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2006

Volume 5, Issue 4, pp. 1045-1366

† Special Section On Multiscale Modeling In Biology

Homogenization of the Cell Cytoplasm: The Calcium Bidomain Equations

Pranay Goel, James Sneyd, and Avner Friedman

Multiscale Model. Simul. 5, pp. 1045-1062 (18 pages) | Cited 12 times

Online Publication Date: November 24, 2006

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All previous models of the dynamics of intracellular calcium concentration have either made the ad hoc assumption that the cytoplasm and the endoplasmic reticulum (ER) coexist at every point in space or have explicitly separated the cytoplasm and the ER into different spatial domains. The former approach is unjustified, and the dependence on the diffusion coefficients on the geometry of the ER is unclear; the latter approach leads to extreme computational difficulties. To avoid the disadvantages of these approaches, we derive a bidomain model of calcium concentration inside the ER network, and outside it, in the cytosol. The homogenized macroscopic behavior is described in a two‐concentration field model, a formula is derived for the effective diffusion coefficients of calcium in the ER and in the cytoplasm, and the effective diffusion coefficients are numerically computed for different ER geometries.

A Semiclassical Transport Model for Thin Quantum Barriers

Shi Jin and Kyle A. Novak

Multiscale Model. Simul. 5, pp. 1063-1086 (24 pages) | Cited 6 times

Online Publication Date: November 24, 2006

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We present a one‐dimensional time‐dependent semiclassical transport model for mixed state scattering with thin quantum barriers. The idea is to solve a stationary Schrodinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interface condition that connects the two classical domains. We then build in the interface condition to the numerical flux, in the spirit of the Hamiltonian‐preserving scheme introduced by Jin and Wen for a classical barrier. The overall cost is roughly the same as solving a classical barrier. We construct a numerical method based on this semiclassical approach and validate the model using various numerical examples.

A Framework for Modeling Subgrid Effects for Two‐Phase Flows in Porous Media

Thomas Y. Hou, Andrew Westhead, and Danping Yang

Multiscale Model. Simul. 5, pp. 1087-1127 (41 pages)

Online Publication Date: December 05, 2006

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In this paper, we study upscaling for two‐phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two‐phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two‐phase flows in practical applications.

A Concurrent Multiscale Method Based on the Meshfree Method and Molecular Dynamics Analysis

Y. T. Gu and L. C. Zhang

Multiscale Model. Simul. 5, pp. 1128-1155 (28 pages) | Cited 1 time

Online Publication Date: December 18, 2006

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This paper presents a concurrent simulation technique for analyzing the deformation of systems that need the integration of material properties from nanoscopic to macroscopic dimensional scales. In the continuum subdomain, a weak‐form based meshfree method using the radial basis function interpolation was employed, but in the atomic subdomain, molecular dynamics analysis was used. The transition from the atomic to continuum domains was realized by transition particles which are independent of either the nodes in the continuum subdomain or the atoms in the atomic subdomain. A simple penalty method was used to ensure the compatibility of displacements and their gradients in the transition. A virtual cell algorithm was developed using a local quasi‐continuum approach to obtain the equivalent continuum strain energy density based on the atomic potentials and Cauchy–Born rule. Numerical examples showed that the present method is very accurate and stable, and has a promising potential to a wide class of multiscale systems.

Multiscale Coupling of Molecular Dynamics and Hydrodynamics: Application to DNA Translocation through a Nanopore

Maria G. Fyta, Simone Melchionna, Efthimios Kaxiras, and Sauro Succi

Multiscale Model. Simul. 5, pp. 1156-1173 (18 pages) | Cited 14 times

Online Publication Date: December 28, 2006

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We present a multiscale approach to the modeling of polymer dynamics in the presence of a fluid solvent. The approach combines Langevin molecular dynamics (MD) techniques with a mesoscopic lattice Boltzmann (LB) method for the solvent dynamics. A unique feature of the present approach is that hydrodynamic interactions between the solute macromolecule and the aqueous solvent are handled explicitly, and yet in a computationally tractable way due to the dual particle‐field nature of the LB solver. The suitability of the present LB‐MD multiscale approach is demonstrated for the problem of polymer fast translocation through a nanopore. We also provide an interpretation of our results in the context of DNA translocation through a nanopore, a problem that has attracted much theoretical and experimental attention recently.
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Special Section On Multiscale Modeling In Biology

Tamar Schlick and Ken Dill

Multiscale Model. Simul. 5, pp. 1174-1174 (1 page) | Cited 9 times

Online Publication Date: January 02, 2007

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What follows is a special section devoted to multiscale molecular modeling in biology, from atoms and molecules to biological cells.
Molecular, structural, and cellular biology represents an extraordinary opportunity for multiscale modeling. Biologically important processes happen over more than twenty orders of magnitude in time scales from optical excitations and bond vibrations on the scale of femtoseconds to organism lifetimes on the scale of years, and over ten orders of magnitude in space scales from atoms to living organisms. To make biology a more predictive discipline, there are major opportunities for modeling at all levels. At each level, we need improved models for quantifying the elementary interactions that govern the kinetics and equilibria of the system, and we need better ways to sample the options and degrees of freedom that the system explores.
Moreover, if we are ever to understand how the properties of cells arise from the properties of their components, we also need ways to bridge from one level to the next. What is represented in this special section is some of the very best current work in the field, covering almost the full spectrum of problems that are being explored in multiscale biology. Starting from the most molecular and working up to the cellular, here’s an overview of the papers in this section. Scheraga et al. describe how to compute the structures of proteins from the atomic arrangements of the amino acids that comprise them, using a hierarchical strategy: fast low‐resolution modeling first, followed by slower high‐resolution modeling. Baker et al. compare the electrostatic properties of proteins at different scales of resolution. Chodera et al. start with very short‐time and detailed molecular dynamics simulations of small peptide fragments of proteins to project the dynamics out to much longer time scales using master‐equation methods.
At the next level up, Olson and Matsumo show how to coarse‐grain to model the motions of DNA molecules by treating each base‐pair as a rigid body. Case et al. treat huge numbers of atoms in ribosomal complexes of proteins and RNA molecules over 100 nanoseconds using multiscale Langevin dynamics. Harvey and Locker simulate the process by which DNA molecules are packaged into viruses, through coarse‐grained simulations based on simple elastic potentials.
At a still more coarse‐grained level, Tanskanan and Winslow model a protein complex called a diad in order to understand how calcium regulates the process by which heart muscle excites and contracts. Goussis and Najm use a system of differential equations describing the translation of RNA into protein structures, the phosphorylation of proteins, protein dimerization and other biochemical processes to understand circadian rhythms in the fruit fly. Finally, Calvetti and Somersalo show how to estimate large numbers of parameters in nonlinear systems through the assistance of Bayesian methods, a problem that is relevant for models covering all the time and space scales of biology.
We thank Tom Hou for spearheading this special issue and the MMS board and SIAM for their support. Thanks too to Andrea Bertozzi, Ron Elber, James Glimm, and Robert Kohn, who volunteered as review editors. We also thank Mitch Chernoff and Heather Blythe of SIAM’s publications staff for their editorial assistance and very effective handling of manuscripts. In addition, we are deeply grateful to all the authors who contributed to this special section.

A Hierarchical Multiscale Approach to Protein Structure Prediction: Production of Low‐Resolution Packing Arrangements of Helices and Refinement of the Best Models with a United‐Residue Force Field

M. Chinchio, C. Czaplewski, S. Ołdziej, and H. A. Scheraga

Multiscale Model. Simul. 5, pp. 1175-1195 (21 pages)

