Download Probing noise in gene expression and protein production

PHYSICAL REVIEW E 80, 031916 共2009兲
Probing noise in gene expression and protein production
Sandro Azaele,1 Jayanth R. Banavar,2 and Amos Maritan3,4
1
Department of Civil and Environmental Engineering, E-Quad, Princeton University, Princeton, New Jersey 08544, USA
Department of Physics, The Pennsylvania State University, 104 Davey Laboratory, University Park, Pennsylvania 16802, USA
3
Dipartimento di Fisica “Galileo Galilei,” Università di Padova, via Marzolo 8, I-35131 Padova, Italy
4
INFN, via Marzolo 8, I-35131 Padova, Italy
共Received 30 January 2009; revised manuscript received 27 July 2009; published 24 September 2009兲
2
We derive exact solutions of simplified models for the temporal evolution of the protein concentration within
a cell population arbitrarily far from the stationary state. We show that monitoring the dynamics can assist in
modeling and understanding the nature of the noise in gene expression. We analyze the dispersion of the
process, i.e., the ratio of the variance to the mean at arbitrary time, and introduce a measure, the fractional
protein distribution, which can be used to probe the phase of transcription of DNA into mRNA.
DOI: 10.1103/PhysRevE.80.031916
PACS number共s兲: 82.39.Rt, 02.50.⫺r, 87.17.Aa
II. MODEL
I. INTRODUCTION
Advances in experimental techniques, which enable the
direct observation of gene expression in individual cells,
have demonstrated the importance of stochasticity in gene
expression, the translation into proteins of the information
encoded within DNA 关1–5兴. Such variability can lead to deleterious effects in cell function and cause diseases 关6兴. On
the positive side, stochasticity in gene expression confers on
cells the ability to be responsive to unexpected stresses and
may augment growth rates of bacterial cells compared to
homogeneous populations 关7兴. Disentangling the various
contributions to production fluctuations is complicated by the
recent finding that different stochastic processes yield the
same response in the variance in protein abundance at stationarity 关8兴. A population of isogenic cells growing under
the same environmental conditions can exhibit protein abundances that vary greatly from cell to cell. The sources of
variability have been identified at multiple levels 关9–13兴,
with transcription and translation playing a major role under
certain circumstances 关14–16兴.
The low concentration of reactants potentially has two
important consequences: the first is that fluctuations around
the mean can be large; the second is that the nature of the
stochastic noise should be taken into account in some detail
because one may not simply invoke the central limit theorem
关17兴, which leads to the universal and ubiquitous Gaussian
noise. Thus, two genes expressed at the same average abundance can produce protein populations with different Fano
factors F共t兲 = var关x共t兲兴 / 具x共t兲典, where var关x共t兲兴 and 具x共t兲典 are
the variance and the mean of the protein concentration x共t兲,
respectively 关18兴. We show here that two distinct models,
one taking into account the detailed nature of the noise and
the other following from an application of the central limit
theorem, yield exactly the same stationary solution for the
distribution of proteins in isogenic cells under the same environmental conditions. The applicability of the central limit
theorem arises from the fact that the stochasticity in gene
expression results from the large number of available components which entangle a lot of different mechanisms within
a cell. The exact dynamical solution of these two simplified
models demonstrates the value of monitoring the dynamics
for understanding the nature of the noise in a cell.
1539-3755/2009/80共3兲/031916共8兲
We make the simplified assumption that the kinetics of
gene expression can be described approximately by four rate
constants: k1 and k2 are the transcription and translation
rates, respectively, and ␥1 and ␥2 are the degradation rates
for mRNA and proteins, respectively. It has been found experimentally that proteins are produced in bursts 关5,18–20兴
with an exponential distribution of the protein concentration
produced in a given event. We will assume that transcription
pulses are Poisson events and that the probability distribution
that in a single event I ⬎ 0 proteins are produced, w共I兲, is
␥
approximated by w共I兲 = k21 e−共␥1/k2兲I, where k2 / ␥1 is the translation efficiency, i.e., the mean number of proteins produced
in a given burst. Here we consider a simple model for the
production of proteins without memory and aging of molecules. Using the specific burst distribution given above allows one to obtain the shape of the protein distribution even
far from stationarity. Under these assumptions, the stochastic
equation that governs the single-variable dynamics of gene
expression can be written as
ẋ共t兲 = ␦ − ␥2x共t兲 + ⌳共t兲.
共1兲
This pseudoequation describes the real-time stochastic
evolution of gene expression through a deterministic part and
a stochastic term ⌳共t兲, which will be defined later on. Here x
is a continuous variable that represents the protein concentration within a cell. For the sake of generality, we have
added the constant ␦, which can be incorporated into the
average noise. However, although ␦ ⬎ 0 can account for important effects in ecological systems 关25兴, we will show in
the following that it does not play a significant role under the
burstlike production of proteins.
In order to understand the nature of the noise for the gene
expression case, let us consider the random variable, Ik, that
is a measure of the number of proteins in the kth transcription event, where k = 1 , 2 , . . . , n. A key quantity of interest is
n共t兲
Ik ⬅ ⌳共t兲⌬t, where n共t兲, the number of events in the time
兺k=1
interval 共t , t + ⌬t兲, is a random variable independent of both x
and the Iks. As in the experiment, let us postulate that: 共i兲 the
Iks are independent and identically distributed with exponential distribution; and 共ii兲 the probability of n events occurring
during the time interval ⌬t is given by the Poisson distribu-
031916-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 031916 共2009兲
AZAELE, BANAVAR, AND MARITAN
tion qn共⌬t兲 = 共k1⌬t兲nexp共−k1⌬t兲 / n!. The distribution of ⌳共t兲
that we use in Eq. 共1兲 can be explicitly calculated 共see Appendix A兲 and leads to the following expression for the cun
␦共ti − t1兲 for n
mulants: 具具⌳共t1兲 ¯ ⌳共tn兲典典 = n ! k1共k2 / ␥1兲n兿i=2
ⱖ 2 and 具⌳共t兲典 = k1k2 / ␥1, independent of time.
Because the cumulants are delta functions, the noise is
still white 共events are uncorrelated if they occur at different
times兲; however the noise is no longer Gaussian because cumulants with n greater than two are nonzero.
The master equation that describes this burstlike process
is 关17兴
⳵ p共x,t兲
⳵
= − 关共␦ − ␥2x兲p共x,t兲兴 + k1
⳵t
⳵x
冕
w共x − y兲p共y,t兲dy
0
共2兲
where p共x , t兲 ⬅ p共x , t 兩 x0 , 0兲 is the conditional probability that
the protein concentration has a value x at time t given that it
␥
has a value x0 at time 0; and w共x兲 = k21 e−共␥1/k2兲x.
Equation 共2兲 can be easily understood as follows. The first
term in the right-hand side is simply related to the deterministic motion given by the first two terms in Eq. 共1兲 and it is
independent of the nature of the noise term. The second and
third terms are related to the probability per unit time to
jump from a population of size y to a population of size x,
w共x − y兲, given that k1 is the transition rate for a transcription
event. The stationary solution of this model was already
known 关21,22兴 with ␦ = 0. For arbitrary ␦ ⬎ 0, the stationary
solution is
冉 冊
␥1
k2
k1/␥2
冉 冊
⌰共x − ␦/␥2兲
␦
x−
⌫共k1/␥2兲
␥2
number of cells
300
200
100
0
0
100
200
300
400
500
600
number of proteins fluorescence units
700
x
− k1 p共x,t兲,
ps共x兲 =
400
k1/␥2−1
e−共␥1/k2兲共x−␦/␥2兲 ,
共3兲
where ⌰共x兲 is the step function equal to 1 when x ⬎ 0 and
zero otherwise. This distinctive feature is a sharp signature of
the nature of the noise even in the stationary solution but is
present only when ␦ ⫽ 0. However, as shown in the fit to the
stationary solution in Fig. 1, the singularity, if it exists, is
easily masked by other noise effects, leading to a rounding
effect.
III. ALTERNATIVE MODEL
Although experiments on gene expression 关5,18兴 are consistent with a burstlike protein production, steady-state distributions of protein abundances are equally compatible with
alternative explanations. In fact, because mRNA is unstable
compared to protein lifetime 共␥1 Ⰷ ␥2兲, one can assume that
transcripts give rise to a constant flux of proteins f and subsequently any protein degrades at a constant rate ␥2. Because
of the great amount of available molecules, one can apply the
central limit theorem and suppose that the amplitude of fluctuations is simply proportional to 冑x. Within this framework
there is no burstlike production; nevertheless the stationary
solutions that one obtains for a burstlike process, including
that of the extended autoregulation model 关23,24兴, are also
obtained in models with appropriately chosen random multiplicative Gaussian noise 共see Appendix C兲. Within this sce-
FIG. 1. The stationary distribution of proteins in a prokaryotic
cell population taken from Ref. 关18兴 fitted to Eq. 共3兲 with ␦ = 0 or
␦ / ␥2 = 60.3 共dashed兲. The best fit parameters are ␥1 / k2 = 0.038,
k1 / ␥2 = 12.88 共␹2 ⯝ 6100兲, and ␥1 / k2 = 0.030, k1 / ␥2 = 8.33 共␹2
⯝ 8700兲, respectively. From the experimental data it is hard to distinguish between the steady-state distributions predicted by Eq. 共3兲
with ␦ = 0 and ␦ ⬎ 0.
nario the stochastic evolution of the protein concentration
x共t兲 is governed by the equation
ẋ共t兲 = f − ␥2x共t兲 + 冑Dx共t兲␩共t兲,
共4兲
where ␩共t兲 is a Gaussian white noise with autocorrelation
具␩共t兲␩共t⬘兲典 = 2␦共t − t⬘兲. Note that the same equation could be
obtained on setting 具⌳共t兲典 = f
and 具具⌳共t兲⌳共t⬘兲典典
⬅ 具⌳共t兲⌳共t⬘兲典 − 具⌳共t兲典具⌳共t⬘兲典 = 2Dx共t兲␦共t − t⬘兲 in Eq. 共1兲 with
all higher-order cumulants being identically zero. Note that
the square root of the multiplicative noise in Eq. 共4兲 is not
introduced ad hoc. It has its roots in the central limit theorem
and can also be justified on the basis of other general considerations about the discrete nature of the process. In fact,
on temporal scales much larger than the mRNA lifetime, the
protein production can be suitably described by a birth and
death process whose rates are proportional to the number of
available molecules. Accordingly, the fluctuations of this discrete Markov process can be well described by the multiplicative noise term present in Eq. 共4兲 in the continuum limit,
i.e., when a large number of molecules is present. In addition, because the noise goes to zero when x共t兲 = 0, it also
prevents the random variable x共t兲 from becoming negative.
