Polymer science and biology: structure and dynamics at multiple scales

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Original Entry by Holly McIlwee, AP225 Fall 09

Overview

Polymer science and biology: structure and dynamics at multiple scales (Opening Lecture) L. Mahadevan, Faraday Discussions, 139, 9, 2008.

Abstract

Studying biological systems at a molecular and cellular level can help probe two types of questions: What are the underlying processes affecting life on a larger scale?, and How can we translate what is going on at the cellular level in biological systems to polymer systems? In this review Mahadevan chose to focus on two elements of biological structures: filaments and membranes. Filamentous aggregates and their affinity to form ordered bundles, which move relative to one another passively and actively, or disordered aggregates, which display behavior not seen in individual filaments, are studied at a fundamental level in order to relate their function to life on a supramolecular scale. Simple cell dynamics are also examined to probe questions relating to the microstructure of the cytoplasm, cell attachment to a substrate or other species, and the cell's ability to spread disease and quickly and effectively communicate with the entire biological system. The intent is to learn more about a highly complex system by looking simply at it, and to start to think about conclusions that can be drawn and ultimately how this can relate to polymeric systems.

Soft Matter

Cells have heterogeneous structures made up of solid and liquid phases containing structural filaments, proteins, the cytoskeleton, and microtubules as well as water, ions, and soluble proteins. Cells are particularly interesting to study because they are autonomous; replicating, repairing, traveling, communicating with their environment, and evolving on their own. The cell is a complex example of a structured fluid.

The microstructure of the cytoplasm is of particular interest to the author. Because of the porous membrane the cell experiences dilatational movement when a load is applied to the membrane. When the load is applied the structural network dilates and experiences unequilibrium. As the fluid leaves the cell the load is bore by more of the solid and it can then relax and regain equilibrium. What is interesting to study is the rate at which equilibrium is reached. It is ultimately related to the poroelasticity of the system, the viscosity or the fluid crossing the membrane, and the size of its pores. Because of the size of the system and different membranes in the cytoplasm and the cell, unequilibrium may cause blebbing in the cell which has been typically brought on by a stimuli such as polarization. It is vital that there is a relationship between the membrane flow in the cytoplasm and the entire cell.

In order for tissues and organisms to form cell adhesion to other cells and to specific substrates must occur. This involves physical, chemical, and mechanical processes. Neglecting all of this and also the differences between different types of cells, the author chose to focus on how a cell responds to adhesion. It is determined that cell contact with a surface is related to the formation of interactions and bonds formed with the new surface. There exists a dynamic balance between the contact area, density of bonds, and the energy, and the strain on the shell or membrane. All in all though it has been found that regardless of cell type, the contact area grows linearly with time.

In conclusion it is seen that there are obviously parallels among polymer science and these studies of biology, particularly when studying the limit to life's evolution, packaging of macromolecular assemblies, and disease mechanisms and ultimately prevention strategies. As Mahadevan states in this field the "challenges are as great as the opportunities".





New entry by Emily Russell, AP225 Fall 2010


“... Polymer science naturally complements biology, since both ultimately involve the study of soft, warm, wet systems that are out of equilibrium.” This paper introduces just a few of the ways in which polymer science can be applied to biological systems, in particular, to cells, and gives some back-of-the-envelope scaling arguments describing properties and behaviors of cells and their contents. In particular, Mahadevan discusses the bending of ordered bundles of polymers (e.g. actin bundles in cells); the stress-strain behavior of ‘floppy’ networks (such as are found in the cytoskeleton or in blood clots); the poroelastic behavior of the cell and cell adhesion; and rheology and flow of a suspension of cells as it applies to sickle-cell anemia.

“A cell ... consists of a heterogenous, yet structured assembly of filamentous and membranous polymers bathed in water.” The first part of the paper therefore discusses the properties of polymers in ordered bundles; because ordered bundles are thicker than the polymer filaments composing them, they have a longer persistence length, and are less susceptible to thermal fluctuations, making them more useful as scaffolding in the cell. Mahedevan points out that a bundle of diameter d made of a material of bulk modulus E has a bending stiffness <math>B \approx Ed^4</math>. Also of interest in cells is the adhesion of the protein filaments to one another to create the bundles; with an adhesion energy per unit length of <math>\gamma</math>, the characteristic adhesion length of a bundle is <math>L_a \approx (B/\gamma)^{1/2}</math>, arrived at simply by scaling arguments. The combination of bending and adhesion introduces kinks into many bundles -- observed in actin, sickle-cell hemoglobin, and amyloid fibers -- when an extra monomer is inserted into the polymer filament on the outside of a bend. Mahadevan also briefly discusses the morphology of bundles of small numbers of filaments, and in particular the possibility of rearrangement of filaments within a bundle.

