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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.

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control theoretic approaches to systems biology

Control-Theoretic Approaches to Systems Biology

Brian Ingalls

Applied Mathematics

University of Waterloo

Waterloo, Ontario, Canada

bingalls@math.uwaterloo.ca

slide5

p53 and Mdm2 logic elements

Kohn & Pommier, Biochem. Biophys. Res. Comm., 2005

slide6

Eric Davidson's Lab at Caltech (http://sugp.caltech.edu/endomes/)

endomesoderm specification in the sea urchin Strongylocentrotus purpuratus

chemical vs biochemical networks
Chemical vs. Biochemical Networks

Chemical Network (all possible reactions)

chemical vs biochemical networks10
Chemical vs. Biochemical Networks

Chemical Network (all possible reactions)

+ Enzyme catalysis (specific reactions)

chemical vs biochemical networks11
Chemical vs. Biochemical Networks

Chemical Network (all possible reactions)

+ Enzyme catalysis (specific reactions)

+ in vivo conditions (open system)

chemical vs biochemical networks12
Chemical vs. Biochemical Networks

Chemical Network (all possible reactions)

+ enzyme catalysis (specific reactions)

+ in vivo conditions (open system)

+ enzyme regulation (allostery)

enzyme catalysed reactions
Enzyme-Catalysed Reactions

http://www.uyseg.org/catalysis/principles/images/enzyme_substrate.gif

competitive inhibition
Competitive Inhibition

catalytic complex

substrate

enzyme

product

catalytic site

competitive inhibitor

inactive complex

allosteric regulation
Allosteric Regulation

enzyme

catalytic site

allosteric site

substrate

allosteric inhibitor

conformational change

integration of allosteric signals

how can enzyme activity be chemically regulated by inducing conformational changes
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/

outline
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
Section 1:Static Negative Feedback: Robustness and Trade-offs in Sensitivity

arXiv:0710.5195v1

mapk pathway negative feedback
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

but increased robustness comes at a price
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

slide27

Section 2: The Frequency Response: the Spectral Density as Sensitivity Analysis

J. Phys. Chem B 2004

slide28

Dynamic Sensitivity

Asymptotic (long time) Response

Perturbation

????

frequency 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 twonumbers for each frequency:

the amplitude (A) - System Gain

the phase () - Phase Shift

slide30

Perturbation

Asymptotic Response

y1 + y2 + y3 +...

Fourier Transform

Inverse Fourier Transform

sum of sinusoids u1 + u2 + u3 + ...

sum of responses y1 + y2 + y3 +...

plotting frequency response
Plotting Frequency Response

Bode plot: gain and phase-shift plotted separately

Gain

Frequency

steady state sensitivity =

zero frequency gain

EE jargon:

DC gain

slide32

Frequency Response of MAPK system

sensitivity of MAPK to ligand

No Feedback

Gain (dB)

Feedback

Frequency

slide34

Application to Glycolysis

J. Doyle, J. Gonçalves, BI

H. M. Sauro, and T.-M. Yi

model details
Model details:

Dynamics based on conservation of mass

rate of production

rate of consumption

Reaction rates: (Michaelis-Menten kinetics)

bode s sensitivity integral a performance constraint
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
Sustained Glycolytic Oscillations

Hess and Boiteux, 1968

slide41

Glycolysis:

Turbo-charged positive feedback

slide42

Bode’s Integral Formula follows from Jensen’s formula:

Right hand side terms may aggravate or alleviate the trade-off

extended model of glycolysis positive feedback
Extended Model of Glycolysis – Positive feedback

F

Disturbance

Cellular

Activity

ATPase

Lower

+

+

Glucose

PFK

ATP

HK

Glycol.

n

ATP

conclusions
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).
slide46

Synthetic Biology:

Forward Engineering of Biochemical and Genetic networks

genetic toggle switch
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
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
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