Online Publication Date: December 28, 2006

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A hierarchical, two‐stage approach to ab initio protein structure prediction is presented and applied to four α‐helical proteins. In the first stage, a bank of low‐resolution models is generated using a highly simplified protein representation and energy function, coupled with a Conformation‐Family Monte Carlo (CFMC) search for the energy minimum. For helical proteins, this procedure (referred to as REPACK) produces a set of plausible packed arrangements of the helices, given their positions in the amino acid sequence. Secondary structure prediction methods such as JPRED can be used to provide the secondary structure assignment. In the second stage, these packing arrangements are used as starting points for a new search method (Local Search), based on the Monte Carlo‐with‐Minimization (MCM) algorithm and a united‐residue (UNRES) energy function. The focus of the Local Search is mainly on improving loop conformations and side‐chain positions, with minor modifications to the overall packing of the helices. By reducing the size of the conformational space that must be sampled with the UNRES energy function, which is much more expensive to compute than the REPACK energy function, this prediction scheme can be applied to much larger proteins than were tractable in the past with other UNRES‐based search methods. It was applied successfully to a 224–residue protein (target T0198, PDB code 1SUM) in the sixth community‐wide blind‐prediction experiment on the Critical Assessment of techniques for protein Structure Prediction (CASP6).

Application of New Multiresolution Methods for the Comparison of Biomolecular Electrostatic Properties in the Absence of Global Structural Similarity

Xiaoyu Zhang, Chandrajit L. Bajaj, Bongjune Kwon, Todd J. Dolinsky, Jens E. Nielsen, and Nathan A. Baker

Multiscale Model. Simul. 5, pp. 1196-1213 (18 pages) | Cited 10 times

Online Publication Date: December 28, 2006

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In this paper we present a method for the multiresolution comparison of biomolecular electrostatic potentials without the need for global structural alignment of the biomolecules. The underlying computational geometry algorithm uses multiresolution attributed contour trees (MACTs) to compare the topological features of volumetric scalar fields. We apply the MACTs to compute electrostatic similarity metrics for a large set of protein chains with varying degrees of sequence, structure, and function similarity. For calibration, we also compute similarity metrics for these chains by a more traditional approach based upon 3D structural alignment and analysis of Carbo similarity indices. Moreover, because the MACT approach does not rely upon pairwise structural alignment, its accuracy and efficiency promise to perform well on future large‐scale classification efforts across groups of structurally diverse proteins. The MACT method discriminates between protein chains at a level comparable to the Carbo similarity index method; i.e., it is able to accurately cluster proteins into functionally relevant groups which demonstrate strong dependence on ligand binding sites. The results of the analyses are available from the linked web databases http://ccvweb.cres.utexas.edu/MolSignature/ and http://agave.wustl.edu/similarity/. The MACT analysis tools are available as part of the public domain library of the Topological Analysis and Quantitative Tools (TAQT) from the Center of Computational Visualization at the University of Texas at Austin (http://ccvweb.csres.utexas.edu/software). The Carbo software is available for download with the open‐source APBS software package at http://apbs.sf.net/.

Long‐Time Protein Folding Dynamics from Short‐Time Molecular Dynamics Simulations

John D. Chodera, William C. Swope, Jed W. Pitera, and Ken A. Dill

Multiscale Model. Simul. 5, pp. 1214-1226 (13 pages) | Cited 49 times

Online Publication Date: December 28, 2006

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Protein folding involves physical timescales—microseconds to seconds—that are too long to be studied directly by straightforward molecular dynamics simulation, where the fundamental timestep is constrained to femtoseconds. Here we show how the long‐time statistical dynamics of a simple solvated biomolecular system can be well described by a discrete‐state Markov chain model constructed from trajectories that are an order of magnitude shorter than the longest relaxation times of the system. This suggests that such models, appropriately constructed from short molecular dynamics simulations, may have utility in the study of long‐time conformational dynamics.

Predicted Effects of Local Conformational Coupling and External Restraints on the Torsional Properties of Single DNA Molecules

Atsushi Matsumoto and Wilma K. Olson

Multiscale Model. Simul. 5, pp. 1227-1247 (21 pages)

