2.2.1. The case of two competing probe sets
In this section, we consider the simplest case of two probe sets targeting the same transcript. Much like the microarray technology in general, our theory will be based on the following assumption:
Assumption 1. No saturation effects are present and the true signal intensity (disregarding the technical measurement error) is proportional to the amount of RNA bound to a given probe set with a constant (non-random) coefficient of proportionality. The coefficients for different probe sets do not have to be the same.
Remark 1. We tested Assumption 1 using the data set on various spiked-in probes (GeneChip HG_U133A) provided by the file HG_U133A_tag_Latin_Square.zip on the Affymetrix website. For some spiked-in probes, the dependence of the fluorescence intensity on the molar concentration was indeed linear, but deviations from linearity were also noted. The small sample size (n = 3) in this study and its sub-optimal design that uses a non-linear scale for the independent variable still leave room for speculations. A more rigorous metrological study is required to validate Assumption 1 governing the overall usefulness of modern high-density oligonucleotide microarray technology.
Proceeding from Assumption 1, suppose that two probe sets compete for the same (common) transcript so that the first probe set binds to this transcript with probability p, and, given the transcript remains unbound, the second probe set binds to it with probability 1 - p. Denote the amount of the common transcripts (RNA molecules) by ν. The model considered by Okoniewski and Miller 1 assumes that the signal intensities Z1 and Z2 for the two probe sets can be represented as Z1 = pν, Z2 = (1 - p)ν, which is why its simulation counterpart always displays a positive covariance. In mechanistic terms, these relationships imply that each probe set "knows" exactly what proportion of the random amount ν it must "catch" in the course of hybridization, thereby making this essentially deterministic model highly implausible.
A more natural model can be constructed if one proceeds from a stochastic nature of cross-hybridization and represents the first probe set signal as
Z
1
=
∑
j
=
1
ν
ξ
j
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aaSbaaSqaaiabigdaXaqabaGccqGH9aqpdaaeWbqaaGGaciab=57a4naaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiab=17aUbqdcqGHris5aOGaeiilaWcaaa@3B04@
where ξj are independent and identically distributed (i.i.d.) indicator variables taking on values 1 and 0 with the probabilities p and 1 - p, respectively. For the second probe set we have
Z
2
=
∑
j
=
1
ν
(
1
−
ξ
j
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aaSbaaSqaaiabikdaYaqabaGccqGH9aqpdaaeWbqaaiabcIcaOiabigdaXiabgkHiTGGaciab=57a4naaBaaaleaacqWGQbGAaeqaaOGaeiykaKIaeiOla4caleaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWF9oGBa0GaeyyeIuoaaaa@3EA4@
The mutually dependent random variables (r.v.s) Z1 and Z2 satisfy the condition: Z1 + Z2 = ν. It is easy to verify that the covariance between Z1 and Z2 can be of any sign under this model. Indeed,
Cov
(
Z
1
,
Z
2
)
=
E
{
∑
j
=
1
ν
ξ
j
∑
k
=
1
ν
(
1
−
ξ
k
)
}
−
E
{
∑
j
=
1
ν
ξ
j
}
E
{
∑
k
=
1
ν
(
1
−
ξ
k
)
}
=
p
(
1
−
p
)
E
{
ν
(
ν
−
1
)
}
−
p
(
1
−
p
)
(
E
{
ν
}
)
2
=
p
(
1
−
p
)
(
Var
{
ν
}
−
E
{
ν
}
)
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaiabboeadjabb+gaVjabbAha2jabcIcaOiabdQfaAnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaemOwaO1aaSbaaSqaaiabikdaYaqabaGccqGGPaqkcqGH9aqptuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=ri8fjabcUha7naaqahabaacciGae4NVdG3aaSbaaSqaaiabdQgaQbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGae4xVd4ganiabggHiLdGcdaaeWbqaaiabcIcaOiabigdaXiabgkHiTiab+57a4naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiyFa0haleaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqGF9oGBa0GaeyyeIuoakiabgkHiTiab=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ri8fjabcUha7jab+17aUjabc2ha9jabcMcaPiabcYcaSaaaaaa@C37D@
so that the sign of Cov (Z1, Z2) depends on the relationship between Var{ν} and E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B3@{ν}, both parameters being unobservable, of course. In particular, Cov (Z1, Z2) = 0 if ν has a Poisson distribution. We find it very interesting that the sign of correlation is entirely determined by probabilistic characteristics of the common transcript and not by its affinity to the competing probe sets, the latter being quantified by the probability p.
