Modelling stochastic biochemical
networks
Peter Swain
Centre for Systems Biology at Edinburgh
University of Edinburgh
Outline
1. Introduce the concept of intrinsic and extrinsic
fluctuations.
2. Modelling intrinsic fluctuations.
3. Modelling extrinsic fluctuations.
4. Unexpected consequences of extrinsic fluctuations.
One stage model of gene expression
mRNA
v0
Protein
v1
d0
v0: transcription
v1: translation
d0, d1: degradation
d1
Two types of stochasticity
Intrinsic stochasticity arises because the dynamics of the system is
stochastic.
Extrinsic stochasticity arises because the system interacts with other
stochastic systems, generating stochastic parameters.
v0 fluctuates
because it is
function of the
number of RNA
polymerases and
generates
extrinsic
fluctuations.
mRNA
v0
Protein
v1
d0
d1
Measuring stochasticity
Extrinsic
fluctuations equally
affect both network
copies.
Intrinsic fluctuations
differ between the
copies of the network.
Network output has both
intrinsic and extrinsic
fluctuations.
Other
stochastic
system
Network of
interest
Network
output
Other
stochastic
system
Copy of network
of interest
Network
output
For gene expression, we have two copies of the gene of
interest.
Extrinsic
Intrinsic
Extrinsic variables affect each copy equally
e.g. number of free RNAPs, cellular environment
Intrinsic variables are different for each copy
e.g. number of transcribing RNAPs, number of mRNAs
Extrinsic fluctuations cause correlations in the proteins expressed
from each gene. Intrinsic fluctuations uncorrelate protein levels.
cfp
yfp
time
Dominating intrinsic fluctuations
fluorescence
fluorescence
Dominating extrinsic fluctuations
time
Definitions
Define a fluctuation in an intrinsic variable, such as protein numbers,
by
If I1 is the intrinsic variable for one copy of the system and I2 is the
same intrinsic variable for the second copy, then
extrinsic
noise
correlation
coefficient
intrinsic
noise
normalized
difference
the square of the total noise is the coefficient of variation of the
output of the network
total
noise
coefficient
of
variation
Experimental measurements of stochasticity in
bacteria
cfp
yfp
Intrinsic noise depends on expression level
yfp
cfp
WT (RPR37) + IPTG
WT (RPR37)
decrease
expression
32-fold
Intensity
ηint
ηext
=1
= 0.063 (0.058-0.069)
= 0.098 (0.09-1.1)
Intensity
ηint
ηext
= 0.03
= 0.25 (0.22-0.27)
= 0.32 (0.3-0.35)
Modelling intrinsic fluctuations
Master equation approach
mRNA
v0
v1
d0
probability
of m
mRNAs
and n
proteins at
time t
Protein
d1
REPORTS
Probing Gene Expression in Live Cells,
One Protein Molecule at a Time
Because the tsr gene is highly expressed (30), a
small amount of exogenous Tsr-Venus poses
minimal perturbation to cells_ normal functions.
Western assay of induced SX4 cells showed the
presence of Venus only in the membrane fraction
1
1
1
2
1
and not in the cytoplasmic fraction, suggesting
Ji Yu, * Jie Xiao, * Xiaojia Ren, Kaiqin Lao, X. Sunney Xie †
efficient membrane localization of Tsr-Venus.
We also compared the levels of induced expresWe directly observed real-time production of single protein molecules in individual Escherichia
sion of Tsr-Venus and Venus in two strains, both
coli cells. A fusion protein of a fast-maturing yellow fluorescent protein (YFP) and a membraneunder the control of the lac promoter ESupporting
targeting peptide was expressed
under
a
repressed
condition.
