Canalyzing Functions Research Articles

Preponderance of generalized chain functions in reconstructed Boolean models of biological networks

Boolean networks (BNs) have been extensively used to model gene regulatory networks (GRNs). The dynamics of BNs depend on the network architecture and regulatory logic rules (Boolean functions (BFs)) associated with nodes. Nested canalyzing functions (NCFs) have been shown to be enriched among the BFs in the large-scale studies of reconstructed Boolean models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. We build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose two other sub-types of NCFs, namely, the class of chain-1 functions and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 (also holds for chain-1) functions decreases exponentially with the number of inputs. We provide analytical treatment for this and other observations on BFs. Then, by analyzing three different datasets of reconstructed Boolean models we find that generalized chain functions are significantly enriched within the NCFs. Lastly we illustrate that upon imposing the constraints of generalized chain functions on three different GRNs we are able to obtain biologically viable Boolean models.

Relative importance of composition structures and biologically meaningful logics in bipartite Boolean models of gene regulation

Boolean networks have been widely used to model gene networks. However, such models are coarse-grained to an extent that they abstract away molecular specificities of gene regulation. Alternatively, bipartite Boolean network models of gene regulation explicitly distinguish genes from transcription factors (TFs). In such bipartite models, multiple TFs may simultaneously contribute to gene regulation by forming heteromeric complexes, thus giving rise to composition structures. Since bipartite Boolean models are relatively recent, an empirical investigation of their biological plausibility is lacking. Here, we estimate the prevalence of composition structures arising through heteromeric complexes. Moreover, we present an additional mechanism where composition structures may arise as a result of multiple TFs binding to cis-regulatory regions and provide empirical support for this mechanism. Next, we compare the restriction in BFs imposed by composition structures and by biologically meaningful properties. We find that though composition structures can severely restrict the number of Boolean functions (BFs) driving a gene, the two types of minimally complex BFs, namely nested canalyzing functions (NCFs) and read-once functions (RoFs), are comparatively more restrictive. Finally, we find that composition structures are highly enriched in real networks, but this enrichment most likely comes from NCFs and RoFs.

On the Distribution of Successor States in Boolean Threshold Networks.

We study the distribution of successor states in Boolean networks (BNs). The state vector y is called a successor of x if y = F(x) holds, where x, y ∈ {0,1}n are state vectors and F is an ordered set of Boolean functions describing the state transitions. This problem is motivated by analyzing how information propagates via hidden layers in Boolean threshold networks (discrete model of neural networks) and is kept or lost during time evolution in BNs. In this article, we measure the distribution via entropy and study how entropy changes via the transition from x to y , assuming that x is given uniformly at random. We focus on BNs consisting of exclusive OR (XOR) functions, canalyzing functions, and threshold functions. As a main result, we show that there exists a BN consisting of d -ary XOR functions, which preserves the entropy if d is odd and , whereas there does not exist such a BN if d is even. We also show that there exists a specific BN consisting of d -ary threshold functions, which preserves the entropy if [Formula: see text]. Furthermore, we theoretically analyze the upper and lower bounds of the entropy for BNs consisting of canalyzing functions and perform computational experiments using BN models of real biological networks.

Minimum complexity drives regulatory logic in Boolean models of living systems.

The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd "bias", that is they produce "on" outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

Data-Driven Modeling of Breast Cancer Tumors Using Boolean Networks.

Cancer is a genomic disease involving various intertwined pathways with complex cross-communication links. Conceptually, this complex interconnected system forms a network, which allows one to model the dynamic behavior of the elements that characterize it to describe the entire system’s development in its various evolutionary stages of carcinogenesis. Knowing the activation or inhibition status of the genes that make up the network during its temporal evolution is necessary for the rational intervention on the critical factors for controlling the system’s dynamic evolution. In this report, we proposed a methodology for building data-driven boolean networks that model breast cancer tumors. We defined the network components and topology based on gene expression data from RNA-seq of breast cancer cell lines. We used a Boolean logic formalism to describe the network dynamics. The combination of single-cell RNA-seq and interactome data enabled us to study the dynamics of malignant subnetworks of up-regulated genes. First, we used the same Boolean function construction scheme for each network node, based on canalyzing functions. Using single-cell breast cancer datasets from The Cancer Genome Atlas, we applied a binarization algorithm. The binarized version of scRNA-seq data allowed identifying attractors specific to patients and critical genes related to each breast cancer subtype. The model proposed in this report may serve as a basis for a methodology to detect critical genes involved in malignant attractor stability, whose inhibition could have potential applications in cancer theranostics.

