Trevor I. Fenner

Algorithms for computing parsimonious evolutionary scenarios for genome evolution, the last universal common ancestor and dominance of horizontal gene transfer in the evolution of prokaryotes

BackgroundComparative analysis of sequenced genomes reveals numerous instances of apparent horizontal gene transfer (HGT), at least in prokaryotes, and indicates that lineage-specific gene loss might have been even more common in evolution. This complicates the notion of a species tree, which needs to be re-interpreted as a prevailing evolutionary trend, rather than the full depiction of evolution, and makes reconstruction of ancestral genomes a non-trivial task.ResultsWe addressed the problem of constructing parsimonious scenarios for individual sets of orthologous genes given a species tree. The orthologous sets were taken from the database of Clusters of Orthologous Groups of proteins (COGs). We show that the phyletic patterns (patterns of presence-absence in completely sequenced genomes) of almost 90% of the COGs are inconsistent with the hypothetical species tree. Algorithms were developed to reconcile the phyletic patterns with the species tree by postulating gene loss, COG emergence and HGT (the latter two classes of events were collectively treated as gene gains). We prove that each of these algorithms produces a parsimonious evolutionary scenario, which can be represented as mapping of loss and gain events on the species tree. The distribution of the evolutionary events among the tree nodes substantially depends on the underlying assumptions of the reconciliation algorithm, e.g. whether or not independent gene gains (gain after loss after gain) are permitted. Biological considerations suggest that, on average, gene loss might be a more likely event than gene gain. Therefore different gain penalties were used and the resulting series of reconstructed gene sets for the last universal common ancestor (LUCA) of the extant life forms were analysed. The number of genes in the reconstructed LUCA gene sets grows as the gain penalty increases. However, qualitative examination of the LUCA versions reconstructed with different gain penalties indicates that, even with a gain penalty of 1 (equal weights assigned to a gain and a loss), the set of 572 genes assigned to LUCA might be nearly sufficient to sustain a functioning organism. Under this gain penalty value, the numbers of horizontal gene transfer and gene loss events are nearly identical. This result holds true for two alternative topologies of the species tree and even under random shuffling of the tree. Therefore, the results seem to be compatible with approximately equal likelihoods of HGT and gene loss in the evolution of prokaryotes.ConclusionsThe notion that gene loss and HGT are major aspects of prokaryotic evolution was supported by quantitative analysis of the mapping of the phyletic patterns of COGs onto a hypothetical species tree. Algorithms were developed for constructing parsimonious evolutionary scenarios, which include gene loss and gain events, for orthologous gene sets, given a species tree. This analysis shows, contrary to expectations, that the number of predicted HGT events that occurred during the evolution of prokaryotes might be approximately the same as the number of gene losses. The approach to the reconstruction of evolutionary scenarios employed here is conservative with regard to the detection of HGT because only patterns of gene presence-absence in sequenced genomes are taken into account. In reality, horizontal transfer might have contributed to the evolution of many other genes also, which makes it a dominant force in prokaryotic evolution.

On key storage in secure networks

We consider systems where the keys for encrypting messages are derived from the pairwise intersections of sets of private keys issued to the users. We give improved bounds on the storage requirements of systems of this type for secure communication in a large network.

An algorithm for finding Hamilton paths and cycles in random graphs

This paper describes a polynomial time algorithm HAM that searches for Hamilton cycles in undirected graphs. On a random graph its asymptotic probability of success is that of the existence of such a cycle. If all graphs withn vertices are considered equally likely, then using dynamic programming on failure leads to an algorithm with polynomial expected time. The algorithm HAM is also used to solve the symmetric bottleneck travelling salesman problem with probability tending to 1, asn tends to ∞.Various modifications of HAM are shown to solve several Hamilton path problems.

On the connectivity of randomm-orientable graphs and digraphs

We consider graphs and digraphs obtained by randomly generating a prescribed number of arcs incident at each vertex.We analyse their almost certain connectivity and apply these results to the expected value of random minimum length spanning trees and arborescences.We also examine the relationship between our results and certain results of Erdős and Rényi.

A stochastic model for the evolution of the Web

Abstract Recently several authors have proposed stochastic models of the growth of the Web graph that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the “rich get richer” phenomenon. However, these models fail to explain several distributions arising from empirical results, due to the fact that the predicted exponent is not consistent with the data. To address this problem, we extend the evolutionary model of the Web graph by including a non-preferential component, and we view the stochastic process in terms of an urn transfer model. By making this extension, we can now explain a wider variety of empirically discovered power-law distributions provided the exponent is greater than two. These include: the distribution of incoming links, the distribution of outgoing links, the distribution of pages in a Web site and the distribution of visitors to a Web site. A by-product of our results is a formal proof of the convergence of the standard stochastic model (first proposed by Simon).

On the existence of Hamiltonian cycles in a class of random graphs

A digraph with n vertices and fixed outdegree m is generated randomly so that each such digraph is equally likely to be chosen. We consider the probability of the existence of a Hamiltonian cycle in the graph obtained by ignoring arc orientation. We show that there exists m (=<23) such that a Hamiltonian cycle exists with probability tending to 1 as n tends to infinity.

High Degree Vertices and Eigenvalues in the Preferential Attachment Graph

The preferential attachment graph is a random graph formed by adding a new vertex at each time step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the world wide web [BA99]. For any constant k, let Δ 1 ≥ Δ 2 ≥ ... ≥ Δ k be the degrees of the k highest degree vertices. We show that at time t, for any function f with f(t) → ∞ as t → ∞, t /f(t) ≤ Δ 1 ≤ t 1/2 f(t), and for i = 2,..., k, t /f(t) ≤ Δ i ≤ Δ i-1 - t /f(t), with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1 ± o(1))Δ 1/2 k whp.

Hamiltonian Cycles in Random Regular Graphs

Abstract The existence of Hamiltonian cycles in random vertex-labelled regular graphs is investigated. It is proved that there exists r0≤796 such that for r≥r0 almost all vertex-labelled r-regular graphs with n vertices have Hamiltonian cycles as n → ∞.

An algorithm for finding Hamilton cycles in random graphs

This paper describes a polynomial time algorithm HAM that searches for Hamilton cycles in undirected graphs. On a random graph its asymptotic probability of success is that of the existence of such a cycle. If all graphs with n vertices are considered equally likely, then using dynamic programming on failure leads to an algorithm with polynomial expected time. Finally, it is used in an algorithm for solving the symmetric bottleneck travelling salesman problem with probability tending to 1, as n tends to ∞.

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

Abstract 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.