Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines

Dimensional reduction and adaptation-development-evolution relation in evolved biological systems

Living systems are complex and hierarchical, with diverse components at different scales, yet they sustain themselves, grow, and evolve over time. How can a theory of such complex biological states be developed? Here we note that for a hierarchical biological system to be robust, it must achieve consistency between micro-scale (e.g., molecular) and macro-scale (e.g., cellular) phenomena. This allows for a universal theory of adaptive change in cells based on biological robustness and consistency between cellular growth and molecular replication. Here, we show how adaptive changes in high-dimensional phenotypes (biological states) are constrained to low-dimensional space, leading to the derivation of a macroscopic law for cellular states. The theory is then extended to evolution, leading to proportionality between evolutionary and environmental responses, as well as proportionality between phenotypic variances due to noise and due to genetic changes. The universality of the results across several models and experiments is demonstrated. Then, by further extending the theory of evolutionary dimensional reduction to multicellular systems, the relationship between multicellular development and evolution, in particular, the developmental hourglass, is demonstrated. Finally, the possibility of collapse of dimensional reduction under nutrient limitation is discussed.

Dynamically driven correlations in elastic net models reveal sequence of events and causality in proteins

An explicit analytic solution is given for the Langevin equation applied to the Gaussian Network Model of a protein subjected to both a random and a deterministic periodic force. Synchronous and asynchronous components of time correlation functions are derived and an expression for phase differences in the time correlations of residue pairs is obtained. The synchronous component enables the determination of dynamic communities within the protein structure. The asynchronous component reveals causality, where the time correlation function between residues i and j differs depending on whether i is observed before j or vice versa, resulting in directional information flow. Driver and driven residues in the allosteric process of cyclophilin A and human NAD-dependent isocitrate dehydrogenase are determined by a perturbation-scanning technique. Factors affecting phase differences between fluctuations of residues, such as network topology, connectivity, and residue centrality, are identified. Within the constraints of the isotropic Gaussian Network Model, our results show that asynchronicity increases with viscosity and distance between residues, decreases with increasing connectivity, and decreases with increasing levels of eigenvector centrality.

Pair dynamics of active force dipoles in an odd-viscous fluid

We discuss the lateral dynamics of two active force dipoles, which interact with each other via hydrodynamic interactions in a thin fluid layer that is active and chiral. The fluid layer is modeled as a two-dimensional (2D) compressible fluid with an odd viscosity, while the force dipole (representing an active protein or enzyme) induces a dipolar flow. Taking into account the momentum decay in the 2D fluid, we obtain analytically the mobility tensor that depends on the odd viscosity and includes nonreciprocal hydrodynamic interactions. We find that the particle pair shows spiral behavior due to the transverse flow induced by the odd viscosity. When the magnitude of the odd viscosity is large as compared with the shear viscosity, two types of oscillatory behaviors are seen. One of them can be understood as arising from closed orbits in dynamical systems, and its circular trajectories are determined by the ratio between the magnitude of the odd viscosity and the force dipole. In addition, the phase diagrams of the particle dipolar angles are obtained numerically. Our findings reveal that the nonreciprocal response leads to complex dynamics of active particles embedded in an active fluid with odd viscosity.

The Statistical Trends of Protein Evolution: A Lesson from AlphaFold Database

The recent development of artificial intelligence provides us with new and powerful tools for studying the mysterious relationship between organism evolution and protein evolution. In this work, based on the AlphaFold Protein Structure Database (AlphaFold DB), we perform comparative analyses of the proteins of different organisms. The statistics of AlphaFold-predicted structures show that, for organisms with higher complexity, their constituent proteins will have larger radii of gyration, higher coil fractions, and slower vibrations, statistically. By conducting normal mode analysis and scaling analyses, we demonstrate that higher organismal complexity correlates with lower fractal dimensions in both the structure and dynamics of the constituent proteins, suggesting that higher functional specialization is associated with higher organismal complexity. We also uncover the topology and sequence bases of these correlations. As the organismal complexity increases, the residue contact networks of the constituent proteins will be more assortative, and these proteins will have a higher degree of hydrophilic-hydrophobic segregation in the sequences. Furthermore, by comparing the statistical structural proximity across the proteomes with the phylogenetic tree of homologous proteins, we show that, statistical structural proximity across the proteomes may indirectly reflect the phylogenetic proximity, indicating a statistical trend of protein evolution in parallel with organism evolution. This study provides new insights into how the diversity in the functionality of proteins increases and how the dimensionality of the manifold of protein dynamics reduces during evolution, contributing to the understanding of the origin and evolution of lives.

