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# Journal Club Theme of January 2007: Biomechanics and Non-Affine Kinematics

Biological materials are frequently constructed of hydrated biopolymer networks. Examples include fibrous collagen in the extracellular matrix and actin within the cell's cytoskeleton. There are differences in the molecular composition of the biopolymer subunits as well as differences in the network density and organization. Images can be seen here and here for dense collagen networks and for portions of actin networks look at images here and here.

The mechanical response of these biopolymer network-based materials is nonlinear and concave-up (exhibiting strain-stiffening). The fundamental mechanics of this response has been widely studied and examined since the 1960s, when Viidik and others started characterizing responses of collagenous tissues to uniaxial loading as analogous to the sequential recruitment of linear spring elements.

More recently, attention has shifted to the nonlinear response of the molecules themselves. The individual biopolymer molecules are typically characterized using a worm-like chain (WLC) model or a more complicated model based on the WLC (such as the Marko-Siggia model or bead-spring chains). Armed with information on this response, a network can be conceivably "built up" from individual WLC elements to represent networks with different densities and different molecular orientation characteristics.

Thus there are two primary possibilities for mechanical stiffening in these networks: stiffening due to a structural effect in the network or stiffening due to a stiffening effect in the individual molecules themselves. Obviously both effects could be at play in real non-idealized networks.

A key question in the construction of "bottom-up" models of biopolymer network mechanics regards the reorganization of the networks under applied mechanical loading. In many cases, the deformation has been modeled as affine, or preserving parallelism (see, for example, the work of Storm et al. mentioned previously on iMechanica). However, recently this assumption has been questioned. There are also interesting, and frequently ignored, possibilities for viscous drag on the networks since the reorganization occurs in a fluid environment. Finally, there are likely length-scale effects in terms of the observation length-scale compared with the material length-scale, especially when the hierarchical structure of biological materials is considered.

In this month's inaugural iMechanica journal club, we examine mechanical deformation in biopolymer networks by first considering three papers that all argue for non-affine network behavior based on experimental and modeling results on collagen-type extracellular networks and actin-type cellular networks.

The three papers are included here for discussion are:

Initial points to consider in discussing these papers are the fidelity of the experimental and modeling approaches, the different assumptions made, and the strength of the conclusions of non-affine behavior based on the results presented by the authors. However, please feel free to initiate discussions on any aspect of the papers including forward-thinking ideas about the future of biopolymer network modeling.

## How does the Journal Club work?

If you are wondering how the Journal Club works, you might want to take a look at previous discussions and tentative operating notes:

## great first post for jClub!

Michelle,

You're definitely setting the bar high for jClub! Nice job!

-John

## It is a pleasure to see a wonderful post for J-Club

It is a pleasure to see a wonderful posting for J-Club. Especially, I think that it would be very good to consider polymer/biopolymer networks and their mechanical response in biomechanics issue in J-Club. The three papers you suggested for J-Club would guide many people (including me) in further interests in biopolymers and protein networks. Furthermore, I have known that van der Giessen, who wrote the paper you provided in this issue, is a member of iMechanica. So I hope that we may have a chance in communication and/or discussion on this topic with one of the authors who wrote the paper. Anyway, I am looking forward to seeing the fruitful discussions here for learning and broadening the biomechanics area. Dr. Oyen, thank you for your efforts in J-club for guiding the biomechanica area that I am also interested in.

Kilho

## Discussions across disciplines

First of all, I want to complement Michelle with having taken the lead on this endeavour and to congratulate her with a job well done.

