1. Extracting Essential Features of Biological Networks Natalie Arkus, Michael P. Brenner School of Engineering and Applied Sciences Harvard University

2. Model Explanations Predictions Empirical System Biological System

3. Biological System Model A B A B

4. Courtesy of http://www.london-nano.com, Guillaume Charrashttp://www.london-nano.com Map Kinase Pathway Nerve growth factor signaling Importin nuclear protein import p53 Pathway

5. Biological System Model A B A B

6. Many nonlinear coupled equations → can’t solve analytically Many unknown parameters → many possible solutions Biological System Complicated Model Explanations Predictions ? Analysis? Current Methods Numerical simulation B = f(A) A B not falsifiable! X

7. Complicated Model Simple Model X Current Methods: Another Option Input Output Explanations! Predictions! Biological System

8. Captures everything Knowingly ignores biology Too complicated to fully analyze Can be fully analyzed AC B

9. Complicated Model Explanations Predictions Simple Model ? math Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk e. Coli heat shock response system El Samad et al., PNAS, 102, 2736 (2005) What is the role of feedback loops in heat shock response? Biological System

10. Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk Heat Shock Response (HSR): Proteins unfold/misfold and malfunction σ32 is upregulated Heat shock gene (hsg) transcription ↑ Heat shock proteins (hsp’s) Ex. DnaK, FtsH Refold and degrade unfolded proteins

11. Feedback Loop: DnaK (chaperone) sequesters σ32 (transcription factor) → decreases rate of hsg transcription

12. Another Feedback Loop: Proteases (FtsH, HslVU) degrade σ32 (transcription factor) → decreases rate of hsg transcription

13. El Samad et al., PNAS, 102, 2736 (2005) 1 st Feedback Loop 2 nd Feedback Loop 1)2 feedback loop model 23 ODEs, 8 AEs, 60 parameters 2) 1 feedback loop model 14 ODEs, 5 AEs, 39 parameters 3) 0 feedback loop model 13 ODEs, 5 AEs, 37 parameters → 11 ODEs, 20 AEs, 48 parameters → 5 ODEs, 14 AEs, 33 parameters → 5 ODEs, 13 AEs, 32 parameters They reduced these systems a priori by assuming that all binding reactions were fast Differential Equations = ODEs Algebraic Equations = AEs

14. What is the response time? How do feedback loops ([σ 32 :DnaK], [FtsH t ],…) effect the response time? but are not equipped to answer such questions… Can ask such questions…

15. 1) Separation of scales → Reduction in the # of differential equations 2) Dominant Balance ≈ 0 3) Let us focus on 1 feedback loop model as an example… Differential Equations (ODEs) Algebraic Equations (AEs) Reduction Method:

16. Michaelis-Menten Equation:

17. 1Feedback Loop Model Transcription & Translation Equations Algebraic Binding Equations Mass Balance (Conservation) Equations

18. 1) Separation of scales → Reduction in the # of differential equations 2) Dominant Balance ≈ 0 3) Reduction Method

19. Look for a separation of time scales: Transcription & Translation Equations 0.5 0.03 0.5 1.4 ~100 Only 1 slow variable!

20. → 1 ODE, 18 AEs, 29 parameters Temperature upshift

21. 1) Separation of scales → Reduction in the # of differential equations 2) Dominant Balance ≈ 0 3) Reduction Method

22. Solving Algebraic Components Algebraic System:

23. One Example → σ32 sequestration hardly effects DnaKf levels!

24. X X

25. Further Dominant Balance 2 term balance: Further low [DnaKt] dominant balance:

27. 1) Separation of scales → Reduction in the # of differential equations 2) Dominant Balance ≈ 0 3) Reduction Method

28. . (after many dominant balances)..

30. ✓ How do feedback loops ([σ 32 :DnaK], [FtsH t ],…) effect the response time? With reduced system, are equipped to answer questions of interest…

31. Reduced Model for all Feedback Loops: Effect of 2 feedback loops Effect of 1 st feedback loop

32. Substituting [FtsHt] into d[DnaKt]/dt yields:

33. Substituting in the quadratic solution for DnaKf then yields: where

34. Complicated Model Explanations Predictions Simple Model math Biological System

35. What Sets the Time of Heat Shock Response? El Samad et al.'s conclusion: Response time decreases as number of feedback loops increase. Is response time feedback- or parameter-dependent? Temperature upshift

36. High [DnaKt] Limit: Low [DnaKt] Limit: (using linear [DnaKf] approximation) Response of folded proteins is a feedback-loop independent property Response time set by when [DnaK t ] = 1.9*10^4

37. Reduced Model for all Feedback Loops: Feedback loops → slower response time

38. Reduced Model for all Feedback Loops: Feedback loops → slower response time

39. Reduced Model for all Feedback Loops: Feedback loops → slower response time

40. Reduced Model for all Feedback Loops: Feedback loops → slower response time How can the response time decrease with additional feedback loops? Production Term Degradation Term B > 0 → smaller production term → slower response time C > 0 → smaller production term → slower response time A = effect of 0F loop B = effect of 1F loop C = combined effect of 1F and 2F loops

41. Changes in Network Topology and Parameter Values Cause Models with More Feedback Loops to Respond Faster For the same value of A, feedback loops  slower response time However, the topology of the σ 32 t equation changes in the 2 feedback loop model  a different expression for the effective parameter A (the 0F term) in the 2 feedback loop model Will be encompassed within C

42. Parameter changes across the feedback loop models Translation of [mRNA(DnaK)] Degradation of [σ 32 ] Effect of parameter changes is unclear in full model

43. 0 feedback loop:1 feedback loop:2 feedback loop: Effect of Parameter Changes Is Apparent in Reduced Model Reduced Model for all Feedback Loops:

44. * *

45. Change in topology leads to different expressions for the effective parameter A

46. If is the same over the 3 feedback loop models and in a certain parameter regime  1 and 0 feedback loop models respond quicker.

47. Cost Feedback loop or parameter dependent? El Samad et al.'s conclusion: Cost decreases as number of feedback loops increase. Cost = steady state value of [DnaK t ] Feedback loops  lower cost x x x 1/nonlinear term dominates 1/A dominates Cost is also parameter dependent

48. Constructing Reduced Models Allows One to Extract Essential Biological Components Here, the effect of topology and parameters were decoupled And it was shown, for example, that response time is a parameter dependent and not a feedback loop dependent property Is this system special, were we just lucky?

49. Wnt signaling pathway Lee et al, PLoS Biology, 1, 116 (2003) (Protein network involved in embryogenesis and cancer) System Is Not Special…

50. Curves a-d: Curve d:

51. The heat shock response system is not special – this works for other systems also… Wnt Signaling Pathway

52. Conclusions simple models with all relevant biological components Back and forth with experiments Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk 31 equations  1 equation 14 equations  3 equations Yeast Cell Cycle (Tyson et al, 2004) 62 equations  17 equations testable, falsifiable!

53. Future Directions { Reduced Model 1, Reduced Model 2,Reduced Model 3, …}…} f(dimenionless parameters) ?

54. Courtesy of cancerworld.org Can we explain a biological system in a way that experiments alone can not?