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

Clarifying mechanisms and kinetics of programmable catalysis

Programmable catalysis-the purposeful oscillation of catalytic potential energy surfaces (PES)-has emerged as a promising method for the acceleration of catalyzed reaction rates. However, theoretical study of programmable catalysis has been limited by onerous computational demands of integrating the stiff differential equations that describe periodic cycling between PESs. This work details methods that reduce the computational cost of finding the limit cycle by ≳108×. These methods produce closed-form analytical solutions for didactic case studies, examination of which provides physical insights of programmable catalysis mechanisms. Generalization of these analyses to more complex reaction networks, including CO oxidation on Pt (111) surfaces, exposes the key catalyst properties required to achieve enhanced rates and conversions via one of two programmable catalysis mechanisms: quasi-static (high frequency) and stepwise (intermediate frequency). Analytical description of each mechanism is critical in understanding the consequences of the Sabatier principle on achievable rate enhancement through programmed catalysis.

Universal slow dynamics of chemical reaction networks

Understanding the emergent behavior of chemical reaction networks (CRNs) is a fundamental aspect of biology and its origin from inanimate matter. A closed CRN monotonically tends to thermal equilibrium, but when it is opened to external reservoirs, a range of behaviors is possible, including transition to a new equilibrium state, a nonequilibrium state, or indefinite growth. This study shows that slowly driven CRNs are governed by the conserved quantities of the closed system, which are generally far fewer in number than the species. Considering both deterministic and stochastic dynamics, a universal slow-dynamics equation is derived with singular perturbation methods and is shown to be thermodynamically consistent. The slow dynamics is highly robust against microscopic details of the network, which may be unknown in practical situations. In particular, nonequilibrium states of realistic large CRNs can be sought without knowledge of bulk reaction rates. The framework is successfully tested against a suite of networks of increasing complexity and argued to be relevant in the treatment of open CRNs as chemical machines.

Temporal (Dis)Assembly of Peptide Nanostructures Dictated by Native Multistep Catalytic Transformations

Emergence of complex catalytic machinery via simple building blocks under non-equilibrium conditions can contribute toward the system level understanding of the extant biocatalytic reaction network that fuels metabolism. Herein, we report temporal (dis)assembly of peptide nanostructures in presence of a cofactor dictated by native multistep cascade transformations. The short peptide can form a dynamic covalent bond with the thermodynamically activated substrate and recruit cofactor hemin to access non-equilibrium catalytic nanostructures (positive feedback). The neighboring imidazole and hemin moieties in the assembled state rapidly converted the substrate to product(s) via a two-step cascade reaction (hydrolase-peroxidase like) that subsequently triggered the disassembly of the catalytic nanostructures (negative feedback). The feedback coupled reaction cycle involving intrinsic catalytic prowess of short peptides to realize the advanced trait of two-stage cascade degradation of a thermodynamically activated substrate foreshadows the complex non-equilibrium protometabolic networks that might have preceded the chemical emergence of life.

Spontaneous Vibrations and Stochastic Resonance of Short Oligomeric Springs

There is growing interest in molecular structures that exhibit dynamics similar to bistable mechanical systems. These structures have the potential to be used as two-state operating units for various functional purposes. Particularly intriguing are the bistable systems that display spontaneous vibrations and stochastic resonance. Previously, via molecular dynamics simulations, it was discovered that short pyridine-furan springs in water, when subjected to stretching with power loads, exhibit the bistable dynamics of a Duffing oscillator. In this study, we extend these simulations to include short pyridine-pyrrole and pyridine-furan springs in a hydrophobic solvent. Our findings demonstrate that these systems also display the bistable dynamics, accompanied by spontaneous vibrations and stochastic resonance activated by thermal noise.

Kinetic Asymmetry versus Dissipation in the Evolution of Chemical Systems as Exemplified by Single Enzyme Chemotaxis

Single enzyme chemotaxis is a phenomenon by which a nonequilibrium spatial distribution of an enzyme is created and maintained by concentration gradients of the substrate and product of the catalyzed reaction. These gradients can arise either naturally through metabolism or experimentally, e.g., by flow of materials through microfluidic channels or by use of diffusion chambers with semipermeable membranes. Numerous hypotheses regarding the mechanism of this phenomenon have been proposed. Here, we discuss a mechanism based solely on diffusion and chemical reaction and show that kinetic asymmetry, a difference in the transition state energies for dissociation/association of substrate and product, and diffusion asymmetry, a difference in the diffusivities of the bound and free forms of the enzyme, are the determinates of the direction of chemotaxis and can result in either positive or negative chemotaxis, both of which have been demonstrated experimentally. Exploration of these fundamental symmetries that govern nonequilibrium behavior helps to distinguish between possible mechanisms for the evolution of a chemical system from initial to the steady state and whether the principle that determines the direction a system shifts when exposed to an external energy source is based on thermodynamics or on kinetics with the latter being supported by the results of the present paper. Our results show that, while dissipation ineluctably accompanies nonequilibrium phenomena, including chemotaxis, systems do not evolve to maximize or minimize dissipation but rather to attain greater kinetic stability and accumulate in regions where their effective diffusion coefficient is as small as possible. The chemotactic response to the chemical gradients formed by other enzymes participating in a catalytic cascade provides a mechanism for forming loose associations known as metabolons. Significantly, the direction of the effective force due to these gradients depends on the kinetic asymmetry of the enzyme and so can be nonreciprocal, where one enzyme is attracted to another enzyme, but the other enzyme is repelled by the one, in seeming contradiction to Newtons third law. This nonreciprocity is an important ingredient in the behavior of active matter.

Nonequilibrium Amyloid Polymers Exploit Dynamic Covalent Linkage to Temporally Control Charge-Selective Catalysis

Extant proteins exploit thermodynamically activated negatively charged coenzymes and hydrotropes to temporally access mechanistically important conformations that regulate vital biological functions, from metabolic reactions to expression modulation. Herein, we show that a short amyloid peptide can bind to a small molecular coenzyme by exploiting reversible covalent linkage to polymerize and access catalytically proficient nonequilibrium amyloid microphases. Subsequent hydrolysis of the activated coenzyme leads to depolymerization, realizing a variance of the surface charge of the assembly as a function of time. Such temporal change of surface charge dynamically modulates catalytic activities of the transient assemblies as observed in highly evolved modern-day biocatalysts.

Nonequilibrium Catalytic Supramolecular Assemblies of Melamine- and Imidazole-Based Dynamic Building Blocks

The development of synthetic nonequilibrium systems has gathered increasing attention due to their potential to illustrate the dynamic, complex, and emergent traits of biological systems. Simple building blocks capable of interacting via dynamic covalent chemistry and physical assembly in a reaction network under nonequilibrium conditions can contribute to our understanding of complex systems of life and its origin. Herein, we have demonstrated the nonequilibrium generation of catalytic supramolecular assemblies from simple heterocycle melamine driven by a thermodynamically activated ester. Utilizing a reversible covalent linkage, an imidazole moiety was recruited by the assemblies to access a catalytic transient state that dissipated energy via accelerated hydrolysis of the activated ester. The nonequilibrium assemblies were further capable of temporally binding to a hydrophobic guest to modulate its photophysical properties. Notably, the presence of an exogenous aromatic base augmented the lifetime of the catalytic microphases, reflecting their higher kinetic stability.

Emergence of a Promiscuous Peroxidase Under Non-Equilibrium Conditions

Herein, we report the substrate induced generation of a transient catalytic microenvironment from a single amino acid functionalized fatty acid in presence of a cofactor hemin. The catalytic state accessed under non-equilibrium conditions showed acceleration of peroxidase activity resulting in degradation of the substrate and subsequently led to disassembly. Equilibrated systems could not access the three-dimensional microphases and showed substantially lower catalytic activity. Further, the assembled state showed latent catalytic function (promiscuity) to hydrolyze a precursor to yield the same substrate. Consequently, the assembly demonstrated protometabolism by exploiting the peroxidase-hydrolase cascade to augment the lifetime and the mechanical properties of the catalytic state.

