Given a diploid genome (h = 2), the probability of coverage is

P 2 , φ = ∑ j = φ N − φ C N , j δ 2 j ( 1 − δ 2 ) N − j [ 1 − ∑ k = 0 φ − 1 C N − j , k δ 2 k ( 1 − δ 2 ) N − j − k ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7442@

where δ₂ = L/(2G) is the diploid Bernoulli probability and C_{N, k} are the binomial coefficients. Eq. 1 also gives the expected fraction of a set of locations that are covered (Methods). This equation relies on the standard IID assumption, but is exact in the sense that it accounts for the fact that the coverings of two corresponding alleles on homologous chromosomes are not strictly independent of one another (Methods). However, parameters in an actual project are such that alleles are almost independent. Moreover, asymptotic approximation can be applied (Methods), in which case

P 2 , φ ≈ ( 1 − ∑ k = 0 φ − 1 1 k ! ( ρ 2 ) k e − ρ / 2 ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabikdaYiabcYcaSiabeA8aMbqabaGccqGHijYUdaqadaqaaiabigdaXiabgkHiTmaaqahabaqcfa4aaSaaaeaacqaIXaqmaeaacqWGRbWAcqGGHaqiaaGcdaqadaqcfayaamaalaaabaGaeqyWdihabaGaeGOmaidaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqWGRbWAaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTiabeg8aYjabc+caViabikdaYaaaaeaacqWGRbWAcqGH9aqpcqaIWaamaeaacqaHgpGzcqGHsislcqaIXaqma0GaeyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaaa@50AC@

is a very good approximation of Eq. 1. Here, e is the Euler Number (≈ 2.71828) and ρ is again the conventional haploid redundancy. Note the basis in a Poisson distribution having a rate ρ/2. Eq. 2 is straightforward to evaluate for any project because φ is typically not very large. This stands in contrast to Eq. 1, which sports an enormous number of terms, as well as tendencies for numerical overflow and underflow of its various components. For convenience, we expand the first three expressions

P 2 , 1 ≈ ( 1 − e − ρ / 2 ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabikdaYiabcYcaSiabigdaXaqabaGccqGHijYUdaqadaqaaiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0IaeqyWdiNaei4la8IaeGOmaidaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaaaaa@3C89@

P 2 , 2 ≈ ( 1 − e − ρ / 2 ( 1 + ρ 2 ) ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabikdaYiabcYcaSiabikdaYaqabaGccqGHijYUdaqadaqaaiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0IaeqyWdiNaei4la8IaeGOmaidaaOWaaeWaaeaacqaIXaqmcqGHRaWkdaWcaaqaaiabeg8aYbqaaiabikdaYaaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaaaaa@42A8@

P 2 , 3 ≈ ( 1 − e − ρ / 2 ( 1 + ρ 2 + ρ 2 8 ) ) 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabikdaYiabcYcaSiabiodaZaqabaGccqGHijYUdaqadaqaaiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0IaeqyWdiNaei4la8IaeGOmaidaaOWaaeWaaeaacqaIXaqmcqGHRaWkjuaGdaWcaaqaaiabeg8aYbqaaiabikdaYaaakiabgUcaRKqbaoaalaaabaGaeqyWdi3aaWbaaeqabaGaeGOmaidaaaqaaiabiIda4aaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaOGaeiOla4caaa@498C@

Under the assumption of independence, Eq. 2 for homologous chromosomes is readily generalized to an arbitrary number of chromosomes, h, specifically

P h , φ ≈ ( 1 − ∑ k = 0 φ − 1 1 k ! ( ρ h ) k e − ρ / h ) h . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabdIgaOjabcYcaSiabeA8aMbqabaGccqGHijYUdaqadaqaaiabigdaXiabgkHiTmaaqahabaqcfa4aaSaaaeaacqaIXaqmaeaacqWGRbWAcqGGHaqiaaGcdaqadaqcfayaamaalaaabaGaeqyWdihabaGaemiAaGgaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqWGRbWAaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTiabeg8aYjabc+caViabdIgaObaaaeaacqWGRbWAcqGH9aqpcqaIWaamaeaacqaHgpGzcqGHsislcqaIXaqma0GaeyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaemiAaGgaaOGaeiOla4caaa@5336@

Note the Poisson basis having a rate ρ/h, for example ρ/3 for chromosomal trisomies. Like Eq. 2, this expression is readily evaluated and straightforward to expand for given values of φ.

