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2. Mathematically Controlled Comparisons Rui Alves Ciencies Mediques Basiques Universitat de Lleida

3. Outline • Design Principles • Classical Mathematically Controlled Comparisons • Statistical Mathematically Controlled Comparisons

4. Operon Gene 1 Gene 2 Gene 3 What are design principles? • Qualitative or quantitative rules that explain why certain designs are recurrently observed in similar types of systems as a solution to a given functional problem • Exist at different levels Nuclear Targeting Sequences

5. S S* S S* R* R R* R Q2 Q2 Q1 Q1 Alternative sensor design in two component systems Monofunctional Sensor Bifunctional Sensor

6. X3 X1 X3 X1 X2 X4 X2 X4 X6 X6 X5 X5 Alternative sensor design in two component systems Monofunctional Sensor Bifunctional Sensor

7. Why two types of sensor? • Why do two types of sensor exist? • Hypothesis: • Random thing • Alternative hypothesis: • There are physiological characteristics in the systemic response that are specific to each type of sensor and that offer selective advantages under different functionality requirements

8. X3 X1 X2 X4 X6 X5 How do we test the alternative hypothesis? 1 – Identify functional criteria that have physiological relevance X2 i) Appropriate fluxes & concentrations ii) High signal amplification iii) Appropriate response to cross-talk iv) Low parameter sensitivity v) Fast responses vi) Large stability margins [X2] X5 Decrease in X5 Fluctuation in X2 Time

9. Functionality criteria for effectiveness • Appropriate fluxes & concentrations • High signal amplification • Appropriate response to cross-talk • Low parameter sensitivity • Fast responses • Large stability margins

10. How to test the alternative hypothesis? 1 – Identify functional criteria that have physiological relevance 2 – Create Mathematical models for the alternatives S-system has analytical steady state solution Analytical solutions → General features of the model that are independent of parameter values

11. X3 X1 X2 X4 X6 X5 A model with a monofunctional sensor Monofunctional Sensor

12. X3 X1 X2 X4 X6 X5 A model with a bifunctional sensor Bifunctional Sensor

13. X3 X1 X2 X4 X6 X5 Approximating the conserved variables Monofunctional Sensor

14. The S-system equations Monofunctional Sensor Bifunctional Sensor

15. S-systems have analytical solutions

16. Analytical solutions are nice!! • Calculating analytical expressions for the gains of the dependent variables with respect to independent variables (Signal amplification) is possible • The same for sensitivity to parameters • The same for other magnitudes

17. Calculating gains is taking derivatives

18. Functionality criteria for effectiveness • Appropriate fluxes & concentrations • High signal amplification • Appropriate response to cross-talk • Low parameter sensitivity • Large stability margins • Fast responses

19. Outline • Design Principles • Classical Mathematically Controlled Comparisons • Statistical Mathematically Controlled Comparisons

20. How to test the alternative hypothesis? 1 – Identify functional criteria that have physiological relevance 2 – Create Mathematical models for the alternatives S-system has analytical steady state solution Analytical solutions → General features of the model that are independent of parameter values 3 – Compare the behavior of the two models with respect to the functional criteria defined in 1 Comparison must be made appropriately, using Mathematically Controlled Comparisons

21. X3 X1 X3 X1 X2 X4 X2 X4 X6 X6 X5 X5 How to compare the inherent differences between designs? Internal Contraints: Corresponding parameters in processes that are identical have the same values in both designs

22. X3 X1 X3 X1 X2 X4 X2 X4 X6 X6 X5 X5 How to compare the inherent differences between designs? Reference System External constraints: b’2 and h’22 are degrees of freedom that the system can use to overcome the loss of bifunctionality.

