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Probing noise in gene expression and protein production
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Azaele, S, Banavar, JR and Maritan, A (2009) Probing noise in gene expression and
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Probing noise in gene expression and protein production
Sandro Azaele,1Jayanth R. Banavar,2and Amos Maritan3,4
1
Department of Civil and Environmental Engineering, E-Quad, Princeton University, Princeton, New Jersey 08544, USA
2
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
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
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 del-eterious 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 sta-tionarity 关8兴. A population of isogenic cells growing under the same environmental conditions can exhibit protein abun-dances 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 abun-dance 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 en-vironmental 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 com-ponents 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.
II. MODEL
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 ex-perimentally 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 byw共I兲=␥k1
2e
−共␥1/k2兲I, wherek
2/␥1 is the trans-lation 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 mol-ecules. Using the specific burst distribution given above al-lows 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
x˙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. Herex is a continuous variable that represents the protein concen-tration 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 im-portant 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 thekth transcrip-tion event, wherek= 1 , 2 , . . . ,n. A key quantity of interest is
兺k=1
n共t兲I
k⬅⌳共t兲⌬t, wheren共t兲, the number of events in the time
interval共t,t+⌬t兲, is a random variable independent of bothx and theIks. As in the experiment, let us postulate that:共i兲the
Iks are independent and identically distributed with
exponen-tial distribution; and共ii兲the probability ofnevents occurring during the time interval⌬t is given by the Poisson
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 Ap-pendix A兲and leads to the following expression for the cu-mulants: 具具⌳共t1兲¯⌳共tn兲典典=n!k1共k2/␥1兲n兿i=2
n ␦共
ti−t1兲 for n
ⱖ2 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 cu-mulants withn greater than two are nonzero.
The master equation that describes this burstlike process is关17兴
p共x,t兲
t = −
x关共␦−␥2x兲p共x,t兲兴+k1
冕
0
x
w共x−y兲p共y,t兲dy
−k1p共x,t兲, 共2兲
wherep共x,t兲⬅p共x,t兩x0, 0兲is the conditional probability that the protein concentration has a valuexat timetgiven that it has a valuex0 at time 0; andw共x兲=
␥1 k2e
−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 determin-istic 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 yto a population of size x, w共x−y兲, given thatk1is 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
ps共x兲=
冉
␥1 k2
冊
k1/␥2⌰共x
−␦/␥2兲
⌫共k1/␥2兲
冉
x− ␦␥2
冊
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 con-sistent with a burstlike protein production, steady-state dis-tributions of protein abundances are equally compatible with alternative explanations. In fact, because mRNA is unstable compared to protein lifetime 共␥1Ⰷ␥2兲, one can assume that transcripts give rise to a constant flux of proteins f and sub-sequently 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 fluc-tuations 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 multi-plicative Gaussian noise共see Appendix C兲. Within thissce-nario the stochastic evolution of the protein concentration x共t兲is governed by the equation
x˙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 introducedad hoc. It has its roots in the central limit theorem and can also be justified on the basis of other general con-siderations 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 dis-crete Markov process can be well described by the multipli-cative noise term present in Eq. 共4兲in the continuum limit, i.e., when a large number of molecules is present. In addi-tion, because the noise goes to zero when x共t兲= 0, it also prevents the random variablex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 na-ture 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 isk2/␥1, rela-tions that are consistent with previous findings关18兴. In order to take into account the effects of feedback in a system un-dergoing 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-0 100 200 300 400 500 600 700
number of proteinsfluorescence units 0
100 200 300 400
num
b
er
o
f
ce
ll
[image:3.609.332.539.66.201.2]s
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 dis-tinguish between the steady-state distributions predicted by Eq.共3兲
with␦= 0 and␦⬎0.
AZAELE, BANAVAR, AND MARITAN PHYSICAL REVIEW E80, 031916共2009兲
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. De-spite 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 sta-tionary 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 so-lutions derive from a careful consideration of the bursty na-ture of the noise. In order to circumvent the indistinguish-ability of steady states, one can look into empirical protein abundances far from stationarity, for which we provide ana-lytical formulas. Thus we turn now to a study of the dynam-ics 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
k1␥1
k2␥2
共e␥2t− 1兲e−k1t
⫻exp
冋
−␥1 k2e␥2t共x−
t兲
册
⫻1F1
冉
k1 ␥2+ 1,2;␥1 k2
共e␥2t− 1兲共x−
t兲
冊
, 共5兲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
de-terministic 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/␥2Ⰶ1. In this case, the solution in Eq.共5兲can be approximated by replac-ing 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 de-pendence of the initial dynamics ofF共t兲on the nature of the noise. Note that, at stationarity, the distinct types of stochas-ticity 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 trans-lation rates and limt→⬁F共t兲=k2/␥1⯝1.