We agree with Dr. Gaile that one can easily derive the correlation coefficient between Z1 and Z2 by the same straightforward argument. However, we resort to another derivation based on the notion of probability generating function (p.g.f.), which is instructive to obtain the needed results in Section 2.2.2. Introduce the p.g.f. of the random variable (r.v.) ν:
P
ν
(
u
)
=
E
u
ν
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeeiuaa1aaSbaaSqaaGGaciab=17aUbqabaGccqGGOaakcqWG1bqDcqGGPaqkcqGH9aqptuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+ri8fjabdwha1naaCaaaleqabaGae8xVd4gaaOGaeiOla4caaa@4388@
Then the Laplace transform f(s, t) of the joint distribution of the vector (Z1, Z2) assumes the form
f
(
s
,
t
)
=
E
{
e
−
s
Z
1
−
t
Z
2
}
=
E
{
e
−
s
∑
j
=
1
ν
ξ
j
−
t
∑
j
=
1
ν
(
1
−
ξ
j
)
}
=
P
ν
(
p
e
−
s
+
(
1
−
p
)
e
−
t
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8873@
Differentiating log f(s, t) twice (with respect to both s and t), one can derive the covariance between Z1 and Z2. The variance of Z1 can be obtained by differentiating log f(s, t) twice with respect to s. In like manner, one obtains the variance of Z2 by differentiating log f(s, t) twice with respect to t. This leads to the following formula for the correlation coefficient between Z1 and Z2:
![]()
Introducing the notation κ = Var{ν}/E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B3@{ν} and taking into account that
P
′
ν
(
1
)
=
E
ν
,
P
″
ν
(
1
)
=
E
ν
2
−
E
ν
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiqbbcfaqzaafaWaaSbaaSqaaGGaciab=17aUbqabaGccqGGOaakcqaIXaqmcqGGPaqkcqGH9aqptuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+ri8fjab=17aUjabcYcaSaqaaiqbbcfaqzaagaWaaSbaaSqaaiab=17aUbqabaGccqGGOaakcqaIXaqmcqGGPaqkcqGH9aqpcqGFecFrcqWF9oGBdaahaaWcbeqaaiabikdaYaaakiabgkHiTiab+ri8fjab=17aUjabcYcaSaaaaaa@530B@
formula (3) assumes the form:
C
o
r
r
(
Z
1
,
Z
2
)
=
p
(
1
−
p
)
(
κ
−
1
)
p
2
κ
+
p
(
1
−
p
)
(
1
−
p
)
2
κ
+
p
(
1
−
p
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6185@
Figure 3 shows the behavior of Corr(Z1, Z2) as a function of p for different values of the parameter k. We thank Dr. Gaile for this figure that illustrates the parametric family given by formula (4). The right-hand side of (4) attains a maximum at p = 0.5 and consequently
<p>Figure 3</p>
The behavior of Corr(Z1, Z2) as a function of p for different values of the parameter k
The behavior of Corr(Z1, Z2)as a function of p for different values of the parameter k. This figure was provided by Dr. Gaile in his review.
![]()
κ
≥
1
+
C
o
r
r
(
Z
1
,
Z
2
)
1
−
C
o
r
r
(
Z
1
,
Z
2
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8NUdSMaeyyzImBcfa4aaSaaaeaacqaIXaqmcqGHRaWkieaacqGFdbWqcqGFVbWBcqGFYbGCcqGFYbGCcqGGOaakcqWGAbGwdaWgaaqaaiabigdaXaqabaGaeiilaWIaemOwaO1aaSbaaeaacqaIYaGmaeqaaiabcMcaPaqaaiabigdaXiabgkHiTiab+neadjab+9gaVjab+jhaYjab+jhaYjabcIcaOiabdQfaAnaaBaaabaGaeGymaedabeaacqGGSaalcqWGAbGwdaWgaaqaaiabikdaYaqabaGaeiykaKcaaOGaeiOla4caaa@4DBC@
For example, if the model is valid for a particular pair (Z1, Z2) and Corr(Z1, Z2) ≥ 0.9, then it follows from inequality (5) that Var{ν} ≥ 19 E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B3@{ν}. In Data Set 1, 53.6% of bad probe sets and 39.4% of good probe sets satisfy this condition. However, this does not mean that the process of cross-hybridization plays a substantial role in the correlation structure of microarray data for the following reasons:
1. Inequality (5) provides a necessary but not a sufficient condition for the correlation Corr(Z1, Z2) to exceed a certain level.
