Simulation of F-actin filaments of several microns

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

Looking "Under the Hood" of Cellular Mechanotransduction with Computational Tools: A Systems Biomechanics Approach across Multiple Scales

Signal modulation has been developed in living cells throughout evolution to promote utilizing the same machinery for multiple cellular functions. Chemical and mechanical modules of signal transmission and transduction are interconnected and necessary for organ development and growth. However, due to the high complexity of the intercommunication of physical intracellular connections with biochemical pathways, there are many missing details in our overall understanding of mechanotransduction processes, i.e., the process by which mechanical signals are converted to biochemical cascades. Cell-matrix adhesions are mechanically coupled to the nucleus through the cytoskeleton. This modulated and tightly integrated network mediates the transmission of mechanochemical signals from the extracellular matrix to the nucleus. Various experimental and computational techniques have been utilized to understand the basic mechanisms of mechanotransduction, yet many aspects have remained elusive. Recently, in silico experiments have made important contributions to the field of mechanobiology. Herein, computational modeling efforts devoted to understanding integrin-mediated mechanotransduction pathways are reviewed, and an outlook is presented for future directions toward using suitable computational approaches and developing novel techniques for addressing important questions in the field of mechanotransduction.

Steered molecular dynamics analysis of the role of cofilin in increasing the flexibility of actin filaments

Cofilin is one of the most essential regulatory proteins and participates in the process of disassembling actin filaments. Cofilin induces conformational changes to actin filaments, and both the bending and torsional rigidity of the filament. In this study, we investigate the effects of cofilin on the mechanical properties of actin filaments using computational methods. Three models defined by their number of bound cofilins are constructed using coarse-grained MARTINI force field, and they are then extended with steered molecular dynamics simulation. After obtaining the stress-strain curves of the models, we calculate their Young's moduli and other mechanical properties that have not yet been determined for actin filaments. We analyze the cause of the different behaviors of the three models based on their atomistic geometrical differences. Finally, it is demonstrated that cofilin binding causes changes in the distances, angles, and stabilities of the residues in actin filaments.

Advances in the mechanical modeling of filamentous actin and its cross-linked networks on multiple scales

The protein actin is a part of the cytoskeleton and, therefore, responsible for the mechanical properties of the cells. Starting with the single molecule up to the final structure, actin creates a hierarchical structure of several levels exhibiting a remarkable behavior. The hierarchy spans several length scales and limitations in computational power; therefore, there is a call for different mechanical modeling approaches for the different scales. On the molecular level, we may consider each atom in molecular dynamics simulations. Actin forms filaments by combining the molecules into a double helix. In a model, we replace molecular subdomains using coarse-graining methods, allowing the investigation of larger systems of several atoms. These models on the nanoscale inform continuum mechanical models of large filaments, which are based on worm-like chain models for polymers. Assemblies of actin filaments are connected with cross-linker proteins. Models with discrete filaments, so-called Mikado models, allow us to investigate the dependence of the properties of networks on the parameters of the constituents. Microstructurally motivated continuum models of the networks provide insights into larger systems containing cross-linked actin networks. Modeling of such systems helps to gain insight into the processes on such small scales. On the other hand, they call for verification and hence trigger the improvement of established experiments and the development of new methods.

Optimization of an elastic network augmented coarse grained model to study CCMV capsid deformation

The major protective coat of most viruses is a highly symmetric protein capsid that forms spontaneously from many copies of identical proteins. Structural and mechanical properties of such capsids, as well as their self-assembly process, have been studied experimentally and theoretically, including modeling efforts by computer simulations on various scales. Atomistic models include specific details of local protein binding but are limited in system size and accessible time, while coarse grained (CG) models do get access to longer time and length scales but often lack the specific local interactions. Multi-scale models aim at bridging this gap by systematically connecting different levels of resolution. Here, a CG model for CCMV (Cowpea Chlorotic Mottle Virus), a virus with an icosahedral shell of 180 identical protein monomers, is developed, where parameters are derived from atomistic simulations of capsid protein dimers in aqueous solution. In particular, a new method is introduced to combine the MARTINI CG model with a supportive elastic network based on structural fluctuations of individual monomers. In the parametrization process, both network connectivity and strength are optimized. This elastic-network optimized CG model, which solely relies on atomistic data of small units (dimers), is able to correctly predict inter-protein conformational flexibility and properties of larger capsid fragments of 20 and more subunits. Furthermore, it is shown that this CG model reproduces experimental (Atomic Force Microscopy) indentation measurements of the entire viral capsid. Thus it is shown that one obvious goal for hierarchical modeling, namely predicting mechanical properties of larger protein complexes from models that are carefully parametrized on elastic properties of smaller units, is achievable.

