Fingerprint mapping continues to play an important role in large-scale DNA sequencing efforts 12345. The procedure is challenging in terms of both its laboratory and computational demands. Indeed, most of the computational steps involve non-trivial algorithmic aspects. While reasonable solutions have been found for many of these, one task that remains particularly problematic is assessing postulated clone overlaps based on their fingerprint similarity.

The "overlap problem", as this is often referred to, basically involves examining all pairwise clone comparisons in order to identify overlaps. For a map consisting of λ clones, there are C_{λ, 2} = λ (λ - 1)/2 such comparisons. In each one, the number of matching fragment lengths between the two associated fragment lists is established. A case having μ > 0 matches indicates a possible overlap because the mutual length(s) may represent the same DNA. Lengths are not unique, so such matches are not conclusive indicators of overlap. Instead, the problem is largely one of probabilistic classification. One or more quantitative metrics are used to evaluate the authenticity of each such case. For example, an apparent overlap might be judged against its likelihood α of arising by chance. Methodologies of varying degrees of rigor have been proposed for this task 67891011. However, the so-called Sulston score, or Sulston probability P_S has emerged as a de facto standard 12, in part because of its integration in the widely-used FPC program 1314. A liability of a number of these methodologies, including P_S, is they assume fragment length comparisons are independent when, in fact, they are not 1015.

Recently, the exact distribution characterizing the overlap problem was determined 1617. Comparisons reveal that the assumption of independence is usually a poor one and that the Sulston score systematically over-predicts actual overlap probabilities, often by orders of magnitude. Consequently, a bias arises in projects that utilize P_S (Table 1). One chooses the significance threshold α to minimize erroneous decisions according to what is presumed to be the actual probabilistic description of the problem, P_E. The alternative result using the Sulston score is an overall increase of false-negatives (Case 1). Clones having significant overlap will still be correctly detected (Case 3). Moreover, false-positives would not be increased because P_S errs on the conservative side with respect to non-overlapping clones (Case 6). Miscalls can obviously be expected when poor values of α are chosen (Cases 4 and 5). However, if α is set too high, there will still be circumstantial cases where the correct decision is made (Case 2). These will presumably be more than offset by a higher rate of false-positives (Case 4). In summary, P_S is not an especially good discriminant for the overlap assessment problem.

The drawback of P_E is that it is rather difficult to compute and cannot be used directly in most cases. For example, current resources are not sufficient to evaluate it for most BAC comparisons or for capillary-based fingerprinting 18. A suitable method of approximating P_E is therefore required. Here, we propose a straightforward correlation-based approach that derives correction factors for the Sulston score. This procedure dramatically increases accuracy without incurring much additional computational effort.

The overlap problem is formally cast in terms of two clones having m and n "bands", respectively, where m ≥ n. Each band represents an individual clone fragment, with its position on a gel image providing an estimate of the fragment's length. Multiple bands of roughly the same length often appear. Finite measurement resolution ± R allows an image of length L to be subdivided into t = 0.5 L/R discrete bins. The Sulston score P_S = P_S (μ, m, n, t) is taken as a provisional estimate of the probability that at least μ fragment matches between the two clones arise by chance. Note here that the variables (μ, m, n) correspond to (M, nH, nL), respectively, in notations used by the FPC program 14. The corresponding exact probability is P_E = P_E (μ, m, n, t), as given in refs. 1617. We formulate a corrected value, P_C, that can be both efficiently calculated and that gives substantially better estimates of P_E than the Sulston score, i.e. |P_E - P_C| ≪ |P_E - P_S|.

The simple log-log plot in Fig. 1 shows good correlation (Pearson's coefficient of ρ ≈ 0.9938), suggesting that standard regression might be a reasonable basis for correction. Note the characteristic over-prediction of P_S. (Points representing the exact probability consistently fall below the hypothetical line of agreement between P_S and P_E.) These particular data are computed for t = 236, which describes traditional settings for fragment length measurements and comparisons, i.e. ± 7 pixels over a 3300 pixel gel image 1319. Considerations of coverage usually dictate a large number of clones in a map 2, so that values substantially above 10^-7 are not usually of interest 20. The data range over 0 ≤ μ ≤ n for a number of different fingerprint comparison sizes: 2 ≤ n ≤ m for 5 ≤ m ≤ 12, 2 ≤ n ≤ 10 for m of 13 and 14, 2 ≤ n ≤ 9 for m of 15 and 16, 2 ≤ n ≤ 8 for m of 17 and 18, and finally 2 ≤ n ≤ 7 for 19 ≤ m ≤ 25. The exact solution becomes difficult to evaluate beyond these ranges using readily-available resources. Specifically, the computational effort increases according to a factor that exceeds m!/(m - n)! 17.

