Modeling and Simulations of the Dynamic Behaviors of Actin-Based

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Modeling and simulations of the dynamic behaviors of actin-based cytoskeletal networks Bo Gong, Xi Wei, Jin Qian, and Yuan Lin ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/ acsbiomaterials.8b01228 • Publication Date (Web): 16 Jan 2019 Downloaded from http://pubs.acs.org on January 20, 2019

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ACS Biomaterials Science & Engineering

Modeling and simulations of the dynamic behaviors of actin-based cytoskeletal networks

Bo Gong,1,a Xi Wei,1,b Jin Qian,*,a Yuan Lin*,b a

Department of Engineering Mechanics, Key Laboratory of Soft Machines and Smart

Devices of Zhejiang Province, Zhejiang University, Hangzhou, Zhejiang 310027, China b

Department of Mechanical Engineering, The University of Hong Kong, Hong Kong,

China 1These

authors contributed equally to the work.

*To whom correspondence should be addressed. E-mail: [email protected] (J.Q.) or [email protected] (Y.L.).

Abstract The cytoskeleton, a dynamic network of biopolymers with their associated crosslinking and motor proteins, is responsible for stabilizing cell shape and driving cell movement. This paper aims to provide an overview of the theoretical and computational approaches that have been developed to understand the dynamic behaviors and underlying mechanisms of actin-based cytoskeletal networks, connecting their microscopic structure to macroscopic performance across various scales, with implications for the observed nonlinear stress-strain relation, viscoelastic properties, stiffening induced by active motors as well as their biological functions in important 1

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processes such as cell adhesion, motility and mechanosensing. In the future, more sophisticated constitutive theories, continuum level and molecular dynamics-based simulations of biopolymer networks are expected to provide critical insights for understanding the material-structure-function relation in the cytoskeleton of cells, and guiding the development of active biomimetic materials.

Keywords cytoskeletal network, mechanical behavior, theory, modeling, finite element method, molecular dynamics simulation

1. Introduction The mechanical response of living cells is largely determined by the cytoskeleton, which is a filamentous network consisting of three main types of biopolymers: actin filaments, microtubules and a group of polymers known collectively as intermediate filaments.1-4 Together, these biopolymers are intricately interconnected to form the dynamic and heterogeneous cytoskeleton, which plays an essential role in maintaining the integrity of intracellular compartments, generating internal forces to drive cell movement, and enabling cells to resist deformations as well as change shapes during a variety of physiological processes.5-8 Actin is one of the most abundant proteins in eukaryotic cells, and the architecture

2


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of the cytoskeletal networks relies on the assembly of actin with different cross-linking proteins.7 Specifically, actin monomers are globular units (also called G-actin) with a diameter of 6-7 nm that can polymerize into double-stranded helical filaments,9,10 as depicted in Figure 1(a) and 1(b). These slender helices of filamentous actin (F-actin) show semiflexible behaviors with a persistence length of 10~17 µm and a contour length up to 20 µm,11 which are essential for important biological functions such as the mobility and contraction of cells during cell division.12 Aided by various actin-binding proteins (ABPs), individual F-actin can be cross-linked into distinct types of actin structures, forming different cytoskeletal networks such as actin cortex, stress fiber, lamellipodium and filopodium.12-14 Figures 1(c)-1(f) show representative locations within a cell body where different actin architectures are formed. Extensive efforts have been made to uncover how these network architectures are related to their deformation characteristics, load-bearing capabilities and corresponding biological duties. For representative actin architectures, it has been found that highly branched Factin networks will be formed at the leading edge of crawling cells whose continuous polymerization provides the driving force for the cells to extend their periphery during locomotion (Figure 1(c)).2,15 Bundles of parallelly aligned actin filaments are responsible for generating filopodial protrusions,16 which are thought to serve as sensors for cells to explore their microenvironment (Figure 1(d)).17 Randomly crosslinked F-actins constitute the main cortex beneath the plasma cell membrane, which is under tension in multiple directions18 (Figure 1(e)) and heavily involved in cellular processes such as immune response,15,19 wound healing20 and cytokinesis.21 The stress 3



fibers, composing of bundled actin filaments and myosin motors, can actively generate contractile forces within the cytoskeleton and transmit intracellular forces to extracellular matrix through focal adhesions (Figure 1(f)).22 Different experimental techniques have been developed to quantify the mechanical response and dynamic characteristics of individual bio-filaments and the entire cytoskeletal network, particularly on measuring the persistence length11,23,24 and stretching behavior25-28 of F-actin, the nonlinear elasticity29-31 and viscoelasticity32-34 of actin gels, network stiffness modulated by active motors,35,36 and diffusivity of nanoparticles embedded in the cytoskeleton or in vitro networks.37-40 Figure 2 illustrates a variety of experimental techniques that are commonly used in characterizing the mechanical properties and behaviors of these networks, performed under both in vivo and in vitro conditions.35,41-49 One generic feature of biopolymer networks revealed by different experiments is their pronounced nonlinear elastic response. In particular, their bulk moduli can undergo orders of magnitude increase as the imposed strain grows.29,50 Studies have shown that such nonlinear stiffening is mainly caused by the transition of network behavior from being dominated by filament bending at small strains to that dictated by filament stretching at high strain levels. Interestingly, it has also been found that reconstituted actin networks can exhibit reversible strain softening through filament buckling under localized compressive forces.51,52 In addition to nonlinearity, cytoskeletal actin networks are also known to respond to imposed deformation in an intriguing time-dependent manner,53-55 resulting in stress relaxation and energy dissipation within the networks under dynamic loading.56,57 From the rheological tests, 4


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it was found that the behavior of permanently cross-linked F-actin networks under high oscillating frequency is dominated by thermal fluctuations, resulting in a universal power-law dependence of both storage (G’) and loss modulus (G’’) on the driving frequency ω.58-61 On the other hand, such networks exhibited a nearly constant level of elasticity at low frequencies while the viscous response became negligible.62 Extensive theoretical and computational efforts have also been made to understand the load bearing and force generation capabilities of bio-filament networks. It is natural to believe that the bulk response of a cytoskeletal network reflects the collective behavior of different architectures involved at various time and length scales. In this regard, theoretical and computational approaches play an indispensable role in connecting the microscopic structure of a network to its macroscopic behavior across the scales that are involved. In particular, theoretical models aim to capture the physical essence of the complex and time-varying system in a concise way; rigorous computational models can implicitly or explicitly incorporate multiscale mechanisms with empirical rules, and provide an effective way to accurately simulate different phenomena observed on biological gels or live cells. Figure 3 depicts the prevailing computational approaches that are commonly adopted at different time and length scales, from molecular dynamics (MD) model for G-actin at angstrom scale63,64 to continuum level cell model65,66. Specifically, full-atom MD approach67-71 can be adopted at nanometer scale to understand the mechanical response of a single actin filament. However, due to the exceedingly high computational cost of full-atom simulations, coarse-grained molecular dynamics (CGMD)72-74 and Brownian dynamics 5



(BD)57,75-90 models have been developed to simulate the behavior of complex F-actin networks in a much affordable manner. This paper overviews key advances in the theoretical and computational investigations on the physical response of biopolymer/cytoskeletal networks, with particular emphases on important constitutive theories, continuum level and molecular dynamics-based simulation approaches that have been developed in the past few decades.

