Motor protein with nonequilibrium potential: Its thermodynamics and efficiency

Application of Advanced Light Microscopy to the Study of HIV and Its Interactions with the Host

This review highlights the significant observations of human immunodeficiency virus (HIV) assembly, release and maturation made possible with advanced light microscopy techniques. The advances in technology which now enables these light microscopy measurements are discussed with special emphasis on live imaging approaches including Total Internal Reflection Fluorescence (TIRF), high-resolution light microscopy techniques including PALM and STORM and single molecule measurements, including Fluorescence Resonance Energy Transfer (FRET). The review concludes with a discussion on what new insights and understanding can be expected from these measurements.

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.

Energetic Costs, Precision, and Transport Efficiency of Molecular Motors

An efficient molecular motor would deliver cargo to the target site at a high speed and in a punctual manner while consuming a minimal amount of energy. According to a recently formulated thermodynamic principle, referred to as the thermodynamic uncertainty relation, the travel distance of a motor and its variance are, however, constrained by the free energy being consumed. Here we use the principle underlying the uncertainty relation to quantify the transport efficiency of molecular motors for varying ATP concentration ([ATP]) and applied load (f). Our analyses of experimental data find that transport efficiencies of the motors studied here are semioptimized under the cellular condition. The efficiency is significantly deteriorated for a kinesin-1 mutant that has a longer neck-linker, which underscores the importance of molecular structure. It is remarkable to recognize that, among many possible directions for optimization, biological motors have evolved to optimize the transport efficiency in particular.

Allocating and Splitting Free Energy to Maximize Molecular Machine Flux

Biomolecular machines transduce between different forms of energy. These machines make directed progress and increase their speed by consuming free energy, typically in the form of nonequilibrium chemical concentrations. Machine dynamics are often modeled by transitions between a set of discrete metastable conformational states. In general, the free-energy change associated with each transition can increase the forward rate constant, decrease the reverse rate constant, or both. In contrast to previous optimizations, we find that in general flux is maximized neither by devoting all free-energy changes to increasing forward rate constants nor by solely decreasing reverse rate constants. Instead, the optimal free-energy splitting depends on the detailed dynamics. Extending our analysis to machines with vulnerable states (from which they can break down), in the strong driving corresponding to in vivo cellular conditions, processivity is maximized by reducing the occupation of the vulnerable state.

Quantifying the Heat Dissipation from a Molecular Motor's Transport Properties in Nonequilibrium Steady States

Theoretical analysis, which maps single-molecule time trajectories of a molecular motor onto unicyclic Markov processes, allows us to evaluate the heat dissipated from the motor and to elucidate its dependence on the mean velocity and diffusivity. Unlike passive Brownian particles in equilibrium, the velocity and diffusion constant of molecular motors are closely inter-related. In particular, our study makes it clear that the increase of diffusivity with the heat production is a natural outcome of active particles, which is reminiscent of the recent experimental premise that the diffusion of an exothermic enzyme is enhanced by the heat released from its own catalytic turnover. Compared with freely diffusing exothermic enzymes, kinesin-1, whose dynamics is confined on one-dimensional tracks, is highly efficient in transforming conformational fluctuations into a locally directed motion, thus displaying a significantly higher enhancement in diffusivity with its turnover rate. Putting molecular motors and freely diffusing enzymes on an equal footing, our study offers a thermodynamic basis to understand the heat-enhanced self-diffusion of exothermic enzymes.

Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces

Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this context is elucidated. Chemomechanical, thermomechanical, and enzyme-catalyzed thermochemical energy transduction processes are discussed. It is argued that mesoscopic stochastic NET in phase space provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology. This theory is also relevant for nanoscale technological advances.

Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether

Here we generalize our previous model of molecular motors trafficking subdiffusing cargos in viscoelastic cytosol by (i) including mechano-chemical coupling between cyclic conformational fluctuations of the motor protein driven by the reaction of ATP hydrolysis and its translational motion within the simplest two-state model of hand-over-hand motion of kinesin, and also (ii) by taking into account the anharmonicity of the tether between the motor and the cargo (its maximally possible extension length). It is shown that the major earlier results such as occurrence of normal versus anomalous transport depending on the amplitude of binding potential, cargo size and the motor turnover frequency not only survive in this more realistic model, but the results also look very similar for the correspondingly adjusted parameters. However, this more realistic model displays a substantially larger thermodynamic efficiency due to a bidirectional mechano-chemical coupling. For realistic parameters, the maximal thermodynamic efficiency can transiently be about 50% as observed for kinesins, and even larger, surprisingly also in a novel strongly anomalous (sub)transport regime, where the motor enzymatic turnovers become also anomalously slow and cannot be characterized by a turnover rate. Here anomalously slow dynamics of the cargo enforces anomalously slow cyclic kinetics of the motor protein.

