Stratification and enumeration of Boolean functions by canalizing depth

Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively picked off, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, 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

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

Boolean Network Topologies and the Determinative Power of Nodes

Boolean networks have been used extensively for modeling networks whose node activity could be simplified to a binary outcome, such as on-off. Each node is influenced by the states of the other nodes via a logical Boolean function. The network is described by its topological properties which refer to the links between nodes, and its dynamical properties which refer to the way each node uses the information obtained from other nodes to update its state. This work explores the correlation between the information stored in the Boolean functions for each node in a property known as the determinative power and some topological properties of each node, in particular the clustering coefficient and the betweenness centrality. The determinative power of nodes is defined using concepts from information theory, in particular the mutual information. The primary motivation is to construct models of real world networks to examine if the determinative power is sensitive to any of the considered topological properties. The findings indicate that, for a homogeneous network in which all nodes obey the same threshold function under three different topologies, the determinative power can have a negative correlation with the clustering coefficient and a positive correlation with the betweenness centrality, depending on the topological properties of the network. A statistical analysis on a collection of 36 Boolean models of signal transduction networks reveals that the correlations observed in the theoretical cases are suppressed in the biological networks, thus supporting previous research results

Tight Bounds for Communication-Assisted Agreement Distillation

Suppose Alice holds a uniformly random string X in {0,1}^N and Bob holds a noisy version Y of X where each bit of X is flipped independently with probability epsilon in [0,1/2]. Alice and Bob would like to extract a common random string of min-entropy at least k. In this work, we establish the communication versus success probability trade-off for this problem by giving a protocol and a matching lower bound (under the restriction that the string to be agreed upon is determined by Alice\u27s input X). Specifically, we prove that in order for Alice and Bob to agree on a common string with probability 2^{-gamma k} (gamma k >= 1), the optimal communication (up to o(k) terms, and achievable for large N) is precisely (C *(1-gamma) - 2 * sqrt{ C * (1-C) gamma}) * k, where C := 4 * epsilon * (1-epsilon). In particular, the optimal communication to achieve Omega(1) agreement probability approaches 4 * epsilon * (1-epsilon) * k. We also consider the case when Y is the output of the binary erasure channel on X, where each bit of Y equals the corresponding bit of X with probability 1-epsilon and is otherwise erased (that is, replaced by a "?"). In this case, the communication required becomes (epsilon * (1-gamma) - 2 * sqrt{ epsilon * (1-epsilon) * gamma}) * k. In particular, the optimal communication to achieve Omega(1) agreement probability approaches epsilon * k, and with no communication the optimal agreement probability approaches 2^{- (1-sqrt{1-epsilon})/(1+sqrt{1-epsilon}) * k}. Our protocols are based on covering codes and extend the approach of (Bogdanov and Mossel, 2011) for the zero-communication case. Our lower bounds rely on hypercontractive inequalities. For the model of bit-flips, our argument extends the approach of (Bogdanov and Mossel, 2011) by allowing communication; for the erasure model, to the best of our knowledge the needed hypercontractivity statement was not studied before, and it was established (given our application) by (Nair and Wang 2015). We also obtain information complexity lower bounds for these tasks, and together with our protocol, they shed light on the recently popular "most informative Boolean function" conjecture of Courtade and Kumar

Information Theoretic Analysis of the Structure-Dynamics Relationships in Complex Biological Systems

Complex systems and networks is an emerging scientific field, with applications in every area of human enquiry, for which a solid theoretical, computational and experimental foundation is lacking. As our technological capability of generating and gathering vast amounts of data from such systems is increasing, precise methods are needed to describe, analyse and synthesize such systems. Systems biology is a prime example of an interdisciplinary field aiming at tackling the complexity of biological organisms and dedicated to understanding their organizing principles and to devising efficient intervention strategies for curing diseases.A very important topic in the study of complex systems and networks is to uncover the laws that govern their structure-dynamics relationships. A complete description of the system’s behaviour as a whole can only be achieved if the structure and the dynamics are investigated together, as well as the intricate ways in which they influence each other. The understanding of structure-dynamics relationships is a key step in the control of complex systems and networks. For example, in biology, understanding these relationships in organisms would enable us to find more precise drug targets and to design better drugs to cure diseases. In gene regulatory networks, it would help devise control strategies to change the network from faulty states that correspond to disease states, to normal states that give the healthy phenotype. When we observe a dynamical behaviour that is different from the normal, healthy one, the knowledge about the structure-dynamics relationships would help us identify which part of the structure gives rise to such behaviour. Then, we would know where and how to change the structure, to return the system to its normal dynamics, that is, to obtain a desired dynamical behaviour.A feasible way of investigating the structure-dynamics relationships is by measuring the amount of information that is communicated in the system and by analysing the patterns of information propagation within its elements. These objectives can be achieved by means of information theory. To this end, with concepts from Kolmogorov complexity and from Shannon’s information theory, we create novel analysis methods of the structure-dynamics relationships in two models of complex systems: an executable model of the human immune systems and the random Boolean network model of gene regulatory networks.In these endeavours, the information-theoretic means of identifying and measuring the information propagation in complex systems and networks needs to be improved and extended. Research is needed into the theoretical foundations of information theory, to refine existing equations and to introduce new ones that can give more accurate results in the investigation of the propagation of information and its applications to the structure-dynamics relationships. To this end, we bring analytical contributions to the generalization of Shannon’s information theory, named Rényi’s information theory. Thus, we continue the development of the theoretical foundations of information theory, for new and better applications in complex systems science and engineering.The goal of this thesis is to characterize various aspects of the structure-dynamics relationships in models of complex biological systems, by means of information theory. Moreover, our goal is to prove that information theory is a model independent analysis framework that can be applied to any class of models. We pursue our objective, by analysing two different classes of models: an executable model of the human immune system and the random Boolean network model of gene regulatory networks.In the executable model of the regulation of cytokines within the human immune system, our aim is to develop computationally feasible analysis methods that can extract meaningful biological information from the complex encoding of the dynamical behaviour of different perturbations of the wild type system. We aim at classifying several structural perturbations of the system, using only their dynamical information. We endeavour to create methods that can make predictions about the structural parameters that should be changed in order to obtain a desired dynamical behaviour. These conclusions have direct applications to the fine-tuning of the real-world biological experiments performed on the system, of whose computational model we analyse. The benefits of our predictions would be increased efficiency and increased reduction of the time required to optimize the parameters of the real-world biological experiments.In the random Boolean network model of gene regulatory networks, our goal is to develop an experimental order parameter that can characterize the dynamical regime of the network, from the dynamical behaviour that simulates that obtained from the measurements of real-world biological experiments. Moreover, we aim at proving that structural information is hidden in the dynamics of random Boolean networks and that it can be extracted with methods from information theory. We study ensembles of random Boolean networks from two distinct structural classes, which take into account the stochasticity present in real biological systems.Another goal of this study is to bring analytical contributions to the field of Rényi’s information theory, which is a generalization of Shannon’s information theory. Recently, it has found novel applications in the study of the structure-dynamics relationships in complex systems and networks