Lecture11 Introduction to signaling pathways Glycolytic Oscillation Reverse Engineering of biological networks.

Lecture11

Introduction to signaling pathwaysGlycolytic OscillationReverse Engineering of biological networks

Signaling networks involves the transduction of “signal” usually from outside to the inside of the cellOn molecular level signaling involves the same type of processes as metabolism such as production and degradation of substances, molecular modifications (mainly phosphorylation but also methylation and acetylation) and activation or inhibition of reactions.But signaling pathways serve for information processing or transfer of information while metabolism provide mainly mass transfer

Introduction to signaling pathways

Introduction to signaling pathways

Signal transduction often involves:

•The binding of a ligand to an extracellular receptor

•The subsequent phosphorylation of an intra cellular enzyme

•Amplification and transfer of the signal

•The resultant change in the cellular function e.g. increase /decrease in the expression of a gene

Signaling paradiam

Usually a signaling network has three principal parts:Events around the membraneReactions that link sub-membrane events to the nucleusEvents that leads to transcription

Source: Systems biology in practice by E. klipp et. al.

Schematic representation of receptor activation

Source: Systems biology in practice by E. klipp et. al.

Steroids Not always a receptor exists at the membrane for example the steroid receptors.Sterol lipids include hormones such as cortisol, estrogen, testosteron and calcitriol.These steroids simply cross the membrane of the target cell and then bound the intracellular receptor which results in the release of the inhibitory molecule from the receptor.The receptor then traverses the nuclear membrane and binds to its site on the DNA to trigger the transcription of the target gene.

Source: Systems biology by Bernhard O. Palsson

G-protein signalingG-protein coupled receptor (GPCR) represents important components of signal transduction networkThis class of receptor comprises 5% of the genes in C. elegansThe G-protein complex consists of three subunits (α, β and λ) and in its inactive state bound to guanosine diphosphate(GDP) When a ligand binds to the GPCR, the G-protein exchanges its GDP for a guanosine trihosphate(GTP) This exchange leads to the dissociation of the G-protein from the receptor and its split into a βλ complex and a GTP-bound α subunit which is its active state initiating other downstream processes

Source: Systems biology by Bernhard O. Palsson

G-protein signaling model

Source: Systems biology in practice by E. klipp et. al.

G-protein signaling model

Time course of G protein activation. The total number of molecules is 10000. The concentration of GDP-bound Gα is low for the whole period due to its fast complex formation with the heterodimer Gβλ

Source: Systems biology in practice by E. klipp et. al.

The JAK-STAT network

A cell surface receptor often dimerizes upon binding to a cytokineThe monomeric form of the receptor is associated with a kinase called JAKWhen the receptor dimerizes the JAKs induce phosphorylation of themselves and the receptor which is the active state of the receptor.The active complex phosphorylates the STAT(signal transducer and activator of transcription) molecules STAT molecules then dimerizes, go to nucleus and trigger transcription

The JAK-STAT signaling system is an important two-step process that is involved in multiple cellular functions including cell growth and inflammatory response

Source: Systems biology in practice by E. klipp et. al.

Schematic representation of the MAP kinase cascade. An upstream signal causes phosphorylation of the MAPKKK. The phosphorylation of the MAPKKK in turn phosphorylates the protein at the next level. Dephosphorylation is assumed to occur continuously by phosphatases or autodephosphorylation

Source: Systems biology in practice by E. klipp et. al.

Signaling pathways in Baker’s yeast

HOG pathway activated by osmotic shock, pheromone pathway activated by pheromones from cells of opposite mating type and pseudohyphal growth pathway stimulated by starvation conditionA MAP kinase cascade is a particular part of many signalling pathways . In this figure its components are indicated by bold border Source: Systems biology in practice by E. klipp et. al.

Glycolytic Oscillation

In living organism we see many periodic changes or oscillations:Pulse of the heartRespirationOvulation in mammalsAnnual flowering of the plantsSleeping at nightLifecycle of cells

Actually after starting or being affected by some perturbationmany systems go through oscillations before becoming stable or unstable (collapsing)

Glycolytic Oscillation

Phosphofructokinase-1 (PFK-1) catalyzes the important step of glycolysis, the conversion of fructose 6-phosphate and ATP to fructose 1,6-bisphosphate and ADP.

The ADP then exert a positive feedback on PFK-1

This system can be represented as follows:

For a large range of parameter values the system moves to a stable steady state but beyond a critical parameter value the system becomes unstable.

Glycolytic Oscillation

The Temporal behavior of the concentration of substrate and product can be described as follows:

Here the supply rate of S is v0 and k1 and k2 are mass action rate constants.

The function R(p) represents the autocatalytic effect of the product P on its own production

Glycolytic OscillationGlycolytic Oscillation

Assuming r(P) = P2, The Temporal behavior of the concentration of substrate and product can be described as follows:

Glycolytic Oscillation

The dynamic behavior of the above system with a particular set of parameter values is represented as follows:

The above solution corresponds to v0 = 1, and k1 =1, and k2 = 1.00001 S(0)=2, P(0)=1 in arbitrary unit

By using dS/dt =0 and dP/dt = 0, The steady state solutions for the above system can be determined as follows:

However the steady state is achievable or not depends on the parameter values

Glycolytic Oscillation

The stability of the steady state can be analyzed by inspection of the Jacobian matrix J.

