Models Of Molecular Motion Research Articles

Bidirectional Molecular Motors by Controlling Threading and Dethreading Pathways of a Linked Rotaxane

AbstractArtificial molecular motors have been presented as models for biological molecular motors. In contrast to the conventional artificial molecular motors that rely on covalent bond rotation, molecular motors with mechanically interlocked molecules (MIMs) have attracted considerable attention owing to their ability to generate significant rotational motion by dynamically shuttling macrocyclic components. The topology of MIM‐type rotational molecular motors is currently limited to catenane structures, which require intricate synthetic procedures that typically produce a low synthetic yield. In this study, we develop a novel class of MIM‐type molecular motors with a rotaxane‐type topology. The switching of the threading/dethreading pathways of the linked rotaxane by protecting/deprotecting the bulky stopper group and changing the solvent polarity enables a net unidirectional rotation of the molecular motor. The threading/dethreading reaction rates were quantitatively evaluated through detailed spectroscopic investigations. Repeated net unidirectional rotation and switching of the direction of rotation were also achieved. Our findings demonstrate that linked rotaxanes can serve as MIM‐type molecular motors with reversible rotational direction controlled by threading/dethreading reactions. These motors hold potential as components of molecular machinery.

Statistical uncertainty principle in Markov kinetics

A reciprocality between the statistical variance of observables of a thermodynamic state and that of their conjugate variables, as entropic forces, originates from the thermodynamic conjugacy with respect to an entropy function. This thermodynamic uncertainty principle in equilibrium can be derived from the Maximum Entropy principle and is independent upon underlying mechanistic details. We present, based on the Maximum Caliber principle as the dynamic generalization of Maximum Entropy, the formalism of the uncertainty principle in kinetics in time homogeneous Markov processes between transitional observables and their conjugate path entropic forces. A stochastic biophysical model for molecular motors is used as an illustrating example. The present work generalizes the phenomenological thermodynamics of uncertainties/fluctuations and is applicable to data ad infinitum.

Bidirectional Molecular Motors by Controlling Threading and Dethreading Pathways of a Linked Rotaxane.

Artificial molecular motors have been presented as models for biological molecular motors. In contrast to the conventional artificial molecular motors that rely on covalent bond rotation, molecular motors with mechanically interlocked molecules (MIM) have attracted considerable attention owing to their ability to generate significant rotational motion by dynamically shuttling macrocyclic components. The topology of MIM-type rotational molecular motors is currently limited to catenane structures, which require intricate synthetic procedures that typically produce a low synthetic yield. In this study, we develop a novel class of MIM-type molecular motors with a rotaxane-type topology. The switching of the threading/dethreading pathways of the linked rotaxane by protecting/deprotecting the bulky stopper group and changing the solvent polarity enables a net unidirectional rotation of the molecular motor. The threading/dethreading reaction rates were quantitatively evaluated through detailed spectroscopic investigations. Repeated net unidirectional rotation and switching of the direction of rotation were also achieved. Our findings demonstrate that linked rotaxanes can serve as MIM-type molecular motors with reversible rotational direction controlled by threading/dethreading reactions. These motors hold potential as components of molecular machinery.

Unlocking the potential of information flow: Maximizing free-energy transduction in a model of an autonomous rotary molecular motor.

Molecular motors fulfill critical functions within all living beings. Understanding their underlying working principles is therefore of great interest. Here we develop a simple model inspired by the two-component biomolecular motor F_{o}-F_{1} ATP synthase. We analyze its energetics and characterize information flows between the machine's components. At maximum output power we find that information transduction plays a minor role for free-energy transduction. However, when the two components are coupled to different environments (e.g., when in contact with heat baths at different temperatures), we show that information flow becomes a resource worth exploiting to maximize free-energy transduction. Our findings suggest that real-world powerful and efficient information engines could be found in machines whose components are subjected to fluctuations of different strength, since in this situation the benefit gained from using information for work extraction can outweigh the costs of information generation.

Brownian motors powered by nonreciprocal interactions.