We point out that in ecology Eq. 共4兲 is useful for studying
the evolution of tropical forests 关25兴, where the detailed nature of the stochastic noise is not important because of the
relatively large numbers of trees of a given species. In the
field of finance, Eq. 共4兲 has been used to study the evolution
of interest rates 共the Cox-Ingersoll-Ross model 关26兴兲, where
analogous considerations on fluctuations can be made. On
defining f ⬅ k1k2 / ␥1 and D ⬅ ␥2k2 / ␥1, Eq. 共4兲 yields the
same stationary state as in Eq. 共2兲 with ␦ = 0, i.e., Equation
共3兲. The mean protein concentration at stationarity is
k1k2 / ␥1␥2 and the Fano factor at stationarity is k2 / ␥1, relations that are consistent with previous findings 关18兴. In order
to take into account the effects of feedback in a system undergoing autoregulation, one can introduce the physically
transparent modification f → Dc共x兲, where c is a response
function which can be modeled as having two distinct limit-
031916-2
PROBING NOISE IN GENE EXPRESSION AND PROTEIN …
px,t0,0
t 5 min
px,t0,0
0.035
0.03
0.025
0.02
0.015
0.01
0.005
PHYSICAL REVIEW E 80, 031916 共2009兲
t 20 min
0.01
0.008
0.006
0.004
0.002
100 200 300 400 500 600 700
px,t0,0
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0.5
50 100 150 200 250
100 200 300 400 500 600 700
x
20
10
50
x
t 120 min
100 200 300 400 500 600 700
px,t0,0
0.004
0.003
0.003
0.002
0.002
x
t 240 min
0.001
0.001
100 200 300 400 500 600 700
x
100 200 300 400 500 600 700
x
FIG. 2. Protein distribution dynamics for different types of noise
and with the same initial condition, i.e., x0 = 0 proteins at t = 0. The
dashed curve is for the multiplicative Gaussian noise, i.e., Eq. 共4兲
with f ⬅ k1k2 / ␥1 and D ⬅ ␥2k2 / ␥1 共see Appendix C兲; whereas the
solid curve is for the non-Gaussian noise, i.e., for Eq. 共5兲. In both
cases the parameters are ␦ = 0, ␥2 / D = ␥1 / k2 = 0.038, f / D = k1 / ␥2
−1
= 12.88 and we have set ␥−1
2 = 40 min, ␥1 = 2 min.
ing values at zero and at infinity with the latter being smaller
than the former. Even in this situation, we obtain the same
stationary distribution with bistability as in Ref. 关22兴. Despite this much more realistic analysis, the final stationary
protein distribution is experimentally indistinguishable from
Eq. 共3兲 with ␦ = 0. Thus, a theoretical modeling of the stationary state of protein production provides little insight into
the microscopic nature of the noise that leads to stationarity.
IV. TEMPORAL EVOLUTION
These results raise the question whether the agreement
between the stationary solutions of the theoretical models
and experiments are in fact a direct probe of the nature of the
microscopic noise and whether the asymmetric stationary solutions derive from a careful consideration of the bursty nature of the noise. In order to circumvent the indistinguishability of steady states, one can look into empirical protein
abundances far from stationarity, for which we provide analytical formulas. Thus we turn now to a study of the dynamics of Eq. 共2兲, which is a powerful probe of the noise effects.
We have derived 共see Appendix B兲 the solution at arbitrary
time 共Fig. 2兲,
p共x,t兲 = e−k1t␦共x − ␰t兲 + ⌰共x − ␰t兲
冋
⫻exp −
⫻ 1F 1
冉
␥1 ␥ t
e 2 共x − ␰t兲
k2
tmin
30
t 60 min
0.005
0.004
0.003
0.002
0.001
0.004
1.0
40
px,t0,0
100 200 300 400 500 600 700
px,t0,0
1.5
50
x
t 40 min
Ft
Ft
册
k 1␥ 1 ␥ t
共e 2 − 1兲e−k1t
k 2␥ 2
冊
k1
␥1
+ 1,2; 共e␥2t − 1兲共x − ␰t兲 ,
␥2
k2
共5兲
100
150
200
250
tmin
FIG. 3. Fano factor dynamics with the same initial condition,
i.e., x0 = 0 proteins at t = 0. The solid curves are for the non-Gaussian
noise case, i.e., F共t兲 = var关x共t兲兴 / 具x共t兲典 is obtained from Eq. 共5兲 with
␦ = 0; the dashed curves are for the multiplicative Gaussian noise,
i.e., F共t兲 is calculated with the analytical solution of Eq. 共4兲. The
parameters 共k1 = 0.01, k2 = 15 min−1兲 in the main figure correspond
to an infrequent production of large bursts, whereas the ones 共k1
= 0.3, k2 = 0.5 min−1兲 used in the inset give rise to small bursts
−1
produced frequently. In both cases we have set ␥−1
1 = 2 and ␥2
= 40 min.
where 1F1共a , b ; x兲 is the confluent hypergeometric function
关27兴 and ␰共t兲 ⬅ x0e−␥2t + ␥␦2 共1 − e−␥2t兲 is the solution of the deterministic part of the equation, i.e., without the noise.
Interestingly, one obtains a distribution of proteins with a
cutoff along the interval 关0 , ␰t兲 at any time whenever ␦ ⬎ 0.
The stochasticity plays a major role when the mean number
of bursts per cell cycle is very small, i.e., k1 / ␥2 Ⰶ 1. In this
case, the solution in Eq. 共5兲 can be approximated by replacing 1F1共1 , 2 ; z兲 with 共ez − 1兲 / z 共see Ref. 关27兴 and Appendix
B兲. On using Eq. 共5兲, one can calculate the evolution of the
Fano factor, F共t兲, starting from an arbitrary initial amount of
proteins. This time evolution is highly sensitive to the nature
of the stochasticity. Figure 3 vividly shows the distinct dependence of the initial dynamics of F共t兲 on the nature of the
noise. Note that, at stationarity, the distinct types of stochasticity are indistinguishable. Interestingly, within the temporal
transient, the fluctuations deviate from Poisson behavior,
whereas, at stationarity, both models predict a Poissonian
Fano factor for genes with high transcription and low translation rates and limt→⬁ F共t兲 = k2 / ␥1 ⯝ 1.
