Complex temporal patterns in molecular dynamics: A direct measure of the phase-space exploration by the trajectory at macroscopic time scales (original) (raw)
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Quantifying long time memory in phase space trajectories of molecular liquids
Journal of Molecular Liquids, 2011
A trajectory of liquid water simulated using classical molecular dynamics has been analysed in the framework of symbolic dynamics. The behaviour of symbolic subsequences (words) of nine symbols long has been studied at a very long time of 1 μs. Contrary to naive expectations, the molecular trajectory behaves very differently compared to both a random signal and a random surrogate with spectral properties identical to the molecular trajectory. The molecular system characteristics resemble those of a chaotic map, the Standard map. We conclude that the most probable reason for deviations from randomness in the molecular system is its deterministic dynamics, in particular, the stickiness of periodic islands in the bulk of chaotic motion.
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A framework that connects computational mechanics and molecular dynamics has been developed and described. As the key parts of the framework, the problem of symbolising molecular trajectory and the associated interrelation between microscopic phase space variables and macroscopic observables of the molecular system are considered. Following Shalizi and Moore, it is shown that causal states, the constituent parts of the main construct of computational mechanics, the -machine, define areas of the phase space that are optimal in the sense of transferring information from the micro-variables to the macro-observables. We have demonstrated that, based on the decay of their Poincaré return times, these areas can be divided into two classes that characterise the separation of the phase space into resonant and chaotic areas. The first class is characterised by predominantly short time returns, typical to quasi-periodic or periodic trajectories. This class includes a countable number of areas corresponding to resonances. The second class includes trajectories with chaotic behaviour characterised by the exponential decay of return times in accordance with the Poincaré theorem.
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The simulated classical dynamics of a small molecule exhibiting self-organizing behavior via a fast transition between two states is analyzed by calculation of the statistical complexity of the system. It is shown that the complexity of molecular descriptors such as atom coordinates and dihedral angles have different values before and after the transition. This provides a new tool to identify metastable states during molecular self-organization. The highly concerted collective motion of the molecule is revealed. Low-dimensional subspaces dynamics is found sensitive to the processes in the whole, high-dimensional phase space of the system.
Dynamical frustration of protein's environment at the nanoseconds time scale
Journal of Molecular Liquids, 2009
A 21-residue peptide in explicit water has been simulated using classical molecular dynamics. The system's trajectory has been analysed with a novel approach that quantifies the process of how atom's environment trajectories are explored. The approach is based on the measure of Statistical Complexity that extracts complete dynamical information from the signal. The introduced characteristic quantifies the system's dynamics at the nanoseconds time scale. It has been found that the peptide exhibits nanoseconds long periods that significantly differ in the rates of the exploration of the dynamically allowed configurations of the environment. During these periods the rates remain the same but different from other periods and from the rate for water. Periods of dynamical frustration are detected when only limited routes in the space of possible trajectories of the surrounding atoms are realised.
Physical Review E, 2007
The present work examines the applicability and efficacy of recurrence plots and recurrence quantification analysis in interpreting statistical-mechanics-based simulations of classical fluids and solids. We analyze temperature time series obtained from molecular dynamics simulations of a Lennard-Jones system at various fluid and solid states. It turns out that the structure of the recurrence plots reflects the different regimes of atomic motion as well as the degree of atomic diffusivity as the system density and temperature are varied. Recurrence plots ͑RPs͒ can help to localize a region where a phase transition occurs, while recurrence quantitative analysis descriptors confirm in a more clear way the results of RPs. The trends identified in our results are in qualitative agreement with direct computation of Lyapunov exponents for liquid Lennard-Jones systems reported in the literature.
Statistical Complexity of Low- and High-Dimensional Systems
Journal of Atomic, Molecular, and Optical Physics, 2012
We suggest a new method for the analysis of experimental time series that can distinguish high dimensional dynamics from stochastic motion. It is based on the idea of statistical complexity, i.e. the Shannon entropy of the so-called -machine (a Markov-type model of the observed time series). This approach has been recently demonstrated to be efficient for making a distinction between a molecular trajectory in water and noise. In this paper we analyse the difference between chaos and noise using the Chirikov-Taylor Standard map as an example in order to elucidate the basic mechanism that makes the value of complexity in deterministic systems high. In particular, we show that the value of statistical complexity is high for the case of chaos, and attains zero value for the case of stochastic noise. We further study the Markov property of the data generated by the Standard map to clarify the role of long time memory in differentiating the cases of deterministic systems and stochastic motion.
Proteins: Structure, Function, and Genetics, 2002
Central to the study of a complex dynamical system is knowledge of its phase space behavior. Experimentally, it is rarely possible to record a system's (multidimensional) phase space variables. Rather, the system is observed via one (or few) scalar-valued signal(s) of emission or response. In dynamical systems analysis, the multidimensional phase space of a system can be reconstructed by manipulation of a one-dimensional signal. The trick is in the construction of a (higher-dimensional) space through the use of a time lag (or delay) on the signal time series. The trajectory in this embedding space can then be examined using phase portraits generated in selected subspaces. By contrast, in computer simulation, one has an embarrassment of riches: direct access to the complete multidimensional phase space variables, at arbitrary time resolution and precision. Here, the problem is one of reducing the dimensionality to make analysis tractable. This can be achieved through linear or nonlinear projection of the trajectory into subspaces containing high information content. This study considers trajectories of the small protein crambin from molecular dynamics simulations. The phase space behavior is examined using principal component analysis on the Cartesian coordinate covariance matrix of 138 dimensions. In addition, the phase space is reconstructed from a one dimensional signal, representing the radius of gyration of the structure along the trajectory. Comparison of low-dimensional phase portraits obtained from the two methods shows that the complete phase space distribution is well represented by the reconstruction. The study suggests that it may be possible to develop a deeper connection between the experimental and simulated dynamics of biomolecules via phase space reconstruction using data emerging from recent advances in singlemolecule time-resolved biophysical techniques.
Long Time Dynamics of Complex Systems
Accounts of Chemical Research, 2002
Molecular dynamics trajectories of large biological molecules are restricted to nanoseconds. We describe a computational method, based on optimization of a functional, to extend the time of molecular simulations by orders of magnitude. Variants of our technique have already produced microsecond and millisecond trajectories. The large steps enable feasible computations of atomically detailed approximate trajectories. Numerical examples are provided: (i) a conformational change in blocked glycine peptide and (ii) helix formation of an alanine-rich peptide.