Several people involved in initializing the Journal Club have emphasized the importance of allowing for discussion across disciplines. As a simple example of this, let me share with you a bit if my own "head-scratching" when I first looked into the behaviour of semi-flexible polymers. In the WLC model, axial deformation is enabled by pulling out undulations in the filament, i.e. by internal bending (much like post-buckling compression, but in the opposite direction). To my initial surprise, however, the WLC model says that the axial stiffness scales with the bending stiffness

squared. Why was I surprised? Because I was not familiar enough with statistical mechancis to know (or immediately see) that axial stiffness is also inversely proportional to the amplitude of the undulations. And, for a given temperature, the amplitude is inversely proportional with the bending stiffness. Hence, one factor of bending stiffness arises from the stiffness of the filament itself, the other one from the magnitude of the undulations. A definite "eureka" for me when I saw this. Many have followed once I got into the subject that is referenced here as Ref. (2).More to the point of affinity (or not), I cannot agree more with Michelle in that this is a major issue in network modeling. One problem I see with it, in general, is that it depends on many factors, in particular the ratio between bending (and torsional) stiffness and axial stiffness. This ratio varies quite a bit in biological systems; in the actin networks that Patrick Onck and I have focused on so far, this ratio is extremely small (one can get a feeling for the behavior of an actin filament by noting that it has an aspect ratio similar to that of a human hair of 1 meter long). MacKintosh and co-workers have developed a map between affinity and stiffness properties in two-deimensional networks -- in three dimensions it is still an open issue.

## Michelle, This is a great

Michelle,

This is a great debut! I enjoyed reading the works you selected. It seems, also from Erik's post, that rigorously developed non-affine field theory of elasticity would be quite fitting for such systems (as opposed to discrete network type models). Does one exist? In a different context (amorphous metals), I came across one paper by DiDonna and Lubensky who made a nice attempt. I am wondering perhaps something else also exists within the mechanics community?

## What do you mean by a non-affine field theory of elasticity?

Pradeep: Your comment intrigues me. The other day I had a lunch with Joost and we had a conversation about the work of Patrick and Erik, and talked about adding extra variables to a field theory. For example, one readily adds strain gradient and electric polarization. By constructing a non-affine field theory, do you mean adding other variables to describe the state of the matter?

## In fact, this is exactly

In fact, this is exactly what I meant. If you look at the paper by DiDonna and Lubensky; the final coarse-grained equations have a distinct nonlocal character reminscent of the strain gradient theories. However to retain the departure from affinity suggests also adding internal variables. In short, the non-affine elasticity theory I am envisioning will be nonlocal in character with degrees of freedom beyond just the conventional displacement field.

## Towards a non-affine generalized continuum theory

The non-affine behaviour in cross-linked networks of initially straight filaments enters from the competition between filament bending and stretching. At low densities the connectivity of the network is so low that additional (non-affine) deformation modes are active that are dominated by bending. As the density increases the additonal degrees of freedom are constrained, leading to behaviour that is governed by filament stretching. A network that is governed by filament stretching only, is affine in nature (think of a fully triangulated lattice; equilibrium dictates that all bending moments are zero). The microstructural parameter that governs this is the node connectivity (the number of filaments that connect in each cross-link). High node connectivity favours stretching leading to a stiffer response. For a nice discussion on this topic see the paper of Deshpande, Ashby and Fleck, Acta Materialia, 2001, Pages 1035-1040.

It is interesting to note that the additional non-affine deformation modes have a "vortex- like" behaviour. We found this in cross-linked networks by analyzing the displacements of the cross-links, which showed rotational, vortex-like behaviour similar to the fields that Didonna and Lubensky found for their amorphous metal-like model system. I envisage that a succesful generalized continuum theory should take these additional rotational modes into account (such as done in Cosserat or Couple Stress type continuum theories in which rotations are added as additonal degrees of freedom).

## Continuum theory of linear response

Thanks to the organizers for a very nice discussion of the affine / non-affine question. I think Patrick summarized well the current understanding of the affine to non-affine transition as a function of network connectivity. This is to say that for more highly connected networks, the small strain, linear elasticity is dominated by the bending of filaments for sparse networks and the stretching of filaments for dense networks. This is a distinct effect from strain stiffening at large extension, when all the bending undulations have been pulled out of a stiff polymer network. I also think that Onck and Van der Giessen's work in this area is very nice - where they show that the network is more affine or stretch dominated after bending degrees of freedom have been pulled out, even for sparsely connected networks.