Traditional models for molecular (Brownian) motors predominantly depend on nonequilibrium driving, while particle interactions rigorously adhere to Newton's third law. However, numerous living and natural systems at various scales seem to defy this well-established law. In this study, we investigated the transport of mixed Brownian particles in a two-dimensional ratchet potential with nonreciprocal interactions. Our findings reveal that these nonreciprocal interactions can introduce a zero-mean nonequilibrium driving force. This force is capable of disrupting the thermodynamic equilibrium and inducing directed motion. The direction of this motion is determined by the asymmetry of the potential. Interestingly, the average velocity is a peaked function of the degree of nonreciprocity, while the effective diffusion consistently increases with the increase of nonreciprocity. There exists an optimal temperature or packing fraction at which the average velocity reaches its maximum value. We share a mechanism for particle rectification, devoid of particle-autonomous nonequilibrium drive, with potential usage in systems characterized by nonreciprocal interactions.

Sterically driven current reversal in a molecular motor model

Simulations can help unravel the complicated ways in which molecular structure determines function. Here, we use molecular simulations to show how slight alterations of a molecular motor's structure can cause the motor's typical dynamical behavior to reverse directions. Inspired by autonomous synthetic catenane motors, we study the molecular dynamics of a minimal motor model, consisting of a shuttling ring that moves along a track containing interspersed binding sites and catalytic sites. The binding sites attract the shuttling ring while the catalytic sites speed up a reaction between molecular species, which can be thought of as fuel and waste. When that fuel and waste are held in nonequilibrium steady-state concentrations, the free energy from the reaction drives directed motion of the shuttling ring along the track. Using this model and nonequilibrium molecular dynamics, we show that the shuttling ring's direction can be reversed by simply adjusting the spacing between binding and catalytic sites on the track. We present a steric mechanism behind the current reversal, supported by kinetic measurements from the simulations. These results demonstrate how molecular simulation can guide future development of artificial molecular motors.

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

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

Martini 3 Coarse-Grained Model for Second-Generation Unidirectional Molecular Motors and Switches.

Artificial molecular motors (MMs) and switches (MSs), capable of undergoing unidirectional rotation or switching under the appropriate stimuli, are being utilized in multiple complex and chemically diverse environments. Although thorough theoretical work utilizing QM and QM/MM methods have mapped out many of the critical properties of MSs and MMs, as the experimental setups become more complex and ambitious, there is an ever increasing need to study the behavior and dynamics of these molecules as they interact with their environment. To this end, we have parametrized two coarse-grained (CG) models of commonly used MMs and a model for an oxindole-based MS, which can be used to study the ground state behavior of MMs and MSs in large simulations for significantly longer periods of time. We also propose methods to perturb these systems which can allow users to approximate how such systems would respond to MMs rotating or the MSs switching.

Obtaining efficient collisional engines via velocity-dependent drivings.

Brownian particles interacting sequentially with distinct temperatures and driving forces at each stroke have been tackled as a reliable alternative for the construction of engine setups. However, they can behave very inefficiently depending on the driving used for the work source and/or when temperatures of each stage are very different from each other. Inspired by some models for molecular motors and recent experimental studies, a coupling between driving and velocities is introduced and detail investigated from stochastic thermodynamics. Exact expressions for thermodynamic quantities and distinct maximization routes have been obtained. The search of an optimal coupling provides a substantial increase of engine performance (mainly efficiency), even for large ΔT. A simple and general argument for the optimal coupling can be estimated, irrespective of the driving and other model details.

A dynamical model for Brownian molecular motors driven by inelastic electron tunneling

In recent years, several artificial molecular motors driven and controlled by electric currents have been proposed. Similar to Brownian machines, these systems work by turning random inelastic tunneling events into a directional rotation of the molecule. Despite their importance as the ultimate component of future molecular machines, their modeling has not been sufficiently studied. Here, we develop a dynamical model to describe these systems. We illustrate the validity and usefulness of our model by applying it to a well-known molecular motor, showing that the obtained results are consistent with the available experimental data. Moreover, we demonstrate how to use our model to extract some difficult-to-access microscopic parameters. Finally, we include an analysis of the expected effects of current-induced forces (CIFs). Our analysis suggests that, although nonconservative contributions of the CIFs can be important in some scenarios, they do not seem important in the analyzed case. Despite this, the conservative contributions of CIFs could be strong enough to significantly alter the system's dynamics.

