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

To reduce the number of equations and parameters we considered two different approaches for the reaction dynamics.

Model 1 (LMA)

The first approach only neglects linear transport processes and describes the receptor binding and dimerization in terms of the law of mass action (LMA).

Model 2 (MM)

The second approach simplifies the scheme to 3 reactions assuming Michaelis-Menten kinetics (MM) in the quasi steady state approximation (Briggs, Haldane 1925) without explicit receptor dimerization.

Differential Equations

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.

Model 1 (LMA)

Model 2 (MM)

get the .m-File (MATLAB source code)

Results and Discussion

Model 1 (LMA)

Model 2 (MM)