The Autoregulation of Gene
Transcription
01/24/2012
Biological networks vs. random networks
¾ Molecular interaction networks can be also
divided into the following not necessarily
mutually exclusive categories according to the
nature of interactions and functions:
1. Protein-protein interaction networks
2 Metabolic networks
2.
3. Gene transcription networks
4 Signal transduction networks
4.
¾ Random networks: nodes are randomly
connected Different models have been
connected.
developed to generate random networks:
1 Erdos
1.
Erdos-Renyi
Renyi random networks
2. Degree-preserving random network models
Biological networks vs. random networks
¾ Erdos-Renyi random network model: Nodes in the graph are
connected with equal probability.
¾ If an Erdos-Renyi random network has N nodes and E edges,
then there are N2 possible ways to connect any two nodes
nodes. The
probability to connect any two nodes is
p=
E
.
2
N
¾ Given a real network with N nodes and E edges, we can
construct a corresponding ER random network with the same
number of nodes and edges, but the edges are connected
randomly with the probability
p=
E
.
N2
¾ The properties of a real network can be studied by comparing a
set of parameters of the real network to those of the
corresponding ER random network.
A real network and one of its corresponding ER
random networks
A real network
N=10 nodes
E= 14 edges
From Figure. 3.1 of Uri Alon
A randomized network
(Erdos – Renyi model)
N=10 nodes
E= 14 edges
What differentiate a biological network from a
random one?
¾ Connection sparsity: only a very small portion of all possible
edges are formed in biological networks, i.e., the probability that
the two nodes are connected is small:
p=
E
<< 1.
Emax
¾ Distribution of connections:
Power law distribution of out-degree of biological networks:
N ( d ) = d − r , log N (d ) = − r log d .
logN(d)
where d is the out-degree of a node, N(d) is the number of node
with out-degree d, and r is a constant.
A yeast protein interaction network
log d
What differentiate a biological network from a
random one?
A network with a power law degree distribution is also called a
scale free network.
Hub: nodes that have many more out
out-degree
degree than average
average.
Small world behavior: each node can be reached from any
other nodes through a short path in a scale free network.
¾ Patterns of connections:
Network motifs: some subgraphs
in biological networks are highly
enriched compared to those in
random networks.
networks
¾ Modularity: biological
networks can be separated
p
into nearly independent sub
networks.
Large networks
Sub-networks
Pathways
Molecules
Autoregulation of gene transcription
¾ Some genes are regulated by its own product
product, this
phenomenon is called autogenous regulation, or autoregulation.
¾ Autoregulation can be positive or negative.
X
A
A
G
Gene
X is
i simply
i l regulated
l t db
by A
A.
kd
From Figure. 3.2 of Uri Alon
X
Gene X is negatively autoregulated.
egu ated Repressor
ep esso X b
binds
ds a ssite
te
in its own promoter and thus acts to
repress its own transcription. The
symbol --|| stands for repression. The
repression threshold is kd.
Autoregulation in the known transcription network
of E. coli
¾ The known gene transcription network in E. coli has N = 420
nodes, E = 520 edges and 40 self edges.
Blue nodes have
self-edges,
which is not
shown for
simplicity.
From Figure. 3.1 of Uri Alon
Is self-edge a network motif?
¾ To test if self-edges,
g , and thus autoregulation
g
are significantly
g
y
more frequent in the transcription network than in a random
network, let’s compute the probability of the occurrence of k selfedges in an ER random network with N nodes and E edges
edges.
The probability that a node forms a self-edge is:
pself = 1 / N
N.
The probability to form k self edges therefore follow a binomial
distribution:
p( N self
⎛E⎞
k
= k ) = ⎜⎜ ⎟⎟ p self (1 − pself ) E −k .
⎝k ⎠
Thus, the average number of self-edge in a ER random network
is
<N
>
= Ep = E / N ,
self
rand
self
and its standard deviation is:
σ rand = (1 − 1 / N ) E / N .
Self-edge is a network motif
¾ The E. coli transcription
p
network contains N=424,, E=519,,
therefore, the corresponding ER random network should have
about one self-edges on average:
< N self > rand = E / N = 519 / 424 = 1.2
In contrast, the real network has 40 self-edges.
