DDP 2 Report

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Mechanics of Filament Networks | 1

Mechanics of Filament Networks

Dual Degree Dissertation

Submitted in fulfillment of the requirements of the degree ofBachelor of Technology and Master of Technology

Tarun Meena

(08D10048)

Under the supervision of

Prof. Mandar M. Inamdar

Department of Civil Engineering

Indian Institute of Technology Bombay

Powai, Mumbai - 400 076

April, 2013


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Mechanics of Filament Networks | ACKNOWLEDGEMENT 2

ACKNOWLEDGEMENT

I would like to express deep gratitude and sincere thanks to my guide Prof. Inamdar for his

invaluable support and guidance throughout the work

.Tarun Meena

May, 2013


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ContentsACKNOWLEDGEMENT ............................................................................................................................. 2

Table of Figure ........................................................................................................................................ 5

Abstract ................................................................................................................................................... 6

Chapter 1: Introduction .......................................................................................................................... 7

1.1 Motivation ..................................................................................................................................... 7

1.2 Objectives...................................................................................................................................... 7

1.3 Organization of the report ............................................................................................................ 7

Chapter 2: The F-Actin Network ............................................................................................................. 8

2.1 The Cytoskeleton .......................................................................................................................... 8

2.2 Composition .................................................................................................................................. 8

2.3 Actin Filament ............................................................................................................................... 9

2.4 F-Actin Network ............................................................................................................................ 9

2.5 Cross Linked F-actin network ...................................................................................................... 10

2.6 Mechanics of F-actin networks in vitro ....................................................................................... 11

2.6.1 Linear Viscoelastic response ................................................................................................ 11

2.6.2 Nonlinear elastic response ................................................................................................... 11

Chapter 3: Existing Models ................................................................................................................... 11

3.1 Deformation of Cross-Linked Semiflexible Polymer Networks ................................................... 11

3.2 Elasticity of Stiff Polymer Networks ............................................................................................ 12

3.3 Elasticity of Planar Network ........................................................................................................ 13

Chapter 4: Modelling and analysis ........................................................................................................ 14

4.1 Finite Element Analysis. .............................................................................................................. 14

4.2 ABAQUS Finite Modelling and Analysis....................................................................................... 16

4.2.1 Assumptions made in model ................................................................................................ 17

4.2.2 The 2D-Network ................................................................................................................... 17

4.3 Procedure .................................................................................................................................... 17

4.4 Random Network formulation .................................................................................................... 17

4.5 Analysis and Calculations ............................................................................................................ 18

Chapter 5: Results and Discussion ........................................................................................................ 19

5.1 Effective elasticity of the filament Network ........................................................................... 19

5.1.1Overview ............................................................................................................................... 19

5.1.2 Results and Discussion ......................................................................................................... 20

5.2 Effect of variation in density on the elasticity of the network ................................................... 20


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5.2.1Overview ............................................................................................................................... 20


5.2 Comparison of Network with long filaments and short filaments ............................................. 22

5.3.1Overview ............................................................................................................................... 22


Conclusion and future scope of work ................................................................................................... 25

We observed the mechanical properties of Filament network. We showed that short filaments

influence the connectivity of the network structure resulting in a reduced elasticity. Similarly, we

observed the effect on elasticity due to density variation of the filament network............................... 25

Bibliography .......................................................................................................................................... 26

ABAQUS Input Code .............................................................................................................................. 27


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Table of Figure

Figure 1: The cytoskeleton Network ....................................................................................................... 8

Figure 2: F-actin Network...................................................................................................................... 10

Figure 3: cross linked filament network................................................................................................ 10

Figure 4: Example of energy distribution throughout networks of cross link densities ....................... 12

Figure 5: Typical networks at high and low density .............................................................................. 13

Figure 6: Deformation of typical fiber network under uniaxial tension ............................................... 14

Figure 7: Eight Nodal Element .............................................................................................................. 15

Figure 8: FEA representation of various engineering problems ........................................................... 16

Figure 9: Model of the Network with vertical displacement and support condition ........................... 18

Figure 10: Model of the Network with vertical displacement and support condition(A) and its

deformed shape(B) ............................................................................................................................... 19

Figure 11: Plot of Reaction Force (N) Vs Displacement(m) ................................................................... 20