A mechanochemical model of actin filaments

In eukaryotic cells, actin filaments are involved in important processes such as motility, division, cell shape regulation, contractility, and mechanosensation. Actin filaments are polymerized chains of monomers, which themselves undergo a range of chemical events such as ATP hydrolysis, polymerization, and depolymerization. When forces are applied to F-actin, in addition to filament mechanical deformations, the applied force must also influence chemical events in the filament. We develop an intermediate-scale model of actin filaments that combines actin chemistry with filament-level deformations. The model is able to compute mechanical responses of F-actin during bending and stretching. The model also describes the interplay between ATP hydrolysis and filament deformations, including possible force-induced chemical state changes of actin monomers in the filament. The model can also be used to model the action of several actin-associated proteins, and for large-scale simulation of F-actin networks. All together, our model shows that mechanics and chemistry must be considered together to understand cytoskeletal dynamics in living cells.

PIM: phase integrated method for normal mode analysis of biomolecules in a crystalline environment

In this study, a normal mode analysis, named phase integrated method (PIM), is developed for computing modes of biomolecules in a crystalline environment. PIM can calculate low-frequency modes on one or a few asymmetric units (AUs) and generate exact modes of a whole unit cell according to space group symmetry, while the translational symmetry between unit cells is maintained via the periodic boundary condition. Therefore, the method can dramatically reduce computational cost in mode calculation in the presence of crystal symmetry. PIM also has an option to map modes onto a single AU to form an orthonormalized mode set, which can be directly applied to normal-mode-based thermal parameter refinement in X-ray crystallography. The performance of PIM was tested on all 65 space groups available in protein crystals (one protein for each space group) and on another set of 83 ultra-high-resolution X-ray structures. The results showed that considering space group symmetry in mode calculation is crucial for accurately describing vibrational motion in a crystalline environment. Moreover, the optimal inter-AU packing stiffness was found to be about 60% of that of intra-AU interactions (non-bonded interaction only).

fSUB: normal mode analysis with flexible substructures

In this paper, we report a novel normal-mode analysis for supramolecular complexes, named fSUB. The method models a complex as a group of flexible substructures. The low-frequency substructure modes are first determined with substructures in isolation, and the motions of the whole complex are then calculated on the basis of substructure modes and substructure-substructure interactions. The calculation of modes in fSUB requires modal analysis without initial energy minimization, which is essential for maintaining energetic and structural consistency between substructures and whole complex. Compared with other coarse-grained methods, such as the RTB method, fSUB delivers much more accurate modes for the complex and allows for the choice of much larger substructures. The method can also accommodate any type of substructure arrangement including covalent bonds across the interface. In tests on molecular chaperonin GroEL (7350 residues) and HK97 capsid complex (118,092 residues), fSUB was shown to be much more efficient in terms of combined accuracy and demand of computing resources. Our results clearly demonstrated the vital importance of including substructure flexibility in complex modal analysis, as the deformational patterns of substructures were found to play an important role even in the lowest frequency modes of the whole complex.