Short Pyridine-Furan Springs Exhibit Bistable Dynamics of Duffing Oscillators

The intensive development of nanodevices acting as two-state systems has motivated the search for nanoscale molecular structures whose dynamics are similar to those of bistable mechanical systems, such as Euler arches and Duffing oscillators. Of particular interest are the molecular structures capable of spontaneous vibrations and stochastic resonance. Recently, oligomeric molecules that were a few nanometers in size and exhibited the bistable dynamics of an Euler arch were identified through molecular dynamics simulations of short fragments of thermo-responsive polymers subject to force loading. In this article, we present molecular dynamics simulations of short pyridine-furan springs a few nanometers in size and demonstrate the bistable dynamics of a Duffing oscillator with thermally-activated spontaneous vibrations and stochastic resonance.

Substrate induced generation of transient self-assembled catalytic systems

Living matter is sustained under non-equilibrium conditions via continuous expense of energy which is coordinated by complex organized events. Spatiotemporal control over exquisite functions arises from chemical complexity under non-equilibrium conditions. For instance, extant biology often uses substrate binding events to access temporally stable protein conformations which show acceleration of catalytic rates to subsequently degrade the substrate. Furthermore, thermodynamically activated but kinetically stable esters (GTP) induce the change of conformation of cytoskeleton proteins (microtubules) which leads to rapid polymerization and triggers an augmentation of catalytic rates to subsequently degrade the ester. Importantly, high-energy assemblies composed of non-activated building blocks (GDP-tubulin) are accessed utilizing the energy dissipated from the catalytic conversion of GTP to GDP from the assembled state. Notably, some experimental studies with simple self-assembled systems have elegantly mimicked the phenomena of substrate induced transient generation of catalytic conformations. Through this review, we endeavour to highlight those select studies which have used simple building blocks to demonstrate substrate induced self-assemblies that subsequently show rate acceleration to convert the substrate into waste. The concept of substrate induced self-assembly of building blocks and rate acceleration from the assembled state has the potential to play a predominant role in the preparation of non-equilibrium systems. The design strategies covered in this review can inspire the possibilities of accessing high energy self-assembled structures that are seen in living systems.

This review highlights the studies which show substrate induced generation of transient catalytic moieties. Examples have been discussed with keeping an eye on the design strategies for development of non-equilibrium high energy assemblies as seen in Nature.

Substituent Effects on Transient, Carbodiimide-Induced Geometry Changes in Diphenic Acids

Nucleotide-induced conformational changes in motor proteins are key to many important cell functions. Inspired by this biological behavior, we report a simple chemically fueled system that exhibits carbodiimide-induced geometry changes. Bridging via transient anhydride formation leads to a significant reduction of the twist about the biaryl bond of substituted diphenic acids, giving a simple molecular clamp. The kinetics are well-described by a simple mechanism, allowing structure-property effects to be determined. The kinetic parameters can be used to derive important characteristics of the system such as the efficiencies (anhydride yields), maximum anhydride concentrations, and overall lifetimes. Transient diphenic anhydrides tolerate steric hindrance ortho to the biaryl bond but are significantly affected by electronic effects, with electron-deficient substituents giving lower yields, peak conversions, and lifetimes. The results provide useful guidelines for the design of functional systems incorporating diphenic acid units.

How Was Nature Able to Discover Its Own Laws-Twice?

The central thesis of the modern scientific revolution is that nature is objective. Yet, somehow, out of that objective reality, projective systems emerged-cognitive and purposeful. More remarkably, through nature's objective laws, chemical systems emerged and evolved to take advantage of those laws. Even more inexplicably, nature uncovered those laws twice-once unconsciously, once consciously. Accordingly, one could rephrase the origin of life question as follows: how was nature able to become self-aware and discover its own laws? What is the law of nature that enabled nature to discover its own laws? Addressing these challenging questions in physical-chemical terms may be possible through the newly emergent field of systems chemistry.