Figure 12: Low density Network(A) and High density Network(B) ...................................................... 21Figure 13: Deformed shapes of the low Density network (A) and High Density Network (B) .............. 21

Figure 14: Plot of Reaction Force Vs Displacement for low Density network (A) and High Density

Network (B) ........................................................................................................................................... 22

Figure 15: F-actin Network organized by long (A) and short (B) actin filaments at identical

concentration of filaments and intersections ....................................................................................... 23

Figure 16: Deformed shape of F-actin Network organized by long (A) and short (B) actin filaments at

identical concentration of filaments and intersections ........................................................................ 23

Figure 17: Plot of Reaction Force Vs Displacement for of F-actin Network organized by long (A) and

short (B) actin filaments ........................................................................................................................ 24


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Abstract

The structural integrity of the cell depends upon its cytoskeleton and for small deformation

the elasticity of the cell depends on its actin network, a major constituent of the cytoskeleton.

Actin filaments combine to form one of the predominant cytoskeletal networks important to

these biological processes. The purpose of this study is to capture the elasticity of the

Network for different microscopic properties of such networks. In this study a computer

model of actin dynamics yields networks structure which can be directly fed into simulationsof network elasticity.

Fibers are idealised as a circular beam of diameter 1mm and 2-dimensional finite element

model of fiber network is generated and analyzed in ABAQUS. The 2-dimensional Model

has been analyzed for variation in microscopic properties such as different density of the

network and Network of different fiber length.


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Chapter 1: Introduction

1.1 Motivation

Numerous experiments have shown mechanical loading to be an important factor in the

development and maintenance of a wide variety of tissues such as muscles, cartilage, tendon

and bone and organs. The deformation of the cells within these tissues and organs dictated by

their mechanical behaviour under loading. A critical component governing the mechanical

behaviour of adherent cells is the actin cytoskeleton.

Mechanical forces play an essential role in living cells. Cells sense and transmit external

forces in the formation of tissue, protrusive and contractile forces are generated in cell

motility and division. In all of these processes, it is believed that cellular cytoskeleton is

essential to a quantitative understanding of processes in living cells.

1.2 Objectives

The objective of this thesis is to understand the mechanics of filament network by developing

a 2-D finite element modelling of the F-actin network. In this report, The 2-dimensional

Model has been analyzed for variation in microscopic properties such as different density of

the network and Network of different fiber length.

1.3 Organization of the report

The content covered in this thesis is outlined as follows:

Chapter 2 consists of brief introduction to F-actin network and its composition Chapter 3 consists of the literature review of finite element analysis Chapter 4 consists briefing about the existing Chapter 5 consists of the results and discussion of analysis

Reference cited are included at the end of thesis In the end Abaqus input code has been cited


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Mechanics of Filament Networks | Chapter 2: The F-Actin Network 8

Chapter 2: The F-Actin Network

2.1 The Cytoskeleton

The cytoskeleton is a cellular scaffolding or skeleton. It is contained within a cells cytoplasm

and is made out of protein. The cytoskeleton is present in cells.

Cells contain elaborate arrays of protein fibers that serve such functions as:

Establishing cell shape Providing mechanical strength Locomotion and muscle fiber contraction Chromosome separation in mitosis and meiosis Intracellular transports of organelles Cell motility Ensures proper division of cells during cellular reproduction.

2.2 Composition

The cytoskeleton is an organized network of three primary protein filaments:

microtubules, actin filaments, and intermediate fibers. A key player in this cytoskeleton isF-actin, which exhibits significant rigidity on the cellular scale. The complexity of the

cytoskeleton can be seen in the abundant F-actin stress fibers (green) in the endothelial

cell shown below:

Figure 1: The cytoskeleton Network


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These small proteins join together and polymerise in large numbers to form the

cytoskeleton, which may be more than 2000 times larger in linear dimensions than the

average size of an individual protein molecule. The cytoskeleton, by forming an internal

framework, supports the cell like a framework of beams and columns supporting a

building.

