Optimizing reaction and transport fluxes in temperature-gradient-driven chemical reaction-diffusion systems

Temperature gradients represent energy sources that can be harvested to generate steady reaction or transport fluxes. Technological developments could lead to the transfer of free energy from heat sources and sinks to chemical systems for the purpose of extraction, thermal batteries, or nonequilibrium synthesis. We present a theoretical study of one-dimensional chemical systems subjected to temperature gradients, for sustaining nonequilibrium chemical fluxes. A complete theoretical framework describes the behavior of the system induced by various temperature profiles. An exact mathematical derivation was established for a simple two-compartment model and was generalized to arbitrary reaction-diffusion systems based on numerical models. An experimental system was eventually scaled and tuned to optimize either nonequilibrium chemical transport or reaction. The relevant parameters for this description were identified; they focused on the system symmetry for chemical reaction and transport. Nonequilibrium thermodynamic approaches lead to a description analogous to electric circuits. Temperature gradients lead to the onset of a steady chemical force, which maintains steady reaction-diffusion fluxes moderated by chemical resistance. The system activity was then assessed using the entropy production rate as a measure of its dissipated power. The chemical characteristics of the system can be tuned for general optimization of the nonequilibrium state or for the specific optimization of either transport or reaction processes. The shape of the temperature gradient can be tailored to precisely control the spatial localization of active processes, targeting either precise spatial localization or propagation over large areas. The resulting temperature-driven chemical system can in turn be used to drive secondary processes into either nonequilibrium reaction fluxes or concentration gradients.

Catalytic resonance theory: the catalytic mechanics of programmable ratchets

Catalytic reaction networks of multiple elementary steps operating under dynamic conditions via a programmed input oscillation are difficult to interpret and optimize due to reaction system complexity. To understand these dynamic systems, individual elementary catalytic reactions oscillating between catalyst states were evaluated to identify their three fundamental characteristics that define their ability to promote reactions away from equilibrium. First, elementary catalytic reactions exhibit directionality to promote reactions forward or backward from equilibrium as determined by a ratchet directionality metric comprised of the input oscillation duty cycle and the reaction rate constants. Second, catalytic ratchets are defined by the catalyst state of strong or weak binding that permits reactants to proceed through the transition state. Third, elementary catalytic ratchets exhibit a cutoff frequency which defines the transition in applied frequency for which the catalytic ratchet functions to promote chemistry away from equilibrium. All three ratchet characteristics are calculated from chemical reaction parameters including rate constants derived from linear scaling parameters, reaction conditions, and catalyst electronic state. The characteristics of the reaction network's constituent elementary catalytic reactions provided an interpretation of complex reaction networks and a method of predicting the behavior of dynamic surface chemistry on oscillating catalysts.

Kinetic Asymmetry and Directionality of Nonequilibrium Molecular Systems

Scientists have long been fascinated by the biomolecular machines in living systems that process energy and information to sustain life. The first synthetic molecular rotor capable of performing repeated 360° rotations due to a combination of photo- and thermally activated processes was reported in 1999. The progress in designing different molecular machines in the intervening years has been remarkable, with several outstanding examples appearing in the last few years. Despite the synthetic accomplishments, there remains confusion regarding the fundamental design principles by which the motions of molecules can be controlled, with significant intellectual tension between mechanical and chemical ways of thinking about and describing molecular machines. A thermodynamically consistent analysis of the kinetics of several molecular rotors and pumps shows that while light driven rotors operate by a power-stroke mechanism, kinetic asymmetry-the relative heights of energy barriers-is the sole determinant of the directionality of catalysis driven machines. Power-strokes-the relative depths of energy wells-play no role whatsoever in determining the sign of the directionality. These results, elaborated using trajectory thermodynamics and the nonequilibrium pump equality, show that kinetic asymmetry governs the response of many non-equilibrium chemical phenomena.

Temporal Self-Regulation of Mechanical Properties via Catalytic Amyloid Polymers of a Short Peptide

We report a short peptide that accessed dynamic catalytic polymers to demonstrate four-stage (sol-gel-weak gel-strong gel) temporal self-regulation of its mechanical properties. The peptide exploited its intrinsic catalytic capabilities of manipulating C-C bonds (retro-aldolase-like) that resulted in a nonlinear variation in the catalytic rate. The seven-residue sequence exploited two lysines for binding and cleaving the thermodynamically activated substrate that subsequently led to the self-regulation of the mechanical strengths of the polymerized states as a function of time and reaction progress. Interestingly, the polymerization events were modulated by the different catalytic potentials of the two terminal lysines to cleave the substrate, covalently trap the electrophilic products, and subsequently control the mechanical properties of the system.

A [3]-catenane non-autonomous molecular motor model: Geometric phase, no-pumping theorem, and energy transduction

We study a model of a synthetic molecular motor-a [3]-catenane consisting of two small macrocycles mechanically interlocked with a bigger one-subjected to time-dependent driving using stochastic thermodynamics. The model presents nontrivial features due to the two interacting small macrocycles but is simple enough to be treated analytically in limiting regimes. Among the results obtained, we find a mapping into an equivalent [2]-catenane that reveals the implications of the no-pumping theorem stating that to generate net motion of the small macrocycles, both energies and barriers need to change. In the adiabatic limit (slow driving), we fully characterize the motor's dynamics and show that the net motion of the small macrocycles is expressed as a surface integral in parameter space, which corrects previous erroneous results. We also analyze the performance of the motor subjected to step-wise driving protocols in the absence and presence of an applied load. Optimization strategies for generating large currents and maximizing free energy transduction are proposed. This simple model provides interesting clues into the working principles of non-autonomous molecular motors and their optimization.