Chemical Fueling Enables Molecular Complexification of Self-Replicators*

Unravelling how the complexity of living systems can (have) emerge(d) from simple chemical reactions is one of the grand challenges in contemporary science. Evolving systems of self-replicating molecules may hold the key to this question. Here we show that, when a system of replicators is subjected to a regime where replication competes with replicator destruction, simple and fast replicators can give way to more complex and slower ones. The structurally more complex replicator was found to be functionally more proficient in the catalysis of a model reaction. These results show that chemical fueling can maintain systems of replicators out of equilibrium, populating more complex replicators that are otherwise not readily accessible. Such complexification represents an important requirement for achieving open-ended evolution as it should allow improved and ultimately also new functions to emerge.

The Transient Covalent Bond in Abiotic Nonequilibrium Systems

Biochemical systems accomplish many critical functions with by operating out-of-equilibrium using the energy of chemical fuels. The formation of a transient covalent bond is a simple but very effective tool in designing analogous reaction networks. This Minireview focuses on the fuel chemistries that have been used to generate transient bonds in recent demonstrations of abiotic nonequilibrium systems (i.e., systems that do not make use of biological components). Fuel reactions are divided into two fundamental classifications depending on whether the fuel contributes structural elements to the activated state, a distinction that dictates how they can be used. Reported systems are further categorized by overall fuel reaction (e.g., hydrolysis of alkylating agents, carbodiimide hydration) and illustrate how similar chemistry can be used to effect a wide range of nonequilibrium behavior, ranging from self-assembly to the operation of molecular machines.

Chemically Fueled Transient Geometry Changes in Diphenic Acids

Transient changes in molecular geometry are key to the function of many important biochemical systems. Here, we show that diphenic acids undergo out-of-equilibrium changes in dihedral angle when reacted with a carbodiimide chemical fuel. Treatment of appropriately functionalized diphenic acids with EDC (N-(3-(dimethylamino)propyl)-N'-ethylcarbodiimide hydrochloride) yields the corresponding diphenic anhydrides, reducing the torsional angle about the biaryl bond by ∼45°, regardless of substitution. In the absence of steric resistance, the reaction is well-described by a simple mechanism; the resulting kinetic parameters can be used to derive important properties of the system, such as yields and lifetimes. The reaction tolerates steric hindrance ortho to the biaryl bond, although the competing formation of (transient) byproducts complicates quantitative analysis.

Enhanced catalytic activity under non-equilibrium conditions

The development of non-equilibrium synthetic systems provides access to innovative materials with life-like properties. Non-equilibrium systems require a continuous input of energy to retain their functional state, which makes for a fundamental difference to systems that operate at thermodynamic equilibrium. Kinetic asymmetry in the energy consumption pathway is required to drive systems out of equilibrium. This understanding has permitted chemists to design dissipative synthetic molecular machines and high-energy materials. Here we show that kinetic asymmetry also emerges at the macroscopic level by demonstrating that local energy delivery in the form of light to a hydrogel containing gold nanoparticles installs a non-equilibrium steady state. The instalment and maintenance of the macroscopic non-equilibrium state is facilitated by the gel matrix in which motion is governed by diffusion rather than convection. The non-equilibrium state is characterized by a persistent gradient in the surface composition of the nanoparticles embedded in the gel, which affects the fluorescent and catalytic properties of the system. We show that the overall catalytic performance of the system is enhanced under these non-equilibrium conditions. In perspective it will be possible to develop out-of-equilibrium matrices in which functional properties emerge as a result of spatially controlled energy delivery and spatially controlled chemistries.

