System and method for cleaning noisy genetic data from target individuals using genetic data from genetically related individuals

[0001] This application is a continuation of U.S. Utility application Ser. No. 16/288,690, filed Feb. 28, 2019. U.S. Utility application Ser. No. 16/288,690 is a continuation of U.S. Utility application Ser. No. 15/187,555 (now U.S. Pat. No. 10,227,652), filed Jun. 20, 2016. U.S. Utility application Ser. No. 15/187,555 (now U.S. Pat. No. 10,227,652) is a continuation of U.S. Utility application Ser. No. 14/092,457, filed Nov. 27, 2013, now U.S. Pat. No. 9,430,611. U.S. Utility application Ser. No. 14/092,457, now U.S. Pat. No. 9,430,611, is a continuation of U.S. Utility application Ser. No. 13/793,133, filed Mar. 11, 2013, now U.S. Pat. No. 9,424,392, and U.S. Utility application Ser. No. 13/793,186, filed Mar. 11, 2013, now U.S. Pat. No. 8,682,592. U.S. Utility application Ser. No. 13/793,133, now U.S. Pat. No. 9,424,392, is a continuation of U.S. Utility application Ser. No. 11/603,406, filed Nov. 22, 2006, now U.S. Pat. No. 8,532,930. U.S. Utility application Ser. No. 13/793,186, now U.S. Pat. No. 8,682,592, is a continuation of U.S. Utility application Ser. No. 11/603,406, filed Nov. 22, 2006, now U.S. Pat. No. 8,532,930. U.S. Utility application Ser. No. 11/603,406, now U.S. Pat. No. 8,532,930, claims the benefit under 35 U.S.C. § 119(e) of the following U.S. Provisional Patent Applications: Ser. No. 60/739,882, filed Nov. 26, 2005; Ser. No. 60/742,305, filed Dec. 6, 2005; Ser. No. 60/754,396, filed Dec. 29, 2005; Ser. No. 60/774,976, filed Feb. 21, 2006; Ser. No. 60/789,506, filed Apr. 4, 2006; and Ser. No. 60/817,741, filed Jun. 30, 2006; and Ser. No. 60/846,610, filed Sep. 22, 2006; the disclosures of the patent applications cited in this paragraph are each incorporated by reference herein in their entirety.

[0002] The invention relates generally to the field of acquiring, manipulating and using genetic data for medically predictive purposes, and specifically to a system in which imperfectly measured genetic data is made more precise by using known genetic data of genetically related individuals, thereby allowing more effective identification of genetic irregularities that could result in various phenotypic outcomes.

[0003] Current methods of prenatal diagnosis can alert physicians and parents to abnormalities in growing fetuses. Without prenatal diagnosis, one in 50 babies is born with serious physical or mental handicap, and as many as one in 30 will have some form of congenital malformation. Unfortunately, standard methods require invasive testing and carry a roughly 1 percent risk of miscarriage. These methods include amniocentesis, chorion villus biopsy and fetal blood sampling. Of these, amniocentesis is the most common procedure; in 2003, it was performed in approximately 3% of all pregnancies, though its frequency of use has been decreasing over the past decade and a half. A major drawback of prenatal diagnosis is that given the limited courses of action once an abnormality has been detected, it is only valuable and ethical to test for very serious defects. As result, prenatal diagnosis is typically only attempted in cases of high-risk pregnancies, where the elevated chance of a defect combined with the seriousness of the potential abnormality outweighs the risks. A need exists for a method of prenatal diagnosis that mitigates these risks.

[0004] It has recently been discovered that cell-free fetal DNA and intact fetal cells can enter maternal blood circulation. Consequently, analysis of these cells can allow early Non-Invasive Prenatal Genetic Diagnosis (NIPGD). A key challenge in using NIPGD is the task of identifying and extracting fetal cells or nucleic acids from the mother's blood. The fetal cell concentration in maternal blood depends on the stage of pregnancy and the condition of the fetus, but estimates range from one to forty fetal cells in every milliliter of maternal blood, or less than one fetal cell per 100,000 maternal nucleated cells. Current techniques are able to isolate small quantities of fetal cells from the mother's blood, although it is very difficult to enrich the fetal cells to purity in any quantity. The most effective technique in this context involves the use of monoclonal antibodies, but other techniques used to isolate fetal cells include density centrifugation, selective lysis of adult erythrocytes, and FACS. Fetal DNA isolation has been demonstrated using PCR amplification using primers with fetal-specific DNA sequences. Since only tens of molecules of each embryonic SNP are available through these techniques, the genotyping of the fetal tissue with high fidelity is not currently possible.

[0005] Much research has been done towards the use of pre-implantation genetic diagnosis (PGD) as an alternative to classical prenatal diagnosis of inherited disease. Most PGD today focuses on high-level chromosomal abnormalities such as aneuploidy and balanced translocations with the primary outcomes being successful implantation and a take-home baby. A need exists for a method for more extensive genotyping of embryos at the pre-implantation stage. The number of known disease associated genetic alleles is currently at 389 according to OMIM and steadily climbing. Consequently, it is becoming increasingly relevant to analyze multiple embryonic SNPs that are associated with disease phenotypes. A clear advantage of pre-implantation genetic diagnosis over prenatal diagnosis is that it avoids some of the ethical issues regarding possible choices of action once undesirable phenotypes have been detected.

[0006] Many techniques exist for isolating single cells. The FACS machine has a variety of applications; one important application is to discriminate between cells based on size, shape and overall DNA content. The FACS machine can be set to sort single cells into any desired container. Many different groups have used single cell DNA analysis for a number of applications, including prenatal genetic diagnosis, recombination studies, and analysis of chromosomal imbalances. Single-sperm genotyping has been used previously for forensic analysis of sperm samples (to decrease problems arising from mixed samples) and for single-cell recombination studies.

[0007] Isolation of single cells from human embryos, while highly technical, is now routine in in vitro fertilization clinics. To date, the vast majority of prenatal diagnoses have used fluorescent in situ hybridization (FISH), which can determine large chromosomal aberrations (such as Down syndrome, or trisomy 21) and PCR/electrophoresis, which can determine a handful of SNPs or other allele calls. Both polar bodies and blastomeres have been isolated with success. It is critical to isolate single blastomeres without compromising embryonic integrity. The most common technique is to remove single blastomeres from day 3 embryos (6 or 8 cell stage). Embryos are transferred to a special cell culture medium (standard culture medium lacking calcium and magnesium), and a hole is introduced into the zona pellucida using an acidic solution, laser, or mechanical drilling. The technician then uses a biopsy pipette to remove a single visible nucleus. Clinical studies have demonstrated that this process does not decrease implantation success, since at this stage embryonic cells are undifferentiated.

[0008] There are three major methods available for whole genome amplification (WGA): ligation-mediated PCR (LM-PCR), degenerate oligonucleotide primer PCR (DOP-PCR), and multiple displacement amplification (MDA). In LM-PCR, short DNA sequences called adapters are ligated to blunt ends of DNA. These adapters contain universal amplification sequences, which are used to amplify the DNA by PCR. In DOP-PCR, random primers that also contain universal amplification sequences are used in a first round of annealing and PCR. Then, a second round of PCR is used to amplify the sequences further with the universal primer sequences. Finally, MDA uses the phi-29 polymerase, which is a highly processive and non-specific enzyme that replicates DNA and has been used for single-cell analysis. Of the three methods, DOP-PCR reliably produces large quantities of DNA from small quantities of DNA, including single copies of chromosomes. On the other hand, MDA is the fastest method, producing hundred-fold amplification of DNA in a few hours. The major limitations to amplifying material from a single cell are (1) necessity of using extremely dilute DNA concentrations or extremely small volume of reaction mixture, and (2) difficulty of reliably dissociating DNA from proteins across the whole genome. Regardless, single-cell whole genome amplification has been used successfully for a variety of applications for a number of years.

[0009] There are numerous difficulties in using DNA amplification in these contexts. Amplification of single-cell DNA (or DNA from a small number of cells, or from smaller amounts of DNA) by PCR can fail completely, as reported in 5-10% of the cases. This is often due to contamination of the DNA, the loss of the cell, its DNA, or accessibility of the DNA during the PCR reaction. Other sources of error that may arise in measuring the embryonic DNA by amplification and microarray analysis include transcription errors introduced by the DNA polymerase where a particular nucleotide is incorrectly copied during PCR, and microarray reading errors due to imperfect hybridization on the array. The biggest problem, however, remains allele drop-out (ADO) defined as the failure to amplify one of the two alleles in a heterozygous cell. ADO can affect up to more than 40% of amplifications and has already caused PGD misdiagnoses. ADO becomes a health issue especially in the case of a dominant disease, where the failure to amplify can lead to implantation of an affected embryo. The need for more than one set of primers per each marker (in heterozygotes) complicate the PCR process. Therefore, more reliable PCR assays are being developed based on understanding the ADO origin. Reaction conditions for single-cell amplifications are under study. The amplicon size, the amount of DNA degradation, freezing and thawing, and the PCR program and conditions can each influence the rate of ADO.

