Action-derived molecular dynamics in the study of rare events

We present a practical method to generate classical trajectories with fixed initial and final boundary conditions. Our method is based on the minimization of a suitably defined discretized action. The method finds its most natural application in the study of rare events. Its capabilities are illustrated by non-trivial examples. The algorithm lends itself to straightforward parallelization, and when combined with molecular dynamics (MD) it promises to offer a powerful tool for the study of chemical reactions.Comment: 7 Pages, 4 Figures (3 in color), submitted to Phys. Rev. Let

Efficient Dynamic Importance Sampling of Rare Events in One Dimension

Exploiting stochastic path integral theory, we obtain \emph{by simulation} substantial gains in efficiency for the computation of reaction rates in one-dimensional, bistable, overdamped stochastic systems. Using a well-defined measure of efficiency, we compare implementations of ``Dynamic Importance Sampling'' (DIMS) methods to unbiased simulation. The best DIMS algorithms are shown to increase efficiency by factors of approximately 20 for a $5 k_B T$ barrier height and 300 for $9 k_B T$, compared to unbiased simulation. The gains result from close emulation of natural (unbiased), instanton-like crossing events with artificially decreased waiting times between events that are corrected for in rate calculations. The artificial crossing events are generated using the closed-form solution to the most probable crossing event described by the Onsager-Machlup action. While the best biasing methods require the second derivative of the potential (resulting from the ``Jacobian'' term in the action, which is discussed at length), algorithms employing solely the first derivative do nearly as well. We discuss the importance of one-dimensional models to larger systems, and suggest extensions to higher-dimensional systems.Comment: version to be published in Phys. Rev.

Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.Comment: 33 pages, 17 figure

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Deep Sequencing the Transcriptome Reveals Seasonal Adaptive Mechanisms in a Hibernating Mammal

Mammalian hibernation is a complex phenotype involving metabolic rate reduction, bradycardia, profound hypothermia, and a reliance on stored fat that allows the animal to survive for months without food in a state of suspended animation. To determine the genes responsible for this phenotype in the thirteen-lined ground squirrel (Ictidomys tridecemlineatus) we used the Roche 454 platform to sequence mRNA isolated at six points throughout the year from three key tissues: heart, skeletal muscle, and white adipose tissue (WAT). Deep sequencing generated approximately 3.7 million cDNA reads from 18 samples (6 time points ×3 tissues) with a mean read length of 335 bases. Of these, 3,125,337 reads were assembled into 140,703 contigs. Approximately 90% of all sequences were matched to proteins in the human UniProt database. The total number of distinct human proteins matched by ground squirrel transcripts was 13,637 for heart, 12,496 for skeletal muscle, and 14,351 for WAT. Extensive mitochondrial RNA sequences enabled a novel approach of using the transcriptome to construct the complete mitochondrial genome for I. tridecemlineatus. Seasonal and activity-specific changes in mRNA levels that met our stringent false discovery rate cutoff (1.0×10−11) were used to identify patterns of gene expression involving various aspects of the hibernation phenotype. Among these patterns are differentially expressed genes encoding heart proteins AT1A1, NAC1 and RYR2 controlling ion transport required for contraction and relaxation at low body temperatures. Abundant RNAs in skeletal muscle coding ubiquitin pathway proteins ASB2, UBC and DDB1 peak in October, suggesting an increase in muscle proteolysis. Finally, genes in WAT that encode proteins involved in lipogenesis (ACOD, FABP4) are highly expressed in August, but gradually decline in expression during the seasonal transition to lipolysis