Theoretical studies of the kinetics of mechanical unfolding of cross-linked polymer chains and their implications for single molecule pulling experiments
We have used kinetic Monte Carlo simulations to study the kinetics of unfolding of cross-linked polymer chains under mechanical loading. As the ends of a chain are pulled apart, the force transmitted by each crosslink increases until it ruptures. The stochastic crosslink rupture process is assumed to be governed by first order kinetics with a rate that depends exponentially on the transmitted force. We have performed random searches to identify optimal crosslink configurations whose unfolding requires a large applied force (measure of strength) and/or large dissipated energy (measure of toughness). We found that such optimal chains always involve cross-links arranged to form parallel strands. The location of those optimal strands generally depends on the loading rate. Optimal chains with a small number of cross-links were found to be almost as strong and tough as optimal chains with a large number of cross-links. Furthermore, optimality of chains with a small number of cross-links can be easily destroyed by adding cross-links at random. The present findings are relevant for the interpretation of single molecule force probe spectroscopy studies of the mechanical unfolding of load-bearing proteins, whose native topology often involves parallel strand arrangements similar to the optimal configurations identified in the study.
We present a dynamic coarse-graining technique that allows to simulate the mechanical unfolding of biomolecules or molecular complexes on experimentally relevant time scales. It is based on Markov state models (MSM), which we construct from molecular dynamics simulations using the pulling coordinate as an order parameter. We obtain a sequence of MSMs as a function of the discretized pulling coordinate, and the pulling process is modeled by switching among the MSMs according to the protocol applied to unfold the complex. This way we cover seven orders of magnitude in pulling speed. In the regio…
We discuss the use of a structure based C$α$-Go model and Langevin dynamics to study in detail the mechanical properties and unfolding pathway of the titin I27 domain. We show that a simple Go-model does detect correctly the origin of the mechanical stability of this domain. The unfolding free energy landscape parameters $x_u$ and $ΔG^{\ddagger}$, extracted from dependencies of unfolding forces on pulling speeds, are found to agree reasonably well with experiments. We predict that above $v=10^4$ nm/s the additional force-induced intermediate state is populated at an end-to-end extension of abo…
We consider a simple model for the unfolding of RNA tertiary structure under dynamic loading. The opening of such a structure is regarded as a two step process, each corresponding to the overcoming of a single energy barrier. The resulting two-barrier energy landscape accounts for the dependence of the unfolding kinetics on the pulling rate. Furthermore at intermediate force, the two barriers cannot be distinguished by the analysis of the opening kinetic, which turns out to be dominated by a single macro-barrier, whose properties depend non-trivially on the two single barriers. Our results sug…
We analyse the unfolding pathway of biomolecules comprising several independent modules in pulling experiments. In a recently proposed model, a critical velocity $v_{c}$ has been predicted, such that for pulling speeds $v>v_{c}$ it is the module at the pulled end that opens first, whereas for $v<v_{c}$ it is the weakest. Here, we introduce a variant of the model that is closer to the experimental setup, and discuss the robustness of the emergence of the critical velocity and of its dependence on the model parameters. We also propose a possible experiment to test the theoretical predictions of …
A theoretical analysis of the unfolding pathway of simple modular proteins in length- controlled pulling experiments is put forward. Within this framework, we predict the first module to unfold in a chain of identical units, emphasizing the ranges of pulling speeds in which we expect our theory to hold. These theoretical predictions are checked by means of steered molecular dynamics of a simple construct, specifically a chain composed of two coiled-coils motives, where anisotropic features are revealed. These simulations also allow us to give an estimate for the range of pulling velocities in …
We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states (Pfold = 0.5). Using results for the forced-unfolding of a RNA hairpin, an exactly soluble model and an analytic theory, we show that Pfold is solely determined by s, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. …
Heterogeneity in biological molecules, resulting in molecule-to-molecule variations in their dynamics and function, is an emerging theme. To elucidate the consequences of heterogeneous behavior at the single molecule level, we propose an exactly solvable model in which the unfolding rate due to mechanical force depends parametrically on an auxiliary variable representing an entropy barrier arising from fluctuations in internal dynamics. When the rate of fluctuations, a measure of dynamical disorder, is comparable to or smaller than the rate of force-induced unbinding, we show that there are tw…
Single-molecule pulling experiments on unstructured proteins linked to neurodegenerative diseases have measured rupture forces comparable to those for stable folded proteins. To investigate the structural mechanisms of this unexpected force resistance, we perform pulling simulations of the amyloid β-peptide (Aβ) and α-synuclein (αS), starting from simulated conformational ensembles for the free monomers. For both proteins, the simulations yield a set of rupture events that agree well with the experimental data. By analyzing the conformations right before rupture in each event, we find that the…
Using force as a probe to map the folding landscapes of RNA molecules has become a reality thanks to major advances in single molecule pulling experiments. Although the unfolding pathways under tension are complicated to predict studies in the context of proteins have shown that topology plays is the major determinant of the unfolding landscapes. By building on this finding we study the responses of RNA molecules to force by adapting Gaussian network model (GNM) that represents RNAs using a bead-spring network with isotropic interactions. Cross-correlation matrices of residue fluctuations, whi…
