Two methods are developed to enhance the stability, efficiency, and robustness of reaction path optimization using a chain of replicas. First, distances between replicas are kept equal during path optimization via holonomic constraints. Finding a reaction path is, thus, transformed into a constrained optimization problem. This approach avoids force projections for finding minimum energy paths (MEPs), and fast-converging schemes such as quasi-Newton methods can be readily applied. Second, we define a new objective function - the total Hamiltonian - for reaction path optimization, by combining the kinetic energy potential of each replica with its potential energy function. Minimizing the total Hamiltonian of a chain determines a minimum Hamiltonian path (MHP). If the distances between replicas are kept equal and a consistent force constant is used, then the kinetic energy potentials of all replicas have the same value. The MHP in this case is the most probable isokinetic path. Our results indicate that low-temperature kinetic energy potentials (<5 K) can be used to prevent the development of kinks during path optimization and can significantly reduce the required steps of minimization by 2-3 times without causing noticeable differences between a MHP and MEP. These methods are applied to three test cases, the C7eq-to-Cax isomerization of an alanine dipeptide, the (4)C1-to-(1)C4 transition of an α-d-glucopyranose, and the helix-to-sheet transition of a GNNQQNY heptapeptide. By applying the methods developed in this work, convergence of reaction path optimization can be achieved for these complex transitions, involving full atomic details and a large number of replicas (>100). For the case of helix-to-sheet transition, we identify pathways whose energy barriers are consistent with experimental measurements. Further, we develop a method based on the work energy theorem to quantify the accuracy of reaction paths and to determine whether the atoms used to define a path are enough to provide quantitative estimation of energy barriers.

The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

To extract mechanistic information of activated processes, we propose to decompose potential energy and free energy differences between configurations into contributions from individual atoms, functional groups, or residues. Decomposition is achieved by calculating the mechanical work associated with the displacements and forces of each atom along a path that connects two states, i.e., following the flow of work. Specifically, we focus on decomposing energy or free energy differences along representative pathways such as minimum energy paths (MEPs) and minimum free energy paths (MFEPs), and a numerical metric is developed to quantify the required accuracy of the reaction path. A statistical mechanical analysis of energy decomposition is also presented to illustrate the generality of this approach. Decomposition along MEP and MFEP is demonstrated on two test cases to illustrate the ability to derive quantitative mechanistic information for different types of activated processes. First, the MEP of alanine dipeptide isomerization in vacuum and the MFEP of isomerization in explicit water is studied. Our analysis shows that carbonyl oxygen and amide hydrogen contribute to most of the energetic cost for isomerization and that explicit water solvation modulates the free energy landscape primarily through hydrogen bonding with these atoms. The second test case concerns the formation of tetrahedral intermediate during a transesterification reaction. Decomposition analysis shows that water molecules not only have strong stabilization effects on the tetrahedral intermediate but also constitute a sizable potential energy barrier due to their significant structural rearrangement during the reaction. We expect that the proposed method can be generally applied to develop mechanistic understanding of catalytic and biocatalytic processes and provide useful insight for strategies of molecular engineering.

"Ten blue links" have defined web search results for the last fifteen years -snippets of text combined with document titles and URLs. In this paper, we establish the notion of enhanced search results that extend web search results to include multimedia objects such as images and video, intentspecific key value pairs, and elements that allow the user to interact with the contents of a web page directly from the search results page. We show that users express a preference for enhanced results both explicitly, and when observed in their search behavior. We also demonstrate the effectiveness of enhanced results in helping users to assess the relevance of search results. Lastly, we show that we can efficiently generate enhanced results to cover a significant fraction of search result pages.

We propose to quantify the trajectory entropy of a dynamic system as the information content in excess of a free-diffusion reference model. The space-time trajectory is now the dynamic variable, and its path probability is given by the Onsager-Machlup action. For the time propagation of the overdamped Langevin equation, we solved the action path integral in the continuum limit and arrived at an exact analytical expression that emerged as a simple functional of the deterministic mean force and the stochastic diffusion. This work may have direct implications in chemical and phase equilibria, bond isomerization, and conformational changes in biological macromolecules as well transport problems in general.