Prestrain-induced contraction in one-dimensional random elastic chains

Prestrained elastic networks arise in a number of biological and technological systems ranging from the cytoskeleton of cells to tensegrity structures. Motivated by this observation, we here consider a minimal model in one dimension to set the stage for understanding the response of such networks as a function of the prestrain. To this end we consider a chain [one-dimensional (1D) network] of elastic springs upon which a random, zero mean, finite variance prestrain is imposed. Numerical simulations and analytical predictions quantify the magnitude of the contraction as a function of the variance of the prestrain, and show that the chain always shrinks. To test these predictions, we vary the topology of the chain, consider more complex connectivity and show that our results are relatively robust to these changes.

Simulating a chemically fueled molecular motor with nonequilibrium molecular dynamics

Most computer simulations of molecular dynamics take place under equilibrium conditions-in a closed, isolated system, or perhaps one held at constant temperature or pressure. Sometimes, extra tensions, shears, or temperature gradients are introduced to those simulations to probe one type of nonequilibrium response to external forces. Catalysts and molecular motors, however, function based on the nonequilibrium dynamics induced by a chemical reaction's thermodynamic driving force. In this scenario, simulations require chemostats capable of preserving the chemical concentrations of the nonequilibrium steady state. We develop such a dynamic scheme and use it to observe cycles of a particle-based classical model of a catenane-like molecular motor. Molecular motors are frequently modeled with detailed-balance-breaking Markov models, and we explicitly construct such a picture by coarse graining the microscopic dynamics of our simulations in order to extract rates. This work identifies inter-particle interactions that tune those rates to create a functional motor, thereby yielding a computational playground to investigate the interplay between directional bias, current generation, and coupling strength in molecular information ratchets.

Dynamics-Evolution Correspondence in Protein Structures

The genotype-phenotype mapping of proteins is a fundamental question in structural biology. In this Letter, with the analysis of a large dataset of proteins from hundreds of protein families, we quantitatively demonstrate the correlations between the noise-induced protein dynamics and mutation-induced variations of native structures, indicating the dynamics-evolution correspondence of proteins. Based on the investigations of the linear responses of native proteins, the origin of such a correspondence is elucidated. It is essential that the noise- and mutation-induced deformations of the proteins are restricted on a common low-dimensional subspace, as confirmed from the data. These results suggest an evolutionary mechanism of the proteins gaining both dynamical flexibility and evolutionary structural variability.

Mechanochemical enzymes and protein machines as hydrodynamic force dipoles: the active dimer model

Mechanochemically active enzymes change their shapes within every turnover cycle. Therefore, they induce circulating flows in the solvent around them and behave as oscillating hydrodynamic force dipoles. Because of non-equilibrium fluctuating flows collectively generated by the enzymes, mixing in the solution and diffusion of passive particles within it are expected to get enhanced. Here, we investigate the intensity and statistical properties of such force dipoles in the minimal active dimer model of a mechanochemical enzyme. In the framework of this model, novel estimates for hydrodynamic collective effects in solution and in lipid bilayers under rapid rotational diffusion are derived, and available experimental and computational data is examined.

Detection of Single Molecules Using Stochastic Resonance of Bistable Oligomers

Ultra-sensitive elements for nanoscale devices capable of detecting single molecules are in demand for many important applications. It is generally accepted that the inevitable stochastic disturbance of a sensing element by its surroundings will limit detection at the molecular level. However, a phenomenon exists (stochastic resonance) in which the environmental noise acts abnormally: it amplifies, rather than distorts, a weak signal. Stochastic resonance is inherent in non-linear bistable systems with criticality at which the bistability emerges. Our computer simulations have shown that the large-scale conformational dynamics of a short oligomeric fragment of thermosrespective polymer, poly-N-isopropylmethacrylamid, resemble the mechanical movement of nonlinear bistable systems. The oligomers we have studied demonstrate spontaneous vibrations and stochastic resonance activated by conventional thermal noise. We have observed reasonable shifts of the spontaneous vibrations and stochastic resonance modes when attaching an analyte molecule to the oligomer. Our simulations have shown that spontaneous vibrations and stochastic resonance of the bistable thermoresponsive oligomers are sensitive to both the analyte molecular mass and the binding affinity. All these effects indicate that the oligomers with mechanic-like bistability may be utilized as ultrasensitive operational units capable of detecting single molecules.

Physical Constraints on Epistasis

Living systems evolve one mutation at a time, but a single mutation can alter the effect of subsequent mutations. The underlying mechanistic determinants of such epistasis are unclear. Here, we demonstrate that the physical dynamics of a biological system can generically constrain epistasis. We analyze models and experimental data on proteins and regulatory networks. In each, we find that if the long-time physical dynamics is dominated by a slow, collective mode, then the dimensionality of mutational effects is reduced. Consequently, epistatic coefficients for different combinations of mutations are no longer independent, even if individually strong. Such epistasis can be summarized as resulting from a global nonlinearity applied to an underlying linear trait, that is, as global epistasis. This constraint, in turn, reduces the ruggedness of the sequence-to-function map. By providing a generic mechanistic origin for experimentally observed global epistasis, our work suggests that slow collective physical modes can make biological systems evolvable.

