A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.

Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm to compute efficiently a sequence of isocommittor surfaces. These surfaces are considered an optimal reaction coordinate. The algorithm analyzes Milestoning results to determine the committor function. It requires only the transition probabilities between the milestones, and not transition times. We discuss the following numerical examples: (i) a transition in the Mueller potential; (ii) a conformational change of a solvated peptide; and (iii) cholesterol aggregation in membranes.

Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral aspects of otherwise intractable models. High model dimensionality and complexity makes symbolic, pen-and-paper model reduction tedious and impractical, a difficulty addressed by recently developed frameworks that computerize reduction. Symbolic work has the benefit, however, of identifying both reduced state variables and parameter combinations that matter most (effective parameters, "inputs"); whereas current computational reduction schemes leave the parameter reduction aspect mostly unaddressed. As the interest in mapping out and optimizing complex input-output relations keeps growing, it becomes clear that combating the curse of dimensionality also requires efficient schemes for input space exploration and reduction. Here, we explore systematic, data-driven parameter reduction by means of effective parameter identification, starting from current nonlinear manifoldlearning techniques enabling state space reduction. Our approach aspires to extend the data-driven determination of effective state variables with the data-driven discovery of effective model parameters, and thus to accelerate the exploration of high-dimensional parameter spaces associated with com- * These two authors contributed equally to this work.Given access to input-output information (black-box function evaluation) but no formulas, one might not even suspect that only the single parameter combination p eff = p 1 p 2 matters. Fitting the model to data f * = (1, 0, 1) in the absence of such information, one would find an entire curve in parameter space that fits the observations. A data fitting algorithm based only on function evaluations could be "confused" by such behavior in declaring convergence. As seen in Fig. 1(a), different initial conditions fed to an optimizer with a practical fitting tolerance δ ≈ 10 −3 (see figure caption for details) converge to many, widely different results tracing a level curve of p eff . The subset of good fits is effectively 1−D; more importantly, and moving beyond the fit to this particular data, the entire parameter space is foliated by such 1−D curves (neutral sets), each composed of points indistinguishable from the model output perspective. Parameter non-identifiability is therefore a structural feature of the model, not an artifact of optimization. The appropriate, intrinsic way to describe parameter space for this problem is through the effective parameter p eff and its level sets. Consider now the inset of Fig. 1(a), corresponding to the perturbed model f ε (p 1 , p 2 ) = f 0 (p 1 , p 2 ) + 2ε(p 1 − p 2 , 0, 0) and fit to the same data. Here, the parameters are identifiable and the minimizer (p 1 , p 2 ) unique: a perfect fit exists. However, the foliation observed for ε = 0 is loosely remembered in the shape of the residual level curves, and the optimizer would be co...

We give a mathematical framework for Exact Milestoning, a recently introduced algorithm for mapping a continuous time stochastic process into a Markov chain or semi-Markov process that can be efficiently simulated and analyzed. We generalize the setting of Exact Milestoning and give explicit error bounds for the error in the Milestoning equation for mean first passage times.

We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig.