Internal Friction and Nonequilibrium Unfolding of Polymeric Globules

Abstract

The stretching response of a single collapsed homopolymer is studied using Brownian dynamic simulations. The irreversibly dissipated work is found to be dominated by internal friction effects below the collapse temperature, and the internal viscosity grows exponentially with the effective cohesive strength between monomers. These results explain friction effects of globular DNA and are relevant for dissipation at intermediate stages of protein folding. DOI: 10.1103/PhysRevLett.103.028102 PACS numbers: 87.15.Àv, 62.40.+i, 82.37.Àj Conformational kinetics are crucial for the function of biopolymers: e.g., the muscle protein titin unfolds at a particular loading force in a highly dissipative manner, irreversibly converting most of the mechanical work into heat [1], while in myoglobin ligand dissociation induces a global conformational change of the protein [2][3][4]. Such transitions involve spatial protein reorganization, and thus internal dissipation mechanisms on different conformational levels in addition to solvent viscosity become important in determining the dynamic response to a given stimulus.Different contributions to internal polymeric friction have experimentally been distinguished [5]: On the smallest length scale are conformational molecular transitions involving torsional bond degrees of freedom [6,7]. For polymer solutions above the overlap concentration or for polymers in confined geometries, entanglement effects become important and contribute significantly [8]. Finally, for collapsed polymers or folded proteins, the continuous breakage and reformation of cohesive bonds gives rise to an extra contribution to the viscosity inside a globule [9,10]. The significant consequence particularly for protein science is that internal friction may dictate the rate of conformational kinetics and thus protein function dynamics. In all of these experimental studies, care is taken to isolate internal friction effects from the (in the present context uninteresting) hydrodynamic drag of the solvent by, for example, variation of the solvent viscosity [7].Coarse-grained stochastic models that involve activated hopping events in smooth and idealized energy landscapes nicely explain experimental titin unfolding force curves and provide insight into the dissipation mechanism involving two-step unfolding [11]. A different mechanism is expected for globular homopolymers, proteins in the molten globular state [12], and some disordered intermediates that occur during conformational protein transformations [13]. Here many near-optimal competing states exist, the energy landscape is rough, and structural changes occur gradually through a whole spectrum of intermediate states [14,15]. Many cohesive bonds are broken and reformed repeatedly during unfolding, and the concept of an internal effective viscosity naturally arises [9,10]. There have been quite a few simulation studies on the forced unfolding of bead-spring models for proteins [16] and globular polymers [17,18], but the concept of an internal viscosity has not been...

How the diffusivity profile reduces the arbitrariness of protein folding free energies

“…the number of possible connecting paths), the energetic ruggedness of such paths [20], as well as orthogonal degrees of freedom [21]. As folding progresses, internal friction starts to play a more dominating role [22,23], while solvent friction becomes less important as more and more peptide groups lose solvent contact [5]. Recently, the simplification of a constant diffusivity was abandoned and a diffusivity profile was extracted from simulations of peptides: these works either considered proteins without solvent (and thus exclude variations of the solvent friction) [24][25][26] or considered exclusively short-time dynamics and thus are not applicable to global folding kinetics [27].…”

Wet and dry internal friction can be measured with the Jarzynski equality

“…The choice of the bead radius, a, is motivated by Alexander-Katz et al's [14] work, where it is suggested that the monomeric radius may be taken as the persistence length of the molecule. We choose a = 30 nm, which is identical to the choice made by Alexander-Katz et al [14] for comparing the results of BD simulations against experiments on DNA. Other values of b and l H , of the same order-ofmagnitude as obtained for the λ-phage DNA case, have been used in this study and in addition to a = 30 nm, bead radii of 80 nm and 100 nm have also been used.…”

“…(1) represents an average with respect to the probability distribution of work values. Prior studies [13][14][15] estimate ∆A from the work done in the quasi-static limit [41] and calculate W dis at finite pulling rates by subtracting ∆A from the total work done, rather than estimating both components of work simultaneously, as is done here. In essence, the proposed protocol consists of calculating W dis at fixed values of both the pulling velocity v and distance d over which the molecule is stretched, but at various values of η s .…”

Internal friction can be measured with the Jarzynski equality

“…3A of the SI, the appropriate values in this case are b = 800 and l H = 500 nm. Alexander-Katz et al 8 suggest that the monomeric radius may be taken as the persistence length of the molecule. A bead radius of a = 30 nm is chosen as a representative value.…”

“…On the other hand, stretch-relaxation experiments on condensed DNA by Murayama et al, 7 and molecular dynamics simulations of polypeptide-stretching by Schulz, Miettinen and Netz 9 provide an estimate for the internal friction coefficient. We are motivated by the protocol proposed by Netz and coworkers, 8,9 wherein the work done in stretching a polymer molecule is partitioned into a reversible component, and an irreversible, dissipative component. The former goes into reversibly increasing the free energy, and is stored in the extension of the molecule.…”