JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
1. A gene exists in an unbound state, G, and also binds to two activating transcription factors,
X and Y . ODE models for binding are
G + nX
GXn
G + nY
GYn
with forward and reverse rate constants k f and kr and fixed total gene concentration GT =
G + GXn + GYn . Assume that steady state has been reached and that the concentrations of X
and Y are sufficiently large to be unaffected by binding.
(a) What is the concentration ratio GXn /G?
kf n
X
kr
(b) What is the concentration ratio GYn /G?
kf n
Y
kr
(c) What is the fraction f of active DNA, f = (GXn + GYn )/GT ?
f
=
kf
kr
k
X n + krf Y n
k
k
1 + krf X n + krf Y n
1n
kr
K =
kf
=
(X/K)n + (Y /K)n
1 + (X/K)n + (Y /K)n
(d) Your answer to the previous question should have the form (xn + yn )/(1 + xn + yn ),
where x and y are related to concentrations X and Y and other model parameters. In
the xy-plane for the range x ∈ 0 . . . 5, y ∈ 0 . . . 5, plot the lines corresponding to f = 1/2
(solid), f = 0.01 (dotted), and f = 0.99 (dotted) for n = 2 (thin lines) and n = 4 (thick
lines).
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
(e) Defining logical inputs as a = Θ(x > 1) and b = Θ(y > 1), with Θ(u) = 1 if u is true
and 0 if false, what logical function of a and b does the active region resemble?
a OR b
2. Now consider the case that X and Y bind together. The ODE model for binding is
G + nX + nY
GXnYn
(a) What is the concentration ratio GXnYn /G?
kf n n
X Y
kr
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
(b) What is the fraction f of active DNA, f = (GXnYn )/GT ?
f=
kf
kr
X nY n
k
1 + krf X nY n
=
K=
(X/K)n (Y /K)n
1 + (X/K)n (Y /K)n
kr
kf
2n1
(c) Your answer to the previous question should have the form (xn yn )/(1 + xn yn ), where
x and y are related to concentrations X and Y and other model parameters. In the xyplane for the range x ∈ 0 . . . 5, y ∈ 0 . . . 5, plot the lines corresponding to f = 1/2 (solid),
f = 0.01 (dotted), and f = 0.99 (dotted) for n = 1 (thin lines) and n = 2 (thick lines).
(d) What logical function does the active region resemble in terms of a = Θ(x > 1) and
b = Θ(y > 1)?
a AND b
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
(e) How does the match to the logical function differ for n = 2 and n = 4? Is one response
sharper than the other (smaller distance between contour lines for f = 0.01 and f =
0.99, or closer to the logical function?
The response is sharper when n = 4.
3. Now consider the case that X is an activator and Y is a repressor that competes for the same
binding site. The ODE models for binding are the same as independent activators,
G + nX
GXn
G + nY
GYn ,
but now the active fraction is f = GXn /GT .
(a) What is the fraction f of active DNA in terms of model parameters?
We can use the chemical kinetics equations above to write a sytem of ODEs:
d
[GXn ] = k f [G] · [X]n − kr [GXn ]
dt
d
[GYn ] = k f [G] · [Y ]n − kr [GYn ]
dt
From here on out we will write all concentration without the brackets (ie we will write
[G], [GXn ] as G, GXn . We can now solve for the steady state values of GXn and GYn by
setting the derivatives in the above ODEs to zero. Doing some algebra we conclude:
GXn =
kf
G · Xn
kr
(1)
Now we solve for GT :
GT
= G + GXn + GYn
kf n kf n
= G 1+ X + Y
kr
kr
We are told in the problem statement that the active fraction is:
f=
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GXn
GT
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
Plugging in we get:
f
=
GXn
GT
=
f
n
kr GX
kf n
kf n
G 1 + kr X + kr Y
=
f
n
kr X
k
k
1 + krf X n + krf Y n
k
k
(b) Your answer should have the form xn /(1 + xn + yn ), where x and y are related to concentrations and model parameters. In the xy-plane for the range x ∈ 0 . . . 5, y ∈ 0 . . . 5,
plot the lines corresponding to f = 1/2 (solid), f = 0.01 (dotted), and f = 0.99 (dotted)
for n = 2 (thin lines) and n = 4 (thick lines).