2.3 Actin F il ament

Monomers of the protein actin polymerize to form long, thin fibers. These are about 8 nm

in diameter and being the thinnest of the cytoskeletal filaments, are also called

microfilaments. (IN skeletal muscle fibers they are called thin filaments.) Some

functions of actin filaments are as follows:

Form a band just beneath the plasma membrane thato Provides mechanical strength to the cello Links transmembrane proteins (e.g., cell surface receptors) to cytoplasmic

proteins

o Pinches dividing animal cells apart during cytokine-sis Generate cytoplasmic streaming in some cells Generate locomotion in cells such as white blood cells and the amoeba Interact with myosin(thick) filaments in skeletal muscle fibers to provide the

force of muscular contraction

2.4 F-Actin Network

The spatiotemporal regulation of the mechanical behaviours of the filamanteous actin(F-

actin) cytoskeleton networks may regulate cellular shape change and force generation in cellmigration and division. In homogeneous networks of F-actin formed with a single cross-

linking protein in vitro. F-actin formed in vitro demonstrate a broad diversity of mechanical

behaviours that, in principle, could be harnessed to regulate the mechanical properties of cells

over a variety of time and length scales.


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Figure 2: F-actin Network

2.5 Cross L inked F-actin network

In the cytoskeleton, the local microstructure and connectivity of F-actin is controlled by

actin-binding proteins. These binding proteins control the organization of F-actin into mesh-

like gels, branched dendritic networks, or parallel bundles and it is these large-scale

cytoskeletal structures that determine force transmission at cellular level.

The cross-linking proteins found inside most cells are quite different from simple rigid,

permanent cross-link in two important ways. Cross-links have a compliance that depends on

their detailed molecular structure and determines network mechanical response. Thus the

kinetics and mechanics of F-actin-binding proteins can have a significant impact on the

mechanical response of cytoskeletal networks.

Figure 3: cross linked filament network


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Mechanics of Filament Networks | Chapter 3: Existing Models 11

2.6 Mechanics of F -actin networks in vitr o

Isotropic, three-dimensional networks of F-actin can be formed in vitro by polymerizing F-

actin in the presence of cross-linking proteins. Although the structure of these networks is

quite different from those found in many live cells, these systems have served as models with

which to identify basic mechanisms of mechanical response of F-actin networks. Here we,briefly discuss two ubiquitous features of the mechanics of F-actin networks: linear

viscoelasticity and non linear elastic response.

2.6.1 Linear Viscoelastic response

Simple elastic materials are describe by a linear stress-strain relationship that relates the

deformation (strain) of a material to the force per unit area(stress) exerted; the ratio between

the stress and strain is called the elastic modulus. By contrast, in a Newtonian fluid, an

applied stress results in a constant deformation rate; the ratio between the stress and strain is

the measure of viscosity. Generally, networks formed from biopolymers often show

mechanical properties that are in between that of a pure fluid or elastic solid are said to beviscoelastic.

2.6.2 Nonlinear elastic response

For most F-actin networks, the mechanical response becomes nonlinear at large stress or

strain, such that stiffness depends on the magnitude of the applied stress or strain. Generally,

networks with a small degree of cross-linking are observed to soften at large strains. In

contrast, dense(high F-actin density) networks with a high concentration of cross-links are

observed to stiffen at intermediate strains.

Chapter 3: Existing Models

3.1 Deformation of Cross-L inked Semifl exible Polymer Networks

(David A. Head, 2003)

Here, a simple model for cross-linked rods has been examined. It does not only allow to

quantitatively test the relationship between microscopic and macroscopic coefficients of

randomly cross-linked network, but also sheds light on the intimately related issue of the

spatial distribution of the network strain. Among the most fundamental properties of

polymer networks is the way in which they deform under stress.

The assumption here is that the deformation field is affine down to length scales

comparable to the smallest microscopic scales in the material is great simplification that

allows one to construct quantitative theories relating the macroscopic elastic constants of a

gel to the microscopic properties of its constituent polymers. We show that the degree of

nonaffine strain is a function of length scale and degree of cross-linking.


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Figure 4: Example of energy distribution throughout networks of cross lin k densities

Through the analysis it is found that networks become increasingly affine even down to thesmallest scales of the network e.g. the mesh size, at high cross-link density, high molecular

weight, or for rigid elements. We find that the bulk elastic moduli converge to those

predicted from affine theory.

Here, model has been studied in detail the dependence of the bulk shear modulus of the

material upon the cross-link density of the polymer gel as well as the bending and extension

moduli of the individual filament. The Network was modelled via the Hamiltonian per unit

length for a filament

3.2 Elasticity of Stif f Polymer Networks

(Frey, 2008)

This model is about the elasticity of a two-dimensional network of rigid rods. The essential

features incorporated into the model are anisotropic elasticity of the rods and the random

geometry of the network. It is shown that there are three distinct regimes, characterized by

two distinct length scales on the elastic backbone.


