Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville Equation can be decomposed via an expansion in terms of a smallness parameter epsilon, wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to order epsilon squared for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker-Planck equation up to the same order in epsilon. This approach applies to a broad range of problems in the nanosciences.Comment: 23 page

Self-Assembly of Nanocomponents into Composite Structures: Derivation and Simulation of Langevin Equations

The kinetics of the self-assembly of nanocomponents into a virus, nanocapsule, or other composite structure is analyzed via a multiscale approach. The objective is to achieve predictability and to preserve key atomic-scale features that underlie the formation and stability of the composite structures. We start with an all-atom description, the Liouville equation, and the order parameters characterizing nanoscale features of the system. An equation of Smoluchowski type for the stochastic dynamics of the order parameters is derived from the Liouville equation via a multiscale perturbation technique. The self-assembly of composite structures from nanocomponents with internal atomic structure is analyzed and growth rates are derived. Applications include the assembly of a viral capsid from capsomers, a ribosome from its major subunits, and composite materials from fibers and nanoparticles. Our approach overcomes errors in other coarse-graining methods which neglect the influence of the nanoscale configuration on the atomistic fluctuations. We account for the effect of order parameters on the statistics of the atomistic fluctuations which contribute to the entropic and average forces driving order parameter evolution. This approach enables an efficient algorithm for computer simulation of self-assembly, whereas other methods severely limit the timestep due to the separation of diffusional and complexing characteristic times. Given that our approach does not require recalibration with each new application, it provides a way to estimate assembly rates and thereby facilitate the discovery of self-assembly pathways and kinetic dead-end structures.Comment: 34 pages, 11 figure

Stochastic Dynamics of Bionanosystems: Multiscale Analysis and Specialized Ensembles

An approach for simulating bionanosystems, such as viruses and ribosomes, is presented. This calibration-free approach is based on an all-atom description for bionanosystems, a universal interatomic force field, and a multiscale perspective. The supramillion-atom nature of these bionanosystems prohibits the use of a direct molecular dynamics approach for phenomena like viral structural transitions or self-assembly that develop over milliseconds or longer. A key element of these multiscale systems is the cross-talk between, and consequent strong coupling of, processes over many scales in space and time. We elucidate the role of interscale cross-talk and overcome bionanosystem simulation difficulties with automated construction of order parameters (OPs) describing supra-nanometer scale structural features, construction of OP dependent ensembles describing the statistical properties of atomistic variables that ultimately contribute to the entropies driving the dynamics of the OPs, and the derivation of a rigorous equation for the stochastic dynamics of the OPs. Since the atomic scale features of the system are treated statistically, several ensembles are constructed that reflect various experimental conditions. The theory provides a basis for a practical, quantitative bionanosystem modeling approach that preserves the cross-talk between the atomic and nanoscale features. A method for integrating information from nanotechnical experimental data in the derivation of equations of stochastic OP dynamics is also introduced.Comment: 24 page

Block network mapping approach to quantitative trait locus analysis

BACKGROUND: Advances in experimental biology have enabled the collection of enormous troves of data on genomic variation in living organisms. The interpretation of this data to extract actionable information is one of the keys to developing novel therapeutic strategies to treat complex diseases. Network organization of biological data overcomes measurement noise in several biological contexts. Does a network approach, combining information about the linear organization of genomic markers with correlative information on these markers in a Bayesian formulation, lead to an analytic method with higher power for detecting quantitative trait loci? RESULTS: Block Network Mapping, combining Similarity Network Fusion (Wang et al., NM 11:333-337, 2014) with a Bayesian locus likelihood evaluation, leads to large improvements in area under the receiver operating characteristic and power over interval mapping with expectation maximization. The method has a monotonically decreasing false discovery rate as a function of effect size, unlike interval mapping. CONCLUSIONS: Block Network Mapping provides an alternative data-driven approach to mapping quantitative trait loci that leverages correlations in the sampled genotypes. The evaluation methodology can be combined with existing approaches such as Interval Mapping. Python scripts are available at http://lbm.niddk.nih.gov/vipulp/ . Genotype data is available at http://churchill-lab.jax.org/website/GattiDOQTL . BMC Bioinformatics 2016 Dec 22; 17(1):544

