Lecture 5: Chemical Clocks. Chemical Clock: BZ reaction

Lecture 5: Chemical Clocks

Chemical Clock: BZ reaction

Purely chemical, isothermal oscillations in a single phase system

Belousov and Zhabotinsky

1951 and 1964

Complex chemistry

At the core of the process is autocatalytic chemistry

8

8

9

.3 2 2 2

.2 2

3 2 2 2

/

2

_________________________

3

______

are reduced and oxidized forms of a

2 2 2

redox catalyst (e.g., Ce

k

k

kred

red o

ox

red o

x

x

BrO HBrO H

BrO HBrO H BrO

M

H

HBrO

O

BrO M H HBrO M

M

M

H O

. . 8 2 32 2

9

8 2 3

2

.28 2 3 9 2

(III)/Ce(IV))

2 [ ][ ]Steady state approximation for : [ ]

[ ]

Balance equation for :

[ ][ ][ [ ][ ][][ ] [ ] ]][ ][

red

red

k HBrO BrOBrO BrO

M k

HBrO

d HBrOk HBrO Br k HBrOO H Brk M H

dH

tOBrO

Simple model: the “Brusselator”1

2

3

4

21 2 3 4

22 3

0 0 20 0

0 2 2

2 3

________________

, , pick ,

putting equations in the simplest possible form:

k

k

k

k

A X

B X Y D dXk A k BX k X Y k X

Y X X dtdYX E k BX k X Ydt

X Y tx y X Y tk B

X Y T

dyX k B k BX

A B

d

E D

3 2 22

3 0

2

23 0

2

0 3 0

20 0

3

ma 1ke

dyx k X x y x x y

d

k BX Y

k X

k B

X

B k

k

k

Dimensionless problem

2

1 4

2 222

3

2 2

2 2

(1 )where ,

Steady state

( ) 1

s: ,

1 det( ) 0

1

s s

dxx x y

k A kddy k Bk Bx x y k Bd k

x y

J

J

tr J

Linear stability analysis

2

2

1,2

2

det 0

Necessary condition for stability:

( )<0, det(J)>0 (always)

St

( )

eady state unstable when

At this boundary

1 >

:

0t

trJ J

t J

i

J

r

r

2

Locus of Hopf bifurcations

Phase plane analysis

2

(1 10

)0

dywh

dx xwhe enn y

d xy

d x

This leads to a closed orbit

What is this good for?

• Internal timekeeping

• Circadian rhythms in cyanobacteria

• Why do you need a clock?

• Protein phosphorylation

Properties of circadian clocks Fundamental properties of a circadian rhythm.   a | A circadian rhythm can be entrained by external stimuli (for example, light/dark or temperature cycles) and will maintain that rhythm when placed in constant conditions. Features of the curve that are commonly measured are the period, the amplitude, and the phase (the relative positioning of the curve in reference to a specific time-point, such as the time placed in constant light). Negative values denote time during light/dark cycles that entrain the clock. Positive values denote time in constant light. b | The phase of a rhythm can be reset by the same stimulus to which it entrains. A 5-hour dark pulse given to cyanobacteria during the subjective day can alter the phase of the rhythm by as much as 10 hours, whereas the same dark pulse given during subjective night produces only a slight change in phase. c | The period of the rhythm is temperature-compensated. At temperatures within the physiological range of the organism, the rhythm persists at a period very near 24 hours. Modified with permission from Ref. 19 © (1993) National Academy of Sciences.

Cyanobacteria

Cyanobacteria are aquatic and photosynthetic, that is, they live in the water, and can manufacture their own food. Because they are bacteria, they are quite small and usually unicellular, though they often grow in colonies large enough to see. They have the distinction of being the oldest known fossils, more than 3.5 billion years old, in fact! It may surprise

you then to know that the cyanobacteria are still around; they are one of the largest and most important groups of bacteria on earth.

S. elongatus

The clock contains a module that can be reconstituted in vitro

This is a purely biochemical clock2005

Toy mathematical model of sequential phosphorylation

Summary

• Oscillations can exist in isothermal chemical systems

• Kinetic models can be analyzed for stability, bifurcations, etc

• What can be said about the possibility of different dynamic behaviors purely on the basis of mechanism?

• Chemical curiosities are exploited by living systems