Modeling Biological Networks

Transcription

1 Modeling Biological Networks Dr. Carlo Cosentino School of Computer and Biomedical Engineering Department of Experimental and Clinical Medicine Università degli Studi Magna Graecia Catanzaro, Italy Dr. Carlo Cosentino 1

2 Outline Classification of biological networks Modeling metabolic networks Modeling gene regulatory networks Inferring gene regulatory networks Dr. Carlo Cosentino 2

3 Types of Biological Network Several different kinds of biological network can be distinguished at the molecular level Gene regulatory Metabolic Signal transduction Protein protein interaction Moreover other networks can be considered as we move to different description levels, e.g. Immunological Ecological Here we will focus exclusively on molecular processes that take place within the cell Dr. Carlo Cosentino 3

4 Goals A major challenge consists in identifying with reasonable accuracy the complex macromolecular interactions at the gene, metabolite and protein levels Once identified, the network model can be used to simulate the process it represents predict the features of its dynamical behavior extrapolate cellular phenotypes Dr. Carlo Cosentino 4

5 Graphs A very useful formal tool for describing and visualizing biological networks is represented by graphs A graph, or undirected graph, is an ordered pair G=(V,E), where V is the set of the vertices, or nodes, and E is the set of unordered pairs of distinct vertices, called edges or lines For each edge {u,v}, the nodes u and v are said to be adjacent We have a directed graph, or digraph, if E is a set of ordered pairs In digraphs, the in degree, k in, (out degree, k out ) of a node is the number of edges incident to (from) that node Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 5

6 Topological Characteristics The degree distribution, P(k), gives the probability that a selected node has exactly k links It allows us to distinguish between different classes of networks (see next slide) The clustering coefficient of a node I, C I, measures the aggregation of its adjacents (number of triangles passing through node I) C(k) is the average clustering coefficient of all nodes with k links Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 6

7 Erdös Rényi Random Networks The Erdös Rényi model of a random network starts with N nodes and connects each pair of nodes with probability p The degree follows a Poisson distribution, thus many nodes have the same number of links (close to the average degree <k> The tail decreases exponentially, which indicates that nodes with k very different from the average are rare The clustering coefficient is independent of a node s degree Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 7

8 Scale Free Networks Scale free networks are characterized by a power law degree distribution The probability that a node has k links follows P(k)~k -γ, where γ is the degree exponent The probability that a node is highly connected is statistically more significant than in a random graph In the Barabási Albert model, at each time point a node with M links is added to the network, which connects to an already existing node I with probability Π i =k i /Σ j k j The underlying mechanism is that nodes with many links have higher probability of getting more (this is also referred to as preferential attachment) Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 8

9 Hierarchical Networks A hierarchical structure arises in systems that combine modularity and scale free topology The hierarchical model is based on the replication of a small cluster of four nodes (the central ones) The external nodes of the replicas are linked to the central node of the original cluster The resulting network has a power law degree distribution, thus it is scale free The average clustering coefficient scales with the degree following C(k )~k -1 Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 9

10 Graphs of Biological Networks Depending on the kind of biological network, the edges and nodes of the graph have different meaning Metabolic network nodes: metabolic product, edge: a reaction transforming A into B Transcriptional regulation network (protein DNA) nodes: genes and proteins, Protein protein network nodes: proteins, edge: a TF regulates a gene edge: interaction between proteins Gene regulatory networks (functional association network) nodes: genes, edge: expressions of A and B are correlated Dr. Carlo Cosentino 10

11 Topology of Biological Networks An extensive commentary has been published by Albert in 2005, reviewing literature on the topology of different kinds of biological networks Albert, Scale free networks in cell biology, Journal of Cell Science 118(21), , 2005 Experimental evidences are reviewed for metabolic, transcriptional regulatory, signal transduction, functional association networks All of the considered networks approximately exhibit power law degree distribution, at least for the in or for the out degree For instance, transcriptional regulation networks exhibit a scale free out degree distribution, signifying the potential of transcription factors to regulate multiple targets On the other hand, their in degree is a more restricted exponential function, suggesting that combinatorial regulation by several TFs is less frequent Dr. Carlo Cosentino 11