2. Constitutive theory of biopolymer materials As mentioned earlier, biopolymers such as those constituting the cytoskeleton or extracellular matrices are made of large globular proteins.7 Given that the diameter of such molecules is usually of the order of tens of nanometers, they are far more rigid against bending than most common synthetic polymers and can therefore be treated as elastic fibers. Nevertheless, because the contour length of a biopolymer is often two or three orders of magnitude larger than its diameter, it still undergoes significant thermal fluctuations, indicating the crucial role of configurational entropy in the mechanical response of such long-chain molecules.91 The key question in analyzing the mechanics of a single bio-filament is the forceextension relationship,92 describing the axial force required to extend the biopolymer to certain end-to-end distance. Such force-extension curve is usually characterized by a linear region followed by a nonlinear region, exhibiting increasing apparent stiffness

6


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of the biopolymer once a threshold level of extension is reached.44,93 Two well-known models capable of providing this kind of relationship are the so-called freely jointed chain (FJC) and worm-like chain (WLC) models, which will be discussed in detail below. Freely jointed chain model: In this theory, the flexible polymer is treated as a chain consisting of 𝑁 connected rigid segments each with the same length 𝑙, often referred to as the Kuhn length.94,95 In addition, it is assumed that a segment within the chain can assume any orientations (characterized by two bond angles 𝜃 and 𝜙 in the polar coordinates) with equal probability, independent of the configurations of other segments as schematically shown in Figure 4(a). Effectively, this assumption implies that neighboring segments can form arbitrary angles, and may even overlap, with respect to each other. Under such circumstance, it can be shown that the probability distribution function of a three-dimensional (3D) biopolymer, in terms of its end-to-end distance 𝑟, takes the Gaussian form as96 3

3/2

(

3𝑟2

)

exp ― 2𝑁𝑙2 .

𝑝(𝑟) = (2𝜋𝑁𝑙2)

(1)

The entropy of the system can then be calculated as 𝑆 = 𝑘𝐵lnΩ, where 𝑘𝐵 is the Boltzmann’s constant and Ω is the number of available configurations of the chain under given end-to-end distance 𝑟. Since Ω is proportional to 𝑝, the force must be applied on both ends of the chain to maintain their separation r is 𝛿𝑆

𝑓𝐺 = 𝑇𝛿𝑟 =

3𝑘𝐵𝑇 𝑁𝑙2

𝑟.

7


(2)


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Equation (2) essentially suggests that the chain is behaving like a linear spring with the spring constant increasing with temperature 𝑇, in direct contrast to most engineering materials which usually become softer under higher temperature. Equation (2) also indicates that the spring stiffness is always a constant and hence the chain can be infinitely extended. This unrealistic feature is introduced by the fact that Equation (1) is actually a continuous approximation of the actual probability distribution of an FJC. Interesting, after properly modifying that a chain can never be stretched beyond its contour length 𝑁𝑙, the corrected force-extension relationship becomes97

𝑓𝑁𝐺 =

𝑘𝐵𝑇 𝑙

𝑟

ℒ ―1(𝑁𝑙),

(3)

where ℒ ―1( ∙ ) is the inverse Langevin function with ℒ(𝑥) = coth (𝑥) ―1/𝑥. It must be pointed out that Equation (3) suggests that the force diverges as

1 𝑁𝑙 ― 𝑟

when the

biopolymer is close to be fully extended. However, recent stretching experiments on DNAs demonstrated that the extension force actually scales like

1 (𝑁𝑙 ― 𝑟)2

as 𝑟→𝑁𝑙

(i.e., the force diverged faster than what Equation (3) predicts), which led to the development of worm-like chain theory. Worm-like chain model: In this model,98 the resistance of biopolymer against bending deformation is taken into account. To simplify the discussion, it is instructive to begin the analysis on a two-dimensional biopolymer (with contour length 𝑙𝑐 and bending rigidity 𝜅) immersed in a thermal bath (refer to Figure 4(b)). Because of the continuous bombardment from medium molecules, the filament will undergo thermal undulations which can be characterized by the change of the tangential angle 𝜃 8


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(formed between the unit tangent vector 𝑡 and, for example, the x-axis) along the biopolymer. Under such circumstance, the bending energy stored in the deformed filament is simply ∂𝑡 2

𝜅

𝐻𝑏 = 2∫(∂𝑠) 𝑑𝑠,

(4)

where s is the arc-length coordinate and the integrand stands for the square of the local curvature of the deformed polymer. Given that the biopolymer will assume specific configurations according to the Boltzmann distribution, with the energy level of each configuration given by Equation (4), it can be shown that the correlation of tangent vectors from one point of the chain to another decays exponentially with their separation, that is99

〈𝑡(𝑠) ∙ 𝑡(𝑠′)〉 = 〈cos [𝜃(𝑠) ― 𝜃(𝑠′)]〉 = 𝑒 ― |𝑠 ― 𝑠′|/2𝑙𝑝,

(5)

where 𝑙𝑝 = 𝜅/𝑘𝐵𝑇 is the so-called persistence length of the biopolymer.100 Note that, the factor 2 on the right-hand side of Equation (4) will vanish for a 3D WLC because the filament can deflect in two transverse directions in that case. At this point, it is clear that a biopolymer will totally loose orientation correlation when its contour length 𝑙𝑐 is much larger than the persistence length 𝑙𝑝, that is 〈𝑡(0) ∙ 𝑡(𝑙𝑐)〉 ≈ 0 when 𝑙𝑝 ≪ 𝑙𝑐. On the other, the filament will remain more or less straight if 𝑙𝑝 ≫ 𝑙𝑐. An interesting scenario is when the persistence length of the biopolymer (often called semiflexible) is comparable to its contour length (i.e., 𝑙𝑝 ∼ 𝑙𝑐), the filament under such circumstance will undergo significant thermal bending fluctuations around a straight conformation, reflecting the competition between the elasticity that favors a straight filament and the 9



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entropy that tends to crumple the biopolymer. An interpolation formula describing the force-extension behavior of a long inextensible WLC (i.e., with 𝑙𝑐 ≫ 𝑙𝑝) was developed by Marko and Siggia101 as

𝑓=

[

𝑘𝐵𝑇 𝑟 𝑙𝑝 𝑙𝑐

1

+

(

4 1―

where the force is predicted to diverge as 𝑓 ∝

]

1

𝑟 2 𝑙𝑐

)

―4 ,

1

(

𝑟 2

4 1―𝑙

)

(6)

when the biopolymer is close to

𝑐

be fully extended (𝑟→𝑙𝑐). Interestingly, if the chain can be over-extended beyond its contour length, then a study by Hu et al.102 suggested that the force will actually scale as 𝑓 ∝

0.2

.