Photoactivated Artificial Molecular Motors

Accurate control of long-range motion at the molecular scale holds great potential for the development of ground-breaking applications in energy storage and bionanotechnology. The past decade has seen tremendous development in this area, with a focus on the directional operation away from thermal equilibrium, giving rise to tailored man-made molecular motors. As light is a highly tunable, controllable, clean, and renewable source of energy, photochemical processes are appealing to activate molecular motors. Nonetheless, the successful operation of molecular motors fueled by light is a highly challenging task, which requires a judicious coupling of thermal and photoinduced reactions. In this paper, we focus on the key aspects of light-driven artificial molecular motors with the aid of recent examples. A critical assessment of the criteria for the design, operation, and technological potential of such systems is provided, along with a perspective view on future advances in this exciting research area.

Cation pumping against a concentration gradient in conical nanopores characterized by load capacitors

We study the cation transport against an external concentration gradient (cation pumping) that occurs in conical nanopores when zero-average oscillatory and white noise potentials are externally applied. This pumping, based on the electrically asymmetric nanostructure, is characterized here by a load capacitor arrangement. In the case of white noise signals, the conical nanopore acts as an electrical valve that allows extraction of order from chaos. No molecular carriers, specific ion pumps, and competitive ion-binding phenomena are required. The nanopore conductance on/off states mimic those of the voltage-gated ion channels in the cell membrane. These channels allow modulating membrane potentials and ionic concentration gradients along oscillatory pulses in circadian rhythms and the cell cycle. We show that the combination of asymmetric nanostructures with load capacitors can be useful for the understanding of nanofluidic processes based on bioelectrochemical gradients.

Kinetic and energetic insights into the dissipative non-equilibrium operation of an autonomous light-powered supramolecular pump

Natural and artificial autonomous molecular machines operate by constantly dissipating energy coming from an external source to maintain a non-equilibrium state. Quantitative thermodynamic characterization of these dissipative states is highly challenging as they exist only as long as energy is provided. Here we report on the detailed physicochemical characterization of the dissipative operation of a supramolecular pump. The pump transduces light energy into chemical energy by bringing self-assembly reactions to non-equilibrium steady states. The composition of the system under light irradiation was followed in real time by ¹H NMR for four different irradiation intensities. The experimental composition and photon flow were then fed into a theoretical model describing the non-equilibrium dissipation and the energy storage at the steady state. We quantitatively probed the relationship between the light energy input and the deviation of the dissipative state from thermodynamic equilibrium in this artificial system. Our results provide a testing ground for newly developed theoretical models for photoactivated artificial molecular machines operating away from thermodynamic equilibrium.

The PLB measurement for the connector in Phi29 bacteriophage reveals the function of its channel loop

The connector protein, also known as the portal protein, located at the portal vertex in the Phi29 bacteriophage has been found to play a key role in the genome DNA packaging motor. There is a disordered region, composed of 12 sets of 18-residue loops N229-N246, that has been assumed to serve as a "clamp" to retain the DNA within the pressurized capsid when DNA is fully packaged. However, the process remains undefined about how the clamping of DNA occurs and what signal is used to engage the channel loops to clamp the DNA near the end of DNA packaging. In this study, we use the planar lipid bilayer (PLB) membrane technique to study the connector with its loops cleaved. The channel properties are compared with those of the connector with corresponding wild-type loops at different membrane potentials. On the basis of the hypothesis of the Donnan effects in the flashing Brownian ratchet model, we associate the PLB experimental results with the outcomes from the relevant biochemical experiments on the proheads containing the connectors without the loops, which enables us to provide a clear picture about how the DNA clamping occurs. A mathematical relationship between the Donnan potential and the DNA packaging density is established, demonstrating that they are both in essence the same signal that is received and transmitted by the connector to dictate DNA clamping and the termination of DNA packaging. At the end of the study, the PLB technique is proposed as a viral research tool, and its potential use to study the functions of specific domains in a portal protein of the tailed bacteriophages is highlighted.

Escapement mechanisms: Efficient free energy transduction by reciprocally-coupled gating

Conversion of the free energy of NTP hydrolysis efficiently into mechanical work and/or information by transducing enzymes sustains living systems far from equilibrium, and so has been of interest for many decades. Detailed molecular mechanisms, however, remain puzzling and incomplete. We previously reported that catalysis of tryptophan activation by tryptophanyl-tRNA synthetase, TrpRS, requires relative domain motion to re-position the catalytic Mg²⁺ ion, noting the analogy between that conditional hydrolysis of ATP and the escapement mechanism of a mechanical clock. The escapement allows the time-keeping mechanism to advance discretely, one gear at a time, if and only if the pendulum swings, thereby converting energy from the weight driving the pendulum into rotation of the hands. Coupling of catalysis to domain motion, however, mimics only half of the escapement mechanism, suggesting that domain motion may also be reciprocally coupled to catalysis, completing the escapement metaphor. Computational studies of the free energy surface restraining the domain motion later confirmed that reciprocal coupling: the catalytic domain motion is thermodynamically unfavorable unless the PPi product is released from the active site. These two conditional phenomena-demonstrated together only for the TrpRS mechanism-function as reciprocally-coupled gates. As we and others have noted, such an escapement mechanism is essential to the efficient transduction of NTP hydrolysis free energy into other useful forms of mechanical or chemical work and/or information. Some implementation of both gating mechanisms-catalysis by domain motion and domain motion by catalysis-will thus likely be found in many other systems.

