Table of contents

Macro-scale thermodynamic engines convert the random motion of fuel-produced heat into directed motion. Such engines cannot be downsized to the nanometre scale, because thermodynamics does not apply to single atoms or molecules, only large assemblies of them. A great challenge for the field of nanotechnology is the design and construction of microscopic motors that can transform input energy into directed motion and perform useful functions such as transporting of cargo. Today's nanotechnologists can only look in envy at the biological world, where molecular motors of various kinds (linear, rotary) are very common and fulfil essential roles.

Inspired by the fascinating mechanism by which proteins move in the presence of thermal noise, many physicists have been trying to establish novel concepts and strategies that might lead to the construction of man-made motors and machines on mesoscopic to molecular scales. Operating far from thermal equilibrium, molecular motors successfully combine noise and space-time asymmetry to generate useful functions such as transport, pumping, separation or segregation of particles. Such man-made molecular machinery, when realized, will not only be able to perform useful tasks on the atomic and molecular scales, but will also provide fundamentally new ways to manipulate molecules and nanoscale objects. Various mechanisms suitable for converting supplied energy into directed motion are discussed in this special issue of Journal of Physics: Condensed Matter. An important problem that has been raised in this issue, and has still to be resolved, concerns the possibility of controlling induced motion. In particular, a major problem is that of resolving the contradiction between the fascinating idea of feeding the energy by a driving random motion, and yet being able to control that motion; for example: starting the motion, stopping it, changing the velocity, and so on.

This special issue aims to provide an overview of current theoretical and experimental works on molecular motors and possible applications. In selecting the papers we have tried to maintain a balance between new results and review-like aspects, such that the present issue is self-contained and, we hope, readily accessible to non-specialists in the field. We believe that the particular appeal of this collection of papers also lies in the fusion of both experiment and theory, thus providing the connection to reality of the sometimes demanding, mathematically inclined contributions.

Profound thanks go to all our colleagues and friends who have contributed to this special issue. Each has made an effort not only to present recent results in a clear and lucid way, but also to provide an introductory review that helps the reader to understand the different topics.

We study the motion of Brownian steppers, which are objects moving unidirectionally by discrete steps. A single step is composed of two processes. An activation process describing the random attachment of a fuel molecule is followed by a conformational change of the stepper, leading to the forward motion. Whereas activation is given by a Markovian rate process, the forward motion is defined by a gamma distribution. In this paper we propose a periodic modulation of the fuel concentration in order to control the random motion of the stepper. We show that the driving may reduce the fluctuations of the stepper. Corresponding minima of the diffusion coefficient and maxima of the Péclet number prove the regularity of the motion.

Recently, a thermal Brownian motor was introduced (Van den Broeck et al 2004 Phys. Rev. Lett. 93 090601), for which an exact microscopic analysis is possible. The purpose of this paper is to review some further properties of this construction, and to discuss in particular specific issues including the relation with macroscopic response and the efficiency at maximum power.

The ability to rectify Brownian forces with spatially extended time-varying light fields creates new opportunities for studying the statistical properties of thermal ratchet models and to exploit these models' interesting and useful properties for practical applications. This paper describes experimental studies of one-dimensional thermal ratchets implemented with the holographic optical trapping technique applied to fluid-borne colloidal spheres. These studies demonstrate the complementary roles of global spatiotemporal symmetry and local dynamics in establishing the direction of ratchet-induced motion and also highlight avenues for future advances in higher-dimensional systems.

The dynamics of a classical particle in a one-dimensional, randomly driven potential is analysed both analytically and numerically. The potential considered here is composed of two identical spatially periodic saw-tooth-like components, one of which is externally driven by a random force. We show that under certain conditions the particle may travel against the averaged external force, performing a saltatory unidirectional drift with a constant velocity. Such a behaviour persists also in situations when the external force averages out to zero. We demonstrate that the physics behind this phenomenon stems from a particular behaviour of fluctuations in random force: upon reaching a certain level, random fluctuations exercise a locking function creating points of irreversibility which the particle cannot overpass. Repeated (randomly) in each cycle, this results in a saltatory unidirectional drift. This mechanism resembles the work of an escapement-type device in watches. Considering the overdamped limit, we propose simple analytical estimates for the particle's terminal velocity.

