The concept of a nested canalizing Boolean function has been studied over the last decade in the context of understanding the regulatory logic of molecular interaction networks, such as gene regulatory networks . Such networks are predominantly governed by nested canalizing functions . Derrida values are frequently used to analyze the robustness of a Boolean network to perturbations. This paper introduces closed formulas for the calculation of Derrida values of networks governed by Boolean nested canalizing functions, which previously required extensive simulations. Recently, the concept of nested canalizing functions has been generalized to include multistate functions, and a recursive formula has been derived for their number, as a function of the number of variables. This paper contains a detailed analysis of the class of nested canalizing functions over an arbitrary finite field. In addition, the concept of nested canalization is further generalized and closed formulas for the number of such generalized functions , as well as for the number of equivalence classes  under permutation of variables , are derived. The latter is motivated by the fact that two nested canalizing functions that differ only by a permutation of the variables share many important properties .
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 c-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 .
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The concept of a nested canalizing Boolean function has been studied over the last decade in the context of understanding the regulatory logic of molecular interaction networks, such as
This paper provides a collection of mathematical and computational tools for the study of robustness in nonlinear
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. Such networks are predominantly
, represented by time- and state-discrete dynamical systems taking on multiple states. The focus is on networks
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. Derrida values are frequently used to analyze the robustness of a Boolean network to perturbations. This paper introduces closed formulas for the
(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 c-sensitivities of multistate NCFs, which enables the
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Derrida values of networks governed by Boolean nested canalizing functions, which previously required extensive simulations. Recently, the concept of nested canalizing functions has been generalized to include multistate functions, and a recursive formula has been derived for their number, as a function of the number of variables. This paper contains a detailed analysis of the class of nested canalizing functions over an arbitrary finite field. In addition, the concept of nested canalization is further generalized and closed formulas
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
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generalized functions
functions in a given number of variables
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an asymptotic formula. Finally, we compute
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of NCFs
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, are derived. The latter is motivated by the fact that two nested canalizing functions that differ only by a permutation of the variables share many important properties
. Together, the results of the paper represent a useful mathematical framework for the study of NCFs and their dynamic networks