When chemically fueled, molecular self-assembly can sustain dynamic aggregates of polymeric fibershydrogelswith tunable properties. If the fuel supply is finite, the hydrogel is transient, as competing reactions switch molecular subunits between active and inactive states, drive fiber growth and collapse, and dissipate energy. Because the process is away from equilibrium, the structure and mechanical properties can reflect the history of preparation. As a result, the formation of these active materials is not readily susceptible to a statistical treatment in which the configuration and properties of the molecular building blocks specify the resulting material structure. Here, we illustrate a stochastic–thermodynamic and information–theoretic framework for this purpose and apply it to these self-annihilating materials. Among the possible paths, the framework variationally identifies those that are typicalloosely, the minimum number with the majority of the probability. We derive these paths from computer simulations of experimentally-informed stochastic chemical kinetics and a physical kinetics model for the growth of an active hydrogel. The model reproduces features observed by confocal microscopy, including the fiber length, lifetime, and abundance as well as the observation of fast fiber growth and stochastic fiber collapse. The typical mesoscopic paths we extract are less than 0.23% of those possible, but they accurately reproduce material properties such as mean fiber length.

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Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.

Thermodynamic intuition suggests that fast, irreversible processes will dissipate more energy and entropy than slow, quasistatic processes connecting the same initial and final states. Here, we show that this intuition does not necessarily hold for stochastic processes when there are strong nonequilibrium currents. To analyze the relationship between dissipation and speed, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution of the master equation. Applying this formula to a model for nonequilibrium self-assembly, we demonstrate that kinetics can sculpt the nonequilibrium path space so that faster structureforming paths dissipate less. The path-level dissipation is largely controlled through the nonequilibrium currents of a dissipative cycle situated between the initial and final states whose relative energies independently control the time to assemble. Overall, these results suggest that minimally dissipative materials, which respond quickly to their environment, may be designed through the stochastic thermodynamics of their path space.