The average number of new virus particles (virions) produced by a virally-infected
cell during its lifetime is called the burst size. We will call the burst size N. Under reasonable
assumptions, N can be approximated by N = p/u�(p) where p is the rate of production of new
viruses and u�(p) is the cell death rate. Over many generations, the virus population can evolve so
that its production rate p lies anywhere on the range 0 < p < pmax. The upper bound pmax is set by
biological constraints (e.g. nutrient supply to the cells). We expect the virus to evolve to maximize
N.
(a) Suppose that the cell death rate is a constant, u�(p) = �0 > 0. What production rate would
you expect the virus to have after a long period of evolutionary change?
(b) Suppose the cell death rate is a linear function of the production rate, �u(p) = ap+�0, where
a and �0 are positive constants. What production rate would you expect the virus to have
after a long period of evolutionary change? Does this depend on a? If so, how?
(c) Finally, suppose the cell death rate is a quadratic function of the production rate, �u(p) =
bp^2 + u0, where b and �u0 are positive constants. What production rate would you expect the virus to have after a long period of evolutionary change? Does this depend on b? If so, how?
(a) In this case, where the cell death rate is a constant, u(p) = u0, we can determine the production rate that would be expected after a long period of evolutionary change by maximizing the burst size, N = p/u(p).
To find the maximum of N, we can take the derivative of N with respect to p and set it equal to zero. Differentiating N = p/u(p) with respect to p gives:
dN/dp = [u(p) - p * u'(p)] / u(p)^2
Setting dN/dp = 0, we get:
0 = [u(p) - p * u'(p)] / u(p)^2
Multiplying through by u(p)^2:
0 = u(p) - p * u'(p)
Solving this equation for p will give us the production rate that maximizes N.
(b) In this case, where the cell death rate is a linear function of the production rate, u(p) = ap + u0, the production rate that would be expected after a long period of evolutionary change can be determined by maximizing the burst size, N = p/u(p).
We can use the same approach as in part (a) to find the maximum of N. Taking the derivative of N = p/u(p) with respect to p:
dN/dp = [u(p) - p * u'(p)] / u(p)^2
Substituting u(p) = ap + u0:
dN/dp = [(ap + u0) - p * a] / (ap + u0)^2
Setting dN/dp = 0:
0 = [(ap + u0) - p * a] / (ap + u0)^2
Multiplying through by (ap + u0)^2:
0 = (ap + u0) - p * a
Solving this equation for p will give us the production rate that maximizes N. The specific value of a does not affect the critical point, but it may affect the behavior of the function.
(c) In this case, where the cell death rate is a quadratic function of the production rate, u(p) = bp^2 + u0, the production rate that would be expected after a long period of evolutionary change can be determined by maximizing the burst size, N = p/u(p).
Again, we can use the same approach as in parts (a) and (b) to find the maximum of N. Taking the derivative of N = p/u(p) with respect to p:
dN/dp = [u(p) - p * u'(p)] / u(p)^2
Substituting u(p) = bp^2 + u0:
dN/dp = [(bp^2 + u0) - p * (2bp)] / (bp^2 + u0)^2
Setting dN/dp = 0:
0 = [(bp^2 + u0) - p * (2bp)] / (bp^2 + u0)^2
Multiplying through by (bp^2 + u0)^2:
0 = (bp^2 + u0) - p * (2bp)
Solving this equation for p will give us the production rate that maximizes N. The specific value of b will affect the critical point, and different values of b may result in different production rates that maximize N.