Mimicking Nonequilibrium Steady States with Time-Periodic Driving

Under static conditions, a system satisfying detailed balance generically relaxes to an equilibrium state in which there are no currents. To generate persistent currents, either detailed balance must be broken or the system must be driven in a time-dependent manner. A stationary system that violates detailed balance evolves to a nonequilibrium steady state (NESS) characterized by fixed currents. Conversely, a system that satisfies instantaneous detailed balance but is driven by the time-periodic variation of external parameters—also known as a stochastic pump (SP)—reaches a periodic state with nonvanishing currents. In both cases, these currents are maintained at the cost of entropy production. Are these two paradigmatic scenarios effectively equivalent? For discrete-state systems, we establish a mapping between nonequilibrium stationary states and stochastic pumps. Given a NESS characterized by a particular set of stationary probabilities, currents, and entropy production rates, we show how to construct a SP with exactly the same (time-averaged) values. The mapping works in the opposite direction as well. These results establish a proof of principle: They show that stochastic pumps are able to mimic the behavior of nonequilibrium steady states, and vice versa, within the theoretical framework of discrete-state stochastic thermodynamics. Nonequilibrium steady states and stochastic pumps are often used to model, respectively, biomolecular motors driven by chemical reactions and artificial molecular machines steered by the variation of external, macroscopic parameters. Our results loosely suggest that anything a biomolecular machine can do, an artificial molecular machine can do equally well. We illustrate this principle by showing that kinetic proofreading, a NESS mechanism that explains the low error rates in biochemical reactions, can be effectively mimicked by a constrained periodic driving.

Popular Summary

Molecular motors are tiny machines that are crucially important to living organisms. These motors perform essential tasks ranging from the transportation of ingredients inside cells to the replication of DNA. Like macroscopic motors, they consume resources in the form of chemical fuel, and in return they produce useful outcomes, such as mechanical motion or enhanced accuracy of protein assembly. In recent years, advances in technology have enabled the construction of artificial nanomachines, essentially molecular complexes with moving parts that are driven, or “pumped,” by time-dependent changes in their external surroundings (e.g., temperature, chemistry), rather than by the consumption of chemical fuel. Here, we compare the range of outcomes that can be achieved by these two classes of molecular motors—powered either by the consumption of a resource or by external pumping—and the thermodynamic cost of achieving these outcomes.

Molecular motors are commonly modeled using stochastic thermodynamics, a theoretical framework for applying the laws of thermodynamics to systems on molecular length scales. We use this framework to show that molecular motors powered by the steady consumption of chemical fuel are thermodynamically equivalent to those driven by time-dependent pumping, in the sense that both kinds of motors are able to generate the same motion at the same thermodynamic cost. In other words, we determine how one kind of motor can be mapped to the other so that the two produce the same time-averaged probabilities, currents, and entropy production rates.

Our results can be applied to optimize the properties of artificial motors given experimental constraints.

Figure 1

A four-state system described by a graph. Each node represents a state of the system. The edges represent nonvanishing transition rates between states. In this example, direct transitions between states 1 and 3 are not allowed.

Figure 2

A concrete example of a NESS for which we build an equivalent stochastic pump. In this example, ${\mathcal{R}}_{ij}\ne 0$ for all $i$, $j$. The spanning tree was chosen to be the 2-1, 2-3, and 2-4 edges (dashed red lines), and the fundamental currents are the currents along the 1-4, 1-3, and 3-4 edges (solid black lines). In this system, there are two current loops: $1\to 3\to 4\to 1$ and $2\to 3\to 4\to 2$.

Figure 3

The currents of the stochastic pump constructed to mimic the NESS in Fig.(2). The pumped currents during the first (left) and second (right) half-periods are shown. On each of the edges, the current directions in the two half periods are opposite. Note that there are no current loops in both of the half periods.

Figure 4

$\overrightarrow{p}(t)$ of the proposed construction for our four-state example. The blue line is the exact (constructed) solution, and the red dots represent the numerical solution of the master equation.

Figure 5

A three-state system, through which we demonstrate kinetic proofreading. The system has three states: the correct state $c$, the wrong state $w$, and the auxiliary state $x$. Each state is associated with internal energies $E_c$, $E_w$, and $E_x$, respectively. Between any two states, there is a (symmetric) barrier, $B_{cw}=B_{wc}$, $B_{cx}=B_{xc}$, and $B_{wx}=B_{xw}$. The transition rates between the different states are given by the corresponding Arrhenius rates.

Figure 6

Patters of currents in periodically driven kinetic proofreading. In the first half-cycle (left), both currents are out of the state $x$, and in the second half-cycle (right), they are both into the state $x$.