The second part of this section describes disordered, cross-linked networks. Mahadevan pays particular attention to “floppy” or under-constrained networks. When the number of internal degrees of freedom (given by <math>(Nd - N_c)</math> for a network of N nodes in d dimensions and with <math>N_c</math> cross-links or constraints) is above zero -- or, equivalently, the average coordination of each node is smaller than <math>z_c = 2d</math> -- the network can deform without supporting a stress up to some critical strain (see figure). Beyond this critical strain, however, links in the network begin to be stretched, and the network is strain-stiffening. Strain-stiffening is a commonly observed property of many biological networks.

Fig. 2 Mechanics of disordered networks. (a) A two-dimensional network of fibres, with the average coordination number (see text) <math>z < z_c</math>, so that the network is floppy, is subject to a strain at the boundaries. As seen, this leads to the shrinking of the network in one direction and an extension in the direction of the applied strain, coupled to an orientational ordering of the filaments. (b) A plot of the nominal stress (using arbitrary units) as a function of the nominal strain shows that the collective response of the network leads to no stress until the strain reaches a critical value g*. This is because of the presence of floppy modes that allow for the rotation of parts of the network to accommodate the boundary strains up to a critical threshold. Only beyond this critical threshold does the network deform with a finite resistance. The springs are all assumed to be harmonic, i.e. they are linear, with a spring constant inversely proportional to their rest length. The simulations were carried out using a damped molecular dynamics method.

In the next part of the paper, Mahadevan considers the cell as a poroelastic medium, that is, an elastic gel (the polymer cytoskeleton) with pores filled by a viscous fluid (here mostly water). In response to a stress, the gel deforms, while the fluid flows, so that the equations of state involve both gel strain and fluid pressure. He shows that displacements of the cytoskeleton then are solutions of the diffusion equation, with diffusion constant <math>D \approx K*k \approx K*l_p^2 / \eta</math>, with K the bulk compressibility of the gel neglecting the presence of the fluid; <math>l_p</math> the mesh size of the gel; and <math>\eta</math> the viscosity of the fluid. This model allows that different parts of the cell are not always in equilibrium, as the equilibration time <math>L^2/D</math> for a cell of size <math>L \approx 10 \mu m</math> can be quite long.

Mahadevan also considers a cell spreading out upon contact with a surface; by balancing the energy of adhesion with the energy dissipated by the viscous flow of the cell contents, he finds that the contact radius should scale as <math>R \approx (Jwt/\eta)^{1/2}</math>, with J the adhesion energy per unit area, and w the thickness of the cell. Indeed, experimental evidence supports that the contact area is proportional to time over a broad range of cells and substrates.

The final observations are on sickle-cell anemia, in which low oxygen concentrations cause polymerization of hemoglobin, stiffening the red blood cells such that they are unable to pass through thin capillaries. Soft matter science offers insight into both the polymerization process, and into the flow of the suspension of cells. The jamming of sickle-cells is very like the jamming of colloids or grains, or, Mahadevan points out, of automobile traffic.

This article demonstrates that the ideas of soft matter science -- rheology and viscoelasticity, adhesion energies, diffusion, polymer cross-linking -- are directly relevant in cell biology, and that even simple arguments can give qualitative predictions and explanations for the behavior of cells. It is a little weak on providing specific comparisons, although it offers numerous references which do address some of the questions discussed. It also, as Mahadevan indeed admits, does not address the problem of calculating some of the parameters such as adhesion energies or rheological properties, but merely derives time or length scales in terms of these properties, taking them as given. However, the point is well made that biology offers many interesting questions for soft matter scientists to consider, and that soft matter scientists can give great insight into biological processes.



