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Next we turn to the computation of the extinction probabilities [MATH] In this case, it is necessary to solve a non-linear system of polynomial equations: |
[EQUATION] This may be done in a recursive way, since the equation for [MATH] is already solved [MATH] and the equation for [MATH] depends only on [MATH] and [MATH] Thus we get for [MATH] |
[EQUATION] When [MATH] the formulas become very complicated and when [MATH] the equation may not even be solvable by radicals, but in general one may write |
[EQUATION] where [MATH] and [MATH] is a strictly increasing smooth function on [MATH] satisfying: (i) [MATH] , (ii) [MATH] , (iii) [MATH] |
for [MATH] and (iv) [MATH] This expression suggests that the surviving probabilities [MATH] can be interpreted as an order parameter associated to the occurrence of a |
phase transition when the deleterious probability [MATH] attains the critical point [MATH] , which marks the transition from super-criticality to sub-criticality, |
[EQUATION] where [MATH] and thus satisfies: (i) [MATH] , (ii) [MATH] (iii) [MATH] for [MATH] and (iv) [MATH] Observe that for a fixed numerical value of [MATH] , the system of equations ) can be easily solved by numerical approximation using Newton’s method. For instance, in Figure we show the curves for the surviving ... |
The result shows that, with respect to [MATH] , the model has a critical behavior in complete analogy to a second order phase transition (see Figure ). Therefore, the critical properties of the model can be characterized by means of relevant critical exponents. |
Finally, it is not difficult to see that for fixed [MATH] , the numbers [MATH] satisfy [MATH] and therefore the extinction probability for a general initial condition |
[MATH] may be estimated far from the critical deleterious probability [MATH] by [EQUATION] where [MATH] and near [MATH] by [EQUATION] |
It has been demonstrated that large population passages are able to increase the adaptability of virus populations On the other hand, small population passages represented by bottleneck events are capable to increase the risk towards viral extinction. Among the aspects of abrupt population reductions are the exacerbate... |
may lead to the random and progressive lost of the best adapted virus in a population. It also has been suggested that large virus populations bearing a significant phenotypic diversity are more adaptable to environment fluctuations and robust. It is correct to assume that large initial virus populations colonizing new... |
From now on we shall split the analysis of the simple phenotypic model according to which it is sub-critical, super-critical or critical. |
3.1 The Sub-critical Regime: Lethal Mutagenesis The first consequence of our results is a generalization, in the context of the phenotypic model (provided one assumes that all effects are of purely mutational nature), of the conjecture of lethal mutagenesis of Bull, Sanjuán and Wilke |
Recall that Bull et al. assume that all mutations are either neutral or deleterious and write the mutation rate [MATH] where the component [MATH] comprises the purely neutral mutations and the component [MATH] comprises the mutations with a deleterious fitness effect. Let [MATH] denote the maximum reproductive capacity... |
states that a sufficient condition for extinction is [EQUATION] According to Bull et al. , the factor [MATH] is both the mean fitness level and also the proportion of offspring with no non-neutral mutations. In the absence of beneficial mutations the only type of non-neutral mutations are the deleterious mutations and ... |
[EQUATION] More precisely, if [MATH] denotes the mean population at time [MATH] then the offspring at time [MATH] is given by [MATH] and thus |
[EQUATION] Since the maximum reproductive capacity among all particles in the viral population in our model is given by the maximum number of replicative classes [MATH] , it follows that |
[MATH] Therefore, the extinction criterion ( 11 ) is equivalent, in the context of the simple phenotypic model, to [EQUATION] which is exactly the condition for the model to be sub-critical. |
Corollary 3.3 In the simple phenotypic model, the virus population becomes extinct in finite time, with probability [MATH] , if the product of the neutral effect probability |
[MATH] with the maximum replication capacity [MATH] is strictly less than [MATH] In another work, Bull et al. suggest a modification of the extinction threshold eq. ( 11 ) that accounts for beneficial effects as long as they do not couple the deleterious ones (see Antoneli et al. |
for a more general result). The main conclusion here is that the existence of lethal mutagenesis depends on “genetic components” (mutational rates) and other additional deleterious effects (host driven pressures intensifications), as well as on strict “ecological components”, namely, the maximum replication capacity of... |
