Effects of oscillations and energy-driven fluctuations on the dynamics of enzyme catalysis and free-energy transduction

Beyond Single-Cycle Autonomous Molecular Machines: Light-Powered Shuttling in a Multi-Cycle Reaction Network

Biomolecular machines autonomously convert energy into functions, driving systems away from thermodynamic equilibrium. This energy conversion is achieved by leveraging complex, kinetically asymmetric chemical reaction networks that are challenging to characterize precisely. In contrast, all known synthetic molecular systems in which kinetic asymmetry has been quantified are well described by simple single-cycle networks. Here, we report on a unique light-driven [2]rotaxane that enables the autonomous operation of a synthetic molecular machine with a multi-cycle chemical reaction network. Unlike all prior systems, the present one exploits a photoactive macrocycle, which features a different photoreactivity depending on the binding sites at which it resides. Furthermore, E to Z isomerization reverses the relative affinity of the macrocycle for two binding sites on the axle, resulting in a multi-cycle network. Building on the most recent theoretical advancements, this work quantifies kinetic asymmetry in a multi-cycle network for the first time. Our findings represent the simplest rotaxane capable of autonomous shuttling developed so far and offer a general strategy to generate and quantify kinetic asymmetry beyond single-cycle systems.

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.

Molecular Ratchets and Kinetic Asymmetry: Giving Chemistry Direction

Over the last two decades ratchet mechanisms have transformed the understanding and design of stochastic molecular systems-biological, chemical and physical-in a move away from the mechanical macroscopic analogies that dominated thinking regarding molecular dynamics in the 1990s and early 2000s (e.g. pistons, springs, etc), to the more scale-relevant concepts that underpin out-of-equilibrium research in the molecular sciences today. Ratcheting has established molecular nanotechnology as a research frontier for energy transduction and metabolism, and has enabled the reverse engineering of biomolecular machinery, delivering insights into how molecules 'walk' and track-based synthesisers operate, how the acceleration of chemical reactions enables energy to be transduced by catalysts (both motor proteins and synthetic catalysts), and how dynamic systems can be driven away from equilibrium through catalysis. The recognition of molecular ratchet mechanisms in biology, and their invention in synthetic systems, is proving significant in areas as diverse as supramolecular chemistry, systems chemistry, dynamic covalent chemistry, DNA nanotechnology, polymer and materials science, molecular biology, heterogeneous catalysis, endergonic synthesis, the origin of life, and many other branches of chemical science. Put simply, ratchet mechanisms give chemistry direction. Kinetic asymmetry, the key feature of ratcheting, is the dynamic counterpart of structural asymmetry (i.e. chirality). Given the ubiquity of ratchet mechanisms in endergonic chemical processes in biology, and their significance for behaviour and function from systems to synthesis, it is surely just as fundamentally important. This Review charts the recognition, invention and development of molecular ratchets, focussing particularly on the role for which they were originally envisaged in chemistry, as design elements for molecular machinery. Different kinetically asymmetric systems are compared, and the consequences of their dynamic behaviour discussed. These archetypal examples demonstrate how chemical systems can be driven inexorably away from equilibrium, rather than relax towards it.

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.

Kinetic Barrier Diagrams to Visualize and Engineer Molecular Nonequilibrium Systems

Molecular nonequilibrium systems hold great promises for the nanotechnology of the future. Yet, their development is slowed by the absence of an informative representation. Indeed, while potential energy surfaces comprise in principle all the information, they hide the dynamic interplay of multiple reaction pathways underlying nonequilibrium systems, i.e., the degree of kinetic asymmetry. To offer an insightful visual representation of kinetic asymmetry, we extended an approach pertaining to catalytic networks, the energy span model, by focusing on system dynamics - rather than thermodynamics. Our approach encompasses both chemically and photochemically driven systems, ranging from unimolecular motors to simple self-assembly schemes. The obtained diagrams give immediate access to information needed to guide experiments, such as states' population, rate of machine operation, maximum work output, and effects of design changes. The proposed kinetic barrier diagrams offer a unifying graphical tool for disparate nonequilibrium phenomena.

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.

