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Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems
, 2009
"... Dedicated to Avner Friedman, on the occasion of his 75th birthday. Discrete dynamical systems defined on the state space Π = {0, 1,...,p − 1} n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent wellknown theor ..."
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Dedicated to Avner Friedman, on the occasion of his 75th birthday. Discrete dynamical systems defined on the state space Π = {0, 1,...,p − 1} n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent wellknown theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary mdimensional systems cannot necessarily be embedded into ndimensional cooperative systems for n = m + 1, as in the Smale theorem for the continuous case, but we show that this is possible for n = m+2 as long as p is sufficiently large. We also prove that strict cooperativity, a natural weakening of the notion of strong cooperativity, implies nontrivial bounds on the lengths of periodic orbits in discrete systems and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strict cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strict cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.
A VelizCuba, The dynamics of conjunctive and disjunctive Boolean network models
 Bull Math Bio. (2010). doi
"... Abstract. The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean fun ..."
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Cited by 3 (0 self)
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Abstract. The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean functions are constructed with the AND (resp. OR) operator only. The main results of this paper describe network dynamics in terms of the structure of the network dependency graph (topology). For a given such network, all possible limit cycle lengths are computed and lower and upper bounds for the number of cycles of each length are given. In particular, the exact number of fixed points is obtained. The bounds are in terms of structural features of the dependency graph and its partially ordered set of strongly connected components. For networks with strongly connected dependency graph, the exact cycle structure is computed. 1.
Extremely Chaotic Boolean Networks
, 811
"... It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditio ..."
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It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include using at most two inputs for every variable, using biased and canalyzing regulatory functions, and restricting the number of negative feedback loops. We also prove for ndimensional Boolean networks that if in addition the number of outputs for each variable is bounded and there exist periodic orbits of length c n for c sufficiently close to 2, any network with these properties must have a large proportion of variables that simply copy previous values of other variables. Such systems share a structural similarity to a relatively small Turing machine acting on one or several tapes. The concept of a Boolean network was originally proposed in the late 1960’s by Stuart Kauffman to model gene regulatory behavior at the cell level [18, 19]. This type of modeling
NETWORK TOPOLOGY AS A DRIVER OF BISTABILITY IN THE LAC OPERON
, 807
"... Abstract. The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models ha ..."
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Abstract. The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits bistability. Further we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions. 1.
Optimizationbased inference for temporally evolving networks with applications to biology
 In Proceedings of the American Control Conference
, 2011
"... personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires pri ..."
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personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement The first author would like to thank support from STX foundation from Korea. This research was supported by the NIH NCI through the ICBP and PSOC projects. This paper was submitted for consideration as a regular paper at the 2011 American Control
The Generalized FeedForward Loop Motif: Definition, Detection and Statistical Significance
"... Network motifs play an important role in the qualitative analysis and quantitative characterization of networks. The feedforward loop is a semantically important and statistically highly significant motif. In this paper, we extend the definition of the feedforward loop to subgraphs of arbitrary si ..."
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Network motifs play an important role in the qualitative analysis and quantitative characterization of networks. The feedforward loop is a semantically important and statistically highly significant motif. In this paper, we extend the definition of the feedforward loop to subgraphs of arbitrary size. To avoid the complexity of path enumeration, we define generalized feedforward loops as pairs of source and target nodes that have two or more internally disjoint connecting paths. Based on this definition, we formally derive an approach for the detection of this generalized motif. Our quantitative analysis demonstrates that generalized feedforward loops up to a certain path length are statistically significant. Loops of greater size are statistically underrepresented and hence an antimotif.
JeanPaul COMET
, 2011
"... et l’Environnement Problèmes type feedback et comportement dynamique des réseaux régulatoires Thèse soutenue publiquement le 18 Août 2011, devant le jury composé de: Martín MATAMALA ..."
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et l’Environnement Problèmes type feedback et comportement dynamique des réseaux régulatoires Thèse soutenue publiquement le 18 Août 2011, devant le jury composé de: Martín MATAMALA
Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems
, 2009
"... Dedicated to Avner Friedman, on the occasion of his 75th birthday Discrete dynamical systems defined on the state space P {0; 1;...; p 2 1} n have been used in multiple applications, most recently for the modelling of gene and protein networks. In this paper, we study to what extent wellknown theo ..."
Abstract
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Dedicated to Avner Friedman, on the occasion of his 75th birthday Discrete dynamical systems defined on the state space P {0; 1;...; p 2 1} n have been used in multiple applications, most recently for the modelling of gene and protein networks. In this paper, we study to what extent wellknown theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that arbitrary mdimensional systems cannot necessarily be embedded into ndimensional cooperative systems for n m þ 1, as in the Smale theorem for the continuous case, but we show that this is possible for n m þ 2 as long as p is sufficiently large. We also prove that strict cooperativity, a natural weakening of the notion of strong cooperativity, implies nontrivial bounds on the lengths of periodic orbits in discrete systems and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strict cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strict cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.
Nonlinear Dynamics and Systems Theory, volume (issue) (year) 1–10 Exponentially long orbits in Boolean networks with exclusively
"... Abstract: The absence of negative feedback in Boolean networks tends to result in systems with relatively short orbits. We present a construction of Ndimensional Boolean networks that use only AND, OR, COPY gates and nevertheless have an exponentially large orbit (of size c N for arbitrary c < 2). ..."
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Abstract: The absence of negative feedback in Boolean networks tends to result in systems with relatively short orbits. We present a construction of Ndimensional Boolean networks that use only AND, OR, COPY gates and nevertheless have an exponentially large orbit (of size c N for arbitrary c < 2). The construction is based on pseudorandom number generation algorithms. A previously obtained nontrivial upper bound on the orbit length under certain limitations on the number of outputs per node is shown to be optimal.