V. FRACTIONAL PROTEIN DISTRIBUTION
Another measurable quantity that directly probes the protein distribution and its temporal evolution is the fractional
protein distribution 共FPD兲, P共␳ , t兲, i.e., the probability that at
time t the ratio x共t兲 / x共0兲 is equal to ␳, where x共t兲 and x共0兲
are the protein concentrations at time t ⬎ 0 and t = 0, respectively. Unlike the stationary distribution in Eq. 共3兲 with ␦
= 0, the FPD can exhibit a distinctive signature of the nature
of the burstlike noise even under stationary conditions: it has
an interval in which it identically vanishes and this, in principle, can be experimentally observed. This quantity can be
defined at arbitrary times 共see Appendix D兲: if the initial
distribution is the steady state given by Eq. 共3兲 with ␦ = 0,
then at t ⬎ 0 the FPD is
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PHYSICAL REVIEW E 80, 031916 共2009兲
AZAELE, BANAVAR, AND MARITAN
P共␳,t兲 = e−k1t␦共␳ − e−␥2t兲 +
⫻ 2F 1
冉
冉 冊
k1
␥2
e 共e␥2t − 1兲⌰共␳ − e−␥2t兲
关1 + e␥2t共␳ − e−␥2t兲兴k1/␥2+1
2 −k1t
冊
共e␥2t − 1兲共␳ − e−␥2t兲
k1
k1
,
+ 1, + 1,2;
␥2
␥2
1 + e ␥2t共 ␳ − e −␥2t兲
共6兲
where 2F1共a , b , c ; x兲 is the standard hypergeometric function
关27兴. Thus, according to Eq. 共6兲, under the burst process
hypothesis we predict that 共i兲 the FPD vanishes between 0
and e−␥2t even though the system is at stationarity, an effect
which ought to be detectable for time scales less than or of
the order of 1 / ␥2, 共ii兲 the FPD depends on k1 and ␥2 only,
共iii兲 at very large time separation there is only one free parameter, the ratio k1 / ␥2, and the FPDs predicted by the
Gaussian and non-Gaussian noises become the same 共see Appendix D兲.
The analogous time-dependent solutions for the Gaussian
white noise can be compared with Eqs. 共5兲 and 共6兲 共see Appendix D兲.
The closer the system is to its steady state, the more difficult it is to distinguish among the effects of gestation, senescence and burstlike production. Thus an experimental
protocol capable of analyzing the cell population and its time
evolution with different initial conditions would be helpful to
disentangle the nature of stochastic noise. At early times, the
evolution of the distribution is strongly affected by the specific mechanisms involved in the dynamics. At this stage,
different distributions of interarrival times between events or
burst sizes produce nonstationary distributions that are very
different, and the distinctive effects of noise, deterministic
driving forces, or coupling of degrees of freedom can be
elucidated. Different conditions at initial times propagate
into the early temporal evolution in strongly different ways
according to the different effects of involved mechanisms,
but inexorably lead to the same distribution for large time
separation.
Let x1 , x2 , . . . , xn be n-independent one-burst processes
with only positive increments. Let us suppose that they are
identically distributed, so they have the same jump transitions, say a probability density function w共x兲 with x ⱖ 0. Let
us further assume that the burst processes occur in time according to a Poisson distribution qn共t兲 = 共k1t兲nexp共−k1t兲 / n!,
where k1 is the transcription rate.
Because any burst produces xi proteins according to the
distribution w共x兲, we are interested in the distribution of the
n
xi, i.e., the total amount of prorandom variable ␰共t兲 = 兺i=1
teins produced from t = 0 through t ⬎ 0, where we are assuming that n bursts have been occurred in a time t. The characteristic function for the process ␰共t兲 is
⬁
具e
iz␰共t兲
典 = 兺 具eiz␰共t兲兩n events by t典prob关n events by t兴
n=0
⬁
= 兺 关g共z兲兴nqn共t兲,
where we have used the independence of any one-burst process and we have used the definition
g共z兲 ⬅ 具eizx j典 =
冕
⬁
eizxw共x兲dx,
共A2兲
0
for any j = 1 , . . . , n. Because we are using the Poisson distribution qn共t兲, one obtains
具eiz␰共t兲典 = ek1t共g共z兲−1兲 .
共A3兲
Thus the characteristic function of ␰共t兲 in Eq. 共A1兲 defines
the following integral equation for the distribution p共x , t兲
= 具␦共x − ␰共t兲兲典:
冕
⬁
0
VI. CONCLUSION
共A1兲
n=0
⬁
eizx p共x,t兲dx = 兺 关g共z兲兴nqn共t兲.
共A4兲
n=0
By direct substitution one can verify that a solution is
In summary, we have shown that fits to experimental data
of the stationary solution do not discriminate between different types of noise. However, the temporal evolution of the
probability distribution of protein or fractional protein concentration under stationary conditions as well as the evolution of the Fano factor can be used as a powerful probe of the
noise effects of protein production. We have shown that the
full time dependence of the analytical solution of the model
proposed in 关22兴 with burstlike protein production events
presents a singularity. This singularity is absent in the corresponding model with Gaussian multiplicative noise.
p共x,t兲 = q0共t兲␦共x兲 + q1共t兲w共x兲 + q2共t兲
冕
x
w共x − y兲w共y兲dx
0
⬁
+ ¯ = 兺 qn共t兲w共x兲 〫 w共x兲 〫 ¯ 〫 w共x兲 ,
n=0
n-fold convolution
共A5兲
where ␦共x兲 is a Dirac delta and the symbol “〫” stands for a
convolution of w共x兲’s.
