Hierarchical Multiscale Modeling of Macromolecules and their Assemblies.

Soft materials (e.g., enveloped viruses, liposomes, membranes and supercooled liquids) simultaneously deform or display collective behaviors, while undergoing atomic scale vibrations and collisions. While the multiple space-time character of such systems often makes traditional molecular dynamics simulation impractical, a multiscale approach has been presented that allows for long-time simulation with atomic detail based on the co-evolution of slowly-varying order parameters (OPs) with the quasi-equilibrium probability density of atomic configurations. However, this approach breaks down when the structural change is extreme, or when nearest-neighbor connectivity of atoms is not maintained. In the current study, a self-consistent approach is presented wherein OPs and a reference structure co-evolve slowly to yield long-time simulation for dynamical soft-matter phenomena such as structural transitions and self-assembly. The development begins with the Liouville equation for N classical atoms and an ansatz on the form of the associated N-atom probability density. Multiscale techniques are used to derive Langevin equations for the coupled OP-configurational dynamics. The net result is a set of equations for the coupled stochastic dynamics of the OPs and centers of mass of the subsystems that constitute a soft material body. The theory is based on an all-atom methodology and an interatomic force field, and therefore enables calibration-free simulations of soft matter, such as macromolecular assemblies.

Multiscale macromolecular simulation: role of evolving ensembles.

Multiscale analysis provides an algorithm for the efficient simulation of macromolecular assemblies. This algorithm involves the coevolution of a quasiequilibrium probability density of atomic configurations and the Langevin dynamics of spatial coarse-grained variables denoted order parameters (OPs) characterizing nanoscale system features. In practice, implementation of the probability density involves the generation of constant OP ensembles of atomic configurations. Such ensembles are used to construct thermal forces and diffusion factors that mediate the stochastic OP dynamics. Generation of all-atom ensembles at every Langevin time step is computationally expensive. Here, multiscale computation for macromolecular systems is made more efficient by a method that self-consistently folds in ensembles of all-atom configurations constructed in an earlier step, history, of the Langevin evolution. This procedure accounts for the temporal evolution of these ensembles, accurately providing thermal forces and diffusions. It is shown that efficiency and accuracy of the OP-based simulations is increased via the integration of this historical information. Accuracy improves with the square root of the number of historical timesteps included in the calculation. As a result, CPU usage can be decreased by a factor of 3-8 without loss of accuracy. The algorithm is implemented into our existing force-field based multiscale simulation platform and demonstrated via the structural dynamics of viral capsomers.

Simulating microbial systems: addressing model uncertainty/incompleteness via multiscale and entropy methods.

Most systems of interest in the natural and engineering sciences are multiscale in character. Typically available models are incomplete or uncertain. Thus, a probabilistic approach is required. We present a deductive multiscale approach to address such problems, focusing on virus and cell systems to demonstrate the ideas. There is usually an underlying physical model, all factors in which (e.g., particle masses, charges, and force constants) are known. For example, the underlying model can be cast in terms of a collection of N-atoms evolving via Newton's equations. When the number of atoms is 10(6) or more, these physical models cannot be simulated directly. However, one may only be interested in a coarse-grained description, e.g., in terms of molecular populations or overall system size, shape, position, and orientation. The premise of this chapter is that the coarse-grained equations should be derived from the underlying model so that a deductive calibration-free methodology is achieved. We consider a reduction in resolution from a description for the state of N-atoms to one in terms of coarse-grained variables. This implies a degree of uncertainty in the underlying microstates. We present a methodology for modeling microbial systems that integrates equations for coarse-grained variables with a probabilistic description of the underlying fine-scale ones. The implementation of our strategy as a general computational platform (SimEntropics™) for microbial modeling and prospects for developments and applications are discussed.

Order parameters for macromolecules: application to multiscale simulation.

Order parameters (OPs) characterizing the nanoscale features of macromolecules are presented. They are generated in a general fashion so that they do not need to be redesigned with each new application. They evolve on time scales much longer than 10(-14) s typical for individual atomic collisions/vibrations. The list of OPs can be automatically increased, and completeness can be determined via a correlation analysis. They serve as the basis of a multiscale analysis that starts with the N-atom Liouville equation and yields rigorous Smoluchowski/Langevin equations of stochastic OP dynamics. Such OPs and the multiscale analysis imply computational algorithms that we demonstrate in an application to ribonucleic acid structural dynamics for 50 ns.

