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Viruses are infectious agents that can cause epidemics and pandemics. The understanding of virus formation, evolution, stability, and interaction with host cells is of great importance to the scientific community and public health. Typically, a virus complex in association with its aquatic environment poses a fabulous challenge to theoretical description and prediction. In this work, we propose a differential geometry-based multiscale paradigm to model complex biomolecule systems. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum domain of the fluid mechanical description of the aquatic environment from the microscopic discrete domain of the atomistic description of the biomolecule. A multiscale action functional is constructed as a unified framework to derive the governing equations for the dynamics of different scales. We show that the classical Navier-Stokes equation for the fluid dynamics and Newton's equation for the molecular dynamics can be derived from the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows.

Viruses are omnipresent infectious agents that are about 100 times smaller than bacteria. Unlike bacteria, viruses are not able to grow or reproduce outside a host cell [

Recent advances in structural biology and microbiology have led to a rapidly growing body of virus structural data [

Currently, the prevention and control of epidemics and pandemics caused by infective viruses, such as H1N1, HIV, SARS, and bird flu are of paramount importance. As an infection starts with the surface attachment between a virus and a host cell, it is important to construct and visualize the surface topology and morphology of viruses in order to understand the surface attachment and further interaction. This information is also crucial to the understanding of the molecular mechanism that gives rise to the assembly of virus capsids and DNA or RNA packaging. Computer-based visualization is able to represent results of explorations in an easy-to-comprehend form and to facilitate convenient information retrieval. Currently, visualization tools are often developed in close conjunction with imaging, data registration, simulation and/or surface construction. Virus visualization plays a unique role in the understanding of virus infection processes, such as, virus attachment of a host cell, binding and fusion between a virus capsid surface and a host cellular surface, and the penetration of a virus into a host cell. However, viruses are not directly visible because their sizes are at the order of tens of nanometers. The virus images are constructed from virus information, which is either collected from modalities described above or generated by computer simulations. Therefore, surface/image construction is a part of the virus visualization. Yu and Bajaj present a computational algorithm to segment asymmetric units of three-dimensional (3D) density maps of icosahedral viruses [

The difficulty of characterizing a virus complex is not only its massive number of atoms, or data sets, but also its everlasting interactions. Except for envelope type of viruses which typically cover their capsids by envelopes derived from lipids and proteins of their host cell membranes, most viruses use their own capsids to interact with the environment and host cells. A viral capsid usually consists of many identical viral protein subunits that form the capsid by symmetric assembly. There are strong interactions between viral protein subunits so that viral capsids are rigid enough to hold viral genome material and protect its content. Viruses have adapted a number of strategies to maintain the stability and flexibility of viral capsids. For many small viruses, such as one of STMV, their subunit proteins generally only touch each other by their edges. Their capsid stability is achieved by strong nonbonding interactions (i.e., hydrogen bonding and van der Waals interactions) between edges of subunit proteins. Some large viruses, such as BMV, have developed overlapping strategies to increase the capsid stability. Some viruses even use a few intricately intertwining layers to strengthen their capsids [

One of the present authors, Wei, introduced some of the first high-order geometric flow equations for image analysis [

An unsolved problem in structural virology is the detailed molecular mechanism of the assembly of virus capsids with the right size that is able to accommodate virus genetic material in the subsequent virus DNA/RNA packaging. Additionally, the process of virus attachment on its host cell, the movement of virus fusion with cellular membrane, and the dynamics of virus penetration into its host cell remain unrevealed mysteries. Prerequisites to unveiling these mysteries are efficient computer science and mathematical tools for modeling virus surface construction, evolution, and visualization, and for analyzing the virus interactions with its host cell. A typical virus has millions of atoms, while a large virus may have tens of millions atoms. Huge viral data sets pose severe challenges to the theoretical understanding and prediction of virus dynamics and interactions. These challenges are considerably exacerbated by the fact that virus behavior and infectivity depend strongly on the physiological environment, where the water molecules are the most common media. This dramatically increases the number of degrees of freedom of a virus system. The real-time dynamic visualization of viral attachment, fusion, and penetration of a host cell in the aquatic environment requires microsecond or even millisecond simulation time and is technically intractable with full-atom models at present [

