Theoretical Background 

The
basis for cell mechanosensation is a coupling of
mechanical influences to the chemical reactions that occur within the
cell. For example, in order for a
stem cell to differentiate into a bone cell on a stiff surface, that
stiffness must affect the internal chemical reactions that lead to gene
transcription and translation.
Thus, to understand mechanosensation, we are
interested in how mechanics and chemistry are coupled  mechanochemistry. Perhaps
the fundamental basis of mechanochemistry is that
the speed of a chemical reaction changes if the reacting molecules experience
forces. Equations that relate
reaction speed to force are therefore the basis of mechanosensation. One such equation, proposed nearly 50
years ago for metal fracture [1], was shown to apply to cell adhesion in the
late 1970s [2]. This equation is
called Bell's approximation. When a
cell interacts with a surface, it applies forces to aggregates of
molecules. These clusters of
molecules, under force, slide across the surface. To understand such a system, we must
consider both its geometry and its multibody nature. Specifically, we want to derive an
equation for the relative sliding speed of two surfaces at a given force. The
work that serves as the basis for our models of mechanosensation
is 1) a theory of chemical reactions under force that extends Bell's
approximation; and 2) a theory that describes the behavior of two filaments
or sheets that approach each other in the presence of proteins that
transiently bind one to the other. 

Publication 1 

Reference: Walcott, S., The load dependence of rate constants. Journal of Chemical Physics. Volume 128, pages 215101, 2008. PDF Reacting molecules are, fundamentally, mechanical systems. When molecules are large, at approximately room temperature and in water, then their dynamics are described by the Smoluchowski equation, a generalization of the diffusion equation that includes external forces. In 1D, this equation is:
Suppose that we consider a unimolecular chemical reaction, for example an enzyme undergoing a conformational change. At a molecular level, the enzyme would exist in one "state" and then transition to another "state." This description presupposes a form for the potential V(x), where there are two local minima separated by a local maximum. The molecule would spend most of its time near the minima and only rarely transition between them. Under these conditions, one can simplify the Smoluchowski equation and derive expressions for the rate constants that govern the chemical transition. The equation describing these reaction rates is Kramers' theory. If a constant force field, F, is applied to a unimolecular chemical reaction, Kramers' theory gives an exact expression for the reaction rate, k, but it can only be used if V(x) is known. In this paper, I showed that when the force field is weak, we can perform a series of approximations of k that depend on only a few unknown parameters. The simplest of these approximations is Bell's approximation:
Figure 1: Data of bond breaking are not well fit by
Bell's theory, but are fit by the more complex theories. Experiments, published in Marshall et
al. 2003, are of the breaking of selectinligand
bonds with an AFM. Fits with the 3 parameter model are significantly better (Ftest) than
with Bell's theory. The ideas behind this work, particularly using the Smoluchowski equation and Kramers'like theories to understand the coupling between mechanics and chemical reaction rate, are the basis of our other theoretical work. 

Publication 2 

Reference: Srinivasan, M.*, Walcott, S.*, Binding site models of friction due to the formation and rupture of bonds: statefunction formalism, force velocity relations, response to slip velocity transients, and slip stability. Physical Review E, Volume 80, pages 046124, 2009. PDF *Equal author contribution. For a variety of applications, two surfaces or filaments slide relative to one another while longchain molecules form transient connections between them. Muscle contraction and dry friction between two smooth surfaces are a biological and physical example, respectively. The central idea of this work is that the equations that govern muscle contraction should also apply to friction mechanics. We assume that the general form of the governing equations is the same for muscle and friction; however, the mathematical details depend on the behavior of the molecules that transiently link the two filaments or surfaces. In this work, we assume a simplistic model, where springy molecules exist in two states: bound and unbound. As two surfaces slide relative to one another, attached molecules become increasingly stretched and more likely to unbind. As with the constant force reactions from publication 1 above, we can derive equations for stretchdependent unbinding using Kramers'like theories and thereby define the friction equations.
Figure 1: The model is consistent with steady state
and transient friction experiments.
A. Steady state friction force as a function of sliding velocity (in
dimensionless units). B. Force
transients (bottom) following step changes in sliding velocity (top). This model of friction qualitatively describes steadystate friction force as a function of sliding velocity, as well as force transients following a step change in sliding velocity (see Fig. 2). With some assumptions, we derived simple analytic expressions that relate friction force and sliding velocity. These expressions relate molecular properties (e.g. molecular stiffness) to macroscopic observables (e.g. friction force). The multiscale nature and the simplicity of these relationships allowed us to develop cellscale models of mechanosensation.


References 

[1] Zhurkov, S. N., Kinetic concept of the strength of solids, International Journal of Fracture, Volume 1, pages 311323, 1965. [2] Bell, G. I., Models for the specific adhesion of cells to cells, Science,
Volume 200, pages 618627, 1978. 

Collaborators 
