![]() ![]() Embryos of most species develop outside the mother and many have evolved so that they can adapt to wide temperature ranges. One of the most complicated biological processes the Arrhenius equation has been applied to was the development of a single fertilized egg into the canonical body plan of an embryo (Chong et al, 2018). This broad applicability of the Arrhenius equation to complex biological systems is surprising given that these systems involve a myriad of reactions, presumably each with its own activation energy and thus temperature dependence. Modifications to Arrhenius have even been made, positing mass accounts for deviations from a fairly universal Arrhenius fit (Gillooly et al, 2002). More recently, the Arrhenius equation has been investigated for use in describing cell cycle duration (Begasse et al, 2015 Falahati et al, 2021), or, by extension, the Q 10 rule modeling proliferation dynamics in populations of bacteria (Martinez et al, 2013). The Arrhenius equation’s use has also been extended to more complex biological systems, such as frog, beetle, and fly development, occasionally finding non-Arrhenius behavior (Krogh, 1914 Bliss, 1926 Bonnier, 1926 Ludwig, 1928 Powsner, 1935). The pre-exponential “frequency” factor A can be physically interpreted as proportional to the number of molecular collisions with favorable orientations. Based on this theory, the exponential term of the Arrhenius equation is proportional to the fraction of molecules with energy greater than the activation energy (E a) needed to overcome the reaction’s energetically unfavorable transition state. The Arrhenius equation would come to stand out from the rest, in part because it could be intuitively interpreted based on transition-state theory (Evans & Polanyi, 1935 Eyring, 1935a, 1935b Laidler & King, 1983). In the late 19 th century, scientists proposed many relationships between reaction rates and temperature (Berthelot, 1862 Schwab, 1883 Van’t Hoff & Hoff, 1884 Van’t Hoff, 1893 Harcourt & Esson, 1895). Thus, we find that complex embryonic development can be well approximated by the simple Arrhenius equation regardless of non-uniform developmental scaling and propose that the observed departure from this law likely results more from non-idealized individual steps rather than from the complexity of the system. In contrast, we find the two enzymes GAPDH and β-galactosidase show non-linearity in the Arrhenius plot similar to our observations of embryonic development. ![]() When we model multi-step reactions of idealized chemical networks, we are unable to generate comparable deviations from linearity. At low and high temperatures, however, we observed significant departures from idealized Arrhenius Law behavior. We find that the Arrhenius equation provides a good approximation for the temperature dependence of embryogenesis, even though individual developmental intervals scale differently with temperature. Here, we evaluate how well the simple Arrhenius equation predicts complex multi-step biological processes, using frog and fruit fly embryogenesis as two canonical models. The famous Arrhenius equation is well suited to describing the temperature dependence of chemical reactions but has also been used for complicated biological processes. ![]()
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