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Mathematical Modelling of Biological Networks Prof:Rui Alves Dept Ciencies Mediques Basiques, 1st Floor, Room 1.08 Website of the Course: Course:

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalisms Creating and studying a mathematical model

Predicting protein networks using protein interaction data Database of protein interactions Server/ Program Your Sequence (A) A BC D E F Continue until you are satisfied You do not need to go beyond this type of representation in Task 5!!!

What does this means?

Clear network representation is fundamental for clarity of analysis AB What does this mean? Possibilities: A B Function B A A B A B B A

Having precise network representations is important We need to know exactly what is being represented, not just that A and B sort of interact in some way. This means that it is important to develop or use network representations that are accurate and in which a given element has a very specific meaning. Accurate computer representations and human readable representations are not necessarily the same.

Computer readable representations SBML (1999) CELLML (1999) BIOPAX (2002) Etc. There must be representations that are easier for humans to use. Let us take a look at one that chemist have been using for a century.

Defining network conventions A B C Full arrow represents a flux of material between A and B Dashed arrow represents modulation of a flux + Dashed arrow with a plus sign represents positive modulation of a flux - Dashed arrow with a minus sign represents negative modulation of a flux A and B – Dependent Variables (Change over time) C – Independent variable (constant value)

Defining network conventions A B C Stoichiometric information needs to be included Dashed arrow represents modulation of a flux + Dashed arrow with a plus sign represents positive modulation of a flux Dashed arrow with a minus sign represents negative modulation of a flux 2 3 D+ Reversible Reaction

Defining network conventions B C Stoichiometric information needs to be included Dashed arrow represents modulation of a flux + Dashed arrow with a plus sign represents positive modulation of a flux Dashed arrow with a minus sign represents negative modulation of a flux 2 A 3 D

Renaming Conventions C Having too many names or names that are closely related may complicate interpretation and set up of the model. Therefore, using a structured nomenclature is important for book keeping Let us call Xi to variable i A B D X3 X1 X2 X4

New Network Representation X2 X3 + 2 X1 3 X4 C A B D X3 X1 X2 X4

Production and sink reactions X2 X0 Independent Variable Production Reaction Sink Reaction

Test Cases: Metabolic Pathway 1 – Metabolite 1 is produced from metabolite 0 by enzyme 12 – Metabolite 2 is produced from metabolite 1 by enzyme 23 – Metabolite 3 is produced from metabolite 2 by enzyme 34 – Metabolite 4 is produced from metabolite 3 by enzyme 45 – Metabolite 5 is produced from metabolite 3 by enzyme 56 – Metabolites 4 and 5 are consumed outside the system7 – Metabolite 3 inhibits action of enzyme 18 – Metabolite 4 inhibits enzyme 4 and activates enzyme 59 – Metabolite 5 inhibits enzyme 5 and activates enzyme 4

Test Cases: Gene Circuit 1 – mRNA is synthesized from nucleotides2 – mRNA is degraded3 – Protein is produced from amino acids4 – Protein is degraded5 – DNA is needed for mRNA synthesis and it transmits information for that synthesis 6 – mRNA is needed for protein synthesis it transmits information for that synthesis 7 – Protein is a transcription factor that negatively regulates expression of the mRNA 7 – Lactose binds the protein reversibly, with a stoichiometry of 1 and creates a form of the protein that does not bind DNA.

Test Cases: Signal transduction pathway 1 – 2 step phosphorylation cascade2 – Receptor protein can be in one of two forms depending on a signal S (S activates R) 3 – Receptor in active form can phosphorylate a MAPKKK.4 – MAPKKK can be phosphorylated in two different residues; both can be phosphorylated simultaneously 5 – MAPKK can be phosphorylated in two different residues; both can be phosphorylated simultaneously 6 – Residue 1 of MAPKK can only be phosphorylated if both residues of MAPKKK are phosphorylated 7 – Residue 2 of MAPKKK can be phosphorylated if one and only one of the residues of MAPKKK are phosphorylated. 8 – All phosphorylated residues can loose phosphate spontaneously9 – Active R inactivates over time spontaneously

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalisms Creating and studying a mathematical model

In silico networks are limited as predictors of physiological behavior What happens? Probably a very sick mutant?

Dynamic behavior unpredictable in non- linear systems X0X1X2X3 X0 X1 X2 X3 t0t1t2t3 t X3

How to predict behavior of network or pathway? Build mathematical models!!!!

