Thermodynamics and kinetics of protein folding under confinement

Understanding the effects of confinement on protein stability and folding kinetics is important for describing protein folding in the cellular environment. We have investigated the effects of confinement on two structurally distinct proteins as a function of the dimension d(c) and characteristic size R of the confining boundary. We find that the stabilization of the folded state relative to bulk conditions is quantitatively described by R(-gamma(c)), where the exponent gamma(c) is approximately 5/3 independent of the dimension of confinement d(c) (cylindrical, planar, or spherical). Moreover, we find that the logarithm of the folding rates also scale as R(-gamma(c)), with deviations only being seen for very small confining geometries, where folding is downhill; for both stability and kinetics, the dominant effect is the change in the free energy of the unfolded state. A secondary effect on the kinetics is a slight destabilization of the transition state by confinement, although the contacts present in the confined transition state are essentially identical to the bulk case. We investigate the effect of confinement on the position-dependent diffusion coefficients D(Q) for dynamics along the reaction coordinate Q (fraction of native contacts). The diffusion coefficients only change in the unfolded state basin, where they are increased because of compaction.