𝑟 2 𝑙𝑐

(1 ― )

Following a slightly different approach, MacKintosh and coworkers4 examined the problem in the semiflexible regime where thermal fluctuations were assumed to result in a contraction ∆𝑙 of the end-to-end distance of the polymer. For simplicity, if the deflection of filament (denoted by 𝑢) in one transverse direction is considered, then this contraction distance can be expressed as

∆𝑙 = ∫( 1 +

∂𝑢 2

| | ∂𝑥

1

― 1)𝑑𝑥 ≈ 2∫

∂𝑢 2

| | 𝑑𝑥, ∂𝑥

(7)

where the integration is conducted along the biopolymer. Since the tensile force 𝑓 is conjugate to ∆𝑙, the total energy of the system becomes 1

[

∂2𝑢 2

]

∂𝑢 2

𝐻 = 2∫ 𝜅(∂𝑥2) + 𝑓(∂𝑥) 𝑑𝑥.

(8)

If the transverse undulation of filament is decomposed into a Fourier series, then the 10


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amplitude of each deformation mode can be directly estimated by the theory of equipartition, leading to an expression of ∆𝑙 (under a given force 𝑓) as 1

〈∆𝑙〉𝑓 = 𝑘𝐵𝑇∑𝑞(𝜅𝑞2 + 𝑓),

(9)

where 𝑞 = 𝑛𝜋/𝑙𝑐 and 𝑛 = 1,2,3,…. Following this line of reasoning, the force-induced change in the end-to-end distance of a 3D biopolymer is 𝑙𝑐2

∅

𝛿𝑙(𝑓) = 〈∆𝑙〉0 ― 〈∆𝑙〉𝑓 = 𝜋2𝑙 ∑𝑛𝑛2(𝑛2 + ∅), 𝑝

(10)

with ∅ = 𝑓𝑙𝑐2/(𝜅𝜋2) being the dimensionless force. Therefore, the relative extension of the filament can be expressed as103 𝜋coth (𝜋 ∅)

𝛿𝑙

∈≡ 〈∆𝑙〉0 = 1 ― 3

𝜋2∅

,

(11)

which allows one to obtain the force-extension relationship numerically. Constitutive models for biopolymer materials: Once the force-extension relationship of a single chain is obtained, it is possible to derive constitutive laws describing the bulk response of biopolymer materials by making assumptions on the microstructure of the materials or the nature of the deformation. For example, in the well-known Arruda-Boyce model,104 it was assumed that a representative unit cell of the polymer material has eight chains, locating along the diagonals of the cubic cell and being connected at the central junction as shown in Figure 4(c). In addition, if the unit cell is taken to be aligned with three macroscopic principal stretching directions, with stretch amplitudes of 𝜆1, 𝜆2 and 𝜆3 respectively, then the effective stretch experienced by each chain is 𝜆𝑐ℎ𝑎𝑖𝑛 =

1

1/2

(𝜆21 + 𝜆22 + 𝜆23)

3

= 𝐼1/3 with 𝐼1 = 𝜆21 + 𝜆22 + 𝜆23

11



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representing the first stretch invariant. Assuming that each chain behaves like an FJC with N segments, the total strain energy stored in the material can be estimated from Equation (3) as 𝑊 = 𝑛𝑘𝐵𝑇 𝑁[𝛽𝜆𝑐ℎ𝑎𝑖𝑛 ― 𝑁In(

sinh 𝛽 𝛽

)] where 𝑛 is the total number 𝜆𝑐ℎ𝑎𝑖𝑛

of chains in the system, a quantity proportional to the chain density, and 𝛽 = ℒ ―1(

𝑁

).

With an expression of the strain energy at hand, the resistance of the material against imposed deformations can be derived. In must be pointed out that, each chain can also be assumed to behave like a WLC within this framework,105 leading to a different constitutive relationship of the material. On the other hand, the constitutive description of biopolymer materials can also be obtained by invoking the affine assumption of the deformation field, by postulating that all cross-linkers in a polymer network are free to rotate according to the imposed strain and consequently filaments within the network only undergo stretching or compression but no bending.4,50 With this assumption, for a polymer strand between two crosslinkers with an orientation 𝒏 in the unstrained network (Figure 4(d)), the stretch it experiences can easily be deduced from the macroscopic Cauchy deformation tensor 𝑪. Specifically, for a simple shear 𝛾 of the 𝑥𝑧-plane in the x-direction, such tensor takes the form:

(

)

1 0 𝛾 𝑪= 0 1 0 . 0 0 1

(12)

The axial strain of the strand in this case is |𝑪𝒏| ―1 while the corresponding tension 𝑓(|𝑪𝒏| ―1) along it can be evaluated according to different force-extension relationships discussed above. Notice that, the 𝑖-component of this tension is simply 𝑓 12


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C𝑖𝑙𝑛𝑙

(|𝑪𝒏| ―1) ∙ |𝑪𝒏| where Einstein summation of repeated indices is implied. Since the volume is conserved under simple shear, the polymer length density per unit volume crossing the 𝑗-plane (oriented perpendicular to the 𝑗-direction) becomes 𝜌𝐶𝑗𝑘𝑛𝑘, where 𝜌 is the initial polymer length density in the undeformed network. Taking these factors into account, the nonlinear stress tensor can be obtained by averaging the force per area crossing the 𝑗-plane as50,106

〈

𝜎𝑖𝑗 = 𝜌 𝑓(|𝑪𝒏| ― 1) ∙

C𝑖𝑙𝑛𝑙C𝑗𝑘𝑛𝑘 |𝑪𝒏|

〉,

(13)

where the angular brackets represent averaging over all possible strand directions. Therefore, under a small imposed strain 𝛾, the corresponding shear stress becomes 𝜎𝑥𝑧 = 𝜌〈𝑓(𝛾𝑛𝑥𝑛𝑧) ∙ 𝑛𝑥𝑛𝑧〉,

(14)

where 𝛾𝑛𝑥𝑛𝑧 is the axial strain of a polymer strand with initial orientation 𝒏. If one conducts the average uniformly over all orientations and adopts the WLC forceextension relationship, the apparent shear modulus of a semiflexible polymer network is predicted to be 𝐺0 = 6𝜌𝑘

𝜅2

𝐵𝑇𝑙𝑐

.

