complex networks evolutionary dynamics using ...

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International Journal of Bifurcation and Chaos c World Scientific Publishing Company

COMPLEX NETWORKS EVOLUTIONARY DYNAMICS USING GENETIC ALGORITHMS Daniel Aguilar-Hidalgo Departamento de F´ısica de la Materia Condensada, Universidad de Sevilla, Av. Reina Mercedes s/n Sevilla, 41012, Spain [email protected] Antonio Córdoba Zurita Departamento de F´ısica de la Materia Condensada, Universidad de Sevilla, Av. Reina Mercedes s/n Sevilla, 41012, Spain [email protected] Ma Carmen Lemos Fernández Departamento de F´ısica de la Materia Condensada, Universidad de Sevilla, Av. Reina Mercedes s/n Sevilla, 41012, Spain [email protected] Received (to be inserted by publisher)

Gene regulatory networks set a second order approximation to genetics understanding, where the first order is the knowledge at the single gene activity level. With the increasing number of sequenced genomes, including humans, the time has come to investigate the interactions among myriads of genes that result into complex behaviours. These characteristics are included in the novel discipline of Systems Biology. The composition and unfolding of interactions among genes determine the activity of cells and, when is considered during development, the organogenesis. Hence the interest of building representative networks of gene expression and their time evolution, i.e. the structure as the network dynamics, for certain development processes. The complexity of this kind of problems makes imperative to analyse the problem in the field of network theory and the evolutionary dynamics of complex systems. All this has led us to investigate, in a first step, the evolutionary dynamics in generic networks. Thus, the results can be used in experimental researches in the field of Systems Biology. This research aims to decode the transformation rules governing the evolutionary dynamics in a network. To do this, a genetic algorithm has been implemented in which, starting from initial and ending network states, it is possible to determine the transformation dynamics between these states by using simple acting rules. The network description is the following: a) The network node values in the initial and ending states can be active or inactive; b) The network links can act as activators or repressors; c) A set of rules is established in order to transform the initial state into the ending one; d) Due to the low connectivity, frequently observed, in gene regulatory networks, each node will hold a maximum of three inputs with no restriction on outputs. The “chromosomes” of the genetic algorithm include two parts, one related to the node links and another related to the transformation rules. The implemented rules are based in certain genetic interactions behaviour. The rules and their combinations are compound by logic conditions and set the bases to the network motifs formation, which are the building blocks of the network dynamics. The implemented algorithm is able to find appropriate dynamics in complex networks evolution 1

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Daniel Aguilar Hidalgo

among different states for several cases. Keywords: Complex networks, evolutionary dynamics, genetic algorithm, systems biology.

1. Introduction In the recent years, Systems Biology offers a holistic approach of biological processes, in which the interaction among the constituent elements of the process affects its own development. This approach allows going deeply into the emergence of properties and complex functions. In the study of several biological processes such as genetic diseases and studies in developmental Biology and organogenesis, a representation of the problem constituent elements is essential for its study, this is, the regulatory network. A gene regulatory network establishes a relationship among genes that solves a common function. The interaction among the elements of the regulatory network can be so complex that the analysis of the regulated function may need high level mathematics and computer resources. One of the possibilities on studying gene regulatory networks is within the field of network theory and its evolution as complex networks. At this point there are multiple methods for the analysis of complex networks capable of providing information on the evolution of the network. The problem of the evolution of a complex network is too large to be treated as a whole, so it should be divided into the interactions of small modules or blocks which are the building units of the entire network. These building blocks of the network will be based on simple rules, and their interaction within the entire network will lead to complex behaviors. Once defined the operation rules of the network building blocks will be necessary to develop a method to recognize these rules within the entire network and decipher which of them govern the network transformation dynamics.

