown in Equation (6)). The model also requires four parameters that relate to time: the time at which growth starts, the mean time interval between achieving the critical weight and PTTH secretion (called the interval to cessation of growth, or ICG [12
]), and the opening and closing times of the photoperiodic gate. All these parameters are empirically measurable and should be characteristic of a given genetic strain. Changes in only three of these parameters (GR, ICG and CW) have been shown to fully account for the evolution of body size in a laboratory strain of Manduca
The model can be run on a computer by numerical integration of Equations (3) and (8), using time steps of one half hour (or less) and keeping track of the time at which the critical weight is attained (at which time Equation (8) replaces Equation (3)), and the time at which photoperiodic gates open and close. Alternatively, the model can be run by calculating the time for the critical weight and the time of opening of the first gate after the ICG and substituting these values into Equations (4) and (9).
Tests and predictions of the models
Table shows the parameter values and actual peak weights of larvae of four different strains of Manduca, and Figure shows the relationship between the actual peak weights of these strains and their predicted peak weights using these parameter values. The model produces excellent predictions of the sizes of genetic strains with different growth parameters.
Parameter values for four genetic strains of Manduca used to generate model results shown in Figure 11
Predicted body sizes. Model predictions of peak weight of larvae of four different genetic strains of Manduca that differ in their growth parameters. 'Empirical data' are from Table 1. Bars are standard deviations.
In real life no two larvae will have exactly the same parameter values for the determinants of body size, because these are affected by both genetic and environmental variation. We therefore examined the effect of introducing variation in each of the parameters, by allowing them to vary randomly with a mean given by the parameter values for the H strain (Table ) and a standard deviation (arbitrarily chosen) of 8% of the mean. Under these conditions, the peak mass of the larvae is approximately normally distributed, but the time required to reach the peak mass is multimodal (Figure ). This is because the photoperiodic gating of PTTH secretion leads to a periodic distribution of the duration of the growth period. Interestingly, this has no appreciable effect on the size frequency distribution. A few animals reach their peak weight on the fourth day of growth, the majority do so on the fifth day, and the remainder on the sixth day. In each case a so-called 'gating bias' [23
] is evident: the first larvae to reach peak weight do so relatively late in the gate, and for the subsequent days the majority of larvae peak early in the gate. The reason for this is that if larvae become competent to secrete PTTH while the gate is closed, they have to 'wait' until the next gate opens, and will thus release PTTH and achieve their peak weight very soon after the next gate opens. In this simulation, most of the individuals in the last group of larvae evidently became competent sometime during day 6 and therefore stopped growing (and thus reached their peak weight) shortly after the gate opened.
Figure 12 Simulation of population variation in body size and development time. One thousand individuals were generated with small amounts of random variation in all parameter values of the model. (a) Frequency distribution of peak sizes; (b) frequency distribution (more ...)
Variation in food quality alters the growth rate without affecting other parameters (G.D. and H.F.N., unpublished results); how does it affect peak weight? The relationship between growth rate and peak weight is illustrated in Figure . The 'sawtooth' character of this relationship is due to the gating of PTTH secretion. As the growth rate increases, the time of PTTH secretion occurs progressively earlier in a gate, until the beginning of that gate is reached after which all larvae secrete PTTH at the beginning of the gate; then as growth rate increases further there is an abrupt switch to the gate of the previous day. Peak weight increases gradually with growth rate but drops abruptly when larvae shift to an earlier gate, after which the gradual increase continues. This kind of sawtooth relationship is seen in experimental data on larvae that vary in growth rate (G.D., unpublished results).
Figure 13 Model predictions of the effect of growth rate on body size. (a) Predicted effect of variation in growth rate on peak size; (b) predicted effect of variation in growth rate on peak size in the presence of a small amount of random variation in all other (more ...)
As before, in real larvae all the other parameters of size regulation vary among individuals, so in real life we should not necessarily expect to observe the idealized relationships shown in Figure . Imposing normal random variation (with standard deviations 8% of means, as before) on the other parameters, using the T strain parameter values (from Table ), while varying growth rate systematically gives the relationships shown in Figure . Linear regression on the simulated results in Figure gives the following relationship between growth rate and peak weight: peakweight = growthrate*0.58 + 6.31. This is close to the empirically observed relationship for this strain: peakweight = growthrate*0.56 + 6.30. Using the H strain parameter values we obtain the predicted relationship peakweight = growthrate*0.90 + 8.32, whereas the empirical relationship is peakweight = growthrate*1.03 + 8.23. Using the B strain parameter values the model predicts the relationship peakweight = growthrate*0.78 + 5.05, and the empirical relationship is peakweight = growthrate*1.07 + 4.19. Thus, the model predicts the correct slope and intercepts of the linear relationship between growth rate and peak weight very accurately for the T and H strains, but not as accurately for the B strain.
Body size and development time
The equations for the determination of body size are time-dependent and therefore they also embody the relationship between body size and development time (here we assume development time to be equivalent to the duration of the fifth larval instar). Development time and peak weight interact in a complex way because development time is determined, in part, by the time at which the critical weight is reached [12
], which depends on the growth rate; and the growth rate also determines the amount of mass that is added after the critical weight is passed. Figure shows the relationship between peak weight and development time under variation in the three fundamental parameters.
Figure 14 Relationship between body size and development time. Variation was introduced by allowing growth rate, critical weight and the ICG to vary with a standard deviation of 10% of the mean values of strain B in Table 1. The line is a linear regression on the (more ...)
Covariance between body size and the components of the mechanism
The mathematical model we have developed can be used to predict how variation in the three fundamental determinants of body size should affect variation of body size. To do this we need to find the functional relationship between peak weight and each of these three parameters. Because the effect of each determinant on peak weight size is nonlinear, and because the determinants interact with each other nonlinearly, there is no unique relationship between variation in any one of them and the peak weight size.
The relationship between any given parameter and peak weight depends on the specific values at which the other parameters are held constant. Therefore, body size cannot be expressed as a simple mathematical function of the three fundamental parameters, but the relationship between body size and each parameter must be found by solution or numerical simulation of the generative equations. It is possible to compute the peak weight that corresponds to any combination of values of the three fundamental parameters (as was done in Figures , , ). The three parameters can be used as the orthogonal axes of a three-dimensional volume in which each location gives the body size for a specific triplet of parameter values. Such a volume is illustrated in Figure . Such a graphical representation illustrates the complexity and context dependency of the relationship between any given parameter and body size.
Figure 15 Body size as a simultaneous function of the three fundamental parameters. The three parameters describe a volume of parameter space in which body size is depicted on a color scale. The two panels show different views of the same graph. The cutout is made (more ...)