Beyond Arrhenius: Nonlinear and negative temperature scaling of biological rates from multi-step mechanisms

Abstract

Temperature shapes all biological processes, particularly during the early development of ectothermic organisms. A widely used framework for describing temperature dependence is the Arrhenius equation, which predicts an exponential increase in rates with temperature. However, biological rates often deviate from this prediction when measured across broader temperature ranges. While negative apparent activation energies are often attributed to protein denaturation, this cannot explain similar behavior observed at temperatures where enzymes remain stable. These broader scaling patterns remain mechanistically unexplained. Here we present a general Markov chain framework for modeling biological timing as cascades of reversible, temperature-dependent steps. Applying this model to 121 published datasets spanning diverse species, biological timescales, and temperature ranges, we find that a consistent three-zone scaling pattern emerges: Arrhenius-like behavior at low and high temperatures, and a quadratic exponential regime at intermediate temperatures. We show that this pattern arises naturally from differences in activation energies across steps in the network. The quadratic exponential regime is an emergent feature of averaging across many steps and is robust to variation across network realizations. In contrast, Arrhenius-like scaling at the extremes tends to be more variable and originates from smaller sub-networks. Apparent negative activation energies can emerge naturally from the dynamics of multi-step networks, even in the absence of protein denaturation. Our framework provides a unified mechanistic explanation for diverse temperature-scaling behaviors in biology and may help predict how developmental and physiological processes respond to environmental change. Although we focus on development, the model is broadly applicable to biological systems governed by multi-step reaction networks.