Recently, one of the present authors, Wei, has developed a differential geometry-based multiscale paradigm to address some of the aforementioned challenges in the nonequilibrium dynamics of viruses, as well as other complex chemical systems, for example, fuel/solar cells, and biological systems, for example, ion channels [

The objective of the present work is threefold. First, we apply the differential geometry-based multiscale models to the formation and evolution virus capsids where challenges originated from a large number of atoms and a variety of interactions in a virus system, including the aquatic environment. To dramatically reduce the number of degrees of freedom of a virus system, we treat the water molecules as a macroscopic continuum. However, we maintain atomic description of the virus to allow an optimal access to detailed biomolecular information. Secondly, we propose a new scale, the coarse-grained particles, to improve the earlier multiscale formalism [

In this section, differential geometry theory of surfaces and potential driven geometric flows are utilized to establish a multiscale paradigm for modeling and simulation of virus formation and evolution. Then, a coarse-grained virus model is formulated to further reduce the number of degrees of freedom. Finally, the use of symmetry in virus surface construction is discussed.

A fundamental issue in biological modeling, and in data analysis, visualization, and dynamical representation is how to deal with a tremendously large number of degrees of freedom resulting from various interaction. Under physiological condition, a virus and its interacting environment may involve tens of millions of protein atoms and water molecules. In principle, the system can be described entirely in the microscopic scale, that is, atomistic description or more detailed description of electrons and nuclei. However, such an approach cannot be productive and does not provide theoretical predictions of physical properties of the virus complex. It is impossible at present, and formidably expensive in near future to describe in full-atomic detail of all the aforementioned interactions for a large virus system. On the other hand, a macroscopic description of the system is incapable of revealing the molecular and atomic information of the virus particle and its dynamics. We plan to reduce the number of degrees of freedom of the virus complex by a differential geometry-based multiscale model. In our multiscale model, we will describe the aquatic environment by a hydrodynamic continuum, that is, a macroscopic description. As such, we are able to dramatically reduce the number of degrees of freedom of millions surrounding water molecules. However, since the biomolecule or the virus is the objective of interest, we will describe the virus in atomic detail, that is, a microscopic, discrete description. Additionally, we carefully consider the solvation process of the virus molecule. The virus surface tension and mechanical work of virus immersion into the solvent are considered in our model, in addition to the possible interaction between virus atoms and the aquatic environment. Finally, the force resulted from virus and solvent interactions is accounted by fluid motion, which is modeled by a viscous fluid.

In our differential geometry-based multiscale model, we use a hypersurface (characteristic) function

We consider the total action functional for the virus complex [

On the right hand side of (

In the present work, we derive four governing equations by employing the principle of the least action to the total action functional (

Additionally, the virus surface evolution equation can be constructed by requiring the term associated with

Moreover, the requirement of the vanishing of the term associated with

Finally, the Newton's equation for molecular dynamics of the

In this multiscale system, all forces are balanced. The fluid dynamics, the molecular dynamics, the electrostatic subsystem, and the hypersurface function are all coupled.

