Abstract: Hydroxyapatite column chromatography can be used to purify filamentous bacteriophage—the phage most commonly used for phage display. Virions that have been partially purified from culture supernatant by two cycles of precipitation in 2% polyethylene glycol are adsorbed onto the matrix at a density of at least $7.6 \times 10^{13}$ virions (about 3 mg) per milliliter of packed bed volume in phosphate-buffered saline (PBS; 0.15 M NaCl, 5 mM $\text{NaH}_2\text{PO}_4$, pH-adjusted to 7.0 with NaOH). The matrix is washed successively with wash buffer I (150 mM NaCl, 125 mM phosphate, pH 7.0), wash buffer II (2.55 M NaCl, 125 mM phosphate, pH 7.0), and wash buffer I; after which virions are desorbed in desorption buffer (150 mM NaCl, 200 mM phosphate, pH 7.0), and the matrix is stripped with stripping buffer (150 mM NaCl, 1 M phosphate, pH 7.0). About half of the applied virions are recovered in desorption buffer. Western blot analysis shows that they have undetectable levels of host-derived protein contaminants that are present in the input virions and in virions purified by CsCl equilibrium density gradient centrifugation—the method most commonly used to prepare virions in high purity. Hydroxyapatite chromatography is thus an attractive alternative method for purifying filamentous virions, particularly when the scale is too large for ultracentrifugation to be practical.

Abstract: Unusual storage: An organic nonporous material, p-tert-butylcalix[4]arene, sorbs acetylene with high storage density under ambient conditions. It is presumed that gas molecules diffuse through the seemingly nonporous lattice without disrupting the arrangement of the host molecules (see picture; red O, blue C, gray H, yellow void space).

Abstract: A high-throughput method has been developed using a commercial piezoelectric inkjet printer for synthesis and characterization of mixed-metal oxide photoelectrode materials for water splitting. The printer was used to deposit metal nitrate solutions onto a conductive glass substrate. The deposited metal nitrate solutions were then pyrolyzed to yield mixed-metal oxides that contained up to eight distinct metals. The stoichiometry of the metal oxides was controlled quantitatively, allowing for the creation of vast libraries of novel materials. Automated methods were developed to measure the open-circuit potentials $(E \text{oc})$, short-circuit photocurrent densities $(J \text{sc})$, and current density vs. applied potential $(J–E)$ behavior under visible light irradiation. The high-throughput measurement of $E \text{oc}$ is particularly significant because open-circuit potential measurements allow the interfacial energetics to be probed regardless of whether the band edges of the materials of concern are above, close to, or below the values needed to sustain water electrolysis under standard conditions. The $E \text{oc}$ measurements allow high-throughput compilation of a suite of data that can be associated with the composition of the various materials in the library, to thereby aid in the development of additional screens and to form a basis for development of theoretical guidance in the prediction of additional potentially promising photoelectrode compositions.

Abstract: It is shown that standard three- and two-dimensional Ewald summation of point charge electrostatics is naturally extended to Gaussian charge distributions. The Gaussian nature of the charges does not affect the regularisation of the conditionally convergent sums, which are performed with spherical and cylindrical orderings, respectively. A clear connection is made between the summation of Gaussian charges and the summation of the associated point charge system. The application of these sums to a simple classical model of a metal surface is discussed. Calculations on a conducting sphere highlight the importance of the model parameterisation.

Abstract: Molecular dynamics computer simulations of the filling of carbon nanotubes (CNTs) by a generic molten salt to form hexagonal-net-based inorganic nanotubes (INTs) are described. A model is introduced to incorporate CNT metallicity which imposes variable Gaussian charges on each atomic site in order to retain an equipotential. The inclusion of CNT metallicity is observed to have no significant effect on the distribution of the INT morphologies formed as compared with the filling of non-metallic CNTs. The application of a voltage bias to the CNT forms a new class of INTs which can be considered as constructed from concentric layers of pseudo-close-packed anions and cations. Removal of the voltage bias leads to the formation of hexagonal-net-based INTs with a distribution of morphologies different to that observed for the filling of the unbiased CNTs. The differences in distributions are interpreted in terms of the CNTs behaving as effective energy landscape filters, for which the applied voltage acts as an additional control variable. The application of a potential acts to control the distribution of INT morphologies by facilitating alternative mechanistic pathways to nanotube formation.

Abstract: We analyze the probability distribution for entropy production rates of trajectories evolving on a class of out-of-equilibrium kinetic networks. These networks can serve as simple models for driven dynamical systems, where energy fluxes typically result in nonequilibrium dynamics. By analyzing the fluctuations in the entropy production, we demonstrate the emergence, in a large system size limit, of a dynamic phase transition between two distinct dynamical regimes.