2. Model The evolution of a complex network starts from an initial state A, reaching a final state B. It is expected that a rule set will establish the transformation dynamics that turns the network state A into its state B. The network attributes are the following: nodes, that can be active or inactive; links, that can be activators or repressors. This way, the state A can have active nodes that may be inactive at state B and viceversa; there also may be active nodes in both states. The functionality of an activator link is to activate a node or keep it active, and the functionality of a repressor link is to deactivate a node or keep it inactive. The model final goal is to determine the transformation dynamics between the network states, this is, the associated transformation rules set. A genetic algorithm (GA) has been developed. If the network’s initial and final states are known, the transformation dynamics between these states can be determined using simple Boolean rules. The choice of a Boolean paradigm on the analysis of complex networks dynamics is firstly based on the computational cost. By definition, a complex network will be formed by a high number of elements (nodes, links) and even non-linear interactions, so the computational cost of designing a continuos model may be very high. Reducing the network to a Boolean model can facilitate its analysis giving comprehensive and useful information. Moreover, this complexity reduction is enough for the correct application of this model to digital based networks. The application of this model to more complex networks as biological networks is, in the same way, useful. Most biological networks can be simplified to a Boolean perspective and its qualitative output do not differ from continuos considerations [Kauffman et al. , 2003; Chaves et al. , 2005]. To do this study, it was built a random network of 32 nodes in which it was considered a maximum of three input links per node, giving no restriction for the number of output links. The resulting network is formed by 32 nodes and 74 links.

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Complex networks evolutionary dynamics using genetic algorithms

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2.1. Genetic Algorithm A genetic algorithm is a heuristic optimization method inspired on natural biological evolution. This method promotes the evolution of a population of individuals, chromosomes, through analogue techniques to biological evolution, this is, recombination and genetic mutation. Each of the individuals of the population is one possible solution of the problem to optimize. The same way biological evolution of organisms can improve their characteristics in the new generations, the GA can make the population of individuals evolve in order to find optimal, or suboptimal, solutions to the original problem within the new generations. To do this, the GA explores the solutions space. In this case, using a GA seems very appropriate due to the structure of the problem, linked information interacting through certain rules that may work as the coupling of building blocks. The GA, in fact, can be treated as working by discovering, emphasizing, and recombining good building blocks of solutions in a highly parallel fashion. This way, good solutions tend to be built from good building blocks. [Mitchell , 1998] The built GA provides simultaneously the optimization of various parameters; the state of the nodes in B, the links referred to the network nodes and the rules applied over each node. The GA chromosome is formed by two of these parameters, the links among the nodes, that can take values +1, −1, 0, depending on the link being an activator, a repressor or there is no link; and the rules applied to each node, that can take as many values as different rules are possible (Fig. 1).

Fig. 1.

Chromosome construction scheme

Each individual of the initial population inserted in the GA starts from a known state A. This state A goes into a transformation dynamics where, by using certain rules associated to each node, it is built a final state B.

2.1.1. Transformation rules One of the most important points in the model is the use of certain rules that, starting from a known initial state of the network, the model is able to build a pragmatic and consequent final state, with both the rules and the previously determined experimental final state. This way, the election of the rules that will define the evolutionary dynamics of the model is very important. It is well known that in a network we can define small subnetworks, or network motifs, which are patterns of interconnections that recur in many different parts of a network at frequencies much higher than those found in randomized networks[Shen-Orr et al. , 2002]; these motifs can be considered as complex networks building blocks[Milo et al. , 2002], providing the network with a modular functionality[Alon , 2007]. The network motifs characterize a modular functionality by using logic rules and a composition of logic rules, this way, the biological motif is translated into a simple logic circuit; so, a complex logic circuit is obtained from the complete biological network.

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The chosen rules are the following: (1) Rule of most: If the target node inputs consist on activators and repressors, the node’s final state is determined by the action of most, if the number of activators is equal to the number of repressors, the final node state will be the same as in the initial state. (2) Absolute repressor rule: Just one repressor link is enough to inhibit the target and deactivate the node. (3) Joint action of two activators: Logic AND gate with activators. The target node will be activated only if, at least, two activator input links operate at the same time. (4) Joint action of two repressors: Logic AND gate with repressors. The target node will be deactivated only if, at least, two repressor input links operate at the same time. The application of these rules is partially redundant because for some situations, the application of different rules may lead to the same result (see table 1). Table 1. List of applicable rules by number and type of inputs. The symbol + represents an activator link; the symbol − represents a repressor link. Inputs

