Type-dependent irreversible stochastic spin models for genetic regulatory networks at the level of promotion-inhibition circuitry

We describe an approach to model genetic regulatory networks at the level of promotion-inhibition circuitry through a class of stochastic spin models that includes spatial and temporal density fluctuations in a natural way. The formalism can be viewed as an agent-based model formalism with agent behavior ruled by a classical spin-like pseudo-Hamiltonian playing the role of a local, individual objective function. A particular but otherwise generally applicable choice for the microscopic transition rates of the models also makes them of independent interest. To illustrate the formalism, we investigate (by Monte Carlo simulations) some stationary state properties of the repressilator, a synthetic three-gene network of transcriptional regulators that possesses oscillatory behavior.

💡 Research Summary

The paper introduces a novel theoretical framework that maps genetic regulatory networks onto stochastic spin‑type models, thereby incorporating spatial and temporal density fluctuations, non‑equilibrium dynamics, and intrinsic noise in a natural and tractable way. Each gene is represented by a discrete “spin” variable whose two states correspond to transcriptional activation (+1) or repression (‑1). Inter‑gene interactions are encoded in a pseudo‑Hamiltonian of the form
( H_i(\sigma)= -\sum_j J_{ij}\sigma_i\sigma_j - h_i\sigma_i ),
where the coupling constants (J_{ij}) can be positive (promotion) or negative (inhibition) and are allowed to be asymmetric, reflecting the type‑dependent nature of real regulatory interactions.

The dynamics are governed by irreversible transition rates that do not satisfy detailed balance. A Glauber‑type rule is adopted:
( w_i(\sigma!\to!\sigma’) = \frac{1}{1+\exp