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2. Mathematical Modelling of Biological Networks Prof:Rui Alves ralves@cmb.udl.es 973702406 Dept Ciencies Mediques Basiques, 1st Floor, Room 1.08 Website of the Course:http://web.udl.es/usuaris/pg193845/Courses/Bioinformatics_2007/ Course: http://10.100.14.36/Student_Server/

3. 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

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

5. What does this means?

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

7. 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.

8. 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.

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

10. Defining network conventions C + 3 D+ A B 2 Reversible Reaction 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

11. Defining network conventions C + 2 A B 3 D 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

12. Renaming Conventions 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 X1 B X2 C X3 D X4

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

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

15. 1 – Metabolite 1 is produced from metabolite 0 by enzyme 1 2 – Metabolite 2 is produced from metabolite 1 by enzyme 2 3 – Metabolite 3 is produced from metabolite 2 by enzyme 3 4 – Metabolite 4 is produced from metabolite 3 by enzyme 4 5 – Metabolite 5 is produced from metabolite 3 by enzyme 5 6 – Metabolites 4 and 5 are consumed outside the system 7 – Metabolite 3 inhibits action of enzyme 1 8 – Metabolite 4 inhibits enzyme 4 and activates enzyme 5 9 – Metabolite 5 inhibits enzyme 5 and activates enzyme 4 Test Cases: Metabolic Pathway

16. 1 – mRNA is synthesized from nucleotides 2 – mRNA is degraded 3 – Protein is produced from amino acids 4 – Protein is degraded 5 – 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: Gene Circuit

17. Test Cases: Signal transduction pathway 1 – 2 step phosphorylation cascade 2 – 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 spontaneously 9 – Active R inactivates over time spontaneously

18. 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

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

20. X3 t Dynamic behavior unpredictable in non-linear systems X0 X1 X2 X3 t0 t1 t2 t3 X0 X1 X2 X3

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

22. 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

23. 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

24. 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.

25. 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.

26. 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

27. Simple Finite State Gene 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

28. + + + 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

29. 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

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

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

32. 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

33. 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).

34. 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.

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

36. 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

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

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

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

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

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

42. 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?!?!?!

43. 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!!!

44. 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

45. 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

46. 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

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

48. 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

49. Representing the time behavior of your system C + A B

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