STEADY STATE ANALYSIS BY PROBABILTY

BOOLEAN NETWORK

• Motivated by human genome project a new view of biology called system biology is emerging.

• System biology dosnot investigate individual gene, protein or cells in isolation . rather it studies the behaviour and relation ship of all cells,proteins , DNA and RNA in a biological system called a CELLULAR NETWORK.

• The main active network may be those associated with gene regulation, which regulates the growth ,replication and death of cells in response to change in the environment.

• In early 1960’s JACOB and MONOD showed that any cells contain a number of REGULATORY GENE that act as switch which can turn each other ON and OFF .thos shows genetic network is of “ON-OFF’’ type.

PROBABILY BOOLEAN NETWORK:

• PBNs have recently been introduced as a promising classes of models of genetic regulatory network.

• It was first introduced by KAUFFMAN.

• The dynamic behaviour of PBN can be analysed in the context of MARKOV CHAIN.

• A key gaol is the determination of STEADY STATE{long run} bahaviour og PBN by analyzing corresponding MORKOV CHAIN.

• This allow one to compute the long term influence of a gene on another gene to dertmine the long term joint probalistic behaviour of few selected genes.

• Because matrix based method quickly become prohibitine for large size of networks , we propose that we use of MONTE CARLO method.

• Using a rently introduced method based on the thery of two MORKOV CHAIN , we illustrate the approach on the sub network designed for HUMAN GELIAMA GENOME EXPRESSION and determine the joints steady state probabilities for several groups of genes.

• PBN has become a most powerful tool for describing, analyzing , and stimulating cellular network .

• Hence they have received much attention not only from biology communoity but also from researcher with background in physics system science etc..

• In this approach a logical relationship is expresses as an algerraic equation and logical dynamical system ,such as a BOOLEAN NETWORK.

• Boolean network is converted into discrete time linear system.

• Similary a Boolean network can be converted into discrete time bilinear system, in this way various tools for solving conventional algebraic equation and dealing with difference or differential equation can be used to solve logic based problem. Under this frame work the tropographical of Boolean are reavealed via the structures of their network transition matrix.

• The state space, subspaces etc…are then defined as sets of logical fuctional. This frame work makes the state-space approach to dynamical(control) system applicable to Boolean network. Using this new technology we investigate the properties and control desgn of Boolean network . many basic problem in control theory are studied, such as controllability , observability realization , stabilization , disturbance decoupling and optimal control.

• The fundamrntal tool in this approach in a new matrix product called SEMI-TENSOR PRODUCT(STP). The stp of matrices is a generalization of conventional matrix product to the case where the dimenion- matching condition is not satisfied.

• That is , we extend the matrix product AB to the case where the column number of A and the row number of B are different . this generalization preserves all major properties of the conventional matrix product.

• Using the stp a logical function can be converted into a multi linaer mapping called the matrix expression of logaical relations. Under the construction the dynamic of a Boolean network can be expressed a certain majot features of the topology of a Boolean network such as fixed points cycles , transient time and basins attractors can be easily reavlead via a set of formulas.

• When the control of a Boolean network is considered the bilinear system representation system representation of a Boolean control network makes it possible to apply most techniques developed in modern control theory to the analysis & developed of Boolean control network.

Steady-state analysis:

Most approaches to steady-state analysis use the state transition matrix in some form or another. For the case of PBNs, this would consist of constructing the state-transition matrix A given in Equation (4)in the Supplementary Material, and then applying numerical methods. A variety of approaches using iterative, projection, decompositional and other methods could potentially be used (Stewart, 1994).Unfortunately, however, in the case of PBNs, the size of the state space grows exponentially inthe number of genes and becomes prohibitive for matrix-based numerical analysis of large networks.Even matrix-geometric methods, which have been successfully used for similar problems in non-linear signal processing (Shmulevich et al., 1999), are unsuitable due to thepossiblyirregular structureof the transition matrix.On a more positive note, it should be recognizedthat even larger state spaces are commonly encountered

in Markov chain Monte Carlo (MCMC) methodsfor many applications, including Markov randomfield modelling in image processing (Winkler,1995), where efficient simulation and estimationare routinely performed. Thus, MonteCarlo methods represent a viable alternative to numerical matrix-based methodsfor obtaining steady-state distributions. Informally speaking, this consists of running the Markov chain for a sufficiently longtime, until convergence to the stationary distributionis reached, and observing the proportion oftime the process spends in the parts of the statespace that represent the information of interest,such as the joint stationary distribution of several

specific genes. A key factor is convergence, which to a large extent depends on theperturbation probability,p. In general, a larger p results in quicker convergence, butmaking p too large is not biologically meaningful.In order for us to perform long-term analysisof the Markov chain corresponding to a PBN using Monte Carlo methods, we need to be ableto estimate the convergence rate of the process.Only after we aresufficiently sure that the chain has reached its stationary distribution can we

begin to collect information of interest. Typical approaches for assessing convergence are basedon the second-largest eigenvalue of the transition probability matrix A. Unfortunately, as mentionedabove, for even a moderate number of genes, obtainingthe eigenvalues of the transition matrixmay be impractical. Thus, it is advantageous to be able to determine the number of iterations necessary until satisfactory convergence is reached. One approach for obtaining a priori bounds on the number of iterations is basedon the so-called minorization condition for Markov chains (Rosenthal,

1995). Unfortunately, as we show in the Supplementary Material, the usefulness of such aresult is rather limited. Even for a moderately small number of genes, the number of iterations, k, predicted by the bound is much too large to be useful in practice. We have also found (see Supplementary Material) that making any assumptions about the relative magnitudes of the probabilities of perturbation and transition via the selected Boolean functions is not likely to significantly improve the bound on convergence, and that it is only by having some information about the structure of the PBN itself that the bound can potentially be lowered.Thus, with limited ability to obtain a priori bounds, we now turn to diagnosing convergence to the steady-state distribution.

## Tuesday, February 1, 2011

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