p . I

Week 7- Monday, 2/26

More IVP 's.

(A)

Circadian

rhythms .

"

Circadian

"

means

"

occurring naturally on a 24 - hour

cycle .

"

One model ( see Tyson et al.( 1999 )

,on our web page)

ssttsbsiprroatdeingnpraknttnsnhosrtareo

.ie#tEe9cnbsHIebFYu1 short for"

timeless" ), on the "Her

' '

and"

tim"

mRNA that produces them .

Variables !

M = perltim mRNA

P±= PER 1 TIM monomers ( basic proteinbuildingblocks )

Pz= PERHTMdinners ( a diner is built up fromtwo monomers )

( A more detailed study .

.

have six variables,

by not grouping per anmd' #tm

, or PER and TIM,

together .But the above three are enough for Us

.

)

The model says:

p. 2

Week 7- Monday, 2/26

l i ) CidDM = a *CMIt 1 + bPz2

C iii ) C

iv) ( v ) ( vi ) ( vii )

( §dd}= DM . §Ygngpz- h B - 21432+2lb

( viii )

Cix) ( × ) ( x i )

dPz =- m Pz - n Pz + KP

,

2- lpz .

dt

ftp.tgpzAnalysis of terms :

( i ) Pz inhibits production of M : ( i ) tells us that more 13means slower growth of M

.

( ii ) M is degraded / used up at a rate proportional tothe amount of M present .

liii ) Ps is produced at a rate proportional to the amount ofM present .

(

iv) and (viii) represent phosphorylation : Pg and Pzcombine with phosphates and are inactivated

.Note

that the decay implied by the numerators of ( iv ) and(

viii) is tempered by the denominators

.

( v ) and ( ix ) represent proteolysis : Pg and Pz decay( proteins have half - lives)

.

p.3

Week 7- Monday, 2/26

( vi ) and ( x ) represent the reactionmonomer A monomer → dinner

.

[ Note that ( vi ) cancels twice ( x ).]

( vii ) and ( x i ) represent the reactiondinner → monomer A monomer .

[ Note that ( vii ) cancels twice ( xil. ]

COOL

FACTS:

( 1 ) Solutions to ( ( R ) are normallycircadian,but

(2)A certain mutation perh ofper mRNA changes

the "dinner ration rate " K in ( vi ) and ( x ) above

,

resulting In M, Pg , Pz having periods > 24 hours !

(3) Equations ( CR ) can further be reduced,

under

certain conditions, by combining Pg and Pz into a single

protein variable P.

See Sage worksheet Circadian . sws

( on the Sage Page ) .

(B) Other phenomena .

Some other things we can model with IVP 's ( and solve usingEuler 's method 1Sage ) are :

( i ) Monomers klimersltrimers :see tutorial , 2/22

.

Iii ) Fermentation :see Mini Project 3 .

( iii ) Genetic toggle switch :see HWG .

( iv ) Neurons : see paper by Nelson, on our course page ,

and section3.4 In our text

.