# Computer Simulation of Protein Materials at Multiple Length Scales: From Single Proteins to Protein Assemblies

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## Abstract

Computer simulation of protein materials and their dynamic or mechanical behavior is of high significance, as proteins perform their functions through their structural changes in response to a force (or stimulus). The computer simulation enables the detailed insight into the structure and behavior of proteins at atomistic resolution, which is inaccessible with experimental toolkits such as single-molecule experiments. With the advancement of computing resources, the computer simulation has recently played a vital role as a virtual microscopy in understanding and characterizing the structure and behaviors of protein materials at atomistic resolution. For examples, computer simulations allow for gaining insight into how some protein domains can exhibit the remarkable mechanical properties and functions. In this article, we would like to address the current state-of-arts of computer simulations that have been employed for studying the structure and properties (or behaviors) of proteins at multiple length scales ranging from single proteins to protein assemblies. Specifically, we summarize various computational modeling techniques, ranging from atomistic models to coarse-grained models and continuum models, applicable for modeling protein structures at multiple length scales from single proteins to protein assemblies. This paper discusses how such various computational modeling/simulation techniques can be employed for studying the structure and properties of protein materials at multiple length scales. This paper sheds light on computer simulations, which are able to unveil the hidden, complex mechanisms related to the structure and properties of protein materials at multiple length scales.

## Keywords

Computer simulation Protein materials Multiple length scales Atomistic simulation Coarse-grained modeling Protein mechanics## Introduction

It is of high importance to study the dynamic (or mechanical) behavior of a single protein molecule, as a protein molecule performs its function through its structural changes upon a chemical stimulus (e.g. ligand binding) and/or a force. For example, hemoglobin transports the oxygen when the oxygen is bound to the hemoglobin. The ability of hemoglobin to bind with oxygen is attributed to the existence of the hemoglobin’s various conformations (structures), few of which may correspond to the conformation of oxygen-bound hemoglobin. In other words, a protein molecule can explore many conformations, some of which corresponds to the ligand-bound state, because the energy landscape of a protein is very rugged in that there are multiple local minima (or equilibria), and that the energy barrier between these minima is comparable to the amount of thermal energy [1, 2, 3]. This unique dynamic feature of a protein molecule enables it to perform its function such as ligand binding or unbinding. Moreover, some protein domains are able to perform remarkable mechanical functions and to exhibit excellent mechanical properties. For instance, an immunoglobulin (Ig) domain in a muscle protein chain titin has a high resilience against a mechanical force in such a way that hydrogen bonds formed between the β-strands of an Ig domain are able to sustain a mechanical force [4, 5, 6, 7]. In addition, the high resilience of a protein domain is also attributed to its unfolding behavior, that is, a force-driven unfolding behavior of a protein folded domain results in its energy dissipation, and consequently, the high mechanical resilience [8]. These two representative examples suggest the necessity of studying the dynamic (or mechanical) behavior of proteins in order to gain a fundamental insight into their functions.

Protein assemblies have recently received significant attention due to their important role in disease pathology [9, 10, 11, 12] as well as their applicability in bionanotechnology [13, 14, 15]. As a representative example, amyloid fibrils have been highlighted due to their role in the pathogenesis of various diseases ranging from neurodegenerative diseases [16, 17] to type 2 diabetes [18] and cardiovascular diseases [19]. The amyloid fibrils are formed through self-assembly process, that is, amyloidogenic proteins (e.g. denatured proteins) are aggregated so as to form the fibril structure [10]. Once amyloid fibrils are formed by self-assembly process, they are quite rigid and structurally stable in a physiological condition, resulting in the formation of amyloid plaques. These fibrils have recently suggested to play a role in disease pathologies due to their remarkable mechanical properties. Specifically, the elastic modulus of amyloid fibrils is in a range of 1–10 GPa [20, 21, 22], which is a few orders of magnitude higher than the elastic modulus of cell membrane. The growth of an amyloid fibril on a cell membrane results in the mechanical disruption of cell membrane [23] due to the difference between elastic moduli of amyloid fibril and cell membrane [24]. In addition, a spider silk crystal, whose molecular structure is similar to that of amyloid fibril, possesses anomalous mechanical properties whose elastic modulus is in the order of 1–10 GPa [25]. The remarkable properties of these protein assemblies (e.g. amyloid fibril, and spider silk crystal) led researchers to study the underlying principles showing how the excellent mechanical properties of protein assemblies can be determined.

For last two decades, single-molecule experimental techniques [26] based on atomic force microscopy (AFM) [27, 28, 29] and/or optical tweezer [30] have played a central role in probing the properties and functions of protein molecules. As an example, AFM-based single-molecule pulling experiments have allowed for gaining a quantitative insight into the response of a protein domain to a mechanical force. Specifically, AFM-based pulling experiments are able to capture the force-driven unfolding event of a protein domain based on measuring the force-extension (equivalent to force–displacement) relationship of a protein domain during its unfolding process. Despite the ability of single-molecule experimental techniques to quantitatively probe the properties of protein molecules, they are very restrictive in that they are unable to provide how the molecular structure of a protein molecule is changed (e.g. deformed) due to a force. In addition, these experimental techniques cannot unveil the underlying principles showing how the properties and behaviors of protein materials are determined. Due to these limitations of single-molecule experimental techniques, computer simulations can play an alternative role in revealing the underlying mechanisms of the dynamic (or mechanical) behaviors and properties of protein materials at atomistic resolution. In particular, computer simulations based on atomistic models are also able to provide a quantitative insight into the relationship between the molecular structure of protein materials and their properties [31, 32]. Furthermore, computer simulations are capable of quantitatively understanding how the molecular structure of protein materials (e.g. protein assemblies) can be determined. The computer simulations can be implemented based on different modeling techniques at different length scales, ranging from atomistic model to coarse-grained model and continuum model. Here, we note that this paper excludes the details of a continuum mechanics-based simulation, which has recently been applied for studying the dynamics of proteins [33, 34], while we briefly review the continuum model that is useful in analyzing the result of computer simulations based on atomistic or coarse-grained models.

In this work, we address the current state-of-arts of computational simulations of protein materials at multiple length scales, ranging from a protein domain to a protein assembly, for unveiling the hidden, complex mechanisms of how the remarkable properties (or functions) of protein materials are determined. In particular, we describe the current state-of-arts of computational simulations at multiple length scales based on atomistic model, coarse-grained model, and/or continuum modeling, which are applicable for understanding the structure–property-function relationship of protein materials at different length scales ranging from a single protein domain to protein assembly. In addition, we present the computer simulation-based analyses of single proteins such as their unfolding mechanics and/or conformational changes. Moreover, we introduced the computer simulation-based characterization of the structures and properties of protein assemblies such as amyloid fibrils and viral capsids. Finally, we conclude our paper with some remarks and outlooks for future works in computational modeling of protein materials.