¾ The probabilistic significance of the real network surpassing the
random network in the number of self-edges can be evaluated
by the Z-score:
< N self > real − < N self > rand
40 − 1.2
=
Z=
= 35.9.
σ rand
1.2(1 − 1 / 40)
¾ Thus, self-edges in the transcription network occur far more
frequently than in random networks.
¾ We conclude that self-edges, and in particular negative
autoregulation are a network motif.
Negative autoregulation of gene
transcription
_
¾ Given a negative autoregulation circuit
X
, that is
is, the
TF X represses its own expression when it binds to its own
promoter to inhibit the production of mRNA.
mRNA
kd
X
Let’s derive the production dynamics equation of [X].
¾ The dynamics
y
of X is described byy ((for simplicity,
p y, we use X to
mean [X] ):
dX
= f X ( X ) − αX .
dt
where fX(X) is the input function, and α is the
degradation/dilution rate.
Dynamics of negative autoregulation using logic
approximation
¾ If we approximate
i t the
th input
i
t function
f
ti with
ith a logic
l i function:
f
ti
f X ( X ) = βθ ( X < k d ),
At the early times,
times X is low
low, so X<kd, therefore,
therefore
dX
= β − aX ,
dt
When X is low, we can even neglect degradation (αX << β), then,
X = β t.
¾ Thus
Thus, the level of X rapidly (linearly) approaches to kd, which
will lead to stopping the production of X. Degradation of X will
bring X< kd , leading to full speed production of X again. Small
oscillations will occur around X = kd, if there is any delay in the
system.
¾ In reality,
reality fX(X) is not a strict logic function
function, rather a smooth
function such as the Hill function, so the oscillation will be
damped.
Dynamics of negative autoregulation using logic
approximation
¾ Eventually, X will lock into a steady state level equal to the
repression coefficient of X on its own promoter,
X(tt) / K
X st = k d .
2
1.8
1.6
1.4
12
1.2
1
0.8
0.6
0.4
02
0.2
0
0
0.1
0.2
From Figure. 3.3 of Uri Alon
0.3
0.4
0.5
0.6
Time (α t)
0.7
0.8
0.9
1
β =5,
kd =1,
=1
α =1.
Dashed line:
D
Dynamics
i off th
the same
gene if auto-regulation
is removed, resulting in
simple
i l regulation
l ti
that approaches a
higher, unrepressed
steady-state:
Xst=β /α =5.
Dynamics of negative autoregulation using logic
pp
approximation
¾ The response time T1/2 can be found using the linear
approximation of X: X (t ) = β t ,
kd
, then,
let X (t ) = β t =
2
kd
(n.a.r)
T1 / 2 = t =
.
2β
¾ The stronger the maximal promoter activity, the shorter the
response time. Negative autoregulation can therefore use a
stronger promoter to give an initial fast production and then use
autoregulation to stop the production at the desired steady
state — Uiri Alon.
¾ Evolution can easily tune β and kd independently by changing
the affinity of RNAP and X to DNA, respectively.
Comparison of the design of simple regulation
g
autoregulation
g
circuits
and negative
¾ To make the comparison meaningful, we assume that the two
designs have as many the same parameters as possible. This
is called mathematically controlled comparison.
comparison
¾ For our comparison, let’s assume both regulations
1 achieve
1.
hi
th same steady
the
t d state
t t llevels,
l Xst, which
hi h d
determine
t
i
the optimal function of a protein.
2 have the same degradation/dilution rate
2.
rate, α.
α
¾ For a simple regulation, the steady state level is Xst=βsimple/α,
for a negative autoregulation,
autoregulation Xst= kd.
In order for them to have the same steady state levels, let
k d = β simple / α .
Comparison of the design of simple regulation
and negative autoregulaton circuits
¾ The response time for the simple regulation is:
T1simple
/2
=
ln 2
α
,
That for a negative autoregulation is:
)
T1(/n.a.r
2
β simple
kd
=
=
. ( k d = β simple / α )
β
2β
2αβ
¾ The ratio of the response times for the two designs is:
)
T1(/n.a.r
2
T1simple
/2
β simple
i l
=
.
