The non-equilibrium thermodynamics and kinetics of focal adhesion dynamics

Abstract

Background: We consider a focal adhesion to be made up of molecular complexes, each consisting of a ligand, an integrin molecule, and associated plaque proteins. Free energy changes drive the binding and unbinding of these complexes and thereby controls the focal adhesion's dynamic modes of growth, treadmilling and resorption.

Principal findings: We have identified a competition among four thermodynamic driving forces for focal adhesion dynamics: (i) the work done during the addition of a single molecular complex of a certain size, (ii) the chemical free energy change associated with the addition of a molecular complex, (iii) the elastic free energy change associated with deformation of focal adhesions and the cell membrane, and (iv) the work done on a molecular conformational change. We have developed a theoretical treatment of focal adhesion dynamics as a nonlinear rate process governed by a classical kinetic model. We also express the rates as being driven by out-of-equilibrium thermodynamic driving forces, and modulated by kinetics. The mechanisms governed by the above four effects allow focal adhesions to exhibit a rich variety of behavior without the need to introduce special constitutive assumptions for their response. For the reaction-limited case growth, treadmilling and resorption are all predicted by a very simple chemo-mechanical model. Treadmilling requires symmetry breaking between the ends of the focal adhesion, and is achieved by driving force (i) above. In contrast, depending on its numerical value (ii) causes symmetric growth, resorption or is neutral, (iii) causes symmetric resorption, and (iv) causes symmetric growth. These findings hold for a range of conditions: temporally-constant force or stress, and for spatially-uniform and non-uniform stress distribution over the FA. The symmetric growth mode dominates for temporally-constant stress, with a reduced treadmilling regime.

Significance: In addition to explaining focal adhesion dynamics, this treatment can be coupled with models of cytoskeleton dynamics and contribute to the understanding of cell motility.

Figure 1. Physical and mathematical model of a FA.

The FA is the grey-colored parallelogram. Binders are shown as double ellipses, either free or bound in complexes. The bound complexes have been depicted to be larger, of length in the -direction and their elastic response is represented by the springs. The elastic elements have length along the -direction. (Note that is an arbitrary length and has no effect on the potential; see the section titled “Driving force due to elasticity”.) The dotted arrows are actin stress fibers, which transfer force to the attached complexes. The bundle of actin stress fibers transmits total force . Also shown are the proximal and distal ends, , the centroid , length and domain boundaries .

Figure 2. Schematic of the bound and unbound states represented by corresponding reaction coordinates, and the energy barriers to transitions between them.

Figure 3. Schematic.

(a) FA geometry and loading. (b) Addition of a complex at the distal end makes the center of the FA move opposite to the direction of the horizontal force component, increasing its potential. (c) Addition of a complex at the proximal end makes the center of the FA move in the direction of the horizontal force component, decreasing its potential.

Figure 4. State diagram of the final position of the focal adhesion's centroid, , and length, at s as a function of force, .

Also shown are the normalized chemical potentials, and at the distal and proximal ends, respectively. The schematic diagrams indicate the dynamics corresponding to each regime. Note the various modes attained as is varied.

Figure 5. Effect of a crack-like force distribution, Eq. (19).

State diagram of the final position of the focal adhesion's centroid, , and length, at s as a function of total force, . The other parameters are as in Fig. 4, with which this state diagram should be compared. Also shown are the normalized chemical potentials, and at the distal and proximal ends, respectively.

Figure 6. Sensitivity of the model to changing the complex size, from the upper bound nm to the lower bound nm.

The other parameters are as in Fig. 4, with which this state diagram should be compared.

Figure 7. Sensitivity of the model to variation in the force-indpendent part of the potential, .

Here this combination has been varied between (a) and (b) . In comparison, in Fig. 4. The other parameters are as in Fig. 4, with which this state diagram should be compared.

Figure 8. Sensitivity of the model to variation in the size of the conformational change, .

Here this parameter has been changed to nm from nm assumed in Fig. 4. The other parameters are as in Fig. 4, with which this state diagram should be compared.

Figure 9. State diagram of FA dynamics for stress-control (see Appendix).

The strains used were , , with other parameters are as in Fig. 4, with which this state diagram could be compared. The main effect of stress-control on FA dynamics is that the symmetric growth mode is strongly favored over most low stress values, as seen by the growth of . However, the Treadmilling Mode does hold over a narrow range for kPa, which compares well with the stress of kPa reported by Balaban and co-workers . The Treadmilling Mode gets suppressed for less polarized strains relative to .