Correlation in Fig. 1 is clearly not perfect. Specifically, the points show some amount of lateral scatter. Accuracy of the correction can be further enhanced to the degree that dispersion within the window can be minimized. Here, we can apply a simple power-law data reduction model to obtain a transformed Sulston score

P T = P S ν μ ξ m η n ζ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaWgaaWcbaGaemivaqfabeaakiabg2da9iabdcfaqnaaDaaaleaacqWGtbWuaeaaiiGacqWF9oGBaaGccqWF8oqBdaahaaWcbeqaaiab=57a4baakiabd2gaTnaaCaaaleqabaGae83TdGgaaOGaemOBa42aaWbaaSqabeaacqWF2oGEaaGccqGGUaGlaaa@3FAF@

The four power values can be chosen empirically such that the data locally collapse into a more highly correlated set. For example, selecting (v, ξ, η, ζ) = (1.2,4,0.8,-3.4) in Eq. 1 leads to the curve-fit

P C ≈ 9.855   P T 1.171 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaWgaaWcbaGaem4qameabeaakiabgIKi7kabiMda5iabc6caUiabiIda4iabiwda1iabiwda1iabbccaGiabdcfaqnaaDaaaleaacqWGubavaeaacqaIXaqmcqGGUaGlcqaIXaqmcqaI3aWncqaIXaqmaaaaaa@3D9B@

and the associated Pearson's coefficient ρ ≈ 0.9980.

We propose Eqs. 1 and 2 as a correction to the standard Sulston score for typical fingerprint mapping conditions 1319. Although shown as two separate equations so as to clarify the concept, these can clearly be combined into a single equation for actual computations. Pearson's coefficient is not especially sensitive to the parameters in Eq. 1 and there are many combinations of (v, ξ, η, ζ) that elevate ρ into the ~0.998 range. Other methods for reducing the data do not perform as well as the model in Eq. 1. For example, standard dimensional analysis 21, which involves correlating variables such as P_E/P_S, m/n, and μ/n, cannot adequately resolve the fact that values of the individual variables relative to one another remain important.

We have calibrated Eq. 2 according to the traditional parameters used in the FPC mapping program 13. Similar corrections can readily be constructed for different parameters. For example, protocols and software sizing methods now allow for band resolutions higher than the customary value of t = 236. Table 2 shows correction parameters for several such cases. Similarity of the correlation coefficients suggests that results would be comparable to that shown in Fig. 2. Although the accuracies derived from this approach are probably acceptable in the correlation range, they could, in principle, be further increased by using multiple corrections calibrated for specific "bins" of the m/n parameter.

Clone overlap assessment is sometimes framed as a statistical testing problem 10. Here, α is the probability of erroneously concluding that two clones overlap, when in fact they do not. (This casually implies that α C_{λ, 2} false positives can be expected for a project containing λ clones.) Consequently, corrections are most immediately relevant in the neighborhood surrounding α (Table 1). The overlaps here are the most valuable to detect in the sense that they are the smallest, and consequently contribute most effectively to a minimum tiling path 8. A large fraction of the comparisons will be either far above or below the threshold, so their assessments will not ultimately be affected. However, correction is still important for these cases. For example, Branscomb et al. 8 have pointed out that the ability to accurately rank all overlaps according to their associated probabilities is useful in the assembly phase of mapping.

Ascertaining the degree to which a particular mapping project would actually be improved by using Sulston score correction is difficult. Aside from the usual factors that complicate comparisons 24, there are special considerations for this kind of evaluation. For example, established Sulston-based mapping projects may have obtained their best results using threshold values that would not necessarily be considered "correct" from the standpoint of the exact probability distribution (Table 1). Biologists have historically viewed selection of the Sulston threshold to be a non-trivial, library-dependent problem and often resort to empirical sampling and iteration 2526. Consequently, one probably cannot obtain an objective comparison by just replacing P_S with P_C for these cases. Another avenue, perhaps more pragmatic, would be to assess corrections on a simulated project. For example, digesting finished sequences in silico 27 enables one to use the resulting simulated fingerprints to see how well a map could be reconstructed. Several variations on this method are possible 2829. Of course, use of correction for new projects is certainly recommended.

Other issues remain unresolved. With the exception of the conditional nature of match trials, the correction in Eq. 2 is based on the same set of assumptions as the Sulston score. Neither consider, for example, possible non-IID distribution of fragment lengths or length-dependent measurement accuracy. Consequently, we feel that the simple correction procedure proposed here represents a reasonable, though admittedly provisional advance in DNA mapping methodology.

Parameters in Eq. 1 were chosen empirically to minimize dispersion (maximize Pearson's coefficient) over a scoring range of roughly 10^-7 to 10^-19. The former is often the maximum value used in a mapping project and is dictated by the need to limit false-positive overlap declarations for the associated libraries, which are typically quite large 20. The latter is set by computational limitations.

Correction of a probability score P_T is implemented as a so-called "power-law" algebraic expression

P C = β   P T φ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaWgaaWcbaGaem4qameabeaakiabg2da9GGaciab=j7aIjabbccaGiabdcfaqnaaDaaaleaacqWGubavaeaacqWFvpGAaaGccqGGSaalaaa@37C7@

where φ and β are regression constants. Eq. 3 can be transformed into log-log form as

ln P_C = ln β + φ ln P_T.

Standard linear regression 30 can be used to determine φ and β in this equation. Specifically, we analyze the transformed system (x', y') = (ln P_T, ln P_C) to obtain the slope s and y-intercept y_o of the straight-line equation y' = sx' + y_o. The desired correction in Eq. 3 is then recovered by substituting φ = s and β = exp(y_o).