Shortcuts in stochastic systems and control of biophysical processes

TL;DR

The paper develops a universal, graph-theoretic framework for counterdiabatic (CD) driving of discrete-state stochastic systems, enabling precise control of probability distributions in biological networks. By casting the master equation in current-graph form with an incidence matrix $\nabla$, it derives a CD current equation whose solution decomposes into a spanning-tree (tree basis) part and a cycle-basis part, yielding multiple physically realizable CD protocols for the same target trajectory. It generalizes CD driving to non-stationary targets and local control, deriving explicit constructions for CD currents and rates under partial control and nonstationary targets, and establishes graphical criteria for when global vs. local control is possible. The authors demonstrate global CD control in a repressor-corepressor genetic switch and local CD control in a chaperone-assisted protein folding model, finding that generated protocols resemble experimentally observed heat-shock responses and can be used to minimize thermodynamic costs under realistic constraints. Together, the work provides a practical, testable framework for steering stochastic biological processes in finite time using accessible control knobs like concentrations and ATP, with broad implications for synthetic biology and evolution.

Abstract

The biochemical reaction networks that regulate living systems are all stochastic to varying degrees. The resulting randomness affects biological outcomes at multiple scales, from the functional states of single proteins in a cell to the evolutionary trajectory of whole populations. Controlling how the distribution of these outcomes changes over time -- via external interventions like time-varying concentrations of chemical species -- is a complex challenge. In this work, we show how counterdiabatic (CD) driving, first developed to control quantum systems, provides a versatile tool for steering biological processes. We develop a practical graph-theoretic framework for CD driving in discrete-state continuous-time Markov networks. Though CD driving is limited to target trajectories that are instantaneous stationary states, we show how to generalize the approach to allow for non-stationary targets and local control -- where only a subset of system states are targeted. The latter is particularly useful for biological implementations where there may be only a small number of available external control knobs, insufficient for global control. We derive simple graphical criteria for when local versus global control is possible. Finally, we illustrate the formalism with global control of a genetic regulatory switch and local control in chaperone-assisted protein folding. The derived control protocols in the chaperone system closely resemble natural control strategies seen in experimental measurements of heat shock response in yeast and E. coli.

Figures (9)

• Figure 1: Schematic of the biological control problem: A) A system of interest (here a membrane receptor protein) is described via a network of transition rates between discrete states. Certain rates may be influenced by factors external to the system, which we denote as control parameters. For biochemical systems these are often concentrations of chemical species (ligands, ATP, etc.) or environmental factors like temperature. B) We consider two types of control: global control, where we require the probability of every state to follow a target trajectory over a finite time interval; and local control, where we impose this requirement on only a subset of states. C) In either case, the goal is to find whether control is possible for a given target, and if so calculate the control parameter protocol that forces the system to follow the target trajectory.
• Figure 2: Overview of the graphical approach for deriving CD solutions. We start with a Markov model defined by a transition matrix $\Omega(\lambda_t)$ dependent on the control protocol $\lambda_t$. Associated with this is a graph with $N$ states, $E$ edges, and a target trajectory $\bm{\rho}(\lambda_t)$ consisting of instantaneous stationary states of $\Omega(\lambda_t)$. The eventual goal is to find the CD transition matrix $\widetilde{\Omega}(\lambda_t,\dot\lambda_t)$ where $\bm{\rho}(\lambda_t)$ is the solution to the associated master equation, Eq. \ref{['3']}. To facilitate this, we must first find the CD currents ${\bm{\mathcal{\widetilde{J}}}}(t)$, the main goal of the graphical approach. The most general form of the solution for ${\bm{\mathcal{\widetilde{J}}}}(t)$ is given by Eq. \ref{['gs10']}, and consists of two components: (i) a spanning tree CD solution $\delta {\bm{\mathcal{J}}}^{(1)}(t)$, given by Eq. \ref{['gs3']} and derived via the procedure outlined in Sec. \ref{['sec:graphsol']}; (ii) a linear combination of the fundamental basis cycle vectors $\bm{c}^{(\gamma)}$, $\gamma = 1,\ldots,\Delta$, where $\Delta = E-N+1$, as described in Sec. \ref{['sec:cycbasis']}. The coefficient functions $\Phi_\gamma(t)$ are arbitrary. Once the CD currents ${\bm{\mathcal{\widetilde{J}}}}(t)$ are known, we can use Eq. \ref{['sol3']} to solve for the CD transition rates $\bm{\widetilde{k}}^\pm(t)$ that determine $\widetilde{\Omega}(\lambda_t,\dot\lambda_t)$.
• Figure 3: A two-loop discrete state Markov model, with $N=4$ states and $E=5$ edges. A) The black arrows correspond to entries in the transition matrix $\Omega(\lambda_t)$: transition rates $k^\pm_i(\lambda_t)$ that depend on an external protocol $\lambda_t$. B) The red arrows labeled $\alpha$ correspond to the oriented stationary currents $\mathcal{J}_\alpha(\lambda_t)$, defined in Eq. \ref{['f5']}. C) On the left, one of the spanning trees of the graph, chosen to be a reference for constructing the tree basis. Edges deleted to form the tree are shown in faint red. On the right, two trees in this set derived from the reference one. Each such derived tree has a one-to-one correspondence with a fundamental cycle of the graph (highlighted in green).
• Figure 4: A) Partition of the graph matrices $\nabla$ and $\widetilde{G}(t)$ into submatrices to facilitate solving the local control problem. B) A network with $N_T = 4$ target states and $E_C = 6$ controllable edges, depicted as indicated by the legend. The three target subgraphs, each containing at least one target state and all other states connected to it via controllable edges, are outlined in dashed curves. Because each target subgraph includes at least one non-target state, the local control condition is satisfied. C) Same network as B, except that the set of $N_T = 4$ target states is different. Here neither of the two target subgraphs contain any non-target states, and hence local control is impossible.
• Figure 5: A) Biochemical network of a repressor-corepressor model, showing an operator site on DNA in three different states: 1) free; 2) bound to a bare repressor protein; 3) bound to a represssor-corepressor complex. Transition rates between the states are shown in green. The binding reaction rates depend on three concentrations of molecules in solution: $R(t)$ for bare repressors, $C(t)$ for corepressors, and $X(t)$ for the complexes. B) One of the spanning trees for the associated network graph, with the edge deleted to form the tree shown in faint red. We take this to be the reference spanning tree for the tree basis. C) The other tree in the basis, with the corresponding fundamental cycle in green.