In our paper Tom Lubensky and I exclusively consider linear response, and we start with the assumption that our material has a well defined coarse grained linear elastic constant at each point. We then connect spatial fluctuations in the elastic constants to the non-affine component of the elastic response. We made several simplifying assumptions and did not fully consider the tensorial nature of the general elastic constants. Still, we worked out several cases of interest, including materials with spatial correlations of given length scales and/or internal stresses. Our goal was to provide a framework for describing non-affinity and connecting measured quantities to internal or microscopic degrees of freedom. Incidentally, we did generically witness vorticity in the particle displacement fields and related it to our internal parameters. I don't think of our work as exclusively describing metal-like systems, although the simple simulations we did to validate our theory was done on highly connected central force lattices. In unpublished work, I have made the same sorts of measurements on "mikodo" lattices build from rods with finite bending rigidity. I think one of our successes was to demonstrate what quantities are good to measure and what types of measurements give equivalent information.

## What makes biopolymer networks unique?

A question that often arises in this context is, to what degree can we apply mechanics principles developed in the context of traditional materials to biological materials? Or put another way, what makes mechanics in the context of biology fundamentally different from, say, metallurgy, or is there no difference at all?

The thing that I always come back to is the heterogeneity that is intrinsic in biological materials. At a very local scale one might be able to use a continuum theory to describe a biopolymer network, but it may be that in the structure itself (say a cell or a tissue) the differences observed from point to point are extremely large. For this reason I think it is very difficult to ever consider these materials as continuua, or to use mechanics principles derived in the context of say an ordered lattice structure as analogous to the description of behavior found in biological networks.

I am struck by a similar discussion in reasonably recent paper concerning indentation of inhomogeneous composites, in which it was discussed that perhaps no homogenization scheme can ever capture the behavior of an inhomogeneous material system, even when loading is taking place at length scales substantially greater than those of the material microstructure.

All together, this presents some interesting challenges in modeling for biological materials, in that perhaps we need to start "fresh" outside of a continuum framework to really capture the physics of what is taking place during mechanical loading and how the non-continuum aspects are perhaps of the utmost importance in the overall system behavior.

## The mechanics of biological and engineering materials

Michelle, that's a key question that you are posing. The point that you are raising above with respect to heterogenous materials addresses the validity of a continuum theory. Using a continuum theory is valid when each material point of the macroscopic structure represents a volume of material that has a size

Dthat is (i) much smaller than the specimen sizeL, and (ii) much larger than the characteristic material length scaled(L>>D>>d) When one of these conditions is violated, continuum theory should not be used and one should resort to methods that take the discrete nature of the materials' microstructure into account.In addition there is the issue on whether we can use a classical (local) continuum theory or whether higher-order (non-local) theories should be used. Classical (i.e. local) continuum theory is well-suited for situations where the variations in stresses and strains (with wavelength

λ) are smooth enough so that they can be approximated as being uniform on the scale of the material points (L>λ>>D>>d). However, in many situations this is not necessarily the case, e.g., near notch, crack and indenter tips, so that often higher-order theories are used.Clearly, whether the theories that we are used to work with can be used for biological materials depends on whether we can separate the scales or not (i.e. whether the above conditions are violated). Biological materials often have an exotic microstructural hierarchy that sets them apart from the materials that we know. This can lead to superior properties, see e.g. recent work of Huajian Gao: H. Gao, X. Wang, H. Yao, S. Gorb and E. Arzt, "Mechanics of hierarchical adhesion structure of gecko," 2005, Mechanics of Materials, Vol. 37(2-3), pp. 275-285 and H. Gao, B. Ji, I. L. Jäger, E. Arzt, and P. Fratzl, "Materials become insensitive to flaws at nanoscale: Lessons from nature," Proceedings of the National Academy of Sciences of USA, 2003, Vol.100(10), pp. 5597-5600.