V. FRACTIONAL PROTEIN DISTRIBUTION
Another measurable quantity that directly probes the pro-tein distribution and its temporal evolution is the fractional protein distribution共FPD兲,P共,t兲, i.e., the probability that at timet the ratio x共t兲/x共0兲 is equal to , where x共t兲 andx共0兲 are the protein concentrations at time t⬎0 andt= 0, respec-tively. 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 prin-ciple, 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 att⬎0 the FPD is
100 200 300 400 500 600 700x 0.001
0.002 0.003 0.004
px,t0,0 t120 min
100 200 300 400 500 600 700x 0.001
0.002 0.003 0.004
px,t0,0 t240 min 100 200 300 400 500 600 700x
0.001 0.002 0.003 0.004 0.005 0.006 0.007
px,t0,0 t40 min
100 200 300 400 500 600 700x 0.001
0.002 0.003 0.004 0.005
px,t0,0 t60 min 100 200 300 400 500 600 700x
0.005 0.01 0.015 0.02 0.025 0.03 0.035
px,t0,0 t5 min
100 200 300 400 500 600 700x 0.002
0.004 0.006 0.008 0.01
[image:4.609.53.294.67.297.2]px,t0,0 t20 min
FIG. 2. Protein distribution dynamics for different types of noise and with the same initial condition, i.e.,x0= 0 proteins att= 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
= 12.88 and we have set␥2−1= 40 min,␥1−1= 2 min.
50 100 150 200 250 tmin
10 20 30 40 50 Ft
50 100 150 200 250 tmin
[image:4.609.328.536.68.203.2]0.5 1.0 1.5 Ft
FIG. 3. Fano factor dynamics with the same initial condition, i.e.,x0= 0 proteins att= 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
produced frequently. In both cases we have set ␥1−1= 2 and ␥2−1
P共,t兲=e−k1t␦共−e−␥2t兲+
冉
k1␥2
冊
2e−k1t共e␥2t
− 1兲⌰共−e−␥2t兲
关1 +e␥2t共−e−␥2t兲兴k1/␥2+1
⫻2F1
冉
k1 ␥2+ 1,k1 ␥2
+ 1,2;共e
␥2t− 1兲共−e−␥2t兲
1 +e␥2t共−e−␥2t兲
冊
, 共6兲where2F1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 ande−␥2teven 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 onk1 and␥2 only,
共iii兲at very large time separation there is only one free pa-rameter, the ratio k1/␥2, and the FPDs predicted by the Gaussian and non-Gaussian noises become the same共see Ap-pendix 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 Ap-pendix D兲.
The closer the system is to its steady state, the more dif-ficult it is to distinguish among the effects of gestation, se-nescence 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 spe-cific 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.
VI. CONCLUSION
In summary, we have shown that fits to experimental data of the stationary solution do not discriminate between differ-ent types of noise. However, the temporal evolution of the probability distribution of protein or fractional protein con-centration under stationary conditions as well as the evolu-tion 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 corre-sponding model with Gaussian multiplicative noise.
ACKNOWLEDGMENT
This work was supported by PRIN 2007.
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.
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 transi-tions, say a probability density functionw共x兲withxⱖ0. Let us further assume that the burst processes occur in time ac-cording to a Poisson distribution qn共t兲=共k1t兲nexp共−k1t兲/n!, wherek1is the transcription rate.
Because any burst produces xi proteins according to the
distributionw共x兲, we are interested in the distribution of the random variable 共t兲=兺i=1
n x
i, i.e., the total amount of
pro-teins produced fromt= 0 throught⬎0, where we are assum-ing thatnbursts have been occurred in a timet. The charac-teristic function for the process 共t兲is
具eiz共t兲典=
兺
n=0⬁
具eiz共t兲兩n events by t典prob关n events by t兴
=
兺
n=0
⬁
关g共z兲兴nqn共t兲, 共A1兲
where we have used the independence of any one-burst pro-cess and we have used the definition
g共z兲 ⬅ 具eizxj典=
冕
0
⬁
eizxw共x兲dx, 共A2兲
for any j= 1 , . . . ,n. Because we are using the Poisson distri-bution 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⬁eizxp共x,t兲dx=
兺
n=0
⬁
关g共z兲兴nqn共t兲. 共A4兲
By direct substitution one can verify that a solution is
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
0x
w共x−y兲w共y兲dx
+ ¯ =
兺
n=0
⬁
qn共t兲w共x兲〫w共x兲〫 ¯ 〫w共x兲
n-fold convolution
,
共A5兲
where␦共x兲is a Dirac delta and the symbol “〫” stands for a convolution ofw共x兲’s.