On the beneficent thickness of water

In the 1930s, Lars Onsager published his famous 'reciprocal relations' describing free energy conversion processes. Importantly, these relations were derived on the assumption that the fluxes of the processes involved in the conversion were proportional to the forces (free energy gradients) driving them. For chemical reactions, however, this condition holds only for systems operating close to equilibrium-indeed very close; nominally requiring driving forces to be smaller than k B T. Fairly soon thereafter, however, it was quite inexplicably observed that in at least some biological conversions both the reciprocal relations and linear flux-force dependency appeared to be obeyed no matter how far from equilibrium the system was being driven. No successful explanation of how this 'paradoxical' behaviour could occur has emerged and it has remained a mystery. We here argue, however, that this anomalous behaviour is simply a gift of water, of its viscosity in particular; a gift, moreover, without which life almost certainly could not have emerged. And a gift whose appreciation we primarily owe to recent work by Prof. R. Dean Astumian who, as providence has kindly seen to it, was led to the relevant insights by the later work of Onsager himself.

Explaining leak states in the proton pump of heme-copper oxidases observed in single-molecule experiments

Heme-copper oxidases couple the exergonic oxygen reduction with the endergonic proton translocation. Redox-linked structural changes have been localized in deeply buried regions of the protein, near the low-potential heme. How these movements can modulate distant gating events along the intramolecular proton path, where the entry (exit) of pumped proton occurs, is a major concern for the proton pump models. Generally, these models associate, more or less directly, all translocation events with redox transitions. Although they can account for many phenomenological aspects of the pump, evidences from single-molecules experiments about leak states of the pump represent a formidable challenge. Disconnecting the redox-linked pK_a shifts of the proton loading site from the external barriers, we obtain a simple stochastic mechanism which behaves similarly to the real enzyme, able to reverse the flow of the proton transfer.

Kinetic asymmetry allows macromolecular catalysts to drive an information ratchet

Molecular machines carry out their function by equilibrium mechanical motions in environments that are far from thermodynamic equilibrium. The mechanically equilibrated character of the trajectories of the macromolecule has allowed development of a powerful theoretical description, reminiscent of Onsager’s trajectory thermodynamics, that is based on the principle of microscopic reversibility. Unlike the situation at thermodynamic equilibrium, kinetic parameters play a dominant role in determining steady-state concentrations away from thermodynamic equilibrium, and kinetic asymmetry provides a mechanism by which chemical free-energy released by catalysis can drive directed motion, molecular adaptation, and self-assembly. Several examples drawn from the recent literature, including a catenane-based chemically driven molecular rotor and a synthetic molecular assembler or pump, are discussed.

The mechanism by which macromolecular catalysts use energy from exergonic reactions to move, adapt, and assemble has been unclear. In this Perspective article, R. Dean Astumian shows that in addition to disequilibrium of the catalyzed reaction, kinetic asymmetry is the essential feature required to drive non-equilibrium response by an information ratchet mechanism.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Motor-like Properties of Nonmotor Enzymes

Molecular motors are thought to generate force and directional motion via nonequilibrium switching between energy surfaces. Because all enzymes can undergo such switching, we hypothesized that the ability to generate rotary motion and torque is not unique to highly adapted biological motor proteins but is instead a common feature of enzymes. We used molecular dynamics simulations to compute energy surfaces for hundreds of torsions in three enzymes-adenosine kinase, protein kinase A, and HIV-1 protease-and used these energy surfaces within a kinetic model that accounts for intersurface switching and intrasurface probability flows. When substrate is out of equilibrium with product, we find computed torsion rotation rates up ∼140 cycles s^-1, with stall torques up to ∼2 kcal mol^-1 cycle^-1, and power outputs up to ∼50 kcal mol^-1 s^-1. We argue that these enzymes are instances of a general phenomenon of directional probability flows on asymmetric energy surfaces for systems out of equilibrium. Thus, we conjecture that cyclic probability fluxes, corresponding to rotations of torsions and higher-order collective variables, exist in any chiral molecule driven between states in a nonequilibrium manner; we call this the "Asymmetry-Directionality" conjecture. This is expected to apply as well to synthetic chiral molecules switched in a nonequilibrium manner between energy surfaces by light, redox chemistry, or catalysis.

Cytochrome c oxidase structures suggest a four-state stochastic pump mechanism

A simple stochastic model for a cytochrome c oxidase proton pump.

Trajectory and Cycle-Based Thermodynamics and Kinetics of Molecular Machines: The Importance of Microscopic Reversibility