Dimensional Reduction in Evolving Spin-Glass Model: Correlation of Phenotypic Responses to Environmental and Mutational Changes

The evolution of high-dimensional phenotypes is investigated using a statistical physics model consisting of interacting spins, in which phenotypes, genotypes, and environments are represented by spin configurations, interaction matrices, and external fields, respectively. We found that phenotypic changes upon diverse environmental change and genetic variation are highly correlated across all spins, consistent with recent experimental observations of biological systems. The dimension reduction in phenotypic changes is shown to be a result of the evolution of the robustness to thermal noise, achieved at the replica symmetric phase.

Allosteric communication in molecular machines via information exchange: what can be learned from dynamical modeling

Allosteric regulation is crucial for the operation of protein machines and molecular motors. A major challenge is to characterize and quantify the information exchange underlying allosteric communication between remote functional sites in a protein, and to identify the involved relevant pathways. We review applications of two topical approaches of dynamical protein modeling, a kinetic-based single-molecule stochastic model, which employs information thermodynamics to quantify allosteric interactions, and structure-based coarse-grained modeling to characterize intra-molecular couplings in terms of conformational motions and propagating mechanical strain. Both descriptions resolve the directionality of allosteric responses within a protein, emphasizing the concept of causality as the principal hallmark of protein allostery. We discuss the application of techniques from information thermodynamics to dynamic protein elastic networks and evolutionary designed model structures, and the ramifications for protein allostery.

Shear viscosity of two-state enzyme solutions

We discuss the shear viscosity of a Newtonian solution of catalytic enzymes and substrate molecules. The enzyme is modeled as a two-state dimer consisting of two spherical domains connected with an elastic spring. The enzymatic conformational dynamics is induced by the substrate binding and such a process is represented by an additional elastic spring. Employing the Boltzmann distribution weighted by the waiting times of enzymatic species in each catalytic cycle, we obtain the shear viscosity of dilute enzyme solutions as a function of substrate concentration and its physical properties. The substrate affinity distinguishes between fast and slow enzymes, and the corresponding viscosity expressions are obtained. Furthermore, we connect the obtained viscosity with the diffusion coefficient of a tracer particle in enzyme solutions.

Analyzing Fluctuation Properties in Protein Elastic Networks with Sequence-Specific and Distance-Dependent Interactions

Simple protein elastic networks which neglect amino-acid information often yield reasonable predictions of conformational dynamics and are broadly used. Recently, model variants which incorporate sequence-specific and distance-dependent interactions of residue pairs have been constructed and demonstrated to improve agreement with experimental data. We have applied the new variants in a systematic study of protein fluctuation properties and compared their predictions with those of conventional anisotropic network models. We find that the quality of predictions is frequently linked to poor estimations in highly flexible protein regions. An analysis of a large set of protein structures shows that fluctuations of very weakly connected network residues are intrinsically prone to be significantly overestimated by all models. This problem persists in the new models and is not resolved by taking into account sequence information. The effect becomes even enhanced in the model variant which takes into account very soft long-ranged residue interactions. Beyond these shortcomings, we find that model predictions are largely insensitive to the integration of chemical information, at least regarding the fluctuation properties of individual residues. One can furthermore conclude that the inherent drawbacks may present a serious hindrance when improvement of elastic network models are attempted.

Simple mechanics of protein machines

While belonging to the nanoscale, protein machines are so complex that tracing even a small fraction of their cycle requires weeks of calculations on supercomputers. Surprisingly, many aspects of their operation can be however already reproduced by using very simple mechanical models of elastic networks. The analysis suggests that, similar to other self-organized complex systems, functional collective dynamics in such proteins is effectively reduced to a low-dimensional attractive manifold.

Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings

The activity of biological cells is primarily based on chemical reactions and typically modeled as a reaction-diffusion system. Cells are, however, highly crowded with macromolecules, including a variety of molecular machines such as enzymes. The working cycles of these machines are often coupled with their internal motion (conformational changes). In the crowded environment of a cell, motion interference between neighboring molecules is not negligible, and this interference can affect the reaction dynamics through machine operation. To simulate such a situation, we propose a reaction-diffusion model consisting of particles whose shape depends on an internal state variable, for crowds of nano- to micromachines. The interference between nearby particles is naturally introduced through excluded volume repulsion. In the simulations, we observed segregation and flow-like patterns enhanced by crowding out of relevant molecules, as well as molecular synchronization waves and phase transitions. The presented model is simple and extensible for diverse molecular machinery and may serve as a framework to study the interplay between the mechanical stress/strain network and the chemical reaction network in the cell. Applications to more macroscopic systems, e.g., crowds of cells, are also discussed.

