Nonlinearity of mechanochemical motions in motor proteins

Irreversibility of stochastic state transitions in Langevin systems with odd elasticity

Active microscopic objects, such as an enzyme molecule, are modeled by the Langevin system with the odd elasticity, in which energy injection from the substrate to the enzyme is described by the antisymmetric part of the elastic matrix. By applying the Onsager-Machlup integral and large deviation theory to the Langevin system with odd elasticity, we can calculate the cumulant generating function of the irreversibility of the state transition. For an N-component system, we obtain a formal expression of the cumulant generating function and demonstrate that the oddness λ, which quantifies the antisymmetric part of the elastic matrix, leads to higher-order cumulants that do not appear in a passive elastic system. To demonstrate the effect of the oddness under the concrete parameter, we analyze the simplest two-component system and obtain the optimal transition path and cumulant generating function. The cumulants obtained from expansion of the cumulant generating function increase monotonically with the oddness. This implies that the oddness causes the uncertainty of stochastic state transitions.

Nonreciprocality of a micromachine driven by a catalytic chemical reaction

We propose a model that describes cyclic state transitions of a micromachine driven by a catalytic chemical reaction. We consider a mechanochemical coupling of variables representing the degree of a chemical reaction and the internal state of a micromachine. The total free energy consists of a tilted periodic potential and a mechanochemical coupling energy. We assume that the reaction variable obeys a deterministic stepwise dynamics characterized by two typical timescales, i.e., the mean first passage time and the mean first transition path time. To estimate the functionality of a micromachine, we focus on the quantity called "nonreciprocality" and further discuss its dependence on the properties of catalytic reaction. For example, we show that the nonreciprocality is proportional to the square of the mean first transition path time. The explicit calculation of the two timescales within the decoupling approximation model reveals that the nonreciprocality is inversely proportional to the square of the energy barrier of catalytic reaction.

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.

Oscillating Piconewton Force Manipulation on Single-Molecule Enzymatic Conformational and Reaction Dynamics

Oscillation force has been demonstrated in theoretical studies as a critical role in unraveling the comprehensive enzymatic dynamics and addressing its regulation on enzyme activity. Utilizing the imposed external mechanical oscillation force by our newly developed magnetic tweezers coupled with a single-molecule photon-stamping imaging spectroscopic microscope, we experimentally studied a millisecond-scale oscillation force manipulation on single horseradish peroxidase enzymatic conformational and reaction dynamics. We have studied the enzymatic reaction dynamics and found that the enzyme activity changes under the real-time oscillatory force manipulation. Moreover, the oscillation force shows the capability of manipulating the enzyme active-site conformational state as well as the nascent-formed product's interaction with the active site of the enzyme, which impacts on the product release pathways. Specifically, we have identified that there are two product releasing pathways, the solvation-mediated diffusion releasing pathway and the spilling-out releasing pathway. We have observed that the spilling-out pathway can be significantly perturbed by the oscillatory force manipulation. Our correlated interpretation of enzymatic conformational and reaction dynamics provides a new insight into the comprehensive understanding of the complex conformational dynamics evolved in an enzymatic reaction. Technically, we have also demonstrated a novel approach capable of unfolding an enzyme under an enzymatic reaction condition in real time and, furthermore, by using an oscillatory mechanical weak piconewton force to manipulate enzyme conformations, and the enzyme thermal fluctuation is fully maintained. The real-time in situ fluorescence probe at the enzymatic active site reports the active-site conformational dynamics through each enzymatic reaction turnovers.

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.

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.

Identification of tail binding effect of kinesin-1 using an elastic network model

Kinesin is a motor protein that delivers cargo inside a cell. Kinesin has many different families, but they perform basically same function and have same motions. The walking motion of kinesin enables the cargo delivery inside the cell. Autoinhibition of kinesin is important because it explains how function of kinesin inside a cell is stopped. Former researches showed that tail binding is related to autoinhibition of kinesin. In this work, we performed normal mode analysis with elastic network model using different conformation of kinesin to determine the effect of tail binding by considering four models such as functional form, autoinhibited form, autoinhibited form without tail, and autoinhibited form with carbon structure. Our calculation of the thermal fluctuation and cross-correlation shows the change of tail-binding region in structural motion. Also strain energy of kinesin showed that elimination of tail binding effect leads the structure to have energetically similar behavior with the functional form.

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.

Comparing the intrinsic dynamics of multiple protein structures using elastic network models

BACKGROUND

Elastic network models (ENMs) are based on the simple idea that a protein can be described as a set of particles connected by springs, which can then be used to describe its intrinsic flexibility using, for example, normal mode analysis. Since the introduction of the first ENM by Monique Tirion in 1996, several variants using coarser protein models have been proposed and their reliability for the description of protein intrinsic dynamics has been widely demonstrated. Lately an increasing number of studies have focused on the meaning of slow dynamics for protein function and its potential conservation through evolution. This leads naturally to comparisons of the intrinsic dynamics of multiple protein structures with varying levels of similarity.

SCOPE OF REVIEW

We describe computational strategies for calculating and comparing intrinsic dynamics of multiple proteins using elastic network models, as well as a selection of examples from the recent literature.

MAJOR CONCLUSIONS

The increasing interest for comparing dynamics across protein structures with various levels of similarity, has led to the establishment and validation of reliable computational strategies using ENMs. Comparing dynamics has been shown to be a viable way for gaining greater understanding for the mechanisms employed by proteins for their function. Choices of ENM parameters, structure alignment or similarity measures will likely influence the interpretation of the comparative analysis of protein motion.

GENERAL SIGNIFICANCE

Understanding the relation between protein function and dynamics is relevant to the fundamental understanding of protein structure-dynamics-function relationship. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

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.

Coarse-grained modeling of conformational transitions underlying the processive stepping of myosin V dimer along filamentous actin

To explore the structural basis of processive stepping of myosin V along filamentous actin, we have performed comprehensive modeling of its key conformational states and transitions with an unprecedented residue level of details. We have built structural models for a myosin V monomer complexed with filamentous actin at four biochemical states [adenosine diphosphate (ATP)-, adenosine diphosphate (ADP)-phosphate-, ADP-bound or nucleotide-free]. Then we have modeled a myosin V dimer (consisting of lead and rear head) at various two-head-bound states with nearly straight lever arms rotated by intramolecular strain. Next, we have performed transition pathway modeling to determine the most favorable sequence of transitions (namely, phosphate release at the lead head followed by ADP release at the rear head, while ADP release at the lead head is inhibited), which underlie the kinetic coordination between the two heads. Finally, we have used transition pathway modeling to reveal the order of structural changes during three key biochemical transitions (phosphate release at the lead head, ADP release and ATP binding at the rear head), which shed lights on the strain-dependence of the allosterically coupled motions at various stages of myosin V's work cycle. Our modeling results are in agreement with and offer structural insights to many results of kinetic, single-molecule and structural studies of myosin V.

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.

Mechanism of processive movement of monomeric and dimeric kinesin molecules

Kinesin molecules are motor proteins capable of moving along microtubule by hydrolyzing ATP. They generally have several forms of construct. This review focuses on two of the most studied forms: monomers such as KIF1A (kinesin-3 family) and dimers such as conventional kinesin (kinesin-1 family), both of which can move processively towards the microtubule plus end. There now exist numerous models that try to explain how the kinesin molecules convert the chemical energy of ATP hydrolysis into the mechanical energy to "power" their processive movement along microtubule. Here, we attempt to present a comprehensive review of these models. We further propose a new hybrid model for the dimeric kinesin by combining the existing models and provide a framework for future studies in this subject.