Transport in a one-dimensional symmetric device can be activated by the combination of thermal noise and a bi-harmonic drive. The results of extensive simulations allow us to distinguish between two apparently different bi-harmonic regimes: (i) harmonic mixing, where the two drive frequencies are commensurate but not too high; (ii) vibrational mixing, where one harmonic drive component possesses a high frequency but finite amplitude-to-frequency ratio. A comparison with the earlier theoretical predictions shows that at present the analytical understanding of nonlinear frequency mixing is still not satisfactory.

Deterministic ratchets, in the inertial and also in the overdamped limit, have a very complex dynamics, including chaotic motion. This deterministically induced chaos mimics, to some extent, the role of noise, changing, on the other hand, some of the basic properties of thermal ratchets; for example, inertial ratchets can exhibit multiple reversals in the current direction. The direction depends on the amount of friction and inertia, which makes it especially interesting for technological applications such as biological particle separation. We overview in this work different strategies to control the current of inertial ratchets. The control parameters analysed are the strength and frequency of the periodic external force, the strength of the quenched noise that models a non-perfectly-periodic potential, and the mass of the particles. Control mechanisms are associated with the fractal nature of the basins of attraction of the mean velocity attractors. The control of the overdamped motion of noninteracting particles in a rocking periodic asymmetric potential is also reviewed. The analysis is focused on synchronization of the motion of the particles with the external sinusoidal driving force. Two cases are considered: a perfect lattice without disorder and a lattice with noncorrelated quenched noise. The amplitude of the driving force and the strength of the quenched noise are used as control parameters.

In order to optimize the directed motion of an inertial Brownian motor, we identify the operating conditions that both maximize the motor current and minimize its dispersion. Extensive numerical simulation of an inertial rocked ratchet displays that two quantifiers, namely the energetic efficiency and the Péclet number (or equivalently the Fano factor), suffice to determine the regimes of optimal transport. The effective diffusion of this rocked inertial Brownian motor can be expressed as a generalized fluctuation theorem of the Green–Kubo type.

Biomolecular motors are often described in mechanical terms, with analogy to cars, turbines, judo throws, levers, etc. It is important to remember however that because of their small size, and because of the aqueous environment in which molecular motors move, viscous drag and thermal noise dominate the inertial forces that drive macroscopic machines. The sequence of motions—conformational changes—by which a motor protein moves can best be described as a random walk, with transitions from one state to another occurring by thermal activation over energy barriers. In this paper I will address the question of how this random walk is biased by a non-equilibrium chemical reaction (ATP hydrolysis) so that the motor molecule moves preferentially (with almost unit certainty) in one direction, even when an external force is applied to drive it in the opposite direction. I will also discuss how these 'soft matter' motors can achieve thermodynamic efficiencies of nearly 100%.

This work seeks to apply the laser optimal control technique to light-driven molecular motors. Taking a recently proposed molecular locomotive as a model system, a control loop is developed specifically for it, and concrete schemes for experimentally closing the loop are devised. A list of unique control objectives is rigorously formulated from the nanomachinery perspective, and corresponding optimization is made feasible by an innovative application of the established technique of closed-loop learning control. The optimization may be pursued for individual laser operational steps as well as for the overall nanolocomotion performance of the entire work cycle. The locomotive optimal control, capable of co-adapting the laser procedure and the periodically driven molecular dynamics, essentially leads to an optimally performing optomechanical work cycle for the locomotive beyond any model-based pre-designed version. These findings reveal a great potential of laser optimally controlled nanowork cycles in the emerging field of nanomachinery.

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.