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalisms Creating and studying a mathematical model

Types of Model Finite State Models –Bolean Network Models Stoichiometric Models –Flux balance analysis models Deterministic Models –Homogeneous –Spatial Detail Stochastic Models –Homogeneous –Spatial Detail

Finite State models A Finite state model is composed of –a set of nodes that are connected by –a set of edges. –Each node can have a finite number of states and the –rules for changing these states with time are transmitted through the edges and based on the state of the neighbors.

Boolean Networks A Boolean network model is composed of –a set of nodes that are connected by –a set of edges. –each node can have TWO states –the rules for changing these states with time are transmitted through the edges and based on the state of the neighbors.

Boolean Networks are usefull They can give you information about the connectivity of your metabolism or gene circuit What you organism can or can not do may also depend on the connectivity of the regulation

Simple Finite State Gene Circuit ABC A – Positively regulates itself and b Negatively regulates C B – Positively regulates itself and c C – Positively regulates itself and b Negatively regulates A

Regulation of the Circuit A B C A – Positively regulates itself and b Negatively regulates C B – Positively regulates itself and c C – Positively regulates itself and b Negatively regulates A ++ _ _ +

Threshold of expression in circuit A,B, C can have three levels of expression (0,1,2) Regulation of A or C occurs whenever a gene is in or above level 1 Regulation of B occurs whenever a gene is in or above level 1

Logical rules for time change Level of expression in the absence of all regulators Level of expression in the presence of A, B or C Level of expression of A,B, C in the presence of A and B, A and C or B and C Level of expression of A,B, C in the presence of A, B and C

Possible states 5 Steady States 2 Oscilatory states Everything else a transient state

Stoichiometric Models A Stoichiometric model is composed of –a stoichiometric matrix that informs on the number of molecules that are transformed –a flux vector that describes the rates of change in the system

A simple stoichiometric model A model system comprising three metabolites (A, B and C) with three reactions (internal fluxes, vi including one reversible reaction) and three exchange fluxes (bi).

Mass balance Stoichiometric matrix S Flux matrix v S · v = 0 in steady state. Mass balance equations accounting for all reactions and transport mechanisms are written for each species. These equations are then rewritten in matrix form. At steady state, this reduces to S · V=0.

Things to do Apply graph theory (or other) to derive information from the model No effect No Flux

Deterministic Models A deterministic model is an extension of a stoichiometric model. It is formed of –a stoichiometric matrix that informs on the number of molecules that are transformed –a flux vector that describes the rates of change in the system. The functions are continuous: dX/dt= S · v

A Bifunctional two component system Bifunctional Sensor S S* R* R Q1 Q2

X3 X1 X2 X4 X5 X6 A model with a bifunctional sensor Bifunctional Sensor O(rdinary)D(iferential)E(quation)s

Homogenous Deterministic Models Assume system is spatially homogeneous If not use P(artial)D(ifferential)E(quations)s

Partial Differential Equations - Diffusion  ∂∂ accumulation of [S] due to transport SJ Rate of change of [S] local production of [S] S f From the definition of flux For one-dimensional space, E.g. Extending the Hodgkin-Huxley model to voltage spread:

Non-Homogenous Deterministic Models Numerical solution is effectively done by coupling many systems of ODEs Much heavier computationally

What do deterministic models assume? That the number of particle changes in a continuum Is this true? Can we have 1 and half molecules of A? –NO!!!!! Now what?!?!?!

What do deterministic models assume? Law of large numbers (statistics) –The larger the population, the better the mean value is as a representation of the population –What if there are 10 TFs in a cell? –Stochastic models, either homogeneous or non-homogeneous!!!

Stochastic Models Replace continuous assumptions by discrete events Use rate constants as measures of probability Assume that at any give sufficiently small time interval only one event occurs

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalisms Creating and studying a mathematical model

Goals of the model (I) Large scale modeling –Reconstructing the full network of the genome –Red Blood Cell Metabolism Modeling Specific Pathways/Circuits –Non-catalytic lipid peroxidation –MAPK Pathways Generating alternative hypothesys for the topology of the model. –ISC Reconstruction –Phosphate metabolism reconstruction

Goals of the model (II) Estimating parameter values –Estimating parameter values in the purine metabolism Identifying Design Principles –Latter

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalism Creating and studying a mathematical model

Representing the time behavior of your system A B C +

What is the form of the function? A B C + A or C Flux Linear Saturating Sigmoid

What if the form of the function is unknown? A B C + Taylor Theorem: f(A,C) can be written as a polynomial function of A and C using the function’s mathematical derivatives with respect to the variables (A,C)

Are all terms needed? A B C + f(A,C) can be approximated by considering only a few of its mathematical derivatives with respect to the variables (A,C)

Linear approximation A B C + Taylor Theorem: f(A,C) is approximated with a linear function by its first order derivatives with respect to the variables (A,C) Linear

What if system is non-linear? Use a first order approximation in a non-linear space.