A molecular machine is a nanoscale device that provides a mechanism for coupling energy from two (or more) processes that in the absence of the machine would be independent of one another. Examples include walking of a protein in one direction along a polymeric track (process 1, driving "force" X₁ = - F⃗· l⃗) and hydrolyzing ATP (process 2, driving "force" X₂ = Δμ_ATP); or synthesis of ATP (process 1, X₁ = -Δμ_ATP) and transport of protons from the periplasm to the cytoplasm across a membrane (process 2, X₂ = Δμ_H⁺); or rotation of a flagellum (process 1, X₁ = -torque) and transport of protons across a membrane (process 2, X₂ = Δμ_H⁺). In some ways, the function of a molecular machine is similar to that of a macroscopic machine such as a car that couples combustion of gasoline to translational motion. However, the low Reynolds number regime in which molecular machines operate is very different from that relevant for macroscopic machines. Inertia is negligible in comparison to viscous drag, and omnipresent thermal noise causes the machine to undergo continual transition among many states even at thermodynamic equilibrium. Cyclic trajectories among the states of the machine that result in a change in the environment can be broken into two classes: those in which process 1 in either the forward or backward direction ([Formula: see text]) occurs and which thereby exchange work [Formula: see text] with the environment; and those in which process 2 in either the forward or backward direction ([Formula: see text]) occurs and which thereby exchange work [Formula: see text] with the evironment. These two types of trajectories, [Formula: see text] and [Formula: see text], overlap, i.e., there are some trajectories in which both process 1 and process 2 occur, and for which the work exchanged is [Formula: see text]. The four subclasses of overlap trajectories [(+1,+2), (+1,-2), (-1,+2), (-1,-2)] are the coupled processes. The net probabilities for process 1 and process 2 are designated π₊₂ - π_-2 and π₊₁ - π_-1, respectively. The probabilities [Formula: see text] for any single trajectory [Formula: see text] and [Formula: see text] for its microscopic reverse [Formula: see text] are related by microscopic reversibility (MR), [Formula: see text], an equality that holds arbitrarily far from thermodynamic equilibrium, i.e., irrespective of the magnitudes of X₁ and X₂, and where [Formula: see text]. Using this formalism, we arrive at a remarkably simple and general expression for the rates of the processes, [Formula: see text], i = 1, 2, where the angle brackets indicate an average over the ensemble of all microscopic reverse trajectories. Stochastic description of coupling is doubtless less familiar than typical mechanical depictions of chemical coupling in terms of ATP induced violent kicks, judo throws, force generation and power-strokes. While the mechanical description of molecular machines is comforting in its familiarity, conclusions based on such a phenomenological perspective are often wrong. Specifically, a "power-stroke" model (i.e., a model based on energy driven "promotion" of a molecular machine to a high energy state followed by directional relaxation to a lower energy state) that has been the focus of mechanistic discussions of biomolecular machines for over a half century is, for catalysis driven molecular machines, incorrect. Instead, the key principle by which catalysis driven motors work is kinetic gating by a mechanism known as an information ratchet. Amazingly, this same principle is that by which catalytic molecular systems undergo adaptation to new steady states while facilitating an exergonic chemical reaction.

Energy consumption in chemical fuel-driven self-assembly

Nature extensively exploits high-energy transient self-assembly structures that are able to perform work through a dissipative process. Often, self-assembly relies on the use of molecules as fuel that is consumed to drive thermodynamically unfavourable reactions away from equilibrium. Implementing this kind of non-equilibrium self-assembly process in synthetic systems is bound to profoundly impact the fields of chemistry, materials science and synthetic biology, leading to innovative dissipative structures able to convert and store chemical energy. Yet, despite increasing efforts, the basic principles underlying chemical fuel-driven dissipative self-assembly are often overlooked, generating confusion around the meaning and definition of scientific terms, which does not favour progress in the field. The scope of this Perspective is to bring closer together current experimental approaches and conceptual frameworks. From our analysis it also emerges that chemically fuelled dissipative processes may have played a crucial role in evolutionary processes.

Redox-Driven Proton Pumps of the Respiratory Chain

In aerobic cells, the proton gradient that drives ATP synthesis is created by three different proton pumps-membrane enzymes of the respiratory electron transport chain known as complex I, III, and IV. Despite the striking dissimilarity of structures and apparent differences in molecular mechanisms of proton pumping, all three enzymes have much in common and employ the same universal physical principles of converting redox energy to proton pumping. In this study, we describe a simple mathematical model that illustrates the general principles of redox-driven proton pumps and discuss their implementation in complex I, III, and IV of the respiratory chain.

Optical vs. chemical driving for molecular machines

Molecular machines use external energy to drive transport, to do mechanical, osmotic, or electrical work on the environment, and to form structure. In this paper the fundamental difference between the design principles necessary for a molecular machine to use light or external modulation of thermodynamic parameters as an energy source vs. the design principle for using an exergonic chemical reaction as a fuel will be explored. The key difference is that for catalytically-driven motors microscopic reversibility must hold arbitrarily far from equilibrium. Applying the constraints of microscopic reversibility assures that a coarse grained model is consistent with an underlying model for motion on a single time-independent potential energy surface. In contrast, light-driven processes, and processes driven by external modulation of the thermodynamic parameters of a system cannot in general be described in terms of motion on a single time-independent potential energy surface, and the rate constants are not constrained by microscopic reversibility. The results presented here call into question the value of the so-called power stroke model as an explanation of the function of autonomous chemically-driven molecular machines such as are commonly found in biology.

Three-stage stochastic pump: Another type of Onsager-Casimir symmetry and results far from equilibrium

The stochastic thermodynamic analysis of a time-periodic single particle pump sequentially exposed to three thermochemical reservoirs is presented. The analysis provides explicit results for flux, thermodynamic force, entropy production, work, and heat. These results apply near equilibrium as well as far from equilibrium. In the linear response regime, a different type of Onsager-Casimir symmetry is uncovered. The Onsager matrix becomes symmetric in the limit of zero dissipation.

