Summary

1. promoter gene p2 B1 a12, d12 determine transcription rate p3 a23, d23 Fk B2 pk a24, d24 p4 Expression = Fk(Q1, Q2, Q3),where Qi is state of binding site Bi attach/detach influence p1 B3 a31, d31 Legend: pj Bi binding site i protein j association and dissociation constants for binding site i and protein j controller of gene k expression Fk aij, dij concentration conc(p1)  ai1 Bi bound by p1 conc(p1)  di1 … Bi unbound conc(pk)  aik time Bi bound by pk conc(pk)  dik Finite State Linear Model and Reverse Engineering of Gene Regulatory Networks Dace Rukliša Institute of Mathematics and Computer Science University of Latvia Dace.Ruklisa@mii.lu.lv Juris Vīksna Institute of Mathematics and Computer Science University of Latvia jviksna@mii.lu.lv Alvis Brāzma EMBL, EBI brazma@ebi.ac.uk Summary Various mathematical models describing gene regulatory networks as well as algorithms for network reconstruction from experimental data have been a subject of intense studies, largely motivated by the current availability of high-throughput experimental data. Boolean network models currently are among the best studied. Also models based on differential equations are sufficiently common, especially in simulation of biological networks. From the viewpoint of biology even more interesting model class is models that unite both discrete and continuous components. However, it is difficult to analyze these models, mainly due to large amount of parameters used to describe a network. We consider finite state linear model (FSLM) proposed in [1, 2]. The model incorporates biologically intuitive gene regulatory mechanism similar to that in Boolean networks, and describes also the continuous changes in regulatory protein concentrations. Using FSLM a biologically adequate network for phage was constructed [1]. Although the model incorporates subtle biological details and is more complex than many other models, we were able to design an efficient network reconstruction algorithm that requires the number of measurements similar to what is required for Boolean networks [3]. We have extended this algorithm for reconstruction of networks, where several transcription factors compete for occupation of one binding site and gene expression has more than two distinct levels. Reconstruction of gene regulatory networks A finite set of time points T={t0,..,tm−1} is representative for a network N, if T contains all time pointst0ttm-1 at which activity of some gene changes. Network reconstruction problem.Time-series measurements M(T) of protein concentrations at representative time points are given. Find a network that produces these concentrations and network’s initial state. Network reconstruction algorithm: foreach gene idofor k=1to maximal regulator countdoforeach candidate setof k regulatorsdoComputeGrowthRatesfor protein iComputePotentialThresholdsfrom measurementsConstructRegulatoryFunctionif for some set of threshold pairs exists function consistent with M(T)thenadd gene i together with growth rates, binding sites, thresholds, regulators and regulatory functionto network Nadd the part of initial state s to initial state S go to next genereturnnetwork N and its initial state Main objects and processes in model • Algorithms and complexity • O(2KnK+1M2Km log m) • (simpler case, where each binding site is either unbound or bound by a specific protein, each gene is either expressed or repressed and regulatory function Fk is a Boolean function), • where K– maximal number of regulators; m – measurement count;M – the number of potential thresholds • O(pBnK+1BKM2B  pmp log (mp)) • (more general case, where several transcription factors compete for occupation of one binding site and gene expression can have more than two different levels), • where p– maximal number of proteins that can bind to one site;B– maximal number of binding sites for one gene • Necessary measurement count similar to Boolean networks. • The algorithm can reconstruct networks with up to 100 genes, when number of regulators  5. • Model properties • For a given network with given initial state and for a given gene i it is algorithmically undecidable whether the gene will ever become active • Network equivalence and behavioural equivalence are algorithmically undecidable • It is possible to define networks with non-periodic behaviour • It is impossible to tell whether the reconstructed network is equivalent to the original network that produced all measurements • Finite State Linear Model: assumptions • Gene activity can assume a finite number of levels, depending on a state of promotor region • The concentration of protein produced by the gene is changing with rate dependent on gene transcription level • Binding site can be in a finite number of states (free or bound by some regulatory protein) • Concentrations of regulatory proteins are continuous and change linearly in time • Binding of a protein pj to a binding site bi depends on the association constant aij and dissociation constant dij (0 < dij < aij): • Phage : lysogeny References [1] Brazma, A., Schlitt, T. 2003. Reverse engineering of gene regulatory networks: a finite state linear model. Genome Biology 4:P5:1-31. [2] Ruklisa, D., Brazma, A., Viksna, J. 2005. Reconstruction of gene regulatory networks under the finite state linear model. Genome Informatics 16:2. [3] Schlitt, T., Brazma, A. 2006. Modelling in molecular biology: describing transcription regulatory networks at different scales. Phil. Trans. R. Soc. B (in press).