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 multi-body 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:

Where \rho(x,t) is the probability density of finding the molecule at position x at time t, kB is Boltzmann's constant, T the absolute temperature, \gamma the viscous drag on the molecule and V(x) the potential.


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 pre-supposes 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:

Bell's approximation (top) has two parameters, k0 and \lambda; the former is the reaction rate in the absence of force, the latter is a length parameter.  The next approximation (middle) has an additional stiffness parameter \kappa, and the next approximation (bottom) includes a force parameter a.  The more complex theories sometimes provide better parameter estimates and are necessary to fit some experimental data (Fig. 1).






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 selectin-ligand bonds with an AFM.  Fits with the 3 parameter model are significantly better (F-test) 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: state-function 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 long-chain 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 stretch-dependent 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 steady-state 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 multi-scale nature and the simplicity of these relationships allowed us to develop cell-scale models of mechanosensation.




[1] Zhurkov, S. N., Kinetic concept of the strength of solids, International Journal of Fracture, Volume 1, pages 311--323, 1965.

[2] Bell, G. I., Models for the specific adhesion of cells to cells, Science, Volume 200, pages 618--627, 1978.



Sean Sun

Manoj Srinivasan