Online Publication Date: December 28, 2006

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A newly developed, coarse‐grained treatment of the low‐frequency normal modes of DNA has been adapted to study the torsional properties of fully extended, double‐helical molecules. Each base pair is approximated in this scheme as a rigid body, and molecular structure is described in terms of the relative position and orientation of successive base pairs. The torsional modulus $C$ is computed from the lowest‐frequency normal twisting mode using expressions valid for a homogeneous, naturally straight elastic rod. Fluctuations of local dimeric structure, including the coupled variation of conformational parameters, are based on the observed arrangements of neighboring base pairs in high‐resolution structures. Chain ends are restrained by an elastic energy term. The calculations show how the end‐to‐end constraints placed on a naturally straight DNA molecule, in combination with the natural conformational features of the double helix, can account for the substantially larger torsional moduli determined with state‐of‐the‐art, single‐molecule experiments compared to values extracted from solution measurements and/or incorporated into theories to account for the force‐extension properties of single molecules. The computed normal‐mode frequencies and torsional moduli increase substantially if base pairs are inclined with respect to the double‐helical axis and the deformations of selected conformational variables follow known interdependent patterns. The changes are greatest if the fluctuations in dimeric twisting are coupled with parameters that directly alter the end‐to‐end displacement. Imposed restraints that mimic the end‐to‐end conditions of single‐molecule experiments then impede the twisting of base pairs and increase the torsional modulus. The natural inclination of base pairs concomitantly softens the Young’s modulus, i.e., ease of duplex stretching. The analysis of naturally curved DNA points to a drop in the torsional modulus upon imposed extension of the double‐helical molecule.

Low‐Resolution Molecular Dynamics Simulations of the 30S Ribosomal Subunit

Qizhi Cui, Robert K.‐Z. Tan, Stephen C. Harvey, and David A. Case

Multiscale Model. Simul. 5, pp. 1248-1263 (16 pages) | Cited 4 times

Online Publication Date: December 28, 2006

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Low‐resolution molecular models can provide appropriate and efficient ways for studying large biomolecular systems such as the ribosome. We have developed computer codes that use the Yammp Under Python modeling package to assemble low‐resolution force fields for RNA‐protein complexes, and that connect these to the Amber molecular simulation package. This pipeline combines many of the complementary strengths of these two packages. Our target here is the 30S ribosomal subunit from Thermus thermophilus. One hundred nanosecond Langevin dynamics simulations were performed for the bound and the unbound 16S RNA, and conformational changes of the 16S RNA and its interaction with the 30S proteins were examined to establish the fidelity of our model. The S7 protein assembly pathway was also examined, and the effects of protein binding order on the 16S RNA were analyzed. The simulations suggest that ribosomal proteins play important roles in maintaining the native 16S RNA structure. “Primary” proteins (in terms of assembly) help more in stabilizing the conformation of the RNA than do secondary and tertiary proteins. Ribosomal proteins appear to bind to the RNA in an organized fashion wherein primary and secondary proteins help to prepare the binding sites for tertiary proteins. The methodology and tools described here should provide useful ways to explore other aspects of ribosomal conformational changes by means of molecular dynamics simulations.

A Model for Viral Genome Packing

C. Rebecca Locker and Stephen C. Harvey

Multiscale Model. Simul. 5, pp. 1264-1279 (16 pages) | Cited 12 times

Online Publication Date: December 28, 2006

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A method is developed to simulate the process of double‐stranded (ds) DNA packing in viral capsids. The dsDNA is modeled as a discrete single‐stranded homopolymer with an all‐elastic potential that is parameterized only by the experimentally measured rise per base pair (bp) and the persistence length of free dsDNA in physiological conditions. No properties from experiments on packaged viral DNA are used in the model, and thus the model does not presuppose any energies or conformations in the viral capsid. While the elastic DNA model is general and may be successfully applied to a wide range of coarse graining (up to about 20bp per monomer), a resolution of 6bp per monomer is chosen for the viral packing problem. The model is simple enough at this resolution to simulate the entire packaging process of viruses as large as bacteriophage λ (with a genome length of 48,502bp). The packing protocol developed here allows for multiple simulations of independent packing events, and the results show that there is substantial variability in the conformational space sampled in individual packing simulations. Packing forces are computed for a model $\phi29$ bacteriophage system, and the force dependence on the amount of genome packaged is similar in functional form to experimentally determined forces, though the magnitude is about 40% lower. These results show that a simple elastic model of dsDNA with no explicit electrostatic interactions and no a priori knowledge of the packaged structure or energy is capable of capturing the main features of genome packaging in viruses. This implies that the final packed structure is largely determined by the elastic properties of the confined nucleic acid chain.

