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The computers used in this experiment were laptops connected via the University of Granada WiFi, they were different models, and were running different operating systems and versions of them. The most powerful computer was the first one; then #2 was the second-best, and finally numbers 3 and 4 were the least powerful o... |
The first thing that was checked with the two problems examined (P-Peaks and MMDP) was whether adding more computers affected the solution rate. For P-Peaks there was no difference, independently of migration rate and number of computers, all experiments found the solution. However, there was a difference for MMDP, sho... |
The evolution with the migration rate can also be observed in figure ; as was advanced in the introduction, the relationship is quite complex and decrease or increase do not lead to a monotonic change of the success rate. In fact, the best success rate corresponds to the highest migration rate (migration after 100 gene... |
. However, it is not clear in this case that migration in 100 generations can be actually considered intermediate and in 200 too high, so more experiments will have to be performed to ascertain the optimum migration rate. |
Thus, having proved that success rate increases with the number of nodes, we will have to study how performance varies with it. Does the algorithm really finds the solution faster when more nodes are added? We have computed time only for the experiments that actually found the solution, and plotted the results in figur... |
As seen above in the case of MMDP, there is not a straightforward relationship between the migration rate and the time to solution; in this case, the relationship between the number of computers and time solution is also complex. If we look first at the P-Peaks experiment in we see that we obtain little time improvemen... |
The situation varies substantially for the MMDP, as seen in figure . In this case, the best result is obtained for four nodes and the smallest mutation gap (every 100 generations, dashed black line). However, it is interesting to observe that trend change for two nodes in all cases, either the solution takes more time ... |
Conclusions and future work In general, and for complex problems like the MMDP, a Dropbox-based system can be configured to take advantage of the paralellization of the evolutionary algorithm and obtain reliably (in a 100% of the cases) solutions in less time than a single computer would. Besides, it has been proved th... |
However, several issues remain to be studied. First, more accurate performance measures must be taken to measure how the time needed to find the solution in all occasions scales when new machines are added. We will have to investigate how parameter settings such as population size and migration gap (time passed between... |
, where it was found out that migrating the best one might not be the best policy. An important issue too is how to interact with Dropbox so that information is distributed optimally and with a minimal latency. In this case we had to stop each node for a certain time (which was heuristically found to be 1 second) to le... |
Finally, this framework opens many new possibilities for distributed evolutionary computation: meta-evolutionary computation, artificial life simulations, and big-scale simulation using hundreds or even thousands of clients. The type of problems suitable for this, as well as the design and implementation issues, will h... |
Acknowledgements This work has been supported in part by the CEI BioTIC GENIL (CEB09-0010) MICINN CEI Program (PYR-2010-13) project and the Andalusian Regional Government P08-TIC-03903 and P08-TIC-03928 projects. |
# Source: arxiv 1110.3368 # Title: Viral Evolution and Adaptation as a Multivariate Branching Process # Sections: all # Downloaded: 2026-03-03T01:59:30.431915+00:00 |
Viral Evolution and Adaptation as a Multivariate Branching Process Abstract In the present work we analyze the problem of adaptation and evolution of RNA virus populations, by defining the basic stochastic model as a multivariate branching process in close relation with the branching process advanced by Demetrius, Schu... |
Introduction RNA viruses exhibit a pronounced genetic diversity This variability allows RNA virus to better adapt to environmental challenges as represented by host and therapy pressures |
Due to the lack of a proofreading activity of viral RNA polymerases (average error incorporation rate in the order of [MATH] per nucleotide, per replication cycle |
), short generation times and huge population numbers, RNA viral populations may be viewed as a collection of particles bearing mutant genomes. As a consequence of high mutation rates, frequencies of mutants depend not only on their level of adaptation but on the probability of being faithfully replicated during viral ... |
It has become important to understand the process by which virus acquire diversity and the dynamics and fluctuations of this diversity in time. However, understanding viral evolution in vivo has proven to be a very cumbersome accomplishment due to the so many variables present in the interplay between virus and their h... |
