Kirchhoff polynomials

Kirchhoff polynomials perform graph-based prime factorization and compression to generate Kirchhoff polynomials for deriving symbolic steady-state expressions in non-equilibrium biochemical reaction networks.

Key Features:

• Graph-based prime factorization: Decomposes graphs in Kirchhoff polynomial expressions into prime components to reduce combinatorial complexity.
• Polynomial equivalence testing: Implements criteria for efficiently testing equality of Kirchhoff polynomials and identifying graphs that yield equivalent polynomials.
• Compressed polynomial generation heuristics: Provides two heuristic algorithms to generate individual Kirchhoff polynomials in a compressed form.
• Algebra-inspired graph simplifications: Applies algebraic simplification principles to graph structures rather than to algebraic expressions.
• Mitigation of history dependence and combinatorial explosion: Reduces the super-exponential growth of symbolic steady-state expressions caused by history dependence away from thermodynamic equilibrium.
• Applicability to linear diffusion and biological systems: Targets models governed by linear diffusion on graphs relevant to enzyme kinetics, G-protein coupled receptors, ion channels, and gene regulation.
• Validation on diverse graph sizes: Demonstrated effectiveness across diverse sets of graphs of varying sizes, including non-equilibrium gene regulation analyses.

Scientific Applications:

• Symbolic steady-state derivation: Derivation of symbolic steady-state expressions for models governed by linear diffusion on graphs.
• Non-equilibrium biochemical network analysis: Analysis of non-equilibrium biochemical reaction networks exhibiting history dependence.
• Gene regulation studies: Non-equilibrium gene regulation analysis using Kirchhoff polynomial compression.
• Enzyme kinetics and receptor/channel modeling: Analysis of enzyme kinetics, G-protein coupled receptors, and ion channels using graph-based steady-state methods.

Methodology:

Decompose graphs into prime components, apply criteria to test Kirchhoff polynomial equality, and use two heuristic algorithms that generate compressed Kirchhoff polynomials by applying algebraic-simplification principles to graph structures.