Comparison of newtonian and special-relativistic trajectories with the general-relativistic trajectory for a low-speed weak-gravity system

We show, contrary to expectation, that the trajectory predicted by general-relativistic mechanics for a low-speed weak-gravity system is not always well-approximated by the trajectories predicted by special-relativistic and newtonian mechanics for the same parameters and initial conditions. If the system is dissipative, the breakdown of agreement occurs for chaotic trajectories only. If the system is non-dissipative, the breakdown of agreement occurs for chaotic trajectories and non-chaotic trajectories. The agreement breaks down slowly for non-chaotic trajectories but rapidly for chaotic trajectories. When the predictions are different, general-relativistic mechanics must therefore be used, instead of special-relativistic mechanics (newtonian mechanics), to correctly study the dynamics of a weak-gravity system (a low-speed weak-gravity system).     

Newtonian versus Special-Relativistic Statistical Predictions for Low-Speed Scattering

The statistical predictions of Newtonian and special-relativistic mechanics, which are calculated from an initially Gaussian ensemble of trajectories, are compared for a low-speed scattering system. The comparisons are focused on the mean dwell time, transmission and reflection coefficients, and the position and momentum means and standard deviations. We find that the statistical predictions of the two theories do not always agree as conventionally expected. The predictions are close if the scattering is non-chaotic but they are radically different if the scattering is chaotic and the initial ensemble is well localized in phase space. Our result indicates that for low-speed chaotic scattering, special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to obtain empirically-correct statistical predictions from an initially well-localized Gaussian ensemble.     

Stretching a macromolecule in an atomic force microscope: statistical mechanical analysis

We formulate the proper statistical mechanics to describe the stretching of a macromolecule under a force provided by the cantilever of an Atomic Force Microscope. In the limit of a soft cantilever, the generalized ensemble of the coupled molecule-cantilever system reduces to the Gibbs ensemble for an isolated molecule subject to a constant force in which the extension is fluctuating. For a stiff cantilever, one obtains the Helmholtz ensemble for an isolated molecule held at a fixed extension with the force fluctuating. Numerical examples and predictions for experiments with cantilevers of differing stiffness are given for short and long chains of poly (ethylene glycol), based on parameter-free ab initio calculations.     

The Path Integral Formulation of Climate Dynamics

The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent) attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model.     

Unbiased Rare Event Sampling in Spatial Stochastic Systems Biology Models Using a Weighted Ensemble of Trajectories

The long-term goal of connecting scales in biological simulation can be facilitated by scale-agnostic methods. We demonstrate that the weighted ensemble (WE) strategy, initially developed for molecular simulations, applies effectively to spatially resolved cell-scale simulations. The WE approach runs an ensemble of parallel trajectories with assigned weights and uses a statistical resampling strategy of replicating and pruning trajectories to focus computational effort on difficult-to-sample regions. The method can also generate unbiased estimates of non-equilibrium and equilibrium observables, sometimes with significantly less aggregate computing time than would be possible using standard parallelization. Here, we use WE to orchestrate particle-based kinetic Monte Carlo simulations, which include spatial geometry (e.g., of organelles, plasma membrane) and biochemical interactions among mobile molecular species. We study a series of models exhibiting spatial, temporal and biochemical complexity and show that although WE has important limitations, it can achieve performance significantly exceeding standard parallel simulation—by orders of magnitude for some observables.     

Efficient sampling in collective coordinate space

Collective motions in biological macromolecules have been shown to be important for function. The most important collective motions occur on slow time scales, which poses a sampling problem in dynamic simulation of biomolecules. We present a novel method for efficient conformational sampling. The method combines the simulation of an ensemble of concurrent trajectories with restraints acting on the ensemble of structures as a whole. Two properties of the ensemble may be restrained: (i) the variance of the ensemble and (ii) the average position of the ensemble. Both properties are defined in a subspace of collective coordinate space spanned by an arbitrary number of modes. We show that weak restraints on the ensemble variance suffice for an increase in sampling efficiency along soft modes by two orders of magnitudes. The resulting trajectories exhibit virtually the same structural quality as trajectories generated by restraint-free-molecular dynamics simulation, as judged by standard structure validation tools. The method is used to probe the resistance of a structure against conformational changes along collective modes and clearly distinguishes soft from stiff modes. Further applications are discussed. Proteins 2000;39:82-88.     

