Bistability and hysteresis 9/20 lecture Bistability and hysteresis

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

(k4)

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

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

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.

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

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

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

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

IPTG aTc

Rao et al, Nature 2002.

Qian & Reluga, Phys Rev Letters, 2005.

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?

From Wolfram.com

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