Kinetic distance and kinetic maps from molecular dynamics simulation

Characterizing macromolecular kinetics from molecular dynamics (MD) simulations requires a distance metric that can distinguish slowly-interconverting states. Here we build upon diffusion map theory and define a kinetic distance for irreducible Markov processes that quantifies how slowly molecular conformations interconvert. The kinetic distance can be computed given a model that approximates the eigenvalues and eigenvectors (reaction coordinates) of the MD Markov operator. Here we employ the time-lagged independent component analysis (TICA). The TICA components can be scaled to provide a kinetic map in which the Euclidean distance corresponds to the kinetic distance. As a result, the question of how many TICA dimensions should be kept in a dimensionality reduction approach becomes obsolete, and one parameter less needs to be specified in the kinetic model construction. We demonstrate the approach using TICA and Markov state model (MSM) analyses for illustrative models, protein conformation dynamics in bovine pancreatic trypsin inhibitor and protein-inhibitor association in trypsin and benzamidine

Coarse-graining the Dynamics of a Driven Interface in the Presence of Mobile Impurities: Effective Description via Diffusion Maps

Developing effective descriptions of the microscopic dynamics of many physical phenomena can both dramatically enhance their computational exploration and lead to a more fundamental understanding of the underlying physics. Previously, an effective description of a driven interface in the presence of mobile impurities, based on an Ising variant model and a single empirical coarse variable, was partially successful; yet it underlined the necessity of selecting additional coarse variables in certain parameter regimes. In this paper we use a data mining approach to help identify the coarse variables required. We discuss the implementation of this diffusion map approach, the selection of a similarity measure between system snapshots required in the approach, and the correspondence between empirically selected and automatically detected coarse variables. We conclude by illustrating the use of the diffusion map variables in assisting the atomistic simulations, and we discuss the translation of information between fine and coarse descriptions using lifting and restriction operators.Comment: 28 pages, 10 figure

Hyperspaces with exactly two orbits

Let C(X) be the hyperspace of all subcontinua of a (metric) continuum X. It is known that C(X) is homogeneous if and only if C(X) is the Hilbert cube. We are interested in knowing when C(X) is 1/2-homogeneous, meaning that there are exactly two orbits for the action of the group of homeomorphisms of C(X) onto C(X). It is shown that if X is a locally connected continuum or a nondegenerate atriodic continuum, and if C(X) is 1/2-homogeneous, then X is an arc or a simple closed curve. We do not know whether an arc and a simple closed curve are the only continua X for which C(X) is 1/2-homogeneous

Langevin Trajectories between Fixed Concentrations

We consider the trajectories of particles diffusing between two infinite baths of fixed concentrations connected by a channel, e.g. a protein channel of a biological membrane. The steady state influx and efflux of Langevin trajectories at the boundaries of a finite volume containing the channel and parts of the two baths is replicated by termination of outgoing trajectories and injection according to a residual phase space density. We present a simulation scheme that maintains averaged fixed concentrations without creating spurious boundary layers, consistent with the assumed physics

Ions in Fluctuating Channels: Transistors Alive

Ion channels are proteins with a hole down the middle embedded in cell membranes. Membranes form insulating structures and the channels through them allow and control the movement of charged particles, spherical ions, mostly Na+, K+, Ca++, and Cl-. Membranes contain hundreds or thousands of types of channels, fluctuating between open conducting, and closed insulating states. Channels control an enormous range of biological function by opening and closing in response to specific stimuli using mechanisms that are not yet understood in physical language. Open channels conduct current of charged particles following laws of Brownian movement of charged spheres rather like the laws of electrodiffusion of quasi-particles in semiconductors. Open channels select between similar ions using a combination of electrostatic and 'crowded charge' (Lennard-Jones) forces. The specific location of atoms and the exact atomic structure of the channel protein seems much less important than certain properties of the structure, namely the volume accessible to ions and the effective density of fixed and polarization charge. There is no sign of other chemical effects like delocalization of electron orbitals between ions and the channel protein. Channels play a role in biology as important as transistors in computers, and they use rather similar physics to perform part of that role. Understanding their fluctuations awaits physical insight into the source of the variance and mathematical analysis of the coupling of the fluctuations to the other components and forces of the system.Comment: Revised version of earlier submission, as invited, refereed, and published by journa

A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n be the Lie algebra of the compact dual (real) torus T_\bR^\vee\cong U(1)^n. Then there is a corresponding conical Lagrangian \Lambda \subset T^*M_\bR and an equivalence of triangulated dg categories \Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda), where \Perf_T(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and \Sh_{cc}(M_\bR;\Lambda) is the triangulated dg category of complex of sheaves on M_\bR with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on M_\bR.Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v

Assessing Operational Total Lightning Visualization Products

In May 2003, NASA's Short-term Prediction Research and Transition (SPoRT) program successfully provided total lightning data from the North Alabama Lightning Mapping Array (NALMA) to the National Weather Service (NWS) office in Huntsville, Alabama. The major accomplishment was providing the observations in real-time to the NWS in the native Advanced Weather Interactive Processing System (AWIPS) decision support system. Within days, the NALMA data were used to issue a tornado warning initiating seven years of ongoing support to the NWS' severe weather and situational awareness operations. With this success, SPoRT now provides real-time NALMA data to five forecast offices as well as working to transition data from total lightning networks at Kennedy Space Center and the White Sands Missile Range to the surrounding NWS offices. The only NALMA product that has been transitioned to SPoRT's partner NWS offices is the source density product, available at a 2 km resolution in 2 min intervals. However, discussions with users of total lightning data from other networks have shown that other products are available, ranging from spatial and temporal variations of the source density product to the creation of a flash extent density. SPoRT and the Huntsville, Alabama NWS are evaluating the utility of these variations as this has not been addressed since the initial transition in 2003. This preliminary analysis will focus on what products will best support the operational warning decision process. Data from 19 April 2009 are analyzed. On this day, severe thunderstorms formed ahead of an approaching cold front. Widespread severe weather was observed, primarily south of the Tennessee River with multiple, weak tornadoes, numerous severe hail reports, and wind. This preliminary analysis is the first step in evaluation which product(s) are best suited for operations. The ultimate goal is selecting a single product for use with all total lightning networks to streamline training and science sharing

Coarse-grained dynamics of an activity bump in a neural field model

We study a stochastic nonlocal PDE, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially-localized ``activity bumps''. This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we revisit the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar "coarse" variable. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately-initialized bursts of direct simulation. We demonstrate this approach in terms of (a) an experience-based "intelligent" choice of the coarse observable and (b) an observable obtained through data-mining direct simulation results, using a diffusion map approach.Comment: Corrected aknowledgement

On central tendency and dispersion measures for intervals and hypercubes

The uncertainty or the variability of the data may be treated by considering, rather than a single value for each data, the interval of values in which it may fall. This paper studies the derivation of basic description statistics for interval-valued datasets. We propose a geometrical approach in the determination of summary statistics (central tendency and dispersion measures) for interval-valued variables