modeling and analysis of gene regulatory networks · 2021. 2. 1. · – the regulatory genome...
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Modeling and analysis of
Gene Regulatory Networks
Hamid Bolouri
Division of Human Biology
Fred Hutchinson Cancer Research Center
http://labs.fhcrc.org/bolouri
Woods Hole, October 2011
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Outline: (new approach proposed by Eric Davison)
- Take a specific set of biological observations
- Explore how various computational approaches can help develop insights
• Papers
– Laslo et al, Cell 2006, 126:755–766
– Spooner et al, Immunity 2009, 31:576–586
– Cherry and Adler, J. Theoretical Biology 2000, 203:117-133
– Saka & Smith, BMC Developmental Biology 2007, 7:47.
– Chickarmane, Enver & Peterson PLoS Computational Biology 2009 5(1): e1000268.
• Further reading
– The regulatory Genome (Eric Davidson 2006)
– An Introduction to Systems Biology (Uri Alon, 2006)
– Computational Modeling of Gene
Regulatory Networks – a Primer (Hamid Bolouri, 2008) – R in Action (Robert Kabacoff, 2011)
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Novershtern et al,
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Lab Exercises: See handout for instructions.
Search for tag “MBL”
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Stefan Materna & Paola Oliveri, Nature Protocols 3, -1876 - 1887 (2008)
Does network structure explain all observations?
Discovery
Analysis
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Novershtern et al,
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G1 G2 G3 G4 G5 G6 G7 G8 G9 G10
time1 0.082377 0.38766 0.61257 0.471963 -0.07442 -0.11739 0.51039 0.006912 0.011694 -0.14743
time2 0.710007 0.175795 0.035997 0.332428 0.499605 0.386174 0.171675 0.564456 0.500018 0.076234
time3 -1.0385 -0.83347 -0.92109 -0.81229 -1.35493 -1.01501 -0.74898 -0.6342 -0.69178 -0.9943
time4 -1.19125 -1.354 -0.73608 -1.03199 -1.15046 -0.81708 -1.22163 -0.88932 -0.41835 -0.5339
1 myData <- as.matrix(read.table(inFile,header=TRUE,sep=",",row.names=1)) 2 if (clusterBy=="genes") myData <- t(myData) 3 myData <- sweep(myData,1,apply(myData,1,mean),"-") 4 myData <- sweep(myData,1,apply(myData,1,sd),"/") 5 library(RColorBrewer) 6 library(gplots) 7 print(heatmap.2(myData[,dim(myData)[2]:1],col=brewer.pal(11,"RdYlGn"), trace="none", dendrogram="row", scale="column", Colv=FALSE,Rowv=TRUE,key=TRUE, margins=c(10,10)))
Search scripts & data for tag term “MBL” at CRdata.org
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Observations
• PU.1 & CEPBa expression levels are both low in progenitors
• Progenitors express low levels of both macrophage and neutrophil associated genes
• PU.1 expression is required for macrophage specification
• CEPBa is required for neutrophil specification
• Ratio of PU.1/ CEPBa determines cell fate
• In mature macrophages and neutorphils, PU.1 and CEBPa are both expressed at high
levels & co-regulate cell-type-specific genes
• Multiple cytokines (G-CSF etc) act upstream of PU.1 and CEBPa
• PU.1 upregulates Egr2 and Nab2 expression (which co-regulate macrophage genes)
• Egr2 and Nab2 co-repress Gfi1 expression while Gfi represses Egr2 transcription
• Not included in model
– PU.1 is autoregulatory in macrophages
– CEPBa regulates PU.1 expression
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PU.1
Myeloid cells
B cells
T cells
Hoogenkamp et al,
http://genomequebec.mcgill.ca/PReMod Blanchette lab, McGill
GATA
GATA
PAX
Bold ChIP Light EMSA
1
Leddin et al, Blood 2011, 117(10):2827-2838. Additional CRMs at -8Kbp & within introns.
Runx1
IKAROS
-14Kb -10Kb -9Kb -1Kb
Zarnegar, Chen & Rothenberg,
Enhancer
PU.1 exor GFi (Spooner ’09)
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Logic simulation: need multiple value levels & memory
Chalk board discussion:
1. Do mutually repressing genes always result in 2 mutually excluded states?
2. How do the threshold parameters of a logic model relate to biochemistry?
http://gin.univ-mrs.fr/GINsim/
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distance traveled = current position – starting position = speed x (time)
speed = [(position at time t2) – (position at time t1)] / (t2 – t1)
speed = time1time2
position1position2
d(time)
)d(position
|(time2 - time1) 0
2
2
d(time)
(position)d
d(time)
d(speed)acceleration =
Using Ordinary Differential Equations (ODEs) to model dynamics
speed
time
t1 t2 t3 t4 t5
Chalk board discussion: Integration & differentiation
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Analysis of dynamic network behavior
Consider A
k1 k2
We can write
Which has a simple analytic solution:
(assuming [A](t=0) = 0)
[A]
t
)1()(.
