The number and probability of canalizing functions

Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems, it is informative to know the number of canalizing functions with a given number of input variables. This is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data. To this end, we derive an exact formula for the number of canalizing Boolean functions of n variables. We also derive a formula for the probability that a random Boolean function is canalizing for any given bias p of taking the value 1. In addition, we consider the number and probability of Boolean functions that are canalizing for exactly k variables. Finally, we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables, which is needed for Monte Carlo simulations of Boolean networks

Temporal patterns of gene expression via nonmetric multidimensional scaling analysis

Motivation: Microarray experiments result in large scale data sets that require extensive mining and refining to extract useful information. We have been developing an efficient novel algorithm for nonmetric multidimensional scaling (nMDS) analysis for very large data sets as a maximally unsupervised data mining device. We wish to demonstrate its usefulness in the context of bioinformatics. In our motivation is also an aim to demonstrate that intrinsically nonlinear methods are generally advantageous in data mining. Results: The Pearson correlation distance measure is used to indicate the dissimilarity of the gene activities in transcriptional response of cell cycle-synchronized human fibroblasts to serum [Iyer et al., Science vol. 283, p83 (1999)]. These dissimilarity data have been analyzed with our nMDS algorithm to produce an almost circular arrangement of the genes. The temporal expression patterns of the genes rotate along this circular arrangement. If an appropriate preparation procedure may be applied to the original data set, linear methods such as the principal component analysis (PCA) could achieve reasonable results, but without data preprocessing linear methods such as PCA cannot achieve a useful picture. Furthermore, even with an appropriate data preprocessing, the outcomes of linear procedures are not as clearcut as those by nMDS without preprocessing.Comment: 11 pages, 6 figures + online only 2 color figures, submitted to Bioinformatic

Computing the output distribution and selection probabilities of a stack filter from the DNF of its positive Boolean function

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function. The so called selection probabilities can be computed along the way.Comment: This is the version published in Journal of Mathematical Imaging and Vision, online first, 1 august 201

Entropy of complex relevant components of Boolean networks

Boolean network models of strongly connected modules are capable of capturing the high regulatory complexity of many biological gene regulatory circuits. We study numerically the previously introduced basin entropy, a parameter for the dynamical uncertainty or information storage capacity of a network as well as the average transient time in random relevant components as a function of their connectivity. We also demonstrate that basin entropy can be estimated from time-series data and is therefore also applicable to non-deterministic networks models.Comment: 8 pages, 6 figure

Response of Boolean networks to perturbations

We evaluate the probability that a Boolean network returns to an attractor after perturbing h nodes. We find that the return probability as function of h can display a variety of different behaviours, which yields insights into the state-space structure. In addition to performing computer simulations, we derive analytical results for several types of Boolean networks, in particular for Random Boolean Networks. We also apply our method to networks that have been evolved for robustness to small perturbations, and to a biological example