2. Condition (5) applies only to those isolated gene pairs where the total amount of the target transcript is eventually bound to one of the two probe sets and no other transcripts can contribute to their ultimate expression measures (recall Section 2.1).
While the above model, designed for a single pair of probe sets competing for the same transcript, is instructive to study general regularities in the process of cross-hybridization, it is obviously of limited utility in studying probable causes of the observed long-range correlation that extends over thousands of genes. A more general model of the non-specific binding process for arbitrary pairs formed from multiple probe sets is considered in the next section.
2.2.2. Multiple probe sets
In Section 2.2.1, we confirmed that the process of cross-hybridization can induce spurious positive correlations within certain families of probe sets whose potential for multiple targeting is suggested by the analysis of transcript sequences. At the same time, our analysis of Section 2.1 shows that the overall correlation structure of gene expression data cannot be driven by the minority of non-overlapping pairs of such problematic probe sets. The analysis of transcript sequences and homologies, however, is only suggestive and we have to consider the case where the event of non-specific binding is admissible for virtually all probe sets, even if its probability for each of them is low. Presented below is a simplistic model that attempts to describe the process of massive cross-hybridization involving all probe sets.
Suppose there exists a large pool of common RNA molecules that have the ability to bind to any of the k probe sets under consideration. Under the proposed model, each probe set binds first to its own highly specific transcript without any competition with other probe sets (i.e., with probability 1) and then it binds to a portion of the common transcript in the competitive fashion. More specifically, each molecule of the common transcript is bound to the jth probe set with probability pj ≥ 0, j = 1,...,k, and the probabilities pj satisfy the condition: ∑j=1kpj=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGWbaCdaWgaaWcbaGaemOAaOgabeaakiabg2da9iabigdaXaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdaaaa@3779@
Denote the amount of a specific transcript bound to the jth probe set with probability 1 by Xj. To make our model identifiable, we need additionally the following basic assumption:
Assumption 2. The r.v.s Xj, j = 1,...,k, are mutually independent.
Assumption 2 is not intended for a realistic description of gene expression signals but rather as a reference for measuring cross-hybridization effects. Recall that Question 2 posed at the end of Section 2.1 explicitly refers to this assumption. Therefore, the results that follow should be interpreted under a hypothetical scenario attributing all of the observable correlation to the competition between multiple probe sets for the above-postulated common transcript.
Assuming the same mechanism of competition for the common transcript, and invoking the same independence assumptions as in Section 2.2.1, we write
Z
j
=
X
j
+
∑
l
=
1
ν
ε
l
j
,
j
=
1
,
...