Abstracts: Albany 2007, The 15th Conversation

Abstract

Forty nine molecular dynamics simulations of unfolding trajectories of the segment B1 of streptococcal protein G (GB1) provide a direct demonstration of the diversity of unfolding pathway and give a statistically utmost unfolding pathway under the physical property space. Twelve physical properties of the protein were chosen to construct a 12-dimensional property space. Then the 12-dimentional property space was reduced to a 3-dimentional principle component property space. Under the property space, the multiple unfolding trajectories look like “trees”, which have some common characters. The “root of the tree” corresponds to the native state, the “bole” homologizes the partially unfolded conformations, and the “crown” is in correspondence to the unfolded state. These unfolding trajectories can be divided into three types. The first one has the characters of straight “bole” and “crown” corresponding to a fast two-state unfolding pathway of GB1. The second one has the character of “the standstill in the middle tree bole”, which may correspond to a three-state unfolding pathway. The third one has the character of “the circuitous bole” corresponding to a slow two-state unfolding pathway. The fast two-state unfolding pathway is a statistically utmost unfolding pathway or preferred pathway of GB1, which occupies 53% of 49 unfolding trajectories. In the property space all the unfolding trajectories construct a thermal unfolding pathway ensemble of GB1. The unfolding pathway ensemble resembles a funnel that is gradually emanative from the native state ensemble to the unfolded state ensemble. In the property space, the thermal unfolded state distribution looks like electronic cloud in quantum mechanics. The unfolded states of the independent unfolding simulation trajectories have substantial overlaps, indicating that the thermal unfolded states are confined by the physical property values, and the number of protein unfolded state are much less than that was believed before.     

Study of multiple unfolding trajectories and unfolded states of the protein GB1 under the physical property space

Forty nine molecular dynamics simulations of unfolding trajectories of the segment B1 of streptococcal protein G (GB1) provide a direct demonstration of the diversity of unfolding pathway and give a statistically utmost unfolding pathway under the physical property space. Twelve physical properties of the protein were chosen to construct a 12-dimensional property space. Then the 12-dimensional property space was reduced to a 3-dimensional principle component property space. Under the property space, the multiple unfolding trajectories look like "trees", which have some common characters. The "root of the tree" corresponds to the native state, the "bole" homologizes the partially unfolded conformations, and the "crown" is in correspondence to the unfolded state. These unfolding trajectories can be divided into three types. The first one has the characters of straight "bole" and "crown" corresponding to a fast two-state unfolding pathway of GB1. The second one has the character of "the standstill in the middle tree bole", which may correspond to a three-state unfolding pathway. The third one has the character of "the circuitous bole" corresponding to a slow two-state unfolding pathway. The fast two-state unfolding pathway is a statistically utmost unfolding pathway or preferred pathway of GB1, which occupies 53% of 49 unfolding trajectories. In the property space all the unfolding trajectories construct a thermal unfolding pathway ensemble of GB1. The unfolding pathway ensemble resembles a funnel that is gradually emanative from the native state ensemble to the unfolded state ensemble. In the property space, the thermal unfolded state distribution looks like electronic cloud in quantum mechanics. The unfolded states of the independent unfolding simulation trajectories have substantial overlaps, indicating that the thermal unfolded states are confined by the physical property values, and the number of protein unfolded state are much less than that was believed before.     

Ensemble MD simulations restrained via crystallographic data: Accurate structure leads to accurate dynamics

