Uncovering Key Springs in Elastic Networks: Insights from Global Epistasis and from Hessian Analyses

Epistasis describes the non-additive effects of mutations on phenotypes. Nonlinear genotype–phenotype relationships, termed global epistasis [1,2], capture systematic nonlinearities that can arise from underlying biophysical constraints. To understand the origins of global epistasis, we study it in elastic networks tuned to perform mechanical tasks.

We consider central-force spring networks, in which each individual spring constant can be tuned among a set of discrete stiffnesses. We tune the networks to simultaneously minimize the mechanical energy (at zero temperature) and a cost function that imposes a given relation between strains applied to input nodes and strains applied to designated output nodes. This double optimization yields a set of stiffnesses that satisfies the input-output relation in mechanical equilibrium. From the cost Hessian, corresponding to the curvatures around the minimum of the cost function in the high-dimensional space of the tunable spring constants, we identify springs that affect the cost the most if their stiffnesses are perturbed. We compare these key springs to predictions from a global epistasis model trained to infer key springs from mutation data. We further extract predictions for key springs from the physical Hessian [3]. Our aim is to determine whether Hessian analyses might yield predictions for key residues in functional proteins that are consistent with those obtained by global epistasis analyses of deep mutational scans. 

[1] S. Kryazhimskiy, D.P. Rice, E.R. Jerison, and M.M. Desai, Science 344, 1519 (2014).

[2] J. Otwinowski, D.M. McCandlish, and J.B. Plotkin, Proc. Natl. Acad. Sci. USA 115, E7550 (2018).  

[3] M. Guzman, F. Martins, M. Stern, and A.J. Liu, Phys. Rev. X 15, 031056 (2025).