3b
5
half−sat
active
inactive
4.5
4
3.5
y
3
2.5
2
1.5
1
0.5
0
0
0.5
1
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1.5
2
2.5
x
3
3.5
4
4.5
5
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
(c) What logical function does the active region resemble in terms of a = Θ(x > 1) and
b = Θ(y > 1)?
Answers may vary. Acceptable answers include:
i.
ii.
iii.
iv.
v.
a AND (NOT b)
a
a AND x > y
a AND (a XOR b) (Which reduces to a AND (NOT b)
There is no logic function.
4. Consider the case that X is an activator and Y is a repressor that binds to X rather than to
DNA. The ODE models for binding are the same as independent activators,
G + nX
GXn + nY
GXn
GXnYn ,
with active fraction is f = GXn /GT .
(a) What is the fraction f of active DNA in terms of model parameters?
Just like problem 3 we start by solving for the steady state of the system of ODEs:
d
[GXn ] = k f [G] · [X]n − kr [GXn ]
dt
d
[GXnYn ] = k f [GXn ] · [Y ]n − kr [GXnYn ]
dt
Dropping the brackets and evalulating when the derivatives are zero we can compute:
kf
GXn = G · X n
kr
2
kf
GXnYn =
G · X n ·Y n
kr
Now we solve for GT :
GT
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= G + GXn + GXnYn
#
"
2
kf
kf n
X n ·Y n
= G 1+ X +
kr
kr
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
Solving for the active fraction:
f
=
GXn
GT
kf
kr
=
k
1 + krf X n +
· Xn
2
kf
kr
X n ·Y n
(b) Your answer should have the form xn /(1 + xn + xn yn ), where x and y are related to concentrations and model parameters. In the xy-plane for the range x ∈ 0 . . . 5, y ∈ 0 . . . 5,
plot the lines corresponding to f = 1/2 (solid), f = 0.01 (dotted), and f = 0.99 (dotted)
for n = 2.
4b
5
half−sat
active
inactive
4.5
4
3.5
y
3
2.5
2
1.5
1
0.5
0
0
0.5
1
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1.5
2
2.5
x
3
3.5
4
4.5
5
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JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
(c) What logical function does the active region resemble in terms of a = Θ(x > 1) and
b = Θ(y > 1)?
a AND (NOT b)
(d) Compare this example to the previous repressor model with competitive binding. How
do the active regions differ? Is one response sharper than the other (smaller distance
between contour lines for f = 0.01 and f = 0.99, or closer to the logical function)?
System 3 does not converge to a logic function of a and b as n goes to ∞ whereas system
4 converges to a AND (NOT b). System 4 is sharper away from the origin but they have
approximately the same sharpness around x = 1.
5. Suppose that three transcription factors X, Y , and Z bind to the same promoter, and the
fraction of active DNA is f = (x2 + y2 + x2 y2 )/(1 + x2 + y2 + x2 y2 + z2 + x2 y2 z2 ).
(a) What are the transcriptional complexes that bind to the promoter?
X2 , Y2 , X2Y2 , Z2 , X2Y2 Z2
(b) Which of these are transcriptionally active?
X2 , Y2 , X2Y2
(c) Suppose that the transcriptional rate is (β1 x2 + β2 y2 + β3 x2 y2 )/(1 + x2 + y2 + x2 y2 +
z2 + x2 y2 z2 ). How would you interpret the coefficients β1 , β2 , and β3 ?
β1 → Transcription rate (transcripts/second) when all genes have X bound (ie GT =
GX2 ).
β2 → Transcription rate (transcripts/second) when all genes have Y bound (ie GT =
GY2 ).
β3 → Transcription rate (transcripts/second) when all genes have both X and Y bound
(ie GT = GX2Y2 ).
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