Using Catalysis to Drive Chemistry Away from Equilibrium: Relating Kinetic Asymmetry, Power Strokes, and the Curtin-Hammett Principle in Brownian Ratchets

Chemically fueled autonomous molecular machines are catalysis-driven systems governed by Brownian information ratchet mechanisms. One fundamental principle behind their operation is kinetic asymmetry, which quantifies the directionality of molecular motors. However, it is difficult for synthetic chemists to apply this concept to molecular design because kinetic asymmetry is usually introduced in abstract mathematical terms involving experimentally inaccessible parameters. Furthermore, two seemingly contradictory mechanisms have been proposed for chemically driven autonomous molecular machines: Brownian ratchet and power stroke mechanisms. This Perspective addresses both these issues, providing accessible and experimentally useful design principles for catalysis-driven molecular machinery. We relate kinetic asymmetry to the Curtin-Hammett principle using a synthetic rotary motor and a kinesin walker as illustrative examples. Our approach describes these molecular motors in terms of the Brownian ratchet mechanism but pinpoints both chemical gating and power strokes as tunable design elements that can affect kinetic asymmetry. We explain why this approach to kinetic asymmetry is consistent with previous ones and outline conditions where power strokes can be useful design elements. Finally, we discuss the role of information, a concept used with different meanings in the literature. We hope that this Perspective will be accessible to a broad range of chemists, clarifying the parameters that can be usefully controlled in the design and synthesis of molecular machines and related systems. It may also aid a more comprehensive and interdisciplinary understanding of biomolecular machinery.

Controlling dynamics in extended molecular frameworks

Molecular machines are essential dynamic components for fuel production, cargo delivery, information storage and processing in living systems. Scientists have demonstrated that they can design and synthesize artificial molecular machines that operate efficiently in isolation - for example, at high dilution in solution - fuelled by chemicals, electricity or light. To organize the spatial arrangement and motion of these machines within close proximity to one another in solid frameworks, such that useful macroscopic work can be performed, remains a challenge in both chemical and materials science. In this Review, we summarize the progress that has been made during the past decade in organizing dynamic molecular entities in such solid frameworks. Emerging applications of these dynamic smart materials in the contexts of molecular recognition, optoelectronics, drug delivery, photodynamic therapy and water desalination are highlighted. Finally, we review recent work on a new non-equilibrium adsorption phenomenon for which we have coined the term mechanisorption. The ability to use external energy to drive directional processes in mechanized extended frameworks augurs well for the future development of artificial molecular factories.

Heterogeneous Heat Absorption Is Complementary to Radiotherapy

Simple Summary

This review shows the advantages of heterogeneous heating of selected malignant cells in harmonic synergy with radiotherapy. The main clinical achievement of this complementary therapy is its extreme safety and minimal adverse effects. Combining the two methods opens a bright perspective, transforming the local radiotherapy to the antitumoral impact on the whole body, destroying the distant metastases by “teaching” the immune system about the overall danger of malignancy.

Abstract

(1) Background: Hyperthermia in oncology conventionally seeks the homogeneous heating of the tumor mass. The expected isothermal condition is the basis of the dose calculation in clinical practice. My objective is to study and apply a heterogenic temperature pattern during the heating process and show how it supports radiotherapy. (2) Methods: The targeted tissue’s natural electric and thermal heterogeneity is used for the selective heating of the cancer cells. The amplitude-modulated radiofrequency current focuses the energy absorption on the membrane rafts of the malignant cells. The energy partly “nonthermally” excites and partly heats the absorbing protein complexes. (3) Results: The excitation of the transmembrane proteins induces an extrinsic caspase-dependent apoptotic pathway, while the heat stress promotes the intrinsic caspase-dependent and independent apoptotic signals generated by mitochondria. The molecular changes synergize the method with radiotherapy and promote the abscopal effect. The mild average temperature (39–41 °C) intensifies the blood flow for promoting oxygenation in combination with radiotherapy. The preclinical experiences verify, and the clinical studies validate the method. (4) Conclusions: The heterogenic, molecular targeting has similarities with DNA strand-breaking in radiotherapy. The controlled energy absorption allows using a similar energy dose to radiotherapy (J/kg). The two therapies are synergistically combined.

Polyrotaxanes and the pump paradigm

The year 2022 marks the 30th anniversary of the first reports of polyrotaxanes in the scientific literature.