The
membrane-localized
YFP
can
be
REPORTS
Online Material (SOM) Text and fig. S1^. No
detected with single-molecule sensitivity. We found that the protein molecules are produced in
notable difference was observed,
indicating measured
that
bursts, with each burst originating from a stochastically transcribed single messenger RNA
perimentally
this abundance to be
the introduction of the tsr sequence
not per cell by counting the
molecule, and that protein copy numbers in the bursts follow a geometric distribution. The
4.1 T 1.8does
molecules
change the yield of Venus production,
is individual cells under the
quantitative study of low-level gene expression demonstrates the potential of single-molecule
moleculeswhich
in È300
REPORTS
not the case for many other membrane-targeting
experiments in elucidating the workings of fundamental biological
processes in living cells.
microscope at the same time.
sequences that we tested.
Lastly, the temporal
spreadmeasured
of the expresperimentally
this abundance to
We first show the abilitysion
to bursts
detect can
single
he central dogma of molecular biology cause of its short maturation time (27). Howbe characterized
from the
auto4.1 T 1.8 molecules
per cell
by counting t
states that DNA is transcribed into mRNA, ever, it is difficult to image a single GFP or YFP Tsr-Venus fluorescent protein
molecules
ex- molecules
in È300 individual
correlation
function
of the fluctuation
in proteincells under t
(2)(t) (Fig.
Figure 1A Cshows
which is then translated into protein. Ever molecule in cytoplasm, because its fluorescence pressed in SX4 cells (Fig. 1).expression,
microscope
at the samefrom
time. 30
4C), averaged
Lastly,15thedifferent
temporalmovies.
spread of the expre
since the pioneering work on the lac operon (1), signal spreads to the entire cytoplasm by fast two diffraction-limited fluorescent
Efull from
differentspots
cell lineages
sion
bursts
can
be
characterized
(2)
È 300exponential
nm^ in fit of C (t) gives a decay from the aut
our knowledge of gene expression has come diffusion during the image acquisition time and width at half maximum (FWHM)
The single
correlation function of the fluctuation in prote
primarily from genetic and biochemical studies is overwhelmed by cellular autofluorescence. the left cell. A line cross section
of
the
fluores(2)(t) (Fig. 4C), averaged
time constant of 7.0 expression,
T 2.5 min,Ccorresponding
to
from
long
axes spread
shows ofdifferent
(2–4) conducted with large populations of cells On the other hand, single YFP fusion protein cence image along the cells_the
average
the stochastic
arrival
cell lineages
fromtimes
15 different movie
and molecules. Recently, many in vitro single- molecules on cell membranes can be detected the signal distinctly above the
autofluoThe single
exponential
C(2)(t) gives a dec
of cells_
fluorescent
reporter
proteins
within fitaofburst,
time
constant
of
7.0
T
2.5
min,
We attribute
each
molecule experiments have probed real-time (28, 29) because their diffusion is slowed. There- rescence background (Fig. 1C).despite
the fact
that the polypeptides are gen-corresponding
the average spread of the stochastic arrival tim
dynamics and yielded valuable mechanistic fore, we designed a fusion protein consisting of fluorescent peak to an individual
Tsr-Venus
erated
within molthe short
lifetime of an mRNA
of fluorescent reporter proteins within a bur
of
the
insights into macromolecules (5–8), including Venus and a membrane protein, Tsr, as the re- ecule on the basis of abrupt disappearance
We show
(SOM
that,
(tmRNA 0 1.5 min). despite
the fact
that Text)
the polypeptides
are ge
transcriptional (9) and translational (10) ma- porter for monitoring lac promoter activity. A signal upon photobleaching, under
whichtheiscondition
charac- that
is one
rate-limiting
eratedthere
within
the short
lifetime of an mRN
Figure
1D posttranslation
shows (tmRNA assembly
chineries. In order to understand the workings well-studied methylation-dependent chemotax- teristic of single molecules. step
0 1.5 min).ofWe
(SOM Text) th
for the
theshow
fusion
protein,
of these machineries in their physiological con- is receptor protein (MCP) (30), Tsr contains such a photobleaching time trace.
Had the signal under the condition that there is one rate-limiti
step for the posttranslation assembly of the fusi
texts, we set out to probe gene expression at the two transmembrane domains and is fused to arisen from multiple molecules, its disappearance
!