Activation

+++ ++− ++ + −−− −−+ −− − +−

1,3 1,3 1,3 1

Deactivation

No action

Not applicable

1

2,4 4 2,4 2,3,4 3 3 3 3,4 3,4

2

1,2,4 1,2,4 1,2,4 1,2 2

2.1.2. Dynamics Once defined the GA individuals and the transformation rules, it is time to know how the algorithm leads state A into B. As mentioned before, each individual is formed by the values of the links among all nodes and the rules applied to each node for the result computation. The algorithm goes through each individual counting the number of input links for each node, differentiating positive from negative interactions. Afterwards, the resulting output for each node (active or inactive) will be determined by the applying rule (e.g. two positive and one negative links as node inputs may result in an activation of the target node if the applied rule is 3-coactivators, or in a deactivation if the applied rule is 2-absolute repressor ). The application of this dynamics results in a new network state (B) from a previous state (A). However, due to the non-linear interactions that the network may have, this dynamics is repeated for five times in order to ensure a system stabilization. It was confirmed by several executions that the network takes less than five loops to reach a final network steady state (B).

2.2. Fitness The genetic algorithm fitness function evaluates each population individual staked by scoring them depending on the mentioned evaluation. In this model, to evaluate the fitness function, two functions are considered: The distance between the known final state, B, and the final state provided for the GA, B ∗ ; and the distance between the known links and the provided by the model; introducing in those functions

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a regulatory parameter λ. Here we consider distance as the number of differences between B and B ∗ .

φ = λ · dist[B, B ∗ ] + (1 − λ) · dist[links, links∗ ]

0≤λ≤1

(1)

Each term of the φ function is multiplied by the mentioned λ parameter, which determines the importance that has every term in the global calculation of the evaluation. The φ function must be normalized in order to use the evaluation data in a controlled scale, so that when having two terms with different maximums, the normalization of the φ function is the following:

φnorm = λ

dist[B, B ∗ ] dist[links, links∗ ] + (1 − λ) max(dist[B, B ∗ ]) max(dist[links, links∗ ])

0≤λ≤1

(2)

The φ function is based on the measure of the differences between the population individual and the desired solution, so the better the individual, the lower the numerical value obtained in the evaluation. If it is preferred, the better the individual, the higher the numerical value obtained in the evaluation, the fitness function must be defined as: f itness = 1 − φnorm

(3)

Once all the individuals are evaluated, a new improved population will be created. To do this, an elitist selection of the population will be made, this is, just the best individual from the previous population will be kept; the rest individuals will be generated by crossover and mutation.

2.3. Crossover As previously commented, just the best individual from the previous population will be kept to the next one, so it will be necessary to generate the rest of the individuals to complete the new population. To do this, the individuals from the old population will be used as genetic resource. In order to generate two new individuals, a crossover between two already evaluated individuals will be made, but, how are the parent individuals chosen? If the aim is to get a new population as an enhanced version of the previous one, it is necessary to try to simulate an improvement like in natural selection, i.e., the better individuals have a higher probability of reproduction; or else, the individuals with better evaluation from the fitness function have higher probability to generate new individuals by crossover. There are numerous methods to make the selection of the better individuals, for example, to select a concrete number of better individuals from the previous selection to generate the whole new population; or to choose for the crossover all the individuals over a fitness threshold. However, since in the real world the reproduction probability not only depends on the individual competence, but there is a strong stochastic component, it was decided to choose the roulette method. In the roulette method, all the individuals in the population are capable of generating new individuals, and their probability is directly proportional to the normalize fitness of each individual, this way, the higher the fitness value, the higher the probability of being chosen to generate new individuals by crossover, so in average, the new generations may be more competent than the above, i.e., the average individual may have a higher fitness value as the number of generations increases. To do the crossover, it should be noted that the individual is formed by two parts, rules and links. It is necessary to determine two related crossover points. The election of the links crossover point will be random, and the rules crossover point will be determined by the following expression: K = int

l N

+1

(4)

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Where K is the rules crossover point, l is the links crossover point and N is the number of nodes in the network.

