Detecting robust time-delayed regulation in Mycobacterium tuberculosis Iti Chaturvedi and Jagath C Rajapakse INCOB 2009

Gene Regulatory Networks (GRN) GRN represents the regulatory effects (causal effects) among genes involved in a particular pathway. Signal transduction may is transient – dynamic delayed regulations seem to exist. Distributed nature causes intense cross-talk M. tuberculosis is a bacteria causing TB in man and has a very slow growth rate in vitro. The DNA repair pathway is activated when a damage to DNA occurs. System consists of LexA and RecA and upto 40 other genes that are regulated by these two proteins. linB Rv2719c lexA fadd21 recA dnaB fadd23 ruvC dnaE2 agenda

Methods of Building GRN Bayesian networks (BN): graphical models with regulations denoted by conditional probabilities (Heckerman et al 1995) Dynamic BN (DBN): Transition network over time can model cyclic events (Friedman, N. et al 1998) Higher-order (HDBN): Extended transition network for longer delays (Zheng et al 2006) Skip-chain BN : Can model very long delays of arbitary length using feature functions. (Galley, M. 2006) a i are the parents of gene i

Optimization Using GA Crossover involves swapping several rows between two parents. Each individual is an o-nary interaction matrix where for no interaction and for o-order interaction

GA Algorithm Initialize N individuals with 0.7 similarity using Mutual Information Rank all individuals using fitness function. Elite individual E is sent to next generation. Select two parents using roulette strategy for mating. Swap last few rows of the selected parents to generate two new children. Try another crossover point if population similarity is < 0.7 Invert a random cell to cause mutation If d ga < 1 for 20 generations or number of generations greater than Q No Yes E is optimal network Initialize Selection Mating Crossover Mutate Terminate

Skip-chain Model The likelihood of a gene expression x i is given by a weighted sum of linear and skip edge scores : Linear-chain feature functions represent local dependencies of o-order. The skip-chain features represent long range dependencies in a GRN using a HMM agenda

Viterbi Forward Path The skip-edge score is given as a the normalized MAP interaction: We can use maximum likelihood to estimate state transition and emission probabilities : where denotes number of occurrences for The most probable path is given by MAP estimate using dynamic programming :

Priors for Networks Most higher-order Markov models are sensitive to change in pathways and associated data. –Gibbs prior is used to model target network prior. –Interaction potentials, denotes an interaction in target network and no interaction –Here a small and large will reflect prior more and vice versa. We use adaption to reduce over-fitting due to sparse feature specific data. Adaption model can combine the reliable DBN with a volatile feature specific HMM for long delays. where

Dirichlet Prior over Parameters We extend the MLE to a Bayesian learning. Dirichlet is a conjugate prior for multinomial distribution. We can maximize probability as (MAP) : Using the linear feature as a Dirichlet conjugate prior for the skip feature of a gene we get : Lastly, the interpolated probability of gene based on linear and skip-edges is : where

Experiments : M. tuberculosis Here we looked at the response of bacteria to drug-induced stress. Treatment with Mitocyin C caused DNA damage and hence led to the upregulation of associated repair genes. Eight time points are available at NCBI Gene Expression Omnibus (GSE1642-GPL1396 series) 0.33hr, 0.75hr,1.5hr, 2hr, 4hr, 6hr, 8hr and 12hr after DNA damage. Data was discretized into 0 for down and 1 for up regulation. The corresponding skip probabilities were calculated as described in methods. Upto seven time points of delays were allowed. Firstly, we used 9 genes previously specified. In order to get an expanded dataset, the original dataset was subjected to ICA and the components closest to 9 genes were identified. This gave us a second dataset of 32 genes. agenda

Table 1. Predicted by DBN, HDBN, and skip-chain without priors Higher-order edges(hrs) # GenesModel:oML1(0.75)2(1.5)3(2)4(4)5(6) 9 DBN: HDBN: SKIP-CHAIN:1/ (3) 32 DBN: HDBN: SKIP-CHAIN:2/ (41)(4) Time delays It can be seen that the ML of the underlying skip-chain prediction is much higher than the DBN or HDBN, confirming that the network fits data well.

Time delays : Priors Higher-order edges(hrs) # GenesModel:oML 1(0.75)2(1.5)3(2)4(4)5(6) 9 SKIP-CHAIN: (3) SKIP-CHAIN(Gibbs): (2) SKIP-CHAIN(Dirichlet): (11)(1) SKIP-CHAIN(Gibbs and Dirichlet): (4)(5) 32 SKIP-CHAIN: (41)(4) SKIP-CHAIN(Gibbs): (40)(3) SKIP-CHAIN(Dirichlet): (37)(4) SKIP-CHAIN(Gibbs and Dirichlet): (41)(4) Table 2. Predicted by skip-chain models with priors: Gibbs prior for structures and Dirichlet prior for parameters Using priors further increased likelihood and gave many new time- delayed interactions. The combined use of both Dirichlet and Gibbs priors is optimal.

Networks 9 genes Fig 3 : Time-delayed interactions in predicted network of 9 genes (a) DBN network, (b) HDBN network, (c) Skip- chain network, (d) Skip-chain network with Gibbs prior, (e) Skip-chain network with Dirichlet prior, (f) Skip-chain network with Gibbs and Dirichlet prior A small number of transcription factors (TF) regulate the rest of the repair system. At the same time the in-degree is low, as each gene is regulated by just one TF.

Networks 32 genes Fig 4 : Time-delayed interactions in predicted network of 32 genes (a) DBN network, (b) HDBN network, (c) Skip- chain network, (d) Skip-chain network with Gibbs prior, (e) Skip-chain network with Dirichlet prior, (f) Skip-chain network with Gibbs and Dirichlet prior The second dataset of 32 genes indicated that our method is good for identifying core genes. Use of priors gave better networks with fewer hubs.

Conclusion An organism responds to changes in its environment by altering the level of expression of critical genes. Skip-chain models address the difficulties of a HDBN by easily incorporating long time-delayed regulations. Numerous time-delays are identified between the same pair of genes. The forward Viterbi path determines the best long-distant time delay between two genes. Using priors gave us higher likelihood and improved the over-fitting in building the regulatory networks. The work can be extended to skip-chain conditional random fields and fusion with protein interaction networks.

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