Models of topological barriers and molecular motors of bacterial DNA

ABSTRACT Bacterial genomes are partitioned into kilobases long domains that are topologically independent from each other, meaning that change of DNA superhelicity in one domain does not propagate to neighbours. This is made possible by proteins like the LacI repressor, which behave like topological barriers and block the diffusion of torsion along the DNA. Other proteins, like DNA gyrases and RNA polymerases, called molecular motors, use the energy released by the hydrolysis of ATP to apply forces and/or torques to the DNA and modify its superhelicity. Here, we report on simulation work aimed at enlightening the interplay between DNA supercoiling, topological barriers, and molecular motors. To this end, we developed a coarse-grained Hamiltonian model of topological barriers and a model of molecular motors and investigated their properties through Brownian dynamics simulations. We discuss their influence on the contact map of a model nucleoid and the steady state values of twist and writhe in the DNA. These coarse-grained models, which are able to predict the dynamics of plectonemes depending on the position of topological barriers and molecular motors, should prove helpful to back up experimental efforts, like the development of Chromosome Conformation Capture techniques, and decipher the organisational mechanisms of bacterial chromosomes.

Extreme-value statistics of stochastic transport processes

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema obtained with martingale theory and reveal a symmetry between the distribution of the maxima and minima of entropy production. Furthermore, we show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marčenko–Pastur distribution of random-matrix theory. Using this result, we obtain efficient estimates for the extreme-value statistics of stochastic transport processes from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results with numerical simulations of stochastic models of molecular motors.

A common chemomechanical coupling model for orphan and conventional kinesin molecular motors

Orphan and conventional kinesin dimers represent two families of the kinesin superfamily molecular motors. Conventional kinesin, having a 14-residue neck linker (NL) in each head, can step processively on microtubule (MT), with an ATP hydrolysis being coupled with a mechanical stepping under no load. Orphan kinesin phragmoplast-associated kinesin-related protein 2 (PAKRP2) dimer, despite having a NL of 32 residues in each head, can also step processively on MT and exhibits tight chemomechanical coupling under no load. However, the dynamic properties of the wild type PAKRP2 and the mutant one with each NL truncated to 14 residues are very different from those of the wild type conventional kinesin and the mutant one with each NL being replaced by the 32-residue NL from PAKRP2. Here, based on a common chemomechanical coupling model we study computationally the dynamics of the two families of the kinesin dimers, with the simulated results explaining quantitatively the available experimental data. The large differences in the dynamics between the two families of kinesin dimers arise mainly from different rate constants of NL docking and ATPase activity and different weak affinities of the head in ADP state for MT. The studies indicate that both the orphan kinesin PAKRP2 and conventional kinesin use the same mechanism for processive motility.

Enhanced Diffusion in a Model of Molecular Motors with Potential Switching

Diffusion enhancement is a phenomenon in which the diffusion coefficient of a system is increased by an external force and it becomes larger than that of the force-free diffusion in thermal equilibrium. It is known that this phenomenon occurs for a Brownian particle in a periodic potential under a constant external force. Recently, it was found that diffusion enhancement also occurred in a biological molecular motor, whose moving part could move itself by switching the potentials generated by the other parts. It was shown that the diffusion coefficient exhibited peaks as a function of a constant external force. Here, we report the occurrence of an additional peak and investigate the condition governing its appearance.

Diffusion enhancement of Brownian motors revealed by a solvable model

A solvable model is proposed and analyzed to reveal the mechanism underlying the diffusion enhancement recently reported for a model of molecular motors and predicted to be observed in the biological rotary motor $\rm F_1$-ATPase. It turns out that the diffusion enhancement for the present model can approximately described by a random walk in which the waiting time for a step to occur is exponentially distributed and it takes nonzero time to proceed forward by the step. It is shown that the diffusion coefficient of such a random walk can significantly be increased when the average waiting time is comparable to the average stepping time.

Perfect anomalous transport of subdiffusive cargos by molecular motors in viscoelastic cytosol

Multiple experiments show that various submicron particles such as magnetosomes, RNA messengers, viruses, and even much smaller nanoparticles such as globular proteins diffuse anomalously slow in viscoelastic cytosol of living cells. Hence, their sufficiently fast directional transport by molecular motors such as kinesins is crucial for the cell operation. It has been shown recently that the traditional flashing Brownian ratchet models of molecular motors are capable to describe both normal and anomalous transport of such subdiffusing cargos by molecular motors with a very high efficiency. This work elucidates further an important role of mechanochemical coupling in such an anomalous transport. It shows a natural emergence of a perfect subdiffusive ratchet regime due to allosteric effects, where the random rotations of a “catalytic wheel” at the heart of the motor operation become perfectly synchronized with the random stepping of a heavily loaded motor, so that only one ATP molecule is consumed on average at each motor step along microtubule. However, the number of rotations made by the catalytic engine and the traveling distance both scale sublinearly in time. Nevertheless, this anomalous transport can be very fast in absolute terms.

