Evolutionary Principles in Viral Epitopes - Mathematical Methods and Models in Biomedicine - page 76

distribution is not a function of the peak density of each protein or of its precise
expression pattern. The only thing that affects the distribution is the integral of the
copy number until the virus can bud.
However, one can already see from the general solution that the synchronous
model has a sharper distribution, with some proteins having no epitopes and other
having all the epitopes. In contrast in the asynchronous models, all p i are strictly
positive.
Note that in both model, either increasing the total copy number or decreasing the
initial expression time will increase g i and lead to a lower optimal epitope number
as is indeed observed. An interesting question remaining is the effect of changing
both factors simultaneously in opposite directions. We here checked a couple of
simplified models for the protein dynamics to study these effects. Given more
realistic protein expression patterns, one can simply compute g i for each protein
and assess the optimal epitope distribution. A basic conclusion from the model up
to now is however that given two proteins with different values of g i , the one with
the higher g i is expected to have less epitopes.

Numerical Results

We assume a low initial value for each protein ( x i 0

1 ) and that budding occurs
after the expression of the last protein. If the saturation level is constant (we set
λ i / σ i =

= 0 .

1), A i is an increasing function of λ i ,and p i is a decreasing function

of λ i (Fig. 10 a). The more interesting situation is when early expressed proteins

have lower protein copy numbers. In such a case the low density could have been
expected to balance the risk induced by the early exposure. A simple example would
be to set / λ i , in other words, setting the saturation level to be inversely

proportional to the expression level. In such a case, early expressed proteins would
have a low saturation level. However, even in this model early expressed proteins
have less epitopes in the optimal solution both in the asynchronous (Fig. 10 a) and
the synchronous case (Fig. 10 b).

λ i / σ i ∝ 1

4 Conclusion and Discussion

We have here presented a systematic bioinformatics and theoretical analysis show-
ing the effect of two main elements on viral survival: (a) the protein expression
timing and (b) the protein total expression level. We have shown using the
bioinformatics analysis and the theoretical analysis that indeed if either the protein
copy number is low or its expression stage is late, the virus does not feel a strong
pressure to reduce its epitope number.
The interesting issue is the effect of their combination. We have shown using
the theoretical model that in case where these two are combined, the main element
shaping the viral escape is actually the expression time and not the copy number.