2
1 2 tke
k
ktA
Akkdt
dA.2 1
2
1
k
k
initial slope=k1
[A] 0 max
2
1
k
k
0][
dt
Ad0
][
dt
Ad
Chalk board discussion: Stable and unstable steady states
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GFi
PU.1
C/EBPa
Egr
NAB2
Macrophage genes Neutrophil genes
GFi
PU.1
C/EBPa
Egr
NAB2
Macrophage genes Neutrophil genes
Laslo et al, Cell 2006, 126:755–766 GFi
PU.1
C/EBPa
Egr
NAB2
Macrophage genes Neutrophil genes
GfiEgrCEBP
CEBP
dt
Gfid
EgrGfiPU
PU
dt
Egrd
PUGfi
e
dt
PUd p
4
4
4
1
1.
1
.)(
1
1.
11
1.)(
11
)1(
a
a
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Steady states in feedback networks
stable steady state 1
stable steady state 2
unstable steady state 1
Chalk board:
mono vs bistability
stability, homeostasis
mediocristan
polarized/differentiated states,
extremestan
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Laslo et al, Cell 126, 755–766, August 25, 2006
developmental
trajectory for
macrophages
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Graph: a convenient graphing tool. Free at http://www.padowan.dk/graph/
GfiEgrCEBP
CEBP
dt
Gfid
EgrGfiPU
PU
dt
Egrd
PUGfi
e
dt
PUd p
4
4
4
1
1.
1
.)(
1
1.
11
1.)(
11
)1(
a
a
What do the fractional terms imply ?
Why are all Khalf=kd=1 ?
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Where do the production rate functions come from? 1. Do the fractions in the rates have biochemical meaning? 2. Why use (fraction1)*(fraction2) form for the rates? - a side-trip into modeling the regulation of transcription
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mRNA NTPs degradation
AAs Protein
Y
degradation
A simple 2-step ODE model of transcription and translation
dt
d[mRNA]
dt
d[P]
= kt.Y – kdm.mRNA kt is the maximal rate of transcription
= ks.mRNA – kdp.P ks is the protein synthesis rate/mRNA concentration unit
Y
mRNA
Protein
time
Pss = (ks/ kdp).(kt/kdm).Y
Pss ∝ Y (Y is usually set to the Fractional Saturation of TF complex on its DNA binding site)
At steady-state:
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Gene NTPs, AAs Protein
Y
An even simpler 1-step ODE model of gene expression
dt
d[mRNA]
dt
d[P]
= kt.Y – kdm.mRNA = 0
= ks. – kdp.P
If we assume rapid mRNA equilibrium, then:
Yk
kmRNA
dm
tss .
Yk
k
dm
t .
PkYk
kk
kkL
dg
gdm
ts
..
.
dt
dP
then et
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G
A P
mRNA
mRNA transcription
synthesis P
Activator (A)
degradation
degradation
Fractional DNA occupancy by one activating factor
PkmRNAkdt
PdmRNAkYk
dt
mRNAd
as before:
AK
AYalently: or equiv
AK
AKY
DNAAKDNA
DNAAK
DNAADNA
DNAAYOccupancyDNAFractional
dpsdmAt
DAA
A
AA
A
AA
..)(
..)(
.1
.
..
..
]:[][
]:[
increasing KA
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G
R P
mRNA mRNA transcription
synthesis P
Repressor (R)
degradation
degradation
Transcriptional repression
mRNAkRK
kdt
mRNAd
RKDNARKDNA
DNAY
RK
RKY
dAR
t
ARARRnot
AR
ARR
..1
1.
)(
.1
1
..
.1
.
)(
Increasing KAR
(fraction of DNA occupied by R)
(1-YR) = fraction of DNA not occupied by R:
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DNA occupancy by 2 factors
D
DB DAB
DA ka
k-a
kba
k-ba
kab k-ab kb k-b D
A P
mRNA
B
At equilibrium:
bbaaabbbaaab
bbab
b
ba
ba
ba
bababa
aaba
a
ab
ab
ab
ababab
b
b
a
aaa
KKKKDBAKKDBAKK
DBAKKDBk
kA
k
kDBA
k
kDABDABkDBAk
dt
DABd
DBAKKDAk
kB
k
kDAB
k
kDABDABkDABk
dt
DABd
DBk
kDBDA
k
kDADAkDAk
dt
DAd
dt
DBd
dt
DAd
dt
DABd
..........