that induce errors in the generation process of new viral particles reducing their replication capacity. A straightforward consequence of extinction criterion eq. ( 11 or eq. ( 13 ) is that a single particle showing the maximum replication capacity [MATH] is able to rescue a viral population driven to extinction by mut... |
3.2 The Super-critical Case: Relaxation and Equilibrium In the super-critical regime, the population grows at a geometric pace indefinitely. Nevertheless, there are two distinct phases that occur during this growth: a transient phase (“relaxation”or “recovery time”) and a dynamical stationary phase. |
3.2.1 Relaxation towards equilibrium. An important question concerning the adaptation process of a viral population to the host environment is the typical time needed to achieve the equilibrium state. As the equilibrium is characterized by constant mean replication capacity an obvious criteria to measure the time to ac... |
). Nevertheless, this method is clearly subjected to the limitations of numerical accuracy with evident disadvantages if one wants a sharp and universal criterion to differentiate populations from the point of view of how fast a population can be typically stabilized in a organism. |
Viral populations are commonly submitted to transient regimes. As pointed out earlier the infection transmission process represents the passage of a small number of particles from one organism to another in such a way that in this process the viral population is submitted to a subsequent bottle-neck effect during sprea... |
[MATH] In order to find the characteristic decay rates one should consider the recursive application of the mean matrix [MATH] on the initial population: |
[MATH] In fact, it is enough to consider the canonical initial population [MATH] By direct inspection it is easily verified that the decay of correlations is typically exponential and given by |
[EQUATION] where [MATH] is the malthusian parameter. The decay rate is therefore given by [MATH] Among others, one possible application of this result relates to the very initial phase of the infection process. If we consider that during this phase the host immune system has not been yet stimulated against the virus, o... |
defines the degree of virulence of the infection during the early stage of the infective process. The increment of deleterious effects plays an opposite role on the decay rates. In fact, as it will be shown below the closest the parameter [MATH] is to its critical value [MATH] more time is needed to achieve equilibrium... |
3.2.2 The Dynamical Stationary State. When the simple phenotypic model is super-critical and is initialized with exactly one particle in the class [MATH] [MATH] ) the effective malthusian parameter is |
[MATH] with corresponding normalized right eigenvector [MATH] , where the components [MATH] with [MATH] , are [EQUATION] Therefore, the simple phenotypic model has [MATH] distinct asymptotic distributions of types of particles, describing [MATH] distinct dynamical stationary states characterized by their asymptotic dis... |
Theorem 3.4 If the simple phenotypic model is super-critical with malthusian parameter [MATH] and starts with at least one particle of class [MATH] then, in the long run, the relative number of particles in each class reaches a stable stationary dynamical state and is (up to a random scalar perturbation) distributed ac... |
Binomial Distribution: [MATH] , where [MATH] are the replication classes. Proof 3.5 This is consequence of the generalized Kesten-Stigum |
results about the asymptotic behaviour of decomposable (i.e., with non-primitive mean matrix) super-critical multitype branching processes and the computation of the normalized right eigenvector associated to the malthusian parameter [MATH] given by equation ( 14 ). |
From theorem 3.4 we immediately obtain: The mean replication capacity is [EQUATION] The phenotypic diversity is [EQUATION] It is well accepted that the phenotypic diversity is an important characteristic of the viral population intuitively related to the idea of population robustness |
In fact, a homogeneous population would be less flexible from the point of view of adaptation. The variance associated with the stationary state can be understood as a natural quantity to measure diversity. It shows that the maximum value of the phenotypic diversity [MATH] is reached if [MATH] for any value of [MATH] I... |
Another important consequence of the above results concerns the efficiency of the use of mutagenic drugs. In the region [MATH] the viral population’s most representative particle is the fittest one (class [MATH] ). If we assume that the drug action is deeply influenced by drug transport coefficients in different host t... |
3.3 The Critical Case: Extinction Threshold The clearest way to characterize the time behavior of the viral population at or around the critical point is through the typical time [MATH] to approach equilibrium derived from the decay of correlations described above. |
