Stochastic Evolutionary Model Research Articles

On the origin and evolution of life in the Galaxy

AbstractA simple stochastic model for evolution, based upon the need to pass a sequence of n critical steps is applied to both terrestrial and extraterrestrial origins of life. In the former case, the time at which humans have emerged during the habitable period of Earth suggests a value of n=4. Progressively adding earlier evolutionary transitions gives an optimum fit when n=5, implying either that their initial transitions are not critical or that habitability began around 6 Ga ago. The origin of life on Mars or elsewhere within the Solar System is excluded by the latter case and the simple anthropic argument is that extraterrestrial life is scarce in the Universe because it does not have time to evolve. Alternatively, the timescale can be extended if the migration of basic progenotic material to Earth is possible. If extra transitions are included in the model to allow for Earth migration, then the start of habitability needs to be even earlier than 6 Ga ago. Our present understanding of Galactic habitability and dynamics does not exclude this possibility. We conclude that Galactic punctuated equilibrium, proposed as a way round the anthropic problem, is not the only way of making life more common in the Galaxy.

A stochastic evolution model for residue Insertion–Deletion Independent from Substitution

Abstract We develop here a new class of stochastic models of gene evolution based on residue Insertion–Deletion Independent from Substitution ( IDIS). Indeed, in contrast to all existing evolution models, insertions and deletions are modeled here by a concept in population dynamics. Therefore, they are not only independent from each other, but also independent from the substitution process. After a separate stochastic analysis of the substitution and the insertion–deletion processes, we obtain a matrix differential equation combining these two processes defining the IDIS model. By deriving a general solution, we give an analytical expression of the residue occurrence probability at evolution time t as a function of a substitution rate matrix, an insertion rate vector, a deletion rate and an initial residue probability vector. Various mathematical properties of the IDIS model in relation with time t are derived: time scale, time step, time inversion and sequence length. Particular expressions of the nucleotide occurrence probability at time t are given for classical substitution rate matrices in various biological contexts: equal insertion rate, insertion–deletion only and substitution only. All these expressions can be directly used for biological evolutionary applications. The IDIS model shows a strongly different stochastic behavior from the classical substitution only model when compared on a gene dataset. Indeed, by considering three processes of residue insertion, deletion and substitution independently from each other, it allows a more realistic representation of gene evolution and opens new directions and applications in this research field.

Dynamic phase synchronization characteristics of variable high-order coupled neuronal oscillator population

Under the premise of analysis on the dynamic characteristics of the transmission mechanism among the synapses, this paper has modified the coupling term in the Tass’s stochastic evolution model of neuronal oscillator population, introduced the variable higher-order coupling term. Then, we have performed the numerical simulation on the modified model. The simulation result shows that the variable coupling mechanism can induce the transition between different cluster states of the neuronal oscillator population, without the external stimulation. Another result from the numerical simulation is that, in the transient process between two different synchronization states caused by the variable coupling mechanism, it is allowed to have a full desynchronization state for a period. However, after the period of desynchronization state, the neuronal oscillator population can still reenter a new synchronization state under the action of the coupling term with the order different from initial condition.

A stochastic model for evolution of creep damage in engineering materials

In this work a stochastic model for the creep damage evolution and associated scatter has been developed in terms of a discontinuous Markov process. The magnitude of damage has been described in the form of a probability distribution function whose evolution in time characterizes the nondeterministic nature of the damage accumulation process. The model is able to describe the state of damage along with the associated scatter at a given time at any stress level. The validity of the model has been established by comparing the predicted creep curves generated for a specific loading condition with those experimentally obtained for an austenitic stainless steel (Type 316, 18Cr 8 Ni 2Mo).

Dynamic inversion for hydrological process monitoring with electrical resistance tomography under model uncertainties

We propose an approach for imaging the dynamics of complex hydrological processes. The evolution of electrically conductive fluids in porous media is imaged using time‐lapse electrical resistance tomography. The related dynamic inversion problem is solved using Bayesian filtering techniques; that is, it is formulated as a sequential state estimation problem in which the target is an evolving posterior probability density of the system state. The dynamical inversion framework is based on the state space representation of the system which involves the construction of a stochastic evolution model and an observation model. The observation model that we use in this paper consists of the complete electrode model for ERT, with Archie's law relating saturations to electrical conductivity. The evolution model is an approximate model for simulating flow through partially saturated porous media. Unavoidable modeling and approximation errors in both the observation and evolution models are considered by computing approximate statistics for these errors. These models are then included in the construction of the posterior probability density of the estimated system state. This approximation error method allows the use of approximate, and therefore computationally efficient, observation and evolution models in the Bayesian filtering. We conside7r a synthetic example and show that the incorporation of an explicit model for the model uncertainties in the state space representation can yield better estimates than the frame‐by‐frame imaging approach.