## Computational Methodologies

This Section demonstrates the fundamental principles of computational methodologies that have been applied for modeling protein materials. Here, we present the computational simulation methodologies at different length scales, based on atomistic model or coarse-grained model.

### Atomistic Modeling/Simulation

#### Molecular Dynamics Simulations

The atomistic molecular dynamics (MD) simulation had been dated back to 1950s, when Alder and Wainwright [35] first considered the atomistic modeling for studying the phase transition of hard-sphere gas system. From 1950s to 1960s, the MD simulations were extensively employed for studying the phase transition of gas and solution. Not until 1970s were the MD simulations employed for modeling biological molecules such as DNA, RNA, and protein. In 1977, Karplus and colleagues [36] had utilized the MD simulations for studying the conformational dynamics of bovine trypsin inhibitor (that is a very small protein molecule). In 1970s, Karplus [37], Levitt [38], and Warshel studied the dynamics of biological molecules (that is related to their function) by using the MD simulations, which led them to become a pioneer in the MD simulation-based study of biological molecules and their dynamics. The pioneering works of Karplus, Levitt, and Warshel allowed them to win a Nobel prize in Chemistry 2013 [39]. Since 2000s, the MD simulations have been extensively used for gaining a quantitative insight into the structure, dynamics, properties, and functions of biological molecules such as proteins, DNAs, and RNAs [40, 41, 42, 43].

**r**

_{i}is the position vector of an

*i*-th atom in a molecular system,

*m*

_{i}is the mass of the

*i*-th atom,

*N*is the total number of atoms for the molecular system,

*V*is the potential energy of the molecular system, and a symbol ∇

_{i}represents a spatial gradient with respect to position vector

**r**

_{i}. With a given initial position vector

**r**

_{i}(0) for the molecular system, the time-dependent position vector

**r**

_{i}(

*t*) of the molecular system can be obtained by the numerical integration of the equation of motion given by Eq. (1). The details of numerical integration scheme are well described in ref [44]. The dynamic behavior of a biological molecule (e.g. protein) can be analyzed based on calculating the fluctuation matrix

**Q**that consists of block matrices

**Q**

_{ij}defined as [45]

*i*-th atom can be measured as follows:

*x*

_{i},

*y*

_{i}, and

*z*

_{i}represent the components of a position vector

**r**

_{i}(

*t*) in the

*x*-,

*y*-, and

*z*-directions, respectively. The RMSF of a protein is very useful in not only understanding the fluctuation motion of the protein but also validating the atomistic model by fitting the RMSF obtained from the MD simulation to that acquired from experiments based on nuclear magnetic resonance or X-ray crystallography. In addition, the correlated motion of a protein can be quantitatively described using a correlation factor (CF) defined as

It should be noted that the value of CF is in a range of − 1 to 1. The value of CF being + 1 indicates that two atoms *i* and *j* move collectively (i.e. together) in the same direction, while the value of CF being − 1 means that these two atoms move in the opposite direction, and the value of CF being zero implies that the motion of an atom *i* is not correlated with that of other atom *j*.

*K*is the stiffness of a force probe,

*R*

_{ij}is a distance between two atoms, one of which is fixed and the other is pulled by a force probe,

*v*is a pulling rate, and

*t*is the time. The equation of motion for a protein that is pulled by a force probe is given by

The numerical simulation of solving the Newton’s equation of motion given by Eq. (6) for studying the mechanical response of a protein is referred to as steered molecular dynamics (SMD) simulations [46]. The SMD simulations enables the acquisition of the force–displacement relationship of a biomolecule (e.g. DNA, RNA, and proteins), and consequently, the extraction of the mechanical properties of a biomolecule.

#### Normal Mode Analysis

*V*= (

*γ*/2)

*u*

^{2}, where

*γ*is a force constant, and

*u*is a distance between two atoms. The kinetic energy of the diatomic molecule is given as

*L*= (

*m*/2)

*v*

^{2}, where

*m*is the atomic mass, and

*v*is the velocity defined as

*v*=

*du*/

*dt*. The statistical mechanics theory provides a probability distribution,

*p*(

*u*,

*v*), to find a specific position and velocity (

*u*,

*v*) on the phase space given as

*k*

_{B}and

*T*are the Boltzmann’s constant and absolute temperature, respectively,

*H*is the Hamiltonian of the diatomic molecule, and

*Z*is the partition function defined as

The statistical mechanics theory given by Eq. (9.b) suggests that the fluctuation behavior of a molecular system is dependent on its stiffness in such a way that the mean-square fluctuation of the system is inversely proportional to its stiffness.

*N*atoms. For the dynamic motion of a protein near its equilibrium state, its potential field can be approximated to a quasi-harmonic potential such as [48, 49, 50]

**K**is the stiffness matrix of a molecular system,

**r**is the atomic coordinates of the system, and a superscript

*T*indicates a transpose of a vector. With a spectral decomposition of the stiffness matrix

**K**such as

**K**=

**P**

^{T}Λ

**P**, where Λ is a diagonal matrix consisting of eigenvalues

*λ*

_{j}for the stiffness matrix and

**P**is the modal matrix, the potential energy can be rewritten as

**y**is a transformed coordinate defined as

**y**=

**Pr**. It should be noted that summation in Eq. (11) excludes six zero normal modes (i.e. rigid body modes). The statistical mechanics theory provides a probability function

*p*(

**y**) to find a specific state

**y**given by

*p*(

**y**) given by Eq. (12), the ensemble averages of quantities,

*y*

_{i}and

*y*

_{i}·

*y*

_{j}, can be found as follows.

*δ*

_{ij}is a Kronecker delta defined as

*δ*

_{ij}= 1 if

*i*=

*j*, otherwise

*δ*

_{ij}= 0. Here, we note that the repeated index in Eq. (13.b) does not imply the Einstein summation rule. With using a transformation

**r**=

**P**

^{T}

**y**, the fluctuation matrix

**Q**of a molecular system can be written as

**P**

_{j}is the

*j*-th column vector of a modal matrix

**P**, that is,

**P**

_{j}is the

*j*-th normal mode of the system.

### Coarse-Grained Modeling

#### Principle of Coarse-Graining

For more than three decades, coarse-grained (CG) modeling techniques have been established and widely utilized for understanding the dynamic behavior of a large protein structures [51, 52, 53]. Though a full atomistic MD simulations have recently allowed for gaining insight into the dynamics of a small protein (consisting of less than few hundreds of residues) at millisecond time scales [54, 55], the full atomistic MD simulations are still computationally restrictive in characterizing the dynamic behavior of a large protein structures at long timescale [56]. In order to overcome the computational limitation of atomistic MD simulations, CG modeling approaches have been widely developed in such a way that a group of atoms, which move collectively, are replaced by a single bead (particle), and that the motion of such beads (that represent the groups of atoms) is numerically predicted based on Newtonian equation of motion while the interaction potential energies between beads have to be properly defined.