The expression [MATH] shows that at the critical point the equilibrium state is never reached, i.e., the decay to equilibrium is at least non-exponential. A scaling exponent characterizing the behavior of [MATH] in the neighborhood of the critical point [MATH] can be easily obtained. The expansion around [MATH] gives |
[EQUATION] Although it is always possible to calculate intermediate distributions of progeny, it is quite easy to see that at the critical point the time evolution of densities never achieves an invariant density. |
Unlike in the super-critical regime, the relative number of particles in each class/sub-population is never stable. Nevertheless, our preliminary results concerning the dynamics of fluctuations show that the time variation of the numbers of particles in each separated class follows a pattern such that the variation obs... |
In fact, according to Eigen, when mutational rates are increased beyond a threshold, infinite viral populations are not anymore able to retain its best adapted variants. At this critical mutation level, selection is overruled by mutation and all variants share the same fitness status. Moreover, populations at Eigen’s e... |
If in the super-critical case the notion of the mean replication capacity and therefore that of the “mean viral particle” exists defining a typical scale in the system, in the critical case this notion is absent. Therefore, in using branching processes to model the time behavior of viral populations the concept of erro... |
The critical behavior of the model can also be observed through the survival probability function [MATH] for [MATH] as show in Figure Expansion of the survival probability function [MATH] around the critical point [MATH] using the system of equations for the extinction probabilities ( ) gives directly |
[EQUATION] It is interesting to note that the critical exponents of [MATH] and [MATH] are the same found in critical behavior of a large class of dynamical random networks |
However, it is noteworthy that here we talk about criticality of a process taking place in time, and therefore the term critical phenomenon (imported from equilibrium statistical mechanics of space distributed systems) is used to highlight the fact that the survival probability behaves like an order parameter and the a... |
This fact is reminiscent from the deep relation existing between branching process and random network theory, where the survival probability function of a branching process is identified with the order parameter associated to the emergence of the giant cluster in a dynamical random network. This fundamental observation... |
and more recently it has become the central technique in the study of more general models of random networks In this direction, it is worth to also note that there is a correspondence between Eigen’s model of molecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system |
, again an indication of a relation between statistical mechanics, random networks and branching processes. The relation between these theories is certainly expected to bring important new insights to virus evolution in the future. |
Conclusions and Outlook Using the previous theoretical model for virus evolution proposed by Lázaro et al. and Aguirre et al. as a starting point we show that virus evolution can be described by an exact solvable multivariate branching process. By applying our approach we are able to identify crucial aspects of the dyn... |
According to our explicit formulas for the progeny distribution, we demonstrate that virus populations maximize their phenotypic diversity by replicating with [MATH] near |
[MATH] , for any value of [MATH] We speculate that this might be a universal property for RNA viruses that replicate under high mutational rates. In this way by increasing their phenotypic diversity viruses augments their chances of survival escaping and adapting to environmental pressures. Maintenance of high mutation... |
We also demonstrate that by keeping the deleterious effects constant the survival probability of a virus population will depend on its initial population size. By increasing the population size at time zero we push the survival probability curves, in the region before the critical point, towards one (see Figure and equ... |
However, according to the model and as discussed before, the [MATH] parameter determines the success of an incoming virus population because the corresponding value of [MATH] is uniquely given by [MATH] . The present work suggests that minimum innoculums must have at least one particle with replicative capacity large e... |
In fact, the experimental data about viral load in HIV early infected patients strongly suggests that the host deleterious effects over the viral population are minimal and increase after the onset of the immunological response |