"2 #
$
protein,
sr
k
single-molecule level by real-time monitoring of the N terminus of Venus.
would be in multiple steps. In addition,
the
fluoð2Þ
C ðtÞ 0
1 þ ! expðj
#
" ktÞ
$
We constructed an E. coli strain SX4 in rescence intensity of each peak is consistent with
protein production in live cells.
1 j r ð2Þ
s sr 2
k
C ðtÞ 0
1
þ
expð
ktÞ
j
Gene expression is often stochastic (11–14), which a single copy of the chimeric gene tsr- in vitro measurements of purified single Venus
1jr
ð2Þs
because most genes exist at single or low copy venus was incorporated into the E. coli chro- molecules (fig. S2).
ð
where s is istheshown
average rate of the expression burst
numbers in a cell. Some genes are expressed at mosome, replacing the native lacZ gene. The
A sketch of our live-cell experiment
where of
s isTsr-Venus
the average rate
of the expression bu
the rate constant
assembly
andk is
spontaneous
high levels and others at low levels. The mRNA endogenous tsr gene of E. coli was left intact. in Fig. 2. Upon an infrequentand
and k is the rate constant of Tsr-Venus assemb
process, consisting process,
of transcription,
translation,
expression can now be tracked in a single cell
consisting of
transcription, translatio
folding, and chromophore
The maturation.
fitting
with single-molecule sensitivity (15, 16). The profolding,maturation.
and chromophore
The fitti
, T 8 min)j
of
Fig.
4C
with
Eq.
2
gives
s
0
(29
T
8
min)
of
Fig.
4C
with
Eq.
2
gives
s j1
0 (29
tein expression has been traditionally characteragreementnumber
with the of
average
number of expre
in agreement with thein average
expresized et
by al.
averages
of cell populations, in which
Yu
(2006)
sion bursts
perTcell
1.2 T 0.3 (Fig. 4A
sion bursts per cell cycle
of 1.2
0.3cycle
(Fig.of4A);
stochasticity is masked. More information is
r 0 0.7 T 0.1, consistent with the value of 0.8
r 0 0.7 T 0.1, consistent
with the value of 0.8 T
available from both the distribution of expres0.1 determined from Fig. 4B; and 1/k 0 7.0
0.1
determined
from
Fig.
and 1/k 0to 7.0
T
sion levels among a cell population (17–19)
2.5 min,4B;
corresponding
the rate-limiting
st
Fig. 3. Real-time monitoring of the expression of tsr-venus under the control of repressed
lac promoter.
2.5
min,
corresponding
to
the
rate-limiting
step
of
the
protein
assembly
process.
Considering t
and the temporal evolution of Fig.
a single
cell by monitoring of the expression
3. Real-time
control
of repressed
lac promoter.
(A) Sequenceofoftsr-venus
fluorescentunder
imagesthe
(yellow)
overlaid
with simultaneous
DIC images (gray) of E. coli cells
of the
assembly
Considering
the and translati
fast process.
transcription
(È45 bases/s)
expressing
Tsr-Venuswith
on agarose
gel pad DIC
of M9images
medium.(gray)
The cell
is 55
T 10 min
in a protein
temperatureusing fluorescent reporters (20).(A)
However,
Sequencethese
of fluorescent images (yellow)
overlaid
simultaneous
of cycle
E. coli
cells
(È15
residues/s)
rates,
we
tentatively
assign 1
fast
transcription
(È45
bases/s)
and
translation
controlled
on The
a microscope
stage.
