Julio Aracena

University of Concepción · Departamento de Ingeniería Matemática

In this paper, we deal with the following decision problem: given a conjunctive Boolean network defined by its interaction digraph, does it have a limit cycle of a given length k? We prove that this problem is NP-complete in general if k is a parameter of the problem and is in P if the interaction digraph is strongly connected. The case where k is...

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. The Seriation problem is to find...

In this paper, we deal the following decision problem: given a conjunctive Boolean network defined by its interaction digraph, does it have a limit cycle of a given length k? We prove that this problem is NP-complete in general if k is a parameter of the problem and in P if the interaction digraph is strongly connected. The case where $k$ is a cons...

The {\em asynchronous automaton} associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, considered in many applications, is the finite deterministic automaton where the set of states is $\{0,1\}^n$, the alphabet is $[n]$, and the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing...

Boolean networks have been used as models of gene regulation networks and other biological systems. One key element in these models is the update schedule, which indicates the order in which states are to be updated. The presence of any limit cycle in the dynamics of a network depends on the update scheme used. Here, we study the complexity of the...

A matrix is incomplete when some of its entries are missing. A Robinson incomplete symmetric matrix is an incomplete symmetric matrix whose non-missing entries do not decrease along rows and columns when moving toward the diagonal. A Strong-Robinson incomplete symmetric matrix is an incomplete symmetric matrix $A$ such that $a_{k,l} \geq a_{i,j}$ i...

Motivation In the modeling of biological systems by Boolean networks a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the regulatory graph like those proposed by Akutsu et al. (1998b); Zhang et al. (2007) which consider a feedback vertex set of the graph. Howe...

In the modeling of biological systems by Boolean networks a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the interaction graph like those proposed by Akutsu et al. in \cite{akutsu1998system,zhang2007algorithms} which consider a feedback vertex set of the gra...

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. In this work, we provide a charac...

The Sitting Closer to Friends than Enemies (SCFE) problem is to find an embedding in a metric space for the vertices of a given signed graph so that, for every pair of incident edges with different sign, the positive edge is shorter (in the metric of the space) than the negative edge. In this document, we present new results regarding the SCFE prob...

The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing noth...

Given a digraph $G$, a lot of attention has been deserved on the maximum number $\phi(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound $\phi(G)\leq 2^{\tau}$, where $\tau$ is the minimu...

Motivation: Boolean networks (BNs) are commonly used to model genetic regulatory networks (GRNs). Due to the sensibility of the dynamical behavior to changes in the updating scheme (order in which the nodes of a network update their state values), it is increasingly common to use different updating rules in the modeling of GRNs to better capture a...

For a graph $G$, let $\mathcal{C}$ be the set of conjunctive networks with interaction graph $G$, and let $\mathcal{H}$ be the set of graphs obtained from $G$ by contracting some edges. Let $\mathrm{fix}(f)$ be the number of fixed points in a network $f\in \mathcal{C}$, and let $\mathrm{mis}(H)$ be the number of maximal independent sets in $H\in\ma...

We are interested in the number of fixed points in AND-OR-NOT networks, i.e. Boolean networks in which the update function of each component is either a conjunction or a disjunction of positive or negative literals. As main result, we prove that the maximum number of fixed points in a loop-less connected AND-OR-NOT network with n components is at m...

An update digraph corresponds to a labeled digraph that indicates a relative order of its nodes introduced to define equivalence classes of deterministic update schedules yielding the same dynamical behavior of a Boolean network. In Aracena et al. [1], the authors exhibited relationships between update digraphs and the feedback arc sets of a given...

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it "gene...

Deterministic Boolean networks are a type of discrete dynamical systems widely used in the modeling of genetic networks. The dynamics of such systems is characterized by the local activation functions and the update schedule, i.e., the order in which the nodes are updated. In this paper, we address the problem of knowing the different dynamics of a...

Deterministic Boolean networks (BNs) have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states have to be updated. In Aracena et al. (2009) was defined equivalence classes of deterministic update schedules according to the labeled digr...

We consider in this tutorial particular features of many definitions of attractors given in the literature by looking at examples of attracting cycles. We prove by considering specific counterexamples, that there is in fact no order between definitions, only some of them being weaker than others. The chaotic case is also discussed. After which we s...

Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states have to be updated. In Aracena et al. (2009) [1], the authors define equivalence classes that relate deterministic update schedules that yield the same update d...

Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous,...

Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the...

Given a directed graph G=(V,E) and w:E→{−1,+1} a sign function on the arcs of G, we study the positive feedback vertex set problem (PFVS) which consists on finding a minimum cardinality set of vertices that meets all the cycles with an even number of negative arcs. This problem is closely related with the number of steady states of Regulatory Boole...

This paper describes the use of a discrete mathematical model to represent the basic mechanisms of regulation of the bacteria E. coli in batch fermentation. The specific phenomena studied were the changes in metabolism and genetic regulation when the bacteria use three different carbon substrates (glucose, glycerol, and acetate). The model correctl...

Rapid and sequential cell shape changes take place during the formation of the ventral furrow (VF) at the beginning of Drosophila gastrulation. At the cellular level, this morphogenetic event demands close coordination of the proteins involved in actin cytoskeletal reorganization. In order to construct a regulatory network that describes these cell...

We study the length of the limit cycles of discrete monotone functions with symmetric connection graph. We construct a family of monotone functions such that the limit cycles are of maximum possible length, which is exponential in the number of variables. Furthermore, we prove for the class of monotone functions with more than two states and connec...

We study the maximum number of fixed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)/2 for n odd and 2(n−2)/2+1 for n even if the digraph has not loops; and 2n−1+1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching the...

This paper deals with the problem of reconstruction of the intergenic interaction graph from the raw data of genetic co-expression coming with new technologies of bio-arrays (DMA-arrays, protein-arrays, etc.). These new imaging devices in general only give information about the asymptotical part (fixed configurations of co-expression or limit cycle...

We study the relationships between the positive and negative circuits of the connection graph and the fixed points of discrete neural networks (DNNs). As main results, we give necessary conditions and sufficient conditions for the existence of fixed points in a DNN. Moreover, we exhibit an upper bound for the number of fixed points in terms of the...

In this paper, we study the recognition complexity of discrete geometric figures (rectangles, squares, circles, ellipses) on a retina by diameter-limited and order-restricted perceptrons. We construct a diameter-limited recognition perceptron for the family of rectangles, beginning with local configurations, which is different from the one shown by...

We deal in this paper with the concept of genetic regulation network. The genes expression observed through the bio-array imaging allows the geneticist to obtain the intergenic interaction matrix W of the network. The interaction graph G associated to W presents in general interesting features like connected components, gardens of Eden, positive an...

The human genome with its 23 pairs of chromosomes, is the result of evolution. This evolution has been ruled by the mutation process and also by the physiological and pathological reorganization of the genomic material inside or between the chromosomes, which are conditioning the genomic variability. This reorganization is starting at singular poin...

The human genome has evolved from a primitive genome to its present state dispatched along the 23 pairs of chromosomes. This evolution has been ruled by the mutation process and also by the physiological and pathological reorganization of the genomic material inside or between the chromosomes, which condition the genomic variability. This reorganiz...

We introduce a new concept of factorization of finite languages called Generalized Cartesian Factorization with applications in the construction of biological regula-tory networks. In particular, we construct an algorithm that solve the problem of Generalized Cartesian Factorization of a given language in polynomial time.