We note that the characteristic form of this data can be easily reproduced by the model (see Castro et al. ). Finally, it is important to mention that the close relation of the theory of branching processes (as used in the present work) and dynamical Erdös-Renyi type networks indicates that the latter may be brought to... |
Acknowledgments FA wish to acknowledge the support of CNPq through the grant PQ-313224/2009-9 and thanks FAP-UNIFESP and BIOMAT Consortium for the financial support to present this work at the “12th International Symposium on Mathematical and Computational Biology”. FB recieved support from the Brazilian agency FAPESP.... |
Appendix A Review of the Theory of Branching Processes In this section we collect a few definitions and results from the theory of branching process that will be necessary in our analysis of the phenotypic model. |
The Mean Matrix of a Branching Process Consider a multitype branching process [MATH] with offspring probability distribution [MATH] and probability generating function [MATH] Suppose that [MATH] has all its first moments finite and not all zero. Then conditioning on the elementary initial populations [MATH] on may defi... |
mean matrix [MATH] of the multitype branching process [MATH] by [EQUATION] In general, a multitype Galton-Watson branching process can be classified into |
decomposable and indecomposable according to which its mean matrix is reducible or irreducible, respectively. A non-negative matrix [MATH] |
[MATH] is called irreducible if for every pair of indices [MATH] and [MATH] , there exists a natural number [MATH] such that [MATH] and it is called reducible otherwise (see Gantmatcher |
). There is another characterization of irreducibility in terms of the graph of the matrix. The graph [MATH] of [MATH] is defined to be the directed graph on [MATH] nodes [MATH] , each corresponding to a type of particle, in which there is a directed edge leading from node [MATH] to node [MATH] if and only if |
[MATH] A graph [MATH] is called path connected if for each pair of nodes [MATH] there is a sequence of directed edges leading from [MATH] to [MATH] A matrix [MATH] is irreducible if and only if [MATH] is path connected (see Meyer |
). A multitype Galton-Watson branching process is called positively regular if its mean matrix [MATH] is primitive , that is, [MATH] is positive for some positive integer [MATH] In particular, a positively regular branching process is indecomposable, since a primitive matrix is irreducible (see Gantmatcher |
or Meyer ). Positive regularity is a standard assumption in the study of multitype branching processes, as it opens up the way to apply the powerful Perron-Frobenius theory (see Harris |
or Athreya and Ney ). Example A.1 The classification of the phenotypic model according to the irreducibility or reducibility of its mean matrix is the following: |
(i) In the version “with zero class” the mean matrix ( ) or ) will have the first column filled with zeros, that is, they are not primitive matrices and thus the corresponding branching processes are not positively regular. Moreover, a quick look at the graph [MATH] in Figure (b) shows that the process is decomposable ... |
(ii) In the version “without zero class” the mean matrix of both models can be obtained from ( ) and ( ) by removing the first row and the first column. Now the general phenotypic model becomes positively regular, since the node corresponding to particles of class [MATH] no longer exists. The simple phenotypic model st... |
Malthusian Parameter and Extinction Probability Let [MATH] denote the spectral radius of [MATH] that is, if [MATH] are the eigenvalues of [MATH] |
then [EQUATION] Since [MATH] is a non-negative matrix, it has at least one largest non-negative eigenvalue which coincides with its spectral radius (see Gantmatcher |
or Meyer ). When the largest eigenvalue is positive we shall call it, following Kimmel and Axelrod , the malthusian parameter [MATH] of the branching process (see also Jagers et al. |
). The malthusian parameter of a multitype Galton-Watson branching process plays the same role as the mean of the probability distribution of the offspring in a simple Galton-Watson process and its name is motivated by equation ), which implies that [MATH] , the average population size increases or decreases at a geome... |
Finally, it follows from the theory of non-negative matrices that there is a left non-negative eigenvector [MATH] and a right non-negative eigenvector |
[MATH] corresponding to the eigenvalue [MATH] [EQUATION] which can be normalized so that [EQUATION] where [MATH] is the transposed of the vector [MATH] Moreover, when [MATH] is irreducible the left and right eigenvectors are positive (see Gantmatcher |
or Meyer ). Let [MATH] be the vector of extinction probabilities [EQUATION] the probability that the process eventually become extinct given that initially there is exactly one particle of class [MATH] In general, when the initial condition is given by a vector of non-negative integers |