Now we use the particular jump distribution w共x兲 = ␭e−␭x
共␭ ⬅ ␥1 / k2兲. Hence
ACKNOWLEDGMENT
g共z兲 =
This work was supported by PRIN 2007.
␭
,
␭ − iz
共A6兲
and the calculations for the nth convolution 共n ⱖ 1兲 result in
APPENDIX A: DEFINITION OF THE NON-GAUSSIAN
WHITE NOISE
In this section we wish to exploit the burst hypothesis of
protein production in gene expression in order to derive its
probability distribution.
␭n
xn−1 −␭x
e .
共n − 1兲!
Thus we obtain with qn共t兲 = 共k1t兲nexp共−k1t兲 / n!
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共A7兲
PROBING NOISE IN GENE EXPRESSION AND PROTEIN …
PHYSICAL REVIEW E 80, 031916 共2009兲
冉 冊
冕
⬁
p共x,t兲 = e−k1t␦共x兲 +
共k1t␭x兲n
e−k1t−␭x
兺
x n=1 共n − 1兲 ! n!
j共x,t兲 = ␦ −
⳵
t
= e−k1t␦共x兲 + e−k1t−␭x I0共2冑k1t␭x兲
⳵t
x
= e−k1t␦共x兲 +
e−k1t−␭x
冑k1t␭xI1共2冑k1t␭x兲,
x
+
共A8兲
where I␯共z兲 is the modified Bessel function of the first kind
关27兴, whose definition is
⬁
I␯共x兲 = 兺
n=0
␯+2n
共x/2兲
.
n ! ⌫共␯ + n + 1兲
By exploiting this definition, one can see that in Eq. 共A8兲
there is no divergence at x = 0. We can also calculate all the
moments
具␰n共t兲典 =
n!
k1te−k1t 1F1共n + 1,2;k1t兲 + e−k1t␦n,0 ,
␭n
共A9兲
where 1F1共a , b ; x兲 is the confluent hypergeometric function
关27兴 and n = 0 , 1 , 2 , . . ..
Now we can consider the fundamental stochastic differential equation which defines the model in the main text,
冉
冊
x共t兲
dt + d␰共t兲,
dx共t兲 = ␦ −
␶
共A10兲
where ␶ ⬅ ␥−1
2 and ␰共t兲 is the noise whose probability distribution is given by Eq. 共A8兲. Finally, in order to recover Eq.
共1兲 in the main text, one can formally write d␰共t兲 = ⌳共t兲dt.
The master equation which governs the process defined by
Eq. 共A10兲 is Eq. 共2兲 in the main text.
冋冉 冊 册 冕
␦−
1
− p共x,t兲,
␪
1
x
p共x,t兲 +
␶
␪
x
x
z
0
0
w共z − y兲p共y,t兲dzdy
x
共B2兲
p共y,t兲dy,
0
j共0,t兲 = ␦ lim p共x,t兲,
where limx→0+ xp共x , t兲 = 0 due to the integrability of p共x , t兲 at
x = 0. By Laplace transforming Eq. 共B1兲 with respect to x,
one obtains
关␳共s兲 − 1兴␻共s,t兲
⳵␻共s,t兲
s ⳵␻共s,t兲
− bs␻共s,t兲 +
= j共0,t兲 −
,
⳵t
␶ ⳵s
␪
共B4兲
␳共s兲 ⬅ 兰⬁0 w共x兲e−sxdx
where
is supposed to be finite. We are
interested in the time-dependent reflecting solution of Eq.
共B1兲, so j共0 , t兲 = 0 and the characteristic equations for Eq.
共B4兲 get
dt =
␶ds
␪d␻
=
,
s
␻共␳共s兲 − 1 − ␦␪s兲
se−t/␶ = s0 ,
冋冕
␻共s兲 = ␻0 exp
␶
␪
册
共B6兲
␳共␴兲 − 1
d␴ − ␦␶共s − s0兲 , 共B7兲
␴
s0
s
where s0 , ␻0 are arbitrary constants. If we choose an arbitrary
well-behaved function ␾共x兲 such that ␾共s0兲 = ␻0, one can
write down the general solution of Eq. 共B4兲,
冋冕
册
␶
␪
␳共␴兲 − 1
d␴
␴
se−t/␶
s
− ␦␶共s − se−t/␶兲 .
共B8兲
Since the initial condition is p共x , 0兲 = ␦共x − x0兲, we have
␻共s , 0兲 = e−sx0, thereby one can determine the function ␾共x兲,
finally achieving
ln ␻共s,t兲 = − 共␦␶ + x0e−t/␶ − ␦␶e−t/␶兲s +
共B1兲
−1
where ␪ ⬅ k−1
1 , ␶ ⬅ ␥2 , ␦ ⱖ 0 , w共x兲 is a probability density
function and p共x , t兲 ⬅ p共x , t 兩 x0 , 0兲 is the conditional probability that the protein concentration has a value x at time t given
that it has a value x0 at time 0. Note that Eq. 共B1兲 exhibits
two temporal scales, ␪ and ␶, which are related to transcription and protein dilution, respectively.