Traditionally, in an effort to make the viral evolution process more palpable, several groups have addressed this subject from different points of view. There is a substantial amount of publications that studied virus populations during their evolution in experimental settings, for instance, cell cultures |
, by challenging the virus with population bottlenecks , or the introduction of antiviral drugs including mutagens, or another competing viral population. Experiment outcomes were evaluated using viral replication kinetics, the intensity and quality of the observed mutational spectra and virus survival/extinction as fi... |
These models are quite tractable but there is always the risk of oversimplification. To escape from oversimplifying the interplay between virus and hosts a model needs to incorporate a few hard rules based on previous experimental data which has been |
Based on other groups experimental data and previous mathematical models put forward by other investigators as the one presented by Lázaro et al. |
we sought to study a stochastic model for virus evolution that would be able to describe some general aspects of RNA virus evolution. Here, RNA viral evolution is described by a multivariate branching process during which each round of replication is accompanied by the introduction of a single point mutation per genome... |
Drake and Holland back in 1999 have inferred, based on limited data, a central value for the RNA virus mutation rate per genome per replication of |
[MATH] and suggested the rate per round of cell infection of [MATH] In 2010, Sanjuán et al. revisiting this theme by reviewing a list of previous publications encountered RNA virus mutational rates in the order of [MATH] |
to [MATH] with [MATH] for the bacteriophage [MATH] (Batschelet et al. ) and [MATH] for hepatitis C virus (Cuevas et al. ). It has been demonstrated that virus populations may be reduced at the moment of infection, and only a few particles are able to start a new infection process in naive hosts |
Abrupt reductions on RNA viral populations known as population bottlenecks may eliminate population diversity and lead the virus to pathways towards extinction due to the exacerbated effects of genetic drift. An incoming virus population recovering from a transmission bottleneck event may show an asymptotic behavior re... |
As pointed out by Drake and Holland , the basal value of RNA virus mutation rates is so large and RNA virus genomes are so informationally dense, that even a modest rate increase extinguishes the population. The frequent appearance of overlapping reading frames and multifunctional proteins augments the risk of a random... |
). If the introduction of a mutagen to a replicating virus population is able to cause its extinction by increasing mutational rates, the process is known as chemical lethal mutagenesis and has been demonstrated in a number of viruses including the vesicular stomatitis virus (VSV) |
human immunodeficiency virus type 1 (HIV-1) poliovirus type 1 foot-and-mouth disease virus lymphocytic choriomeningitis virus Hanta virus |
and Hepatitis C virus Accordingly, in the model, increases on mutational rates, and more specifically, on the deleterious component of the mutational spectrum are able to push viral populations towards extinction. Our results corroborate with the study from Bull, Sanjuán and Wilke |
by showing that the sufficient condition for lethal mutagenesis involves mutational and ecological aspects. Bull et al. arrived at a conjectural criteria for lethal mutagenesis by a heuristic and intuitive approach of possible general applicability. By applying the branching process theory to the evolution of RNA virus... |
is rigorously proven here. Furthermore, we describe four distinct regimes of RNA virus populations: transient regime, stationary equilibrium, extinction threshold, and extinction through lethal mutagenesis. |
We note that in previous works the properties of phenotypic models are discussed starting from a mean field linear model described by a mean matrix without reference to any underlying stochastic process modeling the microscopic dynamics of particle replication. In fact, almost any stochastic model of asexual replicatio... |
that take a different path and explicitly define the stochastic process in order to bring the mathematical theory of branching processes to bear. This attitude has some virtues, since it provides powerful tools, that have been perfected in the past several decades, allowing one to extract quantitative results on a rigo... |
Structure of the Paper. In section we describe a class of models for viral evolution and show that they define a multitype branching process. We explicitly compute the generating function and derive some elementary properties. In section we solve the spectral problem for the mean matrix of the model which allows us to ... |