As a part of our multiscale framework, we consider a coarse-grained formalism for viral surface formation and evolution. Coarse-grained models are often used to deal with exceptionally large biological systems. In the present treatment, we consider each amino acid residue as a particle, located at the

Coarse-grain radii (

Residue | Radius | Residue | Radius | Residue | Radius | Residue | Radius | Residue | Radius |
---|---|---|---|---|---|---|---|---|---|

GLY | 4.20 | ALA | 4.10 | VAL | 4.30 | LEU | 5.70 | ILE | 5.60 |

PRO | 4.10 | PHE | 7.00 | TYR | 8.30 | TRP | 8.30 | SER | 4.30 |

THR | 4.40 | ASN | 5.50 | GLN | 6.80 | CYS | 4.50 | MET | 7.30 |

ASP | 5.50 | GLU | 6.70 | HIS | 6.40 | LYS | 8.20 | ARG | 9.10 |

Coarse-grained model of a viral protein subunit. Left: the full atomic model of a protein subunit of the Nodamura virus (PDB ID: 1nov), Right: the coarse-grained model of a protein subunit of the Nodamura virus.

Viral data may involve tens of millions of atomic coordinates and radii, and are enormously large for structural modeling, simulation and visualization. Viral dynamical cycles may last from millisecond to days, and real-time full-atom viral dynamical simulations of viruses are intractable to the present computational capability [

Illustration of surface construction from a facet patch by using symmetry. Left: the generating subunit (facet patch) of the Nodamura virus (PDB ID: 1nov), Right: the full surface of the Nodamura virus constructed by symmetric assembly.

Viruses have adapted five point group symmetries, that is, circular, dihedral, tetrahedral, octahedral, and icosahedral, in their biological assemblies. Mathematically, only three types of symmetric operations, that is, rotation, inversion, and translation are involved. Starting with the basic set of coordinates of a protein subunit, the virus capsid data can be obtained by the transformation

Recent advances in structural biology and microbiology have given rise to an increasing body of structural data for over 300 viruses and viral complexes. Quaternary structures of viruses and viral complexes pose many challenges for viral representation, visualization, and the analysis of virus stability and interaction [

We also test two other proposed ideas in this work, that is, the coarse-grained virus model and the use of symmetry assembly for the virus surface construction. In particular, we are interested in examining the effect of the symmetry assembly on the virus surface visualization. As shown in Figure

Illustration of virus surfaces constructed by using the proposed geometric flow approach in conjunction with the coarse-grained model and the symmetry assembly. Upper row: Surfaces generated from a facet patch by using symmetry assembly. Lower row: Surfaces generated without the use of symmetry. From left to right: Cucumber green mottle mosaic virus (CGMMV) with helical symmetry (1cgm), Tobacco mosaic virus coat protein four-layer aggregate with

The control of infective viruses released by terrorists, and the prevention of viral epidemics and pandemics, such as HIV, SARS, H1N1, and bird flu are of tremendous importance. The understanding of viral surface formation, evolution, viral attachment and penetration of host cells are prerequisites to viral disease prevention and control. This problem, as well as many other similar problems in molecular biology, poses pressing challenges to the theoretical community due to their large number of degrees of freedom. The main purpose of the present work is to introduce a differential geometry-based multiscale framework to handle complex biological systems. The present multiscale model couples macroscopic fluid dynamics, microscopic molecular dynamics, and surface dynamics in a unified framework. The differential geometry theory of surfaces is utilized to put continuum description and discrete description in an equal footing. The present work constructs a generalized action functional to self-consistently couple different scales. Governing equations for the fluid dynamics, that is, the generalized Navier-Stokes equation, and molecular dynamics, that is, the Newton's equation, are derived by minimizing the action functional. Additionally, we make use of viral symmetry to dramatically reduce viral data sizes and improve viral visualization. Finally, some of the proposed approaches are demonstrated by the generation of a few virus surfaces.

The proposed differential geometry-based multiscale model can be easily generalized to complex systems with multiple interfaces or many biomolecules. Additionally, the incorporation of continuum solid description into the present model will be published elsewhere. Finally, the inclusion of a quantum mechanical description can also be pursued in a similar way and will be published elsewhere. Numerical experiments that further demonstrate the proposed ideas are under our consideration.

This work was supported in part by NSF Grants DMS-0616704 and CCF-0936830, and NIH Grants CA-127189 and GM-090208.