CLC number: Q66

On-line Access: 2024-08-27

Received: 2023-10-17

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Bao-long Li, Yi-fan Wang, Jing-hai Gong. Using a form-finding model to analyze the effect of actin bundles on the stiffness of a cytoskeleton network[J]. Journal of Zhejiang University Science A, 2014, 15(9): 732-742.

@article{title="Using a form-finding model to analyze the effect of actin bundles on the stiffness of a cytoskeleton network",

author="Bao-long Li, Yi-fan Wang, Jing-hai Gong",

journal="Journal of Zhejiang University Science A",

volume="15",

number="9",

pages="732-742",

year="2014",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.A13b0287"

}

%0 Journal Article

%T Using a form-finding model to analyze the effect of actin bundles on the stiffness of a cytoskeleton network

%A Bao-long Li

%A Yi-fan Wang

%A Jing-hai Gong

%J Journal of Zhejiang University SCIENCE A

%V 15

%N 9

%P 732-742

%@ 1673-565X

%D 2014

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.A13b0287

TY - JOUR

T1 - Using a form-finding model to analyze the effect of actin bundles on the stiffness of a cytoskeleton network

A1 - Bao-long Li

A1 - Yi-fan Wang

A1 - Jing-hai Gong

J0 - Journal of Zhejiang University Science A

VL - 15

IS - 9

SP - 732

EP - 742

%@ 1673-565X

Y1 - 2014

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.A13b0287

**Abstract: **Networks of actin filaments and bundles are ubiquitous in cellular cytoskeletons, but the elasticity of the network is not well understood. In this paper, a computational model based on form-finding analysis is proposed to investigate the stiffness of cytoskeleton networks consisting of actin filaments and bundles. The model shows that networks with parallel bundles aligned in the stretching direction are stiffer than those with randomly distributed bundles. The results provide a mechanical explanation for the experimental observation that cells primarily create parallel rather than disordered bundles during cell adhesion and cell motion. The effect of filament undulations on network stiffness is explored briefly. The results show that undulations can soften the network by increasing the bending-dominated deformations in filaments and bundles. Finally, we find that the effect of the relative density of bundles depends on their orientation. Increasing the density of randomly distributed bundles has no effect on the stiffness of cells, but softens the cytoskeleton network. In contrast, the stiffness of networks of parallel bundles first increases, then reduces as the relative density of bundles increases. The stiffest network is a mixture of actin filaments and bundles.

细胞骨架网络；找形模型；弹性模量；微丝束

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1. Introduction

The actin cytoskeleton is a composite intracellular biopolymer network, usually composed of actin filaments and bundles cross-linked by actin-binding proteins (Fig.

Recently, a number of computational models, designed to complement experimental work from the point of view of mathematics and mechanics, have been proposed to predict the elasticity of cytoskeleton networks. These models include the open-foam model (Satcher and Dewey,

In this paper, we use a 2D form-finding model (Gong et al.,

2. Form-finding model

Parameter | Value | Reference |

Elastic modulus of actin filaments (GPa) | 1.4 | Matsushita et al. (2010) |

Diameter of actin filaments (nm) | 7 | Matsushita et al. (2010) |

Length of actin filaments (μm) | 5±2 | Kasza et al. (2010) |

Length of actin filament segments |
0.3±0.06 | |

Relative density of actin filaments |
0.15–0.30 | Ethier and Simmons (2007) |

Yield tensile force of actin filaments (nN) | 0.25 | Lin et al. (2010) |

Maximum length of cross-linkers (μm) | 0.3 | Furuike et al. (2001) |

Stiffness of cross-linkers (EA) (nN) | 1.35 | Furuike et al. (2001) |

Yield tensile force of cross-linkers (pN) | 60 | Ferrer et al. (2008) |

2Relative density is the amount of actin filaments per unit volume of filament network

In this study, we extend the form-finding model to build up a model of cytoskeletal structure composed of cross-linked networks of filaments and bundles. At this stage, our goal is to demonstrate the suitability of the form-finding method. Thus, we assumed that seven actin filaments form an actin bundle whose cross-section is round (Tseng et al.,

Component | Diameter (nm) | Cross-section area (nm^{2}) |
Moment of inertia (nm^{4}) |

1 actin filament | 7 | 25 | 118 |

1 actin bundle | 21 | 175 | 9546 |

Step 1: Define a square domain (10 μm×10 μm) as the extent of the initial cytoskeleton network.

Step 2: Actin filaments and bundles are placed sequentially within the domain stochastically until their relative densities reach the specified value. For each actin filament, the length is sampled from a truncated Gaussian distribution (negative values are ignored) with mean and standard deviation as shown in Table

Step 3: Actin-binding proteins (filamin) are modeled as cable element cross-linkers to connect the actin filaments and bundles into a network. Networks are generated in two steps: (1) actin filaments and bundles are divided into several segments by actin-binding sites, whose lengths are determined by the pore size in real cytoskeleton networks. (2) cross-linkers are generated by connecting any two actin-binding sites whose distance apart is less than the length of the maximum cross-linker (Table

Step 4: After a model is created by steps 1 to 3, a form-finding analysis is carried out to compute the final equilibrium shape of the actin network. The form-finding analysis is accomplished using nonlinear finite element analysis after a small tensile pre-stress force is applied to the cross-linkers. Iteration is then performed until a self-equilibrium configuration is achieved (Fig.