*j*indicates the

*j*-th group of atoms,

*m*

_{l}and

**r**

_{l}are the molecular weight and position vector of the

*l*-th atom, respectively, and

*n*

_{j}is the number of atoms in the

*j*-th group. It should be noted that the position vector of a bead (representing a group of atoms) is equivalent to the center of mass for the atoms that are grouped. Once the groups of atoms are replaced with CG beads, the effective potential field that can be prescribed to CG beads has to be defined. The effective potential field can be written as

*k*

_{D},

*k*

_{Ω}, and

*k*

_{Φ}are the force constants for strain energies corresponding to the stretching of a pseudo-covalent bond, bending of a pseudo-bond angle, and twisting of a pseudo-dihedral angle, respectively,

*N*is the total number of CG beads,

*D*

_{j}is the distance of a pseudo-covalent bond such as

*D*

_{j}= |

**D**

_{j}| = |

**R**

_{j+1}−

**R**

_{j}|, Ω

_{j}is a pseudo-bond angle defined as cosΩ

_{j}=

**D**

_{j}·

**D**

_{j+1}/

*D*

_{j}

*D*

_{j+1}, Φ

_{j}is a pseudo-dihedral angle given as cosΦ

_{j}= (

**D**

_{j}×

**D**

_{j+1})·

**D**

_{j+2}/|(

**D**

_{j}×

**D**

_{j+1})·

**D**

_{j+2}|,

*Q*

_{j}is an effective charge acting on a

*j*-th CG bead,

*X*

_{jk}is a distance between two CG beads

*j*and

*k*such as

*X*

_{jk}= |

**R**

_{j}−

**R**

_{k}|,

*E*

_{0}is the energy scale of van der Waal’s interaction between CG beads, and \(\Upsilon\) is a length scale for van der Waal’s interaction. The major procedure of CG modeling is to determine the parameters of effective potential field given by Eq. (17). The effective potential field can be validated by comparing the results of CG model-based simulations with those of atomistic MD simulations.

**K**) is composed of (off-diagonal) block matrices

**K**

_{ij}given by

Here, gradient ∇_{i} is defined with respect to the coordinates of CG beads, such as ∇_{i} = ∂/∂**R**_{i}. As a consequence, once the eigen-values and their corresponding normal modes are computed, the statistical mechanics theory given by Eq. (14) enables the quantitation of the fluctuation matrix **Q** for the CG model.

Here, *M*_{i} is the effective molecular weight of the *i*-th CG bead. The time-dependent trajectories of CG beads are obtained by numerically integrating the Newtonian equation of motion given by Eq. (19). This method is referred to as coarse-grained MD (CG-MD) simulations.

#### Go-Like Model

*N*is the total number of CG beads,

*d*

_{ij}is the distance between two CG beads

*i*and

*j*,

*k*

_{1}and

*k*

_{2}are force constants for harmonic and quartic potential, respectively, for covalent bond,

*e*

_{0}and

*σ*are the energy parameter and length scale, respectively, for non-bonded interaction, superscript 0 indicates the equilibrium state, and

*δ*

_{i,j}is a Kronecker delta defined as

*δ*

_{i,j}= 1 if

*i*=

*j*; otherwise,

*δ*

_{i,j}= 0.

Here, *m*_{i} and **r**_{i} are the mass and position vector of an *i*-th CG bead, respectively, *η* is a friction coefficient, **f**_{i}(*t*) is a Gaussian random force acting on the *i*-th CG bead, and *U*(**r**) is an effective potential field defined as *U* = *V* + *V*_{L} with *V* and *V*_{L} being the potential field of the CG protein structure given by Eq. (20) and a potential field for a force probe, respectively. It should be noted that **f**_{i}(*t*) satisfies the fluctuation–dissipation theorem such as < **f**_{i}(*t*)·**f**_{i}(*τ*) > = 2*ηk*_{B}*Tδ*(*t* − *τ*), where *k*_{B} and *T* are the Boltzmann’s constant and absolute temperature, respectively, an angle bracket < > indicates an ensemble average (equivalent to the time average for the sufficient long-time simulation for an ergodic system), and *δ*(*t*) is the Dirac delta function. In addition, we note that the potential field of a force probe is typically assumed to a harmonic potential given as *V*_{L} = (*K*/2)(*d*_{1,N} − *vt*)^{2}, where *v* is a pulling speed with which a protein is extended, and *K* is the stiffness of a force probe. By numerically solving the Langevin equation given by Eq. (21), one is able to obtain the dynamic trajectory of a coarse-grained protein structure in response to a force-driven extension with a given pulling speed, which results in gaining the force-extension (i.e. force–displacement) curve for a protein structure, and consequently, its physical properties (e.g. elastic modulus).

#### Elastic Network Model

Though the Go-like model [57, 58, 59] is able to depict the dynamic and mechanical response of a protein structure, it is still computationally limited for studying the dynamic behavior of a large protein complex (or assembly), whose size is much larger than that of a single protein. In order to reduce a computational cost for simulating the dynamic behavior of large protein complex, Tirion [61] introduced an elastic network model (ENM), which assumes a harmonic potential for neighboring alpha carbon atoms. Bahar and colleagues [62] developed a Gaussian network model (GNM), in which a protein structure is described as a network of Gaussian chains that are widely used in rubber elasticity. Despite the inability of GNM to dictate the directionality of normal modes, GNM was extensively utilized for studying the fluctuation dynamics and its correlated functions of large protein complexes. Atilgan et al. [63] introduced an anisotropic network model (ANM) which is able to depict the anisotropic fluctuation behavior of a protein structure. Here, we note that ANM is conceptually identical to ENM. The normal modes of ENM was found to be useful in predicting the conformational changes of a large protein complex due to a stimulus (such as ligand-binding).

*γ*is a force constant,

*R*

_{c}is a cut-off distance (usually given by

*R*

_{c}= 7–10 Å), and

*H*(

*x*) is a Heaviside unit step function defined as

*H*(

*x*) = 1 if

*x*> 0; otherwise,

*H*(

*x*) = 0.

*u*

_{i}is the fluctuation of the

*i*-th alpha carbon atom (i.e. CG bead) defined as

*u*

_{i}= |

**r**

_{i}–

**r**

_{ i}

^{0}| with

**r**

_{i}being the position vector of the

*i*-th alpha carbon atom. Consequently, as described earlier, the fluctuation matrix of a protein structure can be obtained as follows.

