Coupled Random Boolean Network Forming an Artificial Tissue

Transcription

1 Coupled Random Boolean Network Forming an Artificial Tissue M. Villani, R. Serra, P.Ingrami, and S.A. Kauffman 2 DSSC, University of Modena and Reggio Emilia, via Allegri 9, I-4200 Reggio Emilia villani.marco@unimore.it, serra.roberto@unimore.it, pingrami@gmail.com 2 Institute for Biocomplexity and Informatics, University of Calgary 2500 University Drive NW, Calgary AB T2N N4, Canada skauffman@ucalgary.ca Abstract. Random boolean networks (shortly, RBN) have proven useful in describing complex phenomena occurring at the unicellular level. It is therefore interesting to investigate how their dynamical behavior is affected by cell-cell interactions, which mimics those occurring in tissues in multicellular organisms. It has also been suggested that evolution may tend to adjust the parameters of the genetic network so that it operates close to a critical state, which should provide evolutionary advantage ; this hypothesis has received intriguing, although not definitive support from recent findings. It is therefore particularly interesting to consider how the tissue-like organization alters the dynamical behavior of the networks close to a critical state. In this paper we define a model tissue, which is a cellular automaton each of whose cells hosts a full RBN, and we report preliminary studies of the way in which the dynamics is affected. Introduction A very interesting line of research on the study of biological organization is the ensemble approach, pioneered several years ago by one of us [][2] in the study of genetic networks. According to this line the emphasis is placed on the typical properties of networks which are supposed to capture some characteristics of real biological systems, instead of concentrating upon the study of specific cases. While the detailed study of specific organisms and specific genetic circuits is of the utmost importance, it is claimed here that the ensemble approach provides a useful complement to it. The search for typical (often called universal ) behaviors has proven very useful also in the study e.g. of phase transitions and dynamical systems Random boolean networks (RBN) have been proposed as a model of genetic regulatory networks, precisely with the aim of devising a model which should be manageable enough to draw conclusions about its generic behaviors, which should be compared with experimental data. Many excellent presentations of the model exist [2][3], and we will only very briefly outline it below (section 2). S. El Yacoubi, B. Chopard, and S. Bandini (Eds.): ACRI 2006, LNCS 473, pp , Springer-Verlag Berlin Heidelberg 2006

2 Coupled Random Boolean Network Forming an Artificial Tissue 549 In most cases, the RBN model has been used to model a single cell, or a population of single cells, and it has proven able to capture some of their properties, including the response to perturbations in gene knock-out experiments [4][5]. It has also been suggested that evolution may tend to adjust the parameters of the genetic network so that it operates close to a critical state, which should provide evolutionary advantage; this hypothesis has received intriguing, although not definitive support from recent findings [6][7]. On the other hand, multicellular organisms are organized in tissues composed by similar cells which are often close in space, and it is natural to ask whether the multicellular organization affects the dynamics. Does interaction lead to a higher order, or rather the contrary? Kauffman recently suggested that it is likely that the whole tissue operates close to the critical state, and that the single cells might be slightly more ordered than if they were alone. While some work addressing this issue in the context of scale-free RBN has been performed [8], in this paper we investigate on the effects of interactions among neighboring cells using classical random boolean networks (precisely defined in section 2). In particular, we set up a 2D CA model, described in section 3, where each lattice site is occupied by a RBN, and introduce a mechanism whereby neighboring RBN can influence each other. Section 4 describes the experiments which have been performed with this model, in order to analyze the effects of coupling on the dynamics. Finally, in section 5 we draw some brief conclusions and indications for further work. 2 A Brief Description of RBN There exist some different realizations of the idea of a random boolean network, which may differ in the network topology, the choice of the set of boolean functions, the updating strategies [2][3][9]. We will described here only the model which we used in our study, which is the same as that originally proposed by Kauffman, and which will be briefly called the classical RBN. Let us consider a network composed of N genes, or nodes, which can take either the value 0 (inactive) or (active). Let x i (t) {0,} be the activation value of node i at time t, and let X(t)=[x (t), x 2 (t) x N (t)] be the vector of activation values of all the genes (for simplicity, it will be assumed that activations are boolean). Real genes influence each other through their corresponding products and through the interaction of these products with other chemicals, by promoting or inhibiting the activation of target genes. In the corresponding model network these relationships are lumped in directed links (directed from node A to node B, if the product of gene A influences the activation of gene B) and boolean functions (which model the response of each node to the values of its input nodes). In a classical RBN each node has the same number of incoming connections k in, and its k in input nodes are chosen at random with uniform probability among the remaining N- nodes. The probability that a particular combination of input activities gives the response is the same for all the nodes and is specified by the value p. Both the topology and the boolean function associated to each gene do not change in time (i.e. we use the so-called quenched model). The network dynamics therefore is discrete and synchronous, so all

