For the purpose of this discussion, the unit of observation (location and distance) is kilobasepair (kb). We assume that a number n_is of IS is randomly allocated (with a uniform distribution) to the locations of a genome consisting of g kb. A CIS of order n is an n-tuple of IS such that the maximum distance between the lowest and highest position is no greater than a fixed bound.

Further terminology

We start with the elementary observation that E_n equals P_n times the number of subsets of IS consisting of n elements:(1)Clearly,(2)It remains to determine P(n,d). First note that P(1,d) = 0 for d>0. Furthermore, for all m≥1:(3)

A recursive formula for P(m,d), d>0, can be derived by breaking down the potential CIS of order m into subsets of m–1 elements having a span of d'≤d, to which an m-th IS is added such that the maximum span is exactly d:(4)where r is a negligible correction term that arises because the uncorrected recursion formula is strictly valid only for subsets of IS that have a distance ≥d from the telomeres.

By mounting the recursive ladder (m = 1,...,n), these formulas successively yield P(n,d), P_n, and E_n. In particular, one easily obtains (d>0):Plugging this into equations (2) and (1) yields for the expected value E_n:As shown in Table 1, our mathematical approximation corresponds extremely well to the mean values found in 50000 simulation runs.

Statistical inferences, such as the calculation of p-values, can be based on the observation that, under the null hypothesis (H₀) of random uniform allocation of the IS, the number of CIS of order n is (approximately) Poisson distributed with parameter λ = E_n. Thus, if the random variable X denotes the number of CIS of order n, and X = k is observed in a trial, then the p-value P(X≥k) of this observation calculated under H₀, i.e. from the Poisson distribution P_o(E_n), is given bywhere the random variable χ² has a chi-square distribution with 2 k degrees of freedom [32], [33].

The Poisson approximation to the true random distribution of CIS is exceedingly close. In fact, if the number of simulation runs is sufficiently high, the simulated distribution is virtually undistinguishable from P_o(E_n). In particular, both the expected values and the p-values derived from P_o(E_n) are nearly identical to those obtained in computer simulations. The latter point is apparent from Table 2, where for a final proof of principle of our mathematical calculations, results of the analysis of our integration data set retrieved from two clinical SCID-X1 therapy trials [unpublished data] are given.

The p-value can be calculated by means of either of the following commands (‘R’ code): 1–ppois(lambda = E_n, q = k–1) or pchisq(df = 2k, q = 2E_n). Using the data of Table 2 (first line) 1–ppois(lambda = 0.19, q = 2) or pchisq(df = 6, q = 0.38). In both instances, the result is 0.00099. Alternatively, the table of the chisquare distribution with 6 degrees of freedom can be used to look up the probability P(X≤0.38). One should note that, for low E_n, the p-value of a single observed CIS is virtually identical to E_n. This implies that, for n>5, no p-values need to be calculated (and hence no formulas are required for E_n, n>5), because even with an extremely liberal definition of the CIS (d₅ = 500) and a fairly high number of IS (n_is = 1000) a single CIS of order 5 will be statistically significant (p = 0.027).

Defining non-randomness in the clustering of genomic events often requires additional precautions as sequence structures of interest may already have known specific distribution biases. In the case of our clinical example (unpublished data), it is known that retroviral vectors based on the murine leukaemia virus (MLV) tend to integrate into gene coding regions preferentially near the transcriptional start site (TSS) [34]-[36]. It is also proposed that additional factors, indeed mostly unknown, may influence the accessibility of vectors to certain genomic DNA regions [37]. Thus, the null hypothesis of random uniform allocation of MLV IS distribution may not be adequate according to the current ‘state of the art’, as has recently been argued [31]. In line with this study, we portioned the genome into 2 adequate areas that differ in the likelihood of getting targeted by vectors.