*λ*

_{k}is the

*k*-th eigen-value of the Kirchhoff matrix, and \({\mathbf{q}}_{i}^{k}\) is the

*i*-th component of the

*k*-th normal mode of the Kirchhoff matrix. With obtaining the fluctuation matrix from GNM, one is able to easily evaluate the RMSF and CF with using Eqs. (3) and (4), which are useful in describing the fluctuation motion and correlated motion of a protein structure.

**u**

_{i}is the fluctuation vector of the

*i*-th alpha carbon atom defined as \({\mathbf{u}}_{i} = {\mathbf{r}}_{i} - {\mathbf{r}}_{i}^{0}\) with

**r**

_{i}and superscript 0 being the position vector of the

*i*-th alpha carbon atom and equilibrium state, respectively, and

**K**

_{ij}is the off-diagonal (3 × 3) stiffness matrix given by

**d**

_{ij}is a distance vector (between two alpha carbon atoms

*i*and

*j*) defined as

**d**

_{ij}=

**r**

_{i}−

**r**

_{j}, and a symbol \(\otimes\) indicates a tensor product. Once the off-diagonal stiffness matrices are estimated, it is straightforward to obtain the (3

*N*× 3

*N*) stiffness matrix

**K**for the protein structure. As mentioned earlier, the fluctuation matrix of a protein structure can be acquired as follows.

**r**is the position vector of all alpha carbon atoms, defined as

**r**= [

**r**

_{1}, …,

**r**

_{N}],

*ξ*

_{k}is the

*k*-th eigen-value of the stiffness matrix

**K**, and

**v**

^{k}is the

*k*-th normal mode of the stiffness matrix

**K**. Here, it should be noted that in Eq. (27), six zero modes corresponding to the rigid body motion of a protein structure are excluded for computing the fluctuation matrix

**Q**.

### Continuum Model

Though computer simulations based on atomistic and/or CG models are able to provide the fundamental insight into the dynamic and mechanical behavior of protein structures at molecular level, a continuum model is useful in analyzing the results of atomistic MD simulations of a large protein structure (e.g. protein fibril). For example, while the mechanical response of a protein fibril can be obtained from all-atom MD simulations or CG-MD simulations, its mechanical properties such as elastic modulus and fracture stress can be extracted from a continuum model such as Euler–Bernoulli beam model. This Section briefly reviews the continuum models that can be employed for characterizing the mechanical properties of protein structures ranging from a single protein to protein assembly.

#### Polymer Chain Statistics

*l*

_{p}=

*D*/

*k*

_{B}

*T*, where

*D*is the bending rigidity of the biological structure. Among polymer chain models, wormlike chain (WLC) model has been successfully able to capture the fluctuation behavior of a various biological molecules such as DNA and protein fibril. Statistical mechanics theory based on WLC model provides the mean-squared value of end-to-end distance (

*R*) in the form of [67]

*l*is the contour length of a biological structure. There are two ways to measure the persistent length of a biological molecule—the first approach is to place a biological molecule on the surface, and then its end-to-end distance is measured as a function of the contour length by atomic force microscopy (AFM) imaging [68]. The alternative approach is to fix the one end of a biological molecule, while its other end is labeled with a fluorescence dye, and then the fluctuation behavior of the free end of the biological molecule is estimated [69].

*F*is the force acting on the molecule, and

*u*is the displacement (i.e. end-to-end extension) of the molecule. WLC model is very useful in extracting the mechanical property (i.e. persistent length) of a biological molecule when it is stretched by a single-molecule force spectroscopy.

#### Elastic Beam Model

Though the computer simulations based on atomistic or CG models are able to depict the mechanical response of a protein, a continuum elasticity theory is necessary in characterizing the mechanical properties of the protein. For example, the mechanical property of a one-dimensional structure such as protein fibril [78] (or nanoscale system [79]) can be measured by considering an elastic beam model such as Euler–Bernoulli beam model. In particular, the elastic beam model allows for evaluating the mechanical property (e.g. elastic modulus) of a protein fibril based on its vibrational characteristics (e.g. natural frequencies) or its mechanical deformation behavior (i.e. force–displacement relation), which can be obtained from computer simulations based on atomistic or CG models.

*ρ*,

*A*,

*E*and

*I*represent the density, cross-sectional area, elastic modulus, and cross-sectional moment of inertia for a protein fibril, respectively,

*w*(

*x*,

*t*) is the transverse deflection of the fibril,

*x*is a coordinate defined along the longitudinal direction of the fibril, and

*t*is the time. The frequency of the fibril is written in the form of

*L*is the length of the fibril, and

*λ*is an eigenvalue that depends on the boundary condition. From Eq. (31), one is able to estimate the (bending) elastic modulus of a protein fibril as long as the frequency of the fibril is measured from computer simulations based on atomistic or CG models.The Euler–Bernoulli beam theory provides the relationship between the force and displacement for the fibril such as [81]

*x*

_{0}is the location at which a force

*F*is applied, and

*α*(

*x*

_{0}) is a constant that is a function of

*x*

_{0}and a boundary condition. If the force–displacement relation of a protein fibril is measured based on the bending deformation of the fibril with using computer simulations based on atomistic models, it is straightforward to evaluate the bending elastic modulus of the fibril based on Eq. (32). In addition, for a recent decade, Timoshenko beam model has been highlighted in order to understand the role of the length of a protein fibril in its bending elastic property [69]. Timoshenko beam model provides the persistent length (

*l*

_{p}) of a protein fibril in the form of [69, 82]

*G*

_{S}is the shear modulus of the fibril, and parameters

*a*and

*b*are the constants that depend on the boundary condition, and a parameter

*c*is a constant that is dependent on the cross-sectional shape of the fibril. Here, we note that the bending rigidity of a protein fibril is given by

*EI*=

*l*

_{p}

*k*

_{B}

*T*.

## Mechanical Characterization of a Single Protein

### Mechanical Unfolding of Proteins

The study on the mechanical unfolding of proteins based on all-atom MD simulations or CG-MD simulations has been of high importance, as this study allows for not only fundamental insights into the mechanical stability, unfolding, and folding of protein molecules, but also validation of the physical theory such as reaction-rate theory.

