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Physical BiologyBiophysicsPeer-reviewed journal article

Spanning-Tree Thermostatistics of Protein Allostery: An Exact Kirchhoff Framework with Application to Oncogenic KRAS

Fatma Senguler CiftciiD, Burak ErmaniD

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Abstract

This study introduces a statistical mechanical framework for allosteric communication in proteins based on the spanning-tree ensemble of residue contact networks. By representing C protein 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 on C backbone 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.

In-depth summary

Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (28 Jul 2026) · CC BY 4.0

Kirchhoff matrix-tree evaluation of residue-contact spanning-tree ensembles yields exact allosteric thermodynamics; KRAS G12D conserves global and channel heat capacity but reroutes paths via Q61 and F156.

Classical allostery models (Monod-Wyman-Changeux, Cooper-Dryden) ignore the pathway redundancy of residue contact networks, and centrality-based hotspot methods rank individual residues rather than describing the signaling ensemble thermodynamically. Treating spanning trees as microstates yields sampling-free global and channel-level entropy and heat capacity from Cα coordinates, showing how a point mutation can reroute signaling while bulk thermodynamics stay fixed.

Key results

  1. ShownThe global spanning-tree thermodynamics of wild-type and G12D KRAS are nearly identical. This fits the two crystal structures, which are almost superimposable and share nearly all Cα contacts.

    Mean energy +0.13%, entropy +0.37%, heat capacity 139.6 (WT) vs 140.3 (G12D), +0.50%, at kT = 1.0; Cα superposition RMSD 0.19 Å, TM-score 0.998; 793 of 806 contacting pairs shared (Jaccard 0.984), 796 vs 803 contacts, with all 13 differing pairs at 7.5 to 8.2 Å. · 3. Mutation effects on KRAS; 3.1 Global Thermodynamics

  2. ShownChannel heat capacities are also conserved, but through compensation rather than rigidity. The energetic and topological components move in the same direction while the cross-coupling term moves the opposite way, and the accepted path ensemble of every channel grows under G12D.

    |ΔC| at most 0.08 reduced units (under 1.5% of channel value) across ten channels; 12→61 channel ΔCE +2.345, ΔCT +2.450, ΔCX −4.802; 19–142 inter-lobe channel +0.26, +0.49, −0.82, net −0.07; nG12D > nWT in every channel. · 3.2.2 Changes in channel thermodynamics upon…; Thermodynamic Decomposition and the…; 4.3 Invariant channel capacity with internal…

  3. SuggestedAllosteric importance (the share of channel paths running through a residue) is redistributed. G12 and Switch I residues are bypassed and Q61 and the inter-lobe F156/A155 bridge are recruited; E62 gains in the 12→35 channel but loses in the 55→60 and 12→61 channels. The canonical 12→61 axis changes least.

    19→142 channel: F156 +35.5%, A155 +29.1%, D153 +17.4%; 55→60: Q61 +22.7%, G12 −25.9%, E62 −17.8%; 12→35: Y32 −15.9%, D33 −14.9%, Q61 +18.6%, E62 +19.3%; 12→61: only G13 +15.3% and E62 −12.6% exceed the reporting threshold. · 3.2.3 Redistribution of path probabilities upon…; Detailed Biological and Biophysical…; 4.4 Residue-level redistribution and the…

  4. ShownCutting channel path sums at 9 nodes, and keeping paths up to 99% cumulative probability, reproduces most of the exact effective resistance computed from the Laplacian pseudoinverse. The slower P-loop to C-terminal corridor is the exception.

    Most channels recover 93% to 99% of the exact dynamic distance; P-loop 6→11 reaches r ≥ 0.99 by L = 8; 110→117 and 19→142 reach r ≈ 0.93 to 0.97 at L = 9; 12→170 reaches ≈87% at L = 9 (95% at L = 10); paths kept to η = 0.99, resistance tolerance below 10^−3. · 2.3.4 The principle of minimum surprisal; 3.2.1 Convergence of the truncated channel…; 3.2.3 Redistribution of path probabilities upon…; Figure legends

  5. ShownWeighting contacts by distance alone makes the covalent backbone dominate the weighted spanning-tree ensemble, even though no backbone edge is needed to keep the graph connected.

    Backbone edges are 171 of 796 (WT) and 171 of 803 (G12D) edges, none a bridge at 7.8 Å; unweighted limit: 22% of expected tree edges vs 21% of all edges; at kT = 1.0 Å: 58% of expected edges, roughly ten-fold higher mean inclusion probability than non-bonded contacts. · 2.1.3 Edge weights and interaction parameters

Methods, data and reproducibility

Cα contact graphs (7.8 Å cutoff) receive Boltzmann distance weights; the reduced weighted Laplacian determinant yields the exact spanning-tree partition function, and Burton-Pemantle determinants from the Laplacian pseudoinverse yield path probabilities, channel heat-capacity decompositions and residue allosteric importance over paths of up to 9 nodes.

  • Wild-type KRAS crystal structure, active GNP-bound conformation; chain A Cα contact graph with 796 contacts at a 7.8 Å cutoff · n = 172 residues · PDB 6GOD
  • KRAS G12D mutant crystal structure, active GNP-bound conformation; chain A Cα contact graph with 803 contacts at a 7.8 Å cutoff · n = 172 residues · PDB 6GOF
Methods & tools
Kirchhoff Matrix-Tree Theorem (reduced weighted Laplacian determinant), Burton-Pemantle theorem (edge-set inclusion probabilities), Moore-Penrose pseudoinverse of the graph Laplacian (effective resistance), Gaussian Network Model (recovered in the kT → ∞ limit), Kullback-Leibler and Jensen-Shannon divergence, Participation ratio (second-order Rényi entropy), Rayleigh monotonicity-based truncation convergence test, RMSD and TM-score structural superposition
Reproducibility gaps
No code repository, software versions or compute details are given. The inputs are public PDB entries 6GOD and 6GOF; the sensitivity, truncation and contact-difference analyses are in supplementary material that was not provided.
Limitations (5)
  • Inputs are single static crystal structures reduced to Cα backbones rather than conformational ensembles, so the mutation is detected only through backbone geometry, not side-chain chemistry; the residue-level shifts are hypotheses still to be tested by all-atom MD, mutagenesis or NMR. · stated by the authors
  • Results depend on chosen inputs (7.8 Å contact cutoff, kT = 1.0, path truncation); robustness was tested over cutoffs of 7.5-8.5 Å and kT = 0.8-1.2, with those results reported only in supplementary material. · stated by the authors
  • The 12→170 P-loop to C-terminal channel converges more slowly (≈87% of exact dynamic resistance at L = 9), so its shifts at G60, T58 and K169 are flagged as provisional. · stated by the authors
  • Internal inconsistencies: the limitations section gives path truncation at Lmax = 6, whereas the methods use node length 9; the 12→170 channel falls below the stated 93% to 99% convergence range; the cross term enters the heat-capacity sum both with and without a factor of 2. · noted in this summary
  • The mechanistic and therapeutic readings (Q61 diverted from catalysis, altered exchange kinetics, A155/F156 as drug targets) rest on one wild-type/mutant structure pair, with no uncertainty or null baseline for importance shifts that may be partly driven by 13 near-cutoff contacts. · noted in this summary

Read this if you build network or elastic-network models of allostery and want an exact, sampling-free spanning-tree method for channel entropy, heat capacity and residue importance, applied to KRAS G12D.

Citations
2
FWCI
4.18
Field percentile
top 6%

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