Fig. 2.

Crossover scheme

The new individuals obtained by crossover must keep a connectivity, this is, the number of links, similar to the seeked solution. This way, once a new individual is generated, it undergoes a pref itness which is a filter to select the new generated individuals that will conform the new population. In this pref itness it is counted the number of links, and only if its number is kept inside a prefixed interval including the new conditions, the individual will be accepted to be included in the new population. If not, it is rejected. To be concrete, the connectivity must be kept in an interval of the 10% above and below the number of links from the original network. Due to the low connectivity frequently observed in biological networks [Leclerc et al. , 2008], and in order to resemble to a biological problem, each node will hold a maximum of three inputs with no restriction on outputs.

2.4. Mutation In nature, every individual is susceptible to disruption or change in genetic information that can produce a change in their characteristic, with the possibility of transmitting the change to their offspring. This is what is known as mutation. In evolutionary computation, the aim is to simulate the behaviour of nature in the generation of new and better individuals, in order to optimize a specific process. This is why it is convenient to use the concept of mutation in the process of generating new individuals. The use of mutation attempts to avoid

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inbreeding (in biological terms) or that the system is trapped in a local minimum (from the point of view of the optimization process) away from the absolute minimum. In this model, after generating new individuals by crossover from the better individuals of the previous population, each chromosome has a certain mutation probability for each one of their parts. The application of this mutation to either of the chromosome parts is random made, both in the choice of the mutated element as the final value after mutating. Every element in each chromosome part has the same mutation probability, and all the possible final values after mutation has the same weight in the random choice, this is 1/2 for links (changes to one of two remaining possibilities) and 1/3 for rules (changes to one of 3 remaining possibilities). The mutation probability applied to the links among the nodes is: µl = 0.001

(5)

and for the part of the chromosome formed by the rules applied to each node: µr = 0.5

(6)

3. Execution In order to demonstrate the effectiveness of the presented model, the initial state A of a 32 nodes random network has been built. This initial state is formed by 16 active nodes. The network has two types of links, activators and repressors. The activator link supports the activation of the target node, while the repressor link supports disabling the target node. The model is able to determine the transformation dynamics that takes the initial state of a network to its final state. To conduct the experiment properly it is needed to configure a set of rules associated with each node (Table 2). This set of rules applied to the initial state of the network will determine the final state, B, of the network, which has 22 active nodes. It can be seen that in some cases, as in node N.9, the rule 2 (absolute repressor rule) is applied to that node, and yet the node is not disabled. By observing the node N.9 in figure 3, the node has no input repressor link, so its deactivation is not possible, in this case, the rule applied would have no effect on the target node. The merging of both states A and B results in a random network formed by 32 nodes and 74 links (Fig. 3). Some of the nodes are active just in the network initial state A and others are active just in the network final state B; the rest are active during the whole evolution of the network, from A to B. It should be noted that due to the redundancy that rules may show in some instances, different rules applied on the same node can produce the same result. It is set an initial population of 100 individuals. Each chromosome of the genetic algorithm is formed by 32 rules randomly generated, one for each node, with values: • • • •

1 2 3 4

= = = =

Rule of most Absolute repressor rule Co-activators rule Co-repressors rule

And the links among the nodes, with values: • 1 = Activator • -1 = Repressor • 0 = Disabled The values of the initial population links are created as permutations of the original network links up to 100 individuals, keeping the number of links in a determined interval, as previously shown. This permutation is done by moving a position all the elements of the chromosome for each new individual.