2 β ln 2
¾ Th
Thus a negative
ti autoregulation
t
l ti can b
be severall ffold
ld ffaster
t th
than
a simple regulation design by increasing β.
Comparison of the design of simple regulation
g
autoregulation
g
circuits
and negative
The parameters β=5, α =1, βsimple=1 are used
1.5
X/Xst
negative auto-regulation
auto regulation
1
simple regulation
0.5
0
From Figure. 3.4 of Uri Alon
0
0.2
T1/2 (nar)
0.4
0.6
0.8
T1/2 (simple)
1
1.2
1.4
1.6
1.8
time αt
2
Robustness of negative autoregulation
¾ Robustness of negative autoregulation of the steady state level
of expression with respect to fluctuations of the rate of
production β.
I a simple
In
i l regulation,
l i
Xst is
i lilinearly
l d
dependent
d
on β,
X st = β / α ,
Thus, a change in β leads to a proportional change in Xst.
¾ In contrast, in a negative autoregulation,
Xst = kd,
which is not dependent on the production rate β, and kd is
much more stable than β.
This is because the value of kd is mainly determined by the
chemical bonds between X and its DNA sites
sites, and number of
binding sites in the promoter, these parameters vary much less
from cell to cell than the production rate.
Negative autoregulation of gene transcription
¾ If we model the input function as a decreasing Hill equation
equation,
β
dX
β
, then,
=
− aX ,
fX (X ) =
X n
X n
dt
1+ ( )
1+ ( )
kd
kd
If X is a strong repressor,
n
n dX
n +1
X
=
β
k
−
α
X
,
d
X/k >> 1, then,
dt
d
dX βk d n
=
− αX ,
n
dt
X
At steady state, dX/dt = 0,
βk d n
X
n
= αX , αX
n +1
n
= βk d ,
βk dn 1 /( n +1)
)
.
X st = X = (
α
n
dX
β
k
Xn
= ( d − X n +1 )α ,
dt
α
dX n +1
= ( X stn +1 − X n +1 )α ,
( n + 1)dt
d ( X stn +1 − X n +1 )
= −( n + 1)αdt ,
n +1
X st − X
ln | X stn +1 − X n +1 |= −( n + 1)αt + C ,
Negative autoregulation of gene transcription
n +1
X st
−X
n +1
=e
− ( n +1)αt +C
= C' e
− ( n +1)αt
,
When t = 0, X = 0, therefore, C' = X stn +1 , and
X = X st (1 − e
−( n +1)αt 1 /( n +1)
)
.
¾ The response time T1/2 can be found by setting X=Xst/2:
X st
= X st (1 − e −( n +1)αt )1 /( n +1) ,
2
−( n +1)
−( n +1)αt
2
= 1− e
,
e −( n +1)αt = 1 − 2 −( n +1) ,
− ( n + 1)αt = ln(1 − 2 −( n +1) ),
)
t = T1 / 2
1
1
ln((1 − n +1 ),
=−
( n + 1)α
2
1
2 n +1 − 1 −1
ln(( n +1 ) ,
=
( n + 1)α
2
1
2 n +1
ln n +1 .
=
( n + 1)α 2 − 1
¾ The response time decreases with n and α.
Comparison of the response times of
g
and simple
p regulation
g
circuits
autoregulation
¾ The response time T1/2 for a simple regulation is:
T1simple
/2
=
ln 2
α
,
that of a negative autoregulation is:
T1(/n2.a.r )1 / 2
2 n +1
1
=
ln n +1 .
( n + 1)α 2 − 1
the ratio the two type of regulation is:
T1(/n2.a.r )
T1simple
/2
2 n +1
1
=
ln n +1 .
( n + 1) ln
l 2 2 −1
For n = 1, 2, and 3, the ratio = 0.2, 0.06, 0.02, respectively.
¾ Thus, negative autoregulation can accelerate the response time
dramatically.