A second important difference is the dynamics of biological materials. They have a tendency to continuously adapt to environmental changes (including stress redistributions). As a result, the interaction of mechanics and biochemistry becomes an essential ingredient of materials' modelling. A challenging example is mechanotransduction (the two-way interaction between genetic processes and the sensing and generation of mechanical forces). Understanding the underlying mechanisms can have important biological implications.

A third difference is the fact that biological materials at the relevant small scales operate in a fluid environment, making them susceptible to thermal fluctuations (energy kT). The interaction of thermal and mechanical behaviour can be considerable for relevant physiological processes.

## Biological materials and H-bonds

Great points. I think further to your key point there (especially the third of your list of differences) is that the fundamentals of physical bonding are different. Proteins are stabilized by hydrogen bonds which do give rise to the higher order structures such as alpha-helices (collagen) and beta-sheets (spider silks) and the large numbers of H-bonds make them important in the mechanical performance. They are also clearly associated with water due to all the H-bonding and have mechanical properties that are extremely dependent on hydration state. (This can be manipulated for various hydration states by polar solvents, as we have done in bone and has been done in spider silks.) Thus although the possibility of a length-scale effect (a la the Gao et al. PNAS paper) is interesting, it's not clear to me that size effects are as important as other effects, such as hydrogen bonding, to describe the key differences of what makes biological materials different!

## Mechanical response of biopolymers (spider silk protein)

It is quite interesting to me (maybe also other iMechanician) that affine (or sometimes non-affine) network model works for mechanical response of biopolymers. As far as I know, the biopolymers are really attractive since they exhibit the superior mechanical properties to other composites. For instance, the spider silk protein possesses the yield strength comparable to that of high-tensile steel.

For understanding mechanical properties of spider silk protein, it was generally conjectured that beta-sheet-crystals are embedded in an amorphous matrix (alpha helices). Based on this conjecture, Termonia (Macromolecules, 27, p7378, 1994) provided the micromechanical model such that particles (beta-sheet-crystals) are embedded in an amorphous matrix. That is, the network was constructed such that crystals have a relation of s = Ee (s = stress, E = Young's modulus, e = strain), and that the crystals were connected by a polymer chain which possesses the nonlinear elastic behavior. This network model by Termonia seems to work.

However, in recent study by van Beek, et al. (PNAS, 99, p10266, 2002), it was experimetally found that the molecular structure of spider silk protein is similar, in part, to that of Termonia's model, but it was found that matrix (consisting of alpha helices) is not amorphous at all! In van Beek's work, it was shown that the matrix is ordered structure such that the direction of alpha helices is close to the fiber direction. This may suggest that network model based on amorphous matrix may not be sufficient to represent the real structure of spider silk protein.

What I am thinking for spider silk protein is that we may have to take into account the molecular structure rather than network model (based on amorphous matrix) for deeply understanding of remarkable mechanical properties of spider silk proteins. This means that there are many rooms to everyone for gaining insight into how molecular structure determines the mechanical response, especially for biopolymers such as proteins.

## stress tensor

The concept of stress tensor and stress balance equations in the traditional continuum mechanics no long exist in biopolymer networks at the microscopic scales.

## Continuum mechanics and molecules

I could not agree more. This is consistent with the fact, as discussed by Patrick and me above, that we are probably not at all in the realm of continuum mechanics here.

This is probably why recent work like Markus Buehler has been doing on the molecular simulation of biomolecules has received such attention, although I think there is still a long way to go before we understand how to link the response of an isolated molecule to behavior of a biopolymer network.

## carbon nanotube reinforced composite materials

Similar network structures exist in carbon nanotube reinforced composite materials, where nanotubes are normally about 1000 nanometers long, and around 1nanometer in the tube radius. The nanotubes are usually embedded in a polyethylene matrix.

The nanotube networks looks very similar as actin networks in the links Michelle provided at the very beginning of this forum.

## About collagen

About collagen here too writes http://collageena.com/why_unique.html