Stochastically pumped adaptation and directional motion of molecular machines

Recent developments in synthetic molecular motors and pumps have sprung from a remarkable confluence of experiment and theory. Synthetic accomplishments have facilitated the ability to design and create molecules, many of them featuring mechanically bonded components, to carry out specific functions in their environment-walking along a polymeric track, unidirectional circling of one ring about another, synthesizing stereoisomers according to an external protocol, or pumping rings onto a long rod-like molecule to form and maintain high-energy, complex, nonequilibrium structures from simpler antecedents. Progress in the theory of nanoscale stochastic thermodynamics, specifically the generalization and extension of the principle of microscopic reversibility to the single-molecule regime, has enhanced the understanding of the design requirements for achieving strong unidirectional motion and high efficiency of these synthetic molecular machines for harnessing energy from external fluctuations to carry out mechanical and/or chemical functions in their environment. A key insight is that the interaction between the fluctuations and the transition state energies plays a central role in determining the steady-state concentrations. Kinetic asymmetry, a requirement for stochastic adaptation, occurs when there is an imbalance in the effect of the fluctuations on the forward and reverse rate constants. Because of strong viscosity, the motions of the machine can be viewed as mechanical equilibrium processes where mechanical resonances are simply impossible but where the probability distributions for the state occupancies and trajectories are very different from those that would be expected at thermodynamic equilibrium.

Mastering the non-equilibrium assembly and operation of molecular machines

In mechanically interlocked compounds, such as rotaxanes and catenanes, the molecules are held together by mechanical rather than chemical bonds. These compounds can be engineered to have several well-defined mechanical states by incorporating recognition sites between the different components. The rates of the transitions between the recognition sites can be controlled by introducing steric "speed bumps" or electrostatically switchable gates. A mechanism for the absorption of energy can also be included by adding photoactive, catalytically active, or redox-active recognition sites, or even charges and dipoles. At equilibrium, these Mechanically Interlocked Molecules (MIMs) undergo thermally activated transitions continuously between their different mechanical states where every transition is as likely as its microscopic reverse. External energy, for example, light, external modulation of the chemical and/or physical environment or catalysis of an exergonic reaction, drives the system away from equilibrium. The absorption of energy from these processes can be used to favour some, and suppress other, transitions so that completion of a mechanical cycle in a direction in which work is done on the environment - the requisite of a molecular machine - is more likely than completion in a direction in which work is absorbed from the environment. In this Tutorial Review, we discuss the different design principles by which molecular machines can be engineered to use different sources of energy to carry out self-organization and the performance of work in their environments.

Stochastic pumping of non-equilibrium steady-states: how molecules adapt to a fluctuating environment

In the absence of input energy, a chemical reaction in a closed system ineluctably relaxes toward an equilibrium state governed by a Boltzmann distribution. The addition of a catalyst to the system provides a way for more rapid equilibration toward this distribution, but the catalyst can never, in and of itself, drive the system away from equilibrium. In the presence of external fluctuations, however, a macromolecular catalyst (e.g., an enzyme) can absorb energy and drive the formation of a steady state between reactant and product that is not determined solely by their relative energies. Due to the ubiquity of non-equilibrium steady states in living systems, the development of a theory for the effects of external fluctuations on chemical systems has been a longstanding focus of non-equilibrium thermodynamics. The theory of stochastic pumping has provided insight into how a non-equilibrium steady-state can be formed and maintained in the presence of dissipation and kinetic asymmetry. This effort has been greatly enhanced by a confluence of experimental and theoretical work on synthetic molecular machines designed explicitly to harness external energy to drive non-equilibrium transport and self-assembly.

Stochastic thermodynamics for a periodically driven single-particle pump

We present the stochastic thermodynamic analysis of a time-periodic single-particle pump, including explicit results for flux, thermodynamic force, entropy production, work, heat, and efficiency. These results are valid far from equilibrium. The deviations from the linear (Onsager) regime are discussed.

Stochastic thermodynamics of periodically driven systems: Fluctuation theorem for currents and unification of two classes

Periodic driving is used to operate machines that go from standard macroscopic engines to small nonequilibrium microsized systems. Two classes of such systems are small heat engines driven by periodic temperature variations, and molecular pumps driven by external stimuli. Well-known results that are valid for nonequilibrium steady states of systems driven by fixed thermodynamic forces, instead of an external periodic driving, have been generalized to periodically driven heat engines only recently. These results include a general expression for entropy production in terms of currents and affinities, and symmetry relations for the Onsager coefficients from linear-response theory. For nonequilibrium steady states, the Onsager reciprocity relations can be obtained from the more general fluctuation theorem for the currents. We prove a fluctuation theorem for the currents for periodically driven systems. We show that this fluctuation theorem implies a fluctuation dissipation relation, symmetry relations for Onsager coefficients, and further relations for nonlinear response coefficients. The setup in this paper is more general than previous studies, i.e., our results are valid for both heat engines and molecular pumps. The external protocol is assumed to be stochastic in our framework, which leads to a particularly convenient way to treat periodically driven systems.

Nanoscale Ion Pump Derived from a Biological Water Channel

Biological molecular machines perform the work of supporting life at the smallest of scales, including the work of shuttling ions across cell boundaries and against chemical gradients. Systems of artificial channels at the nanoscale can likewise control ionic concentration by way of ionic current rectification, species selectivity, and voltage gating mechanisms. Here, we theoretically show that a voltage-gated, ion species-selective, and rectifying ion channel can be built using the components of a biological water channel aquaporin. Through all-atom molecular dynamics simulations, we show that the ionic conductance of a truncated aquaporin channel nonlinearly increases with the bias magnitude, depends on the channel's orientation, and is highly cation specific but only for one polarity of the transmembrane bias. Further, we show that such an unusually complex response of the channel to transmembrane bias arises from mechanical motion of a positively charged gate that blocks cation transport. By combining two truncated aquaporins, we demonstrate a molecular system that pumps ions against their chemical gradients when subject to an alternating transmembrane bias. Our work sets the stage for future biomimicry efforts directed toward reproducing the function of biological ion pumps using synthetic components.

