# Model for Virus Infection

As described in the previous model for the virus production we established a model of ordinary differential equation (ODE) based on the law of mass action. The following paragraph depicts the reaction scheme and our model assumptions. In the subsequent paragraphs the system of differential equations is specified and the implementation in MathWorks® MATLAB is discussed. Furthermore reasonable model extensions are presented.
The last section deals with our modeling results and gives an outlook to further modeling applications.

## Reaction Scheme

To simplify the mathematical description of the reaction scheme we divide the cell into four compartments: the extracellular matrix (all quantities with the index ext), the space in endosomes (end), the cytoplasm (cyt) and the nucleus (nuc).
A target cell is transduced by viral particles (V) in the extracellular matrix depending on their degree of modification (m) and thus their specificity they can bind to receptors (R) on the cell surface. Once a receptor has formed a complex with a virus particle (VR) receptor dimerization (R2) occurs and the whole complex (VR2) is internalized and endosomes are formed. The virus is released from the endosome to the cytoplasm and is transported to the nucleus where uncoating of the capsid is initiated and single stranded DNA (ssDNA) is released.
Finally viral mRNA is processed and transported into the cytoplasm and the enzyme for therapeutic approach (E) is produced.

 Figure 1: Schematical overview of virus infection

## Reduced Reaction Scheme

This modeling approach neglects the fastest linear transport processes and describes the receptor binding and dimerization in terms of the law of mass action (LMA).

## Differential Equations

The ODE model consists of 7 equations containing 9 rate constants. To the equation for the extracellular virus concentration a degradation term with rate constant (k7,1) is added corresponding to the immune response of the target system. The temporal behavior of ssDNA is supplemented by a linear degradation term with rate constant (k6,1).

### Model Extensions

Since internalization depends on the degree of modification (m) the model is extended by a non-linear dependency in the rate constants. Additionally a decreasing production efficiency reduces the amount of available ssDNA in the nucleus. The estimated functional dependencies are shown in the figure below.

 Figure 2: Assumed functions for internalization (k(m), blue) and fraction of infectious virus particles (r(m), red) dependant on modification degree (m). Regarding the k(m) we assumed that a minimal modification degree of 10% is necessary for virus receptor complex formation due to the fact that at least two binding sites are required for receptor dimerization. For higher m-values the targeting efficiency saturates because enough binding sites exists that every dimerized receptor can build a complex with a virus. The values of r(m) are normalized to the ratio of infectious particles to empty virus capsids and therefore start at one. Increasing m results in a decreasing amount of infectious particles until a minimal value is reached for the completely modified virus (m=100%). Future experiments could provide a better understanding of those functional dependencies and therefore improve the model predictions concerning the targeting efficiency discussed later in this chapter.

## Methods and Simulation

The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integration routine ode15s with automatic integration step size management.
For initial conditions we took virus concentrations used in the experiments. Rate constants were estimated according to published values. The entire internalization process of viral particles being transported from the extracellular matrix through the cytoplasm into the nuleus happens within seconds (Seisenberger et al. 2001; Bartlett et al. 2000) whereas protein synthetization, capsid formation and virus packaging is taking much more time.
Furthermore experiments have been conducted in order to determine typical time scales corresponding to biological dynamics.

The used parameters are given in the table below.

 Table 1: rate constants for the virus infection model

get the m-File (MATLAB source code)!

## Results and Discussion

Figure 3 shows the time course of the virus infection model. Starting with an initial virus concentration of 107 viral particles per mililiter the curve decays fast as the infectious particles are bound to receptors and imported to endosomes in the cytoplasm. Since recepor-virus complexes are built the amout of free recepors on the cell surface decreases while the complex concentrations are rising.
As soon as the first particles reach the nucleus and release their viral DNA the intranuclear concentration of ssDNA rises and enzyme transcription is initialized. Therefore the enzyme concentration in cytoplasm is increased until a steady state is reached due to the fact that ssDNA is degraded inside the nucleus.

 Figure 3: Time course of concentrations. Extracellular virus particles (blue), free receptors on cell surface (green), virus-receptor complex (red), dimerized receptor-virus complex (turquoise), intranuclear virus (purple), single-stranded DNA (yellow) and enzyme in cytoplasm (black).

As mentioned in the previous paragraph concerning the model extensions the internalization rate as well as the amount of empty and for this reason non-functional virus particles strongly depends on the degree of modification. Hence one of the major goals of simulation was to calculate the optimal ratio between wild type and modified capsid proteins on the virus surface.
The simulation result can be seen in figure 4 B where, depending on the assumed functional behavior, the best targeting efficiency was reached with a 60% modified capsid.

 Figure 4: A: Enzyme concentration for different  degrees of modification (m) - the steady state depends on m. B: Enyme concentration as a function of m. The optimum is reached for m=60%.