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Figure 5: Typical networks at high and low density

For understanding the elasticity of stiff polymer networks a two-dimensional is defines as

above. The random network was generated by placing N line-like objects of equal length l

on a plane with area A=L2 such that both position and orientation of the filaments are

randomly distributed.

For high densities, where compressional stiffness is lower or comparable to the bending

stiffness, the shear modulus scales linearly with the filament compressional modulus and

number of filament per unit area. It is by now established that the elastic modulus can be

described quantitatively in terms of effective medium models. In the high line density regime

the network behaves as a homogeneously elastic medium, dominated by the compressional

modulus of the individual filaments.

The conclusion in this model by the observation that almost all of the energy stored in the

deformed network is accounted for by transverse deformation of rods.

3.3 Elasticity of Planar Network

(X.-F. Wu, 2005)

In this model, wool assembly was treated as a layered system with fiber bending deflection

between neighbouring contacts, while no fiber elongation, contraction, or torsion were

considered. In this model, only the fiber axial deformation was considered. The effective

stiffness of the representative area element (RAE) was obtained by averaging the stiffness

contributed by fibers in all possible directions within the RAE.

The effective stiffness of a planar random FN increases with the increase of arial density,

which can be measure by the number of fiber per unit area (fiber concentration).

Here, a macro -mechanics model has been developed for the elastic stiffness of planar FNs.

The model account for microscopic deformations of fiber segments of all possible lengths

and orientations. Explicit expression for the effective stiffness of random FNs has been

derived based on the equivalency of the strain energy of the anisotropic continuum medium.

The obtained constitutive relations can be used for the prediction of FN mechanical

properties, scaling analysis, and optimization of FNs with


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Mechanics of Filament Networks | / 14

Figure 6: Deformation of typical fiber network under uniaxial tension

Strong contact bonds. The constitutive relation derived in this case is as follow

Where,

Chapter 4: Modelling and analysis

4.1 F ini te Element Analysis.

The Finite Element Analysis (FEA) method is a powerful computational technique for

approximate solutions to a variety of real world engineering problems having complex


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Mechanics of Filament Networks | Chapter 4: Modelling and analysis 15

domains subjected to general boundary conditions. FEA has become an essential step in the

design of modelling of a physical phenomenon. A physical phenomenon usually occurs in

continuum matter (solid, liquid or gas) involving several field variables. The field variables

may vary from point to point, thus possessing an infinite number of solution is the domain. A

continuum with a known boundary is called domain.

The basic idea of the finite element method is to break up a continuum into discrete number

of small elements. These elements can be modelled mathematically by a stiffness matrix

and are connected by nodes that have degrees of freedom. This is the same way we deal with

bending and truss elements. However, beam and truss members have natural locations at

which to define nodes. The elements can be expanded upon in order to model fluid flow and

temperature, by the use of conversion matrices carry out by the software packages.

This enables FEM to solve more complex element behaviour to be modelled. This method is

used to solve a modelling problem (Abed, 2010) by dividing the solution domain into discrete

regions, these are the finite elements. These elements are connected by nodal points.

Depending on the degree of accuracy required, the geometry of mesh can be altered during

analysis.

The basic procedure is to assume shape functions that describe how the nodal displacements

are distributed throughout the element based.

Example of common eight nodal elements.

Figure 7: Eight Nodal Element

This element has two degrees of freedom in this case are displacements at each node. The

number of degrees of freedom becomes finite when each element is defined in terms of nodal

values. Therefore the solution can be interpolated using shape function can the nodal values.

Nodal values given for point 1 in terms of coordinate S and T


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The basis of FEA relies on the decomposition of the domain into a finite number of sub-

domains (elements) for which the systematic approximate solution is constructed by applying

the variational or weighted residual method. In effect, FEA reduces the problem to that of a

finite number of unknown field variables in terms of the assumed approximating functions

within each element.

Figure 8: FEA representation of various engineering problems

The ability to discretize the irregular domains with finite elements makes the methodvaluable and practical analysis tool for the solution of boundary, initial and eigenvalue

problems.