12 P P Interaction Network in Yeast This network is based on yeast two hybrid experiments Few highly connected nodes (hubs) hold the network together The color of a node indicates the phenotypic effect deriving from removing the corresponding protein red: lethal green: non lethal orange: slow growth yellow: unknown Barabasi et al, Nature Review Genetics 101(5), , 2004 Dr. Carlo Cosentino 12

13 Outline Classification of biological networks Modeling metabolic networks Modeling gene regulatory networks Inferring gene regulatory networks Dr. Carlo Cosentino 13

14 Metabolic Reactions Living cells require energy and material for building up membranes storing molecules replenishing enzymes replication and repair of DNA movement Metabolic reactions can be divided in two categories Catabolic reactions: breakdown of complex compounds to get energy and building blocks Anabolic reactions: assembling of the compounds used by the cellular mechanisms Dr. Carlo Cosentino 14

15 Basic Concepts of Metabolism Historically metabolism is the part of cell functioning that has been studied more thoroughly during the last decades This implies that several well assessed mathematical tools exist for describing this kind of networks Enzyme kinetics investigates the dynamic properties of the individual reactions in isolation Stoichiometric analysis deals with the balance of compound production and degradation at the network level Metabolic control analysis describes the effect of perturbations in the network, in terms of changes of metabolites concentrations Most of the tools used in the quantitative study of metabolic networks can also be applied to other types of networks Dr. Carlo Cosentino 15

16 Glycolysis We will exploit the case study of glycolysis in yeast in order to illustrate the theoretical concepts introduced hereafter The pathway shown below is part of the glycolysis process List of Reactions v 1 : hexokinase v 5 : aldolase v 2 : consumption of glucose 6 phosphate v 6 : ATP production in lower glycolysis in other pathways v 7 : ATP consumption in other pathways v 3 : phosphoglucoisomerase v 8 : adenylate kinase v 4 : phosphofructokinase Hynne et al, Full scale model of glycolysis in Saccharomyces cerevisiae (2001) Biophys. Chem. 94, Dr. Carlo Cosentino 16

17 ODE Model of Glycolysis The system of ODEs describing the pathway is Dr. Carlo Cosentino 17

18 ODE Model with Constant Glucose The kinetic rates as functions of reactants can be derived by applying the models presented in the previous lecture Model Parameters Dr. Carlo Cosentino 18

19 Stoichiometric Analysis The basic elements considered in stoichiometric analysis of metabolic networks are The concentrations of the various species The reactions or transport processes affecting such concentrations The stoichiometric coefficients denote the proportion of substrate and product molecules involved in a reaction For instance, if we consider the reaction the stoichiometric coefficients of S 1, S 2, P are 1,1,-2 respectively Dr. Carlo Cosentino 19

20 Stoichiometric Analysis The change of concentrations in time can be described by means of ODEs For the simple reaction above we have This means that the degradation of S 1 with rate v is accompanied by the degradation of S 2 with the same rate and by the production of P with a double rate Dr. Carlo Cosentino 20

21 Stoichiometric Matrix In general, for a system of m substances and r reactions, the system dynamics are described by The number n ij is the stoichiometric coefficient of the i-th metabolite in the j-th reaction For the sake of simplicity, we assume that the changes of concentrations are only due to reactions (i.e. we neglect the effect of convection or diffusion) We can then define the stoichiometric matrix in which columns correspond to reactions and rows to concentration variations Dr. Carlo Cosentino 21

22 Stoichiometric Model The mathematical description of the metabolic network can be given in matrix form as where S=(S 1,,S m ) T is the vector of concentration values v=(v 1,,v r ) T is the vector of reaction rates If the system is at steady state (that is ds i /dt = 0 for i=1,,m) we can also define the vector of steady state fluxes, J=(J 1,,J r ) T Finally, the model involves a certain number of parameters, thus we can define also a parameter vector, p=(p 1,,p η ) T Dr. Carlo Cosentino 22

23 Stoichiometric Model of Glycolysis For the glycolysis model we have Dr. Carlo Cosentino 23

24 Analysis of the Stoichiometric Matrix A relevant information that can be readily derived from the N matrix is which combinations of individual fluxes are possible at steady state The system of algebraic eqs admits a nontrivial solution only if rank(n)<r Every possible set of steady state fluxes can be expressed as a linear combination of the basis of the kernel of N, defined by the matrix K, such that N K=0 Therefore, denoting by k i the i-th column of K, Dr. Carlo Cosentino 24

