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2. The whys and hows of mathematical models for biological networks, with a view to pitfalls and limitations Prof:Rui Alves ralves@cmb.udl.es 973702406 Dept Ciencies Mediques Basiques, 1st Floor, Room 1.08

3. Organization of thetalk • From networks to physiological behavior • Graphical network representations • Types of problems • Typical bottlenecks and assumptions in model building

4. Ambiguous in siliconetworks are limited as predictors of physiologicalbehavior Probably a very sick mutant? What happens?

5. Mayberesolvingambiguity in representationisenoughtopredictbehavior? X0 X1 X2 X3 t0 t1 t2 t3 X0 X1 X2 X3

6. X3 t Dynamicbehaviorunpredictable in non-linear systems X0 X1 X2 X3

7. Howtopredictbehavior of networkorpathway? • Build mathematical models!!!!

8. Organization of thetalk • From networks to physiological behavior • Graphical network representations • Types of problems • Typical bottlenecks and assumptions in model building

9. A B A B A B A B B A A B Function Function Function Function Function Network representation is fundamental for clarity of analysis What does this mean? Possibilities:

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

11. 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

12. 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

13. Renaming Conventions Having too many names or names that are closely related my 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

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

15. Production and sink reactions X0 X2 Sink Reaction Production Reaction

16. Cells have compartments X0 X2 Cell Organel Compartmental models are important, both because compartments exist in the cell and because even in the absence of compartments reaction media are not always homogeneous

17. Consistency is important • Whatever representation is used be sure to be consistent and to know exactly what the different elements of a representation mean.

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

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

20. Organization of thetalk • From networks to physiological behavior • Network representations • Types of problems • Typical bottlenecks and assumptions in model building

21. All right, so now, what type of model should i build to answer my question? • That depends on the question!!!! • It also depends upon the system for which you ask the question!!!!

22. What questions are there? • The big one: • How does a cell work??? • What answers are being given? stoichiometric matrix Map onto cellular circuits chart Usually solved for steady state Create stoichiometric model Genome sequenced and annotated rate vector

23. A simple stoichiometric model

24. Now what? • Assume that cells are growing at steady state with some optimal conversion of input material (flux b1) into biomass (A,B,C) • Assume linear kinetics for each rate equations • Use (linear) optimization methods to find a solution for the distribution of fluxes that allows the cell to fulfill 1.

25. Sucessstories (?) • Accurately predicting a decent fraction of knock out mutants that are lethal in S.cerevisiae and H. pylori. Proc Natl Acad Sci U S A. 100: 13134-13139; J Bacteriol. 184: 4582-4593. • Fail to predict all mutants • Does not account for transient behavior • Does not account for dynamic regulation Whole cell modeling is far from being able to answer the big question; not enough info is available to build the models.

26. What other questions are there? • Well, let us be modest: • How does a simple cell work??? • What is a simple cell? • A cell that is much simpler than what we normally think of as a cell • Red Blood cell; lambda phage • Mathematical models using dynamic equations have been created to study these types of cells. (e.g. Ni & Savageau or Arkin ) • A regular cell that we represent in a simplified maner • E-cell project represents the E. coli cell using linear kinetics.

27. How have simple cells been modeled? Savageau & Ni, 1992 JBC, JTB

28. Howhave simple cellsbeenmodeled?

29. The red bloodcellstory • Model was used to assess how complete our understading of red blood cell metabolism is. • How was this done? • Using the notion that model robustness can be used to identify ill defined parts of the model • Using the notion that biological systems should have stable steady states

30. What is robustness? • Robustness is the notion that the dynamic behavior of a system is fairly insensitive to spurious fluctuations in parameter values Steady state value Parameter (T, kinetic parameters)

31. Why robustness as a quality assessment tool? • Because if biological systems were not robust, we would not be alive, given that fluctuations happen all the time. Steady state value Parameter (T, kinetic parameters)

32. What is stability? • Stability of a steady state is the notion that after spurious fluctuations in parameter values, the system will return to the original steady state it was in X t

33. Why stability as a quality assessment tool? • Again, because if biological systems were not at stable steady states, we would not be alive, given that fluctuations happen all the time.

34. What did the model do? • Found that the steady state was unstable • Identified regulatory interactions that stabilized the steady state • Latter confirmed experimentally • Identified parts of the model that have high sensitivy • Incomplete understanding of the system

35. Are there more questions worth asking? • Well, yes there are. • There is a fair amount of modularity in cells • Organeles, Pathways, Circuits, etc. • Therefore, if one is interested in specific parts of cellular function and response, one can isolate the modules responsible for that function or response • How does the specific part of a cell responsible for a given function works???

36. Thisquestion has twoparts • Howdoesthespecificpart of a cellresponsiblefor a givenfunctionworks??? • Howdoesitworkqualitatively • Network reconstruction (RA) M2 M… P1 P2 Pn P… Mn M1

37. Thisquestion has twoparts • Howdoesthespecificpart of a cellresponsiblefor a givenfunctionworks??? • Howdoesitworkqualitatively • Network reconstruction P1 M1 M2 P2 Pn P… M… Mn

38. Network reconstruction • FeSC biogenesis is a pathway that is conserved over evolution • Proteins involved in the pathway are identified • How these proteins act together to form a pathway is unknown; the reaction topology and the regulatory topology is unknown • How do these proteins work together?

39. How it was done • Create all possible topologies • Scan all possible behaviors using simulation • Compare qualitative dynamic behavior of the different topologies to experimental results • Eliminate topological alternatives that do not reproduce experimental results

40. The proteins and their function Alves & Sorribas 2007 BMC Systems Biology 1:10 Alves et. al. 2004 Proteins 56:354 Alves et. al. 2004 Proteins 57:481 Vilella et. al. 2004 Comp. Func. Genomics 5:328 Alves et al. 2008 Curr. Bioinformatics in press

41. Thisquestion has twoparts • Howdoesthespecificpart of a cellresponsiblefor a givenfunctionworks??? • Howdoesitworkquantitatively • Parameterestimationwhennetworkisknown (PM) P1 M1 M2 P2 Pn P… M… Mn

42. Parameterestimation • Ifyouknowthetopology and/ormechanism, thenone can askhowdoes a systemactunderspecificcircumstances • Toanswersuch a questionweoftenneednumericalvaluesfortheparameters of thesystem so thatsimulations can beran • Numericalvaluesforparameters can beestimatedfrom experimental data

43. Pattern formation in Drosophila • Basedon gene expression data, what are theparametervaluesthatcreate a bestfit of themodeltotheobserved experimental results?

44. How it was done • Collect experimental data • Create a mathematical model • Use optimization/fitting methods to estimate the parameters of the model in such a way that a minimum discrepancy exists betrween model predictions and observed data.

45. The fit was decent

46. So, are we done yet? • Hell, No!!!! • Modularity begs the question: • Are there design principles that explain why cell use specific modules for specific functions? [AS; RA;AS]

47. Why regulation by overall feedback? _ Overall feedback X0 X1 X2 X3 + _ _ _ X4 X0 X1 X2 X3 Cascade feedback + X4

48. How it was done • Create mathematical models for the alternative networks • Compare the behavior of the models with respect to relevant functional criteria • Decide according to those criteria which model performs best

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

50. Organization of thetalk • From networks to physiological behavior • Graphical network representations • Types of problems • Typical bottlenecks and assumptions in model building