Abstract: We solve a simple model that supports a dynamic phase transition and show conditions for the existence of the transition. Using methods of large deviation theory we analytically compute the probability distribution for activity and entropy production rates of the trajectories on a large ring with a single heterogeneous link. The corresponding joint rate function demonstrates two dynamical phases—one localized and the other delocalized, but the marginal rate functions do not always exhibit the underlying transition. Symmetries in dynamic order parameters influence the observation of a transition, such that distributions for certain dynamic order parameters need not reveal an underlying dynamical bistability. Solution of our model system furthermore yields the form of the effective Markov transition matrices that generate dynamics in which the two dynamical phases are at coexistence. We discuss the implications of the transition for the response of bacterial cells to antibiotic treatment, arguing that even simple models of a cell cycle lacking an explicit bistability in configuration space will exhibit a bistability of dynamical phases.

Abstract: In a stochastic heat engine driven by a cyclic non-equilibrium protocol, fluctuations in work and heat give rise to a fluctuating efficiency. Using computer simulations and tools from large deviation theory, we have examined these fluctuations in detail for a model two-state engine. We find in general that the form of efficiency probability distributions is similar to those described by Verley et al (2014 Nat. Commun. 5 4721), in particular featuring a local minimum in the long-time limit. In contrast to the time-symmetric engine protocols studied previously, however, this minimum need not occur at the value characteristic of a reversible Carnot engine. Furthermore, while the local minimum may reside at the global minimum of a large deviation rate function, it does not generally correspond to the least likely efficiency measured over finite time. We introduce a general approximation for the finite-time efficiency distribution, $P(\eta )$, based on large deviation statistics of work and heat, that remains very accurate even when $P(\eta )$ deviates significantly from its large deviation form.

Abstract: Importance sampling of trajectories has proved a uniquely successful strategy for exploring rare dynamical behaviors of complex systems in an unbiased way. Carrying out this sampling, however, requires an ability to propose changes to dynamical pathways that are substantial, yet sufficiently modest to obtain reasonable acceptance rates. Satisfying this requirement becomes very challenging in the case of long trajectories, due to the characteristic divergences of chaotic dynamics. Here, we examine schemes for addressing this problem, which engineer correlation between a trial trajectory and its reference path, for instance using artificial forces. Our analysis is facilitated by a modern perspective on Markov chain Monte Carlo sampling, inspired by non-equilibrium statistical mechanics, which clarifies the types of sampling strategies that can scale to long trajectories. Viewed in this light, the most promising such strategy guides a trial trajectory by manipulating the sequence of random numbers that advance its stochastic time evolution, as done in a handful of existing methods. In cases where this “noise guidance” synchronizes trajectories effectively, as the Glauber dynamics of a two-dimensional Ising model, we show that efficient path sampling can be achieved for even very long trajectories.

Abstract: Near equilibrium, small current fluctuations are described by a Gaussian distribution with a linear-response variance regulated by the dissipation. Here, we demonstrate that dissipation still plays a dominant role in structuring large fluctuations arbitrarily far from equilibrium. In particular, we prove a linear-response-like bound on the large deviation function for currents in Markov jump processes. We find that nonequilibrium current fluctuations are always more likely than what is expected from a linear-response analysis. As a small-fluctuations corollary, we derive a recently conjectured uncertainty bound on the variance of current fluctuations.

Abstract: The development of sophisticated experimental means to control nanoscale systems has motivated efforts to design driving protocols that minimize the energy dissipated to the environment. Computational models are a crucial tool in this practical challenge. We describe a general method for sampling an ensemble of finite-time, nonequilibrium protocols biased toward a low average dissipation. We show that this scheme can be carried out very efficiently in several limiting cases. As an application, we sample the ensemble of low-dissipation protocols that invert the magnetization of a 2D Ising model and explore how the diversity of the protocols varies in response to constraints on the average dissipation. In this example, we find that there is a large set of protocols with average dissipation close to the optimal value, which we argue is a general phenomenon.

Abstract: Complex physical dynamics can often be modeled as a Markov jump process between mesoscopic configurations. When jumps between mesoscopic states are mediated by thermodynamic reservoirs, the time-irreversibility of the jump process is a measure of the physical dissipation. We rederive a recently introduced inequality relating the dissipation rate to current fluctuations in jump processes. We then adapt these results to diffusion processes via a limiting procedure, reaffirming that diffusions saturate the inequality. Finally, we study the impact of spatial coarse-graining in a two-dimensional model with driven diffusion. By observing fluctuations in coarse-grained currents, it is possible to infer a lower bound on the total dissipation rate, including the dissipation associated with hidden dynamics. The tightness of this bound depends on how well the spatial coarse-graining detects dynamical events that are driven by large thermodynamic forces.