4.2 ABAQUS F in ite Modell ing and Analysis

Finite Element modelling provides a simply cost effective way of monitoring and predicting

situations that occurs in any sphere of engineering, medicine, aeronautics, and fusing them in

order to better understand the operational capacity of certain aspects, that would have been

unknown unless physical experiments were conducted. Thus, in this thesis it is most suitable

as we are working living cells and peoples lives would hang in the balance, if experiments

were conducted on patients without a definite outcome of such results.


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FEM package such as ABAQUS is a very useful program as it allows for the material,

geometric and boundary conditions to be set and adjusted as information becomes apparent,

as information is exchanged between professional who employ this program. The results of

the model, such as stress strain curves, pressure displacement curves etc. has a short run over

time. As is feature helps to address the problems that might occur and alter input parameters.Thus, hugely saving on the time factors that would hamper any kind of bioengineering

research or study, due to the overwhelming amount of variables.

4.2.1 Assumptions made in model

While Modelling several assumptions have to be made in order to simplify and practically

model the actual system. They key assumptions are as follows:

The fibers are straight and oriented in the same plane The fibers are randomly distributed and of different lengths

The deformation is linearly elastic Beam cross-section is assumed to be circular with radii of 1 mm. Total elasticity of the network is assumed to be dependent on the elasticity of

filaments and connectors.

Effect of the medium, in which fibre networks, is being assumed negligible. Forces are transferred between elements through intersection only.

4.2.2 The 2D-Network

The actin filament has been idealized as 2D beams of circular cross-section with diameter of

1mm. The elements used for the purpose are modelled by the beam element type B23.

Element type B23 is and their hybrid equivalent use linear interpolation.

4.3 Procedure

For the filament network analysis, step by step procedure is as follows:

1. Sketched the two dimensional geometry of filament networks and created a partrepresenting the network

2. Defined the material properties and section properties of the network3. Assembled the model and created instance of the geometry4. Configured the analysis procedure and output requests5. Applied the loads and Boundary conditions to the frame6. Meshed the frame7. Created job named fil network and submitted for analysis8. Viewed the result of the analysis

The above procedure is repeated for variation in network density and application of

constraint.

4.4 Random Network formul ation

A 2D random filament network us defined as a set of independently deposited short line

segments on a two-dimensional plane. Here, in this case fibers of random length and


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orientations are placed in the modelling. Network has been analyzed for variation in length of

cytoskeleton and denseness of the network.

Network has been idealised as mixture of 2-D beam element of diameter 1 mm. Any two

intersection beam may be joined at their intersections. These joints may be either rigid or

free.

In the network at y=0,x=0 and x=L hinge and roller boundary conditions have been applied.

Displacement have been applied at y=L, i.e. at the top of the surface.

Figure 9: Model of the Network with vertical displacement and support condition

For the fixed value of displacement (strain) reaction forces(stress) have been calculated in the

network for establishing the relationship between stress and strain. In the process, a input file

is generated in Abaqus.

4.5 Analysis and Calculations

The model is analysed using static linear analysis procedure. The reaction is obtained on the

upper surface of the network are summed to the total reaction force in 2 direction, i.e. y

direction, RF2. Elastic modulus of the network, Ec, can be calculated as

Ec = (RF2/A)/(disp/A)


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Chapter 5: Results and Discussion

5.1 Effective elasticity of the filament Network

5.1.1Overview

The load-displacement curve is obtained for a network with Length of fiber, L is 20m and

E=2x1011

N/m2. All intersections are connected rigidly and hinge is provides at the nodes x=0,50 m

and y=0m. Reaction forces have been calculated for the displacement in vertical direction.

(A) (B)Figure 10: Model of the Network with vertical displacement and support condition(A) and its

deformed shape(B)


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Mechanics of Filament Networks | Chapter 5: Results and Discussion 20

Figure 11: Plot of Reaction Force (N) Vs Displacement (m)

5.1.2 Results and Discussion

From the above analysis on the filament network Elasticity of the network comes out to be

2.822E+10 N/m2

and the ratio of Elasticity of the network with the elasticity of filament i.e.

