A Theoretical Model for the Mechanical Unfolding of Repeat Proteins

AbstractWe consider the mechanical stretching of a polypeptide chain formed by multiple interacting repeats. The folding thermodynamics and the interactions among the repeats are described by the Ising model. Unfolded repeats act as soft entropic springs, whereas folded repeats respond to a force as stiffer springs. We show that the resulting force-extension curve may exhibit a pronounced force maximum corresponding to the unfolding of the first repeat. This event is followed by the unfolding of the remaining repeats, which takes place at a lower force. As the protein extension is increased, the force-extension curve of a sufficiently long repeat protein displays a plateau, where the force remains nearly constant and the protein unfolds sequentially so that the number of unfolded repeats is proportional to the extension. Such a sequential mechanical unfolding mechanism is displayed even by the repeat proteins whose thermal denaturation is highly cooperative, provided that they are long enough. By contrast, the unfolding of short repeat progressions can be cooperative

A Kinetic Model for the Enzymatic Action of Cellulase

We develop a mechanochemical model for the dynamics of cellulase, a two-domain enzyme connected by a peptide linker, as it extracts and hydrolyzes a cellulose polymer from a crystalline substrate. We consider two random walkers, representing the catalytic domain (CD) and the carbohydrate binding module (CBM), whose rates for stepping are biased by the coupling through the linker and the energy required to lift the cellulose polymer from the crystalline surface. Our results show that the linker length and stiffness play a critical role in the cooperative action of the CD and CBM domains and that, for a given linker length, the steady-state hydrolysis shows a maximum at some intermediate linker stiffness. The maximum hydrolysis rate corresponds to a transition of the linker from a compressed to an extended conformation, where the system exhibits maximum fluctuation, as measured by the variance of the separation distance between the two domains and the dispersion around the mean hydrolysis speed. In the range of experimentally known values of the parameters of our model, improving the intrinsic hydrolytic activity of the CD leads to a proportional increase in the overall hydrolysis rate

Nonequilibrium statistical mechanics of money/energy exchange models

Many-body dynamical models in which Boltzmann statistics can be derived directly from the underlying dynamical laws without invoking the fundamental postulates of statistical mechanics are scarce. Interestingly, one such model is found in econophysics and in chemistry classrooms: the money game, in which players exchange money randomly in a process that resembles elastic intermolecular collisions in a gas, giving rise to the Boltzmann distribution of money owned by each player. Although this model offers a pedagogical example that demonstrates the origins of Boltzmann statistics, such demonstrations usually rely on computer simulations - a proof of the exponential steady-state distribution in this model has only become available in recent years. Here, we study this random money/energy exchange model, and its extensions, using a simple mean-field-type approach that examines the properties of the one-dimensional random walk performed by one of its participants. We give a simple derivation of the Boltzmann steady-state distribution in this model. Breaking the time-reversal symmetry of the game by modifying its rules results in non-Boltzmann steady-state statistics. In particular, introducing "unfair" exchange rules in which a poorer player is more likely to give money to a richer player than to receive money from that richer player, results in an analytically provable Pareto-type power-law distribution of the money in the limit where the number of players is infinite, with a finite fraction of players in the "ground state" (i.e., with zero money). For a finite number of players, however, the game may give rise to a bimodal distribution of money and to bistable dynamics, in which a participant's wealth jumps between poor and rich states. The latter corresponds to a scenario where the player accumulates nearly all the available money in the game

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

Preface: Special Topic on Single-Molecule Biophysics

Single-molecule measurements are now almost routinely used to study biological systems and processes. The scope of this special topic emphasizes the physics side of single-molecule observations, with the goal of highlighting new developments in physical techniques as well as conceptual insights that single-molecule measurements bring to biophysics. This issue also comprises recent advances in theoretical physical models of single-molecule phenomena, interpretation of single-molecule signals, and fundamental areas of statistical mechanics that are related to single-molecule observations. A particular goal is to illustrate the increasing synergy between theory, simulation, and experiment in single-molecule biophysics

Two-dimensional fluorescence resonance energy transfer as a probe for protein folding: A theoretical study

We describe a two-dimensional (2D), four-color fluorescence resonance energy transfer (FRET) scheme, in which the conformational dynamics of a protein is followed by simultaneously observing the FRET signal from two different donor-acceptor pairs. For a general class of models that assume Markovian conformational dynamics, we relate the properties of the emission correlation functions to the rates of elementary kinetic steps in the model. We further use a toy folding model that treats proteins as chains with breakable cross-links to examine the relationship between the cooperativity of folding and FRET data and to establish what additional information about the folding dynamics can be gleaned from 2D, as opposed to one-dimensional FRET experiments. We finally discuss the potential advantages of the four-color FRET over the three-color FRET technique

Transition path dynamics of a nanoparticle in a bistable optical trap

Many processes in chemistry, physics, and biology involve rare events in which the system escapes from a metastable state by surmounting an activation barrier. Examples range from chemical reactions, protein folding, and nucleation events to the catastrophic failure of bridges. A challenge in understanding the underlying mechanisms is that the most interesting information is contained within the rare transition paths, the exceedingly short periods when the barrier is crossed. To establish a model process that enables access to all relevant timescales, although highly disparate, we probe the dynamics of single dielectric particles in a bistable optical trap in solution. Precise localization by high-speed tracking enables us to resolve the transition paths and relate them to the detailed properties of the 3D potential within which the particle diffuses. By varying the barrier height and shape, the experiments provide a stringent benchmark of current theories of transition path dynamics