The probability flux determined by Eq. 共B1兲 is
共B5兲
which can be easily solved. The solutions may be written as
follows:
w共x − y兲p共y,t兲dy
0
共B3兲
x→0+
␻共s,t兲 = ␾共se−t/␶兲exp
The main goal of this section is to obtain the fundamental
solution of the integrodifferential equation that describes the
evolution of the protein concentration within a population of
isogenic cells under the burst hypothesis 关Eq. 共2兲 in the main
text兴,
冕冕
where we assume that limx→⬁ j共x , t兲 = 0 for any t ⬎ 0. The
flux at x = 0 is simply
APPENDIX B: THE SOLUTION OF THE MASTER
EQUATION
⳵ p共x,t兲
⳵
=−
⳵t
⳵x
1
␪
1
x
p共x,t兲 + −
␶
␪
␶
␪
冕
s
se−t/␶
␳共␴兲 − 1
d␴ .
␴
共B9兲
By using the translation theorem which holds for Laplace
transforms 关28兴, we may study only the function
冋冕
⍀共s,t兲 = exp
␶
␪
s
se−t/␶
册
␳共␴兲 − 1
d␴ .
␴
共B10兲
We are interested in solving Eq. 共B1兲 with w共x兲 = ␭e−␭x,
where ␭ ⬅ ␥1 / k2 is the translation efficiency as defined in the
031916-5
PHYSICAL REVIEW E 80, 031916 共2009兲
AZAELE, BANAVAR, AND MARITAN
main text. So ␳共s兲 = ␭ / 共␭ + s兲 and ⍀共s , t兲 becomes
冉
se−t/␶ + ␭
⍀共s,t兲 =
s+␭
冊
␶/␪
⌽共x兲 ⬅
共B11兲
.
It turns out that one can analytically calculate the inverse
Laplace transform of ⍀共s , t兲 共see below兲, and then one can
finally achieve the fundamental solution of Eq. 共B1兲, which
is
p共x,t兩x0,0兲 = e
−t/␪
冉
冊
␶
⫻ 1F1 + 1,2;␭共et/␶ − 1兲共x − ␰t兲 ,
␪
−t/␶
共B12兲
In Fig. 1 in the main text we show ps共x兲 for different
values of ␦ and we compare it with experimental data obtained in a prokaryotic cell population: the agreement is excellent. When the mean number of bursts per cell cycle is
very small, one obtains ␶ / ␪ Ⰶ 1 and we can expand Eq.
共B12兲 in powers of ␶ / ␪. In this case, the solution in Eq.
共B12兲 reads
冉冊
共B16兲
z̄
2␲i
z⬘+i⬁
z⬘−i⬁
1−
1
z
−c
e共xz̄/a兲zdz,
1
z
−c
⬁
=兺
n=0
共c兲n 1
,
n! zn
共B18兲
when 兩z兩 ⬎ 1 and 共c兲n ⬅ c共c + 1兲 ¯ 共c + n − 1兲 , 共c兲0 ⬅ 1. The
series in Eq. 共B18兲 is absolutely and uniformly convergent if
兩z兩 ⬎ 1; thus it can be integrated term by term
⬁
兺
n=0
共c兲n
n!
冕
z⬘+i⬁
z⬘−i⬁
冉冊
兺 冉 冊 冕
e共xz̄/a兲z
xz̄
n dz = 2␲i␦
z
a
⬁
+
n=1
共c兲n xz̄
n! a
n−1
␴⬘+i⬁
␴⬘−i⬁
e␴
d␴ ,
␴n
共B19兲
␦共x兲 =
1
2␲i
冕
z⬘+i⬁
z⬘−i⬁
exzdz.
In Eq. 共B19兲 the integral may be evaluated along a simple
closed path that encircles the origin within which the integrand has a pole of order n at z = 0. Thus, by applying the
residue theorem, one obtains
冖
2␲i
e␴
.
n d␴ =
␴
共n − 1兲!
Therefore Eq. 共B19兲 gets
⬁
1
共c兲
兺 n
2␲i n=0 n!
冕
z⬘+i⬁
z⬘−i⬁
冉冊
共B20兲
⬁
e共xz̄/a兲z
共c兲n 共xz̄/a兲n−1
xz̄
.
+兺
n dz = ␦
z
a
n=1 n! 共n − 1兲!
共B21兲
␶ ⌰共x − ␰t兲
共exp关− ␭共x − ␰t兲
␪ x − ␰t
− t/␪兴 − exp关− ␭e 共x − ␰t兲 − t/␪兴兲 + O„共␶/␪兲 …,
2
共B15兲
We may use the definition and the properties of the confluent hypergeometric function 关27兴 to rewrite this latter relation as follows:
⬁
兺
n=1
where we have used 共see Ref. 关27兴兲
1F1共1,2;z兲 =
c
冉 冊
共B13兲
␭ ␶/␪
⌰共x − ␦␶兲共x − ␦␶兲␶/␪−1e−␭共x−␦␶兲 . 共B14兲
⌫共␶/␪兲
t/␶
冊
冉冊 冕 冉 冊
e−␭x/a a
a
b
⌽共x兲 ⬅
where ␴⬘ ⬎ 1 and we used
when x Ⰷ 1 and a , b ⫽ 0 , −1 , −2 , . . .. Thus the normalized solution at stationarity is
p共x,t兩x0,0兲 = e−t/␪␦共x − ␰t兲 +
k−i⬁
as + ␭ c sx
e ds,
bs + ␭
where x ⬎ 0, k ⬎ 0, 0 ⬍ a ⬍ b, ␭ ⬎ 0, and c ⬎ 0. On using the
variable z ⬅ 共as + ␭兲 / z̄, with z̄ ⬅ ␭共1 − a / b兲 ⬎ 0, one obtains
1−
where ␰t ⬅ x0e + ␦␶共1 − e 兲, 1F1共a , b ; x兲 is the confluent
hypergeometric function 关27兴, ⌰共x兲 is the step function
which is equal to one for x ⱖ 0 and zero otherwise, and ␦共x兲
is a Dirac delta. It is worth noting that Eq. 共B12兲 exhibits a
cutoff along the interval 关0 , ␰t兲 at any time whenever ␦ ⬎ 0,
thus the probability is generally continuous but not differentiable 共at x = ␰t兲. One may also achieve the steady-state solution of Eq. 共B1兲 by using the leading term of the asymptotic
representation of the confluent hypergeometric function 关27兴,
i.e.,
ps共x兲 =
k+i⬁
where z⬘ ⬎ 1. We can also write
−t/␶
⌫共b兲 x −共b−a兲
,
ex
1F1共a,b;x兲 ⯝
⌫共a兲
冕 冉
共B17兲
␶␭
␦共x − ␰t兲 + ⌰共x − ␰t兲 e−t/␪共et/␶ − 1兲
␪
⫻exp关− ␭et/␶共x − ␰t兲兴
1
2␲i
ez − 1
.
