Biomedical Engineering Reference
In-Depth Information
The virus host dynamics can be treated as an epidemic model, where each cell is
susceptible. The virus can spread among the host cells if each infected cell infects
on average at least one susceptible cell. This is the parallel of having an R0 value of
more than 1 in the SIR model [
26
]. One can assume that on evolutionary timescales,
viruses have evolved to maximize this survival probability. We propose to compute
the distribution of p maximizing
P
. Other factors obviously affect the
survival probability, but they do not depend on
p
and will not affect the optimization
results.
The simplest distribution of
p
would obviously be a constant value of zero
epitopes for all viral proteins. However, given the fitness cost of adaptations, a more
complex picture emerges. Assuming each mutation has a multiplicative cost on the
survival probability, the survival probability is
(
survival
)
⎛
⎞
t
budding
⎝
−
μ
∑
i
⎠
,
)=
∏
i
d
p
i
0
−
p
i
exp
P
(
survival
d
t
p
i
x
i
T
i
(4)
0
0
where
d
represents the contribution of each mutation to the death probability (
d
<
1
p
i
0
—basal number of epitopes of protein
i
). The results of the optimization will
be similar even if a more complex function of the mutations is assumed:
,
⎛
⎞
exp
t
budding
∑
i
⎝
−
μ
0
∑
i
⎠
P
(
survival
)=
ϕ
p
i
d
t
p
i
x
i
T
i
0
as long as
is a positive function with a minimum at 0. We do not consider here the
cost of adaptations increasing the epitopes number.
Instead of maximizing
P
ϕ
(
survival
)
,
−
log
(
P
(
survival
))
is minimized, leading to
the following minimization problem:
⎛
⎞
t
budding
p
∗
)=
⎝
μ
0
∑
i
∑
i
(
p
i
0
−
p
i
)
⎠
,
L
(
min
p
i
≥
0
d
t
p
i
x
i
T
i
+
d
(5)
0
where
d
0. This problem is formally similar to assuming a constraint on
the number of epitopes (
=
−
ln
d
>
∑
i
p
i
>
c
) and minimizing:
t
budding
∑
i
μ
d
t
p
i
x
i
T
i
.
0
0
We here test the effect of two aspects of viral protein dynamics on the optimal
epitope number: the expression time and the total protein copy number. We assume a
simplistic model of viral protein dynamics to test these effects. We focus on the life
cycle of virus in a given cell and compute the optimal epitope density distribution
within this life cycle. This distribution is affected by the T cell distribution. We
present two extreme cases: the first case occurs when a large number of cells are
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