Effective modulus of the elasticity, Ec/E=0.1411195

5.2 Effect of variation in density on the elasticity of the network

5.2.1Overview

All intersections are connected rigidly and Hinges and rollers have been put at y=0,x=0 and

x=100m. Nodes at y=50m are subjected to varying vertical displacements and displacement

of the network in y-direction has been restricted. Length of the fiber is take 20m and

E=2*1011N/m2

The resulting reaction forces were summed up corresponding to different displacements and

the elastic modulus of the network was calculated for the different densities. The results for

the modulus of elasticity of the network obtained for variation in densities are compared.

0.00E+00

5.00E+09

1.00E+10

1.50E+10

2.00E+10

2.50E+10

3.00E+10

3.50E+10

4.00E+10

4.50E+10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Reaction

Force(N)

Displacement(m)

Fig: Reaction Force Vs Displacement the

filament Network


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(A) (B)Figure 12: Low density Network(A) and High density Network(B)

(A) (B)Figure 13: Deformed shapes of the low Density network (A) and High Density Network (B)


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Figure 14: Plot of Reaction Force Vs Displacement for low Density network (A) and High

Density Network (B)


From the analysis on both the networks A & B, we get the elasticity of A, low density

network, 3.00E+09N/m2

while the elasticity of the high density network 2.822E+10 N/m2.As it

can be clearly seen that there is a increase in the elasticity as the density of the network

increases. Since the power law is the form of E= C(-ref) (R.Y. Kwon, 2008)where is the

volume(density), validating the above result, where elasticity is increasing with the increase

in density of the network.

5.3 Compari son of Network with long f il aments and short f il aments

5.3.1Overview

All intersections are connected rigidly and Hinges and rollers have been put at y=0,x=0 and

x=100m. Nodes at y=100m are subjected to varying vertical displacements and displacement

of the network in x-direction has been restricted. Network with long filaments and short

filaments has been modelled separately.

The resulting reaction forces were summed up corresponding to different displacements and

the elastic modulus of the network was calculated for both of the filaments. The results for

the modulus of elasticity of network with long filament and short filament have beencompared.

0.00E+00

5.00E+09

1.00E+10

1.50E+10

2.00E+10

2.50E+10

3.00E+10

3.50E+10

4.00E+10

4.50E+10

0 0.5 1 1.5

Reaction

Force(N)

Displacement(m)

Reaction Force Vs Displacement

Fig: Reaction Force Vs

Displacement of low

density Network


Displacement of High

density Network


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(A) (B)Figure 15: F-actin Network organized by long (A) and short (B) actin filaments at identical

concentration of filaments and intersections

(A) (B)Figure 16: Deformed shape of F-actin Network organized by long (A) and short (B) actin

filaments at identical concentration of filaments and intersections


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Figure 17: Plot of Reaction Force Vs Displacement for of F-actin Network organized by long

(A) and short (B) actin filaments


From the above analysis elasticity in the case of network with long filaments(A) comes out to be3.38E+10 N/m

2Which is higher than the network with short filaments(B) of elasticity 1.14E+10

N/m2.

The network with long filaments (A) are arranged regularly along filaments. In contrast, the network

with short filaments (B) forms incomplete loops with many loose ends and their arrangement is

random compared to the network in (A). The difference in structure would cause the network with

short filaments to be less stiff than the one long filaments.

The above model can be verified with the similar model given by Lee, Ferrer ,Nakamura and Lang in

which they concluded that the reducing the length of the individual filaments lead to more loose

ends in the network configuration, thereby altering the network connectivity. The resulting effect is

that network is less capable of withstanding streeses and therefore exhibits a smaller modulus.

0.00E+00

5.00E+09

1.00E+10

1.50E+10

2.00E+10

2.50E+10

3.00E+10

3.50E+10

4.00E+10

4.50E+10

5.00E+10

0 0.5 1 1.5

Reaction

force(N)

Displacement(m)

Load Vs Displacement


Displacement of Network

with short filaments(B)


Displacement of Network

with long filaments(A)


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Mechanics of Filament Networks | Conclusion and future scope of work 25

Conclusion and future scope of work

We observed the mechanical properties of Filament network. We showed that short filamentsinfluence the connectivity of the network structure resulting in a reduced elasticity. Similarly,

we observed the effect on elasticity due to density variation of the filament network.


26/30


Bibliography

Abed, G. (2010). COMPUTATIONAL MECHANICS TOWARDS IMPROVED UNDERSTANDING OF THEBIOMECHANICS OF MYOCARDIAL INFARCTION.