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Daniel Aguilar Hidalgo Table 2. Set of transformation rules between the states A and B associated with each node, initials (Rules) and determined by the GA (GA Rules). Nodes

A

Rules

B

GA Rules

N.0 N.1 N.2 N.3 N.4 N.5 N.6 N.7 N.8 N.9 N.10 N.11 N.12 N.13 N.14 N.15 N.16 N.17 N.18 N.19 N.20 N.21 N.22 N.23 N.24 N.25 N.26 N.27 N.28 N.29 N.30 N.31

0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0

1 1 2 3 2 3 2 4 1 2 1 2 1 1 1 1 2 2 2 1 3 1 4 1 1 2 1 1 2 3 1 1

1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1

1 1 1 3 1 1 2 1 1 1 1 2 3 1 3 1 2 2 2 1 1 1 3 1 1 1 4 1 1 2 2 1

λ parameter allows giving different importance to each fitness function term. It is supposed that it can be found some possible variability in links rather than in nodes, so the used value for this parameter should be high. The value for the λ parameter is the following: λ = 0.9

(7)

This implies that within the space of solutions, the ones seeked are those that meet, especially, with the same active nodes in each state, giving less importance to the links among the nodes. Thus leaving the possibility that the optimization process return previously undetected new links, that can be essential for the correct network transformation dynamics; false positives may also be detected as inserted in the original network. These differences found will be submitted for validation in a further experimental analysis. It was determined that for λ < 0.8 it is difficult to find results with the same node configuration as in

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Fig. 3. Representation of the random network formed by 32 nodes and 74 links. Nodes: Red = A; Green = B; Yellow = A+B. Links: Green = Activator; Red = Repressor

the original network.

4. Results The model, after a 400.000 generations execution, found an individual inside the solutions space that almost completely fits the original conditions, establishing correct applying rules on the network for its transformation between the two states, A and B. The result, with a fitness value of 99.97%, shows the same active nodes as in the original case (Fig. 4). A higher number of links, 77, has been found to resolve the network, finding 3 new links. In figure 5 it is shown the evolution graph of the best solution fitness over the 400.000 generations. The table 2 shows the rule set, found by the GA, applied to each node for a correct transformation between the states A, initial, and B, final. It is not excluded that in other simulations we could obtain a similar result, although not strictly equal. In fact, the AG provides a range of possible solutions that could be a starting point for experimental validation, since there is no certainty that the maximum found for the fitness function is an absolute maximum.

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Fig. 4. Representation of the network found by the GA, formed by 32 nodes and 77 links. Nodes: Red = A; Green = B; Yellow = A+B. Links: Green = Activator; Red = Repressor; Thick = New links

5. Systems Biology application Genetic algorithms have been widely used in problems of molecular Biology, bioinformatics, genomics and proteomics, among other biological and non biological areas. In this case, the application of the model is done on a problem of developmental Biology, in particular, on the evolution of a gene regulatory network between different stages of mus musculus mouse eye development. These two embryonic stages in which the study focuses on are, the states E9 and E10.5, which means 9 and 10.5 days after fertilization. The choice of these stages has biological significance because they determine concrete development phases: the specification of eye progenitor cells at E9 and the morphogenesis of the optic cup at E10.5. The gene network construction has been done on the basis of experimental retinome gene expression data, in mouse embryonic stages, as well as an extensive literature review ([Pinson et al. , 2006], [Goudreau et al. , 2002], [Kim et al. , 2001], [Arai et al. , 2005], [Clarka et al. , 2008], [Zhou & Anderson , 2002], [Rowan & Cepko , 2005], [Tétreaulta et al. , 2009], [Purcella et al. , 2005], [Maa , 2005], [Planque , 2001], [Heanue et al. , 1999], [Bondurand et al. , 2000], [Ohto et al. , 1999], [Hatakeyama et al. , 2001], [Sun et al. , 2003], [Abe et al. , 2006], [Mu et al. , 2008], [Mao et al. , 2008]). All this has established a network of 91 interactions among 32 genes expressed in the studied stages where genes are the network nodes and the interactions are the links (see Fig. 6). To make the execution of the GA with the gene regulatory network data, 100 individuals were built

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0.08

0.07

0.06

max

0.05

0.04

0.03

0.02

0.01

0.00

0

100000

200000

300000

400000

Generations number

Fig. 5.

