Control-Theoretic Approaches
to Systems Biology
Brian Ingalls
Applied Mathematics
University of Waterloo
Waterloo, Ontario, Canada
bingalls@math.uwaterloo.ca
Engineering Control Theory and
Biology?
Engineering Angle:
“Evolutionary Design” vs. Human Design
http://www.aip.org/pt/jan00/berg.htm
p53 and
Mdm2
logic
elements
Kohn & Pommier, Biochem. Biophys. Res. Comm., 2005
Eric
Davidson's
Lab at
Caltech
(http://sugp.
caltech.edu/
endomes/)
endomesoderm specification in the sea urchin Strongylocentrotus purpuratus
How is biochemical feedback
implemented?
Chemical vs. Biochemical Networks
Chemical Network (all possible reactions)
Chemical vs. Biochemical Networks
Chemical Network (all possible reactions)
+ Enzyme catalysis (specific reactions)
Chemical vs. Biochemical Networks
Chemical Network (all possible reactions)
+ Enzyme catalysis (specific reactions)
+ in vivo conditions (open system)
Chemical vs. Biochemical Networks
Chemical Network (all possible reactions)
+ enzyme catalysis (specific reactions)
+ in vivo conditions (open system)
+ enzyme regulation (allostery)
Enzyme-Catalysed Reactions
http://www.uyseg.org/catalysis/principles/images/enzyme_substrate.gif
Competitive Inhibition
catalytic
complex
substrate
enzyme
product
catalytic
site
competitive
inhibitor
inactive
complex
Allosteric Regulation
enzyme
allosteric
site
allosteric
inhibitor
catalytic
site
substrate
conformational change
integration of
allosteric
signals
How can enzyme activity be chemically regulated?
By inducing conformational changes
http://xray.bmc.uu.se/~mowbr
ay/
http://huntingtonlab.cimr.cam.a
c.uk/movies.html
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
Section 1:
Static Negative Feedback:
Robustness and Trade-offs
in Sensitivity
arXiv:0710.5195v1
A Signal Transduction example:
MAPK pathways as amplifiers
One interpretation: amplifier
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
Amplifiers: Static behaviour
Feedback amplifiers
Feedback amplifiers: effect
of internal variation
Feedback amplifiers: effect
of external disturbance
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
MAPK:
Is the negative
feedback in place to
enhance amplifier
behaviour?
Section 2:
The Frequency Response:
the Spectral Density as
Sensitivity Analysis
J. Phys. Chem B 2004
Dynamic Sensitivity
Perturbation
Asymptotic
(long time)
Response
????
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
two numbers for each frequency:
• the amplitude (A) - System Gain
• the phase () - Phase Shift
Perturbation
Asymptotic
Response
y1 + y2 + y3 +...
Fourier
Transform
sum of sinusoids
u1 + u2 + u3 + ...
Inverse
Fourier
Transform
sum of responses
y1 + y2 + y3 +...
Plotting Frequency Response
Gain
Bode plot: gain and phase-shift plotted separately
Frequency
steady state sensitivity =
zero frequency gain
EE jargon:
DC gain
Frequency Response of MAPK
system
sensitivity
of MAPK
to ligand
Gain (dB)
No Feedback
Feedback
Frequency
Section 3:
Dynamic Negative Feedback:
Robustness and Trade-offs
in Sensitivity
Application
to Glycolysis
J. Doyle, J. Gonçalves, BI
H. M. Sauro, and T.-M. Yi
Basic Model of Glycolysis
Disturbance
F
Cellular
Activity
ATPase
Glucose
PFK
HK
Lower
Glycol.
n
N
ATP
+
+
ATP
Model details:
Dynamics based on conservation of mass
rate of
production
Reaction rates:
(Michaelis-Menten kinetics)
rate of
consumption
Model Simulations
Basic Model: Conservation of
Sensitivity
Bode’s Integral Formula
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,...)
Sustained Glycolytic
Oscillations
Hess and Boiteux, 1968
Glycolysis:
Turbo-charged
positive
feedback
Bode’s Integral Formula follows from
Jensen’s formula:


0
log S ( ) d   log  k   log k  log S ()
Right hand side terms may
aggravate or alleviate the trade-off
Extended Model of Glycolysis –
Positive feedback
Disturbance
F
Cellular
Activity
ATPase
Glucose
PFK
HK
Lower
Glycol.
+
+
n
ATP


0
log S ( ) d   log  k   log k  log S ()
ATP
Aggravated Trade-Off
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).
Synthetic Biology:
Forward Engineering of
Biochemical and Genetic
networks
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
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
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
iGEM: international Genetic
Machine Competition