Genes, Morphogenesis, Evolution: Life and ALife Aspects

Cellular Automata Model for Gene Networks

In framework purposed by de Sales et al. model [17] the genome is represented by a set of n binary genes s_i, i=1...n. When s_i=1 the gene is active for transcription and the specific enzymes or structural proteins it codifies are produced. On the other hand, when si=0 the gene is inactive and the products it codifies are not synthesized. The network state at a given time t is specified by the activity pattern s1(t),s2(t) ,...,sn(t). Each gene i is regulated by K-1 other genes and by itself, through a function of the previous state of its regulatory elements. The gene activity state at the next time step is given by

where

is the coupling constant representing the regulatory action of the j_m(i) (m=1...K-1) input on gene i and J_ii is the autogenic regulation. sgn(x)=1 if x > 0 and vise versa.
All the gene states are simultaneously updated. In order to accomplish this, a given gene evaluates the present stimulus from all its regulatory genes, including itself. If the overall stimulus it receives at time t is positive, the gene activates or stays active if it was already active; otherwise it turns inactive or stays inactive.
The coupling constant J_ij is choose taking into account the following biological features.
1) The products of a determinant gene can activate, inhibit or not affect the transcription of another gene. In this model all the activatory interactions will assume the same value +J and the inhibitory ones -J. When the gene j does not influence the expression of a different gene i, the coupling constant is J_ij=0, corresponding to a diluted bond.
2) The gene interactions are asymmetric, i.e. J_ij < > J_ji. The case in which a given gene i activates another gene j that, in turn, inhibits i, is biologically frequent.
3) Autogenic or self-regulation gene control is frequent in living organisms. In the present CA model the self-control is provided by the J_ii coupling constants.
Since the molecular biologists have elucidated only partially the real connectivity matrix among genes one has choose a random distribution of nonsymmetrical J_ij (valid also for the self-interactions J_ii) described by

where

is Dirac's delta function and J=1.
Therefore, for a particular gene network, each bond J_ij is activatory (+1) or inhibitory (-1), with probability (1-p₁)/2, or diluted (J=0) with probability p₁.
Also, since almost all known regulated genes in prokaryotes and eukaryotes are directly controlled by up to six or ten gene products, the model involves K=9 regulatory inputs per gene, including itself. Of them K-1 inputs are either chosen at random among all the other remaining genes, with probability p₂, or are its neighbor genes with probability 1-p₂. Thus the p₂=1 limit corresponds to an infinite-range model with connectivity K=9 (including the self-interactions J_ii), whereas the p₂=0 limit corresponds to a square lattice in which each site has a Moore neighborhood defined by its eight nearest and next-nearest neighbors. For any other p₂ values the simultaneous presence of short- and long-range coupling reflects the biological fact that a given gene can be regulated by either its nearest neighbors or distant DNA sequences, whose proteins, produced in the cytoplasm, diffuse towards the cell nucleus.

References

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