3

(15)

Furthermore, nonlinear stiffening of the material is expected to occur beyond a critical 𝑙𝑐

strain of 𝛾𝑐 = 6𝑙𝑝 with the differential shear modulus (defined as 𝐾 = 𝑑𝜎𝑥𝑧/𝑑𝛾) scaling with the stress as 𝐾~𝜎𝑥𝑧3/2. Interestingly, these predictions all appear to be in good agreement with the experimental observations.107,108 Although assumptions like that the network has a specific unit cell structure or the 13



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deformation is affine allow us to construct constitutive descriptions of the biopolymer materials, the issue of how realistic/valid these simplifications are remains a serious concern. In particular, it is conceivable that significant bending of individual biopolymers can take place, leading to a non-affine deformation field, when the filament density is low or the macroscopic strain level is small. Various numerical approaches have been developed to address these issues, which will be discussed in the following sections.

3. Continuum-level simulations of biopolymer networks As pointed out above, direct simulations on computer-generated networks have received increasing attention in recent years. For example, the so-called Mikado model has been used by different researchers109-111 in examining the response of filamentous networks. Basically, a Mikado network can be constructed by randomly placing a large amount of filaments (each with the same length 𝑙) inside a square cell of size 𝑊 × 𝑊 where periodic boundary condition is enforced, as shown in Figure 5(a). The deformation of the network can be simulated by tracking the positions of all intersecting points, as well as the mid-points of filament segments between two cross-linking points. Based on the coordinates of these points, the strain energy stored in the network was assumed to take the form 𝜇

𝐻 = 2∑

(

𝛿𝑙2𝑖𝑘 𝑙𝑖𝑘

+

𝛿𝑙2𝑗𝑘 𝑙𝑗𝑘

)

𝜅

𝛿𝜃2𝑖𝑗𝑘

+ 2∑ 𝑙𝑖𝑗𝑘 ,

14


(16)

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where 𝑖 and 𝑗 represent two intersecting points while 𝑘 is the mid-point between them. 𝑙𝑖𝑘 and 𝑙𝑗𝑘 are the length of segment 𝑖𝑘 and 𝑗𝑘, respectively, with 𝛿𝑙𝑖𝑘 and 𝛿 𝑙𝑗𝑘 corresponding to their elongation/compression. Finally, 𝛿𝜃𝑖𝑗𝑘 is the change of the angle between segment 𝑖𝑘 and 𝑗𝑘. Essentially, the first term accounts for the stretching energy with 𝜇 representing the stretching stiffness, while the second term 1

roughly describes the bending energy with 𝑙𝑖𝑗𝑘 = 2(𝑙𝑖𝑘 + 𝑙𝑗𝑘) being the mean length of the two segments and 𝜅 having the physical meaning of the bending stiffness of filament. Interestingly, these simulations revealed that the deformation of such network is indeed non-affine at small strains.109-111 To take into account the bending effect in a more accurate manner, subsequent finite element models (FEMs) and numerical procedures on two-dimensional networks were also carried out (Figure 5(b)).112,113 In these models, filaments with random orientations were assembled into networks via rigid or compliant cross-linkers, and each filament was discretized into equal-sized Euler-Bernoulli beam elements so that bending energy stored in the deformed networks can be accurately computed. A main finding from these studies is that the stress-stiffening of a biopolymer network is actually caused by the bending-to-stretching transition of filament deformation, rather than the entropic effect. Specifically, it was shown that deformation can easily be accommodated by filament bending at small strains, however, more filaments will undergo stretching as strain increases, leading to an elevated resistance against deformation.112 Interestingly, it was found that the addition of myosin motors, modeled as distributed force dipoles, in a soft network can result in an order-of-magnitude 15



increase in its apparent modulus.113 Furthermore, it has also been shown that significant alignment and remodeling of fibrous extracellular matrices can be induced by internal cellular contraction, leading to long-range force transmission and cell interaction.114,115 Similar simulations in 3D configurations have also been conducted recently. For example, by forcing a randomly placed filament to move around until getting into contact with another filament, more realistic 3D biopolymer networks were generated.116 Alternatively, 3D fibrous networks can also be constructed on the basis of a face-centered cubic lattice.117,118 Essentially the same conclusions were obtained from simulations on these 3D networks, i.e., the deformation is not affine at small strains and the dominant deformation mode of filament undergoes bending-to-stretching transition as the imposed strain increases. Nevertheless, it was also found that network architecture plays an important role in determining the degree of the non-affine deformation as well as the cross-over strain for the bending-to-stretching transition to occur. It must be pointed out that thermal undulations of individual filaments have not really been taken into account in the aforementioned continuum level simulations. The only attempt to include this effect was made by Onck et al.112 and Huisman et al.116, where a curved shape (Figure 5(b)) was assigned to each polymer at the beginning of simulations to represent the influence of thermal fluctuations. However, this approach is not very realistic as the filament shape should evolve continuously under thermal excitations. To address this issue, a Langevin dynamics based formulation was

16


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proposed by Lin et al.119 to describe the shape fluctuations of biopolymers. Following Langevin120, it was postulated that the effect of bombardment of medium molecules on the biopolymer can be represented by two parts: a viscous force 𝜁𝑉 acting on the filament where 𝜁 is the viscous coefficient per unit length of the filament and 𝑉 is the local velocity of the moving/deforming biopolymer, and a randomly distributed force 𝑓 along the filament (Figure 5(c)) which is Gaussian with zero mean and must be related to 𝜁 as

〈𝑓(𝑥,𝑡) 𝑓(𝑦,𝜏)〉 = 2𝜁𝑘𝐵𝑇𝛿(𝑥 ― 𝑦)𝛿(𝑡 ― 𝜏),

(17)

where the brackets 〈 ∙ 〉 mean taking the average, 𝑥 and 𝑦 are the position vectors of two points on the filament, 𝑡 and 𝜏 represent time, and 𝛿 stands for the Kronecker delta. This idea was then implemented in the FEM simulations119 where the transverse load 𝑓𝑗𝑖 acting on element i over the time interval [𝑡𝑗 𝑡𝑗 + 1], as shown in Figure 5(c), was taken to be uniform and generated randomly from a Gaussian distribution with zero mean and a variance of 2𝜁𝑘𝐵𝑇/∆𝑥∆𝑡, namely,

