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Positive Feedback and Bistability BIOE 423: 2013

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Stable state Simulation of biochemical network Stable steady state Transient stateStable state

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Multiple stable states Different starting points lead to different steady states

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Positive Feedback v1 = ? v2 = ? dS/dt = ? v1 v2

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Positive Feedback p = defn cell $Xo -> S1; 0.5 + Vmax*S1^n/(15 + S1^n); S1 -> $X1; k1*S1; end; p.Xo = 1; p.X1 = 0; p.S1 = 1; p.n = 4; p.Vmax = 10; p.k1 = 2; 5

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Positive Feedback Time S1 High State Low State 6

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Positive Feedback S1 k1 v2 v1 v2 Perturbations around a stable point

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Positive Feedback S1 k1 v2 v1 v2 Perturbations around a stable point S1

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Positive Feedback S1 k1 v2 v1 v2 v2 > v1 Perturbations around a stable point S1

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Positive Feedback S1 k1 v2 v1 v2 v2 > v1 Therefore: dS1/dt is negative Perturbations around a stable point S1

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Positive Feedback S1 k1 v2 v1 v2 Perturbations around a unstable point S1

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Positive Feedback S1 k1 v2 v1 v2 v1 > v2 Perturbations around a unstable point S1

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Positive Feedback S1 k1 v2 v1 v2 Therefore: dS1/dt is positive Perturbations around a unstable point v1 > v2 S1

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Positive Feedback S1 k1 v2 v1 v2 Therefore: dS1/dt is positive Perturbations around a unstable point v1 > v2 S1

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Where in nature do we find multiple steady states? http://weirdscience.ca/2007/ www.phri.org/research/res_pidubnau.asp Eukaryotic cell differentiation Bacterial differentiation and adaptation

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Bistability of the lac operon Where is the positive feedback?

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Genetic Toggle Switch Where is the positive feedback? dA/dt = ? dB/dt = ? Gardner, T. S. Cantor, C. R. Collins, J. J. Construction of a genetic toggle switch in Escherichia coli. Nature (2000) 6767, pages 339-342 Synthetic toggle switch has been built using lacI and tetR repressors.

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Flip-Flop (Latch) Flip-flops can be made either from NAND or NOR gates. In synthetic biology it is probably easier to construct OR like gates than AND gates. In addition an OR based flip-flop is quiescent when both inputs are low, meaning low protein levels. Latching occurs when one or other of the inputs is brought to a high state. 18

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Flip-Flop 0 0 1 0 0 NOR 1 00 Making NOR gates is relatively easy and requires only two operator sites downstream of the RNA polymerase binding site (promoter). 19

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Flip-Flop 0 0 1 0 0 NOR 1 00 20

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 0 1 0 0 1 00 21

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 0 1 0 0 1 10 22

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 0 0 0 0 1 10 23

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 0 0 0 0 0 10 24

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 1 0 0 0 0 10 25

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 1 0 0 0 0 11 26

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Flip-Flop 0 0 1 0 0 NOR 1 00 1 1 0 0 0 0 11 0 1 0 0 0 0 01 27

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Flip-Flop 0 0 1 0 0 NOR 1 00 0 0 1 1 1 1 00 0 0 1 0 0 1 00 Toggle A to reset P1 Toggle B to set P1 28

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Network structures involving toggle switches Developmental Switch

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Bifurcation Diagram h Steady state value of A StableUnstable Stable

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Bistability with Hysteresis One of the parameters in the model Unstable state Stable state Gianluca M. Guidi, and Albert Goldbeter. Bistability without Histeresis in Chemical Reaction Systems: A Theoretical Analysis of Irreversible Transitions between Multiple Steady States. Journal of Physical Chemistry (1997), 101 (49). State Variable

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Bistability with Irreversibility Gianluca M. Guidi, and Albert Goldbeter. Bistability without Histeresis in Chemical Reaction Systems: A Theoretical Analysis of Irreversible Transitions between Multiple Steady States. Journal of Physical Chemistry (1997), 101 (49).

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