23. How do we implement external contraints? • Identify variables that are important for the physiology of the system • Choose one of those variables • Equal it between the reference system and the alternative system • Calculate what the value that leads to such equivalence is for the primed parameter

24. Partial controlled comparisons • There can be situations where the physiology is not sufficiently known → Not enough external contraints for all parameters • There can be interest in determining the effect of different sets of physiological contrainst upon parameter values→ Alternative sets of external constraints

25. Implementing external constraints Choose Functional Criteria so that the value of the primed parameters can be fixed. External Constraint 1: Both systems can achieve the same steady state concentrations AND fluxes Fixes b2’

26. Implementing external constraints Choose Functional Criteria so that the value of the primed parameters can be fixed. External Constraint 2: Both systems can achieve the same total signal amplification Fixes h22’

27. AM AB AM AB Q Q Q Studying physiological differences of alternative designs 1

28. Comparing concentrations and fluxes • Concentrations and fluxes can be the same in the presence of a bifunctional sensor or of a monofunctional sensor

29. Comparing signal amplification • Signal amplification is larger in the system with bifunctional sensor + - - + + + + - + + + - + Property in Reference system Property in Alternative system

30. Comparing cross-talk • Sensitivity to cross talk is higher in the system with monofunctional sensor - - + + - + + - + + Property in Reference system Property in Alternative system

31. Comparing sensitivities • Sensitivities can be larger in either system, depending on which sensitivity and on parameter values.

32. Comparing stability margins • The system with a monofunctional sensor is absolutely stable and has larger stability margins than the system with a bifunctional sensor

33. Comparing transient times • Undecided Linearize Linearize Calculate analytical solution Calculate analytical solution

34. Comparing transient times • Undecided

35. Functionality criteria for effectiveness • Appropriate Concentrations → Both Systems = • Appropriate Fluxes → Both Systems = • Signal amplification → Bifunctional larger • Cross-talk amplification → Bifunctional smaller • Margins of stability → Bifunctional smaller • Sensitivities to parameter changes → Undecided • Fast transient responses → Undecided

36. Physiological predictions • Bifunctional design lowers X6 signal amplification • prefered when cross-talk is undesirable. • Monofunctional design elevates X6 signal amplification • prefered when cross-talk is desirable.

37. Questions • What happens when ratios depend on parameter values to be larger or smaller than one? • When the ratios are always larger or smaller than one, independent of parameter values, how much larger or smaller are they, on average?

38. A solution to both problems • Statistical Mathematically Controlled Comparisons

39. Outline • Design Principles • Classical Mathematically Controlled Comparisons • Statistical Mathematically Controlled Comparisons

40. X3 X1 X3 X1 X2 X4 X2 X4 X6 X6 X5 X5 Alternative sensor design in Two Component Systems Monofunctional Sensor Bifunctional Sensor

41. Functionality criteria for effectiveness • Appropriate Concentrations → Both Systems = • Appropriate Fluxes → Both Systems = • Signal amplification → Bifunctional larger • Cross-talk amplification → Bifunctional smaller • Margins of stability → Bifunctional smaller • Sensitivities to parameter changes → Undecided • Fast transient responses → Undecided

42. Quantifying the differences Tofindouthowmuchbiggerorsmallerorto decide whetheranundecided ratio isbiggerorsmallerthanonewehavetoplug in numbersintotheequations

43. Statistical controlled comparisons • Interested in a specific system from a specific organism: • Plug in values and calculate the quantitative differences • Interested in large scale analysis • Large scale sampling of parameter and independent variable space followed by calculation of properties and statistical comparison

44. Statistical controlled comparisons • Parameters: • as, bs • gs, hs • Independent Variables • X5, X6, X7, X8

45. Basic sampling Sample in Log space Random number generator 0a1, [-L1,b1,L’1], ... a1, b1, ... Sample in Log space Random number generator [-L’’1,X5,L’’’1], ... X5, X6, ... Random number generator [-5,g1,0], [0,g1,5] ... Sample g1, g2, ...

46. Importance sampling Random number generator Normal, Bessel,… Sample Uniform [-L1,b1,L’1] b1 Calculate Values for systemic properties Filters: Positive Signal Amplification Stable Steady State Fast Response Times No Discard set Yes Keep set

47. Warnings about the filters in sampling • Make sure that both the reference and the alternative systems fullfil the filters • Make sure that the sign for the kinetic orders calculated through the external constraints is as it should be

48. Problems with the sampling • Systems with bifurcations in flux • Systems with conservation relationships

49. X3 X1 X2 X4 X6 X5 Systems with bifurcations in flux v1 v2 The measure of the set of parameter values within parameter space that is consistent (generates a steady state that is consistent with v1 and v2) is 0

50. X3 X1 X2 X4 X6 X5 Systems with moiety conservation The measure of the set of parameter values within parameter space that is consistent (generates a steady state that is consistent with v1 and v2) is 0