〈𝑓𝑗𝑖 𝑓𝑗𝑖〉 =

2𝜁𝑘𝐵𝑇 ∆𝑥∆𝑡

,

(18)

with ∆𝑥 and ∆𝑡 being the element length and the size of time step, respectively. Using this method, it was shown that as deformation progresses, the response of randomly cross-linked F-actin networks undergoes the transitions from being entropy dominated to being governed by filament bending, and eventually to being dictated by filament stretching. Interestingly, the levels of macroscopic stress at which these transitions take place were found to be around 1% and 10% of the initial modulus of 17



the network.119 Another interesting question that has attracted great attention is how the deformability of cross-linking molecules influences the bulk response of a biopolymer network. This issue is of great biological relevance because more than 23 classes of cross-linking proteins, with distinct mechanical properties, have been identified in cells.121 As a first attempt, Sharma and coworkers122 examined the response of networks composing of randomly placed rigid filaments interconnected by flexible cross-linking molecules (each modeled as a worm-like chain). Žagar and coworkers123 also considered the problem by representing each cross-linker as a combination of four springs with their stiffnesses coupled to each other. They found that the large-strain response of the network is accompanied by the development of percolation of axially stressed fibers and cross-linkers. Strain stiffening in the bending-dominated regime is caused by filament undulations that tend to pull out from such stress path, while reorientation of this stress path leads to network stiffening in the stretching-dominated regime.123 Recently, a computational model based on FEM was developed by Wei et al.124 to systematically investigate how the physical properties of cross-linkers affect the deformation and failure of biopolymer networks. In the study, each cross-linker was treated as a combination of linear and angular springs that resist both the separation and relative rotation of two cross-linked filaments, referring to Figure 5(d). Also, it was assumed that the cross-linker will break once the strain energy stored inside is above a

18


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threshold value. Under such circumstance, it was found that the stress-strain curve of biopolymer networks typically exhibits linear increase – strain hardening – stress serration – total fracture transitions due to the interplay between the bending/stretching of individual filaments and the deformation/breakage of cross-linkers. Interestingly, the apparent network modulus was found to scale with the linear and rotational stiffness of the cross-linkers to a power exponent of 0.78 and 0.13, respectively. In addition, the fracture energy of networks will reach its minimum at intermediate values of rotational compliance, reflecting the fact that most of the strain energy will be stored in the distorted filaments with rigid cross-linkers while the imposed deformation will be evenly distributed among significantly more cross-linking molecules with high rotational compliance.124

4. Molecular dynamics simulations of biopolymer networks Molecular dynamics (MD) approaches have been widely employed in studying biopolymer networks by tracking the motions of constituent components as an ensemble of atoms, molecules or fine particles with specified potential energies. After that, macroscopic and microscopic quantities of interest can be obtained from analyzing the collective and individual behaviors of the involved atoms/particles. MD simulations can provide insights into nanoscale structural organizations and mechanical forces of cytoskeletal components that give rise to the bulk properties of the entire networks through a multiscale strategy, which cannot be captured by continuum level models. 19



All-atom molecular dynamics approach: The multiscale modeling of cytoskeletal networks started from the atomic level, e.g., by investigating the structural variation and mechanical performance of actin monomers and filaments at atomic scale using an all-atom MD approach. The molecular structure of actin was first clarified by Holmes and coworkers14 using X-ray crystallographic analysis with oriented F-actin gels (Figure 3(a)). Based on this information, Wriggers and Schulten63 employed steered molecular dynamics (SMD) to investigate the pulling process of actinphosphate complex and identified the critical force level at which the phosphate would be released from the actin. Interestingly, Dalhaimer and coworkers64 showed that subtle structural differences of actin monomers have profound effects on how they bind to ATP, ADP and nucleotide. Oda and coworkers67 developed a molecular model to understand the conformational transition from globular actin to fibrous actin (Figure 3(b)) during the polymerization process. On a slightly different front, Pfaendtner and coworkers68 focused on the structural features and dynamic properties of actin filaments and demonstrated how bounded phalloidin can stiffen F-actin and prevent ADP-Pi hydrolysis. Matsushita and coworkers69-71 also used molecular dynamics method to quantify the apparent extensional and torsional stiffness of actin filaments and their couplings. Coarse-grained molecular dynamics approach: All-atom MD simulations are informative but usually computationally expensive, and therefore coarse-grained molecular dynamics (CGMD) methods were often adopted instead to investigate the complex behaviors of cytoskeletal networks,125,126 where the essential idea is to 20


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represent every important molecular group as one artificial particle. Information obtained from all-atom simulations often provide critical input to CGMD models regarding how such artificial particles behave and interact with each other. For example, Chu and Voth72,73 proposed a CGMD model by mapping one single monomer of Gactin to four subunit particles with prescribed interactions (Figure 3(c)). They used the model to examine the effects of monomer conformational changes on the forceextension curve and persistence length of assembled F-actin structures, and then compared their simulations with those from structural data analysis by X-ray diffraction14,68 or electron cryomicroscopy127,128. Saunders and Voth74 also employed CGMD simulations to compare three representative filament models reported by Holmes et al.14, Oda et al.67 and Fujii et al.127. Interestingly, it was found that the main difference of these three models is the degree of G-actin subunit flattening and the position of subdomain 2 of G-actin. Additional simulations also revealed that the rotation of subdomain 2 in the G-actin to F-actin transition is a key motion which can lead to a range of different structures.74 It must be pointed out, although much more computationally efficient than all-atom molecular dynamics, the cost of tracking the motion and interaction of thousands to millions of particles still prevents CGMD simulations from reaching time and spatial scales comparable to those in actual experiments. Brownian/Langevin dynamics approach: It is well-known that biopolymer networks are formed and functioning in an aqueous environment, and consequently the effects of solvent molecules on the trajectory of coarse-grained particles and the 21



