Does Animal Development Follow the Arrhenius Equation?

Abstract

Since the late 19th century we have described the dependence of simple chemical reaction rates (k) on temperature (T) with the Arrhenius equation: k=A*exp(Ea/kbT), where A is the frequency factor, kb the Boltzmann constant, and Ea the activation energy. Recent evidence suggests that more complicated reaction networks in biology, e.g. simple biological processes also follow this simple relationship. Here I carefully investigate the temperature dependence of animal development in frog model embryos. To this end, I recorded development of the frog X. laevis from the fertilized egg to the onset of muscle movement at various temperatures. My data shows that frog development is well described by the Arrhenius equation in a temperature range from 16-23.5 degrees Celsius, regardless of developmental stage analyzed. One simple explanation for this may be that evolution led all rate-limiting activation energies in frog to converge. However, my data shows that different stages of development exhibit significantly different apparent activation energies. Despite these different activation energies, the summed sequence of development continues to follow the predictions of the Arrhenius equation. This finding seems to contradict the simple underlying individual molecular assumption of the Arrhenius Equation, which starkly contrasts with the complexity of embryonic development. To address this question, I simulated coupled reaction networks wherein the individual steps follow the Arrhenius equation and observed how the entire network rate scaled with temperature. I show analytically, that an entire reaction network cannot be described perfectly with the Arrhenius equation when individual activation energies differ. However, for relevant biological temperature ranges and activation energies, the cumulative behavior of the network can be approximated by the Arrhenius equation, likely well within the achievable error bars of biological experiments. Thus, I was able to observe that the dependence of embryonic development rate on temperature is well described with the Arrhenius equation. Furthermore, I present an analytical framework that demonstrates how highly complex biological networks follow this simple rule.