Radical-pairing-induced molecular assembly and motion

Radical-pairing interactions between conjugated organic π-radicals are relative newcomers to the inventory of molecular recognition motifs explored in supramolecular chemistry. The unique electronic, magnetic, optical and redox-responsive properties of the conjugated π-radicals render molecules designed with radical-pairing interactions useful for applications in various areas of chemistry and materials science. In particular, the ability to control formation of radical cationic or anionic species, by redox stimulation, provides a flexible trigger for directed assembly and controlled molecular motions, as well as a convenient means of inputting energy to fuel non-equilibrium processes. In this Review, we provide an overview of different examples of radical-pairing-based recognition processes and of their emerging use in (1) supramolecular assembly, (2) templation of mechanically interlocked molecules, (3) stimuli-controlled molecular switches and, by incorporation of kinetic asymmetry in the design, (4) the creation of unidirectional molecular transporters based on pumping cassettes powered by fuelled switching of radical-pairing interactions. We conclude the discussion with an outlook on future directions for the field.

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.

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.

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.

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.

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.

Allosteric model of an ion pump

We present a simple model of a free-energy transducer made of allosterically coupled two ratchet subsystems. Each of the subsystems transports particles from one particle reservoir to another. The coupling of the subsystems imposes correlated transitions of the potential profiles of the two subsystems. As a result, a downhill flux in one subsystem with higher chemical-potential difference drives an uphill flux in the other subsystem with lower chemical-potential difference. The direction of the driven flux inverts depending on the direction of the driving flux. The ratio between the fluxes conveyed by the two subsystems is variable and nonstoichiometric. By selecting appropriate parameters, the maximum ratio of the driven flux to driving flux and maximum free-energy transducing efficiency reaches some 90 and 40%, respectively. At a stalled state, the driven flux vanishes while the driving flux remains finite. The allosteric model enables explicit analysis of the timing between binding-unbinding of particles and transitions of potential profile. The behavior of the model is similar to but different from that of the alternate access model, which is a biochemical model for active transport proteins. Our model works also as a regulatory system. We suggest that the correlated transitions of the subsystems (subunits or domains) through allosteric interaction are the origin of the diverse functions of the protein machineries.

Biological proton pumping in an oscillating electric field

Time-dependent external perturbations provide powerful probes of the function of molecular machines. Here we study biological proton pumping in an oscillating electric field. The protein cytochrome c oxidase is the main energy transducer in aerobic life, converting chemical energy into an electric potential by pumping protons across a membrane. With the help of master-equation descriptions that recover the key thermodynamic and kinetic properties of this biological "fuel cell," we show that the proton pumping efficiency and the electronic currents in steady state depend significantly on the frequency and amplitude of the applied field, allowing us to distinguish between different microscopic mechanisms of the machine. A spectral analysis reveals dominant reaction steps consistent with an electron-gated pumping mechanism.

Reciprocal relations for nonlinear coupled transport

Reciprocal relations are derived for transport between two pairs of reservoirs energetically coupled at mesoscopic contact points through second order in the chemical potential differences. In the derivation attention is focussed on the average number of particles transported between the reservoirs during each excursion of the contact point away from, and regression back to, a steady state. All quantities involved are experimentally accessible in the full counting statistics of the transport processes.

Universal geometric theory of mesoscopic stochastic pumps and reversible ratchets

We construct a unifying theory of geometric effects in mesoscopic stochastic kinetics. We demonstrate that the adiabatic pump and the reversible ratchet effects, as well as similar new phenomena in other domains, such as in epidemiology, all follow from very similar geometric phase contributions to the effective action in the stochastic path integral representation of the moment generating function. The theory provides the universal technique for identification, prediction, and calculation of pumplike phenomena in an arbitrary mesoscopic stochastic framework.

Response to a temperature modulation as a signature of chemical mechanisms

We consider n reactive species involved in unimolecular reactions and submitted to a temperature modulation of small amplitude. We determine the conditions on the rate constants for which the deviations from the equilibrium concentrations of each species can be optimized and find the analytical expression of the frequency associated with an extremum of concentration shift in the case n=3. We prove that the frequency dependence of the displacement of equilibrium gives access to the number n of species involved in the mechanism. We apply the results to the case of the transformation of a reactant into a product through a possible reactive intermediate and find the order relation obeyed by the activation energies of the different barriers. The results typically apply to enzymatic catalysis with kinetics of Michaelis-Menten type.