TheTeight
frames
from time-lapse fluorescence movie S1
studies have been restricted to expressing
high expression
Tsr-Venus on agarose gel pad
of M9chamber
medium.
cell cycle
is 55
10 min
in are
a temperatureto the
fluorophore
maturation
process (SO
exposure is rates,
taken over 195 min with 100-ms laser exposures (0.3 kW/cm2) every 3 min. An 1100-ms
(È15
residues/s)
we
tentatively
assign
1/k
controlled
chamber on a microscope applied
stage. The
eight frames are from time-lapse fluorescence movie S1
levels because of the low sensitivity
for protein
Although we can only spatially resolve
after each image collection to photobleach the Venus fluorophores. (B) Time traces of the Text).
maturation
process (SOM
taken
over 195
detection, yet many important
proteins
aremin with 100-ms laser exposures (0.3 kW/cm2) every 3 min. An 1100-ms exposure is to the fluorophore few
molecules within an E. coli cell becau
Occasionally, one mRNA
is transcribed.
Downloaded from www.sciencemag.org on May 15, 2007
T
Bursts of
translated
protein.
expression of Tsr-Venus protein molecules (left) along three particular cell lineages (right) extracted from
d from www.sciencemag.org on May 15, 2007
Following expression of a fluorescent membrane protein in bacteria
over time.
Intuitive solution
v1
translation
d0
degradation
mRNA
probability of a given mRNA being translated is
probability of translating r proteins
is then
geometric “burst”
distribution with
generation function
McAdams & Arkin (1997)
Typical number of proteins is the sum of the number of proteins
translated from each mRNA.
Typical number of mRNAs synthesized during one protein lifetime
As the generating function of a sum of independent variables is a product
of their individual generating functions,
F (z) = [f (z)]a = (1 + b − bz)−a
generating
function for
total number
of proteins
generating function for
number of proteins
translated from a single
mRNA
giving
negative binomial
distribution
See also Friedman et al. (2006)
Comparison with simulation
theoretical
prediction in
red
Theory fails when the ratio of protein to mRNA lifetimes, γ, is atypically
small.
protein lifetime
γ=
mRNA lifetime
Intuitive solution
v1
translation
d0
degradation
mRNA
probability of a given mRNA being translated is
probability of translating r proteins
is then
We have assumed that no
proteins are degraded
during the mRNA lifetime.
Large γ implies that proteins translated from one mRNA appear almost
simultaneously on the time-scale for proteins.
Values of γ for budding yeast
the
[18]
[19]
[20]
but
mRNA
v0
Protein
v1
d0
Is this model appropriate?
d1
Red: protein
Green spots: mRNA
scale bar: 1 µm
Time course of mRNA numbers:
mRNA is produced in bursts
mRNA of
sister cell
mRNA
time (min)
piecewise
linear fit
smoothed version of red
data
Standard model of gene expression
We can also approximately solve the Master equation
Pn
!
"n !
"α
b
b
Γ(α + n)Γ(β + n)Γ(κ0 + κ1 )
1−
=
Γ(n + 1)Γ(α)Γ(β)Γ(κ0 + κ1 + n) 1 + b
1+b
!
"
b
×2 F1 α + n, κ0 + κ1 − β, κ0 + κ1 + n;
1+b
with
if
γ!1
bimodal distribution
reflecting DNA switching
is also possible
Modelling extrinsic fluctuations
Extrinsic fluctuations have a substantial lifetime
correlation in
rates of protein
production
As well as intrinsic fluctuations, we should include parameters
that fluctuate with a non-zero time-scale.
Rosenfeld et al. (2005)
Modelling stochasticity: deterministic approximation
dD
dt
dM
dt
dA
dt
= k0 − (k0 + k1 )D
= v0 D − d0 M
= v1 M − d1 A
law of
mass
action
Modelling stochasticity: intrinsic fluctuations
Langevin
theory
with “white” sources for intrinsic fluctuations obeying
sources of
fluctuations have
no time correlations
with themselves or
each other
Modelling stochasticity: intrinsic and extrinsic fluctuations
dD
dt
dM
dt
dA
dt
= k0 − (k0 + k1 )D + ξ1
= v0 D − d0 M + ξ2
system
becomes
non-linear
e!