Now we use the particular jump distribution w共x兲=e−x
共⬅␥1/k2兲. Hence
g共z兲=
−iz, 共A6兲
and the calculations for thenth convolution共nⱖ1兲result in
n x
n−1
共n− 1兲!e −x
. 共A7兲
Thus we obtain with qn共t兲=共k1t兲nexp共−k1t兲/n!
AZAELE, BANAVAR, AND MARITAN PHYSICAL REVIEW E80, 031916共2009兲
p共x,t兲=e−k1t␦共x兲+ e
−k1t−x
x
兺
n=1⬁
共k1tx兲n
共n− 1兲!n!
=e−k1t␦共x兲+ t
xe −k1t−x
tI0共2
冑
k1tx兲=e−k1t␦共x兲+ e
−k1t−x
x
冑
k1txI1共2冑
k1tx兲, 共A8兲whereI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
⬁
共x/2兲+2n n!⌫共+n+ 1兲.
By exploiting this definition, one can see that in Eq.共A8兲 there is no divergence atx= 0. We can also calculate all the moments
具n共t兲典
= n!
nk1te−k1t1F1共n+ 1,2;k1t兲+e−k1t␦n,0, 共A9兲
where 1F1共a,b;x兲 is the confluent hypergeometric function
关27兴andn= 0 , 1 , 2 , . . ..
Now we can consider the fundamental stochastic differen-tial equation which defines the model in the main text,
dx共t兲=
冉
␦−x共t兲
冊
dt+d共t兲, 共A10兲where⬅␥2−1 and共t兲is the noise whose probability distri-bution 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.
APPENDIX B: THE SOLUTION OF THE MASTER EQUATION
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,
p共x,t兲
t = −
x
冋
冉
␦− x
冊
p共x,t兲册
+ 1 冕
0x
w共x−y兲p共y,t兲dy
−1
p共x,t兲, 共B1兲
where⬅k1−1, ⬅␥2−1, ␦ⱖ0 , w共x兲is a probability density function andp共x,t兲⬅p共x,t兩x0, 0兲is the conditional probabil-ity that the protein concentration has a valuexat timetgiven that it has a valuex0 at time 0. Note that Eq.共B1兲exhibits two temporal scales, and, which are related to transcrip-tion and protein dilutranscrip-tion, respectively.
The probability flux determined by Eq.共B1兲is
j共x,t兲=
冉
␦−x
冊
p共x,t兲+ − 1 冕
0x
冕
0zw共z−y兲p共y,t兲dzdy
+1
冕
0x
p共y,t兲dy, 共B2兲
where we assume that limx→⬁ j共x,t兲= 0 for any t⬎0. The
flux atx= 0 is simply
j共0,t兲=␦ lim
x→0+
p共x,t兲, 共B3兲
where limx→0+xp共x,t兲= 0 due to the integrability ofp共x,t兲at
x= 0. By Laplace transforming Eq. 共B1兲 with respect to x, one obtains
共s,t兲
t =j共0,t兲−
s
共s,t兲
s −bs共s,t兲+
关共s兲− 1兴共s,t兲
,
共B4兲
where 共s兲⬅兰0⬁w共x兲e−sxdx 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
s =
d
共共s兲− 1 −␦s兲, 共B5兲
which can be easily solved. The solutions may be written as follows:
se−t/=s0, 共B6兲
共s兲=0exp
冋
冕
s0
s
共兲− 1
d−␦共s−s0兲
册
, 共B7兲wheres0,0are 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,
共s,t兲=共se−t/兲exp
冋
冕
se−t/s
共兲− 1
d
−␦共s−se−t/兲
册
. 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+
冕
se−t/s
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
⍀共s,t兲= exp
冋
冕
se−t/s
共兲− 1
d
册
. 共B10兲We are interested in solving Eq. 共B1兲 with w共x兲=e−x,
main text. So共s兲=/共+s兲and⍀共s,t兲becomes
⍀共s,t兲=
冉
se −t/+s+
冊
/. 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/␦共x−t兲+⌰共x−t兲
e −t/共et/
− 1兲
⫻exp关−et/共x−
t兲兴
⫻1F1
冉
+ 1,2;共e
t/
− 1兲共x−t兲
冊
, 共B12兲where t⬅x0e−t
/+␦共1 −e−t/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 forxⱖ0 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 differen-tiable共atx=t兲. One may also achieve the steady-state
solu-tion of Eq.共B1兲by using the leading term of the asymptotic representation of the confluent hypergeometric function关27兴, i.e.,
1F1共a,b;x兲 ⯝
⌫共b兲
⌫共a兲e
xx−共b−a兲
, 共B13兲
whenxⰇ1 anda,b⫽0 , −1 , −2 , . . .. Thus the normalized
so-lution at stationarity is
ps共x兲=
/
⌫共/兲⌰共x−␦兲共x−␦兲
/−1e−共x−␦兲
. 共B14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 ob-tained in aprokaryoticcell population: the agreement is ex-cellent. When the mean number of bursts per cell cycle is very small, one obtains /Ⰶ1 and we can expand Eq.
