Bounds on the Average Sensitivity of Nested Canalizing Functions

Nested canalizing Boolean (NCF) functions play an important role in biological motivated regulative networks and in signal processing, in particular describing stack filters. It has been conjectured that NCFs have a stabilizing effect on the network dynamics. It is well known that the average sensitivity plays a central role for the stability of (random) Boolean networks. Here we provide a tight upper bound on the average sensitivity for NCFs as a function of the number of relevant input variables. As conjectured in literature this bound is smaller than 4/3 This shows that a large number of functions appearing in biological networks belong to a class that has very low average sensitivity, which is even close to a tight lower bound.Comment: revised submission to PLOS ON

Collectively canalizing Boolean functions

This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for regulatory networks, and the class of functions studied here captures a key feature of biological network dynamics, namely that a subset of one or more variables, under certain conditions, can dominate the value of a Boolean function, to the exclusion of all others. These functions have rich mathematical properties to be explored. The paper shows how the number and type of such sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function. We further connect the concept of collective canalization with the well-studied concept of the average sensitivity of a Boolean function. The relationship between Boolean functions and the dynamics of the networks they form is important in a wide range of applications beyond biology, such as computer science, and has been studied with statistical and simulation-based methods. But the rich relationship between structure and dynamics remains largely unexplored, and this paper is intended as a contribution to its mathematical foundation.Comment: 15 pages, 2 figure

Nested canalyzing depth and network stability

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.Comment: 13 pages, 2 figure

Difference equation for tracking perturbations in systems of Boolean nested canalyzing functions

This paper studies the spread of perturbations through networks composed of Boolean functions with special canalyzing properties. Canalyzing functions have the property that at least for one value of one of the inputs the output is fixed, irrespective of the values of the other inputs. In this paper the focus is on partially nested canalyzing functions, in which multiple, but not all inputs have this property in a cascading fashion. They naturally describe many relationships in real networks. For example, in a gene regulatory network, the statement “if gene A is expressed, then gene B is not expressed regardless of the states of other genes” implies that A is canalyzing. On the other hand, the additional statement “if gene A is not expressed, and gene C is expressed, then gene B is automatically expressed; otherwise gene B\u27s state is determined by some other type of rule” implies that gene B is expressed by a partially nested canalyzing function with more than two variables, but with two canalyzing variables. In this paper a difference equation model of the probability that a network node\u27s value is affected by an initial perturbation over time is developed, analyzed, and validated numerically. It is shown that the effect of a perturbation decreases towards zero over time if the Boolean functions are canalyzing in sufficiently many variables. The maximum dynamical impact of a perturbation is shown to be comparable to the average impact for a wide range of values of the average sensitivity of the network. Percolation limits are also explored; these are parameter values which generate a transition of the expected perturbation effect to zero as other parameters are varied, so that the initial perturbation does not scale up with the parameters once the percolation limits are reached

Algebraic Geometry Arising from Discrete Models of Gene Regulatory Networks

Discrete models of gene regulatory networks have gained popularity in computational systems biology over the last dozen years. However, not all discrete network models reflect the behaviors of real biological systems. In this work, we focus on two model selection methods and algebraic geometry arising from these model selection methods. The first model selection method involves biologically relevant functions. We begin by introducing k-canalizing functions, a generalization of nested canalizing functions. We extend results on nested canalizing functions and derived a unique extended monomial form of arbitrary Boolean functions. This gives us a stratification of the set of n-variable Boolean functions by canalizing depth. We obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions. We characterize the set of k-canalizing functions as an algebraic variety in F2n. 2 . Next, e propose a method for the reverse engineering of networks of k-canalizing functions using techniques from computational algebra, based on our parametrization of k-canalizing functions. We also analyze binary decision diagrams of k-canalizing functions. The second model selection method involves computing minimal polynomial models using Gröbner bases. We built up the connection between staircases and Gröbner bases. We pro-vided a necessary and sufficient condition for the ideal I(V ) to have a unique reduced Gröbner basis, using the concept of a basic staircase. We also provide a sufficient combinatorial characterization of V ⊂ Nnp that yields a unique reduced Grobner basis