Phenotypic Models for Viral Evolution In this section we describe a model for viral evolution that is naturally represented by a multivariate branching stochastic process generalizing, in a non-trivial way, the single-type branching process studied by Demetrius et al. |
For the sake of motivation we start by recalling a probabilistic model introduced by Lázro et al. We interpret the notion of mutation probability as a general effect of probabilistic nature acting on the replication capability of individual viral particles, considered here as a measure of the particle’s fitness charact... |
) for the temporal evolution of the viral population. This probability distribution gives appropriate parameters to classify the asymptotic behavior of the viral population and to describe some of the non-equilibrium properties of the model. |
In other related publications the concept of mutation is extensively used as the cause of replication capacity change. Understanding that those changes constitute an observable output due to many different factors (of genetic and non-genetic nature), we prefer to use the general term “effect” over the replication capac... |
2.1 Definition of the Model A number of viral infections starts with the transmission of a relatively small number of viral particles from one host organism to another one. The initial viral population starts replicating constrained by the unavoidable interaction with the host organism and evolves in time towards an ev... |
We consider that the whole set of particles composing the viral population replicates at the same time in such a way that the evolution of the population is described as a succession of discrete viral generations. This assumption crucially depends on the clear definition of the time needed for a particle to replicate, ... |
generation time . As it depends on the cell environment it is clear that this time period may vary from particle to particle replicating in different cells in such a way that the meaningful concept is a distribution of replication times with a possible clear mean value. The dispersion of the replication times can be co... |
Suppose that we have a population of viruses that start evolving from an initial set of particles (population at [MATH] ), which is partitioned into classes |
according to the replication capacity of each particle, that is, where each particle of class [MATH] produces no copies of itself, each particle with class [MATH] |
produces one copy of itself, and so on. We assume that there is a maximum replication capacity [MATH] imposed by the natural limiting conditions under which any particle of the population replicates. Moreover, as the process of replication is controlled by chemical reactions involving specific enzymes and the template,... |
In the process of replication of a viral particle errors may occur at each replication cycle in the form of point mutations with possible impact on the replication capacity of the progeny particles. Due to the intrinsic stochastic component of chemical reactions it is natural to treat this point mutational cause as pro... |
deleterious effect : the replication capacity of the copied particle decreases by one. When the particle has capacity of replication equal to [MATH] |
it will not produce any copy of itself. beneficial effect : the replication capacity of the copy increases by one. If the replication capacity is already the maximum allowed then the replication capacity of the copies will stay the same. |
neutral effect : the replication capacity of the copies is the same as the replication capacity of the parental particle. For each type of effect we associate a probability at the particle scale applicable to every single replication event: [MATH] for the probability of the occurrence of a |
deleterious effect, [MATH] for the probability of the occurrence of a beneficial effect and the complementary probability [MATH] is the probability of the occurrence of a neutral effect. In the case of in vitro experiments with homogeneous cell populations the parameters [MATH] [MATH] and [MATH] may be considered as mu... |
The simple phenotypic model is obtained by requiring that there are no beneficial effects in time, that is [MATH] This assumption is justified by several experimental results. The frequencies between beneficial, deleterious and neutral mutations appearing in a replicating population have been already measured by prior ... |
Taking their results together, it is reasonable to conclude that beneficial mutations could be as low as 1000 less frequent than either neutral or deleterious mutations. As a result the viral population would be submitted to a large number of successive deleterious and neutral changes and a comparatively small number o... |
From what is described above it should become clear that the model assumes a scenario where a probabilistic processes at the cellular/viral scale take place in the context of the interaction between the viral particle and the host cell. The combined effect of small scale processes are observed at the viral population s... |
