Stochastic models for gene expression

tentchoirAI and Robotics

Nov 15, 2013 (3 years and 6 months ago)

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Stochastic models for gene expression


Ovidiu Radulescu, DIMNP UMR 5235, Univ. of Montpellier 2





Colloque franco
-
roumain mathématiques appliquées, Poitiers 29/08/10

Summary






Motivation : fluctuations in gene networks



Stochastic models for gene expression



Application to biology: fluctuome


1.

Fluctuations in gene
networks

From genes to gene networks

From genes to proteins

Interactions between genes,

proteins, metabolites

Gene networks, feed
-
back

Fluctuations in molecular networks

Epigenetic variability


Which is the origin of fluctuations?

Is there some order/logics in randomness?

Questions to answer

Becksai et al EMBO J 2001:

S.cerevisiae

Ozbudak et al Nature
2002: one promoter noise
in B
.subtilis

Cai et al. Nature 2006:

beta
-
gal operon


Intermittent protein production

How to tame fluctuations? either all numbers large, or all
small numbers fast


The central limit theorem and the law of large numbers
:
if the number of molecules is large for ALL molecular
species, then the fluctuations are small and Gaussian;
dynamics is close to deterministic


The averaging theorem
: if a slow deterministic system receives rapid
random fluctuations as input then the output has small fluctuations

P

G

k
1

k
2

Fast

Noise in multiscale networks with inverted time hierarchy

Hybrid noise : discrete variations of low numbers species, continuous variation
punctuated by jumps or switching of large numbers species

Low and large numbers : a broad distribution
of abundances from a few to 10
4

per cell

Fast and slow processes : from 10
-
3

to 10
4

s

Multiscale


Inverted time hierarchy: some processes involving low numbers are slow

P

G

k
1

k
2

Slow

2.

Stochastic models for gene
expression

Delbrück
-
Rényi
-
Bartolomay approach

Two main assumptions
:



Reactions are independent
;



Transport is instantaneous.



Dynamics variables are numbers of molecules

of different species



All species evolve by discrete jumps separated

by random waiting times

Which reaction?

Which time?

Generate exponential variable of
parameter (k
1
A+k
2
B)

Gillespie’s algorithm

Disadvantages
:



Time costly



Analytical solutions of the

Chemical Master Equation are rarely

available

Continuous time Markov jump processes








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q
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,.)
(





are n chemical species


jump vector

r
n
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n
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,







n
in
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i
A
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...
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1





nr
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)]
(
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[
)
(






nr
j
j
j
i
i
X
V
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q
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)]
(
)
(
[
)/
(
)
(
intensity

distribution of jumps

biochemical reaction

jump probability

n
Z
X

is the state


Hybrid approach

The hybrid stochastic dynamics is a piecewise deterministic Markov process

Assumption
:



Law of large numbers can be applied to

continuous variables


Gaussian noise neglected



2 types of dynamical variables: discrete and
continuous


Discrete variables undergo random jumps,
continuous variables follow ODE dynamics

Which time?

Generate exponential variable of
parameter (k
1
A+k
2
B)
-
1

Hybrid algorithm

Advantages
:



Slight improvement of execution time


Emphasize the hybrid nature of
fluctuations



Analytical solutions more easily available

Continuous variable : ODE

Discrete variable : Gillespie dynamics

2
3
k
P
k
dt
du



P
k
dt
du
3







3
2
1
1
,
*
*
*,
k
k
k
k
P
P
G
G
G
G
switching

Partial fluid approximation:partition

Given a pure jump model, find its hybrid approximation

Species partition: discrete, continuous

)
,
(
C
D
X
X
C
DC
D
R
R
R


Reaction partition

Discrete transitions

Contributions to continuous flow

coupling

intrinsic






3
2
1
1
,
*
*
*,
k
k
k
k
P
P
G
G
G
G
Coupling between discrete and continuous
only if super
-
reactions of type 1: fast,
change mode XC, rates depend on XD

switching

Partial fluid approximation : expansion

)
/
,
(
~
)
,
(
V
X
x
X
X
X
X
C
C
D
C
D



Rescaling

1
st

order Taylor
expansion in











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(
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(
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V
t
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p
X
t
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t
p
c
D
R
i
c
D
i
D
i
D
D
i









)
,
/
,
(
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,
(
t
V
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p
V
x
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V
C
i
DC
C
i
c
D
R
i
c
D
i








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,
/
,
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t
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V
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/








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i





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t
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p
x
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f
c
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xc


V
x
V
V
x
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V
x
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f
dt
dx
C
C
i
DC
C
i
R
i
c
i
R
i
c
D
i
c
D
c
)/
(
)/
,
(
)
,
(









Chemical Master
Equation

Hybrid Master/Fokker
-
Planck equation

Switching

Drift

of a PDP with switching

Coupling if super
-
reactions of type 1

Partial fluid approximation : breakage

Breakage+drift part in the
master/Fokker
-
Planck equation

V
x
V
V
x
X
V
V
x
X
V
x
X
f
C
C
i
DC
C
i
C
i
R
i
c
i
R
i
c
D
i
S
i
c
D
i
c
D
)/
(
)/
,
(
)/
,
(
1
)
,
(
2














super
-
reactions of type 2:

very fast, act on XC, rates depend on XD

breakage



...
)
,
,
(
)
,
(
1
)
,
~
(
)
~
(
)
,
~
(











t
x
X
p
x
X
V
t
X
p
X
t
X
t
p
c
D
D
R
i
c
D
i
D
i
D
i





fast back to ground discrete reactions

D
R

D
X
C
X
...
)]
,
~
(
)
(
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(
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~
(
))
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t
X
p
dy
y
t
y
x
X
p
X
t
X
p
X
g
x
t
t
X
p
D



breakage frequency


breakage size distribution

Hierarchical model reduction

Reduction of stochastic models
:


Linear sub
-
networks


Needs multi
-
scaleness


Works both for deterministic

and for stochastic models



Eliminate inessential details: pruning


Group together reactions: lumping


Zoom in and out complexity


Drastic decrease of simulation time


Which time?

Generate exponential variable of
parameter (k
1
A+k
2
B)
-
1

Dominance and pruning

Un
-
broken cycle: averaging

A
1

A
2

A
i

A
j

A
t

k
2

k
1

k
lim

k
i

k

k<<k
i

A
j

A

kk
lim
/k
i

i
j
j
i
i
k
k
k
k
p
lim
1
1



Inverted time hierarchies



Stochastic leaks of low mass un
-
broken cycles produces slow
transitions of discrete variables, thus inverted time hierarchies



A un
-
broken cycle can be a source of hybrid noise

A
1

A
2

A
i

A
j

A
t

k
2

k
1

k
lim

k
i

k

k<<k
i

A
j

A

kk
lim
/k
i

k
k
k
k
i

lim


Low intensity shot noise is amplified to bursts by fast transitions of
the continuous variables


A
j

A

B

kk
lim
/k
i

kd

kp

Burst amplification kp>>kd

Continuous variable

Discrete variables

Prot

FoldedProt

D

D.R

D.RNAP

TrRNAP

RBS

Rib.RBS

ElRib

Ø

k1=400

km1=1

km2=10


k2=6

k3=0.1

k4=0.3

k6=60

km6=2.25

k7=0.5

k8=0.015

k10=1e
-
5

k11=1e
-
5

k9=1.3e
-
4

Ø

Ø

k5=0.3

model1

Prot

FoldedProt

D**

TrRNAP

RBS

Rib.RBS

ElRib

0.3

60

2.25

0.5

0.015

1e
-
5

1.3e
-
4

Ø

1.5e
-
4

Prot

FoldedProt

RBS*

ElRib

D**

TrRNAP

0.5

0.015

1e
-
5

1.3e
-
4

Ø

model2

model3

1.5e
-
4

0.3

Ø

Ø

0.3

1e
-
2

Hierarchical model

reduction

Inverted time hierarchy

Continuous variables

Burst amplification of ElRib

Hierarchical model reduction: gain in
computational power

Crudu et al BMC Systems Biology 2009

A simple model : solution of the hybrid master/Fokker
-
Planck equation



a=k1/

㈠㨠湵浢敲e映瑲慮獣物r瑩t渠楮楴楡瑩i湳n灥爠灲t敩渠汩fe瑩浥



b=k2/

1

㨠湵浢敲:映⁰t敩湳n灥爠扵rst

)]
,
(
)
/
exp(
1
)
,
(
))
,
(
(
[
)
,
(
0
2
t
x
ap
dy
b
y
b
t
y
x
p
a
t
x
xp
x
t
x
t
p
x











)
/
exp(
1
)
(
b
x
a
x
x
p



Gamma steady state
distribution

1

a
1

a
3.

Application to biology:
fluctuome

Biological system: central carbon metabolism in B.subtilis

Bacteria are grown in two experimental conditions:
glucose
-
rich and malate
-
rich medium

Experimental method: number and brightness

Poissonian field F map

Histograms

Part of the fluctuome: CggR, gapB, CcpN

CggR

gapB

CcpN

Glucose

Malate

Glucose

Malate

Glucose

Malate

Matthew Ferguson, CBS

10min

60min

MdGFP

D

D.R

RBS

ElRib

k3

k7

k8

kdeg

Ø

Ø

k5

k3’

TrRNAP

k4

k1on

k1off

k4’~0

Ø

k6

RNAP

pRNAP
-
R

RNAP

k1off

k1on

CggR

CggR

TrRNAP

paused

D.R.RNAP

k6

RBS

k4’

RNAP

RNAP

RNAP

RNAP

RNAP

RNAP

TrRNAP

D.RNAP

RBS

bursting

RNAP

RNAP

RNAP

RNAP

CcpN

k3 or k3’

gapB

A

C

TrRNAP

D.RNAP

RBS

bursting

CggR

k4

RNAP

RNAP

RNAP

RNAP

RNAP

RNAP

RNAP

B

D

pcggR

pgapB

RNAP

Take home message



The origin of noise in multiscale molecular network is the inverted
time hierarchy; noise is hybrid



Hierarchical model reduction unravels functional structure:
unbroken cycles, burst amplifiers, integrators



Noise carries information about interactions : fluctuome could be
an important tool




Acknowledgements

Nathalie Declerck, Matthew Ferguson, Catherine Royer

CBS Montpellier


Alina Crudu, Arnaud Debussche, Aurelie Muller

IRMAR Rennes, ENS Cachan


Alexander Gorban

University of Leicester