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.