Abstract: A Stirling engine made of a colloidal particle in contact with a nonequilibrium bath is considered and analyzed with the tools of stochastic energetics. We model the bath by non Gaussian persistent noise acting on the colloidal particle. Depending on the chosen definition of an isothermal transformation in this nonequilibrium setting, we find that either the energetics of the engine parallels that of its equilibrium counterpart or, in the simplest case, that it ends up being less efficient. Persistence, more than non-Gaussian effects, are responsible for this result.

Abstract: Systems driven away from thermal equilibrium constantly deliver entropy to their environment. Determining this entropy production requires detailed information about the system’s internal states and dynamics. However, in most practical scenarios, only a part of a complex experimental system is accessible to an external observer. In order to address this challenge, two notions of partial entropy production have been introduced in the literature as a way to assign an entropy production to an observed subsystem: one due to Shiraishi and Sagawa [Phys. Rev. E 91, 012130 (2015)] and another due to Polettini and Esposito [arXiv:1703.05715 (2017)]. We show that although both of these schemes provide a lower bound on the total entropy production, the latter – which utilizes an effective thermodynamics description– gives a better estimate of the total dissipation. Using this effective thermodynamic framework, we establish a partitioning of the total entropy production into two contributions that individually verify integral fluctuation theorems: an observable partial entropy production and a hidden entropy production assigned to the unobserved subsystem. Our results offer broad implications for both theoretical and empirical systems when only partial information is available.

Abstract: Current is a characteristic feature of nonequilibrium systems. In stochastic systems, these currents exhibit fluctuations constrained by the rate of dissipation in accordance with the recently discovered thermodynamic uncertainty relation. Here, we derive a conjugate uncertainty relationship for the first passage time to accumulate a fixed net current. More generally, we use the tools of large deviation theory to simply connect current fluctuations and first-passage-time fluctuations in the limit of long times and large currents. With this connection, previously discovered symmetries and bounds on the large deviation function for currents are readily transferred to first passage times.

Abstract: Systems coupled to multiple thermodynamic reservoirs can exhibit nonequilibrium dynamics, breaking detailed balance to generate currents. To power these currents, the entropy of the reservoirs increases. The rate of entropy production, or dissipation, is a measure of the statistical irreversibility of the nonequilibrium process. By measuring this irreversibility in several biological systems, recent experiments have detected that particular systems are not in equilibrium. Here we discuss three strategies to replace binary classification (equilibrium versus nonequilibrium) with a quantification of the entropy production rate. To illustrate, we generate time-series data for the evolution of an analytically tractable bead-spring model. Probability currents can be inferred and utilized to indirectly quantify the entropy production rate, but this approach requires prohibitive amounts of data in high-dimensional systems. This curse of dimensionality can be partially mitigated by using the thermodynamic uncertainty relation to bound the entropy production rate using statistical fluctuations in the probability currents.

Abstract: Connections between dynamical fluctuations and dissipation into an environment are well- established in equilibrium thermodynamics. However, few quantitative tools are available for the description of physical systems out-of-equilibrium. Here, we offer our perspective on the recent development of a new class of inequalities known as thermodynamic uncertainty relations, which have revealed that dissipation constrains current fluctuations in steady states arbitrarily far from equilibrium. We discuss the stochastic thermodynamic origin of these inequalities. Efforts to expand the applicability of the results have focused on connections between current fluctuations and the fluctuation theorems. We highlight benefits, but also some challenges.

Abstract: We generalize the link between fluctuation theorems and thermodynamic uncertainty relations by deriving a bound on the variance of fluxes that satisfy an isometric fluctuation theorem. The resulting bound, which depends on the system’s dimension d, naturally interpolates between two known bounds. The bound derived from the entropy production fluctuation theorem is recovered for d = 1, and the original thermodynamic uncertainty relation is obtained in the d → ∞ limit. We show that our result can be generalized to order parameters in equilibrium systems, and we illustrate the results on a Heisenberg spin chain.

Abstract: Diverse physical systems are characterized by their response to small perturbations. Near thermody- namic equilibrium, the fluctuation-dissipation theorem provides a powerful theoretical and experimental tool to determine the nature of response by observing spontaneous equilibrium fluctuations. In this spirit, we derive here a collection of equalities and inequalities valid arbitrarily far from equilibrium that constrain the response of nonequilibrium steady states in terms of the strength of nonequilibrium driving. Our work opens new avenues for characterizing nonequilibrium response. As illustrations, we show how our results rationalize the energetic requirements of common biochemical motifs.