,
k
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdQfaAnaaBaaaleaacqWGQbGAaeqaaOGaeyypa0JaemiwaG1aaSbaaSqaaiabdQgaQbqabaGccqGHRaWkdaaeWbqaaGGaciab=v7aLnaaDaaaleaacqWGSbaBaeaacqWGQbGAaaaabaGaemiBaWMaeyypa0JaeGymaedabaGae8xVd4ganiabggHiLdGccqGGSaalaeaacqWGQbGAcqGH9aqpcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGRbWAcqGGSaalaaaaaa@4A74@
where Zj is the total RNA amount bound to the jth probe set, εlj
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8xTdu2aa0baaSqaaiabdYgaSbqaaiabdQgaQbaaaaa@306C@ are indicators taking on a value of 1 with probability pj and of 0 with probability 1 - pj, ∑j=1kεlj=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaaiiGacqWF1oqzdaqhaaWcbaGaemiBaWgabaGaemOAaOgaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aOGaeyypa0JaeGymaedaaa@3915@, and the random vectors (εl11,...,εl1k)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeiikaGccciGae8xTdu2aa0baaSqaaiabdYgaSnaaBaaameaacqaIXaqmaeqaaaWcbaGaeGymaedaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIae8xTdu2aa0baaSqaaiabdYgaSnaaBaaameaacqaIXaqmaeqaaaWcbaGaem4AaSgaaOGaeiykaKcaaa@3D10@ and (εl21,...,εl2k)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeiikaGccciGae8xTdu2aa0baaSqaaiabdYgaSnaaBaaameaacqaIYaGmaeqaaaWcbaGaeGymaedaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIae8xTdu2aa0baaSqaaiabdYgaSnaaBaaameaacqaIYaGmaeqaaaWcbaGaem4AaSgaaOGaeiykaKcaaa@3D14@ are independent for l1 ≠ l2. The r.v. ν represents the total number of molecules of the common transcript available for the competition between different probe sets. To convert the quantity Zj into the corresponding expression intensity in accordance with Assumption 1, one needs to account for the fact that the specific and common transcripts may vary in size (mass). Without any loss of generality, this can easily be accomplished if the unit of measurement of the variables Xj is taken to be the molecular weight of the common transcript.
Recalling the definition of the p.g.f. P
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeeiuaafaaa@2CFA@ (Section 2.2.1), the joint characteristic function g of the random vector Z1 ,...,Zk can be represented as
g
(
t
1
,
...
,
t
k
)
=
∏
j
=
1
k
g
j
(
t
j
)
P
(
∑
j
=
1
k
p
j
e
i
t
j
)
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5C59@
where gj is the marginal characteristic function of Xj, j = 1,...,k. The covariance between two r.v.s equals a minus mixed second order log-derivative of their characteristic function evaluated at zero. From expression (7), we have
Cov
(
Z
r
,
Z
s
)
=
(
σ
ν
2
−
E
ν
)
p
r
p
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaee4qamKaee4Ba8MaeeODayNaeiikaGIaemOwaO1aaSbaaSqaaiabdkhaYbqabaGccqGGSaalcqWGAbGwdaWgaaWcbaGaem4CamhabeaakiabcMcaPiabg2da9iabcIcaOGGaciab=n8aZnaaDaaaleaacqWF9oGBaeaacqaIYaGmaaGccqGHsisltuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+ri8fjab=17aUjabcMcaPiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaemiCaa3aaSbaaSqaaiabdohaZbqabaaaaa@5447@
for r = 1,...,k, s = 1,...,k, r ≠ = s. Here σν2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae83Wdm3aa0baaSqaaiab=17aUbqaaiabikdaYaaaaaa@306F@ is the variance of the r.v. ν. Using the condition
∑
r
=
1
k
∑
s
=
1
k
p
r
p
s
=
1
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaadaaeWbqaaiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaemiCaa3aaSbaaSqaaiabdohaZbqabaGccqGH9aqpcqaIXaqmaSqaaiabdohaZjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aaWcbaGaemOCaiNaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdGccqGGSaalaaa@4325@
from (8) we obtain the formula
(
σ
ν
2
−
E
ν
)
(
1
−
∑
j
=
1
k
p
j
2
)
=
∑
r
≠
s
Cov
(
Z
r
,
Z
s
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeiikaGccciGae83Wdm3aa0baaSqaaiab=17aUbqaaiabikdaYaaakiabgkHiTmrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4hHWxKae8xVd4MaeiykaKIaeiikaGIaeGymaeJaeyOeI0YaaabCaeaacqWGWbaCdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aOGaeiykaKIaeyypa0ZaaabuaeaacqqGdbWqcqqGVbWBcqqG2bGDcqGGOaakcqWGAbGwdaWgaaWcbaGaemOCaihabeaakiabcYcaSiabdQfaAnaaBaaaleaacqWGZbWCaeqaaOGaeiykaKcaleaacqWGYbGCcqGHGjsUcqWGZbWCaeqaniabggHiLdGccqGGUaGlaaa@6452@
Note that the right-hand side of formula (9) can be estimated from observed expression measures. Minimizing ∑j=1kpj2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGWbaCdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aaaa@3661@ with respect to pj under the constraint ∑j=1kpj=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGWbaCdaWgaaWcbaGaemOAaOgabeaakiabg2da9iabigdaXaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdaaaa@3779@, we have
σ
ν
2
−
E
ν
≥
k
k
−
1
∑
r
≠
s
Cov
(
Z
r
,
Z
s
)
≥
∑
r
≠
s
Cov
(
Z
r
,
Z
s
)
=
γ
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae83Wdm3aa0baaSqaaiab=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@7111@
The latter inequality allows us to estimate a lower bound for σν2−Eν
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae83Wdm3aa0baaSqaaiab=17aUbqaaiabikdaYaaakiabgkHiTmrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4hHWxKae8xVd4gaaa@3EF8@.