Coarse-Grained Protein Dynamics Studies Using Elastic Network Models

Elastic networks have been used as simple models of proteins to study their slow structural dynamics. They consist of point-like particles connected by linear Hookean springs and hence are convenient for linear normal mode analysis around a given reference structure. Furthermore, dynamic simulations using these models can provide new insights. As the computational cost associated with these models is considerably lower compared to that of all-atom models, they are also convenient for comparative studies between multiple protein structures. In this review, we introduce examples of coarse-grained molecular dynamics studies using elastic network models and their derivatives, focusing on the nonlinear phenomena, and discuss their applicability to large-scale macromolecular assemblies.

Designed Elastic Networks: Models of Complex Protein Machinery

Recently, the design of mechanical networks with protein-inspired responses has become increasingly popular. Here, we review contributions which were motivated by studies of protein dynamics employing coarse-grained elastic network models. First, the concept of evolutionary optimization that we developed to design network structures which execute prescribed tasks is explained. We then review what presumably marks the origin of the idea to design complex functional networks which encode protein-inspired behavior, namely the design of an elastic network structure which emulates the cycles of ATP-powered conformational motion in protein machines. Two recent applications are reviewed. First, the construction of a model molecular motor, whose operation incorporates both the tight coupling power stroke as well as the loose coupling Brownian ratchet mechanism, is discussed. Second, the evolutionary design of network structures which encode optimal long-range communication between remote sites and represent mechanical models of allosteric proteins is presented. We discuss the prospects of designed protein-mimicking elastic networks as model systems to elucidate the design principles and functional signatures underlying the operation of complex protein machinery.

Macroscopic Theory for Evolving Biological Systems Akin to Thermodynamics

We present a macroscopic theory to characterize the plasticity, robustness, and evolvability of biological responses and their fluctuations. First, linear approximation in intracellular reaction dynamics is used to demonstrate proportional changes in the expression of all cellular components in response to a given environmental stress, with the proportion coefficient determined by the change in growth rate as a consequence of the steady growth of cells. We further demonstrate that this relationship is supported through adaptation experiments of bacteria, perhaps too well as this proportionality is held even across cultures of different types of conditions. On the basis of simulations of cell models, we further show that this global proportionality is a consequence of evolution in which expression changes in response to environmental or genetic perturbations are constrained along a unique one-dimensional curve, which is a result of evolutionary robustness. It then follows that the expression changes induced by environmental changes are proportionally reduced across different components of a cell by evolution, which is akin to the Le Chatelier thermodynamics principle. Finally, with the aid of a fluctuation-response relationship, this proportionality is shown to hold between fluctuations caused by genetic changes and those caused by noise. Overall, these results and support from the theoretical and experimental literature suggest a formulation of cellular systems akin to thermodynamics, in which a macroscopic potential is given by the growth rate (or fitness) represented as a function of environmental and evolutionary changes.

Three-disk microswimmer in a supported fluid membrane

A model of three-disk micromachine swimming in a quasi-two-dimensional supported membrane is proposed. We calculate the average swimming velocity as a function of the disk size and the arm length. Due to the presence of the hydrodynamic screening length in the quasi-two-dimensional fluid, the geometric factor appearing in the average velocity exhibits three different asymptotic behaviors depending on the microswimmer size and the hydrodynamic screening length. This is in sharp contrast with a microswimmer in a three-dimensional bulk fluid that shows only a single scaling behavior. We also find that the maximum velocity is obtained when the disks are equal-sized, whereas it is minimized when the average arm lengths are identical. The intrinsic drag of the disks on the substrate does not alter the scaling behaviors of the geometric factor.

High-speed atomic force microscopy and its future prospects

Various techniques have been developed and used to investigate how proteins produce complex biological architectures and phenomena. Among these techniques, high-speed atomic force microscopy (HS-AFM) holds a unique position. It is only HS-AFM that allows the simultaneous assessment of structure and dynamics of single protein molecules in action. This new microscopy tool has been successfully applied to a variety of proteins, from motor proteins to membrane proteins, antibodies, enzymes, and even to intrinsically disordered proteins. And yet there still remain many biomolecular phenomena that cannot be addressed by HS-AFM in its current form. Here, I present a brief history of HS-AFM development, describe the current state of HS-AFM, and then discuss which new biological scanning probe microscopy techniques will be coming up next.

High-speed atomic force microscopy and its future prospects

Various techniques have been developed and used to investigate how proteins produce complex biological architectures and phenomena. Among these techniques, high-speed atomic force microscopy (HS-AFM) holds a unique position. It is only HS-AFM that allows the simultaneous assessment of structure and dynamics of single protein molecules in action. This new microscopy tool has been successfully applied to a variety of proteins, from motor proteins to membrane proteins, antibodies, enzymes, and even to intrinsically disordered proteins. And yet there still remain many biomolecular phenomena that cannot be addressed by HS-AFM in its current form. Here, I present a brief history of HS-AFM development, describe the current state of HS-AFM, and then discuss which new biological scanning probe microscopy techniques will be coming up next.