Evolution of the best individual

with connectivity similar to the experimental network one. After 200.000 generations a network with a fitness value of 99.95% was obtained (Fig. 8) and a rule set that correctly transforms the network state E9 in E10.5 (see table 3). The GA returns a network with the same configuration in nodes (genes) as in the experimental case, and a difference in 5 links (interactions). Fig. 7 shows the network found by the GA, where the new interactions can be seen as thick links.

6. Conclusions The study of evolutionary dynamics in complex networks using genetic algorithms is resolved as an appropriate and promising method for future studies. The modelled genetic algorithm provides a way of knowing an appropriate dynamics between different states. Its application in Systems Biology field can determine the evolution of gene regulatory networks and provide experimentally testable data resolving new interactions between nodes, and revealing previously unknown false positives.

Acknowledgements ´ To Dr. Fernando Casares and Ma Angeles Dom´ınguez from CABD (CSIC) for their contribution and advices.

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REFERENCES

Fig. 6. Representation of the gene regulatory network formed by 32 genes and 91 interactions. Nodes: Red = E9; Green = E10.5; Yellow = E9+E10.5. Links: Green = Activator; Red = Repressor; Solid = Transcriptional interaction; Discontinuous = Protein-protein interaction.

This work has been partially financed by project FIS-2008-04120 of the Spanish Government.

References Abe, Y., Chen, W., Huang, W., Nishino, M. & Li YP. [2006] “CNBP regulates forebrain formation at organogenesis stage in chick embryos,” Dev. Biol. 295, pp. 116–127. Alon U. [2007] An introduction to systems biology: design principles of biological circuits, (Chapman & Hall/CRC). Arai, Y., Funatsu, N., Numayama-Tsuruta, K., Nomura, T., Nakamura, S. & Osumi, N. [2005] “Role of Fabp7, a downstream gene of Pax6, in the maintenance of neuroepithelial cells during early embryonic development of the rat cortex,” J. Neurosci. 25, pp. 9752–9761. Bondurand, N., Pingault, V., Goerich, D. E., Lemort, N., Sock, E., Le Caignec, C., Wegner, M. & Goossens,

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Fig. 7. Representation of the gene regulatory network found by the GA, formed by 32 genes and 96 interactions. Nodes: Red = E9; Green = E10.5; Yellow = E9+E10.5. Links: Green = Activator; Red = Repressor; Solid = Transcriptional interaction; Discontinuous = Protein-protein interaction; Thick = New links.

M. [2000] “Interaction among SOX10, PAX3 and MITF, three genes altered in Waardenburg syndrome,” Hum. Mol. Genet. 13, pp. 1907–1917. Chaves, M., Albert R. & Sontag E. D. [2005] “Robustness and fragility of Boolean modelsnext term for previous termgenetic regulatory networks,” J. Theor. Biol. 235, pp. 431–449. Clarka, A. M., Yuna, S., Veienb, E. S., Wua, Y., Y., Chowc, R. L., Dorskyb, R. I. & Levine, E. M. [2008] “Negative regulation of Vsx1 by its paralog Chx10/Vsx2 is conserved in the vertebrate retina,” Brain Research 1192, pp. 99–113. Goudreau, G., Petrou, P., Reneker, L. W. & Graw, J. [2002] “Mutually regulated expression of Pax6 and Six3 and its implications for the Pax6 haploinsufficient lens phenotype,” Proc Natl Acad Sci USA 99, pp. 8719–8724. Hatakeyama, J., Tomita, K., Inoue, T. & Kageyama, R. [2001] “Roles of homeobox and bHLH genes in specification of a retinal cell type,” Development 128, pp. 1313–1322. Heanue, T. A., Reshef, R., Davis, R. J., Mardon, G., Oliver, G., Tomarev, S., Lassar, A. B. & Tabin, C. J. [1999] “Synergistic regulation of vertebrate muscle development by Dach2, Eya2, and Six1, homologs of genes required for Drosophila eye formation,” Genes Dev. 13, pp. 3231-3243. Kauffman S., Peterson C., Samuelsson B., & Troein C. [2003] “Random Boolean network models and the yeast transcriptional network” Proc Natl Acad Sci USA 21, pp. 1–5. Kim, A. S., Anderson, S.A., Rubenstein, J. L. R., Lowenstein, D. H. & Pleasure, S. J. [2001] “Pax-6

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Fig. 8.