3 550 M. Villani et al. the nodes update their values at the same time: once the connections and the boolean functions of each node have been specified, X(t) uniquely determines X(t+). A careful analysis of some known real biological control circuits has shown that a) Boolean functions with a low probability of activation (i.e. a relatively high number of outputs which are 0) are more frequent than the others b) In most cases the functions are limited to those which are canalizing In this preliminary work we take (a) into account, while the set of boolean functions is built, as usual, by choosing one with probability p and 0 with probability -p (therefore these functions are not necessarily canalizing). The model shows two main dynamical regimes: by observing for example how the average number of attractors and the average cycle length scales with the number of nodes N we can note that these variables could increase their values as a power law (ordered region) or could diverge exponentially (disordered region), depending upon the value of the parameter k in and the bias p (see Figure a). Systems near the interface between the two regions (i.e. in the critical region) show a particularly interesting behavior, as described in the introduction. Several observations (summarized in [2][0]) indicate that biological cells, because of this biological constraints, tend to be found in the ordered region not too far from the border between ordered and disordered regimes (the edge of chaos ) thus allowing both control and evolution. In this work we are interested in understanding what happens when the cells are grouped in a higher order organization like a tissue, asking what is the influence (if any) of this grouping on the ordered/chaotic behavior of cells. A priori, it could be argued that cells in tissues should be rather more ordered than isolated ones, thus simplifying system-level control, but also the opposite, i.e. that the additional interactions could introduce more constraints, leading to a more frustrated (and disordered) system. These hypotheses need testing, and this can be done in a disciplined way using particular models. (a) (b) Fig.. (a) Ordered and disordered regions for a single random boolean network; the border between the two region is given by the formula (k in ) - =2p(-p)[3] (b) the mathematical idealization of a tissue utilized in this article: each square cell is a complete random boolean network; a subset of its nodes interacts with the first four neighbors RBNs

4 Coupled Random Boolean Network Forming an Artificial Tissue 55 3 A Model Tissue Now we have to define a mathematical analogue of a generic tissue. This requires to define: (a) a topology of the tissue (b) the kind of random boolean networks present on each cell of the tissue (c) the rules of interaction among the cells of the tissue A simple topology like that of Figure b, where square cells interact with their first four neighbors inside a two dimensional world, represents a schematization of the spatial topology of some tissues, and will be used here. Each tissue is composed by homogeneous cells, and in general all the cells of a given multicellular individual share the same genetic material, therefore in each cell we have to consider a copy of the same random boolean network (same topology of the RBN and same boolean function). The rules of interaction among cells are very important. It is possible to take into account several possibilities, however the physics of the problem provides useful suggestions. A gene in cell A can influence another gene in cell B by synthesizing a protein which (may trigger a cascade of reactions some of whose products) may cross the cell membrane. Therefore, if a gene is active in A (so that its value is ) it may affect B, but if it is inactive it has no effect. Therefore we assume that: only a subset of the total number of nodes that define the RBN can be influenced by neighboring cells (not all the proteins cross membranes) ( described by a parameter fraction of interacting nodes, frin) the effective input given to the other nodes by node a ij (activity of node j belonging to cell i), whose protein can diffuse through the membrane, is if at least one of the four nodes with the same value of j, belonging to the RBNs present in the four neighboring cells of cell i, is the interactions are limited to nearest neighbors (we adopted the von Neumann neighborhood N,S,E,W) The model defined above is clearly a square cellular automaton, where each cell has a fairly complicated behavior, since it hosts a full RBN. 4 Results 4. Description of Parameters and Methods In our initial testing of this system, we concentrated on networks which are close to their critical point. This choice should allow us to better detect the effects of embedding them in a higher order system. Therefore, taking into account the fact that in nature those activation functions which (in the boolean approximation) show a bias towards the value 0 seems to be preferred, we chose k in =3 and p close to 0.2 (the critical value for k in =3, cfr the legend to Figure a). Incidentally, this choice implies a large presence of canalizing functions (as it is found in biology []): also with the

