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We develop a path-based approach to continuous-time random walks on networkswith arbitrarily weighted edges. We describe an efficient numerical algorithmfor calculating statistical properties of the stochastic path ensemble. Afterdemonstrating our approach on two reaction rate problems, we present abiophysical model that describes how proteins evolve new functions whilemaintaining thermodynamic stability. We use our methodology to characterizedynamics of evolutionary adaptation, reproducing several key features observedin directed evolution experiments. We find that proteins generally fall intotwo qualitatively different regimes of adaptation depending on their bindingand folding energetics.

A path-based approach to random walks on networks characterizes how proteins evolve new function