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Register a new userInfinite Switch Simulated Tempering in Force (FISST)
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Many proteins in cells are capable of sensing and responding to piconewton scale forces, a regime in which conformational changes are small but significant for biological processes. In order to efficiently and effectively sample the response of these proteins to small forces, enhanced sampling techniques will be required. In this work, we derive, implement, and evaluate an efficient method to simultaneously sample the result of applying any constant pulling force within a specified range to a molecular system of interest. We start from Simulated Tempering in Force, whereby force is applied as a linear bias on a collective variable to the systems Hamiltonian, and the coefficient is taken as a continuous auxiliary degree of freedom. We derive a formula for an average collective-variable-dependent force, which depends on a set of weights, learned on-the-fly throughout a simulation, that reflect the limit where force varies infinitely quickly. These weights can then be used to retroactively compute averages of any observable at any force within the specified range. This technique is based on recent work deriving similar equations for Infinite Switch Simulated Tempering in Temperature, that showed the infinite switch limit is the most efficient for sampling. Here, we demonstrate that our method accurately and simultaneously samples molecular systems at all forces within a user defined force range, and show how it can serve as an enhanced sampling tool for cases where the pulling direction destabilizes states of low free-energy at zero-force. This method is implemented in, and will be freely-distributed with, the PLUMED open-source sampling library, and hence can be readily applied to problems using a wide range of molecular dynamics software packages.
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Single-molecule force spectroscopy has proven to be a powerful tool for studying the kinetic behavior of biomolecules. Through application of an external force, conformational states with small or transient populations can be stabilized, allowing them to be characterized and the statistics of individual trajectories studied to provide insight into biomolecular folding and function. Because the observed quantity (force or extension) is not necessarily an ideal reaction coordinate, individual observations cannot be uniquely associated with kinetically distinct conformations. While maximum-likelihood schemes such as hidden Markov models have solved this problem for other classes of single-molecule experiments by using temporal information to aid in the inference of a sequence of distinct conformational states, these methods do not give a clear picture of how precisely the model parameters are determined by the data due to instrument noise and finite-sample statistics, both significant problems in force spectroscopy. We solve this problem through a Bayesian extension that allows the experimental uncertainties to be directly quantified, and build in detailed balance to further reduce uncertainty through physical constraints. We illustrate the utility of this approach in characterizing the three-state kinetic behavior of an RNA hairpin in a stationary optical trap.
Loops are essential secondary structure elements in folded DNA and RNA molecules and proliferate close to the melting transition. Using a theory for nucleic acid secondary structures that accounts for the logarithmic entropy c ln m for a loop of length m, we study homopolymeric single-stranded nucleic acid chains under external force and varying temperature. In the thermodynamic limit of a long strand, the chain displays a phase transition between a low temperature / low force compact (folded) structure and a high temperature / high force molten (unfolded) structure. The influence of c on phase diagrams, critical exponents, melting, and force extension curves is derived analytically. For vanishing pulling force, only for the limited range of loop exponents 2 < c < 2.479 a melting transition is possible; for c <= 2 the chain is always in the folded phase and for 2.479 < c always in the unfolded phase. A force induced melting transition with singular behavior is possible for all loop exponents c < 2.479 and can be observed experimentally by single molecule force spectroscopy. These findings have implications for the hybridization or denaturation of double stranded nucleic acids. The Poland-Scheraga model for nucleic acid duplex melting does not allow base pairing between nucleotides on the same strand in denatured regions of the double strand. If the sequence allows these intra-strand base pairs, we show that for a realistic loop exponent c ~ 2.1 pronounced secondary structures appear inside the single strands. This leads to a lower melting temperature of the duplex than predicted by the Poland-Scheraga model. Further, these secondary structures renormalize the effective loop exponent c^, which characterizes the weight of a denatured region of the double strand, and thus affect universal aspects of the duplex melting transition.
We performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional STs applicability to dealing with phase transitions and to study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus, this simulation can be referred to as Simulated Tempering and Magnetizing (STM). We also performed the Simulated Magnetizing (SM) simulations, in which the external field was considered as a dynamical variable and temperature was not. As has been discussed by previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Flipping of the entire magnetization did not occur in the SM simulations under $T_mathrm{c}$ in large lattice-size simulations. However, the phase changed through the high temperature region in the STM simulations. Thus, the dimensional extension let us eliminate the difficulty of the first-order phase transitions and study wide area of the phase space. We then discuss how frequently parameter-updating attempts should be made for optimal convergence. The results favor frequent attempts. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior was clearly observed in the simulations in agreement with the theoretical implications.
We applied the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to perform further investigations of the STMs applicability. The temperature as well as the external field are treated as dynamical variables updated during the STM simulations. After we obtained adequate information for several lattice sizes $L$ (up to $160times 160$), we also performed a number of conventional canonical simulations of large lattices, especially in order to illustrate the crossover behavior of the Potts model in external field with increasing $L$. The temperature and external field for larger lattice size simulations were chosen by extrapolation of the detail information obtained by STM. We carefully analyzed the crossover scaling at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions.
We propose a new method for the determination of the weight factor for the simulated tempering method. In this method a short replica-exchange simulation is performed and the simulated tempering weight factor is obtained by the multiple-histogram reweighting techniques. The new algorithm is particularly useful for studying frustrated systems with rough energy landscape where the determination of the simulated tempering weight factor by the usual iterative process becomes very difficult. The effectiveness of the method is illustrated by taking an example for protein folding.
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