Further terminology

Clearly, the expected value E_n of the number CIS of order n is given by the following sum:(5)In the following it will be shown how to calculate the terms on the right side of (5). We start with the expected value of n_cis,GT5 fore what we assume that vector integration into any T5 occurs with the same probability. Then(6)where X is the number of CIS (among those occurring in GT5) that occur in a fixed T5. Observing that i IS in a fixed T5 yield CIS of order n in this T5 one easily obtains the expected value of X(7)Since X is binomially distributed as ∼ B(n_is,GT5,1/n_TSS),(8)Merging equations (6)–(8) yields the desired formula for E(n_cis,GT5):(9)

If n_is,GT5 is small compared to n_TSS (undoubtedly, this is mostly the case), terms of higher order can be neglected so that, because (n_TSS–1)/n_TSS≈1, formula (9) simplifies to(10)

Notice that formulas (6)–(10) do not depend on the spatial distribution of the IS within the T5. (It is unnecessary to account for the closeness of IS within T5 because any pair – or triple, quadruple etc., for that matter – of IS within a T5 yields a CIS.)

Clearly, the expected value of n_cis,Mix E(n_cis,Mix) is not independent of the distance between the IS and the TSS. Thus, inevitably, assumptions regarding the spatial distribution for the IS will influence its value. In the sequel, a formula for E(n_cis,Mix) shall be derived for the case n = 2. As before, CIS of order 2 are defined by a maximum distance d₂ of 30kb between the IS.

If the TSS are indistiguishable with respect to the probability distribution of the integrations, then(11)where p_Mix denotes the probability that an arbitrary pair of IS (with one element in GT5 and one element in the complement of GT5) forms a CIS of order 2 around a fixed TSS.

We will assume that the distributions of IS within a T5 and within +/-35 kb around a TSS are symmetric. Then, again using kb as unit of distance,(12)

In formula (12) the points x = 0 and y = 0 correspond to the TSS-5; f(x) designates the probability density function of vector integrations in T5; and g(y) designates the corresponding density function in [TSS-35, TSS-5].

Formula (12) shall be evaluated for two special cases:

Case 1: Vector integrations are uniformly distributed in GT5 and in the complement of GT5, respectively. I.e.,Solving the integrals in formula (12) we have(13)

Case 2: As above, vector integrations in the complement of GT5 are assumed to be uniformly distributed. However, a triangular distribution is assumed for f(x). The corresponding formula is easily calculated:By plugging this into (12) we get(14)

It may be surprising that a triangular distribution in T5 results in a higher expected value for n_cis,Mix than a uniform distribution. However, this becomes more plausible if one notes that a higher value is also obtained if the IS are concentrated in an extreme manner within the T5, viz. in a one-point distribution with total mass in the TSS. In this special case (which is particularly easy to evaluate), p_Mix  =  50/(n_TSS(g–n_TSS)).

If, with respect to the formation of CIS, the complement of GT5 could be regarded as a continuum, the expected value of n_cis,Comp would be given by the formulas developed in Part 1 of this contribution. In the case of retroviral (MLV) vectors, however, the complement of GT5 has rather to be viewed as a partitioned set consisting of approximately TSS disjoint intervals. It follows that that the residual term on the right-hand side of equation (4) (Part 1) may no longer be negligible. Note however, the assumption of a continuum clearly tends to lead to an overestimation of the number of CIS, because the boundaries of the components reduce the number of CIS occurring in their neighborhood. It follows that the formulas derived in Part 1 form an upper bound for E(n_cis,Comp). In particular, the true p-values are less or equal to the values calculated by means of the formulas derived in Part 1. Therefore, any positive statements regarding statistical significance remain valid. Moreover, the overestimation is probably fairly small given that the sections of GT5 located between the TSS are mostly rather wide compared to the length defining a CIS.

Indeed, the null hypothesis of non-uniform allocation for IS distribution does not substantially change the results we have obtained based on the hypothesis of a random uniform allocation for CIS formation in our clinical samples (Table 2), as is shown in Table 3.

Our mathematical formulas allow a reliable, straightforward calculation of non-randomness in CIS and other genomic event distributions under the null hypothesis of uniform and non-uniform allocation. Using formula based workspaces (available on request), expected values and p-values can be calculated with ease in real-time. They may be preferable to computer simulations when (routine) high-speed processing of large quantities of analyses is needed. Our approach enables a closely problem-oriented, highly exact evaluation of non-randomness that is useful for assessing IS distribution in clinical trials and for assessing the distribution of any sequence motif of interest in a natural or artificial genome.