25 An Example Let us consider the simple network The stoichiometric matrix is N=(1 1 1) and the steady state fluxes are described by the linear combination Dr. Carlo Cosentino 25

26 Null Rates at Steady State For the glycolysis model we have r=8 and rank(n)=5, thus the base of the null space of N is composed of three vectors Note that the entries in the last row are all zero; this means that the net rate for that reaction is null at steady state Hence, at steady state we can neglect the reaction v 8 Dr. Carlo Cosentino 26

27 Unbranched Pathways Another property that can be readily derived is the presence of unbranched pathways In this case, the net rate of all the reactions in the pathway must be equal The entries for the second and third reaction in the matrix K are always equal This implies that the fluxes through reactions 2 and 3 must be equal at steady state Dr. Carlo Cosentino 27

28 Elementary Flux Modes A pathway can be defined as a set of metabolic reactions linked by common metabolites It is not straightforward to recognize pathways in metabolic maps that have been reconstructed from experimental evidences This problem is formalized in the concept of finding the Elementary Flux Modes (EFMs) The aim is to find which are the admissible direct routes for producing a certain metabolite starting from another one In order to have an idea of the usefulness of such mathematical methods, we can have a glimpse at a typical whole organism scale metabolic network Dr. Carlo Cosentino 28

29 Metabolic Network in Yeast Palsson, Systems Biology: Properties of Reconstructed Networks, 2006 Dr. Carlo Cosentino 29

30 Elementary Flux Modes Without going into the mathematical details, we can have a further insight by looking at the elementary flux modes of two simple networks A factor that greatly influences the EFMs is the reversibility of the single reactions Dr. Carlo Cosentino 30

31 Applications of EFM Analysis EFMs can be used to infer the range of metabolic pathways in the network test a set of enzymes for production of a desired compound, and to find the most convenient pathway reconstruct metabolism from annotated genome sequences and analyze the effects of enzyme deficiency reduce drug effects and identify drug targets Dr. Carlo Cosentino 31

32 Flux Balance Analysis Flux Balance Analysis (FBA) deals with the problem of finding the operative modes of metabolic networks subject to three kinds of constraints 1) The operative mode is assumed to be at steady state 2) The operative mode must respect the (ir)reversibility of the reactions 3) The enzyme catalytic activity in each reaction is limited to an admissible range, i.e. α i v i β i Additional constraints may be imposed by biomass composition or other external conditions Dr. Carlo Cosentino 32

33 Flux Balance Constrained Optimization Such constraints confine the steady state fluxes to a feasible set, but usually do not yield a unique solution Hence, the determination of a particular metabolic flux distribution can be cast as a linear optimization problem Maximize an objective function subject to the constraints given above Dr. Carlo Cosentino 33

34 Conservation Relations If a substance is neither added nor removed from the reaction system, its total concentration remains constant This property can be derived by analyzing the null space of N T, defined by the matrix G such that The latter implies GN =0 GṠ = GNv =0 GS =const The dimension of the null space is m-rank(n) Dr. Carlo Cosentino 34

35 Conservation in Glycolysis For the glycolysis example we have which means the sum of concentrations of AMP, ADP, ATP remains constant The conservation relations can be used to simplified the dynamical model, by exploiting the algebraic equations that express the conservation constraints to express some variables as functions of the others Dr. Carlo Cosentino 35

36 Metabolic Control Analysis Metabolic Control Analysis (MCA) deals with the sensitivity of the steady state properties of the network to small parameter changes It can be also applied to models of other kinds of network, like signaling pathways or gene expression Issues addressed by MCA Predict properties of the network from knowledge of individual components Find which specific step has the greatest influence on a flux or steady state concentration or reaction rate Find which is the best target reaction to treat a metabolic disorder These questions are very relevant in biotechnological production processes and health care Dr. Carlo Cosentino 36

37 Basic Concepts of MCA The relations between steady state properties and model parameters are usually highly nonlinear There is no general theory predicting the effect of large parameter changes The MCA approach deals with small parameter changes Under this assumption, the model can be approximated, in the neighborhood of the steady state, with a linear one Given the linearized model it is possible to derive some indexes describing the properties above mentioned, e.g. elasticity coefficients, control coefficients, response coefficients Dr. Carlo Cosentino 37