Integrative Structurally Detailed Model of Calcium Dynamics in the Cardiac Diad

Antti J. Tanskanen and Raimond L. Winslow

Multiscale Model. Simul. 5, pp. 1280-1296 (17 pages) | Cited 2 times

Online Publication Date: December 28, 2006

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In heart muscle, excitation‐contraction coupling translates electric signals into contraction of the heart. Excitation‐contraction coupling in cardiac myocytes (heart muscle cells) depends crucially on calcium‐induced calcium release (CICR) in a small microdomain known as the diad. In this study we develop methods for describing CICR at the structurally detailed level of the diad as well as develop methods by which these models may be simplified for use at the level of cell and tissue. In particular, the method enables description of diad geometry, spatial configuration of proteins within the diad, the motion and distribution of $\text{Ca}^{2+}$ ions within the diad, and $\text{Ca}^{2+}$ ion binding to receptor sites on L‐type $\text{Ca}^{2+}$ channels and sarcoplasmic $\text{Ca}^{2+}$ release channels. The end result of modeling is a biophysically detailed, computationally efficient model of CICR. Such a model enables investigation of a wide variety of biologically interesting issues in cardiac electrophysiology involving detailed descriptions of excitation‐contraction coupling using multiscale models of heart muscle.

Model Reduction and Physical Understanding of Slowly Oscillating Processes: The Circadian Cycle

Dimitris A. Goussis and Habib N. Najm

Multiscale Model. Simul. 5, pp. 1297-1332 (36 pages) | Cited 4 times

Online Publication Date: December 28, 2006

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A differential system that models the circadian rhythm in Drosophila is analyzed with the computational singular perturbation (CSP) algorithm. Reduced nonstiff models of prespecified accuracy are constructed, the form and size of which are time‐dependent. When compared with conventional asymptotic analysis, CSP exhibits superior performance in constructing reduced models, since it can algorithmically identify and apply all the required order of magnitude estimates and algebraic manipulations. A similar performance is demonstrated by CSP in generating data that allow for the acquisition of physical understanding. It is shown that the processes driving the circadian cycle are (i) mRNA translation into monomer protein, and monomer protein destruction by phosphorylation and degradation (along the largest portion of the cycle); and (ii) mRNA synthesis (along a short portion of the cycle). These are slow processes. Their action in driving the cycle is allowed by the equilibration of the fastest processes; (1) the monomer dimerization with the dimer dissociation (along the largest portion of the cycle); and (2) the net production of monomer+dimmer proteins with that of mRNA (along the short portion of the cycle). Additional results (regarding the time scales of the established equilibria, their origin, the rate limiting steps, the couplings among the variables, etc.) highlight the utility of CSP for automated identification of the important underlying dynamical features, otherwise accessible only for simple systems whose various suitable simplifications can easily be recognized.

Large‐Scale Statistical Parameter Estimation in Complex Systems with an Application to Metabolic Models

Daniela Calvetti and Erkki Somersalo

Multiscale Model. Simul. 5, pp. 1333-1366 (34 pages) | Cited 9 times

Online Publication Date: December 28, 2006

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The estimation of a large number of parameters in a complex dynamic multicompartment model in the presence of insufficient data is a difficult and challenging problem. Such problems arise in many applications, e.g., in biology, physiology, and environmental sciences. The model consists of a large system of coupled nonlinear ordinary differential equations, the data consisting of the values of few components at given observation times. The estimation problems are usually ill‐posed and severely underdetermined, while the quality of the scarce data is far from optimal. Therefore, a successful solution necessarily requires additional information about the parameters. A natural framework to introduce a priori information into the model is the Bayesian paradigm. In this article we develop a Bayesian methodology that is able to utilize various types of prior constraints such as approximate algebraic constraints for the parameters or inequality constraints for the solutions and integrate them into a parametric prior distribution. The subsequent parameter estimation is based on a combination of optimization methods and statistical sampling techniques. We apply the methodology to a skeletal muscle metabolism model, in which we are able to simultaneously estimate more than 100 parameters from one fifth as many measured data points.
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