In recent years, Thirumalai and colleagues [95, 96, 97] developed a CG model, that is, self-organized polymer (SOP) model, whose potential field is conceptually similar to that of Go-like model. Specifically, SOP model considers the potential field that consists of the energies associated with covalent bonds and native contacts, whereas the mathematical form of the potential field of SOP is quite different from that of Go-like model. Thirumalai and coworkers [95] have shown that, by using CG-MD simulations based on SOP model, there are possible multiple unfolding pathways for a protein domain such that the alternative unfolding pathway of green fluorescence protein (GFP), which is inaccessible with conventional all-atom MD simulations, can be observed by SOP model. This is attributed to the fact that a pulling rate used in all-atom MD simulations is a few orders of magnitude larger than that employed in CG-MD simulations based on SOP model and/or single-molecule force spectroscopy experiments. The higher pulling rate used in all-atom MD simulations leads to the deterministic rupture of non-covalent bonds such as hydrogen bonds in a protein domain, while the lower pulling rate results in the stochastic rupture event of hydrogen bonds in the protein domain. In addition, a recent study by Thirumalai et al. [98] has interestingly shown the ability of SOP model-based simulations to unveil the refolding mechanisms of a protein domain under the quench of a mechanical force. Specifically, the refolding dynamics of force-driven unfolded protein under the significant decrease of a force applied to the protein from stretching force (with amount of ~ 100 pN) to quenched force (around 10 pN) was studied with using CG-MD simulations based on SOP model. It is shown that refolding time for a unfolded protein is dependent on the amount of quenched force in such a way that the quench force allows the unfolded protein to explore the intermediate, metastable unfolded states during the refolding of the unfolded protein [98]. In addition, SOP model-based CG-MD simulations enabled Thirumalai and coworkers [96] to study the force-driven hopping of RNA hairpin between two intermediate states. The SOP model-based simulation results provide the force-dependent time evolution of end-to-end distance for RNA hairpin, which is comparable to that obtained from single-molecule force spectroscopy experiments. This SOP model-based simulations allowed for accurate description of the folding kinetics of RNA hairpin with dependence on its amino acid sequence.

### Conformational Change of Proteins

Proteins perform their biological functions through their conformational changes driven by a ligand (or inhibitor). For instance, the conformational change of protein kinases is responsible for their biological function such as signaling pathway (that is relevant for cellular function or dysfunction resulting in disease pathologies such as cancer) [105]. This highlights the importance of predicting (or analyzing) the conformational change of proteins (due to ligand-binding) for not only further understanding of their biological functions but also the effective design of novel drug molecules (that are able to regulate the conformational change of proteins) [106].

Though all-atom MD simulations have recently been extensively employed for simulating the long time-scale dynamic behavior of small proteins [54, 107], they are still unable to simulate the conformational change of proteins, as the time-scale relevant to the conformational change of proteins is still larger than that accessible with currently available all-atom MD simulations. For a recent decade, ENM has been widely utilized for computationally efficient analysis of the conformational transitions of proteins. One may ask why ENM is able to predict the conformational change of proteins despite the simplicity of ENM modeling. This is attributed to the important role of the native topology of proteins in their dynamics [108]. A previous study by Teeter et al. [48] reports that the conformational dynamics of proteins is very insensitive to the details of potential fields, which implies that the conformational dynamics of proteins is not affected by the details of inter-atomic interactions. In a similar spirit, Lu and Ma [109] showed that the conformational fluctuation behavior of protein is unaffected by perturbation of Hessian (stiffness) matrix as long as the native contacts of a protein are maintained. In addition, it has been reported that the thermal fluctuation of an unbound protein structure can render it to explore the conformation of a ligand-bound protein structure because of the feature of free energy landscape for the protein structure in such a way that the free energy landscape has multiple metastable, intermediate conformations, some of which may correspond to the conformation of ligand-bound state [1]. These previous studies [1, 48, 109] validates the robustness of ENM modeling in analyzing the conformational dynamics of proteins such as their fluctuation and/or their conformational changes.

**u**

_{i}is the displacement of the

*i*-th residue of the protein, \({\mathbf{q}}_{i}^{k}\) is the

*k*-th low-frequency normal mode vector for the

*i*-th residue, α

_{k}is a fitting parameter, and

*n*is the number of low-frequency normal modes that are utilized to describe the conformational change. They have also shown that the conformational change of protein kinases upon inhibitor binding can be also predicted using the low-frequency normal mode-based perturbation of the protein structure [111]. These previous studies by Bahar and coworkers [110, 111] have supported the hypothesis of population shift model, which suggests that the conformation of ligand-bound state can be accessible based on the conformation of an unbound protein due to the ruggedness of the free energy landscape of the unbound protein [1]. In a similar spirit, Karplus and colleagues [113] have predicted the conformational change of myosin V based on a projection of displacement vector corresponding to the conformation change into the normal mode space. They have provided that the conformational change of myosin V can be predicted with using few low-frequency normal modes. Furthermore, Wolynes and coworkers [114] had developed a theoretical model to predict the conformational change of proteins in such a way that the low-frequency normal modes of ENM were employed to generate the intermediate conformations of the protein during its conformational transition.

Moreover, Maragakis and Karplus [115] developed a plastic network model (PNM) in order to predict the conformational transition pathway of proteins. Specifically, PNM was developed in such a way that the harmonic potential field of intermediate state is obtained by mixing the harmonic potential fields (i.e. the potential fields of ENMs) corresponding to two end states, that are, unbound and ligand-bound states. The harmonic potential field of intermediate state is assumed to be in the form of *E* = (1/2)[*E*_{1} + *E*_{2} − ((*E*_{1} − *E*_{2})^{2} + 4*ε*^{2})^{1/2}], where *E*_{1} and *E*_{2} represent harmonic potential fields (i.e. the potentials of ENM) corresponding to two end states, and *ε* is a parameter related to the degree of mixing two harmonic potential fields. The concept of PNM by mixing two potential fields was inspired by a quantum mechanical coupling of two potential energy surfaces. In a similar manner, Takagi et al. [116] suggested a dual Go-like model (DGM) by mixing two potential fields corresponding to two end conformations, while the potential field of a protein structure is assumed to be in the form of Go-like model, i.e. Equation (20), rather than the ENM-like harmonic potential. Furthermore, Hummer and colleagues [117] had developed a mixed elastic network model (MENM) to predict the conformational transition pathway of proteins by mixing two ENM harmonic potentials at two end states. The harmonic potential field of intermediate state is presumed to be written as *E* = –*k*_{B}*T*ln[exp((*E*_{1} + *ε*_{1})/*k*_{B}*T*) + exp((*E*_{2} + *ε*_{2})/*k*_{B}*T*)], where *ε*_{1} and *ε*_{2} are parameters that are related to the degree of mixing two potential energies.

