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.

Connection to MCA/BST

),( psNvs

BuAxx

)()()( sUsSsY

Decompose S into P and C

Types of Feedback Control

• Proportional: • Integral Control:• Derivative Control:• PID:

Pu y

C

+

yCKyC

yC ykykKyC PI

uPCPy

PuyPCPCyPuCyuPy

1

)1()(

))(()( sks

kKsYsC P

I

Comparing the Controllers

uPCPy

1

uPKPy

1

uPs

Psy

uPs

Py

1

(P)

(I)

(D)

(P = 1, K = 1)

u = unit step

Bode Plot (Frequency Response)

• Used extensively in control design because it contains information about behavior at all frequencies.

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.

Block diagram for integral control

Bacterial chemotaxis signal transduction pathway

Attractant

Receptor Complex(MCP + CheW + CheA)

CheY-P

Tumbling

CheB(demethylase)

(-CH3)

CheR(methylase)

(+CH3

)

Only demethylates activereceptor complexes.

Evidence of Integral Control: Robust Perfect Adaptation

Y0 Yss

+ Asp

Adaptation precision = 10

Y

YSS

Segall, J. E., Block, S. M. & Berg, H. E. Temporal comparisons in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 83, 8987-8991 (1986).

CheR

Alon, U., Surette, M. G., Barkai, N. & Leibler, S. Robustness in bacterial chemotaxis. Nature 397, 168-171 (1998).

Adaptation precision is robust

Modeling Perfect Adaptation

Spiro-Othmer Model:• No integral control• Non-robust perfect adaptation

0 1 M

1 mMPerfect Adaptation

Perfect Adaptation

Barkai-Leibler Model:• Integral control• Robust perfect adaptation

Chemotaxis and integral control

Error

A

.bybrAbbArx

br

Model of Blood Calcium Regulation

++SetPoint

[Ca]

[Ca]0

Ce u

d (disturbance)

[Ca]

H. El-Samad and M. KhammashJTB 214:17-29 (2002).

Homeostasis and Integral Control: Blood Calcium Regulation

• Problem: Parturient Hypocalcemia.

][CaCa][ 0e

PTH][e

PTH][[VitD] kdt

d

(PI controller)

dtdke

ke

[VitD]

PTH][

2

1

ekeku

kku

ip

[VitD]PTH][ 43

H. El-Samad and M. KhammashJTB 214:17-29 (2002).

Blood Glucose Regulation: Insulin and Glucagon

• Why two hormones?

• Two (integral rein control), one, or zero integral controllers?

[glucose]

[insu

lin]

[glu

cag

on

]

Integral Rein Control

• Two linked integral controllers.

• Benefits: Minimize control action.

• Costs: Set points must be the same.

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.

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.

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.

IMP = Internal Model Counteracts Disturbance

• Consider the input

• contains no unstable poles.

• Then,

U(s)

C(s)

K Y(s)

pole. RHP a is , )(

1)( ii

pps

sU

CsKUsY

1)()(

)()(

)(1)(

sbsa

pssC

i

+ +

tty as 0)(

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.

Approximate IMP

Disturbance0

=

Disturbance

- P

C

+

+

Two Chemotactic StrategiesTemporal Sensing (Differentiator)

tumbleelse run,straight ,0 if dt

dC

dt

dC

tt

tCtC

12

12 )()(t2

t1x1

x2

dx

dC

xx

xCxC

12

12 )()(

Spatial Sensing

22 ofdirection in shmoo ,0)(

if xdx

xdC

Examples

Temporal Sensing:Bacterial Chemotaxis

Spatial Sensing:Yeast Mating

a

A. B.

Building a Robust Differentiator for Temporal Sensing

Differentiator #1

Integrator in feedback loop =integral control

Ksu y

skI

Ku yDifferentiator #2

RobustnessNoise filtering

Non-robust

Robust

uKks

KsyI

Ksuy

Noise Filtering

s

Integral control

Bode Plot

Sources of Noise

• Gradient

• Ligand-receptor binding

• Signaling pathway

• Diffusion of bacteria

Estimation Problem

(noisy)

dtdCmax

dtdx

dtdC and

GC

kss

GsCsFv

)(ˆ

Goal:

dxdCdtdC

dtdx

//

Estimate:

)(filter apply , sFnvdtdx Note that

Optimal filter is first-order:

Integral Control

=

G

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.