[0010] All those techniques, however, depend on the minute DNA amount available for amplification in the single cell. This process is often accompanied by contamination. Proper sterile conditions and microsatellite sizing can exclude the chance of contaminant DNA as microsatellite analysis detected only in parental alleles exclude contamination. Studies to reliably transfer molecular diagnostic protocols to the single-cell level have been recently pursued using first-round multiplex PCR of microsatellite markers, followed by real-time PCR and microsatellite sizing to exclude chance contamination. Multiplex PCR allows for the amplification of multiple fragments in a single reaction, a crucial requirement in the single-cell DNA analysis. Although conventional PCR was the first method used in PGD, fluorescence in situ hybridization (FISH) is now common. It is a delicate visual assay that allows the detection of nucleic acid within undisturbed cellular and tissue architecture. It relies firstly on the fixation of the cells to be analyzed. Consequently, optimization of the fixation and storage condition of the sample is needed, especially for single-cell suspensions.

[0011] Advanced technologies that enable the diagnosis of a number of diseases at the single-cell level include interphase chromosome conversion, comparative genomic hybridization (CGH), fluorescent PCR, and whole genome amplification. The reliability of the data generated by all of these techniques rely on the quality of the DNA preparation. PGD is also costly, consequently there is a need for less expensive approaches, such as mini-sequencing. Unlike most mutation-detection techniques, mini-sequencing permits analysis of very small DNA fragments with low ADO rate. Better methods for the preparation of single-cell DNA for amplification and PGD are therefore needed and are under study. The more novel microarrays and comparative genomic hybridization techniques, still ultimately rely on the quality of the DNA under analysis.

[0012] Several techniques are in development to measure multiple SNPs on the DNA of a small number of cells, a single cell (for example, a blastomere), a small number of chromosomes, or from fragments of DNA. There are techniques that use Polymerase Chain Reaction (PCR), followed by microarray genotyping analysis. Some PCR-based techniques include whole genome amplification (WGA) techniques such as multiple displacement amplification (MDA), and MOLECULAR INVERSION PROBES (MIPs) that perform genotyping using multiple tagged oligonucleotides that may then be amplified using PCR with a single pair of primers. An example of a non-PCR based technique is fluorescence in situ hybridization (FISH). It is apparent that the techniques will be severely error-prone due to the limited amount of genetic material which will exacerbate the impact of effects such as allele drop-outs, imperfect hybridization, and contamination.

[0013] Many techniques exist which provide genotyping data. TAQMAN is a unique genotyping technology produced and distributed by Applied Biosystems. TAQMAN uses polymerase chain reaction (PCR) to amplify sequences of interest. During PCR cycling, an allele specific minor groove binder (MGB) probe hybridizes to amplified sequences. Strand synthesis by the polymerase enzymes releases reporter dyes linked to the MGB probes, and then the TAQMAN optical readers detect the dyes. In this manner, TAQMAN achieves quantitative allelic discrimination. Compared with array based genotyping technologies, TAQMAN is quite expensive per reaction (˜$0.40/reaction), and throughput is relatively low (384 genotypes per run). While only 1 ng of DNA per reaction is necessary, thousands of genotypes by TAQMAN requires microgram quantities of DNA, so TAQMAN does not necessarily use less DNA than microarrays. However, with respect to the IVF genotyping workflow, TAQMAN is the most readily applicable technology. This is due to the high reliability of the assays and, most importantly, the speed and ease of the assay (˜3 hours per run and minimal molecular biological steps). Also unlike many array technologies (such as 500 k AFFMETRIX arrays), TAQMAN is highly customizable, which is important for the IVF market. Further, TAQMAN is highly quantitative, so aneuploidies could be detected with this technology alone.

[0014] ILLUMINA has recently emerged as a leader in high-throughput genotyping. Unlike AFFMETRIX, ILLUMINA genotyping arrays do not rely exclusively on hybridization. Instead, ILLUMINA technology uses an allele-specific DNA extension step, which is much more sensitive and specific than hybridization alone, for the original sequence detection. Then, all of these alleles are amplified in multiplex by PCR, and then these products hybridized to bead arrays. The beads on these arrays contain unique “address” tags, not native sequence, so this hybridization is highly specific and sensitive. Alleles are then called by quantitative scanning of the bead arrays. The Illlumina GOLDEN GATE assay system genotypes up to 1536 loci concurrently, so the throughput is better than TAQMAN but not as high as AFFMETRIX 500 k arrays. The cost of ILLUMINA genotypes is lower than TAQMAN, but higher than AFFMETRIX arrays. Also, the ILLUMINA platform takes as long to complete as the 500 k AFFMETRIX arrays (up to 72 hours), which is problematic for IVF genotyping. However, ILLUMINA has a much better call rate, and the assay is quantitative, so aneuploidies are detectable with this technology. ILLUMINA technology is much more flexible in choice of SNPs than 500 k AFFMETRIX arrays.

[0015] One of the highest throughput techniques, which allows for the measurement of up to 250,000 SNPs at a time, is the AFFMETRIX GeneChip 500K genotyping array. This technique also uses PCR, followed by analysis by hybridization and detection of the amplified DNA sequences to DNA probes, chemically synthesized at different locations on a quartz surface. A disadvantage of these arrays are the low flexibility and the lower sensitivity. There are modified approaches that can increase selectivity, such as the “perfect match” and “mismatch probe” approaches, but these do so at the cost of the number of SNPs calls per array.

[0016] Pyrosequencing, or sequencing by synthesis, can also be used for genotyping and SNP analysis. The main advantages to pyrosequencing include an extremely fast turnaround and unambiguous SNP calls, however, the assay is not currently conducive to high-throughput parallel analysis. PCR followed by gel electrophoresis is an exceedingly simple technique that has met the most success in preimplantation diagnosis. In this technique, researchers use nested PCR to amplify short sequences of interest. Then, they run these DNA samples on a special gel to visualize the PCR products. Different bases have different molecular weights, so one can determine base content based on how fast the product runs in the gel. This technique is low-throughput and requires subjective analyses by scientists using current technologies, but has the advantage of speed (1-2 hours of PCR, 1 hour of gel electrophoresis). For this reason, it has been used previously for prenatal genotyping for a myriad of diseases, including: thalassaemia, neurofibromatosis type 2, leukocyte adhesion deficiency type I, Hallopeau-Siemens disease, sickle-cell anemia, retinoblastoma, Pelizaeus-Merzbacher disease, Duchenne muscular dystrophy, and Currarino syndrome.

[0017] Another promising technique that has been developed for genotyping small quantities of genetic material with very high fidelity is MOLECULAR INVERSION PROBES (MIPs), such as AFFMETRIX's GENFLEX Arrays. This technique has the capability to measure multiple SNPs in parallel: more than 10,000 SNPS measured in parallel have been verified. For small quantities of genetic material, call rates for this technique have been established at roughly 95%, and accuracy of the calls made has been established to be above 99%. So far, the technique has been implemented for quantities of genomic data as small as 150 molecules for a given SNP. However, the technique has not been verified for genomic data from a single cell, or a single strand of DNA, as would be required for pre-implantation genetic diagnosis.

[0018] The MIP technique makes use of padlock probes which are linear oligonucleotides whose two ends can be joined by ligation when they hybridize to immediately adjacent target sequences of DNA. After the probes have hybridized to the genomic DNA, a gap-fill enzyme is added to the assay which can add one of the four nucleotides to the gap. If the added nucleotide (A,C,T,G) is complementary to the SNP under measurement, then it will hybridize to the DNA, and join the ends of the padlock probe by ligation. The circular products, or closed padlock probes, are then differentiated from linear probes by exonucleolysis. The exonuclease, by breaking down the linear probes and leaving the circular probes, will change the relative concentrations of the closed vs. the unclosed probes by a factor of 1000 or more. The probes that remain are then opened at a cleavage site by another enzyme, removed from the DNA, and amplified by PCR. Each probe is tagged with a different tag sequence consisting of 20 base tags (16,000 have been generated), and can be detected, for example, by the AFFMETRIX GENFLEX Tag Array. The presence of the tagged probe from a reaction in which a particular gap-fill enzyme was added indicates the presence of the complimentary amino acid on the relevant SNP.

[0019] The molecular biological advantages of MIPS include: (1) multiplexed genotyping in a single reaction, (2) the genotype “call” occurs by gap fill and ligation, not hybridization, and (3) hybridization to an array of universal tags decreases false positives inherent to most array hybridizations. In traditional 500K, TAQMAN and other genotyping arrays, the entire genomic sample is hybridized to the array, which contains a variety of perfect match and mismatch probes, and an algorithm calls likely genotypes based on the intensities of the mismatch and perfect match probes. Hybridization, however, is inherently noisy, because of the complexities of the DNA sample and the huge number of probes on the arrays. MIPs, on the other hand, uses multiplex probes (i.e., not on an array) that are longer and therefore more specific, and then uses a robust ligation step to circularize the probe. Background is exceedingly low in this assay (due to specificity), though allele dropout may be high (due to poor performing probes).

[0020] When this technique is used on genomic data from a single cell (or small numbers of cells) it will—like PCR based approaches—suffer from integrity issues. For example, the inability of the padlock probe to hybridize to the genomic DNA will cause allele dropouts. This will be exacerbated in the context of in-vitro fertilization since the efficiency of the hybridization reaction is low, and it needs to proceed relatively quickly in order to genotype the embryo in a limited time period. Note that the hybridization reaction can be reduced well below vendor-recommended levels, and micro-fluidic techniques may also be used to accelerate the hybridization reaction. These approaches to reducing the time for the hybridization reaction will result in reduced data quality.

[0021] Once the genetic data has been measured, the next step is to use the data for predictive purposes. Much research has been done in predictive genomics, which tries to understand the precise functions of proteins, RNA and DNA so that phenotypic predictions can be made based on genotype. Canonical techniques focus on the function of Single-Nucleotide Polymorphisms (SNP); but more advanced methods are being brought to bear on multi-factorial phenotypic features. These methods include techniques, such as linear regression and nonlinear neural networks, which attempt to determine a mathematical relationship between a set of genetic and phenotypic predictors and a set of measured outcomes. There is also a set of regression analysis techniques, such as Ridge regression, log regression and stepwise selection, that are designed to accommodate sparse data sets where there are many potential predictors relative to the number of outcomes, as is typical of genetic data, and which apply additional constraints on the regression parameters so that a meaningful set of parameters can be resolved even when the data is underdetermined. Other techniques apply principal component analysis to extract information from undetermined data sets. Other techniques, such as decision trees and contingency tables, use strategies for subdividing subjects based on their independent variables in order to place subjects in categories or bins for which the phenotypic outcomes are similar. A recent technique, termed logical regression, describes a method to search for different logical interrelationships between categorical independent variables in order to model a variable that depends on interactions between multiple independent variables related to genetic data. Regardless of the method used, the quality of the prediction is naturally highly dependent on the quality of the genetic data used to make the prediction.

[0022] Normal humans have two sets of 23 chromosomes in every diploid cell, with one copy coming from each parent. Aneuploidy, a cell with an extra or missing chromosomes, and uniparental disomy, a cell with two of a given chromosome that originate from one parent, are believed to be responsible for a large percentage of failed implantations, miscarriages, and genetic diseases. When only certain cells in an individual are aneuploid, the individual is said to exhibit mosaicism. Detection of chromosomal abnormalities can identify individuals or embryos with conditions such as Down syndrome, Klinefelter syndrome, and Turner syndrome, among others, in addition to increasing the chances of a successful pregnancy. Testing for chromosomal abnormalities is especially important as mothers age: between the ages of 35 and 40 it is estimated that between 40% and 50% of the embryos are abnormal, and above the age of 40, more than half of the embryos are abnormal.

[0023] Karyotyping, the traditional method used for the prediction of aneuploides and mosaicism is giving way to other more high throughput, more cost effective methods. One method that has attracted much attention recently is Flow cytometry (FC) and fluorescence in situ hybridization (FISH) which can be used to detect aneuploidy in any phase of the cell cycle. One advantage of this method is that it is less expensive than karyotyping, but the cost is significant enough that generally a small selection of chromosomes are tested (usually chromosomes 13, 18, 21, X, Y; also sometimes 8, 9, 15, 16, 17, 22); in addition, FISH has a low level of specificity. Using FISH to analyze 15 cells, one can detect mosaicism of 19% with 95% confidence. The reliability of the test becomes much lower as the level of mosaicism gets lower, and as the number of cells to analyze decreases. The test is estimated to have a false negative rate as high as 15% when a single cell is analyzed. There is a great demand for a method that has a higher throughput, lower cost, and greater accuracy.

[0024] Listed here is a set of prior art which is related to the field of the current invention. None of this prior art contains or in any way refers to the novel elements of the current invention. In U.S. Pat. No. 6,720,140, Hartley et al describe a recombinational cloning method for moving or exchanging segments of DNA molecules using engineered recombination sites and recombination proteins. In U.S. Pat. No. 6,489,135 Parrott et al. provide methods for determining various biological characteristics of in vitro fertilized embryos, including overall embryo health, implantability, and increased likelihood of developing successfully to term by analyzing media specimens of in vitro fertilization cultures for levels of bioactive lipids in order to determine these characteristics. In US Patent Application 20040033596 Threadgill et al. describe a method for preparing homozygous cellular libraries useful for in vitro phenotyping and gene mapping involving site-specific mitotic recombination in a plurality of isolated parent cells. In U.S. Pat. No. 5,994,148 Stewart et al. describe a method of determining the probability of an in vitro fertilization (IVF) being successful by measuring Relaxin directly in the serum or indirectly by culturing granulosa lutein cells extracted from the patient as part of an IVF/ET procedure. In U.S. Pat. No. 5,635,366 Cooke et al. provide a method for predicting the outcome of IVF by determining the level of 11β-hydroxysteroid dehydrogenase (11β-HSD) in a biological sample from a female patient. In U.S. Pat. No. 7,058,616 Larder et al. describe a method for using a neural network to predict the resistance of a disease to a therapeutic agent. In U.S. Pat. No. 6,958,211 Vingerhoets et al. describe a method wherein the integrase genotype of a given HIV strain is simply compared to a known database of HIV integrase genotype with associated phenotypes to find a matching genotype. In U.S. Pat. No. 7,058,517 Denton et al. describe a method wherein an individual's haplotypes are compared to a known database of haplotypes in the general population to predict clinical response to a treatment. In U.S. Pat. No. 7,035,739 Schadt at al. describe a method is described wherein a genetic marker map is constructed and the individual genes and traits are analyzed to give a gene-trait locus data, which are then clustered as a way to identify genetically interacting pathways, which are validated using multivariate analysis. In U.S. Pat. No. 6,025,128 Veltri et al. describe a method involving the use of a neural network utilizing a collection of biomarkers as parameters to evaluate risk of prostate cancer recurrence.

[0025] The cost of DNA sequencing is dropping rapidly, and in the near future individual genomic sequencing for personal benefit will become more common. Knowledge of personal genetic data will allow for extensive phenotypic predictions to be made for the individual. In order to make accurate phenotypic predictions high quality genetic data is critical, whatever the context. In the case of prenatal or pre-implantation genetic diagnoses a complicating factor is the relative paucity of genetic material available. Given the inherently noisy nature of the measured genetic data in cases where limited genetic material is used for genotyping, there is a great need for a method which can increase the fidelity of, or clean, the primary data.

[0026] The system disclosed enables the cleaning of incomplete or noisy genetic data using secondary genetic data as a source of information. While the disclosure focuses on genetic data from human subjects, and more specifically on as-yet not implanted embryos or implanted fetuses, it should be noted that the methods disclosed apply to the genetic data of a range of organisms, in a range of contexts. The techniques described for cleaning genetic data are most relevant in the context of pre-implantation diagnosis during in-vitro fertilization, prenatal diagnosis in conjunction with amniocentesis, chorion villus biopsy, and fetal blood sampling, and non-invasive prenatal diagnosis, where a small quantity of fetal genetic material is isolated from maternal blood. The diagnoses may focus on inheritable diseases, increased likelihoods of defects or abnormalities, as well as making phenotype predictions for individuals to enhance clinical and lifestyle decisions. The invention addresses the shortcomings of prior art that are discussed above.

[0027] In one aspect of the invention, methods make use of imperfect knowledge of the genetic data of the mother and the father, together with the knowledge of the mechanism of meiosis and the imperfect measurement of the embryonic DNA, in order to reconstruct, in silico, the embryonic DNA at the location of key SNPs with a high degree of confidence. It is important to note that the parental data allows the reconstruction not only of SNPs that were measured poorly, but also of insertions, deletions, and of SNPs or whole regions of DNA that were not measured at all.

[0028] The disclosed method is applicable in the context of in-vitro fertilization, where a very small number of blastomeres are available for genotyping from each embryo being considered for implantation. The disclosed method is equally applicable to the context of Non-Invasive Prenatal Diagnosis (NIPD) where only a small number of fetal cells, or fragments of fetal DNA, have been isolated from the mother's blood. The disclosed method is more generally applicable in any case where a limited amount of genetic data is available from the target genome, and additional genetic data is available from individuals who are genetically related to the target.

[0029] In one aspect of the invention, the fetal or embryonic genomic data which has been reconstructed can be used to detect if the cell is aneuploid, that is, if fewer or more than two of a particular chromosome is present in a cell. A common example of this condition is trisomy-21, which gives rise to Down syndrome. The reconstructed data can also be used to detect for uniparental disomy, a condition in which two of a given chromosome are present, both of which originate from one parent. This is done by creating a set of hypotheses about the potential states of the DNA, and testing to see which one has the highest probability of being true given the measured data. Note that the use of high throughput genotyping data for screening for aneuploidy enables a single blastomere from each embryo to be used both to measure multiple disease-linked loci as well as screen for aneuploidy.

[0030] In another aspect of the invention, the direct measurements of the amount of genetic material, amplified or unamplified, present at a plurality of loci, can be used to detect for aneuploides, or uniparental disomy. The idea behind this method is simply that the amount of genetic material present during amplification is proportional to the amount of genetic information in the initial sample, and measuring these levels at multiple loci will give a statistically significant result.

[0031] In another aspect of the invention, the disclosed method can clean genetic material of the individual which has been contaminated by foreign DNA or RNA by identifying the data generated by extraneous genetic materials. The spurious signals generated by the contaminating DNA can be recognized in a manner similar to that way that chromosome-wide anomalous signals generated by aneuploides can be detected.

[0032] In another aspect of the invention, target cells are isolated, the genetic data contained in those cells are amplified, and measurements of multiple SNPs are made using a combination of one or more of the following techniques: PCR-based amplification techniques, PCR-based measurement techniques, or detection techniques based on MOLECULAR INVERSION PROBES, or microarrays such as the GENECHIP or TAQMAN systems. This genetic data is then used in the system described herein.

[0033] In another aspect of the invention, the genetic data of an individual can be cleaned using diploid and haploid data from both parents. Alternately, haploid data from a parent can be simulated if diploid and haploid data of the parent's parent can be measured. In another aspect, genetic data from any person of a known genetic relation to the individual can be used to clean the data of the individual, including parents, siblings, grandparents, offspring, cousins, uncles, aunts etc.