Dissipative Assembly of Macrocycles Comprising Multiple Transient Bonds

Dissipative assembly has great potential for the creation of new adaptive chemical systems. However, while molecular assembly at equilibrium is routinely used to prepare complex architectures from polyfunctional monomers, species formed out of equilibrium have, to this point, been structurally very simple. In most examples the fuel simply effects the formation of a single short-lived covalent bond. Herein, we show that chemical fuels can assemble bifunctional components into macrocycles containing multiple transient bonds. Specifically, dicarboxylic acids give aqueous dianhydride macrocycles on treatment with a carbodiimide. The macrocycles are assembled efficiently as a consequence of both fuel-dependent and fuel-independent mechanisms; they undergo slower decomposition, building up as the fuel recycles the components, and are a favored product of the dynamic exchange of the anhydride bonds. These results create new possibilities for generating structurally sophisticated out-of-equilibrium species.

Structure-Property Effects in the Generation of Transient Aqueous Benzoic Acid Anhydrides by Carbodiimide Fuels

The design of dissipative systems, which operate out-of-equilibrium by consuming chemical fuels, is challenging. As yet, there are a few examples of privileged fuel chemistries that can be broadly applied in abiotic systems in the same way that ATP hydrolysis is exploited throughout biochemistry. The key issue is that designing nonequilibrium systems is inherently about balancing the relative rates of coupled reactions. The use of carbodiimides as fuels to generate transient aqueous carboxylic anhydrides has recently been used in examples of new nonequilibrium materials and supramolecular assemblies. Here, we explore the kinetics of formation and decomposition of a series of benzoic anhydrides generated from the corresponding acids and EDC under typical conditions (EDC = N-(3-dimethylaminopropyl)-N'-ethylcarbodiimide hydrochloride). The reactions can be described by a simple mechanism that merges known behavior for the two processes independently. Structure-property effects in these systems are dominated by differences in the anhydride decomposition rate. The kinetic parameters allow trends in concentration-dependent properties to be simulated, such as reaction lifetimes, peak anhydride concentrations, and yields. For key properties, there are diminishing returns with the addition of increasing amounts of fuel. These results should provide useful guidelines for the design of functional systems making use of this chemistry.

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.

Thermodynamic efficiency in dissipative chemistry

Chemical processes in closed systems inevitably relax to equilibrium. Living systems avoid this fate and give rise to a much richer diversity of phenomena by operating under nonequilibrium conditions. Recent experiments in dissipative self-assembly also demonstrated that by opening reaction vessels and steering certain concentrations, an ocean of opportunities for artificial synthesis and energy storage emerges. To navigate it, thermodynamic notions of energy, work and dissipation must be established for these open chemical systems. Here, we do so by building upon recent theoretical advances in nonequilibrium statistical physics. As a central outcome, we show how to quantify the efficiency of such chemical operations and lay the foundation for performance analysis of any dissipative chemical process.

Individual-Molecule Perspective Analysis of Chemical Reaction Networks: The Case of a Light-Driven Supramolecular Pump

The first study in which stochastic simulations of a two‐component molecular machine are performed in the mass‐action regime is presented. This system is an autonomous molecular pump consisting of a photoactive axle that creates a directed flow of rings through it by exploiting light energy away from equilibrium. The investigation demonstrates that the pump can operate in two regimes, both experimentally accessible, in which light‐driven steps can be rate‐determining or not. The number of photons exploited by an individual molecular pump, as well as the precision of cycling and the overall efficiency, critically rely on the operating regime of the machine. This approach provides useful information not only to guide the chemical design of a self‐assembling molecular device with desired features, but also to elucidate the effect of the environment on its performance, thus facilitating its experimental investigation.

