Model for Virus Infection

As in the previous model for the virus production we established a ODE model based on the law of mass action. The following paragraph explains 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.
The last section deals with our modeling results.

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 invaginated 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 the 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.

Reduced Reaction Scheme

The modeling approach only neglects 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 is added corresponding to the immune response of the target system. The temporal behavior of ssDNA is completed by a linear degradation term.

Model Extensions

The model is extended by a non-linear dependency of the internalization rate constant of the degree of modification (m) and the production efficiency of functional virus particles reduces the amount of available ssDNA in the nucleus.
The assumed functional dependency is shown in the two figures below.

For k(m) we assumed that a minimal modification degree of 15% is needed that virus receptor complex formation can take place due to the fact that two binding sites are required for receptor dimerization. For higher m-values the targeting efficiency saturates because enough binding sites exists so that every receptor dimer builds a complex with a virus. The values of r(m) is normalized to the ratio of infectious particles to empty virus capsids and therefore starts at 1. With increasing m the amount of infections particles decreases until a minimum value is reached for the completely modified virus (

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 parameters used are given in the table below. Also you can download the MATLAB source code.

Results and Discussion