5 552 M. Villani et al. highest p value we utilized (p=0.22), more that 78% of boolean functions are canalizing in at least one input. The initial condition is chosen at random for every RBN, independently from those of the other cells. The number of nodes of every RBN (N) is 00, and the dimension of our artificial tissue is a square of 20x20 elements (so the total number of genes in the tissue is ); the global topology is that of a torus. In order to find the attractors of each RBN, we run the system for 600 steps (a step being a complete update of each node of each RBN present in the system), and check the presence of an attractor in each RBN belonging to the tissue during the last 200 steps (therefore, we are not able to find attractors whose period is higher than 200 steps, nor those which are reached after a very long transient). When each RBN reaches an attractor (or when the system reaches 600 step) the search ends. For each level of the intensity of interaction frin (the fraction of nodes whose outcome can affect neighboring cells) we made a series of 000 runs, each run involving a different RBN (same kin and p, but different topology and boolean functions) and different initial conditions. We consider the following variables. For each series of runs: the fraction of runs α where all the cells of the system reach the same attractor (out of 000 runs) the fraction of runs β where all the cells of the system reach an attractor (out of 000 runs) the fraction of runs γ where no cell reaches an attractor (out of 000 runs) and, for each run of each series: the number of different attractors present at the end of the run the number of different periods present at the end of the run the average length of the 20x20 RBN periods at the end of the run the structural factor sfct (see below) at the end of the run sfct is an aggregate variable we utilize as a first indicator of presence of homogeneous zones inside the artificial tissue. For each RBN i, we compute the number of nearest neighboring RBNs that are in the same attractor of RBN i, and sum all the 20x20 quantities. If all the RBN share the same attractor (the idealized situation where all the cells of the system belong to only one kind of tissue) this variable reaches its maximum value 600 (20x20x4), otherwise the cells self organize in more sparse structures. 4.2 Experimental Results First of all, we analyze the behavior of the aggregate variables α, β and γ as function of the interaction intensity. As the strength of this interaction grows, the fraction of runs β where all the cells of the system reaches an attractor decreases; contemporarily, the fraction of runs γ where no cell reaches an attractor increases. Obviously, these measures are influenced by the search parameters we utilized, but this general behavior seems to happen for many sets of parameters. This indicates that the increase of interaction strength introduces more and more disorder into the systems.