Canalyzing minors of Boolean functions

Canalyzing functions are a special type of Boolean functions. For a canalyzing function, there is at least one argument, in which taking a certain value can determine the value of the function. Identification of variables can also shrink the resulting function into constant or function depending on one variable. In this paper, we discuss a particular disjunctive normal form for representation of Boolean function with its identification minors. Then an upper bound of the number of canalyzing minors is obtained. Finally, the number of canalyzing minors for Boolean functions with five essential variables is discussed.

A characterization of nested canalyzing functions with maximum average sensitivity

Nested canalyzing functions (NCFs) are a class of Boolean functions which are used to model certain biological phenomena. We derive a complete characterization of NCFs with the largest average sensitivity, expressed in terms of a simple structural property of the NCF. This characterization provides an alternate, but elementary, proof of the tight upper bound on the average sensitivity of any NCF established by Klotz et al. (2013). We also utilize the characterization to derive a closed form expression for the number of NCFs that have the largest average sensitivity.

Categorizing Attractor-Effective Canalyzing Functions in Boolean Networks

Canalyzing Boolean functions have shown their popularity in various biological networks and established themselves to be biologically meaningful at the system level, marking their importance in the analysis of stability and robustness of such complex systems. Based on a matrix representation of Boolean networks due to the recently developed tool called semi-tensor product, we categorize canalyzing functions in terms of their capabilities of affecting the number of attractors in the Boolean network, which is one key index for the stability and robustness of Boolean networks. We show that there exist only three categories of attractor-effective canalyzing functions for any network size larger than 1, while the number of all the interested canalyzing functions is proportional to the square of the network size. We also give the explicit expression of the mean number of attractors with any length for Boolean networks with a single canalyzing function. Compared with Boolean networks without any canalyzing functions, we are able to show quantitatively how canalyzing functions can affect the mean number and length of attractors in Boolean networks for the first time.

Applying attractor dynamics to infer gene regulatory interactions involved in cellular differentiation

The dynamics of gene regulatory networks (GRNs) guide cellular differentiation. Determining the ways regulatory genes control expression of their targets is essential to understand and control cellular differentiation. The way a regulatory gene controls its target can be expressed as a gene regulatory function. Manual derivation of these regulatory functions is slow, error-prone and difficult to update as new information arises. Automating this process is a significant challenge and the subject of intensive effort. This work presents a novel approach to discovering biologically plausible gene regulatory interactions that control cellular differentiation. This method integrates known cell type expression data, genetic interactions, and knowledge of the effects of gene knockouts to determine likely GRN regulatory functions. We employ a genetic algorithm to search for candidate GRNs that use a set of transcription factors that control differentiation within a lineage. Nested canalyzing functions are used to constrain the search space to biologically plausible networks. The method identifies an ensemble of GRNs whose dynamics reproduce the gene expression pattern for each cell type within a particular lineage. The method's effectiveness was tested by inferring consensus GRNs for myeloid and pancreatic cell differentiation and comparing the predicted gene regulatory interactions to manually derived interactions. We identified many regulatory interactions reported in the literature and also found differences from published reports. These discrepancies suggest areas for biological studies of myeloid and pancreatic differentiation. We also performed a study that used defined synthetic networks to evaluate the accuracy of the automated search method and found that the search algorithm was able to discover the regulatory interactions in these defined networks with high accuracy. We suggest that the GRN functions derived from the methods described here can be used to fill gaps in knowledge about regulatory interactions and to offer hypotheses for experimental testing of GRNs that control differentiation and other biological processes.

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's 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's 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.

An Improved Satisfiability Algorithm for Nested Canalyzing Functions and its Application to Determining a Singleton Attractor of a Boolean Network

We study the problem of finding a 0-1 assignment to Boolean variables satisfying a given set of nested canalyzing functions, a class of Boolean functions that is known to be of interest in biology. For this problem, an extension of the satisfiability problem for a conjunctive normal form formula, an O(min(2(k), 2((k+m)/2))poly(m)) time algorithm has been known, where m and k are the number of nested canalyzing functions and variables, respectively. Here we present an improved O(min(2(k), 1.325(k+m), 2(m))poly(m)) time algorithm for this problem. We also study the problem of finding a singleton attractor of a Boolean network consisting of n nested canalyzing functions. Although an O(1.799(n)) time algorithm was proposed in a previous study, it was implicitly assumed that the network does not contain any positive self-loops. By utilizing the improved satisfiability algorithm for nested canalyzing functions, while allowing for the presence of positive self-loops, we show that the general case can be solved in O(1.871(n)) time.