........].[.]0].[].[.)(
........].[.]0].[].[.)(
..]..]].[..)(
0)(
,0)(
,0)(
[
[
[ :l ikewise , [
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DNA occupancy by 2 factors
D
DB DAB
DA ka
k-a
kba
k-ba
kab k-ab kb k-b
D
A P
mRNA
B
In general: KA.KAB = KB.KBA = Kq.KA.KB
Where Kq = cooperativity factor
= 1 if A and B bind independently
DABkDABkdt
DABd
DAkDAkdt
DAd
abab
aa
...)(
...)(
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qBABA
qBABA
.[B].K.[A] .KK.[B] K .[A] K
.[B].K.[A] .KK.[B] K.[A] KY
1
ionsconfiguratDNA all of levels statesteady the of sum
states activating of level statesteady occupancy
DAB DB DA D
DAB DB DA Y
DBAKKKDAKBKDABKDABk
kDAB
DBKDBDAKDAk
kDA
BAqAABAB
ab
ab
BA
a
a
.............
).. :(likewise ....
If A & B activate independently:
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NNDA
N
NA
NA
AK
A Y or
.A)(K1
.A)(K Y
For 1 TF
multimer
qRARA
A
.[R].K.K .[A]K .[R]K .[A]K1
.[A]K
Y
homodimer) afor (Likewise
1 ,
[A]
[A] Y :lyequivalentor
.[A]1
.[A] Y : thenK.KLet
.[A].KK1
.[A].KK Y :sites ecooperativ- 2For
.[B].K.K .[A]K1
.[B].K.K .[A]K Y factors ecooperativhighly 2For
A
dA22
dA
2
22
A
22
A22
Aq
2
Aq
22
Aq
qBA
qBA
A
For 1 repressor, 1 activator
n=1
n=3
Chalk board: (1-occupancy) for a repressor multimer
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50%~AB Ygives which, /nucleusM) 6-(1.7x10 molecules 3000~B A, 510~RBKRAK
, 2M7E~ND 15L4E ~ volume nuclear sites, 1.6E8~ND
:are numbers typical ,purpuratus S. urchin sea the of cellsembryonic in example, For
.K.B.KA.K.DB.K.DA.KB.DA.DD
.K.B.KA.KY
:gives bottom and top the from D cancelling
D
.K.B.KA.K.DB.K.DA.KB.DA.DD
D
.K.B.KA.K
Y
DNA) boundly specifical-non andDNA naked for terms includes rdenominato (note
D
.K.B.KA.K
D
B.K
D
A.K
D
B
D
A1
D
.K.B.KA.K
Y
:B andby A binding ecooperativ For
state)steady at ionsconfiguratDNA possible all of n(proportio
jointly B &by A occupied regions bindingDNA of proportionY
then sites, bindingspecific their of B &by A occupancy joint YLet
D
K.KKK
D
1K ,
D
1K
[A] ][AD
][A].[D
][ADK
then B,& Afactors for sites bindingDNA specific of number D Let
B & A facors for sites binding DNAspecific -non of number D Let
K
KK Let
qRBRANRBNRANN2N
qRBRAAB
2N
2N
qRBRANRBNRANN2N
2N
qRBRA
AB
2N
qRBRA
N
RB
N
RA
NN
2N
qRBRA
AB
AB
AB
N
RAm_NAequilibriuRAm_SAequilibriu
Nm_NBequilibriu
Nm_NAequilibriu
N
N
Nm_NAequilibriu
S
N
ficm_nonspeciequilibriu
m_specificequilibriuR
Calculating promoter occupancy as a function of
specific and non-specific DNA binding rates for two factors
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50%~AB Ygives which, /nucleusM) 6-(1.7x10 molecules 3000~B A, 510~RBKRAK
, 2M7E~ND 15L4E ~ volume nuclear sites, 1.6E8~ND
:are numbers typical ,purpuratus S. urchin sea the of cellsembryonic in example, For
.K.B.KA.K.DB.K.DA.KB.DA.DD
.K.B.KA.KY
:gives bottom and top the from D cancelling
D
.K.B.KA.K.DB.K.DA.KB.DA.DD
D
.K.B.KA.K
Y
DNA) boundly specifical-non andDNA naked for terms includes rdenominato (note
D
.K.B.KA.K
D
B.K
D
A.K
D
B
D
A1
D
.K.B.KA.K
Y
:B andby A binding ecooperativ For
qRBRANRBNRANN2N
qRBRAAB
2N
2N
qRBRANRBNRANN2N
2N
qRBRA
AB
2N
qRBRA
N
RB
N
RA
NN
2N
qRBRA
AB
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Bolouri H & Davidson EH, PNAS, 5 August 2003, 100(16):9371-9376.