Analysis of an ATP-induced conformational transition of ABC transporter MsbA using a coarse-grained model

Upon the binding of ATP molecules to nucleotide binding domains (NBDs), ATP-binding cassette (ABC) exporters undergo a conformational transition from an inward-facing (IF) to an outward-facing (OF) state. This molecular event is a typical example of chemo-mechanical coupling. However, the underlying mechanism remains unclear. In this study, we analyzed the IF→OF transition of a representative ABC exporter, MsbA, by solving the equation of motion under an elastic network model (ENM). ATP was represented as a single node in ENM or replaced by external forces. When two ATP nodes were added to the ENM of the IF state protein, the two NBDs dimerized; subsequently, the two transmembrane domains opened toward the extracellular side, resulting in the formation of the OF structure. Such a conformational transition was also reproduced by applying external forces, which caused the rotational motion of the NBDs instead of the addition of ATP nodes. The process of the conformational transition was analyzed in detail using cross-correlation maps for node-node interactions. More importantly, it was revealed that the ATP binding energy is converted into distortion energy of several transmembrane helices. These results are useful for understanding the chemo-mechanical coupling in ABC transporters.

Deciphering Intrinsic Inter-subunit Couplings that Lead to Sequential Hydrolysis of F1-ATPase Ring

Rotary sequential hydrolysis of the metabolic machine F₁-ATPase is a prominent manifestation of high coordination among multiple chemical sites in ring-shaped molecular machines, and it is also functionally essential for F₁ to tightly couple chemical reactions and central γ-shaft rotation. High-speed AFM experiments have identified that sequential hydrolysis is maintained in the F₁ stator ring even in the absence of the γ-rotor. To explore the origins of intrinsic sequential performance, we computationally investigated essential inter-subunit couplings on the hexameric ring of mitochondrial and bacterial F₁. We first reproduced in stochastic Monte Carlo simulations the experimentally determined sequential hydrolysis schemes by kinetically imposing inter-subunit couplings and following subsequent tri-site ATP hydrolysis cycles on the F₁ ring. We found that the key couplings to support the sequential hydrolysis are those that accelerate neighbor-site ADP and Pi release upon a certain ATP binding or hydrolysis reaction. The kinetically identified couplings were then examined in atomistic molecular dynamics simulations at a coarse-grained level to reveal the underlying structural mechanisms. To do that, we enforced targeted conformational changes of ATP binding or hydrolysis to one chemical site on the F₁ ring and monitored the ensuing conformational responses of the neighboring sites using structure-based simulations. Notably, we found asymmetrical neighbor-site opening that facilitates ADP release upon enforced ATP binding. We also captured a complete charge-hopping process of the Pi release subsequent to enforced ATP hydrolysis in the neighbor site, confirming recent single-molecule analyses with regard to the role of ATP hydrolysis in F₁. Our studies therefore elucidate both the coordinated chemical kinetics and structural dynamics mechanisms underpinning the sequential operation of the F₁ ring.

Design of Elastic Networks with Evolutionary Optimized Long-Range Communication as Mechanical Models of Allosteric Proteins

Allosteric effects often underlie the activity of proteins, and elucidating generic design aspects and functional principles unique to allosteric phenomena represent a major challenge. Here an approach consisting of the in silico design of synthetic structures, which, as the principal element of allostery, encode dynamical long-range coupling among two sites, is presented. The structures are represented by elastic networks, similar to coarse-grained models of real proteins. A strategy of evolutionary optimization was implemented to iteratively improve allosteric coupling. In the designed structures, allosteric interactions were analyzed in terms of strain propagation, and simple pathways that emerged during evolution were identified as signatures through which long-range communication was established. Moreover, robustness of allosteric performance with respect to mutations was demonstrated. As it turned out, the designed prototype structures reveal dynamical properties resembling those found in real allosteric proteins. Hence, they may serve as toy models of complex allosteric systems, such as proteins. Application of the developed modeling scheme to the allosteric transition in the myosin V molecular motor was also demonstrated.

Passive and active colloidal chemotaxis in a microfluidic channel: mesoscopic and stochastic models

Chemotaxis is the response of a particle to a gradient in the chemical composition of the environment. While it was originally observed for biological organisms, it is of great interest in the context of synthetic active particles such as nanomotors. Experimental demonstration of chemotaxis for chemically-powered colloidal nanomotors was reported in the literature in the context of chemo-attraction in a still fluid or in a microfluidic channel where the gradient is sustained by a specific inlet geometry. In this work, we use mesoscopic particle-based simulations of the colloid and solvent to demonstrate chemotaxis in a microfluidic channel. On the basis of this particle-based model, we evaluate the chemical concentration profiles in the presence of passive or chemically active colloids, compute the chemotactic force acting upon them and propose a stochastic model that rationalises our findings on colloidal chemotaxis. Our model is also able to explain the results of an earlier simulation work that uses a simpler geometry and to extend its interpretation.

