The evolutionary dynamics of viruses: virion release strategies, time delays and fitness minima

Abstract Viruses exhibit a diverse array of strategies for infecting host cells and for virion release after replication. Cell exit strategies generally involve either budding from the cell membrane or killing the host cell. The conditions under which either is at a selective advantage is a key question in the evolutionary theory of viruses, with the outcome having potentially important impacts on the course of infection and pathogenicity. Although a plethora of external factors will influence the fitness of either strategy; here, we focus just on the effects of the physical properties of the system. We develop theoretical approaches to assess the effects of the time delays between initial infection and virion release. We show that the length of the delay before apoptosis is an important trait in virus evolutionary dynamics. Our results show that for a fixed time to apoptosis, intermediate delays lead to virus fitness that is lower than short times to apoptosis — leading to an apoptotic strategy — and long times to apoptosis — leading to a budding strategy at the between-cell level. At fitness minima, selection is expected to be disruptive and the potential for adaptive radiation in virus strategies is feasible. Hence, the physical properties of the system are sufficient to explain the existence of both budding and virus-induced apoptosis. The fitness functions presented here provide a formal basis for further work focusing on the evolutionary implications of trade-offs between time delays, intracellular replication and resulting mutation rates.


Model with fixed time to apoptosis and no virus budding
Model with budding delay and no virus-induced apoptosis The details of the parameters are given in the main text.

Equilibrium conditions
The equilibrium conditions for the model without delays (Eq. 1 in the main text) are: The equilibrium conditions for the model with delays (Eq. 2 in the main text) are: where σ = exp(−µ C τ ) and σ = exp(−µ C τ ) -the terms that account for natural cell death before either virus budding or virus release by apoptosis.
The equilibrium conditions for the model with fixed time to apoptosis and no virus budding (S1) are:Ŝ The equilibrium conditions for the model with budding delay and no apoptosis (S2) are:

Derivation of virus fitness functions
Model assuming constant hazard of apoptosis and immediate budding The virus fitness function for the model without delays (Eq. 1 of the main manuscript) can be determined from the determinant of a matrix of the partial derivatives of the contribution of infected cells (I) and free living virus (V ). This matrix (J) has the form: where ω are eigenvalues of the matrix (and the dominant eigenvalue is a measure of fitness). Taking the determinant of this matrix and setting equal to zero yields: Solving this expression for ω gives: Virus fitness in the absence of delays is positive if the discriminant is greater than zero such that, (α + µ C + µ V ) 2 /4 > µ V (µ C + α) − βŜ(λ + γα) and this virus strategy will evolve.

Models with delays
A similar approach can be used to derive a fitness function for the virus model with fixed delays (Eq. 2 of the main manuscript). Due to the time delays, the matrix now has the form: To solve this expression we begin by using exp(−x) = 1 − x which is valid when x is small, under weak selection. This is a reasonable assumption if the difference between a mutant and a resident virus is small. The virus fitness function when fixed time to apoptosis and budding delay are included is: In the same way as detailed above, we can derive fitness functions for the delay models with virus release from virus-induced apoptosis only (S1) and virus release by budding only (S2). With apoptosis only the fitness function is: With budding only (S2), the fitness function is:

Evolutionary invasion analysis
Model assuming constant hazard of apoptosis and immediate budding As described in the main text, we model the invasion, from rare of a 'mutant' virus (m) that releases virions by apoptosis, in the presence of a 'resident' virus that releases virions by budding. For the model assuming constant hazard of apoptosis and immediate budding, this uses the fitness function in Eq. S9, by setting the budding rate (λ) to zero and incorporating the equation for the steady state level of susceptible cells (Ŝ) as per Eq. S3 where the apoptosis rate (α) and yield at apoptosis (γ) are set to zero, to give:

Models with delays
For the models with delays, we use the virus fitness function for the apoptosis only strategy (S13) and replace the steady-state levels of susceptible cells (Ŝ) with the formula in Eq S6 for the budding only strategy to give:

Comparing fitness between models with and without delays
We can show using a simple example why values for fitness are generally lower for the model including delays. For the model without delays, if: and we set β = S = µ V = 1, then virus fitness is: However, with delays: setting β = S = µ V = 1, then virus fitness is: With time delays, fitness is expected to be reduced due to the inherent 'costs' associated with a delay.

Full derivation of the time delay model
We use the full derivation of the time delay model to investigate the interplay between the time delays, budding rate and yield at apoptosis on virus fitness. This derivation is approached in a similar way to the simpler methods used to approximate virus fitness. As above, we start from the point where the trivial steady state (when V * = 0) is perturbed and now assume that the perturbation lasts from t 0 − τ to t 0 , and if the displacement is δx(t), then, generally, Given that f (x * , x * ) = 0, a Taylor series expansion of Eq. 2 in the main manuscript yields the following matrix: In order to get nonzero solutions, we need that: where I is the identity matrix. This characteristic equation is then of the form: When ω = iθ (as roots cross from the negative complex half-plane to the positive half-plane) then: where χ = βS [cos(θτ ) − isin(θτ )] exp(−µ C τ ). Equating real and imaginary parts yields the following set of simultaneous equations: At the point of spread, when θ = 0, then So the the condition for the virus with budding and apoptosis to spread is: Using this approach we investigate the role of budding rate, yield at apoptosis and time delays on the evolution of virus strategies.

A limiting case -long budding delays
For long budding delays (τ → ∞), exp(−µ I τ )exp(−ωτ ) → 0, so the Jacobian is: The characteristic equation is then: When ω = iθ (as roots cross from the negative complex half-plane to the positive halfplane) then: Equating real and imaginary parts yields the following set of simultaneous equations: A limiting case -long apoptosis delays The characteristic equation is then: When ω = iθ (as roots cross from the negative complex half-plane to the positive halfplane) then: Equating real and imaginary parts yields the following set of simultaneous equations: θ 2 + µ C µ V − βSλexp(−µ I τ )cos(θτ ) = 0 θ(µ C + µ V ) + βSλexp(−µ I τ )sin(θτ ) = 0 If θ = 0, then The relative ratio of virus births to deaths has to be greater than the budding time delay for the virus to spread under long apoptosis delays.