John O. Adeyeye

Boolean nested canalizing functions: A comprehensive analysis

Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being nested canalizing, a concept inspired by the concept of canalization in evolutionary biology. It has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks, namely a small number of (large) attractors, as well as relatively short limit cycles. This paper contains a detailed analysis of this class of functions, based on a novel normal form as polynomial functions over the Boolean field. The concept of layer is introduced that stratifies variables into different classes depending on their level of dominance. Using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables. Additional metrics considered include Hamming weight, the activity number of any variable, and the average sensitivity of the function. It is also shown that the average sensitivity of any nested canalizing function is between 0 and 2. This provides a rationale for why nested canalizing functions are stable, since a random Boolean function in n variables has average sensitivity n/2. The paper also contains experimental evidence that the layer number is an important factor in network stability.

Multistate nested canalizing functions and their networks

This paper provides a collection of mathematical and computational tools for the study of robustness in nonlinear gene regulatory networks, represented by time- and state-discrete dynamical systems taking on multiple states. The focus is on networks governed by nested canalizing functions (NCFs), first introduced in the Boolean context by S. Kauffman. After giving a general definition of NCFs we analyze the class of such functions. We derive a formula for the normalized average

A model for guiding undergraduates to success in computational science

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A Newton–Cotes method for quantum stochastic differential equations

-sensitivities of multistate NCFs, which enables the calculation of the Derrida plot, a popular measure of network stability. We also provide a unique canonical parametrized polynomial form of NCFs. This form has several consequences. We can easily generate NCFs for varying parameter choices, and derive a closed form formula for the number of such functions in a given number of variables, as well as an asymptotic formula. Finally, we compute the number of equivalence classes of NCFs under permutation of variables. Together, the results of the paper represent a useful mathematical framework for the study of NCFs and their dynamic networks.

An Extension of the Generalized Quasilinearization Method of Lakshmikantham

This paper presents a model for guiding undergraduates to success in computational science. A set of integrated, interdisciplinary training and research activities is outlined for use as a vehicle to increase and produce graduates with research experiences in computational and mathematical sciences. The model is responsive to the development of new interdisciplinary curricula in computational biology, chemistry, mathematics and physics.

Generalized Newton Iterative Methods for Nonlinear Operator Equations

Abstract We present an implicit Simpson’s rule for the numerical computation of quantum stochastic differential equations. We derive the method, present convergence results and a numerical example. Our method avoid the high cost of high-order methods as well the cost of obtaining starting values for multi-step methods.