Multiscaling for systems with a broad continuum of characteristic lengths and times: Structural transitions in nanocomposites.

The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.

Thermal nanostructure: an order parameter multiscale ensemble approach.

Deductive all-atom multiscale techniques imply that many nanosystems can be understood in terms of the slow dynamics of order parameters that coevolve with the quasiequilibrium probability density for rapidly fluctuating atomic configurations. The result of this multiscale analysis is a set of stochastic equations for the order parameters whose dynamics is driven by thermal-average forces. We present an efficient algorithm for sampling atomistic configurations in viruses and other supramillion atom nanosystems. This algorithm allows for sampling of a wide range of configurations without creating an excess of high-energy, improbable ones. It is implemented and used to calculate thermal-average forces. These forces are then used to search the free-energy landscape of a nanosystem for deep minima. The methodology is applied to thermal structures of Cowpea chlorotic mottle virus capsid. The method has wide applicability to other nanosystems whose properties are described by the CHARMM or other interatomic force field. Our implementation, denoted SIMNANOWORLD, achieves calibration-free nanosystem modeling. Essential atomic-scale detail is preserved via a quasiequilibrium probability density while overall character is provided via predicted values of order parameters. Applications from virology to the computer-aided design of nanocapsules for delivery of therapeutic agents and of vaccines for nonenveloped viruses are envisioned.

Modeling biogeochemical dynamics in porous media: Practical considerations of pore scale variability, reaction networks, and microbial population dynamics in a sandy aquifer.

Prediction of the fate and environmental impacts of groundwater contaminants requires the identification of relevant biogeochemical processes and necessitates the macroscopic representation of microbial activity occurring at the microscale. Using a well-studied sandy aquifer environment, we evaluate the importance of pore distribution on organic matter respiration in a porous medium environment by performing spatially explicit simulations of microbial metabolism at the sub-millimeter scale. Model results using an idealized porous medium under non-biofilm forming conditions indicate that while some heterogeneity is observed for flow rates, distributions of microbes and dissolved organic substrates remain relatively homogenous at the grain scale. At the macroscale in the same environment, we assess the impact of a comprehensive reaction network description for a phenolic contaminant plume, and compare the findings to a setting describing organic matter breakdown in a coastal marine sediment. This comparison reveals the importance of reactions recycling reduced metabolites at redox interfaces, leading to a competition for oxidants. When the spatio-temporal dynamics of microbial groups are accounted for, our simulations show the importance of reaction energetics and nutrient limitations such as microbial nitrogen demands.

Scaling behavior of electronic excitations in assemblies of molecules with degenerate ground states.

The behavior of long space-time excitations in many-electron systems with ground state degeneracy is explored via multiscale analysis. The analysis starts with an ansatz for the wave function's dual dependence on the N-electron configuration (i.e., both by direct means and by indirect means via a set of order parameters). It is shown that a Dirac-like equation form of the wave equation emerges in the limit where the ratio epsilon (of the average nearest-neighbor distance to the characteristic length of the long-scale phenomenon of interest) is small. Examples of the long scale are the size of a quantum dot, nanotube, or wavelength of a density disturbance. The velocities in the Dirac-like equation are the transition moments of the single-particle momentum operator connecting degenerate ground states. While detailed band structure and the independent quasi-particle picture could underlie the behavior of some systems (as commonly suggested for graphene), the present scaling law results show it is not necessarily the only explanation. Rather, it can follow from the scaling properties of low-lying, long spatial scale excitations and ground state degeneracy, even in strongly interacting systems. The generality of our findings suggests graphene may be just one of many examples of Dirac-like equation behavior. A preliminary validation of our quantum scaling law for molecular arrays is presented. As our scaling law constitutes a coarse-grained wave equation, path integral or other methods derived from it hold great promise for calibration-free, long-time simulation of many-particle quantum systems.

Liquid-crystal transitions: a first-principles multiscale approach.

A rigorous theory of liquid-crystal transitions is developed starting from the Liouville equation. The starting point is an all-atom description and a set of order-parameter field variables that are shown to evolve slowly via Newton's equations. The separation of time scales between that of atomic collision or vibrations and the order-parameter fields enables the derivation of rigorous equations for stochastic order-parameter field dynamics. When the fields provide a measure of the spatial profile of the probability of molecular position, orientation, and internal structure, a theory of liquid-crystal transitions emerges. The theory uses the all-atom/continuum approach developed earlier to obtain a functional generalization of the Smoluchowski equation wherein key atomic details are embedded. The equivalent nonlocal Langevin equations are derived, and the computational aspects are discussed. The theory enables simulations that are much less computationally intensive than molecular dynamics and thus does not require oversimplification of the system's constituent components. The equations obtained do not include factors that require calibration and can thus be applicable to various phase transitions which overcomes the limitations of phenomenological field models. The relation of the theory to phenomenological descriptions of nematic and smectic phase transitions, and the possible existence of other types of transitions involving intermolecular structural parameters are discussed.