Fig. 1 shows the traditional de novo haploid coverage model 1516 versus diploid medical sequencing coverage theory for minimum read coverings of φ ∈ {1, 2, 3, 4, 5} for both alleles. The diploid curves were generated by Eq. 2 for a 3.3 billion base-pair genome and 31 base-pair read lengths. Errors associated with not using Eq. 1 for these particular parameters are significantly less than 1% (data not shown). As one would intuitively expect, the required redundancies for a given coverage fraction increase with φ and are noticeably higher than established values for haploid genome sequencing. Although these cases are within the realm of feasibility for the newest-generation sequencing platforms 11, economic factors would probably still preclude the higher values of φ at the present time. Conversely, these depths are much lower than values that have been discussed elsewhere. For example, Warren et al. 22 report ρ up to 100 and 400 for bacterial and viral genomes, respectively, using 25 bp fragments to simulate data from an Illumina instrument.

Two other notable trends are visible in Fig. 1. First, increasingly large redundancies are required just to obtain non-trivial values of P_{2, φ}. For example, the curve for φ = 1 already exceeds 0.01 at ρ = 0.25, whereas this mark is not met until ρ = 5 at φ = 5. Indeed, one may not see even the "beginnings" of coverage until comparatively high redundancy has been reached, depending on the selected φ. Also, the amount by which each curve is drawn-out over the abscissa increases with φ, signifying a decelerating coverage rate. This is especially clear for what appear to be the linear segments of each curve; their slopes progressively decrease. Again comparing the extremes, the difference between 0.1 and 0.9 on the ordinate is a little over 5 units of redundancy for φ = 1 but almost 11 units for φ = 5. This phenomenon bears on point we make below.

Both of these trends arise strictly as mathematical consequences and can perhaps best be understood by referring to Eqs. 3 through 5. The exponential (Euler Number) term represents the tendency for the coverage rate to decay. For each successive value of φ this term is bolstered by additional factors, which themselves grow progressively faster with ρ, whereby the overall effect is realized.

One of the primary issues facing the investigator is the so-called stopping problem. That is, at what ρ should random processing be halted? This question is, of course, context dependent. Yet, it can be answered, at least approximately, by using the analysis given here. For example, suppose the goal is to design a medical sequencing project such that the expected coverage progress corresponds roughly to standard BAC sequencing. This is a calibration-based way of framing the question and exploits the community's collective empirical experience gained from having sequenced hundreds of thousands of such clones. In particular, 6 ≤ ρ ≤ 10 has been found to be a reasonable balance between cost and coverage, although values nearer to 10 are more typically chosen 14. In this capacity, Eqs. 1 and 2 effectively function as scaling laws.

Scaling can conveniently be demonstrated graphically, for example by picking a point on the haploid curve for a desired redundancy, drawing a horizontal line through this point, and reading the redundancy at the intersection of the chosen diploid curve and the horizontal. The asymptotic nature of the curves depicted in Fig. 1 obscures this process, but it can readily be accomplished using a magnified plot. Fig. 2 shows the example of extrapolating haploid sequencing coverage at ρ = 8 to diploid sequencing (φ = 1), the result being about 17.5× for the diploid project. Table 1 furnishes an expanded set of values for φ ∈ {1, 2, 3} calibrated against haploid sequencing for ρ ∈ {6, 8, 10}. Again, the increase of redundancy with the minimum number of reads required to attain coverage is quite clear. Notice that each of the three rows in the table corresponds to covering more than 99% of the unique sequence. In other words, the covering probabilities change very little over fairly significant increases in depth. Consequently, BAC depth provides much better resolution than BAC coverage for scaling the diploid problem. This observation is also obvious from Fig. 2.