Nonlinear network model analysis of vibrational energy transfer and localisation in the Fenna-Matthews-Olson complex

Collective protein modes are expected to be important for facilitating energy transfer in the Fenna-Matthews-Olson (FMO) complex of photosynthetic green sulphur bacteria, however to date little work has focussed on the microscopic details of these vibrations. The nonlinear network model (NNM) provides a computationally inexpensive approach to studying vibrational modes at the microscopic level in large protein structures, whilst incorporating anharmonicity in the inter-residue interactions which can influence protein dynamics. We apply the NNM to the entire trimeric FMO complex and find evidence for the existence of nonlinear discrete breather modes. These modes tend to transfer energy to the highly connected core pigments, potentially opening up alternative excitation energy transfer routes through their influence on pigment properties. Incorporating localised modes based on these discrete breathers in the optical spectra calculations for FMO using ab initio site energies and excitonic couplings can substantially improve their agreement with experimental results.

Fractal globules: a new approach to artificial molecular machines

The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path.

Diffusional correlations among multiple active sites in a single enzyme

Simulations of the enzymatic dynamics of a model enzyme containing multiple substrate binding sites indicate the existence of diffusional correlations in the chemical reactivity of the active sites. A coarse-grain, particle-based, mesoscopic description of the system, comprising the enzyme, the substrate, the product and solvent, is constructed to study these effects. The reactive and non-reactive dynamics is followed using a hybrid scheme that combines molecular dynamics for the enzyme, substrate and product molecules with multiparticle collision dynamics for the solvent. It is found that the reactivity of an individual active site in the multiple-active-site enzyme is reduced substantially, and this effect is analyzed and attributed to diffusive competition for the substrate among the different active sites in the enzyme.

TALEs from a spring--superelasticity of Tal effector protein structures

Transcription activator-like effectors (TALEs) are DNA-related proteins that recognise and bind specific target sequences to manipulate gene expression. Recently determined crystal structures show that their common architecture reveals a superhelical overall structure that may undergo drastic conformational changes. To establish a link between structure and dynamics in TALE proteins we have employed coarse-grained elastic-network modelling of currently available structural data and implemented a force-probe setup that allowed us to investigate their mechanical behaviour in computer experiments. Based on the measured force-extension curves we conclude that TALEs exhibit superelastic dynamical properties allowing for large-scale global conformational changes along their helical axis, which represents the soft direction in such proteins. For moderate external forcing the TALE models behave like linear springs, obeying Hooke's law, and the investigated structures can be characterised and compared by a corresponding spring constant. We show that conformational flexibility underlying the large-scale motions is not homogeneously distributed over the TALE structure, but instead soft spot residues around which strain is accumulated and which turn out to represent key agents in the transmission of conformational motions are identified. They correspond to the RVD loop residues that have been experimentally determined to play an eminent role in the binding process of target DNA.

Coarse-grain simulations of active molecular machines in lipid bilayers

A coarse-grain method for simulations of the dynamics of active protein inclusions in lipid bilayers is described. It combines the previously proposed hybrid simulations of bilayers [M.-J. Huang, R. Kapral, A. S. Mikhailov, and H.-Y. Chen, J. Chem. Phys. 137, 055101 (2012)], based on molecular dynamics for the lipids and multi-particle collision dynamics for the solvent, with an elastic-network description of active proteins. The method is implemented for a model molecular machine which performs active conformational motions induced by ligand binding and its release after reaction. The situation characteristic for peripheral membrane proteins is considered. Statistical investigations of the effects of single active or passive inclusions on the shape of the membrane are carried out. The results show that the peripheral machine produces asymmetric perturbations of the thickness of two leaflets of the membrane. It also produces a local saddle in the midplane height of the bilayer. Analysis of the power spectrum of the fluctuations of the membrane midplane shows that the conformational motion of the machine perturbs these membrane fluctuations. The hydrodynamic lipid flows induced by cyclic conformational changes in the machine are analyzed. It is shown that such flows are long-ranged and should provide an additional important mechanism for interactions between active inclusions in biological membranes.

Frequency response and gap tuning for nonlinear electrical oscillator networks

We study nonlinear electrical oscillator networks, the smallest example of which consists of a voltage-dependent capacitor, an inductor, and a resistor driven by a pure tone source. By allowing the network topology to be that of any connected graph, such circuits generalize spatially discrete nonlinear transmission lines/lattices that have proven useful in high-frequency analog devices. For such networks, we develop two algorithms to compute the steady-state response when a subset of nodes are driven at the same fixed frequency. The algorithms we devise are orders of magnitude more accurate and efficient than stepping towards the steady-state using a standard numerical integrator. We seek to enhance a given network's nonlinear behavior by altering the eigenvalues of the graph Laplacian, i.e., the resonances of the linearized system. We develop a Newton-type method that solves for the network inductances such that the graph Laplacian achieves a desired set of eigenvalues; this method enables one to move the eigenvalues while keeping the network topology fixed. Running numerical experiments using three different random graph models, we show that shrinking the gap between the graph Laplacian's first two eigenvalues dramatically improves a network's ability to (i) transfer energy to higher harmonics, and (ii) generate large-amplitude signals. Our results shed light on the relationship between a network's structure, encoded by the graph Laplacian, and its function, defined in this case by the presence of strongly nonlinear effects in the frequency response.

