Computational Methods in Systems Biology

Here, RNA production kinetics is controlled bykccandkoc. The probability density
function (pdf) of the distribution of intervals between transcription events is the con-

volution of their pdfs:fDtðÞ¼t kkoccc
kkoccc ekcc
tekoc
t



. To measure asymmetries in this

distribution, we use skewness,S¼m^3
m^32 =^2
, wheremr¼^1 nRðÞxixr[4]. We estimate the

sample skewnessSs¼

ffiffiffiffiffiffiffiffiffiffiffi

nnðÞ 1

p
n 2
S, wherenis the sample number [5]. To obtain con-
fidence boundaries forSswe use non-parametric bootstraps as in [6].
In ( 1 ),kccis the inverse of the mean time forRto bind the promoter and complete a
closed complex (scc), whilekocis the inverse of the mean time for an open complex to
form (soc). The mean time between transcription events:Δt=scc+soc.
To validate the model predictions of skewness, we collected empirical data forΔt
andscc/Δt for various promoters (PTetA,PBAD,PLac-ara-1, and PLac-ara-1under oxidative
stress) [7–9] (Fig. 1 ). Next, given the meanΔt of each promoter, we variedscc/Δt (from
0 to 1) while maintainingΔt constant. Then, for each value ofscc/Δt, we calculated
Sfrom the pdf of the distribution of intervals between transcription events (solid line,
Fig. 1 ). Interestingly, we observed thatSis independent of the mean value ofΔt.
Finally, from Fig. 1 ,wefind that the model predictions ofSfit the empirical data.

Importantly, asSis tunable bysccandsoc, which are sequence dependent and
subject to regulation, we expect it to be evolvable and adaptive to environment shifts.

References

1. Kaern, M., et al.: Stochasticity in gene expression: from theories to phenotypes. Nat. Rev.
   Genet. 6 , 451–464 (2005)


2. McClure, W.R.: Mechanism and control of transcription initiation in prokaryotes. Annu. Rev.
   Biochem. 54 , 171–204 (1985)


3. Uptain, S.M., et al.: Basic mechanisms of transcript elongation and its regulation. Annu. Rev.
   Biochem. 66 , 117–172 (1997)


4. MacGillivray, H.L.: Skewness and asymmetry: measures and orderings. Ann. Stat. 14 , 994–
   1011 (1986)

0 0.2 0.4 0.6 0.8 1

cc/ t

1

1.5

2

2.5

Skewness

BAD

TetA Lac-ara-1

Lac-ara-1

(Oxidative)

Fig. 1. Predicted skewness ofΔt distributions with given scc/Δt (solid line) and sample
skewness of the empiricalΔt distributions (with 95% confidence intervals) for the studied
promoters. For each promoter, 100 or moreΔt intervals were extracted from a total of 100 or
more cells.

328 S. Startceva et al.