Statistical methods for evolutionary trees.

In 1963 and 1964, L. L. Cavalli-Sforza and A. W. F. Edwards introduced novel methods for computing evolutionary trees from genetical data, initially for human populations from blood-group gene frequencies. The most important development was their introduction of statistical methods of estimation applied to stochastic models of evolution.

Towards optimal distance functions for stochastic substitution models

Distance based reconstruction methods of phylogenetic trees consist of two independent parts: first, inter-species distances are inferred assuming some stochastic model of sequence evolution; then the inferred distances are used to construct a tree. In this paper we concentrate on the task of inter-species distance estimation. Specifically, we characterize the family of valid distance functions for the assumed substitution model and show that deliberate selection of distance function significantly improves the accuracy of distance estimates and, consequently, also improves the accuracy of the reconstructed tree. Our contribution consists of three parts: first, we present a general framework for constructing families of additive distance functions for stochastic evolutionary models. Then, we present a method for selecting (near) optimal distance functions, and we conclude by presenting simulation results which support our theoretical analysis.

Equilibria, fixed points, and complexity classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are, respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in two-player normal form games, and (mixed) Nash equilibria in normal form games with three (or more) players. This paper reviews the underlying computational principles and the corresponding classes.

Cell lineage tree models of neurogenesis

The production of neurons to form the mammalian cortex, known as embryonic cortical neurogenesis, is a complex developmental process. Insight into the process of cell division during neurogenesis is provided by murine cortical cell lineage trees, recorded through experimental observation. Recurring patterns within cell lineage trees may be indicative of predetermined cell behaviour. The application of mathematical modelling to this process requires careful consideration and identification of the key features to be incorporated into the model. A biologically plausible stochastic model of evolution of cell lineage trees is developed, based on the most important known features of neurogenesis. Tractable means of measuring lineage tree shape are discussed. Symmetry is identified as a significant feature of shape and is measured using Colless's Index of Imbalance. Distributions of tree size and imbalance for large tree sizes are computed and results compared to experimental data. Several refinements to the model are investigated, when the cell division probabilities are weighted according to cell generation. Two models involving generation-dependent cell division probabilities produce imbalance distributions which are the most consistent with the available experimental results. The results indicate that a stochastic cell division mechanism is a plausible basis of mammalian neurogenesis.

Combining Models of Protein Translation and Population Genetics to Predict Protein Production Rates from Codon Usage Patterns

Genes are often biased in their codon usage. The degree of bias displayed often changes with expression level and intragenic position. Numerous indices, such as the codon adaptation index, have been developed to measure this bias. Although the expression level of a gene and index values are correlated, the heuristic nature of these metrics limits their ability to explain this relationship. As an alternative approach, this study integrates mechanistic models of cellular and population processes in a nested manner to develop a stochastic evolutionary model of a protein's production rate (SEMPPR). SEMPPR assumes that the evolution of codon bias is driven by selection to reduce the cost of nonsense errors and that this selection is counteracted by mutation and drift. Through the application of Bayes' theorem, SEMPPR generates a posterior probability distribution for the protein production rate of a given gene. Conceptually, SEMPPR's predictions are based on the degree of adaptation to reduce the cost of nonsense errors observed in the codon usage pattern of the gene. As an illustration, SEMPPR was parameterized using the Saccharomyces cerevisiae genome and its predictions tested using available empirical data. The results indicate that SEMPPR's predictions are as reliable index based ones. In addition, SEMPPR's output is more easily interpreted and its predictions could be improved through refinements of the models upon which it is built.

Stochastic nonlinear dynamics pattern formation and growth models

Stochastic evolutionary growth and pattern formation models are treated in a unified way in terms of algorithmic models of nonlinear dynamic systems with feedback built of a standard set of signal processing units. A number of concrete models is described and illustrated by numerous examples of artificially generated patterns that closely imitate wide variety of patterns found in the nature.

Stochastic models of evolution in genetics, ecology and linguistics

We give an overview of stochastic models of evolution that have found applications ingenetics, ecology and linguistics for an audience of non-specialists, especially statisticalphysicists. In particular, we focus mostly on neutral models in which no intrinsic advantageis ascribed to a particular type of the variable unit, for example a gene, appearing inthe theory. In many cases these models are exactly solvable and furthermore gosome way to describing observed features of genetic, ecological and linguisticsystems.