Forces from the Portal Govern the Late-Stage DNA Transport in a Viral DNA Packaging Nanomotor

In the Phi29 bacteriophage, the DNA packaging nanomotor packs its double-stranded DNA genome into the virus capsid. At the late stage of DNA packaging, the negatively charged genome is increasingly compacted at a higher density in the capsid with a higher internal pressure. During the process, two Donnan effects, osmotic pressure and Donnan equilibrium potentials, are significantly amplified, which, in turn, affect the channel activity of the portal protein, GP10, embedded in the semipermeable capsid shell. In the research, planar lipid bilayer experiments were used to study the channel activities of the viral protein. The Donnan effect on the conformational changes of the viral protein was discovered, indicating GP10 may not be a static channel at the late stage of DNA packaging. Due to the conformational changes, GP10 may generate electrostatic forces that govern the DNA transport. For the section of the genome DNA that remains outside of the connector channel, a strong repulsive force from the viral protein would be generated against the DNA entry; however, for the section of the genome DNA within the channel, the portal protein would become a Brownian motor, which adopts the flash Brownian ratchet mechanism to pump the DNA against the increasingly built-up internal pressure (up to 20 atm) in the capsid. Therefore, the DNA transport in the nanoscale viral channel at the late stage of DNA packaging could be a consequence of Brownian movement of the genomic DNA, which would be rectified and harnessed by the forces from the interior wall of the viral channel under the influence of the Donnan effect.

Combining multi-mutant and modular thermodynamic cycles to measure energetic coupling networks in enzyme catalysis

We measured and cross-validated the energetics of networks in Bacillus stearothermophilus Tryptophanyl-tRNA synthetase (TrpRS) using both multi-mutant and modular thermodynamic cycles. Multi-dimensional combinatorial mutagenesis showed that four side chains from this "molecular switch" move coordinately with the active-site Mg2+ ion as the active site preorganizes to stabilize the transition state for amino acid activation. A modular thermodynamic cycle consisting of full-length TrpRS, its Urzyme, and the Urzyme plus each of the two domains deleted in the Urzyme gives similar energetics. These dynamic linkages, although unlikely to stabilize the transition-state directly, consign the active-site preorganization to domain motion, assuring coupled vectorial behavior.

High-Dimensional Mutant and Modular Thermodynamic Cycles, Molecular Switching, and Free Energy Transduction

Understanding how distinct parts of proteins produce coordinated behavior has driven and continues to drive advances in protein science and enzymology. However, despite consensus about the conceptual basis for allostery, the idiosyncratic nature of allosteric mechanisms resists general approaches. Computational methods can identify conformational transition states from structural changes, revealing common switching mechanisms that impose multistate behavior. Thermodynamic cycles use factorial perturbations to measure coupling energies between side chains in molecular switches that mediate shear during domain motion. Such cycles have now been complemented by modular cycles that measure energetic coupling between separable domains. For one model system, energetic coupling between domains has been shown to be quantitatively equivalent to that between dynamic side chains. Linkages between domain motion, switching residues, and catalysis make nucleoside triphosphate hydrolysis conditional on domain movement, confirming an essential yet neglected aspect of free energy transduction and suggesting the potential generality of these studies.

Validity of the no-pumping theorem in systems with finite-range interactions between particles

The no-pumping theorem states that seemingly natural driving cycles of stochastic machines fail to generate directed motion. Initially derived for single particle systems, the no-pumping theorem was recently extended to many-particle systems with zero-range interactions. Interestingly, it is known that the theorem is violated by systems with exclusion interactions. These two paradigmatic interactions differ by two qualitative aspects: the range of interactions and the dependence of branching fractions on the state of the system. In this work two different models are studied in order to identify the qualitative property of the interaction that leads to breakdown of no pumping. A model with finite-range interaction is shown analytically to satisfy no pumping. In contrast, a model in which the interaction affects the probabilities of reaching different sites, given that a particle is making a transition, is shown numerically to violate the no-pumping theorem. The results suggest that systems with interactions that lead to state-dependent branching fractions do not satisfy the no-pumping theorem.

How molecular motors work - insights from the molecular machinist's toolbox: the Nobel prize in Chemistry 2016

The Nobel prize in Chemistry for 2016 was awarded to Jean Pierre Sauvage, Sir James Fraser Stoddart, and Bernard (Ben) Feringa for their contributions to the design and synthesis of molecular machines.

The Nobel prize in Chemistry for 2016 was awarded to Jean Pierre Sauvage, Sir James Fraser Stoddart, and Bernard (Ben) Feringa for their contributions to the design and synthesis of molecular machines. While this field is still in its infancy, and at present there are no commercial applications, many observers have stressed the tremendous potential of molecular machines to revolutionize technology. However, perhaps the most important result so far accruing from the synthesis of molecular machines is the insight provided into the fundamental mechanisms by which molecular motors, including biological motors such as kinesin, myosin, F_oF₁ ATPase, and the flagellar motor, function. The ability to “tinker” with separate components of molecular motors allows asking, and answering, specific questions about mechanism, particularly with regard to light driven vs. chemistry driven molecular motors.