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.

In silico Geobacter sulfurreducens metabolism and its representation in reactive transport models.

Microbial activity governs elemental cycling and the transformation of many anthropogenic substances in aqueous environments. Through the development of a dynamic cell model of the well-characterized, versatile, and abundant Geobacter sulfurreducens, we showed that a kinetic representation of key components of cell metabolism matched microbial growth dynamics observed in chemostat experiments under various environmental conditions and led to results similar to those from a comprehensive flux balance model. Coupling the kinetic cell model to its environment by expressing substrate uptake rates depending on intra- and extracellular substrate concentrations, two-dimensional reactive transport simulations of an aquifer were performed. They illustrated that a proper representation of growth efficiency as a function of substrate availability is a determining factor for the spatial distribution of microbial populations in a porous medium. It was shown that simplified model representations of microbial dynamics in the subsurface that only depended on extracellular conditions could be derived by properly parameterizing emerging properties of the kinetic cell model.

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 such as 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. Thus, overall nanoscale features of these systems control the relative probability of atomistic fluctuations, while the latter mediate the average forces and diffusion coefficients that induce the dynamics of these nanoscale features. This feedback loop is overlooked in typical coarse-grained methods. We elucidate the role of interscale cross-talk and overcome bionanosystem simulation difficulties with (1) automated construction of order parameters (OPs) describing suprananometer scale structural features, (2) 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 (3) the derivation of a rigorous equation for the stochastic dynamics of the OPs. As the OPs capture hydrodynamic modes in the host medium, "long-time tails" in the correlation functions yielding the generalized diffusion coefficients do not emerge. Since the atomic-scale features of the system are treated statistically, several ensembles are constructed that reflect various experimental conditions. Attention is paid to the proper use of the Gibbs hypothesized equivalence of long-time and ensemble averages to accommodate the varying 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.

Multiscale theory of collective and quasiparticle modes in quantum nanosystems.

A quantum nanosystem (such as a quantum dot, nanowire, superconducting nanoparticle, or superfluid nanodroplet) involves widely separated characteristic lengths. These lengths range from the average nearest-neighbor distance between the constituent fermions or bosons, or the lattice spacing for a conducting metal, to the overall size of the quantum nanosystem (QN). This suggests the wave function has related distinct dependencies on the positions of the constituent fermions and bosons. We show how the separation of scales can be used to generate a multiscale perturbation scheme for solving the wave equation. Results for electrons or other fermions show that, to lowest order, the wave function factorizes into an antisymmetric (fermion) part and a symmetric (bosonlike) part. The former manifests the short-range/exclusion-principle behavior, while the latter corresponds to collective behaviors, such as plasmons, which have a boson character. When the constituents are bosons, multiscale analysis shows that, to lowest order, the wave function can also factorize into short- and long-scale parts. However, to ensure that the product wave function has overall symmetric particle label exchange behavior, there could, in principle, be states of the boson nanosystem where both the short- and long-scale factors are either boson- or fermionlike; the latter "dual fermion" states are, due to their exclusion-principle-like character, of high energy (i.e., single particle states cannot be multiply occupied). The multiscale perturbation analysis is used to argue for the existence of a coarse-grained wave equation for bosonlike collective behaviors. Quasiparticles, with effective mass and interactions, emerge naturally as consequences of the long-scale dynamics of the constituent particles. The multiscale framework holds promise for facilitating QN computer simulations and novel approximation schemes.

Understanding stem cell differentiation through self-organization theory.

