ABSTRACT
Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not con ned to biomolecular systems, and the development of methods to treat such “rare events” is currently an active eld of research.1-31 If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation.32,33 This approach was used in the last century for a number of studies involving atomic
and molecular clusters, as well as biomolecules.34-53 These applications range from analysis of model energy landscapes to coarse-grained and atomistic models. In the present chapter, the focus is upon networks composed of stationary points of the potential energy surface (PES), particularly local minima and the transition states that connect them (Section 14.2.1). The rst studies of biomolecules employing such techniques include applications to the tetrapeptide IAN53 and the fk506 binding protein.54 Berry and coworkers subsequently analyzed sequences of connected local minima for a coarse-grained model,55 which has since been used in a number of other studies.56-59
More recent work has focused either on building up transition networks from geometry optimization more systematically, or on alternative approaches that use dynamical methods to construct the networks.18,60-69 For smaller systems, it may be possible to capture the pathways of interest without speci cally seeking transition states that improve the overall rate constants.70-72 However, for larger biomolecules some sort of directed sampling is likely to be needed. The discrete path sampling approach described in Section 14.2.2 is designed to locate dynamically relevant stationary points that determine the rate constant for interconverting two given end points, which may be local minima or groups of minima.6,13 Algorithms from graph theory and network analysis are used in sampling and analyzing such stationary point databases, and can also be used for constructing free energies.73 For example, a bracketing procedure, which successively re nes upper and lower bounds on selected energy barriers,74 has been applied to analyze the dynamics of the Ras p21 molecular switch.75 Applications of stationary point databases to biomolecules have generally employed implicit solvent models to avoid the complication of trying to distinguish or group together conformations that include solvent degrees of freedom. The use of implicit solvent can produce a qualitative change in the energy landscape, since a different secondary structure may be favored compared to a gas phase description. Assuming that the chosen implicit solvent description produces a faithful description of the polypeptide, the main difference compared to explicit solvent would be an ensemble average over solvent con gurations. If these con gurations were included explicitly, then every local minimum would split into a spectrum of minima, differing principally in the solvent degrees of freedom. Ideally, the implicit solvent treatment should produce energies for local minima and transition states that reproduce the average values for explicit solvent. Hence this component of the solvent “friction” is included, i.e., shifts in the energy of different con gurations, but barriers corresponding to changes in speci c internal solvent degrees of freedom are not.