David A. Head, A. J. (2003). Deformation of Cross-Linked Semiflexible Polymer Networks. PHYSICAL

REVIEW LETTERS, 91 (10), 108102.

Frey, J. W. (2008). Elasticity of Stiff Polymer Networks.

R.Y. Kwon, A. L. (2008). A microstructurally informed model for the mechanical response of three-

dimensional actin networks. Computer Methods in Biomechanics and Biomedical Engineering, 11

(4), 407418.

X.-F. Wu, Y. A. (2005). Elasticity of planar fiber networks. JOURNAL OF APPLIED PHYSICS, 98, 093501.


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ABAQUS Input Code

*Heading

** Job name: filnet Model name: Model-1

** Generated by: Abaqus/CAE 6.10-1

*Preprint, echo=NO, model=NO, history=NO, contact=NO

** PARTS

*Part, name=Filament

*Node

1, -10.9459457, 1.48648643

2, -13.993289, 2.24832225

3, -15.1519718, 2.53799295

4, -16.4705887, 2.86764717

5, -20., 3.75

1957, -5.75764751, -27.6994648

1958, -5.49635839, -28.6496048

1959, -6.86375523, -26.7191525


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Mechanics of Filament Networks | ABAQUS Input Code 28

*Element, type=B23

1, 1, 395

2, 395, 396

3, 396, 2

4, 2, 3

45, 422, 423

2258, 1914, 237

*Nset, nset=_PickedSet2, internal, generate

1, 1959, 1

*Elset, elset=_PickedSet2, internal, generate

1, 2319, 1


1, 1959, 1


1, 2319, 1


1, 1959, 1


1, 2319, 1

** Section: fil Profile: Beam Section values

*Beam Section, elset=_PickedSet6, material="Material properties", poisson = 0.3,

temperature=GRADIENTS, section=CIRC

1.

0.,0.,-1.

*End Part

** ASSEMBLY

*Assembly, name=Assembly

*Instance, name=Filament-1, part=Filament

*End Instance


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Mechanics of Filament Networks | ABAQUS Input Code 29

*Nset, nset=_PickedSet10, internal, instance=Filament-1

205,

*Nset, nset=_PickedSet11, internal, instance=Filament-1, generate

206, 222, 1

*Nset, nset=_PickedSet14, internal, instance=Filament-1

37, 103, 160

*Nset, nset=Set-1, instance=Filament-1

37, 38, 43, 44, 49, 50, 99, 101, 102, 103, 104, 108, 109, 111, 112, 125

126, 150, 156, 157, 160

*End Assembly

** MATERIALS

*Material, name="Material properties"

*Elastic

2e+11, 0.2

** BOUNDARY CONDITIONS

** Name: Hinge Type: Displacement/Rotation

*Boundary

_PickedSet10, 1, 1

_PickedSet10, 2, 2

** Name: Horizontal Rollers Type: Displacement/Rotation

*Boundary

_PickedSet11, 2, 2

** STEP: Apply load

*Step, name="Apply load", perturbation

apply load

*Static

** BOUNDARY CONDITIONS

** Name: Vertical Rollers Type: Displacement/Rotation


30/30

*Boundary

_PickedSet14, 1, 1

_PickedSet14, 2, 2, 7.

** OUTPUT REQUESTS

** FIELD OUTPUT: F-Output-1

*Output, field

*Node Output

CF, RF, RM, RT, TF, U, UR, UT

VF

*Element Output, directions=YES

ALPHA, BF, CENTMAG, CENTRIFMAG, CORIOMAG, CS11, CTSHR, E, EE, ELEDEN, ELEN,

ENER, ER, ESF1, GRAV, HP

IE, LE, MISESMAX, NE, NFORC, NFORCSO, P, PE, PEEQ, PEEQMAX, PEEQT, PEMAG,

PEQC, PS, ROTAMAG, S

SALPHA, SE, SEE, SEP, SEPE, SF, SPE, SSAVG, THE, TRIAX, TRNOR, TRSHR, TSHR, VE,

VEEQ, VS

** HISTORY OUTPUT: H-Output-1

*Output, history

*Node Output, nset=Set-1

RF1, RF2, TF1, TF2, U1, U2, U3, UR1

UR2, UR3

*End Step

DDP 2 Report

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Transcript of DDP 2 Report