Diploid coverage, as discussed above, is a primary consideration for medical sequencing. Yet, it is also useful for comparison to examine such projects in their haploid context. The stopping problem has been extensively studied from a number of analytical perspectives for traditional de novo genomic sequencing projects, for example using the probability of complete coverage, P_C 23 and the intersection probability, P_⋂ 24. While the meaning of the former is probably clear, the latter characterizes how effective additional redundancy will be for improving coverage in light of increasingly-important stochastic effects. Of these two metrics, P_C is the more conservative and could be thought of as setting an upper bound in the context of traditional haploid sequencing. Before proceeding, let us digress briefly to further explain P_⋂.

Consider two hypothetical sequencing projects, A and A', that are identical in every respect, except that A' is always ahead by one whole unit of redundancy (Fig. 3). Now, examine these projects at two particular instances, specifically for project A at both ρ₁ and ρ₂ (with A' at ρ₁ + 1 and ρ₂ + 1, respectively), where ρ₁ <ρ₂. At ρ₁, there may be little overlap in the densities between the two projects. Accordingly, the probability is extremely high, perhaps close to 1, that A' will be more highly covered than A. The system behaves as if there is a deterministic increase in coverage for A → A' for a unit increase of redundancy. At higher redundancies, say ρ₂, mathematical analysis indicates that the intersection of the A and A' densities will grow 24. Consequently, differences in actual coverage between A and A' become progressively more a function of chance than of differences in the redundancies themselves. The tail value of the intersection, P_⋂, can be taken as an indicator on the diminishing returns of the process.

We can once again take a calibration approach to this problem. That is, we calculate P_C and P_⋂ for a typical BAC sequencing project (150 kb clone length and 600 bp read length) at ρ ∈ {6, 8, 10}. We then match these values to their counterparts in the distributions for medical sequencing, which then provides the corresponding "scaled" redundancy. Table 2 shows these results. The values compliment those in Table 1 in the sense that they suggest redundancies far above the conventional full shotgun standard of ρ = 10.

Several labs are now involved in diploid sequencing projects, which should furnish useful examples of coverage progressions that can be monitored empirically. Although the few stopping redundancies reported in the literature appear to conflict with one another, these can be more properly interpreted according the minimum number of times each allele is observed, φ. For example, we mentioned above that Levy et al. 12 considered ρ = 20 for Sanger-based 21 germline sequencing of a healthy individual. This figure approaches the 10× standard for haploid sequencing 25 at the level of φ = 1 (Table 1). Interestingly, the idealized version of their coverage calculation proves to be a special case of our model precisely for φ = 1 (see Appendix). Conversely, Mardis 8 quotes a redundancy up to 30×, which corresponds to values of φ between 2 and 3 when calibrated to haploid 10×.

Richard Durbin and Aylwyn Scally have also analyzed the diploid medical sequencing coverage problem using a different approach from what is described here (Durbin and Scally, personal communication). Specifically, they employed an "extra-variation" Poisson distribution 2627 having a free-parameter to control variance. Values for this parameter can be chosen to a posteriori tune the theoretical fit with empirical data. In particular, such tuning allows one to implicitly consider, at least approximately, factors such as bias and sequencing errors. (In our method, calibration incorporates the empirics of BAC sequencing, essentially serving the same purpose.) Using their semi-empirical approach, Durbin and Scally concluded that redundancies closer to 30× will be required, which again agrees well with results shown in Tables 1 and 2.

