Model for Virus Production

Reaction Scheme

Reducing the complexity of virus production we divide the cell into three compartments: the extracellular matrix (all quantities with the index ext), the cytoplasm (cyt) and the nucleus (nuc). Four plasmids are transfected - the plasmid coding for the helper proteins (helper), the gene of interest (goi) and two types of plasmids coding for the capsid proteins (capwt [wild type], capmod [modified]).
The plasmids are transported into the nucleus where gene expression is initiated. Processed mRNA is transported into the cytoplasm and proteins (phelper, pcapwt, pcapmod) are produced. Containing a nuclear localization sequence proteins are relocated into the nucleus where capsid assembly occurs. The viral capsid is compose of 60 subunits of viral coat proteins. Titration of the two plasmids coding for the capsid proteins leads to virus surfaces with different ratios of wild type and modified capsid proteins.
The gene of interest is replicated by cellular polymerases and single stranded DNA (ssDNA) is encapsidated into the preformed capsids (capsid) forming infectious viral particles (V).
Finally the recombinant viruses are released into the extracellular matrix and can be harvested for transduction.

Reduced Reaction Scheme

Even the coarse model for virus production described in the previous paragraph would still consist of 24 ODEs containing 39 parameters (35 rate constants and 4 initial plasmid concentrations). Taking into account the linearity of the law of mass action (LMA) for simple transport processes we can neglect these fast reactions and for this reason reduce the model to the rate limiting steps like protein synthetization, capsid formation and virus packaging.

Differential Equations

The 13 reactions for the virus production are represented in a system of 17 coupled ODEs.
In addition to the terms provided by the law of mass action we considered the following terms:

• a linear degradation of ssDNA in the nucleus with the rate constant k_14,1
• replication of ssDNA in the nucleus with the rate constant k_15,1

Methods and Simulation

The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integrator ode15s with automatic integration step size management.
To calibrate the dynamics of the mathematical model to those of biological system we used time lapse data of fluorescence experiments as well as published values for the rate constants.

The used model parameters are given in the table below.

Download the m-File (MATLAB source code).

Results and Discussion