The Physics and Physical Chemistry of Molecular Machines

The concept of a "power stroke"-a free-energy releasing conformational change-appears in almost every textbook that deals with the molecular details of muscle, the flagellar rotor, and many other biomolecular machines. Here, it is shown by using the constraints of microscopic reversibility that the power stroke model is incorrect as an explanation of how chemical energy is used by a molecular machine to do mechanical work. Instead, chemically driven molecular machines operating under thermodynamic constraints imposed by the reactant and product concentrations in the bulk function as information ratchets in which the directionality and stopping torque or stopping force are controlled entirely by the gating of the chemical reaction that provides the fuel for the machine. The gating of the chemical free energy occurs through chemical state dependent conformational changes of the molecular machine that, in turn, are capable of generating directional mechanical motions. In strong contrast to this general conclusion for molecular machines driven by catalysis of a chemical reaction, a power stroke may be (and often is) an essential component for a molecular machine driven by external modulation of pH or redox potential or by light. This difference between optical and chemical driving properties arises from the fundamental symmetry difference between the physics of optical processes, governed by the Bose-Einstein relations, and the constraints of microscopic reversibility for thermally activated processes.

Irrelevance of the power stroke for the directionality, stopping force, and optimal efficiency of chemically driven molecular machines

A simple model for a chemically driven molecular walker shows that the elastic energy stored by the molecule and released during the conformational change known as the power-stroke (i.e., the free-energy difference between the pre- and post-power-stroke states) is irrelevant for determining the directionality, stopping force, and efficiency of the motor. Further, the apportionment of the dependence on the externally applied force between the forward and reverse rate constants of the power-stroke (or indeed among all rate constants) is irrelevant for determining the directionality, stopping force, and efficiency of the motor. Arguments based on the principle of microscopic reversibility demonstrate that this result is general for all chemically driven molecular machines, and even more broadly that the relative energies of the states of the motor have no role in determining the directionality, stopping force, or optimal efficiency of the machine. Instead, the directionality, stopping force, and optimal efficiency are determined solely by the relative heights of the energy barriers between the states. Molecular recognition--the ability of a molecular machine to discriminate between substrate and product depending on the state of the machine--is far more important for determining the intrinsic directionality and thermodynamics of chemo-mechanical coupling than are the details of the internal mechanical conformational motions of the machine. In contrast to the conclusions for chemical driving, a power-stroke is very important for the directionality and efficiency of light-driven molecular machines and for molecular machines driven by external modulation of thermodynamic parameters.

Charging a capacitor from an external fluctuating potential using a single conical nanopore

We explore the electrical rectification of large amplitude fluctuating signals by an asymmetric nanostructure operating in aqueous solution. We show experimentally and theoretically that a load capacitor can be charged to voltages close to 1 V within a few minutes by converting zero time-average potentials of amplitudes in the range 0.5–3 V into average net currents using a single conical nanopore. This process suggests that significant energy conversion and storage from an electrically fluctuating environment is feasible with a nanoscale pore immersed in a liquid electrolyte solution, a system characteristic of bioelectronics interfaces, electrochemical cells, and nanoporous membranes.

Enhanced diffusion, chemotaxis, and pumping by active enzymes: progress toward an organizing principle of molecular machines

Active enzymes diffuse more rapidly than inactive enzymes. This phenomenon may be due to catalysis-driven conformational changes that result in "swimming" through the aqueous solution. Recent additional work has demonstrated that active enzymes can undergo chemotaxis toward regions of high substrate concentration, whereas inactive enzymes do not, and, further, that active enzymes immobilized at surfaces can directionally pump liquids. In this Perspective, I will discuss these phenomena in light of Purcell's work on directed motion at low Reynold's number and in the context of microscopic reversibility. The conclusions suggest that a deep understanding of catalytically driven enhanced diffusion of enzymes and related phenomena can lead toward a general organizing principle for the design, characterization, and operation of molecular machines.

DNA polymerase as a molecular motor and pump

DNA polymerase is responsible for synthesizing DNA, a key component in the running of biological machinery. Using fluorescence correlation spectroscopy, we demonstrate that the diffusive movement of a molecular complex of DNA template and DNA polymerase enhances during nucleotide incorporation into the growing DNA template. The diffusion coefficient of the complex also shows a strong dependence on its inorganic cofactor, Mg2+ ions. When exposed to gradients of either nucleotide or cofactor concentrations, an ensemble of DNA polymerase complex molecules shows collective movement toward regions of higher concentrations. By immobilizing the molecular complex on a patterned gold surface, we demonstrate the fabrication of DNA polymerase-powered fluid pumps. These miniature pumps are capable of transporting fluid and tracer particles in a directional manner with the pumping speed increasing in the presence of the cofactor. The role of DNA polymerase as a micropump opens up avenues for designing miniature fluid pumps using enzymes as engines.

No-pumping theorem for many particle stochastic pumps

Stochastic pumps are models of artificial molecular machines which are driven by periodic time variation of parameters, such as site and barrier energies. The no-pumping theorem states that no directed motion is generated by variation of only site or barrier energies [S. Rahav, J. Horowitz, and C. Jarzynski, Phys. Rev. Lett. 101, 140602 (2008)]. We study stochastic pumps of several interacting particles and demonstrate that the net current of particles satisfies an additional no-pumping theorem.