Complex intramolecular mechanics of G-actin--an elastic network study

Systematic numerical investigations of conformational motions in single actin molecules were performed by employing a simple elastic-network (EN) model of this protein. Similar to previous investigations for myosin, we found that G-actin essentially behaves as a strain sensor, responding by well-defined domain motions to mechanical perturbations. Several sensitive residues within the nucleotide-binding pocket (NBP) could be identified, such that the perturbation of any of them can induce characteristic flattening of actin molecules and closing of the cleft between their two mobile domains. Extending the EN model by introduction of a set of breakable links which become effective only when two domains approach one another, it was observed that G-actin can possess a metastable state corresponding to a closed conformation and that a transition to this state can be induced by appropriate perturbations in the NBP region. The ligands were roughly modeled as a single particle (ADP) or a dimer (ATP), which were placed inside the NBP and connected by elastic links to the neighbors. Our approximate analysis suggests that, when ATP is present, it stabilizes the closed conformation of actin. This may play an important role in the explanation why, in the presence of ATP, the polymerization process is highly accelerated.

Myosin-V as a mechanical sensor: an elastic network study

According to recent experiments, the molecular-motor myosin behaves like a strain sensor, exhibiting different functional responses when loads in opposite directions are applied to its tail. Within an elastic-network model, we explore the sensitivity of the protein to the forces acting on the tail and find, in agreement with experiments, that such forces invoke conformational changes that should affect filament binding and ADP release. Furthermore, conformational responses of myosin to the application of forces to individual residues in its principal functional regions are systematically investigated and a detailed sensitivity map of myosin-V is thus obtained. The results suggest that the strain-sensor behavior is involved in the intrinsic operation of this molecular motor.

In silico investigation of conformational motions in superfamily 2 helicase proteins

Helicases are motor proteins that play a central role in the metabolism of DNA and RNA in biological cells. Using the energy of ATP molecules, they are able to translocate along the nucleic acids and unwind their duplex structure. They have been extensively characterized in the past and grouped into superfamilies based on structural similarities and sequential motifs. However, their functional aspects and the mechanism of their operation are not yet well understood. Here, we consider three helicases from the major superfamily 2 - Hef, Hel308 and XPD - and study their conformational dynamics by using coarse-grained relaxational elastic network models. Specifically, their responses to mechanical perturbations are analyzed. This enables us to identify robust and ordered conformational motions which may underlie the functional activity of these proteins. As we show, such motions are well-organized and have large amplitudes. Their possible roles in the processing of nucleic substrate are discussed.

Automating the search of molecular motor templates by evolutionary methods

Biological molecular motors are nanoscale devices capable of transforming chemical energy into mechanical work, which are being researched in many scientific disciplines. From a computational point of view, the characteristics and dynamics of these motors are studied at multiple time scales, ranging from very detailed and complex molecular dynamics simulations spanning a few microseconds, to extremely simple and coarse-grained theoretical models of their working cycles. However, this research is performed only in the (relatively few) instances known from molecular biology. In this work, results from elastic network analysis and behaviour-finding methods are applied to explore a subset of the configuration space of template molecular structures that are able to transform chemical energy into directed movement, for a fixed instance of working cycle. While using methods based on elastic networks limits the scope of our results, it enables the implementation of computationally lightweight methods, in a way that evolutionary search techniques can be applied to discover novel molecular motor templates. The results show that molecular motion can be attained from a variety of structural configurations, when a functional working cycle is provided. Additionally, these methods enable a new computational way to test hypotheses about molecular motors.

Tracing entire operation cycles of molecular motor hepatitis C virus helicase in structurally resolved dynamical simulations

Hepatitis C virus helicase is a molecular motor that splits duplex DNA while actively moving over it. An approximate coarse-grained dynamical description of this protein, including its interactions with DNA and ATP, is constructed. Using such a mechanical model, entire operation cycles of an important protein machine could be followed in structurally resolved dynamical simulations. Ratcheting inchworm translocation and spring-loaded DNA unwinding, suggested by experimental data, were reproduced. Thus, feasibility of coarse-grained simulations, bridging a gap between full molecular dynamics and reduced phenomenological theories of molecular motors, has been demonstrated.