Currently, the best existing molecular dynamics (MD) force fields cannot accurately reproduce the global free‐energy minimum which realizes the experimental protein structure. As a result, long MD trajectories tend to drift away from the starting coordinates (e.g., crystallographic structures). To address this problem, we have devised a new simulation strategy aimed at protein crystals. An MD simulation of protein crystal is essentially an ensemble simulation involving multiple protein molecules in a crystal unit cell (or a block of unit cells). To ensure that average protein coordinates remain correct during the simulation, we introduced crystallography‐based restraints into the MD protocol. Because these restraints are aimed at the ensemble‐average structure, they have only minimal impact on conformational dynamics of the individual protein molecules. So long as the average structure remains reasonable, the proteins move in a native‐like fashion as dictated by the original force field. To validate this approach, we have used the data from solid‐state NMR spectroscopy, which is the orthogonal experimental technique uniquely sensitive to protein local dynamics. The new method has been tested on the well‐established model protein, ubiquitin. The ensemble‐restrained MD simulations produced lower crystallographic R factors than conventional simulations; they also led to more accurate predictions for crystallographic temperature factors, solid‐state chemical shifts, and backbone order parameters. The predictions for ¹⁵N relaxation rates are at least as accurate as those obtained from conventional simulations. Taken together, these results suggest that the presented trajectories may be among the most realistic protein MD simulations ever reported. In this context, the ensemble restraints based on high‐resolution crystallographic data can be viewed as protein‐specific empirical corrections to the standard force fields.     

The homeostatic ensemble for cells

Cells are quintessential examples of out-of-equilibrium systems, but they maintain a homeostatic state over a timescale of hours to days. As a consequence, the statistics of all observables is remarkably consistent. Here, we develop a statistical mechanics framework for living cells by including the homeostatic constraint that exists over the interphase period of the cell cycle. The consequence is the introduction of the concept of a homeostatic ensemble and an associated homeostatic temperature, along with a formalism for the (dynamic) homeostatic equilibrium that intervenes to allow living cells to evade thermodynamic decay. As a first application, the framework is shown to accurately predict the observed effect of the mechanical environment on the in vitro response of smooth muscle cells. This includes predictions that both the mean values and diversity/variability in the measured values of observables such as cell area, shape and tractions decrease with decreasing stiffness of the environment. Thus, we argue that the observed variabilities are inherent to the entropic nature of the homeostatic equilibrium of cells and not a result of in vitro experimental errors.     

Molecular dynamics-solvated interaction energy studies of protein-protein interactions: the MP1-p14 scaffolding complex

Using the MP1-p14 scaffolding complex from the mitogen-activated protein kinase signaling pathway as model system, we explored a structure-based computational protocol to probe and characterize binding affinity hot spots at protein-protein interfaces. Hot spots are located by virtual alanine-scanning consensus predictions over three different energy functions and two different single-structure representations of the complex. Refined binding affinity predictions for select hot-spot mutations are carried out by applying first-principle methods such as the molecular mechanics generalized Born surface area (MM-GBSA) and solvated interaction energy (SIE) to the molecular dynamics (MD) trajectories for mutated and wild-type complexes. Here, predicted hot-spot residues were actually mutated to alanine, and crystal structures of the mutated complexes were determined. Two mutated MP1-p14 complexes were investigated, the p14(Y56A)-mutated complex and the MP1(L63A,L65A)-mutated complex. Alternative ways to generate MD ensembles for mutant complexes, not relying on crystal structures for mutated complexes, were also investigated. The SIE function, fitted on protein-ligand binding affinities, gave absolute binding affinity predictions in excellent agreement with experiment and outperformed standard MM-GBSA predictions when tested on the MD ensembles of Ras-Raf and Ras-RalGDS protein-protein complexes. For wild-type and mutant MP1-p14 complexes, SIE predictions of relative binding affinities were supported by a yeast two-hybrid assay that provided semiquantitative relative interaction strengths. Results on the MP1-mutated complex suggested that SIE predictions deteriorate if mutant MD ensembles are approximated by just mutating the wild-type MD trajectory. The SIE data on the p14-mutated complex indicated feasibility for generating mutant MD ensembles from mutated wild-type crystal structure, despite local structural differences observed upon mutation. For energetic considerations, this would circumvent costly needs to produce and crystallize mutated complexes. The sensitized protein-protein interface afforded by the p14(Y56A) mutation identified here has practical applications in screening-based discovery of first-generation small-molecule hits for further development into specific modulators of the mitogen-activated protein kinase signaling pathway.     