z
⳵
共c兲n 共xz̄/a兲n−1
=
F1共c,1;xz̄/a兲
n! 共n − 1兲! ⳵ 共xz̄/a兲 1
= c 1F1共c + 1,2;xz̄/a兲,
and finally reaching the equation
Laplace transform
⌽共x兲 =
In this section we calculate the inverse Laplace transform
of Eq. 共B11兲. In order to get it we study the function
031916-6
冉 冊 冋冉 冊
共B22兲
册
xz̄
e−␭x/a a c
z̄ ␦
+ c 1F1共c + 1,2;xz̄/a兲 ,
a
b
a
共B23兲
PROBING NOISE IN GENE EXPRESSION AND PROTEIN …
We can obtain the solution in Eq. 共B12兲 on setting a
= e−t/␶, b = 1, c = ␶ / ␪ and by using the translation theorem
which holds for the Laplace transforms.
APPENDIX C: THE GAUSSIAN WHITE NOISE CASE
Let us consider the fundamental equation of the main text
for the number of proteins at time t, x共t兲, within an isogenic
cell population,
ẋ共t兲 = ␦ − x共t兲/␶ + ⌳共t兲,
where
␶ ⬅ ␥−1
2
共C1兲
and
PHYSICAL REVIEW E 80, 031916 共2009兲
APPENDIX D: THE FRACTIONAL PROTEIN
DISTRIBUTION
The fractional protein distribution 共FPD兲, PFPD共␳ , t兲, is
defined as the probability that at time t the ratio x共t兲 / x共0兲 is
equal to ␳, where x共t兲 and x共0兲 are the protein concentrations
within a population of isogenic cells at time t ⬎ 0 and t = 0,
respectively. The system can be initially prepared according
to any kind of probability density function, pinit共x0兲, nevertheless because we are interested in FPD at stationarity, we
are using Eqs. 共5兲 and 共6兲 of the main text, i.e., the timedependent reflecting solution and the stationary probability
distribution with ␦ = 0, respectively. Thus, by definition the
FPD is
具⌳共t兲典 = f ,
PFPD共␳,t兲 = 具␦共␳ − x/x0兲典
=
共C2兲
Because we suppose that all the other cumulants are negligible, Eq. 共C2兲 defines a multiplicative Gaussian white
noise and on using the Itô prescription 关17兴, one can prove
that Eq. 共C1兲 with Eq. 共C2兲 is equivalent to the following
Fokker-Planck equation:
ṗ = ⳵x关共x/␶ − f − ␦兲p兴 + D⳵2x 共xp兲.
pstat共x兲 = 共D␶兲−f/D⌫共f/D兲−1x f/D−1e−x/D␶ ,
共C4兲
where ⌫共x兲 is the gamma function 关27兴. This solution is
equivalent 共with ␦ = 0兲 to Eq. 共3兲 of the main text, which has
been derived on using a non-Gaussian white noise that embodies the burst hypothesis.
One can also obtain the time-dependent solution of Eq.
共C3兲 with reflecting boundary conditions and with initial
population equal to x0. One can find all the details of the
derivation in 关29兴, here we write down only the final expression
p共x,t兩x0,0兲 =
冉 冊
1
D␶
f/D
冤
x f/D−1e−x/D␶
冋冉 冊
1
D␶
2
x0xe−t/␶
册
1/2−f/2D
1 − e−t/␶
冥 冤
冥
1
2
冑x0xe−t/␶
共x + x0兲e−t/␶
D␶
D␶
⫻ exp −
I f/D−1
.
1 − e−t/␶
1 − e−t/␶
dxpstat共x0兲p共x,t兩x0,0兲␦共␳ − x/x0兲,
0
共D1兲
where ␳ ⬎ 0, t ⬎ 0, and ␦共x兲 is a Dirac delta. Notice that
PFPD共␳ , 0兲 = ␦共␳ − 1兲 and
lim PFPD共␳,t兲 =
t→+⬁
冕
⬁
dx0x0 pstat共x0兲pstat共␳x0兲
0
=
⌫共2␤兲 ␳␤−1
,
⌫2共␤兲 共␳ + 1兲2␤
共D2兲
where ␤ = k1 / ␥2 and ⌫共␤兲 is the standard gamma function
关27兴. Note that Eq. 共D2兲 has a peak for ␳ ⬎ 0 only when ␤
⬎ 1, whereas when 0 ⬍ ␤ ⬍ 1 there is an integrable singularity at ␳ = 0. Furthermore, because the Gaussian and nonGaussian noises have the same steady state, they both have
the same FPD when t → +⬁.