The Number of Canalyzing Functions over Any Finite Set

In this paper, we extend the definition of Boolean canalyzing functions to the canalyzing functions of multi-state case. Namely, f:Qn→Q , where Q={a1,a2,...,aq} . We obtain its cardinality and the cardinalities of its various subsets (They may not be disjoint). When q=2, we obtain a combinatorial identity by equating our result to the formula in [1]. For a better understanding to the magnitude, we obtain the asymptotes for all the cardinalities as either n→∞ or q→∞.

Finding a Periodic Attractor of a Boolean Network

In this paper, we study the problem of finding a periodic attractor of a Boolean network (BN), which arises in computational systems biology and is known to be NP-hard. Since a general case is quite hard to solve, we consider special but biologically important subclasses of BNs. For finding an attractor of period 2 of a BN consisting of n OR functions of positive literals, we present a polynomial time algorithm. For finding an attractor of period 2 of a BN consisting of n AND/OR functions of literals, we present an O(1:985(n)) time algorithm. For finding an attractor of a fixed period of a BN consisting of n nested canalyzing functions and having constant treewidth w, we present an O(n(2p(w+1))poly(n)) time algorithm.

Inferring Biologically Relevant Models: Nested Canalyzing Functions

Inferring dynamic biochemical networks is one of the main challenges in systems biology. Given experimental data, the objective is to identify the rules of interaction among the different entities of the network. However, the number of possible models fitting the available data is huge, and identifying a biologically relevant model is of great interest. Nested canalyzing functions, where variables in a given order dominate the function, have recently been proposed as a framework for modeling gene regulatory networks. Previously, we described this class of functions as an algebraic toric variety. In this paper, we present an algorithm that identifies all nested canalyzing models that fit the given data. We demonstrate our methods using a well-known Boolean model of the cell cycle in budding yeast.

The number of multistate nested canalyzing functions

Identifying features of molecular regulatory networks is an important problem in systems biology. It has been shown that the combinatorial logic of such networks can be captured in many cases by special functions called nested canalyzing in the context of discrete dynamic network models. It was also shown that the dynamics of networks constructed from such functions has very special properties that are consistent with what is known about molecular networks, and that simplify analysis. It is important to know how restrictive this class of functions is, for instance for the purpose of network reverse-engineering. This paper contains a formula for the number of such functions and a comparison to the class of all functions. In particular, it is shown that, as the number of variables becomes large, the ratio of the number of nested canalyzing functions to the number of all functions converges to zero. This shows that the class of nested canalyzing functions is indeed very restrictive. The principal tool used for this investigation is a description of these functions as polynomials and a parametrization of the class of all such polynomials in terms of relations on their coefficients. This parametrization can also be used for the purpose of network reverse-engineering using only nested canalyzing functions.

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.

Detecting controlling nodes of boolean regulatory networks

Boolean models of regulatory networks are assumed to be tolerant to perturbations. That qualitatively implies that each function can only depend on a few nodes. Biologically motivated constraints further show that functions found in Boolean regulatory networks belong to certain classes of functions, for example, the unate functions. It turns out that these classes have specific properties in the Fourier domain. That motivates us to study the problem of detecting controlling nodes in classes of Boolean networks using spectral techniques. We consider networks with unbalanced functions and functions of an average sensitivity less than , where k is the number of controlling variables for a function. Further, we consider the class of 1-low networks which include unate networks, linear threshold networks, and networks with nested canalyzing functions. We show that the application of spectral learning algorithms leads to both better time and sample complexity for the detection of controlling nodes compared with algorithms based on exhaustive search. For a particular algorithm, we state analytical upper bounds on the number of samples needed to find the controlling nodes of the Boolean functions. Further, improved algorithms for detecting controlling nodes in large-scale unate networks are given and numerically studied.

Determining a Singleton Attractor of a Boolean Network with Nested Canalyzing Functions

In this article, we study the problem of finding a singleton attractor for several biologically important subclasses of Boolean networks (BNs). The problem of finding a singleton attractor in a BN is known to be NP-hard in general. For BNs consisting of n nested canalyzing functions, we present an O(1.799(n)) time algorithm. The core part of this development is an O(min(2(k/2) · 2(m/2), 2(k)) · poly(k, m)) time algorithm for the satisfiability problem for m nested canalyzing functions over k variables. For BNs consisting of chain functions, a subclass of nested canalyzing functions, we present an O(1.619(n)) time algorithm and show that the problem remains NP-hard, even though the satisfiability problem for m chain functions over k variables is solvable in polynomial time. Finally, we present an o(2(n)) time algorithm for bounded degree BNs consisting of canalyzing functions.

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution.

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.