dynamic behaviors of actin networks should be explicitly considered. The temporal scales at which these biopolymer networks behave are convoluted with the scale at which the solvent molecules move. However, involving real solvent molecules in MD simulations is very demanding. Therefore, Brownian dynamics (BD) or Langevin dynamics (LD) methods have been developed to investigate the dynamic responses of F-actin and assembled networks in a liquid environment (Figure 3(d)), where individual solvent molecules are omitted and their collective effects are captured by a combination of random forces and frictional terms. Newton’s equations of motion are thus replaced by the Langevin equation. For example, the dynamic actin polymerization, depolymerization and severing have all been simulated using a sophisticated BD model,75 in which each F-actin was represented by a chain of spherical beads interconnected by harmonic springs. Following a similar approach, Inoue and coworkers76 studied the remodeling process of actin filaments under externally imposed shear flow. With the comprehensive understanding of filament-level dynamics from BD/LD simulations, the physical properties of actin networks can then be explored with constituted F-actin, cross-linking molecules, myosin motors and regulatory proteins (Figure 3(e)), providing insights for the construction of cell models at continuum level (Figure 3(f)). For example, Kim and coworkers77 developed a coarse-grained Brownian dynamic model to simulate the behavior of actin networks where each F-actin was coarse-grained into a series of beads interconnected with a combined longitude and rotational potentials. In explaining the rheological characteristics of actin networks from the experiments,29,62 Kim and coworkers57,77 also developed a rod-based BD 22


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model which suggested that the storage modulus of actin networks can be enhanced by increasing the density of cross-linkers. Kurniawan and coworkers78 employed a modified ‘shish-kebab model’ to characterize the nonlinear mechanical responses of 3D biopolymer networks that exhibit distinct deformation mechanisms at different length scales. Based on the hierarchy from actin monomers to cross-linked actin networks, a multiscale computational approach was proposed to illustrate how molecular-level conformational changes of actin monomers, caused by different nucleotide combinations and/or cation binding, can propagate to filament level and ultimately influence the apparent rigidity and strain stiffening behavior of the whole networks.79,80 In a typical scheme of BD/LD, the motions of individual beads (or particles) are governed by the Langevin equation120 𝑑2𝒓𝑖

𝑑𝒓𝑖

𝑚 𝑑𝑡2 = 𝑭𝑖 ―𝜁 𝑑𝑡 + 𝑹𝑖(𝑡),

(19)

where 𝒓𝑖 is the instantaneous location of the i-th bead at time t, m is the bead mass calculated from the molecular weight and the number of actin monomers within one bead (depending on the adopted coarse-graining scheme), 𝑭𝑖 is the resultant force vector acting on the i-th bead through the combined potentials to be described later. 𝜁 = 𝑘𝐵𝑇 𝐷 is the frictional coefficient of each bead where the diffusivity D can be calculated through 𝑘𝐵𝑇 (3𝜋𝜂𝜎) with 𝜎 being the bead diameter and 𝜂 being the environmental viscosity. Lastly, 𝑹𝑖 is the random force originated from thermal excitations from the environment whose magnitude obeys a normal distribution with a 2𝑘𝐵𝑇𝜁/𝛥𝑡 according to the fluctuation-

mean of zero and standard deviation of 23



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dissipation theorem, with 𝛥𝑡 being the time step adopted in the BD/LD simulations. This description in Equation (19) is usually implemented in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)129 to track the movements of all involved beads and consequently the collective response of the entire network. It is well-known that an actin filament forms a double-stranded helix by assembling globular actin monomers,10,14 as depicted in Figure 6(a). In capturing the semiflexible behavior, each actin filament can be discretized into a chain of interconnected beads with mass m. The diameter 𝜎 of these beads can be set to be the same as the crosssectional diameter of F-actin.130 To account for the stretching rigidity of the filament, adjacent beads are assumed to be connected by a linear spring with a harmonic stretching potential given by (Figures 6(b) and 6(c)) 1

𝑈𝑠 = 2𝑘𝑠(𝑟𝑠 ― 𝜎)2,

(20)

where 𝑘𝑠 is the spring constant, 𝑟𝑠 is the distance between the centers of neighboring beads in the filament, and 𝜎 also represents the equilibrium spacing between adjacent beads here. Essentially, each actin filament is treated as an array of Hookean springs in series. As such, the value of 𝑘𝑠 can be selected to match the extensional stiffness of individual actin filaments measured by experiments. For example, it has been shown that an actin filament of 1 μm long, corresponding to a series of 167 springs if 𝜎 = 6 nm, exhibited a stretching stiffness of around 34 pN/nm.26,27,131 Therefore, 𝑘𝑠 can be estimated as 5678 pN/nm in the model. Clearly, this selection of 𝑘𝑠 depends on the coarse-graining scheme adopted at the onset. The bending resistance of F-actin can be described by adopting another harmonic 24


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potential with the form 1

𝑈𝑏 = 2𝑘𝑏(𝜃 ― 𝜋)2,

(21)

where 𝜃 is the angle formed between two lines connecting three neighboring beads in the filament (referring to Figure 6(c)), and 𝑘𝑏 represents the rotational stiffness between the two lines, which can be related to the persistence length 𝐿𝑝 of F-actin through 𝑘𝑏 =

𝑘𝐵𝑇 ⋅ 𝐿𝑝 𝜎

,

(22)

where 𝑘𝐵 is the Boltzmann constant and T is absolute temperature. Given that the persistence length of actin filament has been measured to be 10~17 µm,25,26 the bending stiffness 𝑘𝑏 can be calculated as 8280 pN ⋅ nm (by taking 𝑘𝐵𝑇 = 4.14 pN ⋅ nm, 𝐿𝑝 = 12 μm and 𝜎 = 6 nm). Cross-linker is another key component in determining the mechanical properties and dynamic behaviors of actin-based cytoskeletal networks, given the fact that more than 23 classes of cross-linking proteins with distinct properties have been identified in cells.121 The essential role of cross-linkers was explored by comparing the effects of unbinding and unfolding on the dynamic behaviors of cytoskeletal networks.81 For example, Li and coworkers82 found that cross-linkers contribute to the stability of filament bundles in filopodial protrusion. To capture the essential physics of crosslinkers in a simple manner, the interaction energy between two cross-linked filaments can be assumed as: 1

𝑈cl = 2𝑘cl(𝑟cl ― 𝑟0cl)2, 𝑟cl ≤ 𝑙cr,

(23)

where 𝑘cl represents the stiffness of the cross-linkers, while 𝑟cl and 𝑟0cl are the 25



