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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 209
Probabilistic Regulatory Networks: Modelling Genetic Networks M.A. Avino-Diaz
^{1} and O. Moreno^{2}
M.A. Avino-Diaz, O. Moreno, "Probabilistic Regulatory Networks: Modelling Genetic Networks", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 209, 2006. doi:10.4203/ccp.84.209
Keywords: isomorphism of Markov chain, probabilistic regulatory networks, Boolean network, transition matrix, category, dynamical systems.Summary
We can understand the complex interactions of genes
using simplified models, such as discrete or continuous models of
genes. Developing computational tools permits description of gene
functions and understanding the mechanism of regulation
[3]. We focus our attention in the discrete structure of
genetic regulatory networks instead of continuous models.
The probabilistic gene regulatory network (PRN) is a natural
generalization of the probabilistic boolean network (PBN) model
introduced in [4,1]. This model has
n
functions defined over a finite set X to itself, with
probabilities assigned to these functions. We present here the ideas
of -similar networks, and isomorphism of Markov Chains,
using the concept of -homomorphisms, where
,
is a distance between the probabilities.
A probabilistic regulatory network (PRN) [1] is a triple
where X is a finite set and
is a set of functions from X into itself, with a list
of selection probabilities, where
. The state space is a digraph, whose vertices are the
elements of X, and if
, there is an arrow going
from u to v, and the probability is assigned to this
arrow.
If is a set of selection probabilities we denote by the
characteristic function over ,
such that , if and . Let
and
be two PRN. A map
is an -
For example: suppose we have
two genes with two values that we denote as usual , that is
this PRN is a very simple PBN. The set of boolean functions A mathematical method is constructed here given the dependency graph of a set of genes. These genes could have either two, or three values. The model that we obtain gives the information about the subnetworks, the possible projections, and the fixed points. We present here a methodology for construct discrete networks using the dependency graph and a time series data. For genes with more than two states, we assign three possible values , and using the algorithm introduced in [2] we calculate the model PRN. References
- 1
- M. A. Aviñó, "Homomorphisms of Probabilistic Gene Regulatory Networks", Proceedings of GENSIPS 2006.
- 2
- M. A. Aviñó, E. Green, and O. Moreno, "Applications of Finite Fields to Dynamical Systems and Reverse Engineering Problems",
*Proceedings of ACM Symposium on Applied Computing*, 2004. doi:10.1145/967900.967939 - 3
- E. R. Dougherty, A. Datta, and C. Sima, "Developing therapeutic and diagnostic tools", Gen.Sig. Proc., IEEE Signal Processing Magazine, 46-68, Nov. 2005.
- 4
- I. Shmulevich, E. R. Dougherty, and W. Zhang, "From Boolean to probabilistic Boolean networks as models of genetic regulatory networks", Proc. of the IEEE. 90(11): 1778-1792, 2001. doi:10.1109/JPROC.2002.804686
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