Unraveling the Beautiful Complexity of Simple Lattice Model Polymers and Proteins Using Wang-Landau Sampling

We describe a class of "bare bones” models of homopolymers which undergo coil-globule collapse and proteins which fold into their native states in free space or into denatured states when captured by an attractive substrate as the temperature is lowered. We then show how, with the use of a properly chosen trial move set, Wang-Landau Monte Carlo sampling can be used to study the rough free energy landscape and ground (native) states of these intriguingly simple systems and thus elucidate their thermodynamic complexit

Two-dimensional lattice polymers: adaptive windows simulations

We report a numerical study of self-avoiding polymers on the square lattice, including an attractive potential between nonconsecutive monomers. Using Wang-Landau sampling (WLS) with adaptive windows, we obtain the density of states for chains of up to N=300 monomers and associated thermodynamic quantities. The method enables one to simulate accurately the low-temperature regime, which is virtually inaccessible using traditional methods. Instead of defining fixed energy windows, as in usual WLS, this method uses windows with boundaries that depend on the set of energy values on which the histogram is flat at a given stage of the simulation. Shifting the windows each time the modification factor $f$ is reduced, we eliminate border effects that arise in simulations using fixed windows.Comment: 8 pages, 5 figure

Variational Methods for Biomolecular Modeling

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.Comment: 36 page

Advanced Monte Carlo simulation techniques to study polymers under equilibrium conditions

The advances in materials and biological sciences have necessitated the use of molecular simulations to study polymers. The Markov chain Monte Carlo simulations enable the sampling of relevant microstates of polymeric systems by traversing paths that are impractical in molecular dynamics simulations. Several advances in applying Monte Carlo simulations to polymeric systems have been reported in recent decades. The proposed methods address sampling challenges encountered in studying different aspects of polymeric systems. Tracking the above advances has become increasingly challenging due to the extensive literature generated in the field. Moreover, the incorporation of new methods in the existing Monte Carlo simulation packages is cumbersome due to their complexity. Identifying the foundational algorithms that are common to different methods can significantly ease their implementation and make them accessible to the broader simulation community. The present chapter classifies the Monte Carlo methods for polymeric systems based on their objectives and standard features of their algorithms. We begin the article by providing an overview of advanced Monte Carlo techniques used for polymeric systems and their specific applications. We then classify the above techniques into two broad categories: 1) Monte Carlo moves and 2) Advanced sampling schemes. The former category is further divided to distinguish the Monte Carlo moves in the canonical and other ensembles. The advanced sampling schemes attempt to improve Monte Carlo sampling via approaches other than Monte Carlo moves. We use the above classification to identify common features of the methods and derive general expressions that explain their implementation. Such a strategy can help readers select the methods that are suitable for their study and develop computer programs that can be easily modified to implement new methods.Comment: 22 pages, 4 figures, 2 table