1. Nonlinear Control Systems ECSE-6420
Spring 2009
Lecture 2: 20 January 2009

2. Models and equilibrium points
Consider a dynamical system described by:
Special cases:
F(x,t)  unforced dynamics
F(x)  autonomous
Equilibrium points/set satisfies F(x)=0.
state equation
output equation

3. Tunnel diode

4. Tunnel diode I V

5. Tunnel diode V
E/R
time

6. Tunnel diode V
E/R
time

7. Van der Pol oscillator

8. Van der Pol oscillator

9. Predator – Prey model

10. Binary chemical reactions
A + B AB
A + B AB

11. Central dogma of biology

12. Network of genes Transcription Translation Promoter sites RNAp
Inhibition
mRNA
mRNA
Translation
RNAp
Activation
Protein 2
Protein 1
Ribosomes

13. Effective gene-gene network
auto-activation
Gene
auto-inhibition
Gene
Genes regulate each other by proxy
Gene
Protein
Protein
Protein
Protein
Protein
First, explain about effective gene-gene network.
Gene
activation
inhibition

14. Effective gene-gene network
activation
inhibition
Brazhnik, de la Fuente, and Mendes, Trends in Biotechnology 20, , 2002

15. Models for gene regulation

16. Repressilator simulation

17. Engineering a genetic oscillator
Scientists/bioengineers have been able to build functional synthetic genetic circuits, such as oscillators. Bottom left:compare to oscillator in electr.engr. Pictures of gfp on the next slide.
Michael B. Elowitz and Stanislas Leibler
Nature 403, , 2000

18. Green fluorescent protein
Excitation of gfp by photons

19. Toggle switch

20. Genetic toggle switch
Another celebrated design of synthetic network, toggle switch. Function: switch, memory. Can be coupled to a biosensor to enhance detection.

21. Biological phenomenology
lac Z
lac Y
lac A
lac I
mRNA
mRNA
b galactosidase
repressor
The lac operon is under “negative” control (lac repressor, LacI) and “positive” control (CAP or CAMP receptor protein - CRP)
The ‘original’ lac network
permease
allolactose
external
lactose
Internal lactose
by: Vijay Kumar

22. Biological phenomenology
lac Z
lac Y
lac A
lac I
mRNA
mRNA B M
b galactosidase
repressor
The lac operon is under “negative” control (lac repressor, LacI) and “positive” control (CAP or CAMP receptor protein - CRP)
The ‘original’ lac network P
permease E
Internal TMG
external
TMG
Internal TMG T
by: Vijay Kumar

23. Biochemical reactions
Yildirim & Mackey, Biophys J, 2003.

24. Two stables states
mRNA concentration
High state
Low state
External TMG concentration
Two modes of equilibria, low concentration (uninduced) and high concentration (induced).

25. Bistability in nature: E. coli
Bacteria exploit bistability in adaptation!