Food restriction alters energy allocation strategy during growth in tobacco hornworms (Manducasextalarvae)
Electronic Supplementary Material
The Approximation of the equation linking different scaling laws
We consider three functions of variable x, A(x), B(x), and C(x), and. Functions B and C scale with x as and, where the scaling powers b and c are different and both are between 0 and 1. We are going to show that since the difference between b and c can not be larger than 1, the function A can be approximately expressed as a scaling law of x.
First, we normalize the range of the variable x so that it varies between 1 and 100. (The range of x can be different, but in general it can always be normalized by changing the unit and the normalization coefficients.) If the scaling power b is larger than c, then we can write
Eq. (S1).
We will show that the term in the bracket, , can be expressed approximately as a scaling law of x, and therefore is approximately a scaling law of x.
Since b is larger than c (both between 0 and 1), and x varies between 1 and 100, varies between 1 and 0.01, so varies between C0/B0 and 0.01×C0/B0, and the bracket of Eq. (S1),, varies betweenand .
If C0 is much larger than B0, then is slightly larger than C0/B0. So, the shape of the curve , which decreases from to , will be similar to the shape of the curve , which decreases from C0/B0 to 0.01×C0/B0. In this case, we can write, where D is a constant smaller than C0/B0, and |d| is smaller than . Thus, Eq. (S1) becomes . This means, if C0 is much larger than B0, the scaling power of C is dominant in Eq. (S1), and the function A has a scaling power (b+d), which is between the scaling powers b and c, and close to c, (recalling d is close to c−b, so b+d is close to c).
If C0 is much smaller than B0, then is only slightly larger than 1. So, “1” is the dominant term in . When x increases, the value of will not be too far from “1”, i.e.,only weakly scales with x. In this case, we can write, where D’ is a constant larger than 1, and |d’| is larger than zero. Thus, Eq. (S1) becomes . This means, if C0 is much smaller than B0, the scaling power of B is dominant in Eq. (S1), and the function A has a scaling power (b+d’), which is between the scaling powers b and c, and close to b, (recalling d’ is close to zero, so b+d’ is close to b).
If C0 and B0 are close, e.g., C0 = B0, then varies between 2 and 1 when x varies between 1 and 100. Fig. S1 shows that can be approximately expressed as a scaling law of x. In Fig. S1, we choose four sets of values of , and calculate . We then fit the calculated as a scaling law of x. From Fig. S1 we see that the fitting is good when b and c are close (e.g., ). The R2 values decrease as c and b become further apart. Recalling both b and c are between 0 and 1, so (blue dots in the figure) can be considered as an extreme case, in which the R2 value drops down to 0.774.
Figure S1: Four examples of fitting as scaling laws of x. Four sets of scaling power are chosen to vary between −0.1 to −0.9.
The function A(x) is multiplied by B0xb (Eq. S1). The term B0xb makes the statistic fitting of A as a scaling law better (the R2 value is close to 1), even in the case where the difference between c and b is large. In Fig. S2, we choose three sets of scaling powers and the normalization coefficients of both B and C. We then calculate the sum of B and C (A = B +C). We fit the sum A as a scaling law of x,. The fittings are very good, as the R2 values are larger than 0.98 for all sets of parameters, even when (blue dots). The powers of the fitted scaling laws, a, are between c and b.
Figure S2.Three examples of fitting scaling laws of A.A (the dot) are calculated from the equation. The colors denote different choices of the scaling coefficients, B0 and C0, and the scaling powers, b and c. The values of power b are chosen to be larger than that of power c in all three examples. Black dots:(where B0 is much larger than C0, and ); Red dots: (where B0 = C0, and ); Blue dots: (where B0 is much smaller than C0, and ). The fitted equations () and the R2 values are shown in the figure.