We discuss different quantifiers of stochastic resonance (SR) and how far they are mathematically related with each other. Specifically, we address bona fide SR in terms of the areas of the hysteresis loops and of the first peaks in the residence time distributions. We demonstrate a surprisingly good agreement of these two SR quantifiers experimentally for colloidal particles in periodically modulated laser traps. A simple theoretical model is established, which reproduces the experimental observations very well.

Molecular motors are enzymatic proteins that couple the consumption of chemical energy to mechanical displacement. In order to elucidate the translocation mechanisms of these enzymes, it is of fundamental importance to measure the physical step size. The step size can, in certain instances, be directly measured with single-molecule techniques; however, in the majority of cases individual steps are masked by noise. The step size can nevertheless be obtained from noisy single-molecule records through statistical methods. This analysis is analogous to determining the charge of the electron from current shot noise. We review methods for obtaining the step size based on analysing, in both the time and frequency domains, the variance in position from noisy single-molecule records of motor displacement. Additionally, we demonstrate how similar methods may be applied to measure the step size in bulk kinetic experiments.

Individual processive molecular motors, of which conventional kinesin is the most studied quantitatively, move along polar molecular tracks and, by exerting a force F = (F_x,F_y,F_z) on a tether, drag cellular cargoes, in vivo, or spherical beads, in vitro, taking up to hundreds of nanometre-scale steps. From observations of velocities and the dispersion of displacements with time, under measured forces and controlled fuel supply (typically ATP), one may hope to obtain insight into the molecular motions undergone in the individual steps. In the simplest situation, the load force F may be regarded as a scalar resisting force, F_x<0, acting parallel to the track: however, experiments, originally by Gittes et al (1996 Biophys. J.70 418), have imposed perpendicular (or vertical) loads, F_z>0, while more recently Block and co-workers (2002 Biophys. J.83 491, 2003 Proc. Natl Acad. Sci. USA100 2351) and Carter and Cross (2005 Nature435 308) have studied assisting (or reverse) loads, F_x>0, and also sideways (or transverse) loads .

We extend previous mechanochemical kinetic models by explicitly implementing a free-energy landscape picture in order to allow for the full vectorial nature of the force F transmitted by the tether. The load-dependence of the various forward and reverse transition rates is embodied in load distribution vectors, and , which relate to substeps of the motor, and in next order, in compliance matrices and . The approach is applied specifically to discuss the experiments of Howard and co-workers (1996 Biophys. J.70 418) in which the buckling of partially clamped microtubules was measured under the action of bound kinesin molecules which induced determined perpendicular loads. But in the normal single-bead assay it also proves imperative to allow for F_z>0: the appropriate analysis for kinesin, suggesting that the motor 'crouches' on binding ATP prior to stepping, is sketched. It yields an expression for the velocity, V (F_x,F_z;[ATP]), needed to address the buckling experiments.

The traffic of molecular motors which interact through mutual exclusion is studied theoretically for half-open tube-like compartments. These half-open tubes mimic the shapes of axons. The mutual exclusion leads to traffic jams or density plateaus on the filaments. A phase transition is obtained when the motor velocity changes sign. We identify the relevant length scales and characterize the jamming behaviour using both analytical approximations and Monte Carlo simulations of lattice models.

Helicases are molecular motors which unwind double-stranded nucleic acids (dsNA) in cells. Many helicases move with directional bias on single-stranded (ss) nucleic acids, and couple their directional translocation to strand separation. A model of the coupling between translocation and unwinding uses an interaction potential to represent passive and active helicase mechanisms. A passive helicase must wait for thermal fluctuations to open dsNA base pairs before it can advance and inhibit NA closing. An active helicase directly destabilizes dsNA base pairs, accelerating the opening rate. Here we extend this model to include helicase unbinding from the nucleic-acid strand. The helicase processivity depends on the form of the interaction potential. A passive helicase has a mean attachment time which does not change between ss translocation and ds unwinding, while an active helicase in general shows a decrease in attachment time during unwinding relative to ss translocation. In addition, we describe how helicase unwinding velocity and processivity vary if the base-pair binding free energy is changed.