New entry by Andrew Capulli, AP225 Fall 2011


Introduction

In this review, Mahadevan gives a brief overview of cell dynamics and the complex behavior what is the simplest unit of life. He begins by addressing the aggregation of cellular filaments and membranous structures into ordered bundles of limited size and further continues by describing potential models that incorporate the response of the entire cell in terms of the complex material properties of the cytoplasm (and extracellular matrix as discussed in the previous entry). Mahadevan states about cells, "These low dimensional objects have a large surface to volume ratio and thus serve as substrates for chemical reactions associated with the dynamical processes underlying life while having the ability to encode function in complex dynamic structures." His comments highlight the complex material behavior of the cell, the smallest known form of life; this review begins to peel away at the complexity into useful models for future study.

Filament Bundling

The body, and the cell for that matter, is full of polymer molecules either arranged in ordered, bundled networks such as actin bundles in muscle, microtubules within the cell, and even collagen/elastin in hair or 'disordered' networks such as the extracellular matrix of many organs/connective tissue or the cascade of thombin proteins in blood clotting for example. In particular Mahadevan describes the bundling of filaments in terms of know polymer physics. Because of the natural curvature of some polymers in the body, bundling is limited in size and geometry. Curvature presents an interesting problem for the bundling of polymers; as the bundle grows in radius, the polymer filaments on the outside are stretched or compressed due to the curvature. This is much like a simple stress analysis problem of a fixed beam being bent: the face of the beam that force is applied to compresses while the opposite face stretches to compensate. As Emily gives in the previous entry, Mahadevan discusses that kinks in the formation of large bundles are necessary to allow for the described curvature limiting bundle girth of a particular filament complex but allowing for bundles of varied shape and length. Filament curvature may explain why certain filament bundles such as actin and sickle cell hemoglobin for kinks in their structure. Interestingly, Mahadevan discusses the consequence of sickle cell hemoglobin formation: essentially the unnatural kinks in the ill-formed hemoglobin cause red blood cells to lose their natural doughnut shape which allows for flow in the capillaries, the misshapen diseased cells cannot readily flow in this small vasculature which can result in clotting-embolism-and eventually stroke or heart attack. At the root of this disease is polymer science of soft matter; the physical shape of the aggregated filaments (in this case the folding of the hemoglobin protein) reduces flow in small capillary beds, in particular in the lungs, which not only inhibits flow causing potential embolism but also reducing gas exchange slowing respiration.

Model for Cytoplasm - Soft Matter

Perhaps where Mahadevan really delves into the complex nature of soft matter is when he discusses 'A micro-structural model for the cytoplasm' in section 3.1 of this review. Traditionally, models of the cytoplasm simplified the material as simple visco-elastic or visco-plastic. More appropriately, the cell can be modeled as poroelastic which allows for a model that includes the many components of the cytoplasm (water, ions, soluble proteins,filamentous structures, membranous structures, organelles etc). While the mathematics is summarized above in the previous entry, in terms of soft matter, the physical consequences are very interesting. Cell size, membranous pore size, and the viscous nature of the fluid (cytoplasm) in the cell are what control the mechanical response of the cell to an applied stress or strain. In a simplified explanation, as the cytoplasm is stressed as some point on the cell, water flows out of the pores in the membrane and the remaining cell contents are stressed. This propagates throughout the cell until all free water has evacuated and finally the remaining cytoplasm can equilibrate. While we know cells are not always chemically at equilibrium, as Mahadevan points out, they may not be mechanically at equilibrium either. This is why, for example, endothelial cells that line the inside of blood vessels align in the direction of flow; this alignment along the axis of shear stress they experience may keep the cell in a more mechanical equilibrium state. Figure 3 from the review is an example of a cell cultured without the presence of external forces. It is interesting to note the seemingly random microtubule and actin cytoskeleton not in any particular alignment (the cell is seeded in a circular-spread fashion).

MahadevanFIG3.jpg

Studies have shown the cells align along the axis in which they are being stretched. Below is a picture from a study I contributed to of valvular interstitial cells stretched uni-axially (in this case left to right); click to enlarge. As you can see these cells align along the axis of strain, perhaps as Mahadevan suggests, to become more mechanically equilibrated. These valvular interstitial cells (heart valve cells) are subjected to enormous shear stresses as blood flows past and very large pressures during diastole in the heart right before contraction. Like these valvular cells, other cardiac cells such as the myocytes responsible for contraction also align along the axis of strain (direction of contraction). While it is intuitive that cells align for maximum contraction along the axis, it may be the case the this alignment also results in a more uniform mechanical environment within the cell that as we know, has complex composition of soft matter make up.

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