6 Coupled Random Boolean Network Forming an Artificial Tissue 553 But this is not the whole story of the phenomenon: as the strength of this interaction grows, the fraction of runs α where all the cells of the system reach the same attractor increases. What's more, if we consider the fraction of runs (out of the runs where all the cells of the system reach an attractor) where all the cells of the system reach the same attractor, this increase is even more evident. This is an evidence that the increase of interaction strength introduces more and more order into the systems, if the system is already prone to the order (Figure 2a). (a) (b) Fig. 2. Fraction of runs where all the cells of the system reach the same attractor (α),where all the cells of the system reach an attractor (β) and where no cell reach an attractor (γ). (a) Tissue constituted by ordered RBN (k in =3, p=0.20); (b) tissue constituted by slightly disordered RBN (k in =3, p=0.22). A tentative explanation may be based on the observation that, also for RBN well inside the ordered region, exists a small but finite subset of networks that are chaotic [2]. That is, a possible interpretation of our result is that the increasing strength of interaction among neighboring RBNs amplifies the already present tendencies (or at least the already present tendencies of the majority of RBN present inside the tissue). Networks prone to disorder are more disordered, and networks already prone to order can reinforce their tendency and are more ordered. This description is enforced by a new series of simulations (Figure 2b), where the RBN are more slightly into the chaotic region (this series has p=0.22; we remember that the border between order and chaos for systems with k i n=3 is approximately p=0.2). The runs where all the cells reach an attractor decrease in a more evident way, but the system is still able to increase the fraction of cases where all the cells reach the same attractor (phenomenon again more evident if we consider the fraction of cells that reach the same attractor out the fraction of runs where all the cells reach an attractor). Then, what happens to fairly ordered systems? We have to carefully interpret, or select, the data we produced: how we can compare systems where all the cells reach an attractor and systems where only 30% (5%, 75%, ) of the cells do it? As a first step we decided to take into account only the systems where all the cells reach an attractor (but a survey of some less conservative cases shows that the general conclusions could be quite similar).

7 554 M. Villani et al. Therefore, we extract from our data (k in =3 and p=0.20) all the systems where all the cells reach an attractor; the data are very noisy (the number of possible different RBNs and the number of possible different initial conditions are enormous, and therefore any realistic set of runs is always an undersampling), nevertheless some interesting trends are visible (see Figure 3). The most tangible changes are evident on the distributions of the number of different periods and of the average period (Figure 3b and Figure 3c): the higher the strength of the interaction among neighboring cells, the narrower are the distributions. That is, the system decrease the number of different periods that are present on the artificial tissue at the end of the runs, and their average becomes smaller; on average, the RBNs are compelled to share some characteristics. A second observation is that a large part of the effect is already present at the first switch on of the interaction: the further strengthening of the interaction results in changes of smaller entity. Frequency 0, 0,0 0 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Frequency 0, 0,0 0 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0, , Number of different attractors Number of different periods (a) (b) Frequency 0, 0,0 0, Average period 0 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 (c) (d) Fig. 3. Distribution of the number of different attractors (a), of the number of different periods (b), of the average of periods presents on the artificial tissue (c) and of the structural parameter sfct (d) as function of the strength of interaction coupling (spanning from 0.0 to.0 see the legends). The involved RBNs are ordered networks (k in =3 and p=0.20); the total number of RBNs that normalize the distributions is shown by line α in Figure 2a. Let us now consider the number of attractors present inside the artificial tissue. Figure 3a (the distribution of the number of different attractors present at the end of each run) shows that there is a small effect due to the growing strength among neighboring RBN, but this distribution doesn t allow us to observe, for example, the

