Adaptation of protein stability to thermally heterogeneous environments.

Proper protein folding is essential for biological function, and its disruption can lead to disease, reduced fitness, or death. The ability of a protein to maintain its folded conformation is thus critical for life, making it a key target of adaptive evolution. However, protein stability is sensitive to environmental factors, particularly temperature, which can threaten phenotypic integrity and organismal survival under thermal changes. Despite its importance, the influence of complex thermal environments - characterized here by mean temperature, thermal fluctuations, and environmental heterogeneity - on the evolution of protein stability remains poorly understood. To address this, we developed a mathematical framework that combines two well-established models: a population genetic model describing species distributed across habitats with distinct thermal environments, and a thermodynamic model of protein stability incorporating temperature-dependent enthalpy and entropy contributions. We focus on two-state proteins that alternate between folded and unfolded states and assume that allelic fitness is maximized in proteins that achieve an optimal balance between flexibility and rigidity. Using this framework, we performed an invasion analysis of mutations (sensu adaptive dynamics framework) affecting three thermodynamic parameters that fully determine protein stability profiles. Where possible, we derived analytical expressions for evolutionarily optimal thermodynamic parameters and complemented these with numerical solutions. Our results show that mean temperature and thermal fluctuations have orthogonal effects on thermodynamic parameters, underscoring the need to consider both when studying protein stability adaptation. We further examined thermally heterogeneous environments, where subpopulations connected by migration experience different mean temperatures, identifying conditions that favor either local (specialist) or global (generalist) adaptation. Our results may explain why one thermodynamic parameter shows little association with thermal adaptation and suggest that local adaptation is more likely for proteins with stability profiles limited to narrow temperature ranges. Additionally, our analysis reveals whether a locally adapted protein originated in a colder or warmer habitat. Finally, we identified trade-offs in thermodynamic parameters that influence local or global adaptation. This study offers key predictions about protein evolution in complex thermal environments and lays the groundwork for developing practical tools to understand how temperature shapes adaptation and biodiversity.