Efficient Enumeration of Phylogenetically Informative Substrings

We study the problem of enumerating substrings that are common amongst genomes that share evolutionary descent. For example, one might want to enumerate all identical (therefore conserved) substrings that are shared between all mammals and not found in non-mammals. Such collection of substrings may be used to identify conserved subsequences or to construct sets of identifying substrings for branches of a phylogenetic tree. For two disjoint sets of genomes on a phylogenetic tree, a substring is called a tag if it is found in all of the genomes of one set and none of the genomes of the other set. We present a near-linear time algorithm that finds all tags in a given phylogeny; and a sublinear space algorithm (at the expense of running time) that is more suited for very large data sets. Under a stochastic model of evolution, we show that a simple process of tag-generation essentially captures all possible ways of generating tags. We use this insight to develop a faster tag discovery algorithm with a small chance of error. However, since tags are not guaranteed to exist in a given data set, we generalize the notion of a tag from a single substring to a set of substrings. We present a linear programming-based approach for finding approximate generalized tag sets. Finally, we use our tag enumeration algorithm to analyze a phylogeny containing 57 whole microbial genomes. We find tags for all nodes in the phylogeny except the root for which we find generalized tag sets.

A stochastic evolutionary growth model for social networks

We present a stochastic model for a social network, where new actors may join the network, existing actors may become inactive and, at a later stage, reactivate themselves. Our model captures the evolution of the network, assuming that actors attain new relations or become active according to the preferential attachment rule. We derive the mean-field equations for this stochastic model and show that, asymptotically, the distribution of actors obeys a power-law distribution. In particular, the model applies to social networks such as wireless local area networks , where users connect to access points, and peer-to-peer networks where users connect to each other. As a proof of concept , we demonstrate the validity of our model empirically by analysing a public log containing traces from a wireless network at Dartmouth College over a period of three years. Analysing the data processed according to our model, we demonstrate that the distribution of user accesses is asymptotically a power-law distribution.

On the Elimination of Dominated Strategies in Stochastic Models of Evolution With Large Populations

A stochastic myopic best-reply dynamics is said to have property (W), for a given number of players $n$, if every pure weakly dominated strategy in every $n$-player game is eliminated in the long-run distribution of play induced by the dynamics. In this paper I give a necessary and sufficient condition that a dynamics has to satisfy in order for it to have property (W). The key determinant is found to be the sensitivity of the learning-rate to small payoff differences, inherent in the dynamics. If this sensitivity is higher than a certain cut-off, which depends on the number of players, then the dynamics satisfies property (W). If it is equal to or below that cut-off, then the dynamics does not satisfy property (W).

A stochastic model for the evolution of the Web allowing link deletion

Recently several authors have proposed stochastic evolutionary models for the growth of the Web graph and other networks that give rise to power-law distributions. These models are based on the notion of preferential attachment, leading to the “rich get richer” phenomenon. We present a generalization of the basic model by allowing deletion of individual links and show that it also gives rise to a power-law distribution. We derive the mean-field equations for this stochastic model and show that, by examining a snapshot of the distribution at the steady state of the model, we are able to determine the extent to which link deletion has taken place and estimate the probability of deleting a link. Applying our model to actual Web graph data provides evidence of the extent to which link deletion has occurred. We also discuss a problem that frequently arises in estimating the power-law exponent from empirical data and a few possible methods for dealing with this, indicating our preferred approach. Using this approach our analysis of the data suggests a power-law exponent of approximately 2.15 for the distribution of inlinks in the Web graph, rather than the widely published value of 2.1.

A model for collaboration networks giving rise to a power-law distribution with an exponential cutoff

Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the “rich get richer” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein, e-mail, actor and collaboration networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will become inactive. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model empirically by analysing the Mathematical Research collaboration network, the distribution of which has been shown to be compatible with a power law with an exponential cutoff.

Pigouvian pricing and stochastic evolutionary implementation

We study the implementation of efficient behavior in settings with externalities. A planner would like to ensure that a group of agents make socially optimal choices, but he only has limited information about the agents’ preferences, and can only distinguish individual agents through the actions they choose. We describe the agents’ behavior using a stochastic evolutionary model, assuming that their choice probabilities are given by the logit choice rule. We prove that there is a simple price scheme with the following property: regardless of the realization of preferences, a group of agents subjected to the price scheme will spend the vast majority of time in the long run behaving efficiently. The price scheme defines a game that may possess multiple equilibria, but we are able to obtain a unique and efficient selection from this set because of the stochastic nature of the agents’ choice rule. We conclude by comparing the performance of our price scheme with that of VCG mechanisms.

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.

A stochastic evolutionary model exhibiting power-law behaviour with an exponential cutoff

Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the “rich get richer” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein and e-mail networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will be discarded. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model by analysing a yeast protein interaction network, the distribution of which is known to follow a power law with an exponential cutoff.