modeling biological networksbrunos/lecture3.pdfmodeling biological networks dr. carlo cosentino ......

Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20081

Modeling Biological Networks

Dr. Carlo CosentinoSchool of Computer and Biomedical Engineering

Department of Experimental and Clinical MedicineUniversità degli Studi Magna Graecia

Catanzaro, [email protected]

http://bioingegneria.unicz.it/~cosentino


Outline

Classification of biological networks

Modeling metabolic networks

Modeling gene regulatory networks

Inferring gene regulatory networks


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


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


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, kin, (out–degree, kout) of a node is the number of edges incident to (from) that node

Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004


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, CI, 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



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



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 Πi=ki/Σjkj

The underlying mechanism is that nodes with many links have higher probability of getting more (this is also referred to as preferential attachment)



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



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, edge: a TF regulates a gene

Protein – protein network

nodes: proteins, edge: interaction between proteins

Gene regulatory networks (functional association network)

nodes: genes, edge: expressions of A and B are correlated


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

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

Albert, Scale–free networks in cell biology, Journal of Cell Science 118(21), 4947–4957, 2005


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


Outline






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


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


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

Hynne et al, Full–scale model of glycolysis in Saccharomyces cerevisiae (2001) Biophys. Chem. 94, 121–163

v1: hexokinasev2: consumption of glucose–6–phosphate in other pathwaysv3: phosphoglucoisomerasev4: phosphofructokinase

v5: aldolasev6: ATP production in lower glycolysisv7: ATP consumption in other pathwaysv8: adenylate kinase

List of Reactions


ODE Model of Glycolysis

The system of ODEs describing the pathway is


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


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 S1, S2, P are 1,1,-2 respectively


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 S1 with rate v is accompanied by the degradation of S2 with the same rate and by the production of P with a double rate


Stoichiometric Matrix

In general, for a system of m substances and r reactions, the system dynamics are described by

The number nij 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


Stoichiometric Model

The mathematical description of the metabolic network can be given in matrix form as

where

S=(S1,…,Sm)T is the vector of concentration values

v=(v1,…,vr)T is the vector of reaction rates

If the system is at steady–state (that is dSi /dt = 0 for i=1,…,m) we can also define the vector of steady–state fluxes, J=(J1,…,Jr)T

Finally, the model involves a certain number of parameters, thus we can define also a parameter vector, p=(p1,…,pη )T


Stoichiometric Model of Glycolysis

For the glycolysis model we have


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 ki the i-th column of K,


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


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 v8


Unbranched Pathways

Another property that can be readily derived is the presence of unbranchedpathways

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


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


Metabolic Network in Yeast

Palsson, Systems Biology: Properties of Reconstructed Networks, 2006


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


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


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. αi ≤ vi ≤ βi

Additional constraints may be imposed by biomass composition or other external conditions


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


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 NT, defined by the matrix G such that

The latter implies

The dimension of the null space is m-rank(N)

GS = GNv = 0 GS = const

GN = 0


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


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 productionprocesses and health care


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


Outline






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


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


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


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 Xi.

For each Xi, a conditional distribution p(Xi |parents(Xi)) is defined, where parents(Xi) denotes the direct regulators of i

The graph G and the set of conditional distributions uniquely specify a joint probability distribution p(X)


Independency in BN

If Xi 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,

Hence, the joint probability distribution can be decomposed into

i (Xi;Y|Z)

i (Xi; non− descendant(Xi)|parents(Xi))

p(X) =nYi=1

p(Xi|parents(Xi))


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


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 bayesiannetworks (DBNs)


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, 567 – 580, 2000


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


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


Power–Law Models

The basic concept underlying power–law models is the approximation of classical ODE models by means of a uniform mathematical structure


S–Systems

S–systems are a particular class of power–law models in which fluxes are aggregated

( ) ( ) ( )∏−∏===

n

j

hji

n

j

gji

i jiji tXtXdt

tdX11

,,, βα


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


Piecewise–Linear Models

Another class of approximate models based on ODEs is that of piecewise-linear (PWL) models

The basic idea is to approximate sigmoidal curves through step functions

The model takes the general form

where

and the functions bil(·) are boolean valued regulation functions expressed in terms of step functions

Casey, De Jong, Gouzé, J. Math. Biol. 52, 27–56, 2006


Features of PWL Models

Numerical simulation studies have shown that PWL models properlyapproximate 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


Outline






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

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

P (G|D) = P (D|G)P (G)P (D)


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(http://www.cs.duke.edu/~amink/software/banjo)

Yu et al, Advances to bayesian network inference for generating causal networks from observational biological data, Bioinformatics 20: 3594-3603, 2004


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

where the marginal and joint entropy are defined, respectively, as

H(X) = − Xx∈X

p(x)logp(x)

H(X,Y ) = − Xx∈X ,y∈Y

p(x, y)logp(x, y)

MI(X;Y ) = H(X) +H(Y )−H(X,Y )


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

MIij=MIji 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): 382-90, 2005


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 diBernardo, dealing both with steady–state (NIR, MNI) and time–series data (TSNI)


Time–Series Network Identification

The TSNI algorithm is based on the linearized model

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, aij, and those of the input matrix, bi

A non-zero coefficient aij indicates an edge in the (directed) graph, between nodes i and j, whereas a nonzero bij indicates that the node i is directly affected by the perturbation

i = 1, . . . , N

k = 1, . . . ,M

Bansal, Della Gatta, di Bernardo, Bioinformatics 22: 815–822


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


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 aij, bij, 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

x1 x2

x3 x4???x4

??>x3

?<>x2

???x1

x4x3x2x1

Cosentino et al, IET Systems Biology 1(3): 164–173, 2007

activation

inhibition


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 posteriorievaluation

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


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


Results on Experimental Data Sets

Bansal et al, How to infer gene networks from expression profiles, Molecular Systems Biology 3:78, 2007


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)


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


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.)


References

Klipp et al, Systems Biology in Practice, Wiley-VCH, 2005Palsson, Systems Biology: Properties of Reconstructed Networks, Cambridge University Press, 2006Barabasi, Oltvai, Network Biology: Understanding the Cell’s Functional Organization, Nature Review Genetics 101(5), 101–114 , 2004Hynne et al, Full–scale model of glycolysis in Saccharomyces cerevisiae (2001) Biophys. Chem. 94, 121–163De Jong, Modeling and regulation of genetic regulatory systems, INRIA - RR4032, 2000Smolen, Baxter, Byrne, Mathematical model of gene networks, Neuron 26, 567 – 580, 2000Casey et al, Piecewise linear Models of Genetic Regulatory Networks, Equilibria and their Stability, J. Math. Biol. 52, 27–56, 2006Bansal et al, Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, Bioinformatics 22: 815–822Bansal et al, How to infer gene networks from expression profiles, Molecular Systems Biology 3:78, 2007Cosentino et al, Linear Matrix Inequalities Approach to Reconstruction of Biological Networks, IET Systems Biology 1(3): 164–173, 2007

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