Single-molecule experiments provide new insights into biological processes hitherto not accessible by measurements performed on bulk systems. We report on a study of the kinetics of a triple-branch DNA molecule with four conformational states by pulling experiments with optical tweezers and theoretical modelling. Three distinct force rips associated with different transitions between the conformational states are observed in the folding and unfolding trajectories. By applying transition rate theory to a free energy model of the molecule, probability distributions for the first rupture forces o…
Single-molecule experiments with optical tweezers have become an important tool to study the properties and mechanisms of biological systems, such as cells and nucleic acids. In particular, force unzipping experiments have been used to extract the thermodynamics and kinetics of folding and unfolding reactions. In hopping experiments, a molecule executes transitions between the unfolded and folded states at a preset value of the force (constant force mode -CFM- under force feedback) or trap position (passive mode -PM- without feedback) and the force-dependent kinetic rates extracted from the li…
We apply Kramers theory to investigate the dissociation of multiple bonds under mechanical force and interpret experimental results for the unfolding/refolding force distributions of an RNA hairpin pulled at different loading rates using laser tweezers. We identify two different kinetic regimes depending on the range of forces explored during the unfolding and refolding process. The present approach extends the range of validity of the two-states approximation by providing a theoretical framework to reconstruct free-energy landscapes and identify force-induced structural changes in molecular t…
We analyze free energy estimators from simulation trials mimicking single-molecule pulling experiments on a neck linker of a kinesin motor. For that purpose, we have performed a version of steered molecular dynamics (SMD) calculations. The sample trajectories have been analyzed to derive distribution of work done on the system. In order to induce unfolding of the linker, we have stretched the molecule at a constant pulling force and allowed for a subsequent relaxation of its structure. The use of fluctuation relations (FR) relevant to non-equilibrium systems subject to thermal fluctuations all…
We study the mechanical stiffening behavior in two-dimensional (2D) cross-linked networks of semiflexible biopolymer filaments under simple shear. Filamental constituents immersed in a fluid undergo thermally excited bending motions. Pulling out these undulations results in an increase in the axial stiffness. We analyze this stiffening behavior of 2D semiflexible filaments in detail: we first investigate the average, {static} force-extension relation by considering the initially present undulated configuration that is pulled straight under a tensile force, and compare this result with the aver…
Cooperativity is a hallmark of proteins, many of which show a modular architecture comprising discrete structural domains. Detecting and describing dynamic couplings between structural regions is difficult in view of the many-body nature of protein-protein interactions. By utilizing the GPU-based computational acceleration, we carried out simulations of the protein forced unfolding for the dimer WW-WW of the all-beta-sheet WW domains used as a model multidomain protein. We found that while the physically non-interacting identical protein domains (WW) show nearly symmetric mechanical properties…
One of the most intriguing results of single molecule experiments on proteins and nucleic acids is the discovery of functional heterogeneity: the observation that complex cellular machines exhibit multiple, biologically active conformations. The structural differences between these conformations may be subtle, but each distinct state can be remarkably long-lived, with random interconversions between states occurring only at macroscopic timescales, fractions of a second or longer. Though we now have proof of functional heterogeneity in a handful of systems---enzymes, motors, adhesion complexes-…
We investigate irreversibility and dissipation in single molecules that cooperatively fold/unfold in a two state manner under the action of mechanical force. We apply path thermodynamics to derive analytical expressions for the average dissipated work and the average hopping number in two state systems. It is shown how these quantities only depend on two parameters that characterize the folding/unfolding kinetics of the molecule: the fragility and the coexistence hopping rate. The latter has to be rescaled to take into account the appropriate experimental setup. Finally we carry out pulling ex…
Single molecule force spectroscopy of DNA strands adsorbed at surfaces is a powerful technique used in air or liquid environments to quantify their mechanical properties. Although the force responses are limited to unfolding events so far, single base detection might be possible in more drastic cleanliness conditions such as ultra high vacuum. Here, we report on high resolution imaging and pulling attempts at low temperature (5K) of a single strand DNA (ssDNA) molecules composed of 20 cytosine bases adsorbed on Au(111) by scanning probe microscopy and numerical calculations. Using electrospray…
RNA folding is a kinetic process governed by the competition of a large number of structures stabilized by the transient formation of base pairs that may induce complex folding pathways and the formation of misfolded structures. Despite of its importance in modern biophysics, the current understanding of RNA folding kinetics is limited by the complex interplay between the weak base-pair interactions that stabilize the native structure and the disordering effect of thermal forces. The possibility of mechanically pulling individual molecules offers a new perspective to understand the folding of …
One of the most unique physical features of cell adhesion to external surfaces is the active generation of mechanical force at the cell-material interface. This includes pulling forces generated by contractile polymer bundles and networks, and pushing forces generated by the polymerization of polymer networks. These forces are transmitted to the substrate mainly by focal adhesions, which are large, yet highly dynamic adhesion clusters. Tissue cells use these forces to sense the physical properties of their environment and to communicate with each other. The effect of forces is intricately link…