Symmetry relations for trajectories of a Brownian motor

A Brownian motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion. Because of the input energy, Brownian motors function away from thermodynamic equilibrium and concepts such as linear response theory, fluctuation dissipation relations, and detailed balance do not apply. The generalized fluctuation-dissipation relation, however, states that even under strongly thermodynamically nonequilibrium conditions the ratio of the probability of a transition to the probability of the time reverse of that transition is the exponential of the change in the internal energy of the system due to the transition. Here, we derive an extension of the generalized fluctuation dissipation theorem for a Brownian motor for the ratio between the probability for the motor to take a forward step and the probability to take a backward step.

Equilibrium theory for a particle pulled by a moving optical trap

The viscous drag on a colloidal particle pulled through solution by an optical trap is large enough that on experimentally relevant time scales the mechanical force exerted by the trap is equal and opposite the viscous drag force. The rapid mechanical equilibration allows the system to be modeled using equilibrium theory where the effects of the energy dissipation (thermodynamic disequilibrium) show up only in the coordinate transformations that map the system from the laboratory frame of reference, relative to which the particle is moving, to a frame of reference in which the particle is, on average, stationary and on which the stochastic dynamics is governed by a canonical equilibrium distribution function. The simple equations in the stationary frame can be analyzed using the Onsager-Machlup theory for stochastic systems and provide generalizations of equilibrium and near equilibrium concepts such as detailed balance and fluctuation-dissipation relations applicable to a wide range of systems including molecular motors, pumps, and other nanoscale machines.

Resonance of relaxation time in the temperature modulated Schlögl model

We show the possibility to accelerate-in a resonant way-a nonlinear chemical reaction by imposing a small temperature modulation. This classical resonance, which happens for particular modulation frequencies, is illustrated on the athermic cubic Schlogl model, which allows us to get analytical expressions for both the reaction relaxation time and the frequency-resonant delay.

Design principles for Brownian molecular machines: how to swim in molasses and walk in a hurricane

Protein molecular motors-perfected over the course of millions of years of evolution-play an essential role in moving and assembling biological structures. Recently chemists have been able to synthesize molecules that emulate in part the remarkable capabilities of these biomolecular motors (for extensive reviews see the recent papers: E. R. Kay, D. A. Leigh and F. Zerbetto, Angew. Chem., Int. Ed., 2006, 46, 72-191; W. R. Browne and B. L. Feringa, Nat. Nanotechnol., 2006, 1, 25-35; M. N. Chatterjee, E. R. Kay and D. A. Leigh, J. Am. Chem. Soc., 2006, 128, 4058-4073; G. S. Kottas, L. I. Clarke, D. Horinek and J. Michl, Chem. Rev., 2005, 105, 1281-1376; M. A. Garcia-Garibay, Proc. Natl. Acad. Sci., U. S. A., 2005, 102, 10771-10776)). Like their biological counterparts, many of these synthetic machines function in an environment where viscous forces dominate inertia-to move they must "swim in molasses". Further, the thermal noise power exchanged reversibly between the motor and its environment is many orders of magnitude greater than the power provided by the chemical fuel to drive directed motion. One might think that moving in a specific direction would be as difficult as walking in a hurricane. Yet biomolecular motors (and increasingly, synthetic motors) move and accomplish their function with almost deterministic precision. In this Perspective we will investigate the physical principles that govern nanoscale systems at the single molecule level and how these principles can be useful in designing synthetic molecular machines.

Harmonic generation by yeast cells in response to low-frequency electric fields

We report on harmonic generation by budding yeast cells (Saccharomyces cerevisiae, 10(8) cells/ml) in response to sinusoidal electric fields with amplitudes ranging from zero to 5 V/cm in the frequency range 10-300 Hz. The cell-generated harmonics are found to exhibit strong amplitude and frequency dependence. Sodium metavanadate, an inhibitor of the proton pump known as H+-ATPase, and glucose, a substrate of H+-ATPase, are found to increase harmonic production at low amplitudes while reducing it at large amplitudes. This P-type proton pump can be driven by an oscillatory transmembrane potential, and its nonlinear response is believed to be largely responsible for harmonic production at low frequencies in yeast cells. We find that the observed harmonics show dramatic changes with time and in their field and frequency dependence after perturbing the system by adding an inhibitor, substrate, or membrane depolarizer to the cell suspension.

