Individual Proteins Under Mechanical Stress: Lessons from Theory and Computer Simulations

Abstract

Mechanically induced protein unfolding and other mechanically modulated conformational transitions are implicated in many vital biological processes. Motivated by single-molecule pulling studies that probe the mechanical response of proteins and other biomolecules, this chapter introduces essential theoretical ideas needed to understand how the dynamics and thermodynamics of proteins are affected by mechanical stress. Various computational approaches and theoretical models used to explore the relationship between protein structure and mechanical response are critically reviewed, and computational and theoretical predictions are then contrasted with experimental observations. Finally, recent computational efforts to identify globular protein domains with high mechanical stability are described and, whenever possible, compared with experiments.

Appendix

In the Appendix, explicit expressions for the mechanical compliance of a molecule pulled between two chosen atoms i and j are provided, assuming that the molecule’s free energy $G({\mathbf{r}})$ can be approximated as a Taylor expansion, to second order, in the vicinity of its minimum or saddle point. For example, near the minimum free energy configuration ${{\mathbf{r}}^{{(0)}}}$, we have

$$ G({\mathbf{r}}) \approx G({{\mathbf{r}}^{{(0)}}}) + \frac{1}{2}{({\mathbf{r}} - {{\mathbf{r}}^{{(0)}}})^T}{{\mathbf{h}}^{{(0)}}}({\mathbf{r}} - {{\mathbf{r}}^{{(0)}}}), $$

where ${{\mathbf{h}}^{{(0)}}}$ is the molecule’s Hessian matrix. When the distance ${R_{{ij}}}$ between the two selected atoms is increased, other atoms, of course, also become displaced. Since no external forces act on those atoms, their positions are determined from the condition that they are in mechanical equilibrium (i.e., the total force exerted on each of them by other atoms is zero). Given the assumed linearity of the system, its response to an increase in the distance ${R_{{ij}}}$ is that of a Hookean spring, with a compliance (inverse stiffness) $\chi_{{ij}}^{{(0)}}$. Finding $\chi_{{ij}}^{{(0)}}$ can be viewed as a coarse-graining procedure, in which all of the atomic coordinates, except for the coordinates of the atoms i and j, are eliminated based on the above mechanical equilibrium condition. As the atoms of the molecule can be arbitrarily relabeled, it is convenient to assume that one always pulls on the first two atoms. We then write the molecule’s Hessian matrix in the block-diagonal form:

$$ {{\mathbf{h}}^{{(0)}}} = \left( {\begin{array}{llllllllllllll} {{\mathbf{h}}_{{{\mathbf{11}}}}^{{{\mathbf{(0)}}}}} & {{\mathbf{h}}_{{{\mathbf{12}}}}^{{{\mathbf{(0)}}}}} \\{{\mathbf{h}}_{{{\mathbf{21}}}}^{{{\mathbf{(0)}}}}} & {{\mathbf{h}}_{{{\mathbf{22}}}}^{{{\mathbf{(0)}}}}} \\\end{array} } \right). $$

Here ${\mathbf{h}}_{{{\mathbf{11}}}}^{{{\mathbf{(0)}}}}$, ${\mathbf{h}}_{{{\mathbf{12}}}}^{{{\mathbf{(0)}}}}$, ${\mathbf{h}}_{{{\mathbf{21}}}}^{{{\mathbf{(0)}}}}$, and ${\mathbf{h}}_{{{\mathbf{22}}}}^{{{\mathbf{(0)}}}}$ are, respectively, 6 × 6, 6 × (3N−6), (3N−6) × 6, and (3N−6) × (3N−6) matrices. The (3N−6) degrees of freedom of all the atoms other than the first two are eliminated through the standard coarse-graining procedure, to obtain an effective 6 × 6 Hessian matrix that describes the mechanical response of the pair of atoms one is pulling on. This matrix is given by the Schur complement (Konda et al. 2011; Eom et al. 2007; Soheilifard et al. 2011):

$$ {{\bar {h}}}_{{{\mathbf{11}}}}^{{{\mathbf{(0)}}}}{\mathbf{= h}}_{{{\mathbf{11}}}}^{{{\mathbf{(0)}}}} - {\mathbf{h}}_{{{\mathbf{12}}}}^{{{\mathbf{(0)}}}}{({\mathbf{h}}_{{{\mathbf{22}}}}^{{{\mathbf{(0)}}}})^{{ - {\mathbf{1}}}}}{\mathbf{h}}_{{{\mathbf{21}}}}^{{{\mathbf{(0)}}}}. $$

This matrix should, of course, coincide with the Hessian matrix computed from the assumption that the free energy of the system is that of a simple Hookean spring given by (9.25). This, in particular, means that it has five zero eigenvalues and one nonzero eigenvalue equal to $2/\chi_{{ij}}^{{(0)}}$. Thus diagonalization of the 6 × 6 matrix ${\mathbf{\bar{h}}}_{{{\mathbf{11}}}}^{{{\mathbf{(0)}}}}$ readily solves the problem of finding the effective compliance $\chi_{{ij}}^{{(0)}}$ in terms of the full Hessian matrix of the molecule.

The effective compliance $\chi_{{ij}}^{{{\rm{(TS)}}}}$ of the molecule corresponding to its transition state (as well as to any critical point of the molecule’s potential energy surface) can be computed in an analogous manner, using the Hessian matrix corresponding to the transition state. Of course, stretching the molecule while maintaining its transition-state configuration does not correspond to any experimental scenario. Nevertheless, as discussed in Sect. 9.3, this quantity is expedient in calculations of force-dependent rates.