In addition, Jernigan and coworkers [118] introduced an elastic network interpolation (ENI) to acquire the conformational transition pathways of proteins. In their ENI scheme, a displacement vector \({\mathbf{u}}_{i}^{k}\) corresponding to a conformational transition at the *k*-th intermediate state is obtained by minimizing a cost function (i.e. strain energy for conformational transition) defined as \(E^{k} = \left( {1/2} \right)\sum \left[ {\left\{ {\left( {{\mathbf{r}}_{i}^{k} + {\mathbf{u}}_{i}^{k} } \right) - \left( {{\mathbf{r}}_{j}^{k} + {\mathbf{u}}_{j}^{k} } \right)} \right\} - d_{ij}^{k} } \right]^{2}\), where \({\mathbf{r}}_{i}^{k}\) is the position vector of *i*-th residue at *k*-th intermediate state. Here, \(d_{ij}^{k}\) is an interpolated inter-residue distance given by \(d_{ij}^{k} = \alpha_{k} d_{ij}^{0} + \left( {1 - \alpha_{k} } \right)d_{ij}^{F}\), where *d*_{ij} is a distance between two residues *i* and *j*, superscripts 0 and *F* indicate initial (unbound) state and final (ligand-bound) state, respectively, and *α*_{k} is a parameter to indicate the *k*-th intermediate state obtained by linear interpolation between two states. Moreover, they have extended ENI coupled to a rigid cluster model [119], in which the rigid domains of a protein are modeled as a rigid body, while the interaction between the rigid domains is described by elastic springs. A recent study by Kim et al. [120] has suggested a normal mode-guided elastic network interpolation (NGENI) in such a way that low-frequency normal modes are used in the linear interpolation for obtaining an intermediate conformation. In their work, it is shown that NGENI is more computationally efficient when compared with ENI for predicting the conformational transition pathways of large protein complexes. Kidera and colleagues [121] developed a theoretical model to understand the conformational change of protein structures based on a linear response theory with ENM modeling. In their work [121], a displacement vector **u**_{i} for the conformational change is written as \({\mathbf{u}}_{i} = \left( {k_{B} T} \right)^{ - 1} \mathop \sum \nolimits_{k} {\mathbf{q}}_{i}^{k} \lambda_{k} \mathop \sum \limits_{j} {\mathbf{q}}_{j}^{k} \cdot {\mathbf{f}}_{j}\), where \({\mathbf{q}}_{i}^{k}\) is a *k*-th normal mode vector for *i*-th residue, *λ*_{k} is the *k*-th eigenvalue of a covariance (compliance) matrix, which is a pseudo-inverse of the stiffness (Hessian) matrix of ENM, and **f**_{j} is a force vector acting on *j*-th residue due to ligand-binding. In a similar spirit, Demirel and Lesk [122] employed ENM for evaluating a force associated with ligand-receptor binding. Specifically, a force related to ligand-receptor binding is obtained based on a linear response theory in such a way that the force **f**_{i} is given by \({\mathbf{f}}_{i} = k_{B} T\mathop \sum \nolimits_{l} {\mathbf{q}}_{i}^{l} \lambda_{l}^{ - 1} \mathop \sum \limits_{j} {\mathbf{q}}_{j}^{l} \cdot {\mathbf{u}}_{j}\), where **u**_{j} is the displacement vector of *j*-th residue (for a receptor) corresponding to the conformational change from unbound state to ligand-bound state.

In recent years, there are a few efforts that have been made to couple ENM-based structural perturbation scheme and all-atom molecular simulations for quantitatively understanding the conformational change of proteins. It should be noted that intermediate conformations predicted by the low-frequency normal modes of ENM may be sometimes physically unacceptable, because of the simplicity of ENM, which does not have any information for residue-specific interactions, side chain-dependent interactions, and so on. Bahar and coworkers [123] had coupled the ENM modeling and all-atom MD simulations for predicting the conformational transition of proteins. Specifically, the all-atom MD simulation was used to obtain an equilibrium structure for a protein, while ENM modeling was employed to introduce a structural perturbation of the protein. The iterative computations based on ENM normal mode-based perturbation followed by all-atom MD simulations are able to provide an insight into the conformational transition of proteins. In a similar spirit, Doruker et al. [124] had combined ENM normal mode-based structural perturbation scheme and all-atom Monte-Carlo (MC) simulations for studying the conformational transition of proteins. In their work, the structural perturbation of a protein based on normal modes of ENM followed by all-atom MC simulation was iteratively implemented. These studies [123, 124] show the possibility of coupling two different scale modeling techniques for studying the conformational (structural) changes of proteins in such a way that a CG model (e.g. ENM) was used to induce the structural perturbation of proteins while all-atom MD simulation was considered to gain their equilibrium structures. In other words, two different scale modeling techniques can be coupled so as to develop a novel multi-scale simulation techniques, which can provide an insight into protein dynamics typically inaccessible with only all-atom MD simulations.

## Characterization of Protein Assemblies

Understanding the formation and properties of protein assemblies is of high importance for gaining fundamental insight into the biological function of these assembles such as disease pathologies (or etiologies). For example, amyloid materials are formed by self-assembly process of protein aggregation resulting in the different structural forms of amyloids ranging from nanometer to micrometer (or even larger) scales [10, 20]. These amyloid materials have been known to play a crucial role in the pathogenesis of various diseases from neurodegenerative diseases [17] to type 2 diabetes [18] and cardiovascular diseases [19]. This suggests the necessity of unveiling how the amyloid materials are formed as an assembled structure driven by protein aggregation. That is, it is very essential to characterize the structural feature and formation of amyloid materials for understanding the underlying mechanisms of disease pathologies driven by protein assembly. In addition, it has recently been suggested that the mechanical properties of protein assemblies are responsible for their biological functions. For instance, the formation of mechanically stiff amyloid fibrils on a mechanically soft cell membrane results in the mechanical disruption of the cell membrane [24]. This Section presents the structural, mechanical, and dynamic characterization of protein assemblies such as amyloid materials and viral capsids based on computer simulation techniques at different scales ranging from all-atom MD simulations to CG-MD simulations.