Observations of the Effect of Anionic, Cationic, Neutral, and Zwitterionic Surfactants on the Belousov−Zhabotinsky Reaction

In this paper we report the experimental observations of the effects of various surfactants on the oscillations of the ferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction. The oscillations are followed by observing the change in absorbance at 510 nm due to ferroin in a well-stirred closed BZ reacting system. We have used sodium dodecyl sulfate (SDS) as the anionic surfactant, cetyl trimethylammonium bromide (CTAB) as the cationic surfactant, Triton X-100 as the neutral surfactant, and 3-[(3-cholamidopropyl)dimethylammonio)]-1-propanesulfonate (CHAPS) as the zwitterionic surfactant. In general, we observed that there is a change in the oscillation behavior in the presence of each of these surfactants above their critical micellar concentrations. For different surfactants, the time-dependent evolution of the oscillations is found to be characteristic of the surfactant. The results of our study suggest that the evolution of oscillations is most regular in the presence of micelles of SDS.

Chemical peristalsis

Molecules that emulate in part the remarkable capabilities of protein motors were recently chemically synthesized. A promising approach is based on physically interlocked macromolecular complexes such as rotaxanes and catenanes. Using the latter, Leigh et al. [Leigh, D. A., Wong, J. K. Y., Dehez, F. & Zerbetto, F. (2003) Nature 424, 174-179] constructed a molecular rotor in which two small rings are induced by pulses of light to move unidirectionally around a third, larger ring. The mechanism is similar to that by which a peristaltic pump operates. Unlike macroscopic peristalsis, however, in which a traveling wave forces material through a series of one-way valves, the chemical peristaltic mechanism does not directly cause the small rings to move but only alters the energetics, with the motion itself arising by thermal activation over energy barriers. Engines operating by this mechanism are "Brownian" motors. Here we describe a minimal two-state mechanism for a catenane-based molecular motor. Although fluctuations caused by equilibrium processes cannot drive directed motion, nonequilibrium fluctuations, whether generated externally or by a far-from-equilibrium chemical reaction, can drive rotation even against an external torque. We discuss a possible architecture for input and output of information and energy between the motor and its environment and give a simple expression for the maximum thermodynamic efficiency. The proposed Brownian motor mechanism is consistent with the high efficiency observed by Yasuda et al. [Yasuda, Y., Noji, H., Kinoshita, K. & Yoshida, M. (1998) Cell 93, 1117-1124] for the F(1)-ATP synthase operating as an ATP-powered molecular rotor.

Signal processing times in neutrophil activation: dependence on ligand concentration and the relative phase of metabolic oscillations

Intracellular NAD(P)H oscillations exhibited by polarized neutrophils display congruent with 20 s periods, which are halved to congruent with 10 s upon stimulation with chemotactic peptides such as FNLPNTL (N-formyl-nle-leu-phe-nle-tyr-lys). By monitoring this frequency change, we have measured accurately the time interval between stimulus and metabolic frequency changes. A microscope flow chamber was designed to allow rapid delivery of FNLPNTL to adherent cells. Using fluorescein as a marker, we found delivery to be complete and stable throughout the chamber within approximately 400 ms. Peptides were injected into the chamber at concentrations ranging from 10(-6) to 10(-9) M. Injections also varied with respect to the relative phase of a cell's NAD(P)H oscillations. The time interval between injection of 10(-6) M FNLPNTL and the acquisition of congruent with 10 s period metabolic oscillations was found to be 12.2+/-3.3 s when injections occurred at the NAD(P)H oscillation peak whereas the lag time was 22.5+/-4.8 s when coinciding with a trough. At 10(-8) M FNLPNTL, lag times were found to be 26.1+/-5.2 and 30.5+/-7.3 s for injections at NAD(P)H peaks and troughs, respectively. FNLPNTL at 10(-9) M had no effect on metabolic oscillations, consistent with previous studies. Our experiments show that the kinetics of transmembrane signal processing, in contrast to a simple transmembrane chemical reaction, can depend upon both ligand dose and its temporal relationship with intracellular metabolic oscillations.

Flashing ratchet model with high efficiency

As a simple model of the Brownian motor, we consider hopping motion of a particle in a periodic asymmetric double-well potential which randomly switches between two states. The potential profiles of the states are identical but shifted by half a period. The current and the efficiency are explicitly calculated as functions of the parameters of the model, including also a load force. Such a flashing ratchet is shown to be particularly efficient, with the efficiency tending to unity when the highest peak of the potential is high enough to suppress the backward motion.