Protein electron transfer: Dynamics and statistics

Electron transfer between redox proteins participating in energy chains of biology is required to proceed with high energetic efficiency, minimizing losses of redox energy to heat. Within the standard models of electron transfer, this requirement, combined with the need for unidirectional (preferably activationless) transitions, is translated into the need to minimize the reorganization energy of electron transfer. This design program is, however, unrealistic for proteins whose active sites are typically positioned close to the polar and flexible protein-water interface to allow inter-protein electron tunneling. The high flexibility of the interfacial region makes both the hydration water and the surface protein layer act as highly polar solvents. The reorganization energy, as measured by fluctuations, is not minimized, but rather maximized in this region. Natural systems in fact utilize the broad breadth of interfacial electrostatic fluctuations, but in the ways not anticipated by the standard models based on equilibrium thermodynamics. The combination of the broad spectrum of static fluctuations with their dispersive dynamics offers the mechanism of dynamical freezing (ergodicity breaking) of subsets of nuclear modes on the time of reaction/residence of the electron at a redox cofactor. The separation of time-scales of nuclear modes coupled to electron transfer allows dynamical freezing. In particular, the separation between the relaxation time of electro-elastic fluctuations of the interface and the time of conformational transitions of the protein caused by changing redox state results in dynamical freezing of the latter for sufficiently fast electron transfer. The observable consequence of this dynamical freezing is significantly different reorganization energies describing the curvature at the bottom of electron-transfer free energy surfaces (large) and the distance between their minima (Stokes shift, small). The ratio of the two reorganization energies establishes the parameter by which the energetic efficiency of protein electron transfer is increased relative to the standard expectations, thus minimizing losses of energy to heat. Energetically efficient electron transfer occurs in a chain of conformationally quenched cofactors and is characterized by flattened free energy surfaces, reminiscent of the flat and rugged landscape at the stability basin of a folded protein.

In situ structural analysis of the Yersinia enterocolitica injectisome

Injectisomes are multi-protein transmembrane machines allowing pathogenic bacteria to inject effector proteins into eukaryotic host cells, a process called type III secretion. Here we present the first three-dimensional structure of Yersinia enterocolitica and Shigella flexneri injectisomes in situ and the first structural analysis of the Yersinia injectisome. Unexpectedly, basal bodies of injectisomes inside the bacterial cells showed length variations of 20%. The in situ structures of the Y. enterocolitica and S. flexneri injectisomes had similar dimensions and were significantly longer than the isolated structures of related injectisomes. The crystal structure of the inner membrane injectisome component YscD appeared elongated compared to a homologous protein, and molecular dynamics simulations documented its elongation elasticity. The ring-shaped secretin YscC at the outer membrane was stretched by 30–40% in situ, compared to its isolated liposome-embedded conformation. We suggest that elasticity is critical for some two-membrane spanning protein complexes to cope with variations in the intermembrane distance.

DOI:http://dx.doi.org/10.7554/eLife.00792.001

Humans and other animals can use the five senses—touch, taste, sight, smell, and hearing—to interpret the world around them. Single-celled organisms, however, must rely on molecular cues to understand their immediate surroundings. In particular, bacteria gather information about external conditions, including potential hosts nearby, by secreting protein sensors that can relay messages back to the cell.

Bacteria export these sensors via secretion systems that enable the organism both to receive information about the environment and to invade a host cell. A total of seven separate secretion systems, known as types I–VII, have been identified. These different secretion systems handle distinct cargoes, allowing the bacterial cell to respond to a range of feedback from the external milieu.

The type III secretion system, also known as the ‘injectisome’, is found in bacterial species that are enclosed by two membranes separated by a periplasmic space. The injectisome comprises different components that combine to form the basal body, which spans the inner and outer membranes, and a projection from the basal body, called the hollow needle, that mediates the export of cargo from a bacterium to its host or the local environment.

The distance between the inner and outer membranes may vary across species or according to environmental conditions, so the basal body must be able to accommodate these changes. However, no mechanism has yet been established that might introduce such elasticity into the injectisome. Now, Kudryashev et al. have generated three-dimensional structures for the injectisomes of two species of bacteria, Shigella flexneri and Yersinia enterocolitica, and shown that the size of the basal body can fluctuate by up to 20%.

Kudryashev et al. imaged whole injectisomes in these two species and found that the height of the basal body was proportional to the distance between the inner and outer membranes. To probe how this could occur, the properties of two proteins that are important components of the basal body were studied in greater detail. YscD, a protein that extends across the periplasmic space, was crystallized and its structure was then determined and used to develop a computer model to assess its compressibility: this model indicated that YscD could stretch or contract by up to 50% of its total length. The outer membrane component YscC also appeared elastic: when the protein was isolated and introduced into synthetic membranes, its length was reduced 30–40% relative to that observed in intact bacterial membranes.

A further experiment confirmed the adaptability of the basal body: when the separation of the membranes was deliberately increased by placing bacteria in a high-salt medium, the basal body extended approximately 10% in length. Cumulatively, therefore, these experiments suggest that the in-built flexibility of the basal body of the injectisome allows bacteria to adjust to environmental changes while maintaining their sensory abilities and host-invasion potential.

DOI:http://dx.doi.org/10.7554/eLife.00792.002

Braid group and topological phase transitions in nonequilibrium stochastic dynamics

We show that distinct topological phases of the band structure of a non-Hermitian Hamiltonian can be classified with elements of the braid group. As the proof of principle, we consider the non-Hermitian evolution of the statistics of nonequilibrium stochastic currents. We show that topologically nontrivial phases have detectable properties, including the emergence of decaying oscillations of parity and state probabilities, and discontinuities in the steady state statistics of currents.

Enzyme molecules as nanomotors

Using fluorescence correlation spectroscopy, we show that the diffusive movements of catalase enzyme molecules increase in the presence of the substrate, hydrogen peroxide, in a concentration-dependent manner. Employing a microfluidic device to generate a substrate concentration gradient, we show that both catalase and urease enzyme molecules spread toward areas of higher substrate concentration, a form of chemotaxis at the molecular scale. Using glucose oxidase and glucose to generate a hydrogen peroxide gradient, we induce the migration of catalase toward glucose oxidase, thereby showing that chemically interconnected enzymes can be drawn together.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.