Based on the general aspects of the phenomenon of viral replication it is compelling to to model it in terms of a branching process. In this perspective we define a discrete multitype Galton-Watson branching process |
for the evolution of the initial population, where the classes will be represented by the replication capabilities [MATH] The branching process is described by a sequence of vector-valued random variables |
[MATH] giving the number of particles in each replication class in the [MATH] -th generation. Thus [MATH] are vectors of non-negative integers satisfying the following assumption: if the size of the [MATH] -th generation is known, then the probability laws governing the later generations does not depend on the sizes of... |
[MATH] , which is non-zero and non-random. The temporal evolution of the population is obtained from a vector-valued discrete probability distribution [MATH] defined on the set of vectors with non-negative integer entries called the |
offspring distribution of the branching process. For any vector with non-negative entries [MATH] one has that [EQUATION] where [MATH] , with [MATH] in the [MATH] -th position. Thus, [MATH] is the joint probability that an individual particle of class [MATH] [MATH] ) generates [MATH] progeny particles in the class |
[MATH] [MATH] progeny particles in the class [MATH] , …, [MATH] progeny particles in the class [MATH] Note that any vector [MATH] may be written as a sum |
[MATH] and since each particle in [MATH] may be seen as the initial condition of a new branching process independently of the others, equation ( ) determines the probability laws for a general branching process as follows |
[EQUATION] In order to compute the offspring probability distribution [MATH] for the simple phenotypic model, we start by observing that [MATH] is non-zero only when |
[MATH] is of the form [MATH] since a particle with replication capability [MATH] can only produce progeny particles of the replication capability [MATH] or [MATH] , moreover the entries [MATH] and [MATH] should satisfy |
[MATH] Thus we just need to compute the probabilities [MATH] on the vectors of the form [MATH] Suppose that a viral particle [MATH] with replication capacity [MATH] |
[MATH] ) replicates itself producing new virus particles [MATH] For each new particle [MATH] , there are two possible outcomes regarding the type of change that may occur: neutral or deleterious, with probabilities [MATH] and [MATH] respectively. Representing the result of the [MATH] -th replication event by a variable... |
if the effect is neutral (success), the probability distribution of [MATH] is that of a Bernoulli trial with probability of occurrence of a neutral effect [MATH] |
(success), that is, [EQUATION] The total number of neutral effects that occur when the original virus particle reproduces is a random variable [MATH] given by the sum of all variable [MATH] since each copy is produced independently of the others, |
[EQUATION] That is, [MATH] counts the total number of neutral effects (successes) that occurred in the production of [MATH] virus particles [MATH] It also represents the total number of particles that will have the same replication capacity [MATH] of the original particle [MATH] It is well known (see Feller |
) that a sum of [MATH] independent and identically distributed Bernoulli random variables with probability [MATH] of success has a probability distribution given by the binomial distribution |
[EQUATION] Since this is the probability that a class [MATH] virus particle [MATH] produces [MATH] progeny particles with the same replication capability as itself one has therefore |
[EQUATION] Given the offspring probability distribution [MATH] one may set up a probability generating function [MATH] which is defined by the power series |
[EQUATION] The probability generating function of the simple phenotypic model is [EQUATION] Note that the functions [MATH] are polynomials whose coefficients are exactly |
[MATH] This function completely determines the branching process. Now it is easy to obtain the general case where the beneficial effects have a non-zero contribution [MATH] In this case, the binomial distribution is replaced by a |
trinomial distribution (see Feller ) and the probability generating function of the general phenotypic model is [EQUATION] Remark 2.1 |
It is worth to mention that there are other variations of these models that share the same essential properties and are more adequate in different contexts. |
With Zero Class: In this variation, which is the version deduced above, particles of class [MATH] are [MATH] Without Zero Class: In this variation, the particle class [MATH] is omitted and thus the probability generating function has [MATH] variables and [MATH] |
components: omit the variable [MATH] , the first component [MATH] and define [MATH] Particles of class [MATH] undergoing a deleterious change are eliminated in the next generation. |