The pre-stress force in cross-linkers is used to facilitate the form-finding analysis, and its magnitude can be calibrated by comparing the predicted filament undulations with experimental observations. A large pre-stress equates to a severe actin filament undulation and a small force to a gentle fluctuation. The maximum offset of all the actin-binding sites of filaments and bundles after form-finding, which reflects the undulation index of actin filament network samples, is set to 0.5 μm unless otherwise specified. After form-finding, the elastic modulus (

Since we are going to explore how the orientation of actin bundles in networks influences the stiffness of the cytoskeleton, the angle of orientation of the actin bundles needs to be adjustable. In this study, the parameter ‘spread in bundle orientation’ is defined as the absolute maximum value of the angle between the orientation of bundles and the horizontal direction in which the model is stretched. For instance, if the spread in bundle orientation is set to 20° before modeling, each of the bundles to be generated in the model is sampled with an orientation angle to the horizontal direction from uniform random within [−20°, 20°]. Consequently, all the bundles will align in the horizontal direction if the spread in bundle orientation is equal to 0. Furthermore, bundles have a uniform random distribution in every direction when the spread in bundle orientation is equal to 90° (Fig.

The total amount of actin filaments in a form-finding model is determined by the relative density of actin filaments (Table

Compared with the aforementioned models, the form-finding model provides a means to generate realistic cytoskeletal network topologies incorporating major biological features. Another useful feature of the form-finding model is that it is inherently able to handle large deformations, since most biologically relevant deformations are large. Furthermore, the form-finding model captures the influential roles that actin filaments, actin-binding proteins, and actin bundles play in the stiffness of a cell.

3. Results

The influence of the spread in bundle orientation to the stretching direction on network stiffness was studied using a sets of 1000 samples to obtain more reasonable curves (Fig.

In this study, the MO values changed from 0.1 to 0.6 μm, which represented the variation in filament undulations from gentle to relatively severe. We used six sets of samples (100 in each set), across MO values ranging from 0.1 to 0.6 μm in increments of 0.1 μm, to investigate briefly the effect of filament undulations on the elastic modulus of the cytoskeleton network. The same analysis was performed five times for different bundle relative densities of 0%, 20%, 40%, 60%, and 80%, respectively. The spread in bundle orientation was set to 0 in each model. Fig.

1. Parallel bundles

A group of 10 studies (100 samples in each study) was analyzed where the relative density of bundles varied from 0% to 90% in increments of 10%. For each study, the elastic modulus of each sample was computed six times with different intensities of filament undulations, which were embodied in the MO values. Fig.

2. Randomly distributed bundles

The same studies as above were reanalyzed for the case in which all bundles are distributed uniformly and randomly. Fig.

4. Discussion

The computational model of a cytoskeleton network used in this study contains actin filaments cross-linked and bundled by actin-binding proteins. The effects on network stiffness caused by other cytoskeletal components like microtubules or intermediate filaments were neglected. Also, another assumption in this study was that each actin bundle is modeled as a composite structure of seven actin filaments. In fact, the actin bundle is formed by actin filaments bundled by repeating units of bundling proteins, and the spacing between bundling proteins along the bundle can change over time. Therefore, the bundle is not a composite structure of seven actin filaments and the moment of inertia of bundles calculated in this model is the upper bound. Nevertheless, this is probably not critical to the study of the variation in cytoskeleton network stiffness with the relative density and alignment of bundles.

All the results in this work can be explained by the deformation of the cytoskeleton network model being bending dominated (non-affine) or axial stretching dominated (affine). In the actin networks form-finding model, both bending and axial deformations occur within actin filaments and bundles. The axial stretching stiffness (represented by

Filament undulations cause transverse bending in filaments and bundles, increasing the number of filaments/bundles whose deformation is bending dominated. This is the main reason for the reduction in cytoskeleton network stiffness when filament undulations increase (Fig.

The effects of spread in bundle orientation on the stiffness of the cytoskeleton network (Fig.

In this work, the relative density of filaments was fixed at 0.3‰ in every model, i.e., the total amount of actin material was constant. Thus, the number of individual filaments was negatively correlated to the relative density of bundles. On condition that the bundles are distributed randomly in the network, as actin filaments were expended to generate bundles, the total number of segments of actin filaments and bundles, as well as cross-linkers, decreased, and the deformation of actin networks changed from axial stretching dominated (affine) to bending dominated (nonaffine). As a result, the networks became more and more compliant. In contrast, when bundles aligned along the stretching direction, they were dominated mainly by axial deformation. Therefore, as the relative density of bundles increased, the cytoskeleton network became stiffer. However, as most of the actin filaments were expended to generate the parallel bundles, the density of cross-linkers decreased sharply (Fig.

5. Conclusions

* Project supported by the Shanghai Jiao Tong University-Johns Hopkins University International Cooperation Project (No. TS0520101002)

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