Stochastic resonance in a biological motor under complex fluctuations

The dynamics division approach proposed in this work enables us to handle dynamical equations with complex fluctuations. A Brownian motor with cyclic conformational changes is analyzed to understand effects of noise on its signal transduction, and on condition in which stochastic resonance may take place. The result reproduces several features of the experimental data on the electric activation of ion pumping by Na, K-ATPase.

Minimal Brownian ratchet: an exactly solvable model

We develop an analytically solvable three-state discrete-time minimal Brownian ratchet (MBR), where the transition probabilities between states are asymmetric. By solving the master equations, we obtain the steady-state probabilities. Generally, the steady-state solution does not display detailed balance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensional parameter space, we find the null curve on which the net current vanishes and detailed balance holds. A system on this curve is said to be balanced. On the null curve, an additional source of external random noise is introduced to show that a directional motion can be induced under the zero overall driving force.

Spatial distribution of ion channel activity in biological membranes: the role of noise

A new approach is proposed to model a collective ion channel dynamics. We have assumed that ion channels create a two-component spatio-temporal interaction field. Every channel at its current spatial location in membrane contributes permanently to this field with its state (open or closed) and coupling strength to other channels. This field is described by a reaction-diffusion equation, the transition of ion channel from closed to open state (and vice versa) is described by a master equation, and migration of channels in membrane is described by a set of Langevin equations coupled by the interaction field. Within this model, we have investigated critical conditions for spatial distribution of ion channel activity.

Constructive role of noise in signal transmissions by biomembrane proteins

We discuss new examples of the constructive role of environmental fluctuations in biophysical processes, namely quantitative enhancement and qualitative sharpening of the outgoing signal in the intercellular signal transduction, e.g., in the synaptic links. An experimental check in a chemical flow reactor is suggested.

Correlated conformational fluctuations during enzymatic catalysis: Implications for catalytic rate enhancement

Correlated enzymatic conformational fluctuations are shown to contribute to the rate of enhancement achieved during catalysis. Cytidine deaminase serves as a model system. Crystallographic temperature factor data for this enzyme complexed with substrate analog, transition-state analog, and product are available, thereby establishing a measure of atomic scale conformational fluctuations along the (approximate) reaction coordinate. First, a neural network-based algorithm is used to visualize the decreased conformational fluctuations at the transition state. Second, a dynamic diffusion equation along the reaction coordinate is solved and shows that the flux velocity through the associated enzymatic conformation space is greatest at the transition state. These results suggest (1) that there are both dynamic and energetic restrictions to conformational fluctuations at the transition state, (2) that enzymatic catalysis occurs on a fluctuating potential energy surface, and (3) a form for the potential energy. The Michaelis-Menten equations are modified to describe catalysis on this fluctuating potential energy profile, leading to enhanced catalytic rates when fluctuations along the reaction coordinate are appropriately correlated. This represents a dynamic tuning of the enzyme for maximally effective transformation of the ES complex into EP.

Towards a chemically driven molecular electron pump

Charge can be pumped through a tiny gated portal from a reservoir at low electrochemical potential to one at the same or higher electrochemical potential by cyclically modulating the portal and gate energies. A theoretically and experimentally well established mechanism is Thouless adiabatic pumping, achieved by a precisely timed out-of-phase modulation of at least two parameters of the system. Here we show that stochastic modulation between two configurations of gate and portal energies can drive efficient pumping by a different, nonadiabatic, mechanism that may provide a basis for chemically driven electron pumping through a molecular wire.

How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment

Of all the lifeforms that obtain their energy from glycolysis, yeast cells are among the most basic. Under certain conditions the concentrations of the glycolytic intermediates in yeast cells can oscillate. Individual yeast cells in a suspension can synchronize their oscillations to get in phase with each other. Although the glycolytic oscillations originate in the upper part of the glycolytic chain, the signaling agent in this synchronization appears to be acetaldehyde, a membrane-permeating metabolite at the bottom of the anaerobic part of the glycolytic chain. Here we address the issue of how a metabolite remote from the pacemaking origin of the oscillation may nevertheless control the synchronization. We present a quantitative model for glycolytic oscillations and their synchronization in terms of chemical kinetics. We show that, in essence, the common acetaldehyde concentration can be modeled as a small perturbation on the "pacemaker" whose effect on the period of the oscillations of cells in the same suspension is indeed such that a synchronization develops.