In prokaryotes, the frequency of amino acids in proteomes is strongly correlated with the optimal growth temperature (OGT), but the associated substitutional patterns and their dynamics remain poorly understood. The ThermAdapt project addresses these two important issues through an original approach combining genomics, ancestral sequence reconstructions, structural modeling, experimental biology, and persistent homology. The ThermAdapt projects aims at: 1- Tracing back the evolution of OGTs and associated substitutional patterns in different branches of the Tree of Life. The collected data will allow to determine whether substitutional patterns are universal or lineages-specific. 2- Comparing substitutions associated to similar but independant OGT shifts in proteins families, in order to determine whether substitutions occurr at random or specific positions in protein sequences, and thus whether the thermoadaptation process is mainly stochastic or deterministic. 3- Studying the impact of substitutions on the structure and function of proteins by molecular modeling. These predictions will be experimentally tested. To propose new algorithms to model thermoadaptation processes in the primary sequences of proteins as well as in their 3D structures, and in time, using Bayesian and persistent homology approaches. Altogether, the ThermAdapt project will improve our knowledge of mechanisms and patterns of thermoadaptation in prokaryotes.