1. Bistability and hysteresis
9/20 lecture
Bistability and hysteresis

2. X’ = 0 = k1A– k2BX + k3X2Y – k4X Y’ = 0 = k2BX – k3X2Y - k5Y
Dashed lines represent the nullclines in phase space
Additional sink for Y added to system
X’ = 0 = k1A– k2BX + k3X2Y – k4X
Y’ = 0 = k2BX – k3X2Y - k5Y

3. (k4)

4. Consider f to be the strength of the feedback
SS Response
Stimulus
What happens when f > 0??
Xiong and Ferrell, Nature, 2003.

5. Small changes in stimulus lead to large change in response
Ultrasensitivity
Small changes in stimulus lead to large change in response

6. Memory in system. Bistability.
Hysteresis
Memory in system. Bistability.
Different stimulus needed for turning on system vs. maintaining on
In this case, the range grows broader with increasing f.

7. Stimulus does not have to be maintained to sustain response
Toggle switch
Stimulus does not have to be maintained to sustain response
Turns transient input into irreversible response

8. a = synthesis rate
b, g = cooperativity
cooperative
repression
degradation
Gardner et al, Nature, 2000.

9. Under what condition would a monostable system arise?v
a1  a2

10. marks a boundary between phase curves with different properties
b, g > 1

12. IPTG
aTc

14. Rao et al, Nature 2002.

15. Qian & Reluga, Phys Rev Letters, 2005.

16. Deterministic Chaos
A deterministic system has future behavior fully described by initial conditions
This does not necessarily mean that it is predictable
Small perturbations in initial conditions can give rise to wildly different behavior
Rather than settling on a limit cycle, the phase plane will have a fractal dimension to it
Lorenz first described these as “strange attractors”
What are some examples of chaotic behavior in biology?

17. From Wolfram.com

18. A tool for visualizing chaos is known as a Poincare map
Takes a continuous system and discretizes it by mapping a plane in phase space which orbits are crossing
By definition has one less dimension than phase space