Biomedical Engineering Reference
In-Depth Information
In the following section we develop a model to explain these two effects.
3.7
Mathematical Model
We have recently modelled a virus [
4
], as a set of different proteins expressed in
a host cell. Each protein
i
has its own expression profile
x
i
(
—concentration of
protein
i
at time
t
—and can induce a polyclonal T cell response
T
i
(
t
)
. Each
T
i
represents the number of cells in a group of clones reacting to the epitopes in a
given protein. For the sake of simplicity we denote
T
i
as a clone, although it is not a
single clone in the strict sense of the word, since it is composed of multiple clones
reacting to possibly different epitopes on the same protein. Finally, we denote by
p
i
the number of different epitopes presented by each viral protein.
The viral T lymphocytes dynamics can be simplified by the following multidi-
mensional ODE:
t
)
d
x
d
t
=
d
T
d
t
=
(
)
,
(
,{
},
)
,
h
t
f
p
x
T
(1)
where
x
and
p
are the vectors containing
x
i
(
represents the
dynamics of the viral proteins following cell entry in a given cell, and
f
t
)
,
p
i
and
T
i
,
h
(
x
)
)
represents the dynamics of T cell clones during the course of a disease. The
dynamics of
x
are the dynamics of proteins inside an infected cell with a typical
timescale of hours. These dynamics end at cell death or following budding. The
dynamics of T cells evolve at a much slower timescale and span many viral life
cycles. Moreover, the dynamics of T cell clones are affected by all infected cells.
We denote the set of the
x
i
(
(
p
,{
x
},
T
.
If a given protein has
x
i
copies and
p
i
epitopes per copy, it presents a total of
p
i
x
i
epitopes. One can roughly estimate the total number of epitopes presented by
a virus in a cell as
t
)
values in all infected cells as
{
x
}
, assuming no large differences in the affinity of each
peptide. Further assuming a constant killing rate per T cell and per epitope (
∑
i
p
i
x
i
(
t
)
μ
0
), the
total killing rate before budding of a given virally infected cell can be estimated by
t
budding
∑
i
μ
(
t
)=
μ
0
d
t
p
i
x
i
T
i
.
(2)
0
The probability that this host cell would survive for a long enough time allowing
viral budding is thus:
⎛
⎝
−
μ
0
⎞
⎠
.
t
budding
P
(
survival
)=
exp
p
i
x
i
(
t
)
T
i
(
t
)
d
t
(3)
0
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