Logarithmic space is non-linear A B C + g<0 inhibits flux g=0 no influence on flux g>0 activates flux Use Taylor theorem in Log space

Why log space? Intuitive parameters Simple, yet non-linear Convex representation in cartesian space Linearizes exponential space –Many biological processes are close to exponential → Linearizes mathematics

Why is formalism important? Reproduction of observed behavior –For example, inverse space may be better for some models. Tayloring of numerical methods to specific forms of mathematical equations

Test Cases: Metabolic Pathway X0X1X2 X3 X4 _ _ _ _ + +

Test Cases: Gene circuit X0X1 X2 X3 X4X X6 _ +

Test Cases: Signal transduction X0 X1 X2 X4 X3 X5 X X4 X3 X5 X

Inverse space is non-linear A B C + K<0 inhibits flux K=0 no influence on flux K>0 activates flux Use Taylor theorem in inverse space

Organization of the talk Network representations From networks to physiological behavior Types of models Types of problems Mathematical formalism Creating and studying a mathematical model

A model of a biosynthetic pathway X0X0 X1X1 _ + X2X2 X3X3 X4X4 Constant Protein using X 3

What can you learn? Steady state response Long term or homeostatic systemic behavior of the network Transient response Transient or adaptive systemic behavior of the network

What else can you learn? Sensitivity of the system to perturbations in parameters or conditions in the medium Stability of the homeostatic behavior of the system Understand design principles in the network as a consequence of evolution

Steady state response analysis

How is homeostasis of the flux affected by changes in X 0 ? Log[X 0 ] Log[V] Low g 10 Medium g 10 Large g 10 Increases in X0 always increase the homeostatic values of the flux through the pathway

How is flux affected by changes in demand for X 3 ? Log[X 4 ] Log[V] Large g 13 Medium g 13 Low g 13

How is homeostasis affected by changes in demand for X 3 ? Log[X 4 ] Log[X 3 ] Low g 13 Medium g 13 Large g 13

What to look for in the analysis? Steady state response Long term or homeostatic systemic behavior of the network Transient response Transient of adaptive systemic behavior of the network

Transient response analysis Solve numerically

Specific adaptive response Get parameter values Get concentration values Substitution Solve numerically Time [X 3 ] Change in X 4

General adaptive response Normalize Solve numerically with comprehensive scan of parameter values Time [X 3 ] Increase in X 4 Low g 13 Increasing g 13 Threshold g 13 High g 13 Unstable system, uncapable of homeostasis if feedback is strong!!

Sensitivity analysis Sensitivity of the system to changes in environment –Increase in demand for product causes increase in flux through pathway –Increase in strength of feedback increases response of flux to demand –Increase in strength of feedback decreases homeostasis margin of the system

Stability analysis Stability of the homeostatic behavior –Increase in strength of feedback decreases homeostasis margin of the system

How to do it Download programs/algorithms and do it –PLAS, GEPASI, COPASI SBML suites, MatLab, Mathematica, etc. Use an on-line server to build the model and do the simulation –V-Cell, Basis

Design principles Why are there alternative designs of the same pathway? Dual modes of gene control Why is a given pathway design prefered over others? Overall feedback in biosynthetic pathways

Why regulation by overall feedback? X0X0 X1X1 _ + X2X2 X3X3 X4X4 X0X0 X1X1 _ + X2X2 X3X3 X4X4 __ Overall feedback Cascade feedback

Overall feedback improves functionality of the system TimeSpurious stimulation [C] Overall Cascade Proper stimulus Overall Cascade [C] Stimulus Overall Cascade Alves & Savageau, 2000, Biophys. J.

Dual Modes of gene control

Demand theory of gene control Wall et al, 2004, Nature Genetics Reviews High demand for gene expression→ Positive Regulation Low demand for gene expression → Negative mode of regulation

Summary From networks to physiological behavior Network representations Types of Models Types of Problems Mathematical formalism Studying a mathematical model