The mechanism underling stem cells' key property, the ability to either divide into two replicate cells or a replicate and a differentiated daughter, still is not understood. We tested a hypothesis that stem cell asymmetric division/differentiation is spontaneously created by the coupling of processes within each daughter and the resulting biochemical feedbacks via the exchange of molecules between them during mitotic division. We developed a mathematical/biochemical model that accounts for dynamic processes accompanying division, including signaling initiation and transcriptional, translational and post-translational (TTP) reactions. Analysis of this model shows that it could explain how stem cells make the decision to divide symmetrically or asymmetrically under different microenvironmental conditions. The analysis also reveals that a stem cell can be induced externally to transition to an alternative state that does not have the potentiality to have the option to divide symmetrically or asymmetrically. With this model, we initiated a search of large databases of transcriptional regulatory network (TRN), protein-protein interaction, and cell signaling pathways. We found 12 subnetworks (motifs) that could support human stem cell asymmetric division. A prime example of the discoveries made possible by this tool, two groups of the genes in the genetic model are revealed to be strongly over-represented in a database of cancer-related genes.

Chromosome segregation in Escherichia coli division: a free energy-driven string model.

Although the mechanisms of eukaryotic chromosome segregation and cell division have been elucidated to a certain extent, those for bacteria remain largely unknown. Here we present a computational string model for simulating the dynamics of Escherichia coli chromosome segregation. A novel thermal-average force field accounting for stretching, bending, volume exclusion, friction and random fluctuation is introduced. A Langevin equation is used to simulate the chromosome structural changes. The mechanism of chromosome segregation is thereby postulated as a result of free energy-driven structural optimization with replication introduced chromosomal mass increase. Predictions of the model agree well with observations of fluorescence labeled chromosome loci movement in living cells. The results demonstrate the possibility of a mechanism of chromosome segregation that does not involve cytoskeletal guidance or advanced apparatus in an E. coli cell. The model also shows that DNA condensation of locally compacted domains is a requirement for successful chromosome segregation. Simulations also imply that the shape-determining protein MreB may play a role in the segregation via modification of the membrane pressure.

Cancer onset and progression: a genome-wide, nonlinear dynamical systems perspective on onconetworks.

It is hypothesized that the many human cell types corresponding to multiple states is supported by an underlying nonlinear dynamical system (NDS) of transcriptional regulatory network (TRN) processes. This hypothesis is validated for epithelial cells whose TRN is found to support an extremely complex array of states that we term a "bifurcation nexus", for which we introduce a quantitative measure of complexity. The TRN used is constructed and analyzed by integrating a database of TRN information, cDNA microarray data analyzers, bioinformatics modules, a transcription/translation/post-translation kinetic model, and NDS analysis software. Results of this genome-wide approach suggest that a cell can be induced to persist in one state or to transition between distinct states; apparently irreversible transitions can be reversed when the high dimensional space of extracellular and intracellular parameters is understood. As conditions change, certain cellular states (cell lines) are no longer supported, new ones emerge, and transitions (cell differentiation or death) occur. The accumulation of simulated point mutations (minor changes which individually are insignificant) lead to occasional dramatic transitions. The genome-wide scope of many of these transitions is shown to arise from the cross-linked TRN structure. These notions imply that studying individual oncogenes may not be sufficient to understand cancer; rather, "onconetworks" (subsets of strongly coupled genes supporting multiple cell states) should be considered. Our approach reveals several epithelial onconetworks, each involving oncogenes and anti-tumor and supporting genes.

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.

A plausible model for reversal of neoplastic transformations in plants based on multiple steady states.

We offer a plausible interpretation of some experiments on the reversal of neoplastic transformations in plants. We suggest that normal cells and tumorous cells represent multiple stable-steady states corresponding to a reaction feedback mechanism. The (autocatalytic) feedback loop is constructed from observations on the role played by myo-inositol: it increases the permeability of ions through the membrane and the biosynthetic pathway to myo-inositol is activated by ions. Provided that the permeabilities of nutrients (sugars and salts) are a product-enhanced function of myo-inositol, then we have a (oversimplified) model that can exhibit multiple stationary stable states, one or two depending on the exogenous nutrients and myo-inositol concentrations, and reversible and irreversible transitions from one of these states to the other are possible. From this model, straightforward simple experiments are suggested. We also propose that recent models dealing with the intracellular calcium regulation by hormones, where one key step requires the hydrolysis of inositol phospholipids, take into account free myo-inositol and endogenous hormone concentrations (e.g., auxins).

Oscillatory zoning in plagioclase feldspar.

A kinetic mathematical model of crystal growth from the melt is used to describe quantitatively the phenomenon of oscillatory zoning in plagioclase feldspar. In this model, the functional dependence of crystal growth rate on both melt and crystal surface composition and the transport of material within the melt are explicitly considered. Oscillatory zoning is found to develop for a wide variety of such functional dependence and to be sensitive to the initial composition of the melt.