Abstract: It is well-known that Brownian ratchets can exhibit current reversals, wherein the sign of the current switches as a function of the driving frequency. We introduce a spatial discretization of such a two-dimensional Brownian ratchet to enable spectral methods that efficiently compute those currents. These discrete-space models provide a convenient way to study the Markovian dynamics conditioned upon generating particular values of the currents. By studying such conditioned processes, we demonstrate that low-frequency negative values of current arise from typical events and high-frequency positive values of current arises from rare events. We demonstrate how these observations can inform the sculpting of time-dependent potential landscapes with a specific frequency response.

Abstract: Reaction rates are a complicated function of molecular interactions, which can be selected from vast chemical design spaces. Seeking the design that optimizes a rate is a particularly challenging problem since the rate calculation for any one design is itself a difficult compu- tation. Toward this end, we demonstrate a strategy based on transition path sampling to generate an ensemble of designs and reactive trajectories with a preference for fast reaction rates. Each step of the Monte Carlo procedure requires a measure of how a design con- strains molecular configurations, expressed via the reciprocal of the partition function for the design. Although the reciprocal of the partition function would be prohibitively expensive to compute, we apply Booth’s method for generating unbiased estimates of a reciprocal of an integral to sample designs without bias. A generalization with multiple trajectories introduces a stronger preference for fast rates, pushing the sampled designs closer to the optimal design. We illustrate the methodology on two toy models of increasing complexity: escape of a single particle from a Lennard-Jones potential well of tunable depth and escape from a metastable tetrahedral cluster with tunable pair potentials.

Abstract: Most computer simulations of molecular dynamics take place under equilibrium conditions – in a closed, isolated system, or perhaps one held at constant temperature or pressure. Sometimes, extra tensions, shears, or temperature gradients are introduced to those simulations to probe one type of nonequilibrium response to external forces. Catalysts and molecular motors, however, function based on the nonequilibrium dynamics induced by a chemical reaction’s thermodynamic driving force. In this scenario, simulations require chemostats capable of preserving the chemical concentrations of the nonequilibrium steady state. We develop such a dynamic scheme and use it to observe cycles of a new particle-based classical model of a catenane-like molecular motor. Molecular motors are frequently modeled with detailed-balance-breaking Markov models, and we explicitly construct such a picture by coarse-graining the microscopic dynamics of our simulations in order to extract rates. This work identifies inter-particle interactions that tune those rates to create a functional motor, thereby yielding a computational playground to investigate the interplay between accuracy, current generation, and efficiency of molecular information ratchets.

Abstract: The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet’s current. To surmount the many-body problem, we employ the time-dependent variational principle applied to binary tree tensor networks. Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases.

Abstract: Information thermodynamics offers a route to measure how effectively a light-driven molecular machine converts energy from absorbed photons into pumped motion.

Abstract: We present an approach based upon binary tree tensor network (BTTN) states for computing steady-state current statistics for a many-particle 1D ratchet subject to volume exclusion interactions. The ratcheted particles, which move on a lattice with periodic boundary conditions subject to a time-periodic drive, can be stochastically evolved in time to sample representative trajectories via a Gillespie method. In lieu of generating realizations of trajectories, a BTTN state can variationally approximate a distribution over the vast number of many-body configurations. We apply the density matrix renormalization group (DMRG) algorithm to initialize BTTN states, which are then propagated in time via the time-dependent variational principle (TDVP) algorithm to yield the steady-state behavior, including the effects of both typical and rare trajectories. The application of the methods to ratchet currents is highlighted in a companion letter, but the approach extends naturally to other interacting lattice models with time-dependent driving. Though trajectory sampling is conceptually and computationally simpler, we discuss situations for which the BTTN TDVP strategy could be more favorable.

Abstract: The thermodynamic uncertainty relation (TUR) quantifies a relationship between current fluctuations and dissipation in out-of-equilibrium overdamped Langevin dynamics, making it a natural counterpart of the fluctuation-dissipation theorem in equilibrium statistical mechanics. For under-damped Langevin dynamics, the situation is known to be more complicated, with dynamical activity also playing a role in limiting the magnitude of current fluctuations. Progress on those underdamped TUR-like bounds has largely come from applications of the information-theoretic Cramér-Rao inequality. Here, we present an alternative perspective by employing large deviation theory. The approach offers a general, unified treatment of TUR-like bounds for both overdamped and under-damped Langevin dynamics built upon current fluctuations achieved by scaling time. The bounds we derive following this approach are similar to known results but with differences we discuss and rationalize.