[0034] In another aspect of the invention, the target and/or related individual's genetic data may be partly or wholly known in silico, obviating the need for some direct measurements. Portions of the genetic data can be generated in silico by means of an informatics approach utilizing a hidden Markov model.

[0035] In another aspect of the invention, it is possible to estimate the confidence one has in the determination of those SNPs.

[0036] Note that the techniques described herein are relevant both to measurements of genetic material in one, or a small number of cells, as well as to measurements on smaller amounts of DNA such as that which can be isolated from the mother's blood in the context of Non-invasive Prenatal Diagnosis (NIPD). Also note that this method can be equally applied to genomic data in silico, i.e. not directly measured from genetic material.

[0037] It will be recognized by a person of ordinary skill in the art, given the benefit of this disclosure, aspects and embodiments that may implement one or more of the systems, methods, and features, disclosed herein.

Conceptual Overview of the System

[0064] The goal of the disclosed system is to provide highly accurate genomic data for the purpose of genetic diagnoses. In cases where the genetic data of an individual contains a significant amount of noise, or errors, the disclosed system makes use of the similarities between genetic data of related individuals, and the information contained in that secondary genetic data, to clean the noise in the target genome. This is done by determining which segments of chromosomes were involved in gamete formation and where crossovers occurred during meiosis, and therefore which segments of the secondary genomes are expected to be nearly identical to sections of the target genome. In certain situations this method can be used to clean noisy base pair measurements, but it also can be used to infer the identity of individual base pairs or whole regions of DNA that were not measured. In addition, a confidence can be computed for each reconstruction call made. A highly simplified explanation is presented first, making unrealistic assumptions in order to illustrate the concept of the invention. A detailed statistical approach that can be applied to the technology of today is presented afterward.

[0065] Another goal of the system is to detect abnormal numbers of chromosomes, sections of chromosomes, and origins of chromosomes. In genetic samples that are aneuploid, have unbalanced translocations, uniparental disomy, or other gross chromosomal abnormalities, the amount of genetic material present at a plurality of loci can be used to determine the chromosomal state of the sample. There are multiple approaches to this method, and several of them are described here. In some approaches, the amount of genetic material present in a sample is sufficient to directly detect aneuploides. In other approaches, the method for cleaning the genetic material can be used to enhance the efficiency of detection of chromosomal imbalances. A confidence can be computed for each chromosomal call made.

[0066] FIG. 1 illustrates the process of recombination that occurs during meiosis for the formation of gametes in a parent. The chromosome 101 from the individual's mother is shown in grey. The chromosome 102 from the individual's father is shown in white. During this interval, known as Diplotene, during Prophase I of Meiosis, a tetrad of four chromatids 103 is visible. Crossing over between non-sister chromatids of a homologous pair occurs at the points known as recombination nodules 104. For the purpose of illustration, the example will focus on a single chromosome, and three Single Nucleotide Polymorphisms (SNPs), which are assumed to characterize the alleles of three genes. For this discussion it is assumed that the SNPs may be measured separately on the maternal and paternal chromosomes. This concept can be applied to many SNPs, many alleles characterized by multiple SNPs, many chromosomes, and to the current genotyping technology where the maternal and paternal chromosomes cannot be individually isolated before genotyping.

[0067] Attention must be paid to the points of potential crossing over in between the SNPs of interest. The set of alleles of the three maternal genes may be described as (a_m1, a_m2, a_m3) corresponding to SNPs (SNP₁, SNP₂, SNP₃). The set of alleles of the three paternal genes may be described as (a_p1, a_p2, a_p3). Consider the recombination nodules formed in FIG. 1, and assume that there is just one recombination for each pair of recombining chromatids. The set of gametes that are formed in this process will have gene alleles: (a_m1, a_m2, a_p3), (a_m1, a_p2, a_p3), (a_p1, a_m2, a_p3), (a_p1, a_p2, a_m3). In the case with no crossing over of chromatids, the gametes will have alleles (a_m1, a_m2, a_m3), (a_p1, a_p2, a_p3). In the case with two points of crossing over in the relevant regions, the gametes will have alleles (a_m1, a_p2, a_m3), (a_p1, a_m2, a_p3). These eight different combinations of alleles will be referred to as the hypothesis set of alleles, for that particular parent.

[0068] The measurement of the alleles from the embryonic DNA will be noisy. For the purpose of this discussion take a single chromosome from the embryonic DNA, and assume that it came from the parent whose meiosis is illustrated in FIG. 1. The measurements of the alleles on this chromosome can be described in terms of a vector of indicator variables: A=[A₁ A₂ A₃]^T where A₁=1 if the measured allele in the embryonic chromosome is a_m1, A₁=−1 if the measured allele in the embryonic chromosome is a_p1, and A₁=0 if the measured allele is neither a_m1 or a_p1. Based on the hypothesis set of alleles for the assumed parent, a set of eight vectors may be created which correspond to all the possible gametes describe above. For the alleles described above, these vectors would be a₁=[1 1 1]^T, a₂=[1 1 −1]^T, a₃=[1 −1 1]^T, a₄=[1 −1 −1]^T, a₅=[−1 1 1]^T, a₆=[−1 1 −1]^T, a₇=[−1 −1 1]^T, a₈=[−1 −1 −1]^T. In this highly simplified application of the system, the likely alleles of the embryo can be determined by performing a simple correlation analysis between the hypothesis set and the measured vectors:
i*=arg max_i A^Ta_i, i=1 . . . 8  (1)

[0069] Once i* is found, the hypothesis a_i* is selected as the most likely set of alleles in the embryonic DNA. This process is then repeated twice, with two different assumptions, namely that the embryonic chromosome came from the mother or the father. That assumption which yields the largest correlation A^Ta_i* would be assumed to be correct. In each case a hypothesis set of alleles is used, based on the measurements of the respective DNA of the mother or the father. Note that in a typical embodiment of the disclosed method, one measures a large number of SNPs between those SNPs that are important due to their association with particular disease phenotypes—these will be referred to these as Phenotype-associated SNPs or PSNPs. The Non-phenotype-associated SNPs (NSNPs) between the PSNPs may be chosen a-priori (for example, for developing a specialized genotyping array) by selecting from the NCBI dbSNP database those RefSNPs that tend to differ substantially between individuals. Alternatively, the NSNPs between the PSNPs may be chosen for a particular pair of parents because they differ between the parents. The use of the additional SNPs between the PSNPs enables one to determine with a higher level of confidence whether crossover occurs between the PSNPs. It is important to note that while different “alleles” are referred to in this notation, this is merely a convenience; the SNPs may not be associated with genes that encode proteins.

The System in the Context of Current Technology

[0070] In another more complex embodiment, the a-posteriori probability of a set of alleles is computed given a particular measurement, taking into account the probability of particular crossovers. In addition, the scenario typical of microarrays and other genotyping technologies is addressed where SNPs are measured for pairs of chromosomes, rather than for a single chromosome at a time. The measurements of the genotype at the locus i for the embryonic, paternal and maternal chromosomes may be characterized respectively by random variables representing the pairs of SNP measurements (e_1,i, e_2,i), (p_1,i, p_2,i) and (m_1,i, m_2,i). Since one cannot determine the presence of crossovers in the maternal and paternal chromosomes if all measurements are made as pairs, the method is modified: in addition to genotyping the fertilized embryos and paternal and maternal diploid tissue, one haploid cell from each parent, namely, a sperm cell and an egg cell, is also genotyped. The measured alleles of the sperm cell are represented by p_1,i, i=1 . . . N and the complementary alleles measured from the paternal diploid tissue by p_2,i. Similarly, the measured alleles of the egg cell are represented by m_1,i and their complement in the mother's diploid cell by m_2,i. These measurements provide no information on where the parental chromosomes crossed over in generating the measured sperm and egg cells. However, one can assume that the sequence of N alleles on the egg or sperm was created from the parental chromosomes by a small number of, or no, crossovers. This is sufficient information to apply the disclosed algorithm. A certain error probability is associated with calling the paternal and maternal SNPs. The estimation of this error probability will vary based on the measurements made (p_1,i, p_2,i) and (m_1,i, m_2,i) and the signal-to-noise ratio for the technology used. Although these error probabilities can be uniquely computed for each locus without affecting the disclosed method, the algebra is simplified here by assuming that the probabilities of correctly calling the paternal and maternal SNPs are constant at p_p and p_m respectively.

[0071] Assume that a measurement is performed on the embryonic DNA which is termed measurement M. In addition, the notation is slightly modified so that A is now a set and not a vector: A refers to a particular hypothesis about the combination (or set) of alleles derived from each parent. The set of all possible combinations of alleles A from both parents is denoted as S_A. The goal is to determine the combination of alleles (or that hypothesis) A∈S_A with the highest a-posteriori probability, given the measurement M:
A*=arg max_A P(A|M),∀A∈S_A  (2)
Using the law of conditional probabilities, P(A|M)=P(M|A)P(A)/P(M). Since P(M) is common for all different A's, the optimizing search can be recast as:
A*=arg max_AP(M/A)P(A),∀A∈S_A  (3)

[0072] Now consider the computation of P(M/A). Begin with a single locus i, and let the hypothesis be that this locus on the embryo is derived from the parental SNPs p_t,1,i and m_t,1,i, where the underscore_t is used to denote the true value of these Parental SNPs, as opposed to the measurements performed, p_1,i and m_1,i, which may or may not be correct. The true value of the embryonic SNPs is denoted as (e_t,1,i, e_t,2,i). If hypothesis A is true, then (e_t,1,i, e_t,2,i)=(p_t,1,i, m_t,1,i) or (m_t,1,i, p_t,1,i). Since one cannot differentiate which of the measurements (e_1,i, e_2,i) comes from which parent, both orders must be considered so the hypothesis set A=[(p_t,1,i, m_t,1,i), (m_t,1,i, p_t,2,i)]. The probability of a particular measurement M depends on the true values or the underlying states of the parental SNPs, namely (p_t,1,i, p_t,2,i) and (m_t,1,i, m_t,2,i). Since there are four SNPs, p_t,1,i, p_t,2,i, m_t,1,i, m_t,2,i, and each of these can assume the value of four nucleotide bases, A,C,T,G, there are 4⁴ or 256 possible states. The algorithm is illustrated for one state Si for which it is assumed that p_t,1,i≠p_t,2,i≠m_t,1,i≠m_t,2,i. From this explanation, it will be clear how to apply the method to all 256 possible states, s_k, k=1 . . . 256. Assume a measurement M of embryonic SNPs (e_1,i, e_2,i) is performed, and the result e_1,i=p_1,i, e_2,i=m_1,i is obtained. The a priori probability for this measurement given that hypothesis A and state Si are true is computed:
P(e_1,i=p_1,i,e_2,i=m_1,i|A,s₁)=P(e_t,1,i=p_t,1,i,e_t,2,i=m_t,1,i|A,s₁)P(e_1,i=p_1,i|e_t,1,i=p_t,1,i)P(e_2,i=m_1,i|e_t,2,i=m_t,1,i)+P(e_t,1,i=m_t,1,i,e_t,2,i=p_t,1,i|A,s₁)P(e_1,i=p_1,i|e_t,1,i=m_t,1,i,p_t,1,i≠m_t,1,i)P(e_2,i=m_1,i|e_t,2,i=p_t,2,i,p_t,2,i≠m_t,1,i)   (4)
Consider the first expressions in the first term and second term: P(e_1,i=p_1,i,e_2,i=m_1,i|A,s₁)=P(e_1,i=m_1,i,e_2,i=p_1,i|A,s₁)=0.5 since the hypothesis A=[(p_t,1,i, m_t,1,i), (m_t,1,i, p_t,1,i)] makes two orderings for the embryonic SNPs equally likely. Now consider the second expression of the first term, P(e_1,i=p_1,i|e_t,1,i=p_t,1,i), the probability of measuring e_1,i=p_1,i given the assumption that embryonic SNP e_t,1,i actually is derived from paternal SNP p_t,1,i. The probabilities for correctly measuring the paternal SNPs, maternal SNPs, and embryonic SNPs are p_p, p_m, and p_e. Given the assumption (e_t,1,i=p_t,1,i), the measurement (e_1,i=p_1,i) requires either that both embryonic and paternal SNPs are correctly measured, or that both are incorrectly measured and they happen to be incorrectly measured as the same nucleotide (A, C, T, or G). So, P(e_1,i=p_1,i|e_t,1,i=p_t,1,i)=p_ep_p+(1−p_e)(1−p_p)/3 where it is assumed for simplicity that the probability of incorrectly calling all of the four nucleotides is equally likely—the algorithm can be easily modified to accommodate different probabilities of calling a particular nucleotide (A,C,T,G) given a measurement on another particular nucleotide. The same approach may be applied to the third expression in the first term to obtain P(e_2,i=m_1,i|e_t,2,i=m_t,1,i)=p_ep_m+(1−p_e)(1−p_m)/3. Now consider the second expression of the second term. P(e_1,i=p_1,i|e_t,1,i=m_t,1,i, m_t,1,i≠p_t,1,i) requires either that e_1,i or p_1,i be an incorrect measurement, or that both be incorrect measurements, so that the measured values happen to be equal: P(e_1,i=p_1,i|e_t,1,i=m_t,1,i, m_t,1,i≠p_t,1,i)=p_e(1−p_p)/3+(1−p_e)p_p/3+(1−p_e)(1−p_p)2/9. The same argument can be applied to the last expression of the second term to yield P(e_2,i=m_1,i|e_t,2,i=p_t,2,i, m_t,1,i≠p_t,2,i)=p_e(1−p_m)/3+(1−p_e)p_m/3+(1−p_e)(1−p_m)2/9. Now, combining all of these terms, and making the assumption—merely to simplify the algebra—that p_e=p_p=p_m=p, one can compute:
P(M(e_1,i=p_1,i,e_2,i=m_1,i)|A,s₁)= 1/162(160p⁴−160p³+96p²−28p+13)  (5)

[0073] Although the computation will vary, a similar conceptual approach to that described here would be used for all 256 possible states, s_k, k=1 . . . 256. Computing P(e_1,i=p_1,i, e_2,i=m_1,i|A,s₁) for all 256 states si and summing over the probability of each si one obtains P(e_1,i=p_1,I, e_2,i=m_1,i|A). In other words:

[0074] $\begin{array}{cc}P\left(M❘A\right)=\sum _{i=1\mathrm{\dots 256}}^{}P\left(M❘A,s_i\right)P\left(s_i\right)& \left(6\right)\end{array}$

[0075] In order to compute the probabilities of each state si, P(s_i), one must treat all the separate alleles making up a state as separate events since they are on separate chromosomes, in other words: P(s_i)=P(p_t,1,i, p_t,2,i, m_t,1,i m_t,2,i)=P(p_t,1,i)P(p_t,2,i)P(m_t,1,i)P(m_t,2,i). Bayesian techniques may be applied to estimate the probability distribution for the individual measurements. Every measurement of an allele on the maternal or paternal chromosomes at locus i may be treated as a coin toss experiment to measure the probability of this allele being a particular value (A, C, T or G). These measurements are made on the adult tissue samples and may be treated as being totally reliable, even though pairs of alleles are measured for each SNP, and it is not possible to determine which allele comes from which chromosome. Let w_p,1,i=P(p_t,1,i), corresponding to the probability of the SNP i on the father's chromosome being value p_t,1,i. In the following explanation, w is used instead of w_p,1,i. Let the measurements performed on SNP i of the father's chromosome be characterized as collecting data D. One can create a probability distribution for w, p(w) and update this after the data is measurement according to Bayes Theorem: p(w|D)=p(w)p(D|w)/p(D). Assume n alleles of SNP i are observed and that the particular allele corresponding to w comes up h times—in other words, heads is observed h times. The probability of this observation can be characterized by the binomial distribution

[0076] $\begin{array}{cc}p\left(D❘w\right)=\left(\begin{array}{c}n\\ h\end{array}\right){w^h\left(1-w\right)}^{n-h}& \left(7\right)\end{array}$
Before data is collected, assume there is a prior distribution p(w) which is uniform between 0 and 1. By applying the Bayes theorem, it is straightforward to show that the resulting distribution for p(w|D) will be a beta distribution of the form:

[0077] $\begin{array}{cc}p\left(w❘D\right)=\frac{1}{c}{w^h\left(1-w\right)}^{n-h}\mathrm{where}c={\int }_0^1{w^h\left(1-w\right)}^{n-h}\mathrm{dw}& \left(8\right)\end{array}$
and c is a normalizing constant. However many times p(w|D) is then updated by applying Bayes theorem and new measurements, it will continue to have a beta distribution as above. The estimates of p(w) are updated every time a new measurement is collected. Note that there will be a different function p(w) for different races and different genders, using the same groupings used in the Hapmap project, since the probability of different alleles at particular SNPs is dependent on these groupings of race and gender. For the computation of P(s_i), each allele on each chromosome will be associated with an estimated probability distribution, namely p_p,1,i(w_p,1,i), p_p,2,i(w_p,2,i), p_m,1,i(w_m,1,i) and p_m,2,i(w_m,2,i). One may then compute the maximum a-posteriori (MAP) estimate for P(s_i) according to the MAP estimate for each of the individual distributions. For example, let w_p,1,i* be the argument that maximizes p_p,1,i(w_p,1,i). The MAP estimate of P(s_i) may be found according to
P(s_i)_MAP=w_p,1,i*w_p,2,i*w_m,1,i*w_m,2,i*  (9)

[0078] Since there is a probability distribution for each w, one can also compute conservative estimates of the values P(s_i) to any specified confidence level, by integrating over the probability distribution, rather than simply using the MAP estimates. It is possible to do this, for example, to conservatively estimate P(M|A) to within some confidence level. Whether a conservative estimate or a MAP estimate is used, the estimate of P(s_i) is continually refined for the computation of P(M|A). In what follows, reference to the assumed state will be eliminated to simplify the notation, and state s_i is assumed for all explanations of detailed computation. Bear in mind that in actuality these calculations would be performed for each of 256 states and be summed over the probability of each.

[0079] The method of computing P(M|A) is now extended to multiple SNP loci, assuming that M represents the set of measurements of N pairs of SNPs on the embryo, M=[M₁, . . . , MN]. Assume also that A represents the set of hypotheses for each SNP about which parental chromosomes contributed to that SNP, A=[A₁, . . . , A_N]. Let S_A′ represent the set of all other possible hypotheses that are different from A or are in the set A′. P(M|A) and P(M|A′) may be computed:

[0080] $\begin{array}{cc}P\left(M❘A\right)=\prod _{i=1\dots N}^{}P\left(M_i❘A_i\right),P\left(M❘A^{\prime }\right)=\sum _{A\in S_{A^{\prime }}}^{}P\left(A\right)\prod _{i=1\dots N}^{}P\left(M_i❘A_i\right)& \left(10\right)\end{array}$
Consider the computation of P(A). In essence, this is based on the likelihood of particular crossovers occurring in the formation of the gametes that form the embryo. The probability of a particular allele set depends on two factors, namely the probability that the embryonic chromosome comes from the mother or the father, and the probability of a particular combination of crossovers. For a normal set of embryonic chromosomes that do not suffer from aneuploidy, the a-priori probability that the embryonic chromosome comes from the mother or father is ˜50% and is consequently common for all A. Now, consider the probability of a particular set of recombination nodes. The number of relevant recombination sites R depends on the number of measured SNPS: R=N−1. Since the DNA segment constituting N NSNPs around the PSNP of interest will be relatively short, crossover interference makes it highly improbable that two crossovers on the same chromosome can occur in one region. For reasons of computational efficiency this method assumes that only one crossover will occur in each region for each relevant chromosome, and this can occur at R possible sites. It will be obvious to someone skilled in the art how this method may be extended to include the possibility where there are multiple crossovers in a given region.

[0081] Let the probability of a crossover in each region between SNPs be denoted P_r, r=1 . . . N−1. To first order, the probability of a recombination node in a region r between two SNPs is proportional to the genetic distance between those SNPs (measured in cMorgans). However, much recent research has enabled a precise modeling of the probability of recombination between two SNP loci. Observations from sperm studies and patterns of genetic variation show that recombination rates vary extensively over kilobase scales and that much recombination occurs in recombination hotspots, and causes linkage disequilibrium to display a block-like structure. The NCBI data about recombination rates on the Human Genome is publicly available through the UCSC Genome Annotation Database.

[0082] Various data sets can be used singly or in combination. Two of the most common data sets are from the Hapmap Project and from the Perlegen Human Haplotype Project. The latter is higher density; the former is higher quality. See FIG. 2 for the regional recombination rates from positions 1,038,423 to 4,467,775 of chromosome 1, based on the HapMap Phase I data, release 16a. These rates were estimated using the reversible-jump Markov Chain Monte Carlo (MCMC) method which is available in the package LDHat. The state-space considered is the distribution of piece-wise constant recombination rate maps. The Markov chain explores the distribution of the number and location of rate change-points, in addition to the rates for each segment, 201. These results may be used to generate an estimate of P_r by integrating over the recombination rates times by the length of each constant segment between the SNPS. The cumulative recombination rate over the nucleotides 202 is shown in FIG. 2 in red.

[0083] Let C be a set of indicator variables c_r such that c_r=1 if a crossover occurred in region r and 0 otherwise. c₀=1 if no crossovers occurred and 0 otherwise. Since it is assumed that only one crossover can occur in a region of N SNPs, only one element of the set C is non-zero. Hence, the probability of crossover represented by set C is found to be:

[0084] $\begin{array}{cc}{P}_c={\left(1-\sum _{r=1\dots N-1}^{}{P}_r\right)}^{c_0}\prod _{r=1}^{}{P}_r^{c_r}& \left(11\right)\end{array}$
In the hypothesis A about SNPs 1 . . . N, there are four potential crossovers of relevance. Namely, the potential crossovers in i) the paternal chromosomes that formed the embryo (denoted by set C_pe of indicator variables), ii) the paternal chromosomes that formed the sequenced sperm (set C_ps), iii) the maternal chromosomes that formed the embryo (set C_me) and iv) the maternal chromosomes that formed the sequenced egg (set C_ee). Two additional assumptions are v) whether the first paternal embryonic SNP comes from p_t,1,1 or p_t,2,1 and vi) whether the first maternal embryonic SNP comes from m_t,1,1 or m_t,2,1. Since the probabilities of crossovers between SNPs is found to differ between races and sexes, different crossover probabilities will be denoted as P_p,r for the paternal chromosomes, and P_m,r for the maternal chromosomes. Therefore, the probability of a particular hypothesis A, which subsumes the sets C_pe, C_ps, C_me, C_ee is expressed as:

[0085] $\begin{array}{cc}P\left(A\right)=\frac{1}{4}{\left(1-\sum _{r=1\dots N-1}^{}{P}_{p,r}\right)}^{c_{\mathrm{pe},0}}\prod _{r=1\dots N-1}^{}{P_{p,r}^{c_{\mathrm{ps},r}}\left(1-\sum _{r=1\dots N-1}^{}{P}_{p,r}\right)}^{c_{\mathrm{ps},0}}\prod _{r=1\dots N-1}^{}{P_{p,r}^{c_{\mathrm{ps},r}}\left(1-\sum _{r=1\dots N-1}^{}{P}_{m,r}\right)}^{c_{\mathrm{me},0}}\prod _{r=1\dots N-1}^{}{P_{m,r}^{c_{\mathrm{me},r}}\left(1-\sum _{r=1\dots N-1}^{}{P}_{m,r}\right)}^{c_{\mathrm{ee},0}}\prod _{r=1\dots N-1}^{}{P}_{m,r}^{c_{\mathrm{ee},r}}& \left(12\right)\end{array}$

[0086] Now with the equations for determining P(A) and P(M/A), all the elements necessary to compute A* per Equation 3 above have been defined. Hence, it is possible to determine from the highly error-prone measurements of the embryonic SNPs where crossovers occurred, and to consequently clean the embryonic measurements with a high degree of confidence. It remains to determine the degree of confidence in the best hypothesis A*. To determine this, it is necessary to find the odds ratio P(A*|M)/P(A*′|M). The tools have all been described above for this computation:

[0087] $\begin{array}{cc}\frac{P\left(A^{*}❘M\right)}{P\left(A^{*\prime }❘M\right)}=\frac{P\left(A^{*}❘M\right)}{1-P\left(A^{*}❘M\right)}=\frac{P\left(A^{*}\right)P\left(M❘A^{*}\right)}{P\left(A^{*\prime }\right)P\left(M❘A^{*\prime }\right)}=\frac{P\left(A^{*}\right)P\left(M❘A^{*}\right)}{\left(1-P\left(A^{*}\right)\right)P\left(M❘A^{*\prime }\right)}={\mathrm{OR}}_{A^{*}}& \left(13\right)\end{array}$

[0088] The confidence in A* is then given as P(A*|M)=OR_A*/(1+OR_A*). This computation indicates the confidence in a particular hypothesis A*, but it does not indicate a confidence in a particular determination of a SNP. In order to compute the confidence in a determination of embryonic PSNP n, it is necessary to create the set of all hypotheses A that don't change the value of this SNP. This set will be denoted as S_A*,n, which corresponds to all hypothesis that result in PSNP n on the embryo having the same value as is predicted by hypothesis A*. Similarly, create a set S_A*′,n which corresponds to all hypothesis that result in PSNP n having a different value to that predicted by hypothesis A*. Now, it is possible to compute the odds ratio of the probability that the SNP is correctly called versus the probability that the SNP is incorrectly called:

[0089] $\begin{array}{cc}{\mathrm{OR}}_{A^{*},n}=\frac{\sum _{A\in S_{A^{*},n}}^{}P\left(A❘M\right)}{\sum _{A\in S_{A^{*\prime },n}}^{}P\left(A❘M\right)}=\frac{\sum _{A\in S_{A^{*},n}}^{}P\left(A❘M\right)}{1-\sum _{A\in S_{A^{*},n}}^{}P\left(A❘M\right)}=\frac{\sum _{A\in S_{A^{*},n}}^{}P\left(A\right)P\left(M❘A\right)}{\sum _{A\in S_{A^{*\prime },n}}^{}P\left(A\right)P\left(M❘A\right)}& \left(14\right)\end{array}$
The confidence in the particular call of embryonic SNP n based on the odds ratio OR_A*,n can be computed as:

[0090] $\begin{array}{cc}P\left(\mathrm{correctlycalledSNPn}\right)=\sum _{A\in S_{A^{*},n}}^{}P\left(A❘M\right)=\frac{{\mathrm{OR}}_{A^{*},n}}{1+{\mathrm{OR}}_{A^{*},n}}& \left(15\right)\end{array}$

[0091] Note that this technique could also be used to detect such defects as uniparental disomy (UPD) wherein two of the same chromosomes are from the same parent, while none of that chromosomes from the other parent is present. Upon attempting to deduce the crossovers in the parent chromosomes, there will be no hypothesis which adequately explains the data with a high confidence, and if alternate hypotheses are allowed that include the possibility of UPD, they will found to be more likely.

Bounding the Effect of Uncertainty in Recombination Rates and SNP Measurement Reliability

[0092] The disclosed method depends on: assumptions about the probability of recombination between particular SNPs; assumptions about the probability of the correct measurement of each SNP on the embryonic, sperm, egg, paternal and maternal chromosomes; and assumptions about the likelihood of certain alleles within different population groups. Consider each of these assumptions: the mechanism of recombination is not perfectly understood and modeled, and the crossover probability has been established to vary based on an individual's genotype. Furthermore, the techniques by which the recombination rates are measured show substantial variability. For example, the package LDHat, which implements the reversible-jump Markov Chain Monte Carlo (MCMC) method, makes a set of assumptions and requires a set of user inputs about the mechanism and characterization of recombination. These assumptions can affect predicted recombination rates between SNPs as is evinced by the different results obtained by various studies.

[0093] It is anticipated that the assumptions about recombination rates, out of all assumptions listed above, will have the most impact on Equation 15. The computations described above should be based on the best estimates of the probability for crossover between SNPS, P_r. Thereafter, conservative estimates may be used for P_r using values at, for example, the 95% confidence bounds for the recombination rates, in the direction that reduces the confidence measure P(correctly called SNP n). The 95% confidence bounds may be derived from confidence data produced by various studies of recombination rates, and this may be corroborated by looking at the level of discordance between published data from different groups using different methods.

[0094] Similarly, the 95% confidence bounds may be used for the estimates of the probability that each SNP is correctly called: p_p, p_m, p_e. These numbers can be computed based on the actual measured array intensities included in the genotyping assay output files, combined with empirical data on the reliability of the measurement technique. Note that those NSNPs for which these parameters p_p, p_m and p_e are not well established may be ignored. For example, since the diploid parental data is reliably measured, one may ignore NSNP measurements of the parents' haploid cells and on the embryo that do not correspond to any of the alleles on the relevant SNPs of the parent's diploid tissue.

[0095] Lastly, consider the assumptions about the likelihood of certain alleles within different population groups, which give rise to the computation P(s_i). These assumptions also will not have a large impact on the disclosed method since the measurement of the parental diploid data is reliable i.e. direct measurement of the state s_i from the parental samples typically result in data with high confidence. Nonetheless, it is possible to use the probability distribution for each as described in Equation 8 in order to compute a confidence bound for the probability of each state P(s_i). As above, one may compute the 95% confidence bound for each P(s_i) in the conservative direction that reduces confidence measure P(correctly called SNP n).

[0096] The determination of P(correctly called SNP n) will inform the decision about how many NSNPs need to be measured around each PSNP in order to achieve the desired level of confidence.

[0097] Note that there are different approaches to implementing the concept of the disclosed method, namely combining the measurement of the parent's DNA, the measurement of the DNA of one or more embryos, and the a-priori knowledge of the process of meiosis, in order to obtain a better estimate of the embryonic SNPs. It will be clear to one skilled in the art, how similar methods can be applied when different subsets of the a-priori knowledge are known or not known, or known to a greater or lesser degree of certainty. For example, one can use the measurements of multiple embryos to improve the certainty with which one can call the SNPs of a particular embryo or to accommodate missing data from the parents. Note also that one does not need a PSNP of interest to be measured by the measurement technique. Even if that PSNPs is not determined by the measurement system, it can still be reconstructed with a high degree of confidence by the disclosed method.

[0098] Also note that once the points of crossover that occurred during meiosis have been determined, and the regions of the target genome have been mapped to the pertinent regions of the parental DNA, it is possible to infer not only the identity of individual SNPs of interest, but also whole regions of DNA that may be missing in the measured target genome due to allele drop-out or other errors in measurement. It is also possible to measure insertions and deletions in the parental DNA, and use the disclosed method to infer that they exist in the target DNA.

[0099] Various techniques may be used to improve the computational complexity of the disclosed algorithm described above. For example, one may only or predominantly select those NSNPs that differ between the mother and the father. Another consideration would be to only use NSNPs that are spaced nearby the PSNPs to minimize the chance of crossovers occurring between the NSNPs and PSNPs of interest. One could also use NSNPs that were spaced along the chromosome so as to maximize coverage of multiple PSNPs. Another consideration will be to initially use only a small number of NSNPs to determine roughly where crossovers occurred, and with only a limited degree of certainty. Additional NSNPs can then be used to refine the crossover model and increase the probability of correctly calling the PSNPs. The number of crossover combinations to consider scales roughly as N^C where N is the number of SNPs and C is the maximum number of crossovers. Consequently, for C=4 it is possible to accommodate roughly N=100 for each PSNP while remaining computationally tractable on a Pentium-IV processor. Using the approaches described above and other approaches for increased computational efficiency, N>100, C>4 can be easily accommodated. One such approach is described below.

[0100] Note that there are many other approaches to make a call on a PSNP and generate an estimate of the probability that a PSNPs has been correctly determined, based on a particular set of embryonic data, parent data, and algorithm used, without changing the underlying concept. This probability can be used for individual decision-making, and for implementing a reliable service in the context of IVF or NIPGD.

Recursive Solution to the Genetic Data Cleaning Algorithm

[0101] Another embodiment of the invention involving an algorithm that scales linearly is described here. Given the limited nature of computation power, the length of the computation may be a significant factor in the use of the disclosed method. When running computations, any algorithm that must compute certain values where the number of computations needed rises exponentially with the number of SNPs can become unwieldy. A solution that involves a number of calculations that increase linearly with the number of SNPs will always be preferred from a time standpoint as the number of SNPs gets large. Below this approach is described.

[0102] A simple approach, which is to consider all possible hypotheses must contend with the running time being an exponential function in number of SNPs. Suppose, as before, that measured data are a collection of measured embryo, father and mother chromosome measurements on k SNPs, i.e. M={M₁, . . . , M_k} where M₁=(e_1i,e_2i,p_1i,p_2i,m_1i,m_2i). As before, the hypotheses space is S_H={H¹, . . . , H^q}={set of all the hypotheses}, where each hypothesis is of the format H^j=(H^j₁, . . . H^j_k) where H^j_i is the “mini” hypothesis for snip i, of the format H^j_i=(p_i*,m_i*) where p_i*∈{p_1i,p_2i} and m_i*∈{m_1i,m_2i}. There are 4 different “mini” hypotheses H^j_i, in particular:

H^j_i1: (e_1i,e_2i)={(p_1i,m_1i) or (m_1i,p_1i)}; H^j_i2: (e_1i,e_2i)={(p_1i,m_2i) or (m_2i,p_1i)}

H^j_i3: (e_1i,e_2i)={(p_2i,m_1i) or (m_1i,p_2i)}; H^j_i4: (e_1i,e_2i)={(p_2i,m_2i) or (m_2i,p_2i)}

The goal is to choose the most likely hypothesis H* as:
H*=arg max_H∈SHP(H|M)=arg max_H∈SHF(M,H)
where function F(M,H)=P(H|M)

[0103] There are 4^k different hypotheses in the space S^H. By trying to find the best hypothesis by exhaustively exploring the entire space S^H, the necessary algorithm would be of exponential order in k O(exp(k)), where k is the number of SNPs involved. For large k, even k>5, this is immensely slow and unpractical. Therefore, it is more practical to resort to a recursive solution which solves the problem of size k as a function of the problem of size (k−1) in constant time. The solution shown here is of the linear order in k, O(k).

Recursive Solution Linear in the Number of SNPs

[0104] Begin with F(M,H)=P(H|M)=P(M|H)*P(H)/P(M). Then argmax_H F(M,H)=argmax H P(M|H)*P(H) and the goal is to solve P(M|H)*P(H) in linear time. Suppose that M_(s,k)=measurement on SNPs s to k, H(s,k)=hypothesis on SNPs s to k, and to simplify notation M_(k,k)=M_k, H_(k,k)=H_k=measurement and hypothesis on SNP k. As shown before:

[0105] $P\left(M_{\left(1,k\right)}❘H_{\left(1,k\right)}\right)=\prod _{i=1}^{k}P\left(M_i❘H_i\right)=P\left(M_k❘H_k\right)*\prod _{i=1}^{k-1}P\left(M_i❘H_i\right)=P\left(M_k❘H_k\right)*P\left(M_{\left(1,k-1\right)}❘H_{\left(1,k-1\right)}\right)$ $\text{Also:}$ $P\left(H_{\left(1,k\right)}\right)=1/4*\prod _{i=2}^k\mathrm{PF}\left(H_{i-1},H_i\right)=\mathrm{PF}\left(H_{k-1},H_k\right)*1/4*\prod _{i=2}^{k-1}\mathrm{PF}\left(H_{i-1},H_i\right)=\mathrm{PF}\left(H_{k-1},H_k\right)*P\left(H_{\left(1,k-1\right)}\right)$ $\mathrm{where}$ $\mathrm{PF}\left(H_{i-1},H_i\right)=\{\begin{array}{cc}1-\mathrm{PC}\left(H_{i-1},H_i\right)& H_{i-1}=H_i\\ \mathrm{PC}\left(H_{i-1},H_i\right)& H_{i-1}\ne H_i\end{array}$
and PC(H_i−1,H_i)=probability of crossover between H_i−1, H_i

[0106] Finally, for k SNPs:
F(M,H)=P(M|H)*P(H)=P(M_(1,k)|H_(1,k))*P(H_(1,k))=P(M_(1,k-1)|H_(1,k-1)*P(H_(1,k-1))*P(M_k|H_k)*PF(H_k-1,H_k)
so in short F(M,H)=F(M_(1,k),H_(1,k)))=F(M_(1,k-1),H_(1,k-1))*P(M_k|H_k)*PF(H_k-1,H_k) i.e. we can reduce the calculation of F on k SNPs to the calculation of F on k−1 SNPs.