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deformed and resting length, respectively (Figure 6(c)). Depending on the specific type of cross-linking proteins, the resting length was reported to be in a wide range from 10 to 160 nm,132-134 and the value of 𝑘cl was also measured to vary from 0.01 pN/nm for compliant cross-linkers (such as filamin A) to 6 pN/nm for rigid ones (such as scruin).124,135,136 It was postulated that two cross-linked filaments would disengage from each other once 𝑟cl exceeds a threshold value 𝑙cr,137 a quantity representing crosslinkers with different strength. On the other hand, a broken cross-linker was assumed to reform once the opposing pair are held within a rebinding distance less than 𝑙cr.138,139 Cytoskeletal networks are highly dynamic and active materials driven by various biochemical reactions and active forces induced by molecular motors. Specifically, myosin motors can generate internal contractions between opposing actin filaments and enhance the apparent network stiffness by more than two orders of magnitude.36,140 To capture the essential role of myosin motors, a prescribed number of force dipoles can be randomly added to the intersecting points of a given network in simulations. In this case, a linear potential can be introduced to every bead pair with a motor in between, namely, 𝑈m =

{

𝑓m(𝑟m ― 𝑟0𝑚), 𝑟m > 𝑟0𝑚 , 0, 𝑟m ≤ 𝑟0𝑚

(24)

to effectively exert equal but opposite forces (with the magnitude of 𝑓m) on the opposing beads (Figure 6(c)). It has been reported that the force magnitude induced by a single myosin head falls into the range of 1 to 5 pN, while multiple proteins usually assemble into thick motor bundles.86,140,141 These formed motor bundles have a typical initial length of ~1 µm and can generate a force dipole with magnitude up to 10 pN.39,113 26


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Computational modeling on reconstituted filamentous networks with active motors may provide insights into the connection between biochemical kinetics and mechanical deformation from molecular to network level. For example, Borau and coworkers85 established a three-dimensional BD model of cross-linked actin network with active motors and found that the motor-mediated network can sense substrate stiffness through cell-substrate interactions. Mak and coworkers86 developed a coarse-grained nonequilibrium BD model by incorporating three key processes at the microscopic scale, i.e. filament polymerization and depolymerization, force-dependent cross-linker binding and unbinding, and active motor sliding along filaments and generating contractile forces, and elucidated how these microscopic processes determine the internal stress profiles and the transitional mechanism of nonequilibrium morphology from homogeneous to aggregated networks. Following a similar approach, Freedman and coworkers87 constructed an agent-based BD model to recapture several important network characteristics observed in experiments, such as the spatial and temporal fluctuations of filaments, motor motility and motor-induced remodeling of the network. Excluded volume effect among different filaments could also be taken into account to prevent them from penetrating each other in simulations. In particular, the following shifted Lennard-Jones potential can be introduced to operate among filaments to serve this purpose:

𝑈LJ =

{

𝜎 12

[( )

4𝜀

𝑟

―

( ) ] ― 4𝜀[( ) 𝜎 6

𝜎 12

𝑟

𝑟𝑐

―

( ) ], 𝜎 6

𝑟𝑐

𝑟 ≤ 𝑟𝑐 ,

(25)

0, 𝑟 > 𝑟𝑐

where 𝜀 is a parameter representing the depth of the energy well, and r is the inter27



bead distance for bead pairs from different filaments (Figure 6(c)). Notice that, this energy becomes zero once r is beyond the cut-off distance 𝑟𝑐. Recently, Gong and coworkers90 employed the aforementioned Langevin dynamics strategy, i.e. Equations (19)-(25), to investigate the mechanical response and transport characteristics of actin networks containing fragile cross-linkers and active motors (Figure 6(c)). Simulation snapshots of the deformed network under different levels of shear strain are shown in Figure 6(d). Interestingly, the combination of stretching and bending potentials (Equations (20) and (21)) successfully reproduced the tensile response of a single filament as predicted by Odijk’s worm-like chain model142 as well as reported in the stretching experiment by Liu and Pollack26 (Figure 6(e)). When applied to 3D networks consisting of randomly oriented filaments, breakable crosslinkers and active motors, pronounced strain stiffening of the networks under shear was observed. However, such stiffening effect was found to be attenuated by the successive rupture of cross-linkers (Figure 6(f)). Interestingly, simulations also predicted that the appearance of motor-induced contractile forces within the networks will significantly increase their apparent modulus and shift their response to a different nonlinear regime. Specifically, the differential stiffness (defined as 𝑑𝜏 𝑑𝛾 along the stress-strain curve), increased by nearly two orders of magnitude at small strains when the number of motors became larger (Figure 6(g)). On the other hand, the network behavior under large strains was observed to be dominated by direct stretching of filaments, resulting in a very high bulk stiffness, and consequently, the influence of motors became relatively small (Figure 6(g)), all in quantitative agreement with experimental and theoretical 28


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findings.36,143 Finally, the fluctuating forces generated by active motors were also found to contribute to the enhanced diffusion of nanometer-sized particles within the networks,90 in consistency with recent experiments.39,144 It must be pointed out that, in addition to different network models discussed above, the so-called tensegrity approach has also been vigorously developed in the past decades to study cytoskeletal dynamics.145-148 The main idea there is that the cytoskeleton maintains its structural integrity and stability through a distribution of prestresses that are generated by contractile actomyosin stress fibers, against other structural elements such as microtubules, extracellular matrices and neighboring cells.145 Interestingly, numerous experimental observations of cytoskeleton such as strain hardening,149 prestress-dependent stiffness,150 elasticitc151 or viscoelastic152 response, and stretch-induced reorientation153 have all been satisfactorily explained by such idealized representations of structural tensegrity in terms of bars and strings.

5. Concluding remarks This review summarizes the vast and rapidly increasing amount of theoretical, finite element and molecular dynamics investigations on the mechanical behaviors of biopolymer networks that are essential to cellular physiological functions. The ultimate goal is to connect molecular level information, such as the microstructure of monomeric proteins, elasticity and fluctuation of individual filaments, binding kinetics of crosslinkers and contractility of active motors, to the bulk properties of filamentous actin networks including the nonlinear stress-strain relation, viscoelastic response, motor29



induced stiffening, etc. In this regard, the modeling and simulation studies described here can provide critical insights for advancing our understanding of the materialstructure-function relation in the cytoskeleton of cells, and guiding the development of active biomimetic materials in the future. Despite the tremendous progress achieved in this area, there is still a lot to explore on the mechanosensitivity and active response of cytoskeletal networks because of the intrinsic limitation of each modeling/simulation strategy. For example, many existing theoretical models only account for general physical features, like the semi-flexibility of individual filaments, strain-stiffening of the whole networks, force-bearing or forcegeneration capability of cross-linkers or motors, without incorporating biochemical processes of different protein species involved as well as the associated regulatory signaling pathways, which are known to play key roles in determining the dynamic response of the networks. All-atom molecular dynamics approach can provide molecular-level structural changes of proteins and their interactions, but is too computationally demanding for simulating the whole cytoskeleton/cell in a realistic manner. Coarse-graining methods can spare computational cost, at the expense of reduced information, but the simulated spatial and temporal scales are still far below those in actual experiments. Future efforts are required to coordinate models at different levels (i.e., from atomic to coarse-grained and eventually to continuum scale) to ensure accurate transmission of critical information among them. Although existing experimental evidence is far from allowing definitive validation or benchmarking of different theoretical models and simulations, these efforts do 30


