78 research outputs found
Waiting time models of cancer progression
Cancer progression is an evolutionary process that is driven by mutation and
selection in a population of tumor cells. We discuss mathematical models of
cancer progression, starting from traditional multistage theory. Each stage is
associated with the occurrence of genetic alterations and their fixation in the
population. We describe the accumulation of mutations using conjunctive
Bayesian networks, an exponential family of waiting time models in which the
occurrence of mutations is constrained to a partial temporal order. Two
opposing limit cases arise if mutations either follow a linear order or occur
independently. We derive exact analytical expressions for the waiting time
until a specific number of mutations have accumulated in these limit cases as
well as for the general conjunctive Bayesian network. Finally, we analyze a
stochastic population genetics model that explicitly accounts for mutation and
selection. In this model, waves of clonal expansions sweep through the
population at equidistant intervals. We present an approximate analytical
expression for the waiting time in this model and compare it to the results
obtained for the conjunctive Bayesian networks
Structurama: Bayesian Inference of Population Structure
Structurama is a program for inferring population structure. Specifically, the program calculates the posterior probability of assigning individuals to different populations. The program takes as input a file containing the allelic information at some number of loci sampled from a collection of individuals. After reading a data file into computer memory, Structurama uses a Gibbs algorithm to sample assignments of individuals to populations. The program implements four different models: The number of populations can be considered fixed or a random variable with a Dirichlet process prior; moreover, the genotypes of the individuals in the analysis can be considered to come from a single population (no admixture) or as coming from several different populations (admixture). The output is a file of partitions of individuals to populations that were sampled by the Markov chain Monte Carlo algorithm. The partitions are sampled in proportion to their posterior probabilities. The program implements a number of ways to summarize the sampled partitions, including calculation of the ‘mean’ partition—a partition of the individuals to populations that minimizes the squared distance to the sampled partitions
Central limit theorem for multiplicative class functions on the symmetric group
Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for
the characteristic polynomial of a permutation matrix with respect to the
uniform measure on the symmetric group. We generalize this result in several
ways. We prove here a central limit theorem for multiplicative class functions
on symmetric group with respect to the Ewens measure and compute the covariance
of the real and the imaginary part in the limit. We also estimate the rate of
convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but
there are several improvments in the presentation, including a more intuitve
name for the considered function
Shift in critical temperature for random spatial permutations with cycle weights
We examine a phase transition in a model of random spatial permutations which
originates in a study of the interacting Bose gas. Permutations are weighted
according to point positions; the low-temperature onset of the appearance of
arbitrarily long cycles is connected to the phase transition of Bose-Einstein
condensates. In our simplified model, point positions are held fixed on the
fully occupied cubic lattice and interactions are expressed as Ewens-type
weights on cycle lengths of permutations. The critical temperature of the
transition to long cycles depends on an interaction-strength parameter
. For weak interactions, the shift in critical temperature is expected
to be linear in with constant of linearity . Using Markov chain
Monte Carlo methods and finite-size scaling, we find .
This finding matches a similar analytical result of Ueltschi and Betz. We also
examine the mean longest cycle length as a fraction of the number of sites in
long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial
permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures
which appeared at the end of the PDF
Geometry, Scaling and Universality in the Mass Distributions in Heavy Ion Collisions
Various features of the mass yields in heavy ion collisions are studied. The
mass yields are discussed in terms of iterative one dimensional discrete maps.
These maps are shown to produce orbits for a monomer or for a nucleus which
generate the mass yields and the distribution of cluster sizes. Simple
Malthusian dynamics and non-linear Verhulst dynamics are used to illustrate the
approach. Nuclear cobwebbing, attractors of the dynamics, and Lyapanov
exponents are discussed for the mass distribution. The self-similar property of
the Malthusian orbit offers a new variable for the study of scale invariance
using power moments of the mass distribution. Correlation lengths, exponents
and dimensions associated with scaling relations are developed. Fourier
transforms of the mass distribution are used to obtain power spectra which are
investigated for a behavior.Comment: 29 pages in REVTEX, 9 figures (available from the authors), RU-92-0
A Finitary Characterization of the Ewens Sampling Formula
As the Ewens sampling formula represents an equilibrium distribution satisfying detailed balance, some properties difficult to prove are derived in a simple way
Commitment versus persuasion in the three-party constrained voter model
In the framework of the three-party constrained voter model, where voters of
two radical parties (A and B) interact with "centrists" (C and Cz), we study
the competition between a persuasive majority and a committed minority. In this
model, A's and B's are incompatible voters that can convince centrists or be
swayed by them. Here, radical voters are more persuasive than centrists, whose
sub-population consists of susceptible agents C and a fraction zeta of centrist
zealots Cz. Whereas C's may adopt the opinions A and B with respective rates
1+delta_A and 1+delta_B (with delta_A>=delta_B>0), Cz's are committed
individuals that always remain centrists. Furthermore, A and B voters can
become (susceptible) centrists C with a rate 1. The resulting competition
between commitment and persuasion is studied in the mean field limit and for a
finite population on a complete graph. At mean field level, there is a
continuous transition from a coexistence phase when
zeta=
Delta_c. In a finite population of size N, demographic fluctuations lead to
centrism consensus and the dynamics is characterized by the mean consensus time
tau. Because of the competition between commitment and persuasion, here
consensus is reached much slower (zeta=Delta_c) than
in the absence of zealots (when tau\simN). In fact, when zeta<Delta_c and there
is an initial minority of centrists, the mean consensus time asymptotically
grows as tau\simN^{-1/2} e^{N gamma}, where gamma is determined. The dynamics
is thus characterized by a metastable state where the most persuasive voters
and centrists coexist when delta_A>delta_B, whereas all species coexist when
delta_A=delta_B. When zeta>=Delta_c and the initial density of centrists is
low, one finds tau\simln N (when N>>1). Our analytical findings are
corroborated by stochastic simulations.Comment: 25 pages, 6 figures. Final version for the Journal of Statistical
Physics (special issue on the "applications of statistical mechanics to
social phenomena"
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