To use inequality (10), one needs to compute covariances for all gene pairs. For the GeneChip HG_U133A, this means conducting such computations for all pairs formed from 12340 genes, which is computationally prohibitive. For this reason, we used the St. Jude Hospital Children's Research Hospital Database produced with U95 arrays 17. Specifically, we used a subset of data that reports expression levels of k = 7084 genes in n = 79 patients with childhood leukemia (TELL type). In this data set, probe sets with dubious definitions 18 were removed using the custom CDF file from 19, which gives reason for us to believe that our analysis refers predominantly to "good" probe sets. The estimated value of the lower bound γ in inequality (10) for these probe sets is γ^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf83SdCMbaKaaaaa@2D91@ = 8.74 × 1011.
There are two ways of interpreting the estimate γ^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf83SdCMbaKaaaaa@2D91@ resulted from our analysis. First, we proceed from the obvious inequality: σν2>8.74×1011
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae83Wdm3aa0baaSqaaiab=17aUbqaaiabikdaYaaakiabg6da+iabiIda4iabc6caUiabiEda3iabisda0iabgEna0kabigdaXiabicdaWmaaCaaaleqabaGaeGymaeJaeGymaedaaaaa@3B57@. The average (across genes) variance of gene expression levels in this data set is 3.5 × 104, which is at least seven orders of magnitude lower than that of the common transcript given by the above inequality. This huge variance of ν seems quite implausible if the common transcript is thought of as being produced by one of the typical protein-encoding genes such as those that are measured by the standard microarray technology. Second, we can gain some insight into the magnitude of the mean value of ν. Introducing the notation μ = E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B3@ν, we represent inequality (11) in the form
v
ν
2
μ
2
−
μ
≥
k
k
−
1
∑
r
≠
s
Cov
(
Z
r
,
Z
s
)
≥
∑
r
≠
s
Cov
(
Z
r
,
Z
s
)
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaGGaciab=17aUbqaaiabikdaYaaakiab=X7aTnaaCaaaleqabaGaeGOmaidaaOGaeyOeI0Iae8hVd0MaeyyzImBcfa4aaSaaaeaacqWGRbWAaeaacqWGRbWAcqGHsislcqaIXaqmaaWaaabuaeaacqqGdbWqcqqGVbWBcqqG2bGDcqGGOaakcqWGAbGwdaWgaaqaaiabdkhaYbqabaGaeiilaWIaemOwaO1aaSbaaeaacqWGZbWCaeqaaiabcMcaPiabgwMiZoaaqafabaGaee4qamKaee4Ba8MaeeODayNaeiikaGIaemOwaO1aaSbaaeaacqWGYbGCaeqaaiabcYcaSiabdQfaAnaaBaaabaGaem4CamhabeaacqGGPaqkaeaacqWGYbGCcqGHGjsUcqWGZbWCaeqacqGHris5aaqaaiabdkhaYjabgcMi5kabdohaZbqabiabggHiLdGaeiilaWcaaa@6515@
where v = σν/μ is the variation coefficient of the r.v. ν. Figure 4 displays the histogram of variation coefficients in the TELL data. The histogram is extremely narrow, indicating that the variation coefficient is effectively constant across genes. The same fact was documented by Wu and Irizarry 20. In the TELL data set, the mean variation coefficient is equal to 0.235. Using this value as an estimate of v and solving the quadratic inequality (11) with respect to μ, we have μ > 3.98 × 106. This lower bound for μ is quite close to the mean total expression, i.e., the sum of the mean values of all gene expressions, the latter being equal to 4.22 × 106.