The effect of sequence heterogeneity on the dynamics of molecular motors is reviewed and analysed using a set of recently introduced lattice models. First, we review results for the influence of heterogeneous tracks such as a single strand of DNA or RNA on the dynamics of the motors. We stress how the predicted behaviour might be observed experimentally in anomalous drift and diffusion of motors over a wide range of parameters near the stall force and discuss the extreme limit of strongly biased motors with one-way hopping. We then consider the dynamics in an environment containing a variety of different fuels which supply chemical energy for the motor motion, either on a heterogeneous or on a periodic track. The results for motion along a periodic track are relevant to kinesin motors in a solution with a mixture of different nucleotide triphosphate fuel sources.

The dynamics of motor protein molecules consisting of two subunits is investigated using simple discrete stochastic models. Exact steady-state analytical expressions are obtained for velocities and dispersions for any number of intermediate states and conformations between the corresponding binding states of proteins. These models enable us to provide a detailed description and comparison of two different mechanisms of the motion of motor proteins along the linear tracks: the hand-over-hand mechanism, when the motion of subunits alternate; and the inchworm mechanism, when one subunit is always trailing another one. It is shown that the proteins in the hand-over-hand mechanism move faster and fluctuate more than the molecules in the inchworm mechanism. The effect of external forces on dynamic properties of motor proteins is also discussed. Finally, a quantitative method, based on experimental observations for single motor proteins, is proposed for distinguishing between two mechanisms of motion.

Autonomous oscillations in biological systems may have a biochemical origin or result from an interplay between force-generating and visco-elastic elements. In molecular motor assemblies the force-generating elements are molecular engines and the visco-elastic elements are stiff cytoskeletal polymers. The physical mechanism leading to oscillations depends on the particular architecture of the assembly. Existing models can be grouped into two distinct categories: systems with a delayed force activation and anomalous force–velocity relations. We discuss these systems within phase plane analysis known from the theory of dynamic systems and by adopting methods from control theory, the Nyquist stability criterion.

We describe a thermodynamic principle determining the phenomenon of protein self-assembly controlled by elastic stresses. This principle is based on the Gibbs–Dühem-like relationship between the chemical potential of the aggregated molecules and the stresses produced by forces acting on a protein aggregate. We present two biological systems whose operation can be driven by this principle: actin filament, a polymerizing processive capping by proteins of the formin family, and focal adhesions mediating a mechanical link between the cytoskeleton and extracellular substrates. We describe the major phenomenology of these systems and overview recent models, aimed at understanding the mechanisms of their functioning.

A microscopic model is proposed for the motility of a bead driven by the polymerization of actin filaments. The model exhibits a rich spectrum of behaviours similar to those observed in biomimetic experiments, which include spontaneous symmetry-breaking, various regimes of the bead's motion and correlations between the structure of the actin tail which propels the bead and the bead dynamics. The dependences of the dynamical properties (such as symmetry-breaking time, regimes of motion, mean velocity, and tail asymmetry) on the physical parameters (the bead radius and viscosity) agree well with the experimental observations. We find that most experimental observations can be reproduced taking into account only one type of filaments interacting with the bead: the detached filaments that push the bead. Our calculations suggest that the analysis of mean characteristics only (velocities, symmetry-breaking times, etc) does not always provide meaningful information about the mechanism of motility. The aim should be to obtain the corresponding distributions, which might be extremely broad and therefore not well represented by their mean only. Our findings suggest a simple coarse-grained description, which captures the main features obtained within the microscopic model.

