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Integral Feedback Control: From Homeostasis to Chemotaxis

Integral Feedback Control: From Homeostasis to Chemotaxis. Tau-Mu Yi Developmental and Cell Biology UCI. Outline. Primer on Integral Control. Examples of Integral Control. Homeostasis, Integral Control, and the Internal Model Principle. Integral control and robust chemotaxis.

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Integral Feedback Control: From Homeostasis to Chemotaxis

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  1. Integral Feedback Control: From Homeostasis to Chemotaxis Tau-Mu Yi Developmental and Cell Biology UCI

  2. Outline • Primer on Integral Control. • Examples of Integral Control. • Homeostasis, Integral Control, and the Internal Model Principle. • Integral control and robust chemotaxis.

  3. Connection to MCA/BST Decompose S into P and C

  4. P u y + C Types of Feedback Control • Proportional: • Integral Control: • Derivative Control: • PID:

  5. Comparing the Controllers u = unit step (P = 1, K = 1) (P) (I) (D)

  6. Bode Plot (Frequency Response) • Used extensively in control design because it contains information about behavior at all frequencies.

  7. Primer on Integral Feedback Control • Time integral of system error is fed back. • Ensures that steady-state error approaches zero despite changes in the input or in the system parameters. • Ubiquitous in complex engineered systems.

  8. Block diagram for integral control

  9. Bacterial chemotaxis signal transduction pathway Attractant Receptor Complex (MCP + CheW + CheA) (+CH3) (-CH3) CheB (demethylase) CheR (methylase) CheY-P Only demethylates active receptor complexes. Tumbling

  10. Adaptation precision = Adaptation precision is robust Y0 Yss + Asp Alon, U., Surette, M. G., Barkai, N. & Leibler, S. Robustness in bacterial chemotaxis. Nature397, 168-171 (1998). CheR Evidence of Integral Control: Robust Perfect Adaptation Segall, J. E., Block, S. M. & Berg, H. E. Temporal comparisons in bacterial chemotaxis.Proc. Natl. Acad. Sci. USA83, 8987-8991 (1986).

  11. Barkai-Leibler Model: • Integral control • Robust perfect adaptation 0 1 mM 1 mM Perfect Adaptation Perfect Adaptation Modeling Perfect Adaptation • Spiro-Othmer Model: • No integral control • Non-robust perfect adaptation

  12. Chemotaxis and integral control A Error

  13. + + [Ca] Model of Blood Calcium Regulation d (disturbance) e u Set Point C [Ca] [Ca]0 H. El-Samad and M. Khammash JTB 214:17-29 (2002).

  14. Homeostasis and Integral Control: Blood Calcium Regulation • Problem: Parturient Hypocalcemia. (PI controller) H. El-Samad and M. Khammash JTB 214:17-29 (2002).

  15. Blood Glucose Regulation: Insulin and Glucagon • Why two hormones? • Two (integral rein control), one, or zero integral controllers? [insulin] [glucagon] [glucose]

  16. Integral Rein Control • Two linked integral controllers. • Benefits: Minimize control action. • Costs: Set points must be the same.

  17. Homeostasis is Fundamental to Life • Homeostasis is dynamic self-regulation. • Examples: temperature, energy, key metabolites, blood pressure, immune response, hormone balance, neural functioning, etc. • Sensory adaptation is a type of homeostasis.

  18. Necessity of Integral Control • Integral feedback control is not only sufficient but also necessary for robust perfect adaptation. • Other feedback strategies for achieving robust perfect adaptation must be equivalent to integral control. • If the Barkai-Leibler model is later contradicted, another mechanism implementing integral control is likely to be present.

  19. Internal Model Principle (IMP) • Internal Model Principle is a generalization of the necessity of integral control. • Robust tracking of an arbitrary signal requires a model of that signal in the controller. • Intuitively, the internal model counteracts the external signal.

  20. IMP = Internal Model Counteracts Disturbance + + U(s) K Y(s) • Consider the input • contains no unstable poles. • Then, C(s)

  21. IMP in the Real World • Biological systems are subjected to arbitrary, changing disturbances. • Internal models of these disturbances must exist within the biological system. • Homeostasis entails approximate internal models.

  22. e e + + Disturbance Approximate IMP P 0 - C = Disturbance

  23. x2 t2 Spatial Sensing x1 t1 Two Chemotactic Strategies Temporal Sensing (Differentiator)

  24. Examples B. A. Temporal Sensing: Bacterial Chemotaxis Spatial Sensing: Yeast Mating a a

  25. Building a Robust Differentiator for Temporal Sensing Non-robust Differentiator #1 u y Robust Differentiator #2 u y Integrator in feedback loop = integral control Robustness Noise filtering

  26. Noise Filtering s Integral control Bode Plot

  27. Sources of Noise • Gradient • Ligand-receptor binding • Signaling pathway • Diffusion of bacteria

  28. Estimation Problem Goal: Estimate: (noisy) Note that = Optimal filter is first-order: G Integral Control

  29. Summary • Integral control is a ubiquitous form of feedback control. • Integral control may represent an important strategy for ensuring homeostasis. • A robust differentiator can be implemented through integral control.

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