Evolution of the best individual

Regulates Expression of SFRP-2 and Wnt-7b in the Developing CNS,” J. Neurosci. 21, pp. 1–5. Leclerc, R.D. [2008] “Survival of the sparsest: robust gene networks are parsimonious,” Mol. Syst. Biol. 4:213. Maa, W., Yana, R., Maoa, W. & Wang, Z. [2005] “Neurogenin3 promotes early retinal neurogenesis,” Mol. Cell. Neurosci. 40, pp. 187–198. Mao, C.A., Wang, S. W., Pan, P. & Klein, W. H. [2008] “Rewiring the retinal ganglion cell gene regulatory network: Neurod1 promotes retinal ganglion cell fate in the absence of Math5,” Development 135, pp. 3379–3388. Milo, R., Shen-Orr, Itzkovitz, S., Kashtan, N., Chklovskii, D. & Alon, U. [2002] “Network Motifs: Simple Building Blocks of Complex Networks,” Science 298, pp. 824–827. Mitchell, M. [1998] An Introduction to Genetic Algorithms, (A Bradford Book, The MIT Press. Cambridge, Massachussets. London, Englad). Mu, X., Fu, X., Beremand, P. D., Thomas, T. L. & Klein, W. H. [2008] “Gene-regulation logic in retinal ganglion cell development: Isl1 defines a critical branch distinct from but overlapping with Pou4f2,” Proc Natl Acad Sci USA 105, pp. 6942–6947 . Ohto, H., Kamada, S., Tago, K., Tominaga, S., Ozaki, H., Sato, S. & Kawakami, K. [1999] “Cooperation of Six and Eya in Activation of Their Target Genes through Nuclear Translocation of Eya,” Mol. Cell. Biol. 19, pp. 6815–6824.

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Table 3. Rules applied to each gene to lead state E9 to E10.5. Genes

E9

Rules

E10.5

Pax6 Tbr1 Sfrp2 Neurog3 Fabp7 Sox10 Foxc1 Eomes Mitf Lhx2 Ascl1 Maf Nes Six3 Eya2 Zic2 Cnbp Six2 Eya1 Six1 Six6 Dach2 Mdfi Pax3 Dach1 Olig2 Shh Nk2.2 Isl1 Neurod1 Vsx2 Olig1

1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1

1 1 2 3 2 3 2 4 1 2 1 2 1 1 1 1 2 2 2 1 3 1 4 1 1 2 1 1 2 3 1 1

1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1

Pinson, J., Simpson, T. I., Mason, J. O. & Price, D. J. [2006] “Positive autoregulation of the transcription factor Pax6 in response to increased levels of either of its major isoforms, Pax6 or Pax6(5a), in cultured cells,” Dev. Biology 6, pp. 25–34. Planque, N., Leconte, L., Coquelle, F. M., Martin, P. & Saule, S. [2001] “Specific Pax-6/Microphthalmia Transcription Factor Interactions Involve Their DNA-binding Domains and Inhibit Transcriptional Properties of Both Proteins,” J. Biol. Chem. 276, pp. 29330–29337. Purcella, P., Oliverb, G., Mardonc, G., Donnera A. L. & Maas, R. L. [2005] “Pax6-dependence of Six3, Eya1 and Dach1 expression during lens and nasal placode induction,” Gene Expr. Patterns 6, pp. 110–118. Rowan, S. & Cepko, C. L. [2005] “A POU factor binding site upstream of the Chx10 homeobox gene is required for Chx10 expression in subsets of retinal progenitor cells and bipolar cells,” Dev. Biology 281, pp. 240–255. Shen-Orr, S. S., Milo, R., Manga, S. & Alon, U. [2002] “Network motifs in the transcriptional regulation network of Escherichia coli,” Nature Genetics 31, pp. 64–68.

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REFERENCES

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