Landscape Boolean Functions

In this paper we define a class of Boolean and generalized Boolean functions defined on $\mathbb{F}_2^n$ with values in $\mathbb{Z}_q$ (mostly, we consider $q=2^k$), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page

Relative Stability of Network States in Boolean Network Models of Gene Regulation in Development

Progress in cell type reprogramming has revived the interest in Waddington's concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington's landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system. However, quasi-potential landscapes, originally developed for continuous systems, are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development. We impose biologically motivated structural constraints (corresponding to specific type of Boolean functions) and dynamical constraints (corresponding to stable attractor states) to limit the space of BN models for pancreas development. In addition, we enforce a novel functional constraint corresponding to the relative ordering of attractor states in BN models to restrict the space of BN models to the biological relevant class. We find that BNs with canalyzing/sign-compatible Boolean functions best capture the dynamics of pancreas cell differentiation. This framework can also determine the genes' influence on cell state transitions, and thus can facilitate the rational design of cell reprogramming protocols.Comment: 24 pages, 6 figures, 1 tabl

Evolution of Canalizing Boolean Networks

Boolean networks with canalizing functions are used to model gene regulatory networks. In order to learn how such networks may behave under evolutionary forces, we simulate the evolution of a single Boolean network by means of an adaptive walk, which allows us to explore the fitness landscape. Mutations change the connections and the functions of the nodes. Our fitness criterion is the robustness of the dynamical attractors against small perturbations. We find that with this fitness criterion the global maximum is always reached and that there is a huge neutral space of 100% fitness. Furthermore, in spite of having such a high degree of robustness, the evolved networks still share many features with "chaotic" networks.Comment: 8 pages, 10 figures; revised and extended versio

Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization

Chicano, F., Whitley D., & Tinós R. (2016). Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization. 16th European Conference on Evolutionary Computation for Combinatorial Optimization (LNCS 9595), pp. 88-103Local search algorithms and iterated local search algorithms are a basic technique. Local search can be a stand-alone search method, but it can also be hybridized with evolutionary algorithms. Recently, it has been shown that it is possible to identify improving moves in Hamming neighborhoods for k-bounded pseudo-Boolean optimization problems in constant time. This means that local search does not need to enumerate neighborhoods to find improving moves. It also means that evolutionary algorithms do not need to use random mutation as a operator, except perhaps as a way to escape local optima. In this paper, we show how improving moves can be identified in constant time for multiobjective problems that are expressed as k-bounded pseudo-Boolean functions. In particular, multiobjective forms of NK Landscapes and Mk Landscapes are considered.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Fulbright program, Ministerio de Educación (CAS12/00274), Ministerio de Economía y Competitividad (TIN2014-57341-R), Air Force Office of Scientific Research, Air Force Materiel Command, USAF (FA9550-11-1-0088), FAPESP (2015/06462-1) and CNPq

Learning, Generalization, and Functional Entropy in Random Automata Networks

It has been shown \citep{broeck90:physicalreview,patarnello87:europhys} that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transfer functions are chosen at random initially, can be evolved by using a state-topology evolution to solve simple tasks. We measure the learning and generalization performance, investigate the influence of the average node connectivity $K$, the system size $N$, and introduce a new measure that allows to better describe the network's learning and generalization behavior. We show that the connectivity of the maximum entropy networks scales as a power-law of the system size $N$. Our results show that networks with higher average connectivity $K$ (supercritical) achieve higher memorization and partial generalization. However, near critical connectivity, the networks show a higher perfect generalization on the even-odd task

Identification of control targets in Boolean molecular network models via computational algebra

Motivation: Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system. Experimentally, node manipulation requires technology to completely repress or fully activate a particular gene product while edge manipulations only require a drug that inactivates the interaction between two gene products. Results: This paper presents a method for the identification of potential intervention targets in Boolean molecular network models using algebraic techniques. The approach exploits an algebraic representation of Boolean networks to encode the control candidates in the network wiring diagram as the solutions of a system of polynomials equations, and then uses computational algebra techniques to find such controllers. The control methods in this paper are validated through the identification of combinatorial interventions in the signaling pathways of previously reported control targets in two well studied systems, a p53-mdm2 network and a blood T cell lymphocyte granular leukemia survival signaling network.Comment: 12 pages, 4 figures, 2 table

Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems

Efficient hill climbers have been recently proposed for single- and multi-objective pseudo-Boolean optimization problems. For $k$-bounded pseudo-Boolean functions where each variable appears in at most a constant number of subfunctions, it has been theoretically proven that the neighborhood of a solution can be explored in constant time. These hill climbers, combined with a high-level exploration strategy, have shown to improve state of the art methods in experimental studies and open the door to the so-called Gray Box Optimization, where part, but not all, of the details of the objective functions are used to better explore the search space. One important limitation of all the previous proposals is that they can only be applied to unconstrained pseudo-Boolean optimization problems. In this work, we address the constrained case for multi-objective $k$-bounded pseudo-Boolean optimization problems. We find that adding constraints to the pseudo-Boolean problem has a linear computational cost in the hill climber.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech