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.

Figure 1: Schematic overview of virus production: A production cell line is  transfected with 4 plasmid types. DNA is replicated, transcribed (1) and proteins are synthesized (2). Capsid assembly occurs (3) and single-stranded DNA is packaged into the viral particle (4).

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.
In order to adjust the dynamical model to biological data we extracted the average intensity out of the time lapse recordings of fluorescence experiments as well as published values for the rate constants. For initial conditions we took the plasmid concentrations we used in experiments.

Figure 2: Fluorescence microscopy of transfected cells.  mVenus is included to the modified capsid plasmid i.e. fluorescence intensity reflects capsid protein concentration.	The image on the right shows one snapshot out of the time lapse recorded over a period of 1560 minutes (26 hours) after transfection. The bright spots correspond to the fluorescence intensity of mVenus in the upper and of mCherry in the lower picture. To see the whole time lapse as an animation just click on the picture!
Figure 3: Fluorescence microscopy of transfected cells.  Viral particles containing mCherry as gene of interest are visible.

Figure 4: A shows the average intensity of mCherry recorded using fluorescence microscopy. The curve corresponds to the rising phase of protein concentration and is expected to saturate for longer times as the harvest of viral particles is done after 3 days (4320min). B: time course of the intensity of mCherry. Due to the weak expression of mCherry the signal to noise ratio is quiet low and the functional dependency is not clearly determinable.

The average intensity was extracted from the raw data through a script written in MathWorks® MATLAB. Download the m-File (MATLAB source code).

Table 1: Rate constants for the virus production model. Generally foward reactions were assumed to be faster than reverse ones. Replication of ssDNA is slower than its degradation.

Results and Discussion

Figure 5:

Figure 6:

Figure 7:

Figure 9: Data fitting approach. The model can be fitted perfectely to the data (not shown) but is not meaningfull in a biological sense because exponential increase does not occur in this system. Considering the fact that virus concentration should saturate a sigmoidal shape is expected. Such a fit was not achieved because to many unknown parameters for one single data set.