17 research outputs found
Revealing the canalizing structure of Boolean functions: Algorithms and applications
Boolean functions can be represented in many ways including logical forms,
truth tables, and polynomials. Additionally, Boolean functions have different
canonical representations such as minimal disjunctive normal forms. Other
canonical representation is based on the polynomial representation of Boolean
functions where they can be written as a nested product of canalizing layers
and a polynomial that contains the noncanalizing variables. In this paper we
study the problem of identifying the canalizing layers format of Boolean
functions. First, we show that the problem of finding the canalizing layers is
NP-hard. Second, we present several algorithms for finding the canalizing
layers of a Boolean function, discuss their complexities, and compare their
performances. Third, we show applications where the computation of canalizing
layers can be used for finding a disjunctive normal form of a nested canalizing
function. Another application deals with the reverse engineering of Boolean
networks with a prescribed layering format. Finally, implementations of our
algorithms in Python and in the computer algebra system Macaulay2 are available
at https://github.com/ckadelka/BooleanCanalization.Comment: 13 pages, 1 figur
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
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
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
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
Molecular Network Control Through Boolean Canalization
Boolean networks are an important class of computational models for molecular
interaction networks. Boolean canalization, a type of hierarchical clustering
of the inputs of a Boolean function, has been extensively studied in the
context of network modeling where each layer of canalization adds a degree of
stability in the dynamics of the network. Recently, dynamic network control
approaches have been used for the design of new therapeutic interventions and
for other applications such as stem cell reprogramming. This work studies the
role of canalization in the control of Boolean molecular networks. It provides
a method for identifying the potential edges to control in the wiring diagram
of a network for avoiding undesirable state transitions. The method is based on
identifying appropriate input-output combinations on undesirable transitions
that can be modified using the edges in the wiring diagram of the network.
Moreover, a method for estimating the number of changed transitions in the
state space of the system as a result of an edge deletion in the wiring diagram
is presented. The control methods of this paper were applied to a mutated
cell-cycle model and to a p53-mdm2 model to identify potential control targets