A number of labs, including our own, are now adopting "next generation" short-read sequencing technology 8 and have started to generate human medical sequencing data related to various cancers. However, there is still a dearth of published results from which actual coverage progressions can be derived. For the purposes of comparison, we refer instead to the recently completed pilot resequencing project for C. elegans. Hillier et al. 28 resequenced strain N2 Bristol using the Illumina Genome Analyzer in order to characterize the accuracy and utility of short-read, massively parallel data. We have projected their C. elegans coverage results for φ ∈ {1, 2} onto Fig. 1. Agreement is very good up to about 60% coverage, after which the rate of empirical coverage falls below expectation. This behavior seems to typify theoretical-empirical differences. For example, Wendl and Barbazuk 29 noted precisely this trend for sequencing filtered genomes.

The physical explanation of this phenomenon is straightforward. Specifically, biases are not manifested early in a project because there is not enough information to distinguish unbiased coverage configurations from biased ones. (Think of an extreme case, for example placing a single read of high GC content onto a genome of high AT content. Despite the obvious non-IID nature of this scenario, the predicted coverage will still be identical to the actual coverage.) Given a model based on the IID assumption, as ours is, empirical and theoretical results should start to diverge as sufficient information gathers to expose latent biases. In this case, Hillier et al. 28 note a definite AT bias using the Illumina platform, i.e. remarkably lower coverage in regions of high AT content, which we presume accounts for much of the difference shown in Fig. 1. The proclivities of other methods and platforms are evidently different 30. Consequently, Eq. 2 should also be useful as a yardstick for comparison among these approaches for specific applications.

Finally, Fig. 1 also shows simulation data reported by Smith and Bernstein 20 for φ = 1 on a 20 kb circularized fragment using 250 bp reads. Agreement is once again good up to about 60% coverage, after which the sequencing process seems to grow more efficient for the fragment. This observation is not surprising, given two important aspects of this study. First, the circularized configuration is not subject to the so-called "edge effect", which can dramatically affect coverage rates 2931. Second, distribution theories show that configurations having larger L/G ratios do indeed cover more readily than those having smaller values 24. We presume these two factors account for most of the difference, especially given that L/G = 0.0125 for the simulation is more than a million times larger than values associated with short-read medical sequencing projects. For example, 31 bp reads on a 3.3 billion bp genome yields L/G ≈ 1 × 10^-8.

We expect that many future studies will be based on sequencing DNA derived from matched tumor/normal samples (for example, the latter being obtained from uninvolved skin or blood) from the same patient 34. Here, the whole genome of each sample in a pair is sequenced and mutations are found by comparison to the human reference. Let us call the sets of mutations for a tumor and a normal sample S_T and S_N, respectively, where we generally expect S_N ⊆ S_T. Most of the germline variation in S_N will be polymorphisms not related to pathogenesis 1, whereas S_T will contain a potentially more relevant collection of somatic mutations. In principle, germline sequence variations can be removed from further consideration by taking the difference S_T - S_N 6. Such filtering will appreciably focus subsequent work, since the overwhelming majority of sequence variants should be polymorphisms found in normal tissue 5. How does one efficiently accomplish this from a process-engineering standpoint?

We propose a refinement of simple subtraction 6 in the form of a straightforward differential sequencing strategy. In principle, false-negative errors are controlled by sequencing at least to diploid coverage at the level of φ = 1. However, tumor samples should actually be sequenced as heavily as economically possible in order to minimize false-positive hits for both germline and somatic mutations. These types of mistakes arise, for example, by misinterpreting a random sequencing error as a true mutation. Given current state-of-the-art capabilities, we will assume this condition translates to diploid coverage at the level of φ = 2, but emphasize that future instruments will undoubtedly permit higher φ.