### Structural Characteristics of Amyloid Materials

^{C}) into the scrapie isoform of prion protein (PrP

^{SC}), which is responsible for the formation of prion fibrils involving in various transmissible spongiform encephalopathies. This pioneering work [126] is still limited in quantitative understanding of the kinetics of structural conversion into dimerization (and subsequently, fibrillization). In recent years, with the advancement of computational resources, the all-atom MD simulations have been able to probe the more detailed information of the formation of amyloid materials. For example, Chong and Ham [127] have studied the dimerization of Alzheimer-β (Aβ) monomers using all-atom MD simulations. In their work [127], they are able to unveil the underlying mechanisms of the dimerization process such that a water-mediated hydration force plays a crucial role in the formation of Aβ dimer. In addition, all-atom MD simulations with enhanced sampling methods such as replica exchange [128] or metadynamics [129, 130] have allowed for characterizing not only the formation of amyloid dimers but also the inherent structural feature of amyloid monomer, which may be correlated with self-assembly process of protein aggregation. For instance, a bias-exchanged metadynamics simulations have provided the pathway of the dimerization of islet amyloid polypeptide (IAPP) and the intermediate structural forms of IAPP dimers [131]. A recent study by Shea and coworkers [132] have shown the ability of replica exchange MD (REMD) simulations to probe the inherent structural properties of IAPP monomers with the effect of sequence mutations. In their work [132], it is found that human IAPP (hIAPP) monomer is able to exhibit more contents of β-sheets, which are responsible for protein aggregation, than rat IAPP (Fig. 6b). Their finding [132] is consistent with clinical observation [18] that though IAPP proteins are found in both human and rat, human suffers from type 2 diabetes due to aggregation of IAPP proteins, whereas the type 2 diabetes is not found in a case of rat. In addition, Chakraborty and Das [133] have recently employed REMD to probe the inherent structural properties of Aβ monomer and the effect of inhibitors in the structural properties of Aβ monomers. They found that an inhibitor, which is a six-residue Aβ fragment (referred to as Aβ hexapeptide), makes a critical impact on the secondary structural propensity of Aβ monomers. In particular, Aβ hexapeptide reduces the contents of β-sheets in an Aβ monomer, which implies that Aβ hexapeptide inhibits the aggregation of Aβ monomers. A recent study by Shea and colleagues [134] have also considered REMD simulations for gaining insight into the role of osmolytes (e.g. urea or trimethylamine N-oxide, abbreviated as TMAO) in the structural propensity of tau fragment peptide. In their REMD simulations [134], the osmolytes induces the shift of the populations of secondary structures for tau fragment peptide, which is consistent with experimental observation that the osmolytes inhibits the formation of amyloid fibrils but promotes the formation of helical amyloid oligomers. The previous studies reviewed here suggest the increasing role of all-atom MD simulations in revealing hidden, complex mechanisms that determine the structure and formation of amyloid materials at different length scales ranging from oligomeric structures to fibril structures.

### Mechanical Properties of Amyloid Materials

Mechanical characterization of amyloid materials is a priori requisite for gaining insight into the molecular mechanism of disease pathologies, which is attributed to the fact that once amyloid fibrils are formed by self-assembly process (i.e. protein aggregation), these fibrils are quite stable and are unlikely to be denatured (or ruptured) in physiological conditions so as to deposit onto an organ (e.g. brain, pancreas, etc.) resulting in the disease pathologies. In addition, the mechanical properties of amyloid fibrils are directly linked to the amyloid-driven pathologies. For example, a recent study by Zewail and coworkers [24] suggest that the formation of mechanically stiff amyloid fibrils (or protofibrils) on a mechanically soft cell membrane results in the disruption of the membrane. In addition, the fracture property (i.e. division rate) of prion fibrils are found to be correlated with prion infectivity, that is, cell-to-cell prion propagation [135].

^{2+}ions improve the stability of Aβ oligomers while these ions reduce the stability of Aβ fibrils, which is consistent with clinical observation that high concentration of Zn

^{2+}ions promotes the formation of Aβ oligomers that are known as toxic agent to functional cells (Fig. 7). In addition, a recent study by MacPhee et al. [138] reports the effect of protonation state in the stability of transthyretin (TTR) amyloid oligomers based on all-atom MD simulations. In their work [138], it is shown that the protonation state affects the stability of amyloid oligomers so as to result in the different form (e.g. decamer, hexamer, etc.) of amyloid oligomers. It is found that the decamer is stable regardless of the protonation state of TTR, whereas the protonation results in the stable formation of the hexamer, and de-protonation favors the formation of higher order oligomers. These previous studies [136, 137, 138] show the ability of all-atom MD simulations to characterize the mechanical stability and formation of amyloid structures at different length scales ranging from oligomers to protofibrils. It should be noted that, however, the all-atom MD simulations are still computationally prohibited for simulating the stability of a very long amyloid fibril whose length scale is larger than 100 nm. Until now, the all-atom MD simulations is computationally able to characterize the amyloid protofibrils whose length scale is in the order of 10 nm.

*l*

_{p}∝ [(

*nq*)

^{2}− sin

^{2}(

*nq*)]

^{1/2}[68], which was obtained from a helical model. It is also shown that a H31Y point mutation, where histidine residue locating in 31st residue from the N-terminus is changed into tyrosine residue, results in a change in not only the structure of multi-stranded amyloid fibrils but also their elastic moduli.

### Mechanical and Dynamic Properties of Viral Capsids

Viruses are nano-sized particles that are formed by self-assembly process. For a recent decade, there are efforts that have been conducted for characterizing the mechanical and dynamic properties of viral capsids [149], as these properties are closely related to the biological function of viruses such as gene packaging. Viral capsids are mechanically strong so as to resist disassembly, so that they are able to contain a gene information (i.e. DNA or RNA molecules inside the virus). Here, we review computational efforts that have been made, for a recent decade, to characterize the dynamic and mechanical properties of viral capsids using CG and/or atomistic simulations.

*γ*, which is defined as

*γ*=

*ER*

^{2}/

*D*, where

*E*,

*R*, and

*D*are the Young’s modulus, radius, and bending rigidity of the viral capsid. This FvK number, which quantifies the competition between bending energy and tension energy during the deformation of the viral capsid, determines the shape of the capsid, i.e. whether it exhibits spherical or icosahedral facetted shapes. In recent years, Widom and coworkers [155] have considered ENM-based NMA simulations coupled with continuum elasticity theory, i.e. Lamé equation (for sphere). It is shown that the low-frequency vibration modes of viral capsid, obtained from ENM-based NMA simulations, were comparable to those predicted by Lamé equation, which suggests the robustness of a continuum elastic model in characterizing the mechanical properties of viral capsids. The Young’s modulus (

*E*) and Poisson’s ratio (

*ν*) of viral capsids were measured as

*ν*= 0.2 ~ 0.3 and

*E*/

*γ*= 6.5 ~ 36.2 nm

^{−1}, where

*γ*is a force constant used in ENM (here,

*γ*= ~ 0.5 N/m), i.e.

*E*= 3.2 ~ 18.1 GPa. A previous study by Roos et al. [156] employed a continuum elastic model-based finite element simulations for characterizing the squeezing (i.e. indentation) mechanics of viral capsids. Specifically, the authors considered the finite deformation continuum hyperelasticity (with a neo-Hookean constitutive law) in finite element simulations to probe the indentation mechanism of viral capsids (Fig. 12). It is shown that the mechanical indentation behavior of viral capsids is critically dependent on their shape that can be described by FvK number. For the capsids with their shape of FvK < 150, their indentation response is well dictated by a linear relation between indentation displacement and force. However, for the capsids with 150 < FvK < 800, the indentation response becomes nonlinear, whereas the indentation behavior of capsids with Fv > 800 is well depicted by a linear response as long as the indentation displacement becomes 40% of the radius of the capsid.