共B12兲 in powers of /. In this case, the solution in Eq.
共B12兲reads
p共x,t兩x0,0兲=e−t/␦共x−t兲+
冉
冊
⌰共x−t兲
x−t
共exp关−共x−t兲
−t/兴− exp关−et/共x−t兲−t/兴兲+O„共/兲2…,
共B15兲
where we have used共see Ref.关27兴兲
1F1共1,2;z兲= ez− 1
z .
Laplace transform
In this section we calculate the inverse Laplace transform of Eq.共B11兲. In order to get it we study the function
⌽共x兲 ⬅ 1 2i
冕
k−i⬁k+i⬁
冉
as+bs+
冊
c
esxds, 共B16兲
wherex⬎0,k⬎0, 0⬍a⬍b, ⬎0, andc⬎0. On using the variable z⬅共as+兲/¯z, with¯z⬅共1 −a/b兲⬎0, one obtains
⌽共x兲 ⬅e −x/a
a
冉
a
b
冊
c ¯z
2i
冕
z⬘−i⬁z⬘+i⬁
冉
1 −1 z
冊
−c
e共xz¯/a兲zdz,
共B17兲
wherez
⬘
⬎1. We can also write冉
1 −1z
冊
−c=
兺
n=0
⬁ 共c兲n
n! 1
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⬁e共xz¯/a兲z
zn dz= 2i␦
冉
xz¯a
冊
+
兺
n=1
⬁ 共c兲n
n!
冉
xz¯a
冊
n−1
冕
⬘⬘−+i⬁i⬁end,
共B19兲
where
⬘
⬎1 and we used␦共x兲= 1 2i
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 inte-grand has a pole of order n at z= 0. Thus, by applying the residue theorem, one obtains
冖
e nd=2i
共n− 1兲!. 共B20兲
Therefore Eq. 共B19兲gets
1 2i
兺
n=0⬁ 共c兲n
n!
冕
z⬘−i⬁z⬘+i⬁e共xz¯/a兲z
zn dz=␦
冉
xz¯a
冊
+n兺
=1⬁ 共c兲n
n!
共xz¯/a兲n−1
共n− 1兲!.
共B21兲
We may use the definition and the properties of the con-fluent hypergeometric function 关27兴to rewrite this latter re-lation as follows:
兺
n=1
⬁ 共c兲n
n!