Reliability of rectified transport: Coherence and reproducibility of transport by open-loop and feedback-controlled Brownian ratchets

Brownian ratchets are small-scale systems which rectify thermal fluctuations to produce a net current of particles. They have inspired many models of molecular motors that perform transport in the noisy environment of living cells. For the most common ratchet systems, this rectification is achieved by means of the switching of a periodic and spatially asymmetric potential (flashing ratchets) or by means of a rocking force (rocking ratchets). The rectification mechanism can be applied without information on the state of the system (open-loop ratchets) or using information on the state of the system (feedback or closed-loop ratchets). In order to characterize the transport, the most used quantity is the mean velocity of the center of mass of the system. However, another important transport attribute that has not received much attention is its quality. Here we analyze the quality of transport by studying the coherence and reproducibility of the transport induced by several representative open-and closed-loop rectification protocols under the maximum mean velocity conditions. We find that for few-particle systems, the best protocol is the rocked feedback protocol, producing the transport of particles with the highest coherence and reproducibility per distance traveled at the maximum mean velocity, while for larger systems it is overtaken by its open-loop counterpart. Our results also show that protocols with similar maximum mean velocities can have quite different coherences and reproducibilities. This highlights the importance of studying the reliability of rectified transport to develop performant synthetic rectification devices. These contributions to the emerging field of reliable transport in noisy environments are expected also to provide insight into the performance of natural molecular motors.

Tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor

We present a tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor. Our derivation is based on the continuous diffusion equation for overdamped Brownian motion on a time-independent tilted periodic potential in two dimensions. The free-energy potential is nonseparable to allow coupling between the chemical and mechanical degrees of freedom. We formally discretize the chemical coordinate by expanding in Wannier states that are localized along the chemical coordinate and parametrized along the mechanical coordinate. A discrete-continuous equation is derived that is valid for anisotropic systems with weak mechanochemical coupling and deep potential wells along the chemical coordinate. The discrete-continuous description is consistent with established theoretical models of molecular motors with discrete chemical states but is constrained by the underlying continuous two-dimensional potential. In particular, we derive analytic expressions for the effective potential along the mechanical coordinate and for the rate of thermal hopping between chemical states. We determine the thermodynamic efficiency of energy conversion and find that, for a molecular motor with one chemical state per cycle, the derived discrete-continuous equation can accurately describe mechanochemical coupling but cannot describe energy conversion.

Jump locations of jump-diffusion processes with state-dependent rates

We propose a general framework for studying statistics of jump-diffusion systems driven by both Brownian noise (diffusion) and a jump process with state-dependent intensity. Of particular natural interest in many physical systems are the jump locations: the system evaluated at the jump times. As an example, this could be the voltage at which a neuron fires, or the so-called ‘threshold voltage’. However, the state-dependence of the jump rate provides direct coupling between the diffusion and jump components, making it difficult to disentangle the two to study individually. In this work, we provide an iterative map formulation of the sequence of distributions of jump locations. The distributions computed by this map can be used to elucidate other interesting quantities about the process, including statistics of the interjump times. Ultimately, the limit of the map reveals that knowledge of the stationary distribution of the full process is sufficient to recover (but not necessarily equal to) the distribution of jump locations. We propose two biophysical examples to illustrate the use of this framework to provide insight about a system. We find that a sharp threshold voltage emerges robustly in a simple stochastic integrate-and-fire neuronal model. The interplay between the two sources of noise is also investigated in a stepping model of molecular motor in intracellular transport pulling a diffusive cargo.

Fluctuation theorems for discrete kinetic models of molecular motors

Motivated by discrete kinetic models for non-cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi-1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti–Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in Lacoste et al (2008 Phys. Rev. E 78 011915). We also derive fluctuation theorems for the time-integrated cycle currents and discuss how the matrix approach of Lacoste et al (2008 Phys. Rev. E 78 011915) can be extended to derive the above Gallavotti–Cohen symmetry for any Markov random walk on with periodic jump rates. Finally, we review in the present context some large deviation results of Faggionato and Silvestri (2017 Ann. Inst. Henri Poincaré 53 46–78) and give some specific examples with explicit computations.