Geometry of optimal control in chemical reaction networks

TL;DR

This paper develops a theoretical framework for optimal control in chemical reaction networks, linking dissipation minimization to geodesic paths in probability space and deriving analytical expressions for optimal protocols.

Contribution

It introduces a general approach to determine optimal control protocols in CRNs, including a Kirchhoff's law for probability currents and a metric tensor for dissipation.

Findings

Total dissipation relates to a probability space L2-distance.

Optimal trajectories are geodesics in probability space.

Tighter lower bounds for dissipation are derived.

Abstract

Although optimal control (OC) has been studied in stochastic thermodynamics for systems with continuous state variables, less is known in systems with discrete state variables, such as Chemical Reaction Networks (CRNs). Here, we develop a general theoretical framework to study OC of CRNs for changing the system from an initial distribution of states to a final distribution with minimum dissipation. We derive a ``Kirchhoff's law" for the probability current in the adiabatic limit, from which the optimal kinetic rates are determined analytically for any given probability trajectory. By using the optimal rates, we show that the total dissipation is determined by a $L_2$-distance measure in the probability space and derive an analytical expression for the metric tensor that depends on the probability distribution, network topology, and capacity of each link. Minimizing the total dissipation leads to the geodesic trajectory in the probability space and the corresponding OC protocol is determined by the Kirchhoff's law. To demonstrate our general approach, we use it to find a lower bound for the minimum dissipation that is tighter than existing bounds obtained with only global constraints. We also apply it to simple networks, e.g., fully connected 3-state CRNs with different local constraints and show that indirect pathway and non-functional transient state can play a crucial role in switching between different probability distributions efficiently. Future directions in studying OC in CRNs by using our general framework are discussed.