Entropy-generated power and its efficiency

We propose a simple and analytically solvable model for a motor that generates mechanical motion by exploiting an entropic force arising from the topology of the underlying phase space. We show that the generation of mechanical forces in our system is surprisingly robust to local changes in kinetic and topological parameters. Furthermore, we find that the efficiency at maximum power of the motor may show discontinuities.

Dissipation, generalized free energy, and a self-consistent nonequilibrium thermodynamics of chemically driven open subsystems

Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe." A question remains with regard to what the minimally required description for the surrounding of such an open driven system is so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system or in a quasisteady state (QSS) with an excess amount of adenosine triphosphate (ATP), adenosine diphosphate (ADP), and inorganic phosphate (Pi). We introduce the key notion of minimal work that is needed, W(min), for the external regenerating system to sustain a NESS (e.g., maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the second law in terms of the same positive entropy production rate. The heat dissipation of a NESS without mechanical output is exactly the W(min). This provides a justification for introducing an ideal external regenerating system and yields a free-energy balance equation between the net free-energy input F(in) and total dissipation F(dis) in an NESS: F(in) consists of chemical input minus mechanical output; F(dis) consists of dissipative heat, i.e. the amount of useful energy becoming heat, which also equals the NESS entropy production. Furthermore, we show that for nonstationary systems, the F(dis) and F(in) correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics and identify a relative entropy H as a generalized free energy. We reach a new formulation of Markovian nonequilibrium thermodynamics based on only the internal kinetic equation without further reference to the intrinsic degree of freedom within each Markov state. It includes an extended free-energy balance and a second law which are valid for driven stochastic dynamics with an ideal external regenerating system. Our result suggests new ingredients for a generalized thermodynamics of self-organization in driven systems.

Cooperative rectification in confined Brownian ratchets

We analyze the rectified motion of a Brownian particle in a confined environment. We show the emergence of strong cooperativity between the inherent rectification of the ratchet mechanism and the entropic bias of the fluctuations caused by spatial confinement. Net particle transport may develop even in situations where separately the ratchet and the geometric restrictions do not give rise to particle motion. The combined rectification effects can lead to bidirectional transport depending on particle size, resulting in a different route for segregation. The reported mechanism can be used to control transport in mesostructures and nanodevices in which particles move in a reduced space.

Optimality and thermodynamics determine the evolution of transcriptional regulatory networks

Transcriptional motifs are small regulatory interaction patterns that regulate biological functions in highly-interacting cellular networks. Recently, attempts have been made to explain the significance of transcriptional motifs through dynamic function. However, fundamental questions remain unanswered. Why are certain transcriptional motifs with similar dynamic function abundant while others occur rarely? What are the criteria for topological generalization of these motifs into complex networks? Here, we present a novel paradigm that combines non-equilibrium thermodynamics with multiobjective-optimality for network analysis. We found that energetic cost, defined herein as specific dissipation energy, is minimal at the optimal environmental conditions and it correlates inversely with the abundance of the network motifs obtained experimentally for E. coli and S. cerevisiae. This yields evidence that dissipative energetics is the underlying criteria used during evolution for motif selection and that biological systems during transcription tend towards evolutionary selection of subgraphs which produces minimum specific heat dissipation under optimal conditions, thereby explaining the abundance/rare occurrence of some motifs. We show that although certain motifs had similar dynamical functionality, they had significantly different energetic cost, thus explaining the abundance/rare occurrence of these motifs. The presented insights may establish global thermodynamic analysis as a backbone in designing and understanding complex networks systems, such as metabolic and protein interaction networks.

Interacting molecular motors: efficiency and work fluctuations

We investigate the model of "reversible ratchet" with interacting particles, presented by us earlier [F. Slanina, EPL 84, 50009 (2008)]. We further clarify the effect of efficiency enhancement due to interaction and show that it is of energetic origin, rather than a consequence of reduced fluctuations. We also show complicated structures emerging in the interaction and density dependence of the current and response function. The fluctuation properties of the work and input energy indicate in detail the far-from-equilibrium nature of the dynamics.

Relaxation dynamics near nonequilibrium stationary states in Brownian ratchets

A comprehensive study of the static and dynamical properties of a representative stochastic model of Brownian ratchet effects for molecular motors is reported. The model describes Brownian motions on two periodic potentials under static and time-dependent forces, where there are two distinct locations of chemical reactions coupling the levels with reversible rates within a period. Complete stationary properties have been obtained analytically for arbitrary potentials under external force. Dynamical relaxation properties near nonequilibrium stationary states were examined by considering the response function of velocity upon time-dependent external force, expressed in terms of the conditional probability density of the model. The latter is fully calculated using a systematic numerical method using matrix diagonalization, which is easily generalized to more complicated models for studying both static and dynamical properties. The behavior of the time-dependent response examined for model potentials suggests that the characteristic relaxation time near stationary states generally decreases linearly with respect to increasing velocity as one goes away from equilibrium via an increase in chemical potential of fuel species, a prediction testable in single molecule experiments.

Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations

Gibbsian equilibrium statistical thermodynamics is the theoretical foundation for isothermal, closed chemical, and biochemical reaction systems. This theory, however, is not applicable to most biochemical reactions in living cells, which exhibit a range of interesting phenomena such as free energy transduction, temporal and spatial complexity, and kinetic proofreading. In this article, a nonequilibrium statistical thermodynamic theory based on stochastic kinetics is introduced, mainly through a series of examples: single-molecule enzyme kinetics, nonlinear chemical oscillation, molecular motor, biochemical switch, and specificity amplification. The case studies illustrate an emerging theory for the isothermal nonequilibrium steady state of open systems.

Phosphorylation energy hypothesis: open chemical systems and their biological functions

Biochemical systems and processes in living cells generally operate far from equilibrium. This review presents an overview of a statistical thermodynamic treatment for such systems, with examples from several key components in cellular signal transduction. Open-system nonequilibrium steady-state (NESS) models are introduced. The models account quantitatively for the energetics and thermodynamics in phosphorylation-dephosphorylation switches, GTPase timers, and specificity amplification through kinetic proofreading. The chemical energy derived from ATP and GTP hydrolysis establishes the NESS of a cell and makes the cell--a mesoscopic-biochemical reaction system that consists of a collection of thermally driven fluctuating macromolecules--a genetically programmed chemical machine.

Nonequilibrium potential function of chemically driven single macromolecules via Jarzynski-type Log-Mean-Exponential Heat

Applying the method from recently developed fluctuation theorems to the stochastic dynamics of single macromolecules in ambient fluid at constant temperature, we establish two Jarzynski-type equalities: (1) between the log-mean-exponential (LME) of the irreversible heat dissiption of a driven molecule in nonequilibrium steady-state (NESS) and ln P(ness)(x) and (2) between the LME of the work done by the internal force of the molecule and nonequilibrium chemical potential function mu(ness)(x) identical with U(x) + k(B)T ln P(ness)(x), where P(ness)(x) is the NESS probability density in the phase space of the macromolecule and U(x) is its internal potential function. Psi = integral mu(ness)(x) P(ness)(x) dx is shown to be a nonequilibrium generalization of the Helmholtz free energy and DeltaPsi = DeltaU - TDeltaS for nonequilibrium processes, where S = - kB integralP(x) ln P(x) dx is the Gibbs entropy associated with P(x). LME of heat dissipation generalizes the concept of entropy, and the equalities define thermodynamic potential functions for open systems far from equilibrium.

Kinesin's biased stepping mechanism: amplification of neck linker zippering

A physically motivated model of kinesin's motor function is developed within the framework of rectified Brownian motion. The model explains how the amplification of neck linker zippering arises naturally through well-known formulae for overdamped dynamics, thereby providing a means to understand how weakly-favorable zippering leads to strongly favorable plus-directed binding of a free kinesin head to microtubule. Additional aspects of kinesin's motion, such as head coordination and rate-limiting steps, are directly related to the force-dependent inhibition of ATP binding to a microtubule bound head. The model of rectified Brownian motion is presented as an alternative to power stroke models and provides an alternative interpretation for the significance of ATP hydrolysis in the kinesin stepping cycle.

Cycle kinetics, steady state thermodynamics and motors-a paradigm for living matter physics

An integration of the stochastic mathematical models for motor proteins with Hill's steady state thermodynamics yields a rather comprehensive theory for molecular motors as open systems in the nonequilibrium steady state. This theory, a natural extension of Gibbs' approach to isothermal molecular systems in equilibrium, is compared with other existing theories with dissipative structures and dynamics. The theory of molecular motors might be considered as an archetype for studying more complex open biological systems such as biochemical reaction networks inside living cells.

Analytical theory of the stochastic dynamics of the power stroke in nonprocessive motor proteins

Statistical distributions of the structural states of individual molecules of nonprocessive motor complexes such as actomyosins are examined theoretically by considering a two-state stochastic model coupled by chemical reactions along the reaction coordinate representing the internal conformational states of the motor. The use of a conformational reaction coordinate allows for the approximation of taking the rate constants as local in their dependence on the reaction coordinate, and yields a simple analytic solution of the stationary states. The approximation is also tested against numerical solutions with a nonlocal form of rate constants. The theory is well-suited for computational treatments based on atomic structures of protein constituents using free energy molecular dynamics simulations. With empirical sets of free energy functions, stationary distributions of forces exerted by a motor head compare well with known experimental data.