Conformational energy analysis of melanostatin

Conformational energy calculations were carried out on the hypothalamic hormone melanostatin, a tripeptide with the primary structure H-L-Pro-L-Leu-Gly-NH2. The calculated lowest energy conformation was a type II beta bend, very similar to that reported in an X-ray crystal study. This conformation, however, was only one of 109 low-energy structures (less than or equal to 3 kcal/mol above the global minimum), indicating that the molecule in solution exists as an ensemble of conformations and is very flexible, in agreement with relaxation data from n.m.r. measurements. A statistical analysis yielded an average end-to-end distance of 6.8 A and a bend probability of 0.62, suggesting that, in nonpolar solvents, bend structures predominate within the statistical ensemble. The statistical analysis, however, also yielded a probability of only 0.11 for the occurrence of a 4 leads to 1 hydrogen bond. Hence, the calculations show that, although bend conformations predominate, bends would be difficult to observe in solution if the experiments were designed only to detect 4 leads to 1 hydrogen bonds.     

Coarse-Grained Prediction of RNA Loop Structures

One of the key issues in the theoretical prediction of RNA folding is the prediction of loop structure from the sequence. RNA loop free energies are dependent on the loop sequence content. However, most current models account only for the loop length-dependence. The previously developed “Vfold” model (a coarse-grained RNA folding model) provides an effective method to generate the complete ensemble of coarse-grained RNA loop and junction conformations. However, due to the lack of sequence-dependent scoring parameters, the method is unable to identify the native and near-native structures from the sequence. In this study, using a previously developed iterative method for extracting the knowledge-based potential parameters from the known structures, we derive a set of dinucleotide-based statistical potentials for RNA loops and junctions. A unique advantage of the approach is its ability to go beyond the the (known) native structures by accounting for the full free energy landscape, including all the nonnative folds. The benchmark tests indicate that for given loop/junction sequences, the statistical potentials enable successful predictions for the coarse-grained 3D structures from the complete conformational ensemble generated by the Vfold model. The predicted coarse-grained structures can provide useful initial folds for further detailed structural refinement.     

Statistical decision theory and trade-offs in the control of motor response

We present a novel approach to the modeling of motor responses based on statistical decision theory. We begin with the hypothesis that subjects are ideal motion planners who choose movement trajectories to minimize expected loss. We derive predictions of the hypothesis for movement in environments where contact with specified regions carries rewards or penalties. The model predicts shifts in a subject's aiming point in response to changes in the reward and penalty structure of the environment and with changes in the subject's uncertainty in carrying out planned movements. We tested some of these predictions in an experiment where subjects were rewarded if they succeeded in touching a target region on a computer screen within a specified time limit. Near the target was a penalty region which, if touched, resulted in a penalty. We varied distance between the penalty region and the target and the cost of hitting the penalty region. Subjects shift their mean points of contact with the computer screen in response to changes in penalties and location of the penalty region relative to the target region in qualitative agreement with the predictions of the hypothesis. Thus, movement planning takes into account extrinsic costs and the subject's own motor uncertainty.     

Structural forecasting of species persistence under changing environments

The persistence of a species in a given place not only depends on its intrinsic capacity to consume and transform resources into offspring, but also on how changing environmental conditions affect its growth rate. However, the complexity of factors has typically taken us to choose between understanding and predicting the persistence of species. To tackle this limitation, we propose a probabilistic approach rooted on the statistical concepts of ensemble theory applied to statistical mechanics and on the mathematical concepts of structural stability applied to population dynamics models – what we call structural forecasting. We show how this new approach allows us to estimate a probability of persistence for single species in local communities; to understand and interpret this probability conditional on the information we have concerning a system; and to provide out‐of‐sample predictions of species persistence as good as the best experimental approaches without the need of extensive amounts of data.     

Ability of static and statistical mechanics posturographic measures to distinguish between age and fall risk

