SOME ISSUES ABOUT DISCRETE–TIME REGULATORY NETWORKS

´ SALGADO1 AND EDGARDO UGALDE1 BEATRIZ LUNA1 , AGUSTINO MART´ INEZ2 , RAUL

1. Instituto de F´ısica, Universidad Aut´ onoma de San Luis Potos´ı, Av. Manuel Nava 6, Zona Universitaria, 78290 San Luis Potos´ı, M´ exico. 2. Departamento de Ingenier´ıa Gen´ etica, Centro de Investigaci´ on y de Estudios Avanzados del Instituto Polit´ ecnico Nacional, Campus Guanajuato, Irapuato, 36500. M´ exico. Abstract. Discrete–time regulatory networks are dynamical systems on directed graphs with a structure that is inspired on natural systems of interacting units. We study some issues on these networks. First using a notion of determination between vertices, we define sets of dominant vertices, and we prove that in the asymptotic regime, the dynamics on a dominant set allows us to determine the state of the whole system at all times. Then we study the dynamical complexity of two–dimensional networks, which is quantity related to the proliferation of distinguishable temporal behavior. Finally, by using this modeling strategy we analyze the two–component transduction systems as module networks that regulate the integration of external information in bacterial cells.

the role of dominant vertices in the regulatory dynamics, as well as an example. Then we show a result on the dynamic complexity of a class two–dimensional regulatory networks. Finally, we present the modeling of two– component systems sensing exogenous conditions in E. coli, as open discrete–time regulatory networks.

1. Introduction We consider a class of discrete–time dynamical systems on networks. This models was first introduced in [10], and in [4] their low dimensional representatives were studied in full detail. These systems can be thought as a discrete– time alternative to the systems of piecewise affine differential equations extensively studied in [1, 2, 6], and to finite state models, better known as logical networks, which have been considered in [5, 3, 9]. In each of these modeling strategies, the interacting units have a regular behavior when taken separately, but are capable to generate global complex dynamics when arranged in a complex interaction architecture. In the discrete–time dynamical systems considered here, all interacting units or vertices V evolve synchronously at discrete time steps, and the level of activity of each one of them varies according a rule dictated by its neighboring units.

2. Dominant Vertices An orbit is obtained by letting evolve the system under the action of the transformation F . For a fixed collection of vertices U ⊆ V , a U –controlled trajectory is obtained by assigning any sequence of values to the vertices in U , while letting the rest of the vertices evolve under the iteration of F. We say that vertex set U ⊆ V is dominant if all the interactions in the network can be recursively determined from them. In [8] we rigorously define this notion and propose an algorithm to determine dominant sets.

The interaction from unit i to unit j is weighted by a factor kij ≥ 0, and a sign sij = {−1, 1} attached to it indicates whether this interaction is an inhibition or an activation respectively. If the activity level xi of the unit i exceeds a threshold value Tij ∈ [0, 1), then the interaction from i to j becomes effective.

Our main result in this direction states that given two orbits or U –controlled trajectories, both˛uniformly˛ separated t ˛ = 0, then from the discontinuity set, if limt→∞ ˛xtU − yU limt→∞ |xt − yt | = 0.

The temporal evolution of the activity level of the network is given the action of a piecewise affine transformation xt+1 := F (xt ), defined by the equation xt+1 := aj xtj + (1 − aj ) j

n X

In the sake of illustration we present in Figure 1 a realization of a Barab´ asi–Albert network on 100 vertices, where we distinguish a dominant set of vertices (marked with ×) and non–dominant proper subset of it (marked with *). Then, in Figure 2 we show the distance between the several orbits when we apply a constant control to the dominant set, compared to the situation when the control is applied to the proper non–dominant subset.

kij H(sij (xtj − Tij )).

i=1

The Heaviside function H (H(x) = 1 if x > 0, and H(x) = 0 otherwise) models the abrupt changes in the interactions, and introduces discontinuities in the transformation F . We denote by ∆F this set of discontinuities. In what follows present a brief description of our results in three topics related to the dynamics of discrete time regulatory networks. First we present the result concerning 1

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4. Regulatory dynamics of bacterial two-component systems Here we study modules for signal transduction. The archetypal architecture of these systems consists of two proteins: a sensor and a regulatory component. But since each pair of the genes encoding for these systems are contained in a single transcriptional unit and are expressed from a single promoter, then they could be represented as a single vertex in the network. We classify this systems taking into account whether they need additional regulators to be transcribed or not, then we propose a graphical representation of these systems, and analyze its dynamics. In our study we consider four dynamical stages: the resting state, the turn on, the turn off, and the multi–stability. Figure 1. The dominant vertices determined by our algorithm are marked with either × or ∗. 0.7

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Figure 2. Constant control applied to the dominant (left) and non–dominant (right) vertex set.

3. Dynamical complexity The n–th iteration F n , of the evolution transformation F , is a piecewise affine map. The complexity C(n) of a regulatory network with evolution transformation F is simply the number of continuity domains of F n . This quantity coincides with the number of distinguishable orbits up to time n. Currently we are studying the complexity of the system in the figure 3.

Figure 3. Coupled system on two vertex For this system we obtain an upper bound of the complexity as follows » – (n)(n + 1) C(n) ≤ + 1 [(n)(n + 1) + 1] , 2 and the conditions for to reach this upper bound.

Figure 4. Module for signal transduction

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