Quantitative analysis of extension-torsion coupling of actin filaments

Actin filaments have a double-helix structure consisting of globular actin molecules. In many mechanical cellular activities, such as cell movement, division, and shape control, modulation of the extensional and torsional dynamics of the filament has been linked to regulatory actin-binding protein functions. Therefore, it is important to quantitatively evaluate extension-torsion coupling of filament to better understand the actin filament dynamics. In the present study, the extension-torsion coupling was investigated using molecular dynamics simulations. We constructed a model for the actin filament consisting of 14 actin subunits in an ionic solvent as a minimal functional unit, and analyzed longitudinal and twisting Brownian motions of the filament. We then derived the expected value of energy associated with extension and torsion at equilibrium, and evaluated the extension-torsion stiffness of the filament from the thermal fluctuations obtained from the MD simulations. The results demonstrated that as the analyzed sampling-window duration was increased, the extension-torsion coupling stiffness evaluated on a nanosecond scale tended to converge to a value of 7.6×10(-11) N. The results obtained from this study will contribute to the understanding of biomechanical events, under mechanical tension and torque, involving extension-torsion coupling of filaments.

Virtual interface substructure synthesis method for normal mode analysis of super-large molecular complexes at atomic resolution

Normal mode analysis of large biomolecular complexes at atomic resolution remains challenging in computational structure biology due to the requirement of large amount of memory space and central processing unit time. In this paper, we present a method called virtual interface substructure synthesis method or VISSM to calculate approximate normal modes of large biomolecular complexes at atomic resolution. VISSM introduces the subunit interfaces as independent substructures that join contacting molecules so as to keep the integrity of the system. Compared with other approximate methods, VISSM delivers atomic modes with no need of a coarse-graining-then-projection procedure. The method was examined for 54 protein-complexes with the conventional all-atom normal mode analysis using CHARMM simulation program and the overlap of the first 100 low-frequency modes is greater than 0.7 for 49 complexes, indicating its accuracy and reliability. We then applied VISSM to the satellite panicum mosaic virus (SPMV, 78,300 atoms) and to F-actin filament structures of up to 39-mer, 228,813 atoms and found that VISSM calculations capture functionally important conformational changes accessible to these structures at atomic resolution. Our results support the idea that the dynamics of a large biomolecular complex might be understood based on the motions of its component subunits and the way in which subunits bind one another.

Response of an actin filament network model under cyclic stretching through a coarse grained Monte Carlo approach

Cells are complex, dynamic systems that actively adapt to various stimuli including mechanical alterations. Central to understanding cellular response to mechanical stimulation is the organization of the cytoskeleton and its actin filament network. In this manuscript, we present a minimalistic network Monte Carlo based approach to model actin filament organization under cyclic stretching. Utilizing a coarse-grained model, a filament network is prescribed within a two-dimensional circular space through nodal connections. When cyclically stretched, the model demonstrates that a perpendicular alignment of the filaments to the direction of stretch emerges in response to nodal repositioning to minimize net nodal forces from filament stress states. In addition, the filaments in the network rearrange and redistribute themselves to reduce the overall stress by decreasing their individual stresses. In parallel, we cyclically stretch NIH 3T3 fibroblasts and find a similar cytoskeletal response. With this work, we test the hypothesis that a first-principles mechanical model of filament assembly in a confined space is by itself capable of yielding the remodeling behavior observed experimentally. Identifying minimal mechanisms sufficient to reproduce mechanical influences on cellular structure has important implications in a diversity of fields, including biology, physics, medicine, computer science, and engineering.

A finite element framework for computation of protein normal modes and mechanical response

A computational framework based on the Finite Element Method is presented to calculate the normal modes and mechanical response of proteins and their supramolecular assemblies. Motivated by elastic network models, proteins are treated as continuum elastic solids with molecular volume defined by their solvent-excluded surface. The discretized Finite Element representation is obtained using a surface simplification algorithm that facilitates the generation of models of arbitrary prescribed spatial resolution. The procedure is applied to a mutant of T4 phage lysozyme, G-actin, syntenin, cytochrome-c', beta-tubulin, and the supramolecular assembly filamentous actin (F-actin). Equilibrium thermal fluctuations of alpha-carbon atoms and their inter-residue correlations compare favorably with all-atom-based results, the Rotational-Translational Block procedure, and experiment. Additionally, the free vibration and compressive buckling responses of F-actin are in quantitative agreement with experiment. The proposed methodology is applicable to any protein or protein assembly and facilitates the incorporation of specific atomic-level interactions, including aqueous-electrolyte-mediated electrostatic effects and solvent damping. The procedure is equally applicable to proteins with known atomic coordinates as it is to electron density maps of proteins, protein complexes, and supramolecular assemblies of unknown atomic structure.