Traditional posturographic analysis and four statistical mechanics techniques were applied to center-of-pressure (COP) trajectories of young, older "low-fall-risk" and older "high-fall-risk" individuals. Low-fall-risk older adults were active 3 days per week in a cardiac rehabilitation program, while high-fall-risk older adults were diagnosed with perilymph fistula. Subjects diagnosed with perilymph fistula must have experienced two of the following vestibular findings: constant disequilibrium, positional vertigo and/or a positive fistula test. Non-parametric statistical tests were used to determine whether the posturographic measures could detect differences between the young and older "low-fall-risk" groups (age comparison) and between the older "low-" and "high-risk" groups (risk of falling comparison). The statistical mechanics techniques were more sensitive than the traditional measures: detecting significant differences between the young and older "low-risk" groups, while none of the traditional measures were significantly different. In addition, interpretation of the statistical mechanics techniques may offer more insight into the nature of the process controlling the COP trajectories. However, the methods offered slightly different explanations. For instance, the Hurst rescaled range analysis suggests that the movement of the COP is governed solely by anti-persistent behavior, whereas the stabilogram diffusion analysis suggests a short-term persistence balanced by a long-term anti-persistence. These discrepancies highlight the need for a model that incorporates the biological systems responsible for maintaining balance and experimental methods to directly quantify their status and roles. Until such a model exists, however, the statistical mechanics techniques appear to have some advantages over traditional posturographic measures for studying balance control.     

Bayesian Parameter Inference and Model Selection by Population Annealing in Systems Biology

Parameter inference and model selection are very important for mathematical modeling in systems biology. Bayesian statistics can be used to conduct both parameter inference and model selection. Especially, the framework named approximate Bayesian computation is often used for parameter inference and model selection in systems biology. However, Monte Carlo methods needs to be used to compute Bayesian posterior distributions. In addition, the posterior distributions of parameters are sometimes almost uniform or very similar to their prior distributions. In such cases, it is difficult to choose one specific value of parameter with high credibility as the representative value of the distribution. To overcome the problems, we introduced one of the population Monte Carlo algorithms, population annealing. Although population annealing is usually used in statistical mechanics, we showed that population annealing can be used to compute Bayesian posterior distributions in the approximate Bayesian computation framework. To deal with un-identifiability of the representative values of parameters, we proposed to run the simulations with the parameter ensemble sampled from the posterior distribution, named “posterior parameter ensemble”. We showed that population annealing is an efficient and convenient algorithm to generate posterior parameter ensemble. We also showed that the simulations with the posterior parameter ensemble can, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference. Lastly, we introduced the marginal likelihood in the approximate Bayesian computation framework for Bayesian model selection. We showed that population annealing enables us to compute the marginal likelihood in the approximate Bayesian computation framework and conduct model selection depending on the Bayes factor.     

Conformational Recognition of an Intrinsically Disordered Protein

There is a growing interest in understanding the properties of intrinsically disordered proteins (IDPs); however, the characterization of these states remains an open challenge. IDPs appear to have functional roles that diverge from those of folded proteins and revolve around their ability to act as hubs for protein-protein interactions. To gain a better understanding of the modes of binding of IDPs, we combined statistical mechanics, calorimetry, and NMR spectroscopy to investigate the recognition and binding of a fragment from the disordered protein Gab2 by the growth factor receptor-bound protein 2 (Grb2), a key interaction for normal cell signaling and cancer development. Structural ensemble refinement by NMR chemical shifts, thermodynamics measurements, and analysis of point mutations indicated that the population of preexisting bound conformations in the free-state ensemble of Gab2 is an essential determinant for recognition and binding by Grb2. A key role was found for transient polyproline II (PPII) structures and extended conformations. Our findings are likely to have very general implications for the biological behavior of IDPs in light of the evidence that a large fraction of these proteins possess a specific propensity to form PPII and to adopt conformations that are more extended than the typical random-coil states.     

The Coalescent Process in Models with Selection and Recombination

The statistical properties of the process describing the genealogical history of a random sample of genes at a selectively neutral locus which is linked to a locus at which natural selection operates are investigated. It is found that the equations describing this process are simple modifications of the equations describing the process assuming that the two loci are completely linked. Thus, the statistical properties of the genealogical process for a random sample at a neutral locus linked to a locus with selection follow from the results obtained for the selected locus. Sequence data from the alcohol dehydrogenase (Adh) region of Drosophila melanogaster are examined and compared to predictions based on the theory. It is found that the spatial distribution of nucleotide differences between Fast and Slow alleles of Adh is very similar to the spatial distribution predicted if balancing selection operates to maintain the allozyme variation at the Adh locus. The spatial distribution of nucleotide differences between different Slow alleles of Adh do not match the predictions of this simple model very well.