We investigate the translocation of a stiff polymer consisting of M monomers through a nanopore in a membrane, in the presence of binding particles (chaperones) that bind onto the polymer, and partially prevent backsliding of the polymer through the pore. The process is characterized by the rates: k for the polymer to make a diffusive jump through the pore, q for unbinding of a chaperone, and the rate qκ for binding (with a binding strength κ); except for the case of no binding κ = 0 the presence of the chaperones gives rise to an effective force that drives the translocation process. In more detail, we develop a dynamical description of the process in terms of a (2+1)-variable master equation for the probability of having m monomers on the target side of the membrane with n bound chaperones at time t. Emphasis is put on the calculation of the mean first passage time as a function of total chain length M. The transfer coefficients in the master equation are determined through detailed balance, and depend on the relative chaperone size λ and binding strength κ, as well as the two rate constants k and q. The ratio γ = q/k between the two rates determines, together with κ and λ, three limiting cases, for which analytic results are derived: (i) for the case of slow binding (), the motion is purely diffusive, and for large M; (ii) for fast binding () but slow unbinding (), the motion is, for small chaperones λ = 1, ratchet-like, and ; (iii) for the case of fast binding and unbinding dynamics ( and ), we perform the adiabatic elimination of the fast variable n, and find that for a very long polymer , but with a smaller prefactor than for ratchet-like dynamics. We solve the general case numerically as a function of the dimensionless parameters λ, κ and γ, and compare to the three limiting cases.

Several DNA nanomotors have been recently constructed in laboratories worldwide. These machines are, however, relatively slow and do not perform continuous rotations. We have recently proposed a rotary DNA nanomachine that shows a continuous rotation with a frequency of 10²–10⁴ Hz. This motor is a closed DNA ring whose elastic features are tuned such that it can be externally driven via e.g. periodic temperature changes. As a result, the twirling ring propels itself through the fluid with a speed of tens of nanometres up to a few microns per second. The current paper gives a more detailed presentation of this motor and provides a derivation of the low- and high-frequency asymptotic behaviour of thermal ratchets in general.

Processive molecular motors act as intracellular transporters of a broad range of cargoes varying from organelles to messenger RNAs. Due to the nanometre range movements and complex dynamics of these motors, highly specialized tools are required to study them, in particular at the single-molecule level. Such tools are what physicists are providing for understanding these biological systems. Fluorescence based real-time localization techniques, with 1 nm spatial resolution and down to 1 ms temporal resolution (FIONA: fluorescence imaging with one-nanometre accuracy), and their applications to a group of molecular motors (myosin V, myosin VI, kinesin, and dynein) are the topics of this paper. In addition to the well established in vitro studies, the recent applications of these techniques to the much more challenging, but also more informative, in vivo realm will be discussed.

Molecular motors operate in an environment dominated by viscous friction and thermal fluctuations. The chemical reaction in a motor may produce an active force at the reaction site to directly move the motor forward. Alternatively a molecular motor may generate a unidirectional motion by rectifying thermal fluctuations using free energy barriers established in the chemical reaction. The reaction cycle has many occupancy states, each having a different effect on the motor motion. The average effect of the chemical reaction on the motor motion can be characterized by the motor potential profile. The biggest advantage of studying the motor potential profile is that it can be reconstructed from the time series of motor positions measured in single-molecule experiments. In this paper, we use the motor potential profile to express the Stokes efficiency as the product of the chemical efficiency and the mechanical efficiency. We show that both the chemical and mechanical efficiencies are bounded by 100% and, thus, are properly defined efficiencies. We discuss implications of high efficiencies for motor mechanisms: a mechanical efficiency close to 100% implies that the motor potential profile is close to a constant slope; a chemical efficiency close to 100% implies that (i) the chemical transitions are not slower than the mechanical motion and (ii) the equilibrium constant of each chemical transition is close to one.

Phospholipases are a class of molecular machines that are involved in the active remodelling processes of biological membranes. These lipases are interfacially activated enzymes and in the specific case of phospholipase A₂ (PLA₂) the enzyme catalyses the hydrolysis of di-acyl phospholipids into products of lysolipids and fatty acids, that dramatically change the physical properties of lipid membrane substrates. Using dissipative particle dynamics simulations on a simple coarse-grained bead–spring model of a fluid lipid bilayer in water, the mechanical and diffusive properties of the bilayer in the pure state and after the action of PLA₂ have been calculated. It is found that, in response to hydrolysis, the lipid membrane becomes mechanically softened and the various in-plane and trans-bilayer diffusional modes become enhanced. The results compare favourably with available experimental data.