The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.

Atomic-level information is essential to explain the specific interactions governing protein–protein recognition in terms of structure and dynamics. Of particular interest is a characterization of the time-dependent kinetic aspects of protein–protein association and dissociation. A powerful framework to characterize the dynamics of complex molecular systems is provided by Markov State Models (MSMs). The central idea is to construct a reduced stochastic model of the full system by defining a set of conformational featured microstates and determining the matrix of transition probabilities between them. While a MSM framework can sometimes be very effective, different combinations of input featurization and simulation methods can significantly affect the robustness and the quality of the information generated from MSMs in the context of protein association. Here, a systematic examination of a variety of MSMs methodologies is undertaken to clarify these issues. To circumvent the uncertainties caused by sampling issues, we use a simplified coarse-grained model of the barnase–barstar protein complex. A sensitivity analysis is proposed to identify the microstates of an MSM that contribute most to the error in conjunction with the transition-based reweighting analysis method for a more efficient and accurate MSM construction.

Umbrella sampling (US) simulation is a highly effective method for sampling the conformations of a complex system within a small subspace of predefined coordinates. In a typical US stratification strategy, biasing "window" potentials spanning the subspace of interest are introduced to narrow down the range of accessible conformations and accelerate the sampling. The speed of convergence in each biased window simulation may, however, differ. For example, windows that coincide with a large energetic barrier along a coordinate that is orthogonal to the predefined subspace are often plagued by slow relaxation timescales. Here, we design a method that can quantitatively detect this type of issue and gain further insight into the origin of the slow relaxation timescale. Once the problematic windows affected by slow convergence are identified, additional simulations limited to only these windows can be carried out, thereby reducing the overall computational effort. Several possible approaches aimed at performing US simulations adaptively are discussed, and their respective performance is illustrated using a simple model system. Last, simulations of an atomic deca-alanine system are used to demonstrate the efficacy of analyzing US simulation trajectories using the proposed method.

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