On the Movement of Vertex Fixed Points in the Simple GA

ABSTRACT The Vose dynamical system model of the simple genetic algorithm models the behavior of this algorithm for large population sizes and is the basis of the exact Markov chain model. Populations consisting of multiple copies of one individual correspond to vertices of the simplex. For zero mutation, these are fixed points of the dynamical system and absorbing states of the Markov chain. For proportional selection, the asymptotic stability of vertex fixed points is understood from previous work. We derive the eigenvalues of the differential at vertex fixed points of the dynamical system model for tournament selection. We show that as mutation increases from zero, hyperbolic asymptotically stable fixed points move into the simplex, and hyperbolic asymptotically unstable fixed points move outside of the simplex. We calculate the derivative of local path of the fixed point with respect to the mutation rate for proportional selection. Simulation analysis shows how fixed points bifurcate with larger changes in the mutation rate and changes in the crossover rate

Structure of the Afferent Terminals in Terminal Ganglion of a Cricket and Persistent Homology

We use topological data analysis to investigate the three dimensional spatial structure of the locus of afferent neuron terminals in crickets Acheta domesticus. Each afferent neuron innervates a filiform hair positioned on a cercus: a protruding appendage at the rear of the animal. The hairs transduce air motion to the neuron signal that is used by a cricket to respond to the environment. We stratify the hairs (and the corresponding afferent terminals) into classes depending on hair length, along with position. Our analysis uncovers significant structure in the relative position of these terminal classes and suggests the functional relevance of this structure. Our method is very robust to the presence of significant experimental and developmental noise. It can be used to analyze a wide range of other point cloud data sets

Somitogenesis Clock-Wave Initiation Requires Differential Decay and Multiple Binding Sites for Clock Protein

Somitogenesis is a process common to all vertebrate embryos in which repeated blocks of cells arise from the presomitic mesoderm (PSM) to lay a foundational pattern for trunk and tail development. Somites form in the wake of passing waves of periodic gene expression that originate in the tailbud and sweep posteriorly across the PSM. Previous work has suggested that the waves result from a spatiotemporally graded control protein that affects the oscillation rate of clock-gene expression. With a minimally constructed mathematical model, we study the contribution of two control mechanisms to the initial formation of this gene-expression wave. We test four biologically motivated model scenarios with either one or two clock protein transcription binding sites, and with or without differential decay rates for clock protein monomers and dimers. We examine the sensitivity of wave formation with respect to multiple model parameters and robustness to heterogeneity in cell population. We find that only a model with both multiple binding sites and differential decay rates is able to reproduce experimentally observed waveforms. Our results show that the experimentally observed characteristics of somitogenesis wave initiation constrain the underlying genetic control mechanisms

Multi-valued characteristics and Morse decompositions

A theory of monotone input-output systems is one of a few mathematical approaches that has been successfully applied to complex models of biological and chemical interactions. Replacing some dynamic interactions between variables by a set of static inputs and characterizing the resulting open loop system by an input-output characteristic, the theory establishes convergence results for the original closed loop system. We significantly extend the theory to the situation when the open loop system has multiple stable equilibria and hence a multi-valued characteristic. We show that the information embedded in the characteristic can be used to construct a Morse decomposition of the invariant set of the closed loop system. These results can be used to predict bistability as well as suggest existence of periodic orbits for the closed loop system. The previous theory on global convergence is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory. We apply our tools to a previously studied model of cell cycle maintenance. Our results show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions

Singular boundary value problems via the Conley index

We use Conley index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem

Oscillations in multi-stable monotone systems with slowly varying feedback

varying feedbac