Team:Freiburg Bioware/Modeling/Virus Infection
From 2010.igem.org
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 (R_{2})
occurs and the whole complex (VR_{2}) 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 (k_{7,1})
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 (k_{6,1}).
Model Extensions
Since internalization depends on the degree
of modification (m) the model is extended by a nonlinear
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 mvalues 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 mFile (MATLAB source code)!
Results and Discussion
Figure 3 shows the time course of the virus infection model. Starting
with an initial virus concentration of 10^{7} viral particles
per mililiter
the curve decays fast as the infectious particles are bound to
receptors and imported to endosomes in the cytoplasm. Since
receporvirus 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), virusreceptor complex (red), dimerized receptorvirus complex (turquoise), intranuclear virus (purple), singlestranded 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 nonfunctional 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%. 