Molecular dynamics (MD) simulation is a technique that aids in the understanding of fundamental processes in biology and chemistry, and has important technological ap­plications in pharmacy, biotechnology, and nanotechnology. Many complex molecular processes have cascades of timescales spanning the range from 10-15 s to 1 s, often with no pronounced gap that would permit ef.cient coarse-grained time integration. 
In many applications, the slowest timescales and the associated structural rearrange­ments are the ones of interest. In this project, we study the challenging process of in­duced folding of peptides. The metastable states associated with the slowest timescale of an induced folding problem are shown in Fig. A04-1. Such states may be found when peptide ligands bind to proteins and when membrane-associated proteins anchor into the membrane. Here, association and conformational changes occur on physical timescales of nano-to milliseconds, while dissociation events may require seconds or longer. 
As a root model, we choose classical molecular dynamics with atomistic resolution and explicit solvent. Thus, the simulation system consists of a box, containing typically 
10.000 to 100.000 classical particles, representing solvent, ions, and the solvated pro­tein. The system evolves by a time-stepping scheme that approximates the solution of the classical equations of motion. Additionally, the time-stepping scheme usually con­tains a stochastic term that models the coupling of the molecular system to a heat bath, and thus ensures the desired thermodynamic ensemble (e.g., canonical). In this setting, molecular dynamics is a Markov process in a high-dimensional state space. The domi­nant timescales and their associated structure changes between metastable (long-lived) states are given by the eigenvalues and eigenfunctions of the transfer operator of the Markov process. These dominant eigenvalues and eigenfunctions therefore need to be approximated.

The introduction of Markov state models (MSMs) to molecular simulation in the past few years has been a breakthrough in providing the ability to perform such an approxi­mation. An MSM consists of a discretization of the molecular state space into sets, often found by geometric clustering of available simulation data, and a matrix of transition probabilities between them, estimated from the same simulation data. This is an esti­mation of a set discretization of the transfer operator. Despite their success, the current algorithmic realization of MSMs for high-dimensional system suffers from two funda­mental problems:

1. Discretization Problem: When the initial discretization for the MSM, based on Eu­clidean distances in the data, is poor, the dominant transfer operator eigenvalues (and timescales) will be systematically underestimated, resulting in numerical un­reliability of the approach. When the user is interested in approximating a sizable number (e.g., 10–100) of slow processes with high accuracy, the common practice to use data-driven geometric clustering methods may not be a viable approach.

2. Sampling Problem: MSMs contain only information of states that have been visited and transitions that have occurred in the simulation data. While the slowest events may occur on timescales of seconds, affordable simulation lengths are on the order of microseconds. Thus, MSM construction suffers from a severe sampling problem.

Both problems are coupled. Based on keystones set by recent theoretical results, we now set out to develop a concise numerical and algorithmic framework to address them. 
The long-term aims of this project are to develop efficient modeling and simulation methods for the dominant (slow) timescales of complex biomolecular simulation sys­tems, and apply them to folding-binding problems in biomolecules. 
In contrast to previous conformation-dynamics approaches such as Markov state modeling that are driven by a set-based approach, we attempt a paradigm shift and will focus on developing methods to approximate and sample individual timescales and eigenfunctions one by one.