38 Outline Classification of biological networks Modeling metabolic networks Modeling gene regulatory networks Inferring gene regulatory networks Dr. Carlo Cosentino 38

39 Gene Regulatory Networks A protein synthesized from a gene can serve as a transcription factor for another gene, as an enzyme catalyzing a metabolic reaction, or as a component of a signal transduction pathway Apart from DNA transcription regulation, gene expression may be controlled during RNA processing and transport, RNA translation, and the post translational modification of proteins Therefore, gene regulatory networks (GRNs) involve interactions between DNA, RNA, proteins and other molecules A suitable way to dominate this complexity may consist of using functional association networks In this networks the edges of the corresponding graph do not represent chemical interactions, but functional influences of one gene on the other Dr. Carlo Cosentino 39

40 Example of a GRN A toy regulatory network of three genes is depicted in the cartoon below De Jong, Modeling and regulation of genetic regulatory systems, INRIA - RR4032, 2000 Dr. Carlo Cosentino 40

41 Modeling GRNs In what follows we will present an overview of the models used to describe GRNs Two main issues have to be taken into account when choosing a modeling framework Computational requirements for simulation Available methods for inferring the network topology Dr. Carlo Cosentino 41

42 Bayesian Networks In the formalism of Bayesian Networks, the structure of a genetic regulatory system is modeled by a directed acyclic graph G= V,E The vertices i V, i=1,,n, represent genes expression levels and correspond to random variables X i. For each X i, a conditional distribution p(x i parents(x i )) is defined, where parents(x i ) denotes the direct regulators of i The graph G and the set of conditional distributions uniquely specify a joint probability distribution p(x) Dr. Carlo Cosentino 42

43 Independency in BN If X i is independent of Y given Z, where Y and Z are set of variables, we can state a conditional independency For every node i in G, i (X i ; Y Z) i (X i ; non descendant(x i ) parents(x i )) Hence, the joint probability distribution can be decomposed into p(x) = ny i=1 p(x i parents(x i )) Dr. Carlo Cosentino 43

44 Example of BN Here we illustrate the formulation of the BN model for a simple network Two graphs are said to be equivalent if the imply the same set of independencies; they cannot be distinguished by observation on X De Jong, Modeling and regulation of genetic regulatory systems, INRIA - RR4032, 2000 Dr. Carlo Cosentino 44

45 Features of BNs There is no need to specify a single value for each parameter of the model, but rather a distribution over the admissible range of values is assigned This characteristic helps in avoiding overfitting, which is common in the presence of a small data set and a large number of parameters It is a statistical modeling approach, which nicely fits the stochastic nature of biological systems BNs are static models, although it is possible to take into account dynamical aspects through an extension of this theory, namely dynamical bayesian networks (DBNs) Dr. Carlo Cosentino 45

46 Boolean Networks In the framework of Boolean Networks, the expression level of a gene can attain only two values, that is active (on, 1) or inactive (off, 0) Accordingly, the interactions between elements of the network are represented by Boolean functions Smolen, Baxter, Byrne, Mathematical model of gene networks, Neuron 26, , 2000 Dr. Carlo Cosentino 46

47 Features of Boolean Networks Deterministic description Very easy to build the model and to simulate it, even for very large networks They provide only a coarse grained description of the network behavior, thus not useful for a more detailed analysis of the regulatory mechanisms Dr. Carlo Cosentino 47

48 ODE Models We have seen that the mechanistic ODE approach has been widely exploited since the beginning of the last century for modeling biochemical reactions When the order of the system increases, classical nonlinear ODE models become hardly tractable, in terms of parametric analysis, numerical simulation and especially for identification purposes In order to overcome this limitations, alternative modeling approaches have been devised for application to biological networks Dr. Carlo Cosentino 48

49 Power Law Models The basic concept underlying power law models is the approximation of classical ODE models by means of a uniform mathematical structure Dr. Carlo Cosentino 49

50 S Systems S systems are a particular class of power law models in which fluxes are aggregated dx i dt () t i n = α X j= 1 j n gi, j () t β X () t i j= 1 j h i, j, Dr. Carlo Cosentino 50

51 Features of S - Systems S systems feature low computational requirements Their structural homogeneity allows to easily identify the model parameters from steady state data by means of logarithmic linearization Generalized aggregation may introduce a loss of accuracy Violation of biochemical fluxes concentration It may conceal important structural features of the network Dr. Carlo Cosentino 51