= v1 ! M − d1 A + ξ3
"e #
with “white” sources for intrinsic fluctuations and a “coloured” source
for extrinsic fluctuations with autocorrelation
ηε determines the strength of extrinsic fluctuations
τ determines their lifetime
Modelling extrinsic fluctuations
Ornstein-Uhlenbeck
process
white noise
v1 ξ0
normal white fluctuations
log-normal coloured fluctuations
e!
v1 !
!e "
Simulating intrinsic and extrinsic fluctuations:
modifying Gillespie’s algorithm
Allow reaction rates to change discontinuously in time.
Before running simulation, simulate a time series for extrinsic
fluctuations.
extrinsic parameter
Change the fluctuating reaction rate discontinuously during
the simulation using a piece-wise linear approximation to the
time series generated for extrinsic fluctuations.
time
Example: extrinsic fluctuations in protein synthesis
k
A
d
k fluctuating
k constant
Both magnitude and lifetime of extrinsic fluctuations affect the
extrinsic noise in protein numbers.
ηext
CV of extrinsic variable
τ of extrinsic variable (sec)
Analytical solution (dashed line) calculated by Langevin theory.
Number of proteins
Extrinsic fluctuations can affect mean protein number
CV of extrinsic variable
τ of extrinsic variable (sec)
Analytical solution (dashed line) calculated with the unified coloured noise approximation
(Jung & Hanggi, 1987).
Extrinsic fluctuations can affect the intrinsic noise in protein
numbers
ηint
CV of extrinsic variable
τ of extrinsic variable (sec)
Intrinsic noise is determined by the projection of P(I1,I2,E) onto the I1 and I2
plane. Only have
[not P (I1 |E)P (I2 |E)P (E) ]
because of coloured extrinsic fluctuations.
Extrinsic
fluctuations
I2 contains predictive
information on the history of E
Coloured extrinsic fluctuations can help explain
experimentally observed trends.
Sigal et al. (2006)
Sigal et al. observe that the autocorrelation time of protein numbers varies
with their coefficient of variation.
Coloured extrinsic fluctuations can explain the observed correlation
Green points have extrinsic noise at least
75% of total noise.
We simulate 10,000
networks with
randomly chosen
parameter sets and
with extrinsic
fluctuations on a
randomly chosen
parameter.
Extrinsic fluctuations are non-specific and may combine
constructively or destructively.
(i,j)
(ηext )2
(i)
(j)
(ηext )2 + (ηext )2
Uncorrelated extrinsic fluctuations combine constructively.
Extrinsic fluctuations in parameters with opposing effects on protein
number combine destructively; extrinsic fluctuations on parameters
with similar effects on protein numbers combine constructively.
(i,j)
(ηext )2
(i)
(j)
(ηext )2 + (ηext )2
Constructive and destructive extrinsic fluctuations can be used by
feed-forward loops in genetic networks to enhance or attenuate
extrinsic fluctuations
X and X’ have
uncorrelated
extrinsic
fluctuations.
Incoherent feed-forward
loops attenuate extrinsic
fluctuations in X.
noise in Z
Coherent feed-forward loops amplify extrinsic fluctuations in X
noise in Z
Conclusions
1. Stochasticity in biochemical networks is generated by both
intrinsic and extrinsic fluctuations.
2. Intrinsic fluctuations can be modelled using a Master
equation and the Gillespie algorithm.
3. Extrinsic fluctuations are coloured and non-specific. They
can be simulated with a modification of the Gillespie
algorithm.
4. Extrinsic fluctuations create non-linearities in the
dynamics of the system and generate counter-intuitive
behaviour.
5. Cells both regulate and exploit stochasticity.
6. Acknowledgments
Vahid Shahrezaei
Imperial College, London
Julien Ollivier
McGill University
Nitzan Rosenfeld
Cambridge Research Institute
Michael Elowitz
Caltech
Shahrezaei et al. Mol Syst Biol 4; 196 (2008)
Shahrezaei & Swain, Proc Nat Acad Sci USA 105; 17256 (2008)
Shahrezaei & Swain, Curr Opin Biotechnol 19; 369 (2008)
http://swainlab.bio.ed.ac.uk