Conversely, a germline mutation in a normal sample only has to be detected once in order to be eliminated from S_T. We are also not as concerned about false-positives here because their appearance in S_N does not affect the subtraction S_T - S_N. It is possible that an error could lead to a spurious entry in S_N that precisely matches a true somatic mutation in S_T by pure chance. The somatic mutation would then be erroneously eliminated from further investigation. However, such events seem unlikely, given the low anticipated number of bona fide somatic mutations. These observations collectively imply that normal samples may only need to be sequenced to diploid coverage at the level of φ = 1.

The 10× standard for BAC sequencing is well-established 25 and provides a reasonably conservative basis to translate the above design into actual redundancies for medical sequencing (Table 1). We suggest then that sequencing of tumor samples should not be pursued to less than about 26.5× redundancy, given the 2-read minimum coverage condition. Furthermore, paired normal samples need only to be sequenced to about 21.5× redundancy for the φ = 1 coverage condition.

The observation that the required redundancy for the φ = 1 coverage level is not simply half that of the φ = 2 level may initially seem counter-intuitive. We remarked above that curves for increasing values of φ tend to have progressively smaller slopes. Consequently, there is not, in general, an integer-valued relationship between corresponding points on any two particular curves. In other words, curves are not simply shifted along ρ. Returning to Fig. 1, we show an example that replots the φ = 2 curve, except where the abscissa-value of each of its points has been divided by two. It is clear that the result does not coincide with the φ = 1 curve, as intuition may have suggested. The curves only intersect at a single point, here at an expected coverage slightly more than 0.5, which is well below the > 0.99 calibration points we chose above. In other words, redundancy for φ = 1 would only have been half that for φ = 2 had we chosen to cover 50% instead of > 0.99 of SNPs. This trend holds generally. That is, we do not expect an integer relationship between the required redundancies for two unequal, but otherwise arbitrary values of φ.

Aneuploidy can be manifested in a number of ways: as an autosomal 32 or sex chromosome 33 aberration, and in conjunction with cancer 34. We anticipate the eventual application of DNA sequencing to aneuploid chromosome configurations and offer some early projections based upon Eq. 6. Fig. 4 shows expected coverage for trisomy and tetrasomy for φ = 2 and φ = 3. Required depths are clearly much higher than for diploid sequencing. For example, we find redundancies of ρ = 42 and ρ = 57 for trisomy and tetrasomy, respectively, when scaling to 10× BAC sequencing at the level of φ = 2. Recall that φ = 2 is presumed to be feasible for diploid whole-genome sequencing using current hardware. These redundancies are clearly out of reach at the moment for a whole-genome project, but may be feasible for chromosome-specific projects. In other words, the appreciably higher cost of sequencing aneuploid chromosomes may justify the effort of separating them into their own self-contained projects.

As with the classical theories of sequencing 1617, the main assumption here is that reads are independently and identically distributed (IID). In other words, this analysis does not formally consider biological or instrument-specific biases, software biases and sequencing errors, for example in base-calling, assembly problems, or any other heuristic inputs. The idiosyncrasies of each of these factors are difficult to characterize analytically, although the calibration step does allow some implicit accounting, as noted above. Appreciable differences in levels and types of bias have been noted, for instance in Sanger-style sequencing versus pyrosequencing 30, so any results should be interpreted with these qualifications in mind.

In general, the assumption of allele independence should be valid for most medical sequencing projects since φ will be small and L/G → 0 and N ≫ 1 (see Methods). For example, maximum error for φ ∈ {1, 2} is on the order of 10^-5 percent for diploid sequencing using 650 bp read lengths. The theory further assumes that sequence reads have no preference for either chromosome of a homologous pair and neglects any tendency for reads to align to multiple positions. The latter has been found to occur with some frequency if reads are short enough 35. Read-pairing certainly curtails this phenomenon, but the pairing process itself has negligible effect on coverage unless the target is very small 31. Such is not the case in whole-genome medical sequencing, so the net effect of pairing is simply that the amount of uniquely-alignable sequence one gets to count toward ρ increases commensurately. In plain terms, fewer data will be discarded.