2.2 Basic Properties of the Phenotypic Model We start by recalling that, when calculating probabilities and expectations, there is no loss of generality if one considers only initial populations consisting of just one particle of class [MATH] [MATH] ), since the general case can be decomposed as a sum of independent pr... |
We shall introduce the notation [MATH] for the condition [MATH] which is the initial population consisting of one particle of class [MATH] and zero particles of other classes. Thus [MATH] A basic assumption in the theory of branching processes is that all the first moments are finite and that they are not all zero. The... |
matrix of first moments [MATH] which describes how the averages of the sub-populations of particles in each replication class evolves in time: |
[EQUATION] In terms of the probability generating function one has [EQUATION] Denoting by [MATH] the jacobian matrix of [MATH] one may write |
[EQUATION] The evolution of the averages [MATH] of [MATH] is given by [EQUATION] From the generating functions ( ) and ( it is trivial to compute the mean matrix of the phenotypic model. The mean matrix of the simple phenotypic model is |
[EQUATION] Note that it is an upper triangular matrix. In the case of the general phenotypic model the mean matrix is [EQUATION] |
Interestingly, the mean matrix [MATH] provided by this class of models is a tridiagonal matrix , which is an ubiquitous type of matrix appearing in several fields ranging from statistical signal processing |
information theory , lattice dynamical systems Remark 2.2 In Demetrius et al. a single type branching process is proposed as a model for the evolution of polynucleotides, moreover they state that state that the results they obtained for the single type model are still valid in a multitype situation, provided one exclud... |
[MATH] , can be neglected. This implies, in particular, that the mean matrix is upper triangular In fact, it is easy to see that, in the simplest case where the mutation rates are non-zero only for adjacent classes, it is formally identical to the mean matrix ). Moreover, one of the main assumptions of the basic theory... |
, that is, some power of the matrix is positive. However, an upper triangular mean matrix is never primitive and hence the underlying branching processes, in principle, are out of the reach of the theory. Fortunately, as it will be shown here, there is a generalization of the theory of multitype branching processes whi... |
The mean matrix of the phenotypic model can be viewed as the adjacency matrix of a directed weighted graph where the nodes represent the particle classes according to their replication capacity and the arrows represent the effect of decrease or increase of the replication capacity due to the replication process (see Fi... |
The Simple Phenotypic Model For the simple phenotypic model it is easy to compute the eigenvalues [MATH] of the mean matrix [MATH] |
[EQUATION] In particular, the malthusian parameter is the largest positive eigenvalue [EQUATION] Therefore we have the following immediate result. |
Theorem 3.1 The simple phenotypic model has three distinct regimes. (i) If [MATH] then the branching process is sub-critical That is, with probability [MATH] , the virus population becomes extinct in finite time. |
(ii) If [MATH] then the branching process is super-critical That is, with positive probability, the virus population survives and grows indefinitely at an exponential rate proportional to [MATH] |
when [MATH] (iii) If [MATH] then the branching process is critical That is, with probability [MATH] , the virus population becomes extinct but this may take an infinite time to happen. |
Proof 3.2 This is a straightforward consequence of the classification of multitype branching processes, as generalized by Sevastyanov |
, which is necessary to include the simple phenotypic model, and equation ( ). Theorem 3.1 provides a partition of the parameter space of the simple phenotypic model [MATH] into two regions (see Figure ). The survival region defined by [MATH] and the extinction region |
defined by [MATH] The curve [MATH] gives the extinction threshold It is also important, specially in order to describe the asymptotic behaviour in the super-critical case, to know the left eigenvectors [MATH] |
and right eigenvectors [MATH] corresponding to the eigenvalue [MATH] Let us write the left and right eigenvectors in components as |
[EQUATION] and assume that they are normalized in the following way: [EQUATION] Then we have the following. (i) In the version “with zero class” the left eigenvector [MATH] is given by |
[EQUATION] and the right eigenvector [MATH] have coordinates [MATH] given by [EQUATION] (ii) In the version “without zero class” there is no components [MATH] and [MATH] The left eigenvector [MATH] is given by |
[EQUATION] and the right eigenvector [MATH] have coordinates [MATH] given by [EQUATION] It is interesting to note that the simple phenotypic model is a “completely solvable” branching process in the sense that we may explicitly solve the spectral problem for its mean matrix independently of the numerical values of the ... |
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