52 Piecewise Linear Models Another class of approximate models based on ODEs is that of piecewiselinear (PWL) models The basic idea is to approximate sigmoidal curves through step functions The model takes the general form where and the functions b il ( ) are boolean valued regulation functions expressed in terms of step functions Casey, De Jong, Gouzé, J. Math. Biol. 52, 27 56, 2006 Dr. Carlo Cosentino 52

53 Features of PWL Models Numerical simulation studies have shown that PWL models properly approximate the behavior of the corresponding original nonlinear ones A drawback of this class of systems is that their behavior is very difficult to analyze from a rigorous point of view PWL models, indeed, can exhibit singular steady states, that is equilibrium points lying on the threshold surfaces Moreover it is known that the stability of switching systems cannot be reduced to the analysis of the stability of the linear systems in each sub-space Dr. Carlo Cosentino 53

54 Outline Classification of biological networks Modeling metabolic networks Modeling gene regulatory networks Inferring gene regulatory networks Dr. Carlo Cosentino 54

55 Inferring Bayesian Networks In order to reverse engineering a Bayesian network model of a gene network, we must find the directed acyclic graph that best describes the data To do this, a scoring function is chosen, in order to evaluate the candidate graphs G with respect to the data set D The score can be defined using Bayes rule P (G D) = P (D G)P (G) P (D) If the topology of the network is partially known, the a priori knowledge can be included in P(G) The most popular scores are the Bayesian Information Criterion (BIC) or Bayesian Dirichlet equivalence (BDe) They incorporate a penalty for complexity to cope with overfitting Dr. Carlo Cosentino 55

56 Inferring Bayesian Networks The evaluation of all possible networks involves checking all possible combinations of interactions among the nodes This problem is NP-hard, therefore heuristic methods are used, like the greedy hill climbing approach, the Markov Chain Monte Carlo method, or Simulated Annealing A software tool for inferring both BNs and DBNs is Banjo, developed by the group of Hartemink ( Yu et al, Advances to bayesian network inference for generating causal networks from observational biological data, Bioinformatics 20: , 2004 Dr. Carlo Cosentino 56

57 Information Theoretic Approaches Information theoretic approaches use a generalization of the Pearson correlation coefficient used in hierarchical clustering, namely the Mutual Information (MI), which is computed as MI(X; Y )=H(X)+H(Y ) H(X, Y ) where the marginal and joint entropy are defined, respectively, as H(X) = X H(X, Y )= x X X x X,y Y p(x)logp(x) p(x, y)logp(x, y) Dr. Carlo Cosentino 57

58 Information Theoretic Approaches From the definitions above it follows that MI becomes zero if the two variables are statistically independent A high value of MI indicates that the variables are non randomly associated to each other MI ij =MI ji therefore the resulting reconstructed graph is undirected An important characteristic is that, since the approach is based on the independence of samples, it is not suitable for application to time series (it can applied only to steady state data sets) A software tool based on Mutual Information theory is ARACNE, described in Basso et al, Reverse engineering of regulatory networks in human B cells, Nature Genetics 37(4): , 2005 Dr. Carlo Cosentino 58

59 Inference of ODE Models The identification of the structure and parameters of mechanistic nonlinear ODE models is a very demanding task for non trivial networks, both from a theoretical point of view and in terms of computational requirements A feasible approach is based on the use of linearized dynamical models, which yield good results when applied to data sets obtained through perturbation experiments Several methods have been developed from the groups of Gardner and di Bernardo, dealing both with steady state (NIR, MNI) and time series data (TSNI) Dr. Carlo Cosentino 59

60 Time Series Network Identification The TSNI algorithm is based on the linearized model i =1,...,N k =1,...,M The data set consists of the expression level of N genes, sampled at M time points with a fixed sampling interval The experimental data are derived from perturbation experiments (e.g. by treatment with a compound or gene overexpression/downregulation) A linear regression algorithm is used to estimate the coefficients of the dynamical matrix, a ij, and those of the input matrix, b i A non-zero coefficient a ij indicates an edge in the (directed) graph, between nodes i and j, whereas a nonzero b ij indicates that the node i is directly affected by the perturbation Bansal, Della Gatta, di Bernardo, Bioinformatics 22: Dr. Carlo Cosentino 60