[MATH] the extinction probability is [EQUATION] A basic result of the theory of branching processes is that the vector of extinction probabilities [MATH] is the solution in [MATH] with smallest components of the equation |
[EQUATION] where [MATH] is the probability generating function. Observe that [MATH] is always a fixed point of [MATH] that is, a solution of equation ( 15 ). Therefore, if there is no other solution of equation ( 15 in the unit cube [MATH] then the process always has probability [MATH] to become extinct. |
The main classification result in the indecomposable case, states that there are only three possible regimes (see Harris or Athreya and Ney |
): (i) If [MATH] then [MATH] is the unique stable fixed point of [MATH] in the unit cube [MATH] different than [MATH] and the branching process is called super-critical Therefore, with positive probability, the population will survive indefinitely. |
(ii) If [MATH] then [MATH] is the unique stable fixed point of [MATH] in the unit cube [MATH] and the branching process is called sub-critical Therefore, with probability [MATH] , the process will become extinct in finite time. |
(iii) If [MATH] then [MATH] is the unique marginal fixed point of [MATH] in the unit cube [MATH] and the branching process is called critical Here, the expected time to extinction is infinite, despite the fact that extinction is bound to occur almost surely. |
Unfortunately this theorem does not cover all the interesting cases, one important example for us being the phenotypic model for viral evolution. Nevertheless, one of the earliest results about decomposable branching processes is the generalization of the classification, due to Sevastyanov (see Harris |
and Jiřina ). In the general decomposable case, there is a fourth alternative identified by Sevastyanov and in order to formulate this condition we need to introduce another important concept. |
A multitype Galton-Watson branching process is called singular if its probability generating function is linear without constant term, that is, |
[MATH] In this case, there is no branching since each particle produces exactly one particle that can be of any class and the process is equivalent to an ordinary finite Markov chain. More generally, a decomposable process may have singular path components Two nodes [MATH] and [MATH] are said to be in same path compone... |
singular path component , if any particle whose class is in [MATH] has probability [MATH] of producing, in the next generation, exactly one particle whose class is in [MATH] Equivalently, the component functions of the probability generating function corresponding to the classes in a path component [MATH] are linear fu... |
singular path component then the branching process never become extinct, no matter what is the value of the malthusian parameter. |
Example A.2 The graph corresponding to the general phenotypic model (Figure (b)) have two path components: [MATH] and [MATH] In the simple phenotypic model (Figure (a)), the path components are exactly the sets containing one node, [MATH] [MATH] , …, [MATH] From the expressions of the generating functions ( ) and ) it ... |
positively regular Therefore, the phenotypic model displays only the three regimes determined by the malthusian parameter, which depends on the values of the parameters [MATH] and [MATH] |
It is important to stress that the regime of a multitype branching process can not be read from the mean matrix alone (i.e, the malthusian parameter). Essentially this happens because of the existence of decomposable branching processes with singular components. |
Example A.3 Consider the following generating functions: [EQUATION] where [MATH] and [MATH] They have the same mean matrix given by |
[MATH] and so the malthusian parameter is [MATH] It is easy to solve the fixed point equation ( 15 in both cases and compute the respective extinction probability vectors |
[MATH] : for the function [MATH] we have that [MATH] and [MATH] if [MATH] and [MATH] if [MATH] For the function [MATH] we have that [MATH] Therefore, the branching process defined by [MATH] becomes extinct if and only if |
[MATH] while the branching process defined by [MATH] never becomes extinct irrespective of the value of the malthusian parameter! |
Asymptotic Behaviour of Surviving Populations According to the “Malthusian Law of Growth” it is expected that a super-critical branching process will grow indefinitely at a geometric rate proportional to [MATH] |
and we would like to write [MATH] , where [MATH] is a random vector with a finite “asymptotic distribution of classes” when [MATH] The formalization of this heuristic argument is due to Kesten and Stigum (see Kesten-Stigum |
for the case of indecomposable multitype branching processes and Kesten-Stigum for the case of a general decomposable multitype branching processes). |
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