Nonlinearity of mechanochemical motions in motor proteins

The assumption of linear response of protein molecules to thermal noise or structural perturbations, such as ligand binding or detachment, is broadly used in the studies of protein dynamics. Conformational motions in proteins are traditionally analyzed in terms of normal modes and experimental data on thermal fluctuations in such macromolecules is also usually interpreted in terms of the excitation of normal modes. We have chosen two important protein motors — myosin V and kinesin KIF1A — and performed numerical investigations of their conformational relaxation properties within the coarse-grained elastic network approximation. We have found that the linearity assumption is deficient for ligand-induced conformational motions and can even be violated for characteristic thermal fluctuations. The deficiency is particularly pronounced in KIF1A where the normal mode description fails completely in describing functional mechanochemical motions. These results indicate that important assumptions of the theory of protein dynamics may need to be reconsidered. Neither a single normal mode nor a superposition of such modes yields an approximation of strongly nonlinear dynamics.

Biological cells use a variety of molecular machines representing enzymes, ion channels or pumps, and motors. Motor proteins are nanometer-size devices generating forces and actively moving or rotating under the supply of chemical energy through ATP hydrolysis. They are crucial for many cell functions and promising for nanotechnology of the future. Although such motors represent single molecules, their operation cycles cannot be followed in detail in simulations even on the best modern supercomputers and some approximations need to be employed. It is often assumed that conformational dynamics of motor proteins is well described within a linear response approximation and corresponds to excitation of normal modes. We have checked this assumption for two motor proteins, myosin V and kinesin KIF1A. Our results show that, while both these biomolecules respond by well-defined motions to energetic excitations, these motions are essentially nonlinear. The effect is particularly pronounced in KIF1A where relaxation proceeds through a sequence of qualitatively different conformational changes, which may facilitate complex functional motions without additional control mechanisms.

Dynamics of biomembranes with active multiple-state inclusions

Nonequilibrium dynamics of biomembranes with active multiple-state inclusions is considered. The inclusions represent protein molecules which perform cyclic internal conformational motions driven by the energy brought with adenosine triphosphate (ATP) ligands. As protein conformations cyclically change, this induces hydrodynamical flows and also directly affects the local curvature of a membrane. On the other hand, variations in the local curvature of the membrane modify the transition rates between conformational states in a protein, leading to a feedback in the considered system. Moreover, active inclusions can move diffusively through the membrane so that their surface concentration varies. The kinetic description of this system is constructed and the stability of the uniform stationary state is analytically investigated. We show that, as the rate of supply of chemical energy is increased above a certain threshold, this uniform state becomes unstable and stationary or traveling waves spontaneously develop in the system. Such waves are accompanied by periodic spatial variations of the membrane curvature and the inclusion density. For typical parameter values, their characteristic wavelengths are of the order of hundreds of nanometers. For traveling waves, the characteristic frequency is of the order of a thousand Hz or less. The predicted instabilities are possible only if at least three internal inclusion states are present.

Asymptotic nonequilibrium steady-state operators

We present a method for the calculation of asymptotic operators for nonequilibrium steady-state quantum systems. The asymptotic steady-state operator is obtained by averaging the corresponding operator in Heisenberg representation over infinitely long time. Several examples are considered to demonstrate the utility of our method. The results obtained within our approach are compared to those obtained within the Schwinger-Keldysh nonequilibrium Green's functions.

Towards active microfluidics: Interface turbulence in thin liquid films with floating molecular machines

Thin liquid films with floating active protein machines are considered. Cyclic mechanical motions within the machines, representing microscopic swimmers, lead to molecular propulsion forces applied to the air-liquid interface. We show that when the rate of energy supply to the machines exceeds a threshold, the flat interface becomes linearly unstable. As a result of this instability, the regime of interface turbulence, characterized by irregular traveling waves and propagating machine clusters, is established. Numerical investigations of this nonlinear regime are performed. Conditions for the experimental observation of the instability are discussed.

Nonequilibrium microstructures in reactive monolayers as soft matter systems

Chemical systems provide classical examples of nonequilibrium pattern formation. Reactions in weak aqueous solutions, such as the extensively investigated Belousov-Zhabotinsky reaction, demonstrate a rich variety of patterns, ranging from travelling fronts to rotating spiral waves and chemical turbulence. Pattern formation in such systems is based on interplay between the reactions and diffusion. Intrinsically, this puts a restriction on the minimum length scale of the developing structures, which cannot be shorter than the diffusion length of the reactants. However, much smaller nonequilibrium structures, with characteristic lengths reaching down to nanoscales, are also possible. They are found in reactive soft matter, where energetic interactions between molecules are present as well. In these systems, chemical reactions and diffusion interfere with phase transitions, yielding active, stationary or dynamic microstructures. Nonequilibrium soft-matter microstructures are of fundamental importance for biological cells and may have interesting engineering applications. In this Minireview, we focus on the microstructures found in reactive soft-matter monolayers at solid surfaces or liquid-air interfaces.