共xz¯/a兲n−1
共n− 1兲! =
共xz¯/a兲1F1共c,1;xz¯ /a兲
=c1F1共c+ 1,2;xz¯/a兲, 共B22兲
and finally reaching the equation
⌽共x兲=e −x/a
a
冉
a
b
冊
c
z ¯
冋
␦冉
xz¯a
冊
+c1F1共c+ 1,2;xz¯/a兲册
,共B23兲
AZAELE, BANAVAR, AND MARITAN PHYSICAL REVIEW E80, 031916共2009兲
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 timet,x共t兲, within an isogenic cell population,
x˙共t兲=␦−x共t兲/+⌳共t兲, 共C1兲
where⬅␥2−1and
具⌳共t兲典=f,
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.共C2兲
Because we suppose that all the other cumulants are neg-ligible, 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:
p˙=x关共x/−f−␦兲p兴+Dx2共xp兲. 共C3兲
In this case the constant␦only shifts the average number of proteins, so we can assume that␦= 0 without lack of gener-ality. In Eq.共C3兲p⬅p共x,t兩x0, 0兲is the conditional probabil-ity that the protein concentration has a valuexat timetgiven that it has a valuex0at time 0, i.e.,兰n
n+⌬n
p共x,t兩x0, 0兲dxis the fraction of cells with protein population between n and n +⌬n; furthermore, f⬅k1k2/␥1 and D⬅␥2k2/␥1. Setting p˙ = 0, one obtains the stationary solution of Eq.共C3兲,
pstat共x兲=共D兲−f
/D⌫共f/D兲−1xf/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 em-bodies 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 expres-sion
p共x,t兩x0,0兲=
冉
1 D冊
f/D
xf/D−1e−x/D
冋
冉
1 D
冊
2
x0xe−t/
册
1/2−f/2D
1 −e−t/
⫻exp
冤
− 1D共x+x0兲e −t/
1 −e−t/
冥
If/D−1冤
2 D冑
x0xe−t/
1 −e−t/
冥
.共C5兲
In Fig.2of 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.
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 tthe ratiox共t兲/x共0兲 is equal to, wherex共t兲andx共0兲are the protein concentrations within a population of isogenic cells at time t⬎0 andt= 0, respectively. The system can be initially prepared according to any kind of probability density function, pinit共x0兲, never-theless 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 time-dependent reflecting solution and the stationary probability distribution with ␦= 0, respectively. Thus, by definition the FPD is
PFPD共,t兲=具␦共−x/x0兲典
=
冕
0
⬁
dx0
冕
0⬁
dxpstat共x0兲p共x,t兩x0,0兲␦共−x/x0兲,
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
t→+⬁
PFPD共,t兲=
冕
0
⬁
dx0x0pstat共x0兲pstat共x0兲
=⌫共2兲
⌫2共兲 −1
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 singular-ity at = 0. Furthermore, because the Gaussian and non-Gaussian noises have the same steady state, they both have the same FPD whent→+⬁.
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
/+1
⌫共/兲e
−t/
⫻共et/− 1兲
冕
0⬁
dxx/exp关−x共1 +et/共−e−t/兲兲兴
⫻1F1
冉
+ 1,2;共e
t/
− 1兲共−e−t/兲x
冊
, 共D3兲where ⬅k1−1, ⬅␥2−1
, ⬅␥1/k2. On exploiting the for-mula 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 /Ⰶ1 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+ −
冉
冊
2⌰共−e−t/兲e−t/
−e−t/
⫻ln
冉
1 +−e−t/
1 +et/共−e−t/兲
冊
+O关共/兲3兴, 共D4兲
2F1共1,1,2;z兲= −
ln共1 −z兲
z .
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
⌫
冉
fD+ 1 2
冊
⌫
冉
D冊f共+ 1兲
共et/兲f/2D 1 −e−t/
⫻
冢
sinh
冉
t 2冊
冣
f/D+1
冉
42共+ 1兲2et/− 4
冊
f/D+1/2 ,
共D5兲
where f⬅k1k2/␥1, ⬅␥2 −1
and D⬅␥2k2/␥1. Note that in this casePFPD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.
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0.5 1 1.5 2 2.5 3 Ρ 0.2
0.4 0.6 0.8 1 1.2
FPD t60 min
0.5 1 1.5 2 2.5 3 Ρ 0.2
0.4 0.6 0.8 1
FPD t200 min 0.5 1 1.5 2 2.5 3 Ρ
0.25 0.5 0.75 1 1.25 1.5 1.75
FPD t20 min
0.5 1 1.5 2 2.5 3 Ρ 0.25
0.5 0.75 1 1.25 1.5
FPD t30 min 0.5 1 1.5 2 2.5 3 Ρ
1 2 3 4 5 6
FPD t1 min
0.5 1 1.5 2 2.5 3 Ρ 0.5
1 1.5 2 2.5
[image:9.609.316.557.66.302.2]FPD t10 min
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/␥1and
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 casesk1/␥2= 12.88 and we have set␥2
−1
= 40 min.
AZAELE, BANAVAR, AND MARITAN PHYSICAL REVIEW E80, 031916共2009兲