Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways

Diverse classes of proteins function through large-scale conformational changes; sophisticated enhanced sampling methods have been proposed to generate these macromolecular transition paths. As such paths are curves in a high-dimensional space, they have been difficult to compare quantitatively, a prerequisite to, for instance, assess the quality of different sampling algorithms. The Path Similarity Analysis (PSA) approach alleviates these difficulties by utilizing the full information in 3N-dimensional trajectories in configuration space. PSA employs the Hausdorff or Fr\'echet path metrics---adopted from computational geometry---enabling us to quantify path (dis)similarity, while the new concept of a Hausdorff-pair map permits the extraction of atomic-scale determinants responsible for path differences. Combined with clustering techniques, PSA facilitates the comparison of many paths, including collections of transition ensembles. We use the closed-to-open transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for the assessment enhanced sampling algorithms---to examine multiple microsecond equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free form alongside transition ensembles from the MD-based dynamic importance sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for instance, that differences in DIMS-MD and FRODA paths were mediated by a set of conserved salt bridges whose charge-charge interactions are fully modeled in DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis methods relying on pre-defined collective variables, such as native contacts or geometric quantities, can be used synergistically with PSA, as well as the application of PSA to more complex systems such as membrane transporter proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also available from journal site

Random Feature Models for Learning Interacting Dynamical Systems

Particle dynamics and multi-agent systems provide accurate dynamical models for studying and forecasting the behavior of complex interacting systems. They often take the form of a high-dimensional system of differential equations parameterized by an interaction kernel that models the underlying attractive or repulsive forces between agents. We consider the problem of constructing a data-based approximation of the interacting forces directly from noisy observations of the paths of the agents in time. The learned interaction kernels are then used to predict the agents behavior over a longer time interval. The approximation developed in this work uses a randomized feature algorithm and a sparse randomized feature approach. Sparsity-promoting regression provides a mechanism for pruning the randomly generated features which was observed to be beneficial when one has limited data, in particular, leading to less overfitting than other approaches. In addition, imposing sparsity reduces the kernel evaluation cost which significantly lowers the simulation cost for forecasting the multi-agent systems. Our method is applied to various examples, including first-order systems with homogeneous and heterogeneous interactions, second order homogeneous systems, and a new sheep swarming system

Firefly algorithm for polynomial Bézier surface parameterization

A classical issue in many applied fields is to obtain an approximating surface to a given set of data points. This problem arises in Computer-Aided Design and Manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy

Methods for Shape-Constrained Kernel Density Estimation

Nonparametric density estimators are used to estimate an unknown probability density while making minimal assumptions about its functional form. Although the low reliance of nonparametric estimators on modelling assumptions is a benefit, their performance will be improved if auxiliary information about the density\u27s shape is incorporated into the estimate. Auxiliary information can take the form of shape constraints, such as unimodality or symmetry, that the estimate must satisfy. Finding the constrained estimate is usually a difficult optimization problem, however, and a consistent framework for finding estimates across a variety of problems is lacking. It is proposed to find shape-constrained density estimates by starting with a pilot estimate obtained by standard methods, and subsequently adjusting its shape until the constraints are satisfied. This strategy is part of a general approach, in which a constrained estimation problem is defined by an estimator, a method of shape adjustment, a constraint, and an objective function. Optimization methods are developed to suit this approach, with a focus on kernel density estimation under a variety of constraints. Two methods of shape adjustment are examined in detail. The first is data sharpening, for which two optimization algorithms are proposed: a greedy algorithm that runs quickly but can handle a limited set of constraints, and a particle swarm algorithm that is suitable for a wider range of problems. The second is the method of adjustment curves, for which it is often possible to use quadratic programming to find optimal estimates. The methods presented here can be used for univariate or higher-dimensional kernel density estimation with shape constraints. They can also be extended to other estimators, in both the density estimation and regression settings. As such they constitute a step toward a truly general optimizer, that can be used on arbitrary combinations of estimator and constraint

