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Well-mixed chemical reaction networks (CRNs) contain many distinct chemical species with copy numbers that fluctuate in correlated ways. While those correlations are typically monitored via Monte Carlo sampling of stochastic trajectories, there is interest in systematically approximating the joint distribution over the exponentially large number of possible microstates using tensor networks or tensor trains. We exploit the tensor network strategy to determine when the steady state of a seven-species gene toggle switch CRN model supports bistability as a function of two decomposition rates, both parameters of the kinetic model. We highlight how the tensor network solution captures the effects of stochastic fluctuations, going beyond mean field and indeed deviating meaningfully from a mean-field analysis. The work furthermore develops and demonstrates several technical advances that will allow steady-states of broad classes of CRNs to be computed in a manner conducive to parameter exploration. We show that the steady-state distributions can be computed via the ordinary density matrix renormalization group (DMRG) algorithm, despite having a non-Hermitian rate operator with a small spectral gap, we illustrate how that steady-state distribution can be efficiently projected to an order parameter that identifies bimodality, and we employ excited-state DMRG to calculate a relaxation timescale for the bistability. 

Transition path theory (TPT) offers a powerful formalism for extracting the rate and mechanism of rare dynamical transitions between metastable states. Most applications of TPT either focus on systems with modestly sized state spaces or use collective variables to try to tame the curse of dimensionality. Increasingly, expressive function approximators such as neural networks and tensor networks have shown promise in computing the central object of TPT, the committor function, even in very high-dimensional systems. That progress prompts our consideration of how one could use such a high-dimensional function to extract mechanistic insights. Here, we present and illustrate a straightforward but powerful way to track how individual dynamical coordinates evolve during a reactive event. The strategy, which involves marginalizing the reactive ensemble, naturally captures the evolution of the dynamical coordinate’s distribution, not just its mean reactive behavior. 

The interplay between stochastic chemical reactions and diffusion can generate rich spatiotemporal patterns. While the timescale for individual reaction or diffusion events may be very fast, the timescales for organization can be much longer. That separation of timescales makes it particularly challenging to anticipate how the rapid microscopic dynamics gives rise to macroscopic rates in the nonequilibrium dynamics of many reacting and diffusing chemical species. Within the regime of stochastic fluctuations, the standard approach is to employ Monte Carlo sampling to simulate realizations of random trajectories. Here, we present an alternative numerically tractable approach to extract macroscopic rates from the full ensemble evolution of many-body reaction-diffusion problems. The approach leverages the Doi-Peliti second-quantized representation of reaction-diffusion master equations along with compression and evolution algorithms from tensor networks. By focusing on a Schlögl model with one-dimensional diffusion between L otherwise well-mixed sites, we illustrate the potential of the tensor network approach to compute rates from many-body systems, here with approximately 3 × 10^15 microstates. Specifically, we compute the rate for switching between metastable macrostates, with the expense for computing those rates growing subexponentially in L. Because we directly work with ensemble evolutions, we crucially bypass many of the difficulties encountered by rare event sampling techniques—detailed balance and reaction coordinates are not needed. 

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis.