ALBERT

Description: We aim to construct higher order tau-leaping methods for numerically simulating stochastic chemical kinetic systems in this paper. By adding a random correction to the primitive tau-leaping scheme in each time step, we greatly improve the accuracy of the tau-leaping approximations. This gain in accuracy actually comes from the reduction in the local truncation error of the scheme in the order of τ , the marching time step size. While the local truncation error of the primitive tau-leaping method is O ( τ 2 ) for all moments, our Poisson random correction tau-leaping method, in which the correction term is a Poisson random variable, can reduce the local truncation error for the mean to O ( τ 3 ) , and both Gaussian random correction tau-leaping methods, in which the correction term is a Gaussian random variable, can reduce the local truncation error for both the mean and covariance to O ( τ 3 ) . Numerical results demonstrate that these novel methods more accurately capture crucial properties such as the mean and variance than existing methods for simulating chemical reaction systems. This work constitutes a first step to construct high order numerical methods for simulating jump processes. With further refinement and appropriately modified step-size selection procedures, the random correction methods should provide a viable way of simulating chemical reaction systems accurately and efficiently.

Description: To interpret molecular dynamics simulations of complex systems, systematic dimensionality reduction methods such as principal component analysis (PCA) represent a well-established and popular approach. Apart from Cartesian coordinates, internal coordinates, e.g., backbone dihedral angles or various kinds of distances, may be used as input data in a PCA. Adopting two well-known model problems, folding of villin headpiece and the functional dynamics of BPTI, a systematic study of PCA using distance-based measures is presented which employs distances between C α -atoms as well as distances between inter-residue contacts including side chains. While this approach seems prohibitive for larger systems due to the quadratic scaling of the number of distances with the size of the molecule, it is shown that it is sufficient (and sometimes even better) to include only relatively few selected distances in the analysis. The quality of the PCA is assessed by considering the resolution of the resulting free energy landscape (to identify metastable conformational states and barriers) and the decay behavior of the corresponding autocorrelation functions (to test the time scale separation of the PCA). By comparing results obtained with distance-based, dihedral angle, and Cartesian coordinates, the study shows that the choice of input variables may drastically influence the outcome of a PCA.

Description: The process of protein search for specific binding sites on DNA is fundamentally important since it marks the beginning of all major biological processes. We present a theoretical investigation that probes the role of DNA sequence symmetry, heterogeneity, and chemical composition in the protein search dynamics. Using a discrete-state stochastic approach with a first-passage events analysis, which takes into account the most relevant physical-chemical processes, a full analytical description of the search dynamics is obtained. It is found that, contrary to existing views, the protein search is generally faster on DNA with more heterogeneous sequences. In addition, the search dynamics might be affected by the chemical composition near the target site. The physical origins of these phenomena are discussed. Our results suggest that biological processes might be effectively regulated by modifying chemical composition, symmetry, and heterogeneity of a genome.

Description: We describe the oligosaccharides-exchange dynamics performed by the so-called D-enzymes on polysaccharides. To mimic physiological conditions, we treat this process as an open chemical network by assuming some of the polymer concentrations fixed (chemostatting). We show that three different long-time behaviors may ensue: equilibrium states, nonequilibrium steady states, and continuous growth states. We dynamically and thermodynamically characterize these states and emphasize the crucial role of conservation laws in identifying the chemostatting conditions inducing them.

Description: Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.

Description: Sterols play an essential role in modulating bilayer structure and dynamics. Coarse-grained molecular dynamics parameters for cholesterol and related molecules are available for the Martini force field and have been successfully used in multiple lipid bilayer studies. In this work, we focus on the use of virtual sites as a means of increasing the stability of cholesterol and cholesterol-like structures. We improve and extend the Martini parameterization of sterols in four different ways: 1—the cholesterol parameters were adapted to make use of virtual interaction sites, which markedly improves numerical stability; 2—cholesterol parameters were also modified to address reported shortcomings in reproducing correct lipid phase behavior in mixed membranes; 3—parameters for ergosterol were created and adapted from cholesterols; and 4—parameters for the hopanoid class of bacterial polycyclic molecules were created, namely, for hopane, diploptene, bacteriohopanetetrol, and for their polycyclic base structure.

Description: Predicting conformational changes of proteins is needed in order to fully comprehend functional mechanisms. With the large number of available structures in sets of related proteins, it is now possible to directly visualize the clusters of conformations and their conformational transitions through the use of principal component analysis. The most striking observation about the distributions of the structures along the principal components is their highly non-uniform distributions. In this work, we use principal component analysis of experimental structures of 50 diverse proteins to extract the most important directions of their motions, sample structures along these directions, and estimate their free energy landscapes by combining knowledge-based potentials and entropy computed from elastic network models. When these resulting motions are visualized upon their coarse-grained free energy landscapes, the basis for conformational pathways becomes readily apparent. Using three well-studied proteins, T4 lysozyme, serum albumin, and sarco-endoplasmic reticular Ca 2+ adenosine triphosphatase (SERCA), as examples, we show that such free energy landscapes of conformational changes provide meaningful insights into the functional dynamics and suggest transition pathways between different conformational states. As a further example, we also show that Monte Carlo simulations on the coarse-grained landscape of HIV-1 protease can directly yield pathways for force-driven conformational changes.

Description: We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy “ensemble refinement of SAXS” (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraints should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the “Bayesian inference of ensembles” method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations.

Description: A measure of enzyme efficiency is proposed for an open reaction network that, in suitable form, applies to closed systems as well. The idea originates from the description of classical enzyme kinetics in terms of cycles. We derive analytical expressions for the efficiency measure by treating the network not only deterministically but also stochastically. The latter accounts for any significant amount of noise that can be present in biological systems and hence reveals its impact on efficiency. Numerical verification of the results is also performed. It is found that the deterministic equation overestimates the efficiency, the more so for very small system sizes. Roles of various kinetics parameters and system sizes on the efficiency are thoroughly explored and compared with the standard definition k 2 / K M . Study of substrate fluctuation also indicates an interesting efficiency-accuracy balance.

Description: The forced translocation of homopolymers through αβα sandwich-like compound channels was investigated by Monte Carlo simulation. The interaction between polymer and part α is strongly attractive, whereas that between polymer and part β is purely repulsive. Simulation results show that the translocation is influenced obviously by the length of part β ( L β ) and the starting position of part β ( L α1 ). For small L β , the translocation is mainly governed by the escaping process, and polymer is trapped near the exit of the channel. However, the translocation time can be tuned by varying L α1 and the fastest translocation can be achieved at relatively large L α1 . Whereas for large L β and small L α1 , the translocation is mainly controlled by the filling process. It is difficult for polymer to enter the channel, and polymer is trapped at the first αβ interface. Finally, the dynamics for the filling process and the escaping process are discussed from the view of free-energy landscape, respectively.