Abstract: Biomolecular machines are driven by information ratchet mechanisms, where kinetic asymmetry in the machine’s chemomechanical cycle of fuel-to-waste catalysis induces net directional dynamics. A large-scale energetically downhill conformational change, termed a “power stroke,” has often been erroneously implicated as a mechanistic driving feature in such machines. We investigated the roles of kinetic asymmetry and power strokes in a series of rotaxane-based information ratchets and found that kinetic asymmetry alone determines ratchet directionality such that all ratchets use the same amount of fuel to reach the same normalized steady state. However, power strokes can nonetheless influence ratchet performance, such as how fast the steady state is reached. Moreover, nonequilibrium thermodynamic analysis revealed that power strokes alter the amount and form (information [Shannon entropy] versus intercomponent binding energy) of the free energy stored by ratchets. These findings have implications for both the understanding of biological ratchets and the design principles for artificial nonequilibrium (supra)molecular machines.

Abstract: Simulations can help unravel the complicated ways in which molecular structure determines function. Here, we use molecular simulations to show how slight alterations of a molecular motor’s structure can cause the motor’s typical dynamical behavior to reverse directions. Inspired by autonomous synthetic catenane motors, we study the molecular dynamics of a minimal motor model, consisting of a shuttling ring that moves along a track containing interspersed binding sites and catalytic sites. The binding sites attract the shuttling ring while the catalytic sites speed up a reaction between molecular species, which can be thought of as fuel and waste. When that fuel and waste are held in a nonequilibrium steady-state concentration, the free energy from the reaction drives directed motion of the shuttling ring along the track. Using this model and nonequilibrium molecular dynamics, we show that the shuttling ring’s direction can be reversed by simply adjusting the spacing between binding and catalytic sites on the track. We present a steric mechanism behind the current reversal, supported by kinetic measurements from the simulations. These results demonstrate how molecular simulation can guide future development of artificial molecular motors.

Abstract: 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 3x10¹⁵ 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.

Abstract: Thermodynamic uncertainty relations (TURs) relate precision to the dissipation rate, yet the inequalities can be far from saturation. Indeed, in catenane molecular motor simulations, we record precision far below the TUR limit. We further show that this inefficiency can be anticipated by four physical parameters: the thermodynamic driving force, fuel decomposition rate, coupling between fuel decomposition and motor motion, and rate of undriven motor motion. The physical insights might assist in designing molecular motors in the future.

Abstract: 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 like 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 insight. 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.

Abstract: A defining feature of living matter is the ability to harness energy to self-organize multiscale structures whose functions are facilitated by irreversible nonequilibrium dynamics. While progress has been made in elucidating the underlying principles, what remains unclear is the role that thermodynamics plays in shaping these structures and their ensuing functions. Here, we unravel how a fundamental thermodynamic connection between the physical energy dissipation sustaining a nonequilibrium system and a measure of statistical irreversibility (arrow of time), can provide quantitative insight into the mechanisms of nonequilibrium activity across scales. Specifically, we introduce a multiscale irreversibility metric and demonstrate how it can be used to extract model-independent estimates of dissipative timescales. Using this metric, we measure the dissipation timescale of a multiscale cellular structure - the actomyosin cortex - and further observe that the irreversibility metric maintains a monotonic relationship with the underlying biological nonequilibrium activity. Additionally, the irreversibility metric can detect shifts in the dissipative timescales when we induce spatiotemporal patterns of biochemical signaling proteins upstream of actomyosin activation. Our experimental measurements are complemented by a theoretical analysis of a generic class of nonequilibrium dynamics, elucidating how dissipative timescales manifest in multiscale irreversibility.

Abstract: For several decades, molecular motor directionality has been rationalized in terms of the free energy of molecular conformations visited before and after the motor takes a step, a so-called power stroke mechanism with analogues in macroscopic engines. Despite theoretical and experimental demonstrations of its flaws, the power stroke language is quite ingrained, and some communities still value power stroke intuition. By building a catalysis-driven motor into simulated numerical experiments, we here systematically report on how directionality responds when the motor is modified accordingly to power stroke intuition. We confirm that the power stroke mechanism generally does not predict motor directionality. Nevertheless, the simulations illustrate that the relative stability of molecular conformations should be included as a potential design element to adjust the motor directional bias. Though power strokes are formally unimportant for determining directionality, we show that practical attempts to alter a power stroke have side effects that can in fact alter the bias. The change in the bias can align with what power stroke intuition would have suggested, offering a potential explanation for why the flawed power stroke mechanism can retain apparent utility when engineering specific systems.