Virus Replication as a Phenotypic Version of Polynucleotide Evolution

Abstract

In this paper, we revisit and adapt to viral evolution an approach based on the theory of branching process advanced by Demetrius et al. (Bull. Math. Biol. 46:239–262, 1985), in their study of polynucleotide evolution. By taking into account beneficial effects, we obtain a non-trivial multivariate generalization of their single-type branching process model. Perturbative techniques allows us to obtain analytical asymptotic expressions for the main global parameters of the model, which lead to the following rigorous results: (i) a new criterion for “no sure extinction”, (ii) a generalization and proof, for this particular class of models, of the lethal mutagenesis criterion proposed by Bull et al. (J. Virol. 18:2930–2939, 2007), (iii) a new proposal for the notion of relaxation time with a quantitative prescription for its evaluation, (iv) the quantitative description of the evolution of the expected values in four distinct “stages”: extinction threshold, lethal mutagenesis, stationary “equilibrium”, and transient. Finally, based on these quantitative results, we are able to draw some qualitative conclusions.

Appendix: Spectral Perturbation Theory

In order to carry out the perturbative calculations, we need to solve the spectral problem for the matrix

The eigenvalues $\lambda_{r}^{0}$ of the mean matrix M ₀ are

$$\lambda_r^0=rc=r(1-d) \quad r=0,\ldots,R. $$

In particular, the Malthusian parameter is the largest positive eigenvalue

$$m^0=\lambda_R^0=Rc=R(1-d). $$

The normalized left and right eigenvectors, $\boldsymbol{v}_{r}^{0}$ and $\boldsymbol {u}_{r}^{0}$, associated to the eigenvalue $\lambda_{r}^{0}$, satisfy

$$\bigl(\boldsymbol{v}_s^0 \bigr)^{\mathrm{t}} \boldsymbol{u}_s^0=1 \quad\text{and}\quad \boldsymbol{1}^{\mathrm{t}} \boldsymbol{u}_s^0=1. $$

Writing $\boldsymbol{v}_{r}^{0}$ and $\boldsymbol{u}_{r}^{0}$ in components as

one has that

$$v_r^0(k)= \begin{cases} 0 & \mbox{for}\ k=0,\ldots,r-1, \\ \frac{-d(r+1)}{(1-d)^{r+1}} & \mbox{for}\ k=r+1,\\ \frac{((-1)d)^{k-r}}{(1-d)^{k}} & \mbox{for}\ k=r,r+2,\ldots,R \end{cases} $$

and

$$u_r^0(k)= \begin{cases} \binom{r}{k} (1-d)^k d^{r-k} & \mbox{for}\ k=0,\ldots,r, \\ 0 & \mbox{for}\ k=r+1,\ldots,R. \end{cases} $$

Now we write the eigenvalue λ _r of M as a function of the parameter b, expanded as a power series

$$\lambda_r = \lambda_r^0 + \lambda_r^1 b + \lambda_r^2 b^2 + \cdots, $$

where $\lambda_{r}^{0}$ is the corresponding eigenvalue of M ₀. Clearly, $\lambda_{r}(b)\to\lambda_{r}^{0}$ as b→0. The higher order perturbation coefficients $\lambda_{r}^{i}$, (i=1,2,3,…) are written in terms of all the eigenvalues $\lambda_{s}^{0}$ (s=0,…,R) of M ₀ and their associated left and right normalized eigenvectors $\boldsymbol{v}_{s}^{0}$ and $\boldsymbol{u}_{s}^{0}$ and the perturbation matrix

Note that the operator norm of P is ∥P∥=2(R−1) and so the magnitude of the perturbation is 2b(R−1).

The general expressions for the first and second order coefficients of the perturbation expansion of an eigenvalue are

We want to use these formulas to compute a perturbation approximation (for b around 0) of the Malthusian parameter m(b) of ( M). Recall that, m(0)=m ⁰=R(1−d) and the corresponding left and right eigenvectors are given by

with u _R (k)=binom(k;R,1−d). The first-order coefficient of the Malthusian parameter is

$$m^1 = \bigl(\boldsymbol{v}_R^0 \bigr)^{\mathrm {t}} \boldsymbol{P}\boldsymbol{u}_R^0 = \dfrac {(R-1)u_R(R-1)}{(1-d)^R} = \dfrac{R(R-1)d}{1-d}. $$

For the second-order coefficient, we also need

$$\boldsymbol{v}_{R-1}^0=1/(1-d)^{R-1} \bigl(0, \ldots,0,1,-Rd/(1-d) \bigr). $$

The second-order coefficient is

Analogous perturbative formulas exist for the perturbation expansion of the left and right eigenvector corresponding to an eigenvalue λ _r . The left and right eigenvector v _r and u _r of the matrix M associated with the eigenvalue λ _r can be written as a function of the parameter b, expanded as a vector-valued power series:

where $\boldsymbol{v}_{r}^{0}$ and $\boldsymbol{u}_{r}^{0}$ are, respectively, the left and right eigenvectors of the matrix M ₀ associated to the eigenvalue $\lambda_{r}^{0}$. The perturbation terms $A_{r,s}^{i}$ and $B_{r,s}^{i}$ with i=1,2,3,… , can be written in terms of the eigenvalues of M ₀ and their associated left and right eigenvectors and the perturbation matrix P. Observe that when b→0 we get the normalized left and right eigenvectors $\boldsymbol{v}_{R}^{0}$ and $\boldsymbol{u}_{R}^{0}$ associated to the dominant eigenvalue m ⁰=(1−d)R of the mean matrix M ₀.

We are interested in the normalized eigenvectors v _R and u _R associated to the Malthusian parameter m=(1−d)R

$$\boldsymbol{v}_R=\boldsymbol{v}_R^0 + b \sum _{k=0}^{R} \alpha_k \boldsymbol{v}_k^0 + \mathit {O}\bigl(b^2 \bigr), $$

with $\alpha_{k}=A_{R,k}^{1}$, (0⩽k⩽R−1) and $\alpha_{R}=-\sum_{k=0}^{R-1} \alpha_{k}$,

$$\boldsymbol{u}_R=\boldsymbol{u}_R^0 + b \sum _{k=0}^{R} \beta_k \boldsymbol{u}_k^0 + \mathit {O}\bigl(b^2 \bigr), $$

with $\beta_{k}=B_{R,k}^{1}$, (0⩽k⩽R−1) and $\beta_{R}=-\sum_{k=0}^{R-1} \beta_{k}$.

The first-order coefficients are given by

and the second-order terms are given by

for s=0,…,R−1.

One may use the formula of v _R up to second order

$$\boldsymbol{v}_R = \boldsymbol{v}_R^0 + b \dfrac{(R-1)}{(1-d)^2}\boldsymbol{v}_{R-1}^0 + \mathit {O}\bigl(b^2 \bigr) $$

and the first-order expansion $\boldsymbol{u}_{R} = \boldsymbol{u}_{R}^{0} + \mathit {O}(b)$ in order to estimate the product of coefficients u _R (r)v _R (r) (up to their leading order). For instance, it is easy to find that

In general, using a higher order expansion formula for v _R , one concludes that

$$u_R(R-r)v_R(R-r) = \mathit {O}\bigl(b^r \bigr). $$

This follows form the fact that the coefficient of order b ^r in the perturbation expansion of v _R is a linear combination of the left eigenvectors $\boldsymbol{v}_{R-1}^{0},\ldots,\boldsymbol{v}_{r-R}^{0}$, all of which have zeros in the first r−1 components.