## Conclusion

In this paper, we introduce recent efforts that have been made, for a last decade, to characterize protein materials at multiple length scales ranging from single proteins to protein assemblies, based on various simulation techniques (at different length scales) from atomistic MD simulations to CG simulations.

First, we have shown that CG-MD simulations (using Go-like or SOP potential) have been extensively employed to understand the mechanical unfolding or refolding dynamics of a single protein molecule due to a force. Here, we note that though all-atom SMD simulations are able to provide a useful insight into protein unfolding (or refolding) mechanisms of a single protein, they are still computationally prohibited for directly interpreting the results of single-molecule force spectroscopy experiments. Specifically, the timescale accessible with all-atom SMD simulations is quite different from that used in a single-molecule force spectroscopy experiment such that a pulling rate used in SMD simulations is larger by six orders of magnitude than that considered in a single-molecule force spectroscopy experiment. This discrepancy between these timescales results in the extensive usage of CG-MD simulations for understanding the unfolding (or refolding) mechanisms of single proteins. However, it should be noted that although CG-MD simulations are able to provide the unfolding dynamics of a single protein with a timescale comparable to that used in experiment, these simulations have to be cautiously used for interpreting the unfolding (or refolding) mechanisms of a single protein, because the CG models may not carry the all atomistic information of the protein. For example, a single point mutation (i.e. single amino acid substitution) may affect the structure, dynamic (e.g. unfolding and/or refolding), and properties of a single protein, which typically cannot be captured by CG models. Recently, with the improvement of computing resources as well as the development of enhanced sampling methods, all-atom MD (or SMD) simulations may be more extensively utilized for interpreting the results of a real single-molecule force spectroscopy experiment. For instance, metadynamics scheme has recently been developed and widely utilized to probe the dynamics of a protein with a timescale relevant to the experiment. Here, the principle of metadynamics is to apply a Gaussian penalty function to the potential field of a protein in order to speed up the reaction rate of a protein related to its structural (conformational) changes due to a force or other stimulus (e.g. ligand-binding) [129, 130]. Our recent studies report the unfolding mechanisms of ubiquitin [157] and/or prion protein [158] based on all-atom metadynamics simulations. In our recent studies [157, 158], it is shown that all-atom model-based metadynamics simulations allow for unveiling the unfolding mechanics of a protein with a timescale relevant to a single-molecule force spectroscopy experiment. Despite few studies [157, 158] of protein unfolding based on an enhanced sampling-based all-atom simulations (e.g. all-atom metadynamics), we believe that recent development of enhanced sampling techniques may enable all-atom MD simulations to be practically considered for interpreting the results of a single-molecule force spectroscopy experiment in near future.

For studying the conformational changes of a single protein molecule, we have discussed the extensive usage of CG models such as ENM. All-atom MD simulations have not been still applicable for revealing the underlying mechanisms of the conformational changes, as their available timescale is much shorter than that relevant to the conformational changes. This has led to the development and usage of CG models such as ENM for studying the conformational changes. The robustness of ENM in predicting the conformational changes is attributed to the fact that the conformational changes of a protein are correlated with its low-frequency vibrational modes. Nevertheless, ENM-based NMA simulations are unable to provide insight into the time-evolution dynamics of conformational changes for a protein. One possibility to overcome this restriction is the development of computational techniques based on coupling between CG model (i.e. ENM) and all-atom MD simulations. As discussed in a paper by Bahar and coworkers [123], the low-frequency vibrational modes obtained from ENM were used to explore conformational spaces relevant to the conformational changes, while all-atom MD simulations were utilized to obtain the equilibrium dynamics of a protein during its conformational transitions. We anticipate that the sampling based on the normal modes (obtained from CG models) may be coupled to all-atom MD simulations for gaining a profitable knowledge on the conformational transitions of a protein.

For simulating large protein assemblies such as amyloid materials and viral capsids, we have shown that CG or continuum model-based simulations are able to provide the fruitful insight into the structure, dynamics, and properties of these assemblies. Despite a recent study by Perilla and Schulten [154] reporting the dynamics of viral capsids based on all-atom MD simulations, these atomistic simulations are still computationally unfavorable for predicting and characterizing the structure, dynamics, and properties of protein assemblies. We have presented many recent works on CG model (e.g. ENM) and/or continuum model-based simulations for characterizing the mechanical properties of protein assemblies such as amyloid materials and viral capsids, whose size is typically > 100 nm. However, these simulations based on CG models (e.g. ENM) are unable to provide the time-evolution dynamics of mechanical behavior (e.g. fracture) for protein assemblies such as amyloid materials. Though continuum model-based finite element simulations may provide the time-evolution of mechanical deformation for protein assemblies, they require a priori knowledge on a constitutive law (e.g. parameters related to material properties) for protein assemblies. One of routes to improve the simulation techniques for characterizing the protein assemblies is to combine all-atom MD simulations and CG model-based simulations (or continuum model-based simulations such as finite element simulation). Specifically, as described in our recent work [144], all-atom MD simulations are considered to obtain the equilibrium structure of structural segment for protein materials (e.g. amyloid materials), while simulations based on CG model (e.g. ENM) are applied to a whole protein assembly, which was constructed based on the equilibrium structure of structural segment, in order to characterize the properties of this assembly. Another avenue to improve the simulation techniques is that all-atom MD simulations are taken into account to compute parameters (e.g. parameters for a constitutive law) that are required for a finite element simulation of protein assembly, while finite element simulations are performed to depict the time-evolution of mechanical deformation for protein assembly [159].

In conclusion, computer simulations based on various models at multiple length scales, ranging from atomistic model to CG and continuum models, have recently played invaluable role in predicting, characterizing, and revealing the hidden, complex, underlying mechanisms of proteins’ dynamics and mechanics, which govern their biological functions. These simulations may help to not only interpret the experimental results (e.g. results obtained from single-molecule experiments) but also provide new insights into proteins’ dynamics and mechanics, which are still difficult to be observed in experiments due to their limited spatial and temporal resolutions. In addition, these computer simulations may also allow for providing an important design principles for developing a novel materials such as new-class protein materials [160, 161, 162] and nano-bio composite material [163, 164, 165] (i.e. material formed by coupling protein material and nanomaterial).

## Notes

### Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) under Grant no. NRF-2015R1A2A2A04002453.

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