Modeling

Introduction

Our modeling part consists of two main parts:

• The mathematical modeling of virus production and infection based on a model of ordinary differential equations (ODE).
• The modeling of the three dimensional structure of the virus capsid based on X-ray crystallography data.

The first model describes dynamical behavior of the virus production in terms of the law of mass action (Sidorenko & Reichl 2004). Therefore the temporal change concentrations of titrated plasmids, synthesized proteins, replicated single-stranded DNA and formed virus capsids and finally concentration of viral particles are simulated using MathWorks® MATLAB R2010b.
Virus infection is modeled in a similar way but receptor binding has to be considered (Gurevich 2004). Starting with a given concentration of infectious virus particles complexes with dimerized receptors are built which can be internalized to the cytoplasm and transduction takes place.
The third part follows a different approach: X-ray crystallography data (Xie et al. 2002) has been used to reproduce the three dimensional structure of the virus. The resulting 3D visualization is obtained using the open source software PyMOL (Schrödinger 2010).

Virus Production Model

Virus Infection Model

Structure Modeling

Literature

Mathematical Modeling

• Briggs, G. E., & Haldane, J. B. (1925) A Note on the Kinetics of Enzyme Action, Biochem J 19, 338-339.
• Culshaw, R. V., Ruan, S., & Webb, G. (2003). A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. Journal of mathematical biology, 46(5), 425-44. Springer Berlin / Heidelberg. doi: 10.1007/s00285-002-0191-5.
• Endres, D., & Zlotnick, A. (2002). Model-Based Analysis of Assembly Kinetics for Virus Capsids or Other Spherical Polymers. Biophysical Journal, 83(2), 1217-1230. Elsevier. doi: 10.1016/S0006-3495(02)75245-4.
• Endy, D., Kong, D., & Yin, J. (1997). Intracellular Kinetics of a Growing Virus : A Genetically Structured Simulation for Bacteriophage T7. Biotechnology, 7.
• Friedman, A. (2006). Cancer Models and Their Mathematical Analysis. Cancer, 246, 223- 246.
• Gurevich, K. G. (2004). Application of methods of identifying receptor binding models and analysis of parameters. Theoretical biology & medical modelling, 1, 11. doi: 10.1186/1742-4682-1-11.
• Johnston, I. G., Louis, A. A., & Doye, J. P. K. (2010). Modelling the self-assembly of virus capsids. Journal of Physics: Condensed Matter, 22(10), 104101. doi: 10.1088/0953-8984/22/10/104101.
• Komarova, N. L., & Wodarz, D. (2010). ODE models for oncolytic virus dynamics. Journal of Theoretical Biology, 263(4), 530-543. Elsevier. doi: 10.1016/j.jtbi.2010.01.009.
• Moradpour, D., Penin, F., & Rice, C. M. (2007). Replication of hepatitis C virus. Nature, 5(June), 453-463. doi: 10.1038/nrmicro1645.
• Novozhilov, A. S., Berezovskaya, F. S., Koonin, E. V., & Karev, G. P. (2006). Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models. Biology direct, 1, 6. doi: 10.1186/1745-6150-1-6.
• Sidorenko, Y., & Reichl, U. (2004). Structured Model of Influenza Virus Replication in MDCK Cells. Biotechnology and Bioengineering, 88, 1-14. doi: 10.1002/bit.20096.
• Sweeney, B., Zhang, T., & Schwartz, R. (2008). Exploring the Parameter Space of Complex Self-Assembly through Virus Capsid Models. Biophysical Journal, 94(3), 772-783. Elsevier. doi: 10.1529/biophysj.107.107284.
• Tao, Y., & Guo, Q. (2005). The competitive dynamics between tumor cells, a replication-competent virus and an immune response. Journal of mathematical biology, 51(1), 37-74. doi: 10.1007/s00285-004-0310-6.
• Wu, J T, Byrne, H. M., Kirn, D H, & Wein, L M. (2001). Modeling and analysis of a virus that replicates selectively in tumor cells. Bulletin of mathematical biology, 63(4), 731-68. doi: 10.1006/bulm.2001.0245.
• Wu, Joseph T, Kirn, David H, & Wein, Lawrence M. (2004). Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response. Bulletin of mathematical biology, 66(4), 605-25. doi: 10.1016/j.bulm.2003.08.016.
• Zlotnick, Adam. (1997). To Build a Virus Capsid. Journal of molecular biology.

Structure Modeling