Coarse-grained models reveal functional dynamics--I. Elastic network models--theories, comparisons and perspectives

In this review, we summarize the progress on coarse-grained elastic network models (CG-ENMs) in the past decade. Theories were formulated to allow study of conformational dynamics in time/space frames of biological interest. Several highlighted models and their underlined hypotheses are introduced in physical depth. Important ENM offshoots, motivated to reproduce experimental data as well as to address the slow-mode-encoded configurational transitions, are also introduced. With the theoretical developments, computational cost is significantly reduced due to simplified potentials and coarse-grained schemes. Accumulating wealth of data suggest that ENMs agree equally well with experiment in describing equilibrium dynamics despite their distinct potentials and levels of coarse-graining. They however do differ in the slowest motional components that are essential to address large conformational changes of functional significance. The difference stems from the dissimilar curvatures of the harmonic energy wells described for each model. We also provide our views on the predictability of 'open to close' (open-->close) transitions of biomolecules on the basis of conformational selection theory. Lastly, we address the limitations of the ENM formalism which are partially alleviated by the complementary CG-MD approach, to be introduced in the second paper of this two-part series.

A New Method for Coarse-Grained Elastic Normal-Mode Analysis

In this paper, we report a new method for coarse-grained elastic normal-mode analysis. The purpose is to overcome a long-standing problem in the conventional analysis called the tip effect that makes the motional patterns (eigenvectors) of some low-frequency modes irrational. The new method retains the merits of a conventional method such as not requiring lengthy initial energy minimization, which always distorts structures, and also delivers substantially more accurate low-frequency modes with no tip effect for proteins of any size. This improvement of modes is crucial for certain types of applications such as structural refinement or normal-mode-based sampling.

Energetics and dynamics of constrained actin filament bundling

The formation of filopodia-like bundles from a dendritic actin network has been observed to occur in vitro as a result of branching induced by Arp2/3 complex. We study both the energetics and dynamics of actin filament bundling in such a network to evaluate their relative importance in bundle formation processes. Our model considers two semiflexible actin filaments fixed at one end and free at the other, described using a normal-mode approximation. This model is studied by both Brownian dynamics and free-energy minimization methods. Remarkably, even short filaments can bundle at separations comparable to their lengths. In the dynamic simulations, we evaluate the time required for the filaments to interact and bind, and examine the dependence of this bundling time on the filament length, the distance between the filament bases, and the cross-linking energy. In most cases, bundling occurs in a second or less. Beyond a certain critical distance, we find that the bundling time increases very rapidly with increasing interfilament separation and/or decreasing filament length. For most of the cases we have studied, the energetics results for this critical distance are similar to those obtained from dynamics simulations run for 10 s, suggesting that beyond this timescale, energetics, rather than kinetic constraints, determine whether or not bundling occurs. Over a broad range of conditions, we find that the times required for bundling from a network are compatible with experimental observations.

Coarse-grained modeling of the actin filament derived from atomistic-scale simulations

A coarse-grained (CG) procedure that incorporates the information obtained from all-atom molecular dynamics (MD) simulations is presented and applied to actin filaments (F-actin). This procedure matches the averaged values and fluctuations of the effective internal coordinates that are used to define a CG model to the values extracted from atomistic MD simulations. The fluctuations of effective internal coordinates in a CG model are computed via normal-mode analysis (NMA), and the computed fluctuations are matched with the atomistic MD results in a self-consistent manner. Each actin monomer (G-actin) is coarse-grained into four sites, and each site corresponds to one of the subdomains of G-actin. The potential energy of a CG G-actin contains three bonds, two angles, and one dihedral angle; effective harmonic bonds are used to describe the intermonomer interactions in a CG F-actin. The persistence length of a CG F-actin was found to be sensitive to the cut-off distance of assigning intermonomer bonds. Effective harmonic bonds for a monomer with its third nearest neighboring monomers are found to be necessary to reproduce the values of persistence length obtained from all-atom MD simulations. Compared to the elastic network model, incorporating the information of internal coordinate fluctuations enhances the accuracy and robustness for a CG model to describe the shapes of low-frequency vibrational modes. Combining the fluctuation-matching CG procedure and NMA, the achievable time- and length scales of modeling actin filaments can be greatly enhanced. In particular, a method is described to compute the force-extension curve using the CG model developed in this work and NMA. It was found that F-actin is easily buckled under compressive deformation, and a writhing mode is developed as a result. In addition to the bending and twisting modes, this novel writhing mode of F-actin could also play important roles in the interactions of F-actin with actin-binding proteins and in the force-generation process via polymerization.

