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Mechanics of FilamentNetwork
Tarun Meena(08D10048)
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Introduction Deformation of the cells are dictated by their
mechanical behavior under loading
Critical component which is governing the mechanicalbehavior is actin cytoskeleton
In this thesis filament network has been idealized as2-D network in Abaqus
Behavior of the Network is analyzed due to finitedisplacement in vertical direction
Effect of the variation in basic parameters like densityand length of the fiber has been discussed
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The F-Actin Network The cytoskeleton is a cellular scaffolding or skeleton for
cells.
The cytoskeleton is an organized network of three primary
protein filaments Microtube
Actin Filaments
Intermediate fibers
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Continued F-actin filament plays an important role in the
mechanical behavior of the network
Some Functions of the F-actin are as follows Provides mechanical strength to the cell Generate locomotion in cells
Links transmemberane proteins to cytoplasmic
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Cross linked F-actin Network In the cytoskeleton, the local microstructure and
connectivity of F-actin is controlled by actin bindingproteins.
The binding proteins control the organization of F-actin into mesh like gels, branched network etc.
Cross-links have a compliance that depends on their
detailed molecular structure and determines networkmechanical response.
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Existing ModelsDeformation of Cross-Linked Semiflexible Networks(F.C.Mackintosh,2003)
It is a simple model of cross-linked rods
It is assumed that the deformation field is affine(Strain isuniform)
Through the analysis it is found that network becomeincreasingly affine down to the smallest scale of thenetwork e.g. mesh size, cross link density etc.
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Continued Model was studied for dependence of the bulk shear
modulus on the Cross link density
Bending and Extension of the individual filament
Elasticity of Stiff polymer network
Model is about the elasticity of two-dimensional networkof rigid rods
Essential features of this model are Anisotropic elasticity of the rods
Random geometry of the network
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Continued Typical networks at high density and low density
The random network was generated by placing N line
like objects of equal length l and area L2
It was observed that shear modulus scales vary linearly
with filament compressional modulus and no. offilaments per unit area.
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ContinuedElasticity of Planar Network
In this model wool assembly was treated as a layered
system with fiber bending deflection In the model, only fiber axial deformation was
considered
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Continued It was observed in the model that effective stiffness of
planar random Filament Network increase with theincrease of arial density, which can be measured by thenumber of fibers per unit area
The model account for microscopic deformation offiber segments of all possible lengths and orientations
The constitutive relation is derived is as follows
Where,
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Modeling and analysisFinite Element Analysis
Powerful computational technique for approximate
solution to a variety of engineering problems This method is used to solve a modelling problem by
dividing the solution domain into discrete regions,these are the finite elements
Finite element modelling provides a simply costeffective way of monitoring and predicting situationsthat occurs in any sphere of engineering, medicine,aeronautics etc.
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ABAQUS Finite Modelling and analysis FEM packages such as ABAQUS is a very useful
program as it allows for the material, geometric, andboundary conditions to be set and adjusted asinformation becomes apparent
The result of model such as stress vs strain curves,
pressure displacement diagram has a short run overtime in ABAQUS
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Assumptions made in model In order to simplify and practically model the system
some assumption are taken, are as follows The fibers are straight and oriented in the same plane
The fibers are randomly distributed and of same lengths
Total Elasticity of the network is dependent on the elasticityof filaments
Effect of medium, in which fiber networks, is being assumed
negligible The fiber elasticity used is 2x10^11 N/mm^2 and possions ratio
to be 0.3 ,which are arbitrary properties of the fiber
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Random Network FormulationA 2-D Fiber network defined as a set of independently
short line segments are placed.
Fibers of same length and orientation has been place Network has been idealised as mixture of 2-D beam
element of diameter 1mm.
Any two intersection beams are joined at their
intersections. In the Network at y=0 hinge and roller boundary
conditions have been applied and at y=20mmdisplacement has been applied.
