Review Lecture 2
Michaelis-Menten kinetics
E + S
k1
ES
k2
E + P
k-1
d[S]
= −k 1[E][S] + k −1 [ES]
dt
d[E]
= −k1 [E][S] + (k −1 + k 2)[ES]
dt
d[ES]
= k1 [E][S] − (k −1 + k 2)[ES]
dt
dP
= k 2[ES] ≡ v
dt
Eo = [E] + [ES]
d[S]
= −k 1Eo [S] + (k 1[S] + k −1 )[ES]
dt
d[ES]
= k1Eo[S] − (k1[S] + k −1 + k 2)[ES]
dt
Initial conditions:
[S]t=0 = So
[E]t=0 = Eo
[ES]t=0 = 0
[P]t=0 = 0
v maxSo
v0 =
K m + So
Good approximation if So >> Eo
in this case S0 ~ [S] at the start of
quasi-steady state
Review Lecture 2
Equilibrium binding and cooperativity
S+P
↔P
j− 1
j
Adair’s Equation:
K [S]+ 2K K [S]2 + 3K K K [S]3 +...+ nK K ...Kn[S]n
1 2
1 2 3
1 2
r= 1
1 + K [S]+ K K [S]2 +...+ K K ...Kn[S]n
1
1 2
1 2
[P ]
j
K =
j [P
][S]
j− 1
macroscopic association constant
for transitions between state j-1 and j
Note #1 Detailed balance
Po
k+1
k-1
P1
k+2
k-2
P2
...
Pn-1
k+n
k-n
Pn
d[P ]
o = −k [P ][S]+ k [P ]
0=
+1 o
−1 1
dt
d[P ]
1 = −k [P ][S]+ k [P ]+ k [P ][S]− k [P ]=
0=
+2 1
−2 2
+1 o
−1 1
dt
= −k [P ][S]+ k [P ]
−2 2
+2 1
etc.
k
[P ]
j
K ≡
=
j k
[P
][S]
−j
j− 1
+j
I Identical and independent binding sites
k+
S
k+
k-
kS S
k+
kK=k+/k-
K1=2K
k+
S
kK2=K/2
2K[S]+ 2K2[S]2
2K[S]
=
use Adair: r =
1 + 2K[S]+ K2[S]2 1 + K[S]
II Non-identical and independent binding sites
k+
S
k+*
k-*
k-
S S
k+*
k-*
k+
S
kK=k+/kK*=k+*/k-*
K[S]
K*[S]
Independent binding: r =
+
1 + K[S] 1 + K*[S]
III Identical and interacting binding sites
S
k+
k+*
k-*
k-
S S
k+
k+*
S
kK=k+/kK*=k+*/k-*
use Adair:
K1=2K
k-*
K2=K*/2
2K[S]+ 2KK*[S]2
r=
1 + 2K[S]+ KK*[S]2
Cooperativity
2K[S]+ 2KK*[S]2
r=
1 + 2K[S]+ KK*[S]2
Y=
x(1 + βx)
1 + 2x + βx2
β = K*/K
x = K[S]
β > 1: positive cooperativity
β > 2: sigmoidal curve
β < 1: negative cooperativity
(always: d2Y/dx2 < 0)
Hill number for ‘real’ dimer
Introduction phage biology
Phage genome:
48512 base pairs ~ 12 kB
‘phage.jpg’ ~ 10 kB
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
The lysis-lysogeny decision:
As the phage genome is injected
phage genes are transcribed and
translated by using the host’s
machinery.
Which set of phage proteins are
expressed determines the fate of the
phage: lysis or lysogeny
Image by MIT OCW.
Image removed due to copyright considerations.
A lysogen is immune to
invasion of another phage.
Repressor dimers turn off genes
in the injected phage
chromosome. High concentration
of repressor keeps cell in
lysogenic state.
The lysis-lysogeny decision is a genetic switch
only ‘space’ for one RNA polymerase (mutual exclusion)
Image by MIT OCW. After Ptashne, Mark. A genetic switch : phage lambda. 3rd ed. Cold Spring Harbor, N.Y. :
Cold Spring Harbor Laboratory Press, 2004.
Single repressor dimer bound - three cases:
I
Negative control, dimer binding to OR2 inhibits
RNAp binding to right PR promoter.
Positive control, dimer binding to OR2 enhances
RNAp binding to left PRM promoter.
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
II
Negative control, dimer binding to OR1 inhibits
RNAp binding to right PR promoter.
Negative control, dimer binding to OR1 inhibits
RNAp binding to left PRM promoter (too distant).
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
III
Negative control, dimer binding to OR3 inhibits
RNAp binding to left PRM promoter.
Positive control, dimer binding to OR3 allows
RNAp binding to right PR promoter.
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
Repressor-DNA binding is highly cooperative
intrinsic association constants:
KOR1 ~ 10 KOR2 ~ 10 KOR3
However KOR2* >> KOR2 (positive cooperativity)
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
Flipping the switch by UV:
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.:
Cold Spring Harbor Laboratory Press, 2004.
In lysogenic state, [repressor]
is maintained at constant level
by negative feedback
Image by MIT OCW.
Repressor-DNA binding is highly cooperative
intrinsic association constants:
KOR1 ~ 10 KOR2 ~ 10 KOR3
However KOR2* >> KOR2 (positive cooperativity)
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
Cro dimers bind non-cooperatively to OR sites
KOR3 ~ 10 KOR2 ~ 10 KOR1
Note for repressor:
KOR1 ~ 10 KOR2 ~ 10 KOR3
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
Image removed due to copyright considerations.
See Ptashne, Mark. A genetic switch: phage lambda.
3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004.
Cooperative effects make sharp switch
(‘well defined’ decision)
100
nH=3, positively cooperative
1.0
promoter controlled by a
single repressor-operator system
0.8
nH=1, non cooperative
0.6
Y
% Repression
99.7% repression
λPD
0.4
50
0.2
lysogen
0.0
0
1
2
3
4
[S] (mM)
Repressor concentration
Images by MIT OCW.
Note: several layers of cooperativity:
dimerization, cooperative repressor binding
5
How to create a mathematical model that
captures the essence of the switch ?
Images removed due to copyright considerations. See Arkin, A., J. Ross, and H. H. McAdams.
"Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells."
Genetics 149, no. 4 (Aug, 1998): 1633-48.