61 Features of TSNI For small networks (tens of genes), TSNI is able to correctly infer the network structure Besides topological inference, ODE-based methods are also well suited for uncovering unknown targets of perturbations, even in complex networks It is not possible to exploit prior knowledge about the network topology, because this would require the exact knowledge of non physical parameters Dr. Carlo Cosentino 61

62 LMI-based Inference Approach The basic idea is improving linear ODE based methods by exploiting available prior knowledge about the network topology (as in BNs) The identification of the parameters a ij, b ij, is cast as a convex optimization problem, in the form of linear matrix inequalities (LMIs) This formulation allows to reduce the admissible solution space by assigning sign constraints to the coefficients corresponding to known interactions x 1 x 2 x 3 x 1??? x 4 x 2 > <? x 3 >?? x 4??? x 1 x 2 x 3 x 4 activation inhibition Cosentino et al, IET Systems Biology 1(3): , 2007 Dr. Carlo Cosentino 62

63 Features of the LMI-based Approach Numerical tests show that exploitation of prior knowledge greatly improves the reconstruction performances The method can exploit qualitative a priori knowledge, as well as quantitative information Such knowledge is exploited within the reconstruction, not for a posteriori evaluation The optimization problem is convex, therefore the optimal solution, in terms of data-interpolation, can be always found The latter feature, on other hand, implies a higher tendency to overfitting Hard to apply to large scale networks (more than 100 nodes), due to the computational load deriving from the high number of constraints Dr. Carlo Cosentino 63

64 Choice of the Inference Algorithms In a recent study, Bansal et al have compared the performance obtained using different modeling formalisms (BNs, MI, hierarchical clustering, ODE-based models) Bansal et al, How to infer gene networks from expression profiles, Molecular Systems Biology 3:78, 2007 Dr. Carlo Cosentino 64

65 Results on Experimental Data Sets Bansal et al, How to infer gene networks from expression profiles, Molecular Systems Biology 3:78, 2007 Dr. Carlo Cosentino 65

66 Results Discussion The different techniques considered in the review infer networks that overlap for only 10% in the best case Furthermore, the edges predicted by more than one method are not more accurate than those inferred by a single one On the other hand, taking the union of the interactions found by all the methods would yield an even larger number of false positives Local perturbation experiments (i.e. affecting one or few genes) seems to yield better results than global ones (perturbations on a high number of genes) Dr. Carlo Cosentino 66

67 Remarks on Inference Algorithms A relevant issue, that is common to all inference algorithm, is that the problem is very often over determined All modeling formalisms, indeed, involve a large number of parameters, whereas the number of samples is usually limited (curse of dimensionality) Possible solutions Devise methods to exploit different data sets Reduce the dimensionality of the problem, via data pre processing, e.g. clustering algorithm elimination of statistically non expressed nodes Dr. Carlo Cosentino 67

68 Concluding Remarks Regardless to the adopted formalism, good inference performances can be achieved only by exploiting the available prior knowledge from biological literature Despite the great concern about the topological characterization of biological networks, much has still to be done in terms of exploitation of such features in the inference process Several other approaches exist, both for modeling and inferring biological networks (discrete events, formal languages, machine learning methods, etc.) Dr. Carlo Cosentino 68

69 References Klipp et al, Systems Biology in Practice, Wiley-VCH, 2005 Palsson, Systems Biology: Properties of Reconstructed Networks, Cambridge University Press, 2006 Barabasi, Oltvai, Network Biology: Understanding the Cell s Functional Organization, Nature Review Genetics 101(5), , 2004 Hynne et al, Full scale model of glycolysis in Saccharomyces cerevisiae (2001) Biophys. Chem. 94, De Jong, Modeling and regulation of genetic regulatory systems, INRIA - RR4032, 2000 Smolen, Baxter, Byrne, Mathematical model of gene networks, Neuron 26, , 2000 Casey et al, Piecewise linear Models of Genetic Regulatory Networks, Equilibria and their Stability, J. Math. Biol. 52, 27 56, 2006 Bansal et al, Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, Bioinformatics 22: Bansal et al, How to infer gene networks from expression profiles, Molecular Systems Biology 3:78, 2007 Cosentino et al, Linear Matrix Inequalities Approach to Reconstruction of Biological Networks, IET Systems Biology 1(3): , 2007 Dr. Carlo Cosentino 69