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Journal Club Theme of March 2012: Brownian motion and entropic elasticity of fluctuating filaments and networks
The advance of mechanics towards the frontiers of biology and soft matter necessitates an understanding of entropic forces which manifest themselves as thermal fluctuations at length scales of a micron and below. The effects of these fluctuations can be easily seen in biofilaments at thermodynamic equilibrium under the action of forces and moments as they fluctuate around their minimum energy configuration due to Brownian motion. This remains true of filaments in networks and gels such as those of actin, spectrin, fibrin or other biopolymers. The thermal motion of these filaments at the microscopic scales manifests itself as entropic elasticity at the macroscopic scales. In this post we present a theory to efficiently calculate the thermo-mechanical properties of fluctuating heterogeneous filaments and networks. The central problem is to evaluate the partition function and free energy of heterogeneous filaments and networks under the assumption that their energy can be expressed as a quadratic function in the kinematic variables. We analyze the effects of various types of boundary conditions on the fluctuations of filaments and show that our results are in agreement with recent work on homogeneous rods as well as experiments and simulations. We apply similar ideas to filament networks and calculate the area expansion modulus and shear modulus for triangular and square networks.This post illustrates how we combine the finite element method with path integral techniques from statistical mechanics to solve problems in which configurational entropy due to thermal fluctuations plays a significant role.
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entropic elasticity of polymer and networks
Dear Prof. Purohit,
Thanks for posting the journal club theme for this month.
I think it is pretty amazing that the highly fluctuating structures inside the cells can carry out well-organized work. One can see how much an actin filament fluctuates even in electrial fields in the supplement vedios in the following Biophys J paper (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1989696/bin/biophysj_107.114538_index.html). Yet it is these fluctuating filaments that provide the mechanical support to the our cells. Can anyone imagine the buildings we are living in be supported by some vibrating pillars, or a train running on a constantly vibrating track?
Thermal fluctuation has been used, or measured, to detect small changes in the mechanical properties of these soft filaments for quite some time (http://www.jbc.org/content/270/19/11437.full.pdf). It is also well known that thermal fluctuations can significantly change the mechanical properites of the filaments. If I understand correctly, the nonlinearity of a worlike chain (marko siggia 1995 macromolecules) arises only because of thermal fluctuations. In other words, if we shut down the temperature, a wormlike chain is nothing more than a linear elastic spring.
As far as I know, it is still not very clear how much thermal fluctuation contributes to the initial strain stiffening of a polymer network. Some people argue that the structure and geometry of the network can also contribute to the stiffening. It is also debating when thermal fluctuating is a significant effect in filament networks. Some belive that if the distance between cross-links is comparable to the persistence length of the polymer, fluctuating is not significant while other researchers found that at least for some networks, entropic effect due to thermal undulation is still responsible for the elasticity of the networks even when cross-links distance is smaller than the persistence length. So I think it is important that we have an efficient computational tool to understand these issues.
The 2012 paper Prof. Purohit attached is a framework trying to combine finite element method and statistical mechanics to understand the entropic effects. If some people doing finite element simulations on polymer networks are reading the posts here, I would like to ask them the typical size of networks they are simulating using the finite element packages. I know some typical degrees of freedom is like ~10000 but I am not sure. I myself am not familiar with using finite element packages, I am also wondering what happens in finite element simulations if the stiffness matrix is singular? Like if some filaments in the networks are not well connected to others, or the density of the network is below the percolation threshold, I would imagine the stiffness matrix be singular. How do we inverse the stiffness matrix there to solve the problem?
Thanks for reading.
Sincerely,
Tianxiang Su
Answers to Tianxiang's questions
Thanks for your commentary on my post and the papers.
Indeed our point of view when we were performing this research was that fluctuating filaments in a cell support it mechanically in a way that is a bit different from how a beam frame supports a building. The issue is not completely settled. But one point of view is that the tension generated in actin filaments (and perhaps also intermediate filaments in cells) due to thermal fluctuations is balanced by compression in microtubules. This is the essence of the tensegrity model for cells. As for the analogy of trains moving on vibrating tracks, molecular motors at least partly use thermal fluctuations to their advantage (in the diffusive search step of their cycles) to move along filaments. Similar to real trains moving on tracks microscopic motors convert chemical energy to mechanical work to get directed motion.
The second issue you brought up concerns non-linearity in the force-extension relation of filaments. Thermal fluctuation is one obvious source for this as in DNA, actin, and most polymers but there could be many other effects. One such effect is stress induced structural change. This is seen in many filamentous molecules including DNA, fibrin fibers and also in some intermediate filaments.
Stiffening in the stress-strain response of networks can also arise due to many reasons. Stretching out of thermal fluctuations is one of them. But reorientation of filaments along the principal direction corresponding to tension is another reason. Of course, cross-links are in the picture as well and the constraints they put on the filaments they connect will have an impact on the overall stiffness.
Spring network
Dear Prof. Purohit and Tianxiang,
Thank you for bringing such an interesting topic. I will read more carefully about your papers. I have noted that spring network model has been often used by many people to study biopolymer networks and gels. Do you have any comment on that?
Answer to Cai Shengqiang's comment
You are absolutely right in saying that people have modeled polymers as network of springs in the past. This type of models began to appear in the 1940s and 1950s. The energy of a network of springs can be written as a quadratic form and the partition function and free energy can be easily computed using Gaussian integrals. But, a rods are different from springs. Linear springs behave the same way in tension or compression, whereas rods buckle under compression. Our method is able to capture buckling of rods. Buckling in fiber networks leads to interesting effects such as reversible softening and non-affine deformation which a spring network model cannot pick up. Our method can capture these effects. The price we pay is that our method is semi-analytical whereas spring network models can be treated analytically.
salam
Thermal fluctuation useful in mechanical characterization
Dear. Purohit: Thanks for posting the journal club at iMechanica. Your posting on thermal fluctuation behavior of biological fibers is interesting and useful for understanding the behavior of such biological fibers.
By the way, I would like to point out that measuring the thermal fluctuation of biological fiber can be a useful tool in extracting the mechanical properties of such fiber. Specifically, the relationship between thermal fluctuation and mechanical properties is founded on statistical mechanics theory, which demonstrates that measurement of ensemble average of quantities relevant to the deformation of biological fiber is directly related to the mechanical properties of such fiber. For example, if the thermal fluctuation behavior of a flexible biological fiber (e.g. DNA, RNA, microtubule, etc.) obeys the bending motion, then one can find the bending rigidity (or persistent length) of such fiber by measuring the ensemble average of end-to-end distance or bond angle in order to estimate the bending rigidity of such fiber. The relevant references are listed as below:
[1] Pampaloni, et. al., "Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistent length", PNAS 103, 10248 (2006)
[2] Rivetti, et. al., "Scanning force microscopy of DNA deposited onto mica: Equilibration versus kinetic trapping studied by statistical polymer chain theory", J. Mol. Biol. 264, 919 (1996)
[3] Wiggins, et. al., "High flexibility of DNA on short length scales probed by atomic force microscopy", Nat. Nanotechnol. 1, 137 (2006)
[4] Valdman, et. al., "Spectral analysis methods for the robust measurement of the flexural rigidity of biopolymers", Biophys. J. 102, 1144 (2012)
Reply to Kilho Eom's comment
Dear Kilho,
I completely agree with your remark that thermal motion can be used to extract mechanical properties of long slender molecules. Jonathon Howard is a pioneer in this area. In fact this technique is used heavily in research on molecular motors to determine the stiffness of various states. The references you point to are all very interesting and I have been closely involved with researchers in the Wiggins et al. paper.