# Laplacian Dynamics on General Graphs

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## Abstract

In previous work, we have introduced a “linear framework” for time-scale separation in biochemical systems, which is based on a labelled, directed graph, *G*, and an associated linear differential equation, \(dx/dt = \mathcal{L}(G)\cdot x\), where \(\mathcal{L}(G)\) is the Laplacian matrix of *G*. Biochemical nonlinearity is encoded in the graph labels. Many central results in molecular biology can be systematically derived within this framework, including those for enzyme kinetics, allosteric proteins, G-protein coupled receptors, ion channels, gene regulation at thermodynamic equilibrium, and protein post-translational modification. In the present paper, in response to new applications, which accommodate nonequilibrium mechanisms in eukaryotic gene regulation, we lay out the mathematical foundations of the framework. We show that, for any graph and any initial condition, the dynamics always reaches a steady state, which can be algorithmically calculated. If the graph is not strongly connected, which may occur in gene regulation, we show that the dynamics can exhibit flexible behavior that resembles multistability. We further reveal an unexpected equivalence between deterministic Laplacian dynamics and the master equations of continuous-time Markov processes, which allows rigorous treatment within the framework of stochastic, single-molecule mechanisms.

## Keywords

Time-scale separation Linear framework Graph Laplacian Matrix-Tree theorem Gene regulation Nonequilibrium mechanisms Markov process Master equation## Notes

### Acknowledgements

We thank Arthur Jaffe for historical remarks on citations (Bott and Mayberry 1954; Jaffe 1965), David Perkinson for information about the proof in the Appendix, and an anonymous reviewer for helpful comments. The work undertaken here was supported by the United States NSF under Grant 0856285.

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