<p>Figure 4</p>
Variation coefficients for gene expression levels in the TELL data set
Variation coefficients for gene expression levels in the TELL data set.
![]()
The unrealistically huge mean and variance of the hypothetical common transcript targeted by multiple probe sets make it very unlikely that a massive cross-hybridization manifests itself at the probe set level. Put another way, if such a putative transcript exists and induces the observed long-range correlation between gene expression signals by binding to multiple probe sets, it cannot be a typical protein-encoding transcript, irrespective of whether its affinity to the probe sets is high or low. Nor can the random technical noise cause this kind of correlation between gene expression levels as follows from our analysis of the MAQC data 2. This leads us to conclude that the observed correlation structure of microarray data is of a biological nature. There are several conceivable biological causes of correlations between gene expression signals such as regulatory processes engaged in gene function, composite cellular make-up of tissues, and latent heterogeneity of subjects. We see no way to estimate their contributions from microarray data because of identifiability problems.
Remark 2. Regulatory interactions between genes participating in biochemical pathways or networks may (or may not) cause only a short-range correlation or the so-called clumpy dependence 21. Superimposed on this causal dependence are the effects caused by different species of noncoding RNA implicated in regulation of large sets of genes. The global term "noncoding RNA" (ncRNA) refers to a large class of transcripts that do not encode a protein product. An important subclass of functional ncRNAs is represented by microRNAs (miRNAs). These small, typically 21-25nt long, transcripts have been subject of intense studies in recent years. Using either miRNA transfection into cultured cells 22 or miRNA antagonists in vivo 23, it has been shown that a particular miRNA may affect hundreds of genes by interfering with their transcripts. This is a large-scale effect but it is still doubtful whether the observed long-range correlation between gene expression levels, involving thousands of genes, can be exhaustively explained by this mechanism. The two major modes of miRNA action are mRNA cleavage and translational inhibition. In the latter case, all untranslated mRNAs are eventually fated to degradation as well. While there is some similarity between such effects and those of cross-hybridization (binding to a common transcript), they call for a different stochastic model that would allow for the cognate mRNA degradation. Nevertheless, it is interesting to see whether the expression levels of miRNA are subject to a much higher variation than those mRNAs presented in Figure 4. Figure 5 displays variation coefficients of expression levels for different miRNA in SKBr3 breast cancer cells (untreated controls, n = 38) produced by spotted oligonucleotide microarrays. The data were retrieved from the Gene Expression Omnibus Database (see 24, GSE3798). It is clear that some classes of miRNA are much more variable than any of the protein-coding mRNAs in Figure 4, despite the fact that the inter-sample variability is generally expected to be lower in vitro than in vivo. This suggests that, when trying to explain the nature of long-range correlations in gene expression data involving gigantic sets of genes, one should look more closely at the "dark matter" of ncRNAs and confounding effects caused by heterogeneity of biological tissues or/and subjects 131425 rather than at technical flaws of microarray technology.
<p>Figure 5</p>
Variation coefficients for expression levels of miRNAs in SKBr3 breast cancer cells
Variation coefficients for expression levels of miRNAs in SKBr3 breast cancer cells.
![]()
The above characteristics of the putative common transcript were obtained from a set of 7084 genes. It is conceivable that numerous genes outside of this set may also contribute to the effect of cross-hybridization even if they are not physically present on the array. The question arises as to whether our estimates can be extrapolated to the whole totality of genes. There is no definitive answer to this question. On the one hand, if the covariances between gene expression levels are expected to be predominantly positive, the right hand sides of inequalities (11) and (12) may only increase when additional pairs are formed from the complementary genes. In this special sense, the estimates given by inequalities (10) and (11) are conservative. It is also important that they are independent of specific values of pj. On the other hand, it cannot be entirely ruled out that a heretofore unknown gene (probe set) with the above-described peculiar features or a formidable number of negatively correlated gene