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 representing protein structures as weighted graphs, we identify each spanning tree as a topological microstate. The canonical partition function is evaluated exactly 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 approximation. 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 exact 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) proteins. Results show that while the mutation minimally affects mean internal energy and entropy, it reduces global heat capacity by 27.3%. This indicates a topological stiffening where the mutant occupies a significantly narrower landscape of spanning-tree configurations. At the channel level, the mutation maintains distributional stability across six functional routes but triggers a substantial internal redistribution of allosteric importance. Specific residues, such as Q61 and F156, shift occupancy by up to 35.5%. These findings suggest that the G12D mutation does not destroy communication pathways but reorganizes internal information traffic to favor a catalytically impaired state. This approach provides a rigorous, parameter-free metric for understanding how point mutations perturb distal protein signaling.