Finally, our analysis does not account for what might be called the "uneven coverage" problem of alleles. Mutation detection programs may decline calling out a SNP if one allele is covered much more heavily than the other 36. Because this phenomenon is both software-specific and sequence-specific, it is beyond our scope. Departure from any of these idealizations will tend to reduce coverage, implying that our analysis is best viewed in the context of upper bounds of performance. In other words, required redundancies for specific projects may still exceed what we have advocated here, as the C. elegans data in Fig. 1 illustrate.

A subtle mathematical point is also worth mentioning. Eqs. 1 through 5 represent the probability of covering a specific allele pair, or alternatively, the expected fraction of pairs covered. These expressions do not provide the underlying distribution of the number of covered pairs, which is a more formidable mathematical problem. In other words, this is a model only of coverage expectation, exactly analogous to what classical theories 1617 are for traditional de novo haploid sequencing. Consequently, the results themselves are not strong functions of L/G. In fact, for most applications the results will be completely independent of this ratio and will instead follow a set of "universal" curves, the first 5 of which are shown in Figs. 1 and 2. This point is underscored by Eq. 2, which is a function strictly of ρ. (The observation holds more generally for aneuploidy as described by Eq. 6, as well.) This phenomenon contrasts with distribution-based models, such as P_C and P_⋂ discussed above, which are indeed sensitive to L/G. The basis of this effect is discussed in ref. 24.

A corollary to this observation is that adjusting the fundamental parameters within their biologically-relevant limits will have no effect on the results we have discussed. For example, the haploid genome size G could be adjusted to reflect only that part of the sequence to which read data can be uniquely aligned 35. Yet, the underlying assumptions leading to Eqs. 2 and 6 will still be satisfied in this circumstance, mainly L/G → 0 and N ≫ 1 (Methods). The same holds for varying L in order to represent different kinds of sequencing platforms, e.g. pyrosequencing or Sanger instruments. In summary, the contributions of the three independent variables L, G, and N collapse into the single dimensionless variable ρ, which governs the process exclusively. Formal theory 37 predicts such systematic reductions of variables whenever a unified dimensionless parameter lurks in a problem.

Let B_{i, j} be the event where an allele at position x on chromosome i is "covered", i.e. spanned by at least φ out of any collection of j reads, where j ≥ φ. Given N total reads, our definition of diploid medical sequencing coverage for position x is then B_{1, N} ⋂ B_{2, N} and its probability is P_{2, φ}(B_{1, N} ⋂ B_{2, N}). If β_{i, j, k} is the event whereby the allele on chromosome i is spanned by exactly k of j reads, then B_{i, j} ≡ β_{i, j, φ} ⋃ β_{i, j, φ+1} ⋃ ⋯ ⋃ β_{i, j, j}. Considering two homologous chromosomes, i ∈ {1, 2}, the probability that a single given read spans x on a specific chromosome is δ₂ = L/(2G), where L and G are read length and haploid genome length, respectively. Since the process is binomial (covering or not covering), we immediately have P(βi,j,k)=Cj,kδ2k(1−δ2)j−k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaeqOSdi2aaSbaaSqaaiabdMgaPjabcYcaSiabdQgaQjabcYcaSiabdUgaRbqabaGccqGGPaqkcqGH9aqpcqWGdbWqdaWgaaWcbaGaemOAaOMaeiilaWIaem4AaSgabeaakiabes7aKnaaDaaaleaacqaIYaGmaeaacqWGRbWAaaGccqGGOaakcqaIXaqmcqGHsislcqaH0oazdaWgaaWcbaGaeGOmaidabeaakiabcMcaPmaaCaaaleqabaGaemOAaOMaeyOeI0Iaem4AaSgaaaaa@4AA4@, where C_{j, k} are the binomial coefficients.