Usefulness and limitations of normal mode analysis in modeling dynamics of biomolecular complexes

Various types of large-amplitude molecular deformation are ubiquitously involved in the functions of biological macromolecules, especially supramolecular complexes. They can be very effectively analyzed by normal mode analysis with well-established procedures. However, despite its enormous success in numerous applications, certain issues related to the applications of normal mode analysis require further discussion. In this review, the author addresses some common issues so as to raise the awareness of the usefulness and limitations of the method in the general community of structural biology.

The role of shape in determining molecular motions

We examined the role of molecular shape in determining the patterns of low-frequency deformational motions of biological macromolecules. The low-frequency subspace of eigenvectors in normal mode analysis was found to be robustly similar upon randomization of the Hessian matrix elements as long as the structure of the matrix is maintained, which indicates that the global shape of molecules plays a more dominant role in determining the highly anisotropic low-frequency motions than the absolute values of stiffness and directionality of local interactions. The results provided a quantitative foundation for the validity of elastic normal mode analysis.

Significance of kinetic degrees of freedom in operation of the actomyosin motor

The actomyosin motor as a principal functional component of cell motility is highly coordinated in regulating the participating molecular components. At the same time, it has to be flexible and plastic enough to accommodate itself to a wide variety of operational conditions. We prepared two different types of actomyosin systems. One is a natural intact actomyosin system with no artificial constraint on the kinetic degrees of freedom of the actin filaments, and the other is a regulated one with actin filaments supplemented by intra- and intermolecular crosslinking to suppress the kinetic degrees of freedom to a certain extent. Crosslinked actomyosin systems were found to remain almost insensitive to calcium regulation even when intact troponin-tropomyosin regulatory component was incorporated. Both the ATPase and the motile activities of the actin filaments sliding on myosin molecules were markedly lowered by the crosslinking. In contrast, once the crosslinking was cleaved, both properties returned to the normal as with intact actomyosin systems.

New advances in normal mode analysis of supermolecular complexes and applications to structural refinement

Normal mode analysis is an effective computational method for studying large-amplitude low-frequency molecular deformations that are ubiquitously involved in the functions of biological macromolecules, especially supermolecular complexes. The recent years have witnessed a substantial advance in methodology development in the field. This review is intended to summarize some of the important advances that enable one to simulate deformations of supermolecular complexes at expended resolution- and length-scales, with particular emphasis on the implications in structural refinement against low- to intermediate-resolution structural data such as those from electron cryomicroscopy and fibre diffraction.

Refinement of F-actin model against fiber diffraction data by long-range normal modes

The atomic model of F-actin was refined against fiber diffraction data using long-range normal modes as adjustable parameters to account for the collective long-range filamentous deformations. To determine the effect of long-range deformations on the refinement, each of the four domains of G-actin was treated as a rigid body. It was found that among all modes, the bending modes make the most significant contributions to the improvement of the refinement. Inclusion of only 7-9 bending modes as adjustable parameters yielded a lowest R-factor of 6.3%. These results demonstrate that employing normal modes as refinement parameters has the advantage of using a small number of adjustable parameters to achieve a good fitting efficiency. Such a refinement procedure may therefore prevent the refinement from overfitting the structural model. More importantly, the results of this study demonstrate that, for any fiber diffraction data, a substantial amount of refinement error is due to long-range deformations, especially the bending, of the filaments. The effects of these intrinsic deformations cannot be easily compensated for by adjusting local structural parameters, and must be properly accounted for in the refinement to achieve improved fit of refined models with experimental diffraction data.