8 Coupled Random Boolean Network Forming an Artificial Tissue 555 formation of islands of attractors inside the matrix (which might be an interesting phenomenon). Therefore we need another indicator: as a first attempt, we propose the quantity sfct discussed above. The sfct distribution has an evident peak on it maximum value (that is, the cells tend to reach the same attractor); moreover, this peak grows hardly as the interaction strength increases (Figure 3d and its insert). When the system doesn t reach this so homogeneous situation, it could be found in a very high number of situations (the long tails at the left of the peaks), but the importance of these tails decreases as the strength of interaction becomes more intense. That is, the presence of homogeneous zones inside the system is more and more intense as the interaction strength grows up (see Figure 4 for an example of association between the presence of homogeneous zones and the value of sfct). (a) (b) (c) (d) Fig. 4. Emerging of homogeneous zones inside the artificial tissue; the variable shown is the kind of attractor. (a) interaction strength at 0 and sfct=323; (b) interaction strength at 0. and sfct=308; (c) interaction strength at 0.3 and sfct=38; (d) interaction strength at 0.7 and sfct=600. This last case shows a complete homogeneous tissue, with only one attractor. There is another more subtle and interesting issue: in Figure 4 we are observing the kind of attractors, but this is not NECESSARILY WHAT MATTERS FROM A FUNCTIONAL VIEWPOINT. If we consider one node (whose product can pass through the cell membrane) of one particular RBN and if its activation is 0, this doesn t means that the real effect of this activation is 0. It is enough that one of its neighboring RBNs has the same node with activation, that under all the functional aspects this node behaves as the state. That is, it is possible that a part of differences we are now observing among the cells doesn t exists as functional

9 556 M. Villani et al. difference. And, for a tissue, it is important that all the cells be similar under the functional aspect. This aspect of the problem will be the subject of further analysis. 5 Conclusions This work is a preliminary study of the effects of the interactions among several RBN: an intriguing phenomenon has been observed, i.e. that the interaction, as it has been modeled here, can have different effects on different kinds of RBN. In particular, the fraction of networks which do not reach an attractor increases, indicating a growth of dynamical disorder. But, limiting our considerations to those networks which reach an attractor within the time limits of our simulations, we observe that they tend to more homogeneous attractors. It is interesting to speculate about the possible implications of this finding from the viewpoint of evolution theory; since a certain degree of order is needed to allow robust functionality, those networks which reach an attractor might have been selected, and in this case the ordering effect of interaction would prevail a finding which seems biologically plausible. Further work has to been done in order to investigate, inter alia, the effect of different coupling interactions, different values of parameters, and to investigate the interaction between genomic and functional differences. References. Kauffman, S.A.: Gene Regulation Networks: A Theory of their Global Structure and Behavior. Curr. Top. Dev. Biol 6 (97), Kauffman, S. A.: The origins of order. Oxford University Press (993) 3. M. Aldana, S. Coppersmith, L. P. Kadanoff, Boolean Dynamics with Random Couplings, in E. Kaplan, J.E. Marsden, K.R. Sreenivasan (eds), Perspectives and Problems in Nonlinear Science. Springer Applied Mathematical Sciences Series (2003). Also available at 4. Serra, R., Villani, M. & Semeria, A.: Robustness to damage of biological and synthetic networks. In W. Banzhaf, T. Christaller, P. Dittrich, J.T. Kim & J. Ziegler (eds): Advances in Artificial Life. Berlin: Springer Lecture Notes in Artificial Intelligence 280, (2003) Serra, R., Villani, M. & Semeria, A.: Genetic network models and statistical properties of gene expression data in knock-out experiments. J. Theor. Biol. 227 () (2004) P.Ramo, J.Kesseli, O. Yli-Harja 2005 Perturbation avalanches and criticality in gene regulatory networks Journal of Theoretical Biology, submitted 7. Shmulevich, I. and Kauffman, S.A. Activities and Sensitivities in Boolean Network Models, Phys Rev. Lett. 93(4), (-4) (2004) 8. S.Kauffman, C.Peterson, B.Samuelsson, C.Troein Genetic networks with canalyzing Boolean rules are always stable PNAS vol. 0 no.49 (2004) 9. Harvey, I., and Bossomaier, T. Time out of joint: Attractors in asynchronous random boolean networks. In Husbands, P., and Harvey, I., eds., Proceedings of the Fourth European Conference on Artificial Life (ECAL97) MIT Press (997) Kauffman, S.A.: Investigations. Oxford University Press (2000). Harris, S.E., Sawhill, B.K., Wuensche, A. & Kauffman, S.A.: A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity 7 (2002) U. Bastolla and G. Parisi The modular structure of Kauffman networks Physica D 5 (998)