When substituting Eqs. 共3兲 and 共5兲 of the main text in Eq.
共D1兲 and performing some simple manipulations, one gets
PFPD共␳,t兲 = e−t/␪␦共␳ − e−t/␶兲 + ⌰共␳ − e−t/␶兲
⫻共et/␶ − 1兲
⫻ 1F 1
冉
冕
⬁
␶ ␭␶/␪+1 −t/␪
e
␪ ⌫共␶/␪兲
dxx␶/␪ exp关− ␭x共1 + et/␶共␳ − e−t/␶兲兲兴
0
冊
␶
+ 1,2;␭共et/␶ − 1兲共␳ − e−t/␶兲x ,
␪
共D3兲
−1
where ␪ ⬅ k−1
1 , ␶ ⬅ ␥2 , ␭ ⬅ ␥1 / k2. On exploiting the formula 7.621.4 in 关30兴 for the integral, one can come up with
Eq. 共6兲 of the main text, which is independent on ␭. When
the mean number of bursts per cell cycle is very small, one
obtains ␶ / ␪ Ⰶ 1 and we can expand Eq. 共6兲 of the main text
in powers of ␶ / ␪. In this case, the solution reads
PFPD共␳,t兲 = e−t/␪␦共␳ − e−t/␶兲 + −
共C5兲
In Fig. 2 of the main text we have compared the behavior
of this equation with that in Eq. 共5兲 of the main text. Unlike
this latter, Eq. 共C5兲 has no any cutoff neither at finite times
nor at stationarity.
⬁
dx0
0
共C3兲
In this case the constant ␦ only shifts the average number of
proteins, so we can assume that ␦ = 0 without lack of generality. In Eq. 共C3兲 p ⬅ p共x , t 兩 x0 , 0兲 is the conditional probability that the protein concentration has a value x at time t given
p共x , t 兩 x0 , 0兲dx is the
that it has a value x0 at time 0, i.e., 兰n+⌬n
n
fraction of cells with protein population between n and n
+ ⌬n; furthermore, f ⬅ k1k2 / ␥1 and D ⬅ ␥2k2 / ␥1. Setting ṗ
= 0, one obtains the stationary solution of Eq. 共C3兲,
冕 冕
⬁
具具⌳共t兲⌳共t⬘兲典典 ⬅ 具⌳共t兲⌳共t⬘兲典 − 具⌳共t兲典具⌳共t⬘兲典 = 2Dx共t兲␦共t − t⬘兲.
冉
⫻ ln
冉冊
冊
2
⌰共␳ − e−t/␶兲e−t/␪
␳ − e−t/␶
1 + ␳ − e−t/␶
+ O关共␶/␪兲3兴,
1 + et/␶共␳ − e−t/␶兲
where we have used 共see Ref. 关27兴兲
031916-7
␶
␪
共D4兲
PHYSICAL REVIEW E 80, 031916 共2009兲
AZAELE, BANAVAR, AND MARITAN
2F1共1,1,2;z兲 = −
Along the same lines one can obtain the FPD for the
Gaussian white noise. Thus, substituting the time-dependent
reflecting solution in Eq. 共C5兲 and the steady state in Eq.
共C4兲, one comes up with the final expression
PFPD共␳,t兲 =
2
f/D−1 ⌫
冑␲
冢
冉 冊
冉冊
冉 冊冣
f/D+1
冉
4␳2
共␳ + 1兲2et/␶ − 4␳
2
1.5
1
0.5
0.5
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
1
2
2.5
Ρ
3
0.5
冊
,
共D5兲
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1.5
1
2
2.5
3
Ρ
0.5
2
2.5
3
2
2.5
3
2
2.5
3
Ρ
1
1.5
Ρ
t 200 min
FPD
1.2
1
0.8
0.6
0.4
0.2
1.5
t 30 min
FPD
1.5
1.25
1
0.75
0.5
0.25
t 60 min
FPD
where f ⬅ k1k2 / ␥1 , ␶ ⬅ ␥−1
2 and D ⬅ ␥2k2 / ␥1. Note that in
this case PFPD共␳ , t兲 is much simpler than Eq. 共6兲 in the main
text and depends only on f / D = k1 / ␥2 and ␶. In Fig. 4 we
have compared the FPD for the non-Gaussian white noise
关Eq. 共6兲 in the main text兴 and Eq. 共D5兲, which have the same
steady state.
关5兴
1.5
t 20 min
1
0.8
0.6
0.4
0.2
0.5
关1兴
关2兴
关3兴
关4兴
1
FPD
0.5
f/D+1/2
t 10 min
FPD
2.5
6
5
4
3
2
1
1.75
1.5
1.25
1
0.75
0.5
0.25
f 1
+
D 2 共␳ + 1兲 共et/␶兲 f/2D
f
␳ 1 − e−t/␶
⌫
D
t
sinh
2␶
⫻
␳
t 1 min
FPD
ln共1 − z兲
.
z
1
1.5
2
2.5
3
Ρ
0.5
1
1.5
Ρ
FIG. 4. Fractional protein distribution 共FPD兲. The dashed curve
is for the Gaussian noise case, i.e., Eq. 共D5兲 with f ⬅ k1k2 / ␥1 and
D ⬅ ␥2k2 / ␥1; whereas the solid curve is for the non-Gaussian noise
case, i.e., Eq. 共6兲 in the main text. The arrow indicates the cutoff
point. Note, however, that extrinsic noise could tend to smooth out
the discontinuity. In both cases k1 / ␥2 = 12.88 and we have set ␥−1
2
= 40 min.
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