Comparison of the response times of
g
and simple
p regulation
g
circuits
autoregulation
X / Xstt 1
Negative
Auto-regulation
X / X st = (1 − e − ( n +1)αt )1 /( n +1)
0.8
X / X st = 1 − e
Simple
regulation
0.6
X / X st = 1 − e − ( α − β1 ) t
−αt
Positive
Auto-regulation
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
cell-generations [log(2)/α ]
4
4.5
5
From Figure. 3.5 of Uri Alon
Experiment on negatively auto-regulated and
py g
g
genes
simply-regulated
ffree reprressor co
oncentrattion
[[normaliz
zed]
¾ Thus, the real-life
negative
autoregulation system
shows all the
properties of an
isolated negative
g
circuit,,
autoregulation
even through it is
embedded in the
entire interaction
network of the cell.
X / Xst
¾ The experiment used green-fluorescent protein fused to the
TetR repressor as a reporter and fluorescence intensity was
measured on growing E.
E coli cells.
cells
Source: Rosenfeld, Elowitz, Alon, JMB 323:785 2002
1
0.8
06
0.6
0.5
negative
autoregulation
1- e -α t
-2α t 1/2
(1-e )
no negative
autoregulation
- t
(1-e α )
0.4
Modeled with a Hill input
function with n=1
0.2
0
0 0.21 0.5
1
1.5
2
2.5
cell cycles
Time [cell
generations]
3
Positive autoregulation
¾ Among the 40 self-edges in the known transcription network in
E. coli, 6 are positive autoregulation:
+
X
that is, the transcription factor X activates its own expression
when it binds its own promoter
promoter.
kd
X
¾ Assume that the promoter of X has a basal expression level β,
and the dynamics of X can be modeled by the following linear
equation:
dX
= β + β1 X − αX .
dt
Positive autoregulation
¾ Let’s solve this equation
q
by
y variable separation:
p
dX
= β + β1 X − αX ,
dt
dX
= β − (α − β1 ) X ,
dt
β
At steady state, dX/dt = 0, X st =
.
α − β1
dX
= dt ,
β − (α − β1 ) X
d ( β − (α − β1 ) X )
= −(α − β1 )dt ,
β − (α − β1 ) X
ln | β − (α − β1 ) X |= −(α − β1 )t + C ,
Positive autoregulation
ln | β − (α − β1 ) X |= −(α − β1 )t + C ,
β − (α − β1 ) X = Ce −(α − β1 )t ,
When t = 0, X (0) = 0, therefore, C = β ,
X=
β
(α − β1 )
(1 − e −(α − β1 ) t ),
= X st (1 − e −( α − β1 )t ).
¾ To find response time T1/2, let X
X=X
Xst/2,
/2
X st
= X st (1 − e −( α − β1 ) t ),
2
1
−( α − β1 ) t
e
= ,
2
ln 2
ln 2
( p.a . r )
T1 / 2
=t=
>
, if β1 > 0.
α − β1
α
Positive autoregulation
¾ The ratio of response times for positive autoregulation over
simple regulation is:
=
α ln 2
( α − β1 ) ln 2
¾ Thus positive autoregulation has an effect
that is opposite to that of
negative auto-regulation.
¾ The former slows
response time, whereas
the latter speeds response
times.
From Figure. 3.5 of Uri Alon
α
=
( α − β1 )
> 1 when β1 > 0.
1
X / Xst
( p.a . r )
T1 / 2
T1simple
/2
08
0.8
Negative
Auto-regulation
Simple
regulation
0.6
Positive
Auto-regulation
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
cell-generations [log(2)/α ]
4.5
5
Positive autoregulation
¾ The delayed
y response
p
of p
positive autoregulation
g
can be useful
in a process that takes a relative long time such as
developmental process, where a protein produced in an early
stage is used in a late stage
stage.
¾ Under strong positive autoregulation, in which β1 > α, it can lead
to unchecked production of X in the model.
X = X st (1 − e − ( α − β1 )t ).
¾ IIn reall systems, this
hi instability
i
bili will
ill b
be lilimited
i db
by other
h ffactors
such as saturation of other resources, the X is locked in an ON
state.
¾ In addition, locking X in an ON state of high expression may
occur even after its activating input vanishes. This is a useful
property in the developmental process to lock a cell in its
differentiated state.