Nonlinear Control Systems ECSE-6420

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

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

Tunnel diode

Tunnel diode I V

Tunnel diode V E/R time

Tunnel diode V E/R time

Van der Pol oscillator

Van der Pol oscillator

Predator – Prey model

Binary chemical reactions A + B AB A + B AB

Central dogma of biology

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

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

Effective gene-gene network activation inhibition Brazhnik, de la Fuente, and Mendes, Trends in Biotechnology 20, 467-472, 2002

Models for gene regulation

Repressilator simulation

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, 335-338, 2000

Green fluorescent protein Excitation of gfp by photons

Toggle switch

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

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

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

Biochemical reactions Yildirim & Mackey, Biophys J, 2003.

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

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