The Essence of Systems Chemistry

Systems Chemistry investigates the upkeep of specific interactions of an exceptionally broad choice of objects over longer periods of time than the average time of existence of the objects themselves. This maintenance of a dynamic state focuses on conditions where the objects are thermodynamically not very stable and should be rare or virtually inexistent. It does not matter whether they are homochirally enriched populations of chiral molecules, a specific composition of some sort of aggregate, supramolecules, or even a set of chemically relatively unstable molecules that constantly transform one into another. What does matter is that these specific interactions prevail in complex mixtures and eventually grow in numbers and frequency through the enhancing action of autocatalysis, which makes such systems ultimately resemble living cells and interacting living populations. Such chemical systems need to be correctly understood, but also intuitively described. They may be so complex that metaphors become practically more important, as a means of communication, than the precise and correct technical description of chemical models and complex molecular or supramolecular relations. This puts systems chemists on a tightrope walk of science communication, between the complex reality and an imaginative model world. This essay addresses, both, scientists who would like to read “A Brief History of Systems Chemistry”, that is, about its “essence”, and systems chemists who work with and communicate complex life-like chemical systems. I illustrate for the external reader a light mantra, that I call “to make more of it”, and I charily draw systems chemists to reflect upon the fact that chemists are not always good at drawing a clear line between a model and “the reality”: The real thing. We are in a constant danger of taking metaphors for real. Yet in real life, we do know very well that we cannot smoke with Magritte’s pipe, don’t we?

Controlling the preferential motion of chiral molecular walkers on a surface

Molecular walkers standing on two or more "feet" on an anisotropic periodic potential of a crystal surface may perform a one-dimensional Brownian motion at the surface-vacuum interface along a particular direction in which their mobility is the largest. In thermal equilibrium the molecules move with equal probabilities both ways along this direction, as expected from the detailed balance principle, well-known in chemical reactivity and in the theory of molecular motors. For molecules that possess an asymmetric potential energy surface (PES), we propose a generic method based on the application of a time-periodic external stimulus that would enable the molecules to move preferentially in a single direction thereby acting as Brownian ratchets. To illustrate this method, we consider a prototypical synthetic chiral molecular walker, 1,3-bis(imidazol-1-ylmethyl)-5(1-phenylethyl)benzene, diffusing on the anisotropic Cu(110) surface along the Cu rows. As unveiled by our kinetic Monte Carlo simulations based on the rates calculated using ab initio density functional theory, this molecule moves to the nearest equivalent lattice site via the so-called inchworm mechanism in which it steps first with the rear foot and then with the front foot. As a result, the molecule diffuses via a two-step mechanism, and due to its inherent asymmetry, the corresponding PES is also spatially asymmetric. Taking advantage of this fact, we show how the external stimulus can be tuned to separate molecules of different chirality, orientation and conformation. The consequences of these findings for molecular machines and the separation of enantiomers are also discussed.

How Defects Control the Out-of-Equilibrium Dissipative Evolution of a Supramolecular Tubule

Supramolecular architectures that work out-of-equilibrium or that can change in specific ways when absorbing external energy are ubiquitous in nature. Gaining the ability to create via self-assembly artificial materials possessing such fascinating behaviors would have a major impact in many fields. However, the rational design of similar dynamic structures requires to understand and, even more challenging, to learn how to master the molecular mechanisms governing how the assembled systems evolve far from the equilibrium. Typically, this represents a daunting challenge due to the limited molecular insight that can be obtained by the experiments or by classical modeling approaches. Here we combine coarse-grained molecular models and advanced simulation approaches to study at submolecular (<5 Å) resolution a supramolecular tubule, which breaks and disassembles upon absorption of light energy triggering isomerization of its azobenzene-containing monomers. Our approach allows us to investigate the molecular mechanism of monomer transition in the assembly and to elucidate the kinetic process for the accumulation of the transitions in the system. Despite the stochastic nature of the excitation process, we demonstrate how these tubules preferentially dissipate the absorbed energy locally via the amplification of defects in their supramolecular structure. We find that this constitutes the best kinetic pathway for accumulating monomer transitions in the system, which determines the dynamic evolution out-of-equilibrium and the brittle behavior of the assembly under perturbed conditions. Thanks to the flexibility of our models, we finally come out with a general principle, where defects explain and control the brittle/soft behavior of such light-responsive assemblies.

Chemically fueled covalent crosslinking of polymer materials

Non-equilibrium covalently crosslinked hydrogels are synthesized using carbodiimide fueled coupling of carboxylic acids to anhydrides which eventually dissipate by hydrolysis.

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.

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.