Multiscale Theory of Finite Size Bose Systems: Implications for Collective and Single-Particle Excitations

Boson droplets (i.e., dense assemblies of bosons at low temperature) are shown to mask a significant amount of single-particle behavior and to manifest collective, droplet-wide excitations. To investigate the balance between single-particle and collective behavior, solutions to the wave equation for a finite size Bose system are constructed in the limit where the ratio \varepsilon of the average nearest-neighbor boson distance to the size of the droplet or the wavelength of density disturbances is small. In this limit, the lowest order wave function varies smoothly across the system, i.e., is devoid of structure on the scale of the average nearest-neighbor distance. The amplitude of short range structure in the wave function is shown to vanish as a power of \varepsilon when the interatomic forces are relatively weak. However, there is residual short range structure that increases with the strength of interatomic forces. While the multiscale approach is applied to boson droplets, the methodology is applicable to any finite size bose system and is shown to be more direct than field theoretic methods. Conclusions for Helium-4 nanodroplets are drawn.Comment: 28 pages, 5 figure

Double-Well Optical Lattices with Atomic Vibrations and Mesoscopic Disorder

Double-well optical lattice in an insulating state is considered. The influence of atomic vibrations and mesoscopic disorder on the properties of the lattice are studied. Vibrations lead to the renormalization of atomic interactions. The occurrence of mesoscopic disorder results in the appearance of first-order phase transitions between the states with different levels of atomic imbalance. The existence of a nonuniform external potential, such as trapping potential, essentially changes the lattice properties, suppressing the disorder fraction and rising the transition temperature.Comment: Latex file, 21 pages, 2 figure

The Karyote® Physico-Chemical Genomic, Proteomic, Metabolic Cell Modeling System

Modeling approaches to the dynamics of a living cell are presented that are strongly based on its underlying physical and chemical processes and its hierarchical spatio-temporal organization. Through the inclusion of a broad spectrum of processes and a rigorous analysis of the multiple scale nature of cellular dynamics, we are attempting to advance cell modeling and its applications. The presentation focuses on our cell modeling system, which integrates data archiving and quantitative physico-chemical modeling and information theory to provide a seamless approach to the modeling/data analysis endeavor. Thereby the rapidly growing mess of genomic, proteomic, metabolic, and cell physiological data can be automatically used to develop and calibrate a predictive cell model. The discussion focuses on the Karyote® cell modeling system and an introduction to the CellX® and VirusX® models. The Karyote software system integrates three elements: (1) a model-building and data archiving module that allows one to define a cell type to be modeled through its reaction network, structure, and transport processes as well as to choose the surrounding medium and other parameters of the phenomenon to be modeled; (2) a genomic, proteomic, metabolic cell simulator that solves the equations of metabolic reaction, transcription/translation polymerization and the exchange of molecules between parts of the cell and with the surrounding medium; and (3) an information theory module (ITM) that automates model calibration and development, and integrates a variety of data types with the cell dynamic computations. In Karyote, reactions may be fast (equilibrated) or slow (finite rate), and the special effects of enzymes and other minority species yielding steady-state cycles of arbitrary complexities are accounted for. These features of the dynamics are handled via rigorous multiple scale analysis. A user interface allows for an automated generation and solution of the equations of multiple timescale, compartmented dynamics. Karyote is based on a fixed intracellular structure. However, cell response to changes in the host medium, damage, development or transformation to abnormality can involve dramatic changes in intracellular structure. As this changes the nature of the cellular dynamics, a new model, CellX, is being developed based on the spatial distribution of concentration and other variables. This allows CellX to capture the self-organizing character of cellular behavior. The self-assembly of organelles, viruses, and other subcellular bodies is being addressed in a second new model, VirusX, that integrates molecular mechanics and continuum theory. VirusX is designed to study the influence of a host medium on viral self-assembly, structural stability, infection of a single cell, and transmission of disease