The character of the steady state is determined by the value and signs (positive, negative , zero etc.) of trace and the determinant of the Jacobian matrix

Glycolytic Oscillation

Glycolytic Oscillation

The stability analysis of the above system for fixed value of k1=1 and variable values of V0 and k2

Glycolytic Oscillation

The stability analysis of the above system for fixed value of k1=1 and variable value of vo and k2

With v0 = 1, k1 =1, and k2 = 1.00001 the system is in the stable focus region which means it gradually goes to steady state through decaying oscillation

Reverse Engineering of biological networks

The task of reverse engineering of a genetic network is the reconstruction of the interactions among biological entities ( genes, proteins, metabolites etc.) in a qualitative way from experimental data using algorithm that weight the nature of the possible interactions with numerical values.

In forward modeling network is constructed with known interactions and subsequently its topological and other properties are analyzed

In reverse engineering the network is estimated from experimental data and then it is used for other predictions

Reverse Engineering of gene regulatory networkBy clustering the gene expression data, we can determine co-expressed genes.

Co-expressed genes might have similar regulatory characteristics but it is not possible to get the information about the nature of the regulation.

Here we discuss a reverse engineering method of estimating regulatory relation between genes based on gene expression data from the following paper:

Reverse engineering gene networks using singular value decomposition and robust regressionM. K. Stephen Yeung, Jesper Tegne´ r†, and James J. Collins‡Proc. Natl. Acad. Sci. USA 99:6163-6168

It is assumed that the dynamics i.e. the rate of change of a gene-product’s abundance is a function of the abundance of all other genes in the network.

For all N genes the system of equations are as follows:

In Vector notation

Where f(X) is a vector valued function

Reverse Engineering of gene regulatory network

Under linear assumption i.e. has linear relation with Xi s we can write

Here Aij is the coupling parameter that represents the influence of Xj on the expression rate of Xi . In other words Aij represents a network showing the regulatory relation among the genes.

Target of reverse engineering is to determine A. Solving A requires a large number of measurements of and X

Reverse Engineering of gene regulatory network

Measurement of is difficult and hence can be estimated in several ways.

First, if time series data can be obtained then can be approximated by using the profiles of the expression values for fixed time intervals

Alternatively a cellular system at steady state can be perturbed by external stimulation and then can be determined by comparing the gene expression in the perturbed cellular population and the unperturbed reference population.

Reverse Engineering of gene regulatory network

Now using any method if we can produce matrices and then we can write

Or, (if external perturbation is used)

Here BNxM is the matrix representing the effect of perturbation

The goal of reverse engineering is to use the measured data B, X, and to deduce A i.e. the connectivity matrix of the regulatory relation among the genes.

Reverse Engineering of gene regulatory network

By taking transpose the system can be rewritten as

A is the unknown. If M =N and X is full-ranked, we can simply invert the matrix X to find A. However, typically M<<N mainly because of the high cost of perturbations and measurements. We therefore have an underdetermined problem. Underdetermined problem means the number of linearly independent equations is less than the number of unknown variables. Therefore there is no unique solution One way to get around this is to use SVD to decompose XT into

Reverse Engineering of gene regulatory network

where U and V are each orthogonal which means:

with I being the identity matrix, and W is diagonal:

Without loss of generality, we may assume that all nonzero elements of wk are listed at the end, i.e., w1, w2, . . . , wL =0 and wL+1, wL+2,. . . , wN≠0, where L :=dim(ker(XT)). Then one particular solution for A is:

Reverse Engineering of gene regulatory network

the general solution is given by the affine space

with C = (cij)N×N, where cij is zero if j >L and is otherwise an arbitrary scalar coefficient. This family of solutions in Eq. 3represents all the possible networks that are consistent with the microarray data. Among these solutions, the particular solution A0 is the one with the smallest L2 norm. Now, the question is which one of the solutions of equation 3 is the best.

Reverse Engineering of gene regulatory network

In such cases, we may rely on insights provided by earlier works on gene regulatory networks and bioinformatics databases, which suggest that naturally occurring gene networks are sparse, i.e., generally each gene interacts with only a small percentage of all the genes in the entire genome. Imposing sparseness on the family of solutions given by Eq. 3 means that we need to choose the coefficients cij to maximize the number of zero entries in A. This is a nontrivial problem.

Reverse Engineering of gene regulatory network

The task is equivalent to the problem of finding the exact-fit plane in robust statistics, where we try to fit a hyperplane to a set of points containing a few outliers. Here they have chosen L1 regression where the figure of merit is the minimization of the sum of the absolute values of the errors, for its efficiency. In short, this method of reverse engineering can produce multiple solutions (gene networks) that are consistent with a given microarray data. This paper says among them the sparsest one is the best solution and used L1 regression to detect the best solution.

Reverse Engineering of gene regulatory network