Spanning-tree thermostatistics of protein allostery: an exact Kirchhoff framework with application to oncogenic KRAS.

This study introduces a statistical mechanical framework for allosteric communication in proteins based on the spanning-tree ensemble of residue contact networks. By representingprotein backbones as weighted graphs, we identify each spanning tree as a topological microstate. The canonical partition function is evaluated analytically via the determinant of the reduced weighted Kirchhoff (Laplacian) matrix, allowing for the derivation of global thermodynamic functions (including Helmholtz free energy, internal energy, entropy, and heat capacity) without stochastic sampling. Allosteric channels between specific residue pairs are defined as sub-ensembles containing unique simple paths. Using the Burton-Pemantle theorem and the Moore-Penrose pseudoinverse of the graph Laplacian, we compute path probabilities and channel-specific thermodynamics. This methodology enables a decomposition of channel heat capacity into energetic and topological components and quantifies residue-level allosteric importance through fractional contributions to the channel partition function. The framework was applied to the G12D mutation in KRAS, comparing wild-type (PDB: 6GOD) and mutant (PDB: 6GOF) structures. Results show that while global thermodynamic properties remain highly conserved across the tight structural superposition, channel-level analysis shows a substantial internal redistribution of allosteric importance among intermediate residues, highlighted by the primary 12-61 signaling axis and distal routes (including shifts in residues such as Q61 and F156). Operating onbackbone geometry, these topological shifts provide predictive hypotheses for subsequent molecular dynamics and experimental testing. Overall, this approach offers a rigorous, parameter-robust framework for understanding how point mutations perturb distal signaling networks.