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

abstract: Diverse classes of proteins function through large-scale conformational changes and various sophisticated computational algorithms have been proposed to enhance sampling of these macromolecular transition paths. Because such paths are curves in a high-dimensional space, it has been difficult to quantitatively compare multiple paths, a necessary prerequisite to, for instance, assess the quality of different algorithms. We introduce a method named Path Similarity Analysis (PSA) that enables us to quantify the similarity between two arbitrary paths and extract the atomic-scale determinants responsible for their differences. PSA utilizes the full information available in 3N-dimensional configuration space trajectories by employing the Hausdorff or Fréchet metrics (adopted from computational geometry) to quantify the degree of similarity between piecewise-linear curves. It thus completely avoids relying on projections into low dimensional spaces, as used in traditional approaches. To elucidate the principles of PSA, we quantified the effect of path roughness induced by thermal fluctuations using a toy model system. Using, as an example, the closed-to-open transitions of the enzyme adenylate kinase (AdK) in its substrate-free form, we compared a range of protein transition path-generating algorithms. Molecular dynamics-based dynamic importance sampling (DIMS) MD and targeted MD (TMD) and the purely geometric FRODA (Framework Rigidity Optimized Dynamics Algorithm) were tested along with seven other methods publicly available on servers, including several based on the popular elastic network model (ENM). PSA with clustering revealed that paths produced by a given method are more similar to each other than to those from another method and, for instance, that the ENM-based methods produced relatively similar paths. PSA applied to ensembles of DIMS MD and FRODA trajectories of the conformational transition of diphtheria toxin, a particularly challenging example, showed that the geometry-based FRODA occasionally sampled the pathway space of force field-based DIMS MD. For the AdK transition, the new concept of a Hausdorff-pair map enabled us to extract the molecular structural determinants responsible for differences in pathways, namely a set of conserved salt bridges whose charge-charge interactions are fully modelled in DIMS MD but not in FRODA. PSA has the potential to enhance our understanding of transition path sampling methods, validate them, and to provide a new approach to analyzing conformational transitions.The article is published at http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.100456

Nonparametric inference of interaction laws in systems of agents from trajectory data

Inferring the laws of interaction between particles and agents in complex dynamical systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach to estimate the governing laws of distance-based interactions, with no reference or assumption about their analytical form, from data consisting trajectories of interacting agents. We demonstrate the effectiveness of our learning approach both by providing theoretical guarantees, and by testing the approach on a variety of prototypical systems in various disciplines. These systems include homogeneous and heterogeneous agents systems, ranging from particle systems in fundamental physics to agent-based systems modeling opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics

Abstractions, Analysis Techniques, and Synthesis of Scalable Control Strategies for Robot Swarms

Tasks that require parallelism, redundancy, and adaptation to dynamic, possibly hazardous environments can potentially be performed very efficiently and robustly by a swarm robotic system. Such a system would consist of hundreds or thousands of anonymous, resource-constrained robots that operate autonomously, with little to no direct human supervision. The massive parallelism of a swarm would allow it to perform effectively in the event of robot failures, and the simplicity of individual robots facilitates a low unit cost. Key challenges in the development of swarm robotic systems include the accurate prediction of swarm behavior and the design of robot controllers that can be proven to produce a desired macroscopic outcome. The controllers should be scalable, meaning that they ensure system operation regardless of the swarm size. This thesis presents a comprehensive approach to modeling a swarm robotic system, analyzing its performance, and synthesizing scalable control policies that cause the populations of different swarm elements to evolve in a specified way that obeys time and efficiency constraints. The control policies are decentralized, computed a priori, implementable on robots with limited sensing and communication capabilities, and have theoretical guarantees on performance. To facilitate this framework of abstraction and top-down controller synthesis, the swarm is designed to emulate a system of chemically reacting molecules. The majority of this work considers well-mixed systems when there are interaction-dependent task transitions, with some modeling and analysis extensions to spatially inhomogeneous systems. The methodology is applied to the design of a swarm task allocation approach that does not rely on inter-robot communication, a reconfigurable manufacturing system, and a cooperative transport strategy for groups of robots. The third application incorporates observations from a novel experimental study of the mechanics of cooperative retrieval in Aphaenogaster cockerelli ants. The correctness of the abstractions and the correspondence of the evolution of the controlled system to the target behavior are validated with computer simulations. The investigated applications form the building blocks for a versatile swarm system with integrated capabilities that have performance guarantees

Using wireless sensors and networks program for chemical particle propagation mapping and chemical source localization

Chemical source localization is a challenge for most of researchers. It has extensive applications, such as anti-terrorist military, Gas and oil industry, and environment engineering. This dissertation used wireless sensor and sensor networks to get chemical particle propagation mapping and chemical source localization. First, the chemical particle propagation mapping is built using interpolation and extrapolation methods. The interpolation method get the chemical particle path through the sensors, and the extrapolation method get the chemical particle beyond the sensors. Both of them compose of the mapping in the whole considered area. Second, the algorithm of sensor fusion is proposed. It smooths the chemical particle paths through integration of more sensors\u27 value and updating the parameters. The updated parameters are associated with including sensor fusion among chemical sensors and wind sensors at same positions and sensor fusion among sensors at different positions. This algorithm improves the accuracy and efficiency of chemical particle mapping. Last, the reasoning system is implemented aiming to detect the chemical source in the considered region where the chemical particle propagation mapping has been finished. This control scheme dynamically analyzes the data from the sensors and guide us to find the goal. In this dissertation, the novel algorithm of modelling chemical propagation is programmed using Matlab. Comparing the results from computational fluid dynamics (CFD) software COMSOL, this algorithm have the same level of accuracy. However, it saves more computational times and memories. The simulation and experiment of detecting chemical source in an indoor environment and outdoor environment are finished in this dissertation --Abstract, page iii