Control-Theoretic Approaches to Systems Biology

1. Control-Theoretic Approaches to Systems Biology Brian Ingalls Applied Mathematics University of Waterloo Waterloo, Ontario, Canada bingalls@math.uwaterloo.ca

2. Engineering Control Theory and Biology? Engineering Angle:“Evolutionary Design” vs. Human Design

3. http://www.aip.org/pt/jan00/berg.htm

5. p53 and Mdm2 logic elements Kohn & Pommier, Biochem. Biophys. Res. Comm., 2005

6. Eric Davidson's Lab at Caltech (http://sugp.caltech.edu/endomes/) endomesoderm specification in the sea urchin Strongylocentrotus purpuratus

7. Systems and Control Theory

8. How is biochemical feedback implemented?

9. Chemical vs. Biochemical Networks Chemical Network (all possible reactions)

10. Chemical vs. Biochemical Networks Chemical Network (all possible reactions) + Enzyme catalysis (specific reactions)

11. Chemical vs. Biochemical Networks Chemical Network (all possible reactions) + Enzyme catalysis (specific reactions) + in vivo conditions (open system)

12. Chemical vs. Biochemical Networks Chemical Network (all possible reactions) + enzyme catalysis (specific reactions) + in vivo conditions (open system) + enzyme regulation (allostery)

13. Enzyme-Catalysed Reactions http://www.uyseg.org/catalysis/principles/images/enzyme_substrate.gif

14. Competitive Inhibition catalytic complex substrate enzyme product catalytic site competitive inhibitor inactive complex

15. Allosteric Regulation enzyme catalytic site allosteric site substrate allosteric inhibitor conformational change integration of allosteric signals

16. How can enzyme activity be chemically regulated? By inducing conformational changes http://huntingtonlab.cimr.cam.ac.uk/movies.html http://xray.bmc.uu.se/~mowbray/

17. Outline • 1) Static Negative Feedback: Robustness and Trade-offs in Sensitivity • 2) The Frequency Response • 3) Dynamic Negative Feedback: Robustness and Trade-offs in Sensitivity

18. Section 1:Static Negative Feedback: Robustness and Trade-offs in Sensitivity arXiv:0710.5195v1

19. A Signal Transduction example: MAPK pathways as amplifiers One interpretation: amplifier

20. MAPK Pathway: negative feedback negative feedback Suggested roles of feedback: Enhanced deactivation Adaptation to persistent signalling Generation of oscillations Alternative hypothesis (H. Sauro): negative feedback amplifier

21. Amplifiers: Static behaviour

22. Feedback amplifiers

23. Feedback amplifiers: effect of internal variation

24. Feedback amplifiers: effect of external disturbance

25. But! increased robustness comes at a price: sensitivity to variation in system components: sensitivity to variation in feedback components: Conservation Law: Sensitivity in A + Sensitivity in F = 1

26. MAPK:Is the negative feedback in place to enhance amplifier behaviour?

27. Section 2: The Frequency Response: the Spectral Density as Sensitivity Analysis J. Phys. Chem B 2004

28. Dynamic Sensitivity Asymptotic (long time) Response Perturbation ????

29. Frequency Response The asymptotic response of a linear system to a sinusoidal input is a sinusoidal output of the same frequency. system This input-output behaviour can be described by twonumbers for each frequency: the amplitude (A) - System Gain the phase () - Phase Shift

30. Perturbation Asymptotic Response y1 + y2 + y3 +... Fourier Transform Inverse Fourier Transform sum of sinusoids u1 + u2 + u3 + ... sum of responses y1 + y2 + y3 +...

31. Plotting Frequency Response Bode plot: gain and phase-shift plotted separately Gain Frequency steady state sensitivity = zero frequency gain EE jargon: DC gain

32. Frequency Response of MAPK system sensitivity of MAPK to ligand No Feedback Gain (dB) Feedback Frequency

33. Section 3:Dynamic Negative Feedback: Robustness and Trade-offs in Sensitivity

34. Application to Glycolysis J. Doyle, J. Gonçalves, BI H. M. Sauro, and T.-M. Yi

35. Basic Model of Glycolysis

36. Model details: Dynamics based on conservation of mass rate of production rate of consumption Reaction rates: (Michaelis-Menten kinetics)

37. Model Simulations

38. Basic Model: Conservation of Sensitivity Bode’s Integral Formula

39. Bode's Sensitivity Integral:a performance constraint Biological systems have evolved under the same constraints: tight regulation may result in unwanted behaviours (oscillations, disease states,...)

40. Sustained Glycolytic Oscillations Hess and Boiteux, 1968

41. Glycolysis: Turbo-charged positive feedback

42. Bode’s Integral Formula follows from Jensen’s formula: Right hand side terms may aggravate or alleviate the trade-off

43. Extended Model of Glycolysis – Positive feedback F Disturbance Cellular Activity ATPase Lower + + Glucose PFK ATP HK Glycol. n ATP

44. Aggravated Trade-Off

45. Conclusions • The overall "robustness" of a system is constrained by conservation laws. • Regulation by feedback control has the effect of redistributing the sensitivity of a system. • The redistribution of sensitivity can be in terms of components or time-scales (or both).

46. Synthetic Biology: Forward Engineering of Biochemical and Genetic networks

47. Genetic Toggle Switch Gardner, T.S., Cantor, C.R., and Collins, J.J. (2000). Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339–342. http://www.cellbioed.org/articles/vol4no1/i1536-7509-4-1-19-f02.jpg

48. Genetic Oscillator: the Repressilator Elowitz, M.B., and Leibler, S. (2000). A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338. http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v420/n6912/full/nature01257_r.html

49. Construction of computational elements (logic gates) and cell-cell communication Genetic circuit building blocks for cellular computation, communications, and signal processing, Weiss, Basu, Hooshangi, Kalmbach, Karig, Mehreja, Netravali. Natural Computing. 2003. Vol. 2, 47-84. http://www.molbio.princeton.edu/research_facultymember.php?id=62

50. iGEM: international Genetic Machine Competition