The coverings of two homologous alleles are not independent of one another. For instance, if one allele is already covered by j reads, there are only N - j remaining reads that have a chance to cover the other allele. Consequently,

P 2 , φ ( B 1 , N ∩ B 2 , N ) = P ( B 1 , N ) ⋅ P ( B 2 , N | B 1 , N ) = ∑ j = φ N P ( β 1 , N , j ) ⋅ P ( B 2 , N − j | β 1 , N , j ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7B05@

where

P ( B 2 , N − j | β 1 , N , j ) = ∑ k = φ N P ( β 2 , N − j , k ) = 1 − ∑ k = 0 φ − 1 P ( β 2 , N − j , k ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6C00@

Eq. 1 follows from the observation that P(β_{2, N-j, k}) = 0 for k > N - j.

If we neglect the dependence of alleles, then P_{2, φ}(B_{1, N} ⋂ B_{2, N}) = P (B_{1, N})·P (B_{2, N}). Without loss of generality, this probability is identical to P²(B_{1, N}), from which

P 2 , φ = ( 1 − ∑ k = 0 φ − 1 C N , k δ 2 k ( 1 − δ 2 ) N − k ) 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabikdaYiabcYcaSiabeA8aMbqabaGccqGH9aqpdaqadaqaaiabigdaXiabgkHiTmaaqahabaGaem4qam0aaSbaaSqaaiabd6eaojabcYcaSiabdUgaRbqabaGccqaH0oazdaqhaaWcbaGaeGOmaidabaGaem4AaSgaaOGaeiikaGIaeGymaeJaeyOeI0IaeqiTdq2aaSbaaSqaaiabikdaYaqabaGccqGGPaqkdaahaaWcbeqaaiabd6eaojabgkHiTiabdUgaRbaaaeaacqWGRbWAcqGH9aqpcqaIWaamaeaacqaHgpGzcqGHsislcqaIXaqma0GaeyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaOGaeiOla4caaa@5380@

Parameters for medical sequencing projects are such that L/G → 0 and N ≫ 1. These relations hold for both "short read" platforms and instruments that provide "full-length" Sanger read data. Moreover, φ ≪ N, which implies k ≪ N. Consequently, the binomial coefficients C_{N, k} are well-approximated by N^k/k!. These conditions also imply (1 - δ₂)^N ~ exp(-Nδ₂), i.e. that asymptotic approximation can be used for the power term. Eq. 2 follows directly.

The case of aneuploidy under the assumption of allele independence is a straightforward extension of the proof for Eq. 2. For h homologous chromosomes, P_{h, φ} (B_{1, N} ⋂ B_{2, N} ⋂ ⋯ ⋂ B_{h, N}) = P(B_{1, N})P (B_{2, N}) ⋯ P (B_{h, N}) = P^h(B_{1, N}). Given that all h chromosomes are equally likely to be sampled, δ_h = L/(hG), which is the appropriate Bernoulli probability for P(B_{1, N}). Eq. 6 follows from the same approximation arguments made for Eq. 2.

We can take the coverage status of a specific allele pair as a Bernoulli trial, whereby elementary probability theory shows that the expected number of pairs covered is their total number multiplied by P_{2, φ}(B_{1, N} ⋂ B_{2, N}). Consequently, P_{2, φ}(B_{1, N} ⋂ B_{2, N}) in Eq. 1 and its approximation in Eq. 2 also represent the expected fraction of covered pairs. The same argument holds for Eq. 6.

Hillier et al. 28 used the Illumina Genome Analyzer to resequence the C. elegans N2 Bristol genome. Release ws188 of the genomic sequence 38 was downloaded from http://www.wormbase.org and randomly chosen subsets of the resequence data were aligned against the reference at regular intervals for each chromosome using the maq aligner (http://maq.sourceforge.net). Data that could not be uniquely placed on the reference were discarded. Coverage was calculated for each alignment as the number of corresponding base positions spanned by at least one read on homologous chromosomes (φ = 1) and by at least two reads on homologous chromosomes (φ = 2).