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Continued For different value of displacements the corresponding
reaction force has been calculated
Model of the Network with vertical displacement andsupport condition
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Continued The model is analyzed using static linear analysis
procedure
The reaction is obtained on the upper surface of thenetwork are summed up to the total reaction force, RF2in y-direction
Elastic modulus of elasticity can be calculated as
followsEc = (RF2/A)/(disp/L)
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Results and DiscussionComparison of Beam element B21 and B23 B23 is an Euler-Bernoulli beam, a 2 node cubic beam in
plane
It does not allow transverse shear deformation i.e. planesections initially normal remain plane While B21 beam is a Timoshenko beam and they allow
transverse shear deformation, a 2 node linear beam inplane
The model is analyzed for both the beam elements forcomparison purpose The comparative plot of Force-Displacement for both of
the beam elements is as follows
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Continued
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Continued From the plot it can be seen that results are almost
same i.e.
But as in the model we are considering fiber which isslender, i.e. the beams cross-sectional dimensionsshould be small compared to typical distances alongits axis, that comply with the B23 beam element
Therefore, it would be more suitable to use B23 beamelement than B21 in modelling
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Effect of variation in density on the
elasticity of the network To study the effect of variation in density, we created
two models of different density of fibers under sameloading condition
We keep length of the fiber and orientation same
Low density Network and its deformed shape
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Continued Relatively High Density network and its deformed
shape
The Stress-Strain curve for both of the network is asfollows
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Continued
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Continued Elasticity for Low Density network=6.81E+10 N/mm2
Elasticity for High Density Network=1.20E+11 N/mm2
It can be clearly seen that the Elasticity of the networkincreases with increase in the density of the Network
which comply the power law E= C(-ref) (R.Y. Kwon, 2008)where is the volume(density), validating the aboveresult.
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Effect of variation in Fiber length
on Elasticity of the network To study the effect of variation in length two different
models of different fiber length of 10mm and15mm,keeping the orientation same has been created
In the model orientation of fibers and loadingcondition has been kept same
Network and its deformed shape for fiber length of10mm
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ContinuedNetwork and its deformed shape for fiber length of15mm
The plot for displacement Vs Force is as follow for both ofthe Model
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Continued
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Continue Elasticity in the case of fiber of 10mm length is found
to be 6.81E+10 N/mm2
and in the case of network of fibers of 15 mm length isfound to be 1E+11 N/mm2
The result suggests that Length of the fiber plays asignificant role in the morphology of the network
structure or the linear elasticity.
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References (n.d.). Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Cytoskeleton
ABAQUS CAE documentation. (n.d.). Retrieved fromhttp://abaqus.civil.uwa.edu.au:2080/v6.9/
Abed, G. (2010). COMPUTATIONAL MECHANICS TOWARDS IMPROVED
UNDERSTANDING OF THE BIOMECHANICS OF MYOCARDIALINFARCTION.
David A. Head, A. J. (2003). Deformation of Cross-Linked SemiflexiblePolymer Networks. PHYSICAL REVIEW LETTERS, 91 (10), 108102.
F. C. MacKintosh, J. K. (1995). Elasticity of Semiflexible Biopolymer
Networks. P. A. Janmey, 75 (24). Frey, J. W. (2008). Elasticity of Stiff Polymer Networks.
G. A. Buxton, N. C. (2009). Actin dynamics and the elasticity of cytoskeletalnetworks. eXPRESS Polymer Letters Vol.3, No.9 , 579-587.
Gardel, M. L. (2004). Elasticity of F-actin Networks.
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Jeffrey S. Palmer, M. C. (2008). Constitutive modeling of the stressstrainbehavior of F-actin filament networks.Actabiomaterialia, 4, 597612.
Jonathan Stricker, T. F. (2010). Mechanics of the F-actin cytoskeleton.Journal of Biomechanics, 43, 914.
Madenci, E. (2005). The Finite Element Method and Applications in EngineeringUsing ANSYS(1st ed.). Springer.
Margaret L. Gardel, *. K. (2008). Mechanical Response of Cytoskeletal Networks.In M. L. Gardel,METHODS IN CELL BIOLOGY(Vol. 89, pp. 487-519).
R.Y. Kwon, A. L. (2008). A microstructurally informed model for the mechanicalresponse of three-dimensional actin networks. Computer Methods inBiomechanics and Biomedical Engineering, 11 (4), 407418.
Risler, T. (2011). Cytoskeleton and Cell Motility.
X.-F. Wu, Y. A. (2005). Elasticity of planar fiber networks.JOURNAL OF APPLIEDPHYSICS, 98, 093501.
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Thank You