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provide interesting insights into a number of physical and mechanical aspects of the complex cytoskeletal system. A major direction in the future will be to continue elucidating the relationship between the molecular architecture of the networks and the associated functioning/signaling pathways, which will be extremely useful for subsequent modeling. With continuous progress from the experimental side, it is also conceivable that more biological features can be incorporated into the present theoretical and computational platforms to examine important issues on the transmission and sensing of forces within the cytoskeleton of cells, which are crucial for understanding the role of cytoskeletal dynamics in related phenomena such as cell volume regulation,154-156 cell shape transformation,157,158 and mechanosensitive intracellular diffusion.159-161 On the other hand, the development of new theories and simulations could provide clues for the design of more effective and sophisticated experiments, helping us to understand exactly how the heterogeneous and dynamic cytoskeleton behaves in vivo.

Acknowledgments This work was supported by the National Natural Science Foundation of China (11672268, 11621062 and 91748209), the Zhejiang Provincial Natural Science Foundation of China (LR16A020001), and the Fundamental Research Funds for Central Universities of China (2017FZA4029). Y.L. gratefully acknowledges support from the Research Grants Council (Project No. HKU 17205114, HKU 17211215 and HKU 17257016) of the Hong Kong Special Administration Region. 31


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Figure captions Figure 1. Schematic diagram showing the main structural components of the cytoskeleton in a typical eukaryotic cell, with distinct types of actin architectures. (a) Crystal structure of globular actin9 (PDB code: 3HTB). (b) All-atom structure10 (PDB code: 5MVA) and schematic diagram of a single actin filament. (c) Branched actin networks formed with the promoting factor of branching nucleation (Arp2/3 complex) and the capping protein that terminates filament growth. (d) Bundles of filamentous actin composed of parallelly organized actin filaments and cross-linking proteins (fascin). (e) Random actin networks cross-linked with flexible and extensible crosslinkers (filamin). (f) Stress fibers formed with long orthogonal cross-linkers (α-actinin) and parallel myosin II bundles.

Figure 2. Experimental techniques used to characterize the mechanical response of biopolymer networks. (a) Shear rheology. (b) Stretching rheology. A shear/extensional load or deformation is imposed to the network, and the mechanical properties can be obtained by measuring the responses of the network structure.41-44 (c) Atomic force microscopy. A low-amplitude oscillation is applied to the biopolymer network, where the network displacement and reaction force are measured through the deflection of the cantilever.45,46 An external force is applied to the probe particles embedded in the network using (d) magnetic tweezers or (e) optical tweezers, where the network properties can be derived by the force-displacement relation of the probe particles through generalized Stokes equation.35,47 (f) Passive microrheology, by relating the 52


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local elastic modulus of the network to the excursion of tracer particles under thermal fluctuations.48,49

Figure 3. The spectrum of time and length scales encountered in modeling actin monomers, filamentous actin structures, actin networks and whole cells through a multiscale strategy. (a) Molecular dynamics (MD) model for G-actin at angstrom scale.63,64 (b) Full-atom MD model,67-71 (c) coarse-grained molecular dynamics (CGMD) model,72-74 and (d) Brownian dynamics (BD) model75,76 to investigate the mechanical behaviors of single actin filaments. (e) BD model57,77-90 to characterize the mechanical behaviors of complex actin networks. (f) Continuum level cell model.65,66

Figure 4. (a) Schematic of a freely jointed chain.94,95 (b) Schematic of a worm-like chain.98 (c) Schematic of the eight-chain Arruda-Boyce model with and without principal stretches.104,105 (d) Schematic of the affine model for biopolymer networks under shear.4,50,106

Figure 5. (a) Mikado model.110 (b) An initially undulated filament network.112 (c) The FEM-Langevin dynamics model.119 A filament immersed in a thermal reservoir undergoes shape fluctuations. The effects of bombardment of water molecules on the motion of the filament can be represented by a macroscopic medium viscosity ξ and a distributed random load f. Following the standard FEM procedure, an un-deformed filament is divided into numerous equal-sized segments and a transverse load 𝑓𝑗𝑖, acting 53



on element i over the time interval [𝑡𝑗 𝑡𝑗 + 1], is applied to represent the effect of thermal fluctuations. (d) Filamentous network with compliant cross-linkers, modeled as a combination of linear and rotational springs.124

Figure 6. A coarse-grained Langevin dynamic model of actin-based cytoskeletal networks. (a) Coarse-graining scheme of a single actin filament by a chain of interconnected beads. (b) A representative configuration of two intersecting filaments (green), with a cross-linker (blue) and a contractile motor (pink) in between. (c) The stretching and bending of filaments, cross-linking and motor-induced contraction represented by different interaction potentials in the model. (d) Snapshots of the deformed network under different levels of shear strain  . The color bar indicates the level of combined stretching and bending energy in the filaments. (e) The end-to-end distance of a single filament as a function of the applied force from the simulation, in agreement with the experimental data by Liu and Pollack26 and the worm-like chain model by Odijk142. (f) Nonlinear stress-strain curves of the networks when the crosslinkers break at a threshold distance 𝑙cr of 30 or 40 nm. The case of permanent crosslinkers is also shown for comparison (𝑙cr = + ∞). (g) Differential stiffness 𝑑𝜏 𝑑𝛾 of the networks as a function of shear strain for different numbers of contractile motors (𝑁𝑚).

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Figure 1

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Figure 2

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Figure 3

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Figure 4

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Figure 5

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Figure 6

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For Table of Contents Use Only

Modeling and simulations of the dynamic behaviors of actin-based cytoskeletal networks

Bo Gong, Xi Wei, Jin Qian, Yuan Lin

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Fig-1 169x183mm (300 x 300 DPI)


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Fig-2 169x105mm (300 x 300 DPI)



Fig-3 357x259mm (300 x 300 DPI)


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Fig-4 169x90mm (300 x 300 DPI)



Fig-5 169x134mm (300 x 300 DPI)


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Fig-6 169x167mm (300 x 300 DPI)


Modeling and Simulations of the Dynamic Behaviors of Actin-Based

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