Example model fit to sea urchin data – with added transcription initiation step
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22
2
1
22
11
1
2
11
22
12
22
11
21
11
1
1
1
1
.1
1
.1
1
.G) - k
K
G
.( k dt
dG
.G) - k
K
G
.( k dt
dG
.G) - kG K
.( k dt
dG
.G) - kGK
.( k dt
dG
d
NDiss
Nt
d
NDiss
Nt
dNNA
t
dNNA
t
:as writere
Reducing the number of unknown parameters – a technique to simplify model exploration
22
1
22
11
2
11
1
1
1
1
.G) - kG
.( k dt
dG
.G) - kG
.( k dt
dG
KG
dNt
dNt
Diss
then , of units in measure weIf
Mutual repression
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gene 2 expression
gen
e 1
exp
ress
ion
22
1
22
11
2
11
1
1
1
1
.G) - kG
.( k dt
dG
.G) - kG
.( k dt
dG
dNt
dNt
Mutual repression
1
1
d
t
k
k
cooperativity factor = 2
cooperativity factor = 3
cooperativity factor = 4
2
2
d
t
k
k
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For exploratory exercises, see the Lab Notes handout
Chalk board discussion: - change-direction arrows - how inputs set the state
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x1=off,x2=on
Inputs=0
x1=on,x2=off
Inputs=0
Controlling the state of a mutual repression switch with 2 independent activating inputs
x1=off,x2=on
Input1=0, Input2=0.5
x1=high, x2=low
Input1=0.75, Input2=0.5
k1,k2=5
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Cf. Memory term in our earlier logic model But what feature of the system creates the memory?
Hysteresis
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Two autoregulatory positive feedback loops maintain PU.1
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Steady states in feedback networks
gene G conceptual model: .Gk
GK
G.k
dt
dGdt
rate of clearance
rate of production
rate of clearance > rate of production G will decrease over time
rate of production > rate of clearance G will increase over time
At Steady state,
production rate = clearance rate G
rate
stable steady state
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Steady states in feedback networks
gene G
conceptual model: .GkGK
G.k
dt
dGdNN
N
t
1 2 3 4
0.25
0.5
0.75
1
1.25
G
rate
rate of clearance
rate of production
rate of clearance > rate of production G will decrease over time
rate of production > rate of clearance G will increase over time
stable steady states
unstable steady state
stable steady state 1
stable steady state 2
unstable steady state 1
At Steady states,
production rate = clearance rate
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gene G
Two ways of providing input:
.GkiK
i.k
GK
G.k
dt
dGdnn
i
n
t2NNG
N
t1
gene G
independent
activating
input
Input is (or acts on) the
same protein as feedback
(1) (2)
add a second occupancy term:
.Gki)GK
i)(G.k
dt
dGdNN
N
t
(
add to G in the occupancy term:
rate of clearance
rate of production
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input
mRNASS or PSS Auto-regulation: hysteresis &
bistable lock-on switches
increasing KDiss
G
rate
reducing input
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And:
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PU
.1 a
t st
ead
y st
ate
Gata1 at steady state
Where might the nonlinearities come from?
Additional feedback loops:
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Where might the nonlinearities come from?
Facilitation by ‘pioneer’ factors
Simple model:
where D=DNA, T= transcription factor,
at steady state:
DTTTDTTD
2
1
2
1
2
1
2
2
1
2
11
].[
][][1
].[
][
].[
]].[[
.
]].[[ ,
]].[[
DqD
Dq
DqDqD
KK
T
K
T
KK
T
DTTDTD
DTTY
KK
TD
KK
TDTDTT
K
TDDT
sigmoid DNA
occupancy curve:
Y
T
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At steady state:
=
R RP
S
If 1, 21 T
m
T
m
R
K
R
K
9.0, 21 T
m
T
m
R
K
R
K
1.0, 11 T
m
T
m
R
K
R
K
R RP
S
response is sigmoidal
RPP
r rP
S
R RP
Other cases:
Where might the nonlinearities come from? Se
e al
so h
ttp
://e
n.w
ikip
edia
.org
/wik
i/G
old
bet
er-K
osh
lan
d_k
inet
ics
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steady state locus of A
steady state locus of B
A
B
Time (A.U.)
Portrait of the state space
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2x2 patch of activated cells at time
zero
Activity dies out
Simulation of 100X100 array of cells with autocrine signaling pathway.
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10X10 patch of activated cells at time
zero
Activity restricted to one patch
Simulation of 100X100 array of cells with autocrine signaling pathway.
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There is a minimum cluster-size
requirement for activation
Activity stabilizes
Activity dies out
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Randomly distributed 1% of cells
activated at time zero
Activity dies out
Simulation of 100X100 array of cells with autocrine signaling pathway.
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Randomly distributed 10% of cells
activated at time zero
Activity spreads
Simulation of 100X100 array of cells with autocrine signaling pathway.
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GFi
PU.1
Ikaros
Egr
NAB2
Macrophage genes B cell genes
Spooner et al, Immunity 2009, 31:576–586
Ids
E2A
Early activator