[0107] For H=(H₁, . . . H_k),the hypothesis on k SNPs:

[0108] $\underset{H}{\max }F\left(M,H\right)=\underset{\left(H_{\left(1,k-1\right)},H_k\right)}{\max }F(M,\left(H_{\left(1,k-1\right)},H_k\right)=\underset{H_k}{\max }\underset{H_{\left(1,k-1\right)}}{\max }F(M,\left(H_{\left(1,k-1\right)},H_k\right)=\underset{H_k}{\max }G\left(M_{\left(1,k\right)},H_k\right)\mathrm{where}G\left(M_{\left(1,n\right)},H_n\right)=\underset{H_{\left(1,n-1\right)}}{\max }F(M_{\left(1,n\right)},\left(H_{\left(1,n-1\right)},H_n\right)=\underset{H_{\left(1,n-1\right)}}{\max }F\left(M_{\left(1,n-1\right)},H_{\left(1,n-1\right)}\right)*P\left(M_n❘H_n\right)*\mathrm{PF}\left(H_{n-1},H_n\right)=P\left(M_n❘H_n\right)*\underset{H_{\left(1,n-1\right)}}{\max }F\left(M_{\left(1,n-1\right)},H_{\left(1,n-1\right)}\right)*\mathrm{PF}\left(H_{n-1},H_n\right)=P\left(M_n❘H_n\right)*\underset{H_{n-1}}{\max }\underset{H_{\left(1,n-2\right)}}{\max }F(M_{\left(1,n-1\right)},\left(H_{\left(1,n-2\right)},H_{n-1}\right)*\mathrm{PF}\left(H_{n-1},H_n\right)=P\left(M_n❘H_n\right)*\underset{H_{n-1}}{\max }\mathrm{PF}\left(H_{n-1},H_n\right)*G\left(M_{\left(1,n-1\right)},H_{n-1}\right)$

[0109] To summarize this:

[0110] $\underset{H}{\max }F\left(M,H\right)=\underset{H_n}{\max }G\left(M_{\left(1,k\right)},H_k\right)$
where G can be found recursively: for n=2, . . . , k

[0111] $G\left(M_{\left(1,n\right)},H_n\right)=P\left(M_n❘H_n\right)*\underset{H_{n-1}}{\max }\lfloor \mathrm{PF}\left(H_{n-1},H_n\right)*G\left(M_{\left(1,n-1\right)},H_{n-1}\right)\rfloor$
and G(M_(1,1),H₁)=0.25*P(M₁|H₁)

[0112] The algorithm is as follows:

For n=1: Generate 4 hypotheses H₁i, calculate G(M₁|H₁i) for i=1, . . . , 4.

For n=2: Generate 4 hypothesis for H₂i, calculate G(M_(1,2)|H₂i), i=1, . . . , 4 in constant time using the formula:

[0113] $G\left(M_{\left(1,2\right)},H_{2}i\right)=P\left(M_2❘H_{2}i\right)*\underset{j=1,\dots ,4}{\max }\left[\mathrm{PF}\left(H_1,H_{2}i\right)*G\left(M_1,H_{1}j\right)\right]$
For n=k: Generate 4 hypothesis for H_ki, make G(M_(1,k)|H_ki), i=1, . . . , 4 by

[0114] $G\left(M_{\left(1,k\right)},H_{k}i\right)=P\left(M_k❘H_{k}i\right)*\underset{j=1,\dots ,4}{\max }\lfloor \mathrm{PF}\left(H_{k-1},H_{k}i\right)*G\left(M_{\left(1,k-1\right)},H_{k-1}j\right)\rfloor$
At any time there are only 4 hypotheses to remember and a constant number of operations. So the algorithm is linear in k, number of SNPs, as opposed to exponential.
Solving P(M) in Linear Time

[0115] It is not necessary to solve for P(M) to get the best hypothesis, since it is constant for all H. But in order to get the actual, meaningful number for the conditional probability P(H|M)=P(M|H)*P(H)/P(M), it is also necessary to derive P(M). As above, we can write:

[0116] $P\left(M\right)=P\left(M_{\left(1,k\right)}\right)=\sum _{H_{\left(1,k\right)}}^{}P\left(M_{\left(1,k\right)}❘H_{\left(1,k\right)}\right)*P\left(H_{\left(1,k\right)}\right)=\sum _{H_k}P\left(M_K❘H_k\right)\sum _{H_{\left(1,k-1\right)}}^{}P\left(M_{\left(1,k-1\right)}❘H_{\left(1,k-1\right)}\right)*P\left(H_{\left(1,k-1\right)}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)=\sum _{H_k}P\left(M_K❘H_k\right)*W\left(M_{\left(1,k-1\right)}❘H_k\right)$ $\mathrm{where}$ $W\left(M_{\left(1,k-1\right)}❘H_k\right)=\sum _{H_{\left(1,k-1\right)}}^{}P\left(M_{\left(1,k-1\right)}❘H_{\left(1,k-1\right)}\right)*P\left(H_{\left(1,k-1\right)}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)$
We can solve for W(M,H) by recursion:

[0117] $W\left(M_{\left(1,k-1\right)}❘H_k\right)=\sum _{H_{\left(1,k-1\right)}}^{}P\left(M_{\left(1,k-1\right)}❘H_{\left(1,k-1\right)}\right)*P\left(H_{\left(1,k-1\right)}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)=\sum _{H_{k-1}}^{}P\left(M_{k-1}❘H_{k-1}\right)\sum _{H_{\left(1,k-2\right)}}P\left(M_{\left(1,k-2\right)}❘H_{\left(1,k-2\right)}\right)*P\left(H_{\left(1,k-2\right)}\right)*\mathrm{PF}\left(H_{k-2},H_{k-1}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)=\sum _{H_{k-1}}^{}P\left(M_{k-1}❘H_{k-1}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)*W\left(M_{\left(1,k-2\right)}❘H_{k-1}\right)$
so in short, problem of size k is reduced to the problem of size (k−1) by

[0118] $W\left(M_{\left(1,k-1\right)}❘H_k\right)=\sum _{H_{k-1}}^{}P\left(M_{k-1}❘H_{k-1}\right)*\mathrm{PF}\left(H_{k-1},H_k\right)*W\left(M_{\left(1,k-2\right)}❘H_{k-1}\right)$ $\mathrm{and}$ $W\left(M_{\left(1,1\right)}❘H_2\right)=\sum _{H_1}^{}P\left(M_1❘H_1\right)*0.25*\mathrm{PF}\left(H_1,H_2\right)$
As before, for n=2:k, generate W(2), . . . , W(K)=W(M_(1,k-1)|H_k) recursively, until finally, it is possible to derive

[0119] $P\left(M\right)=\sum _{H_k}P\left(M_K❘H_k\right)*W\left(M_{\left(1,k-1\right)}❘H_k\right).$

[0120] At each level there are only four different hypotheses H_k, so the algorithm is again linear in the number of SNPs k.

Individual SNP Confidence in Linear Time

[0121] Once the best hypothesis H*=(H₁*, . . . , H_k*), has been computed, it now may be desired to derive the confidence in the final answer for each SNP, namely P(H_i*|M), for i=1, . . . , k. As before P(H_i*|M)=P(M|H_i*)P(H_i*)/P(M)=W(H_i*,M)/P(M), where P(M) is already known.

[0122] $W\left(M,H_i^{*}\right)=\sum _{H,H_i=H_i^{*}}^{}P\left(M❘H\right)*P\left(H\right)=\sum _{H=\left(H_{\left(1,i-1\right)},H_i^{*},H_{\left(i+1,k\right)}\right)}^{}P\left(M❘H\right)*P\left(H\right),$
i.e. hypothesis H has been broken up to the hypothesis on first i−1 SNPs, ith SNP, and hypothesis on the i+1 to kth SNP. As before:

[0123] $P\left(M_{\left(1,k\right)}❘H_{\left(1,k\right)}\right)=\prod _{j=1}^{k}P\left(M_j❘H_j\right)=\prod _{j=1}^{i-1}P\left(M_j❘H_j\right)*P\left(M_i❘H_i\right)*\prod _{j=i+1}^{k}P\left(M_j❘H_j\right)=P\left(M_{\left(1,i-1\right)}❘H_{\left(1,i-1\right)}\right)*P\left(M_i❘H_i^{*}\right)*P\left(M_{\left(i+1,k\right)}❘H_{\left(i+1,k\right)}\right)\mathrm{and}$ $P\left(H_{\left(1,k\right)}\right)=1/4*\prod _{j=2}^k\mathrm{PF}\left(H_{j-1},H_j\right)=1/4*\prod _{j=2}^{i-1}\mathrm{PF}\left(H_{j-1},H_j\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*\prod _{j=j+2}^k\mathrm{PF}\left(H_{j-1},H_j\right)=1/4*T\left(H_{\left(1,i-1\right)}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*T\left(H_{\left(i+1,k\right)}\right)$ $\mathrm{So}$ $P\left(H_{\left(1,k\right)}\right)=1/4*T\left(H_{\left(1,k\right)}\right)=1/4*T\left(H_{\left(1,i-1\right)}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*T\left(H_{\left(i+1,k\right)}\right)$ $\mathrm{where}T\left(H_{\left(1,k\right)}\right)=\prod _{j=2}^k\mathrm{PF}\left(H_{j-1},H_j\right).$
From this it is possible to show that

[0124] $W\left(M_{\left(1,k\right)},H_i^{*}\right)=\sum _{H,H_i-H_i^{*}}^{}P\left(M❘H\right)*P\left(H\right)=\sum _{H,H_i-H_i^{*}}^{}P\left(M❘H\right)*1/4*T\left(H\right)=\sum _{H=\left(H_{\left(1,i-1\right)},H_i^{*},H_{\left(i+1,k\right)}\right)}^{}P\left(M_{\left(1,i-1\right)}❘H_{\left(1,i-1\right)}\right)*P\left(M_i❘H_i^{*}\right)*P\left(M_{\left(i+1,k\right)}❘H_{\left(i+1,k\right)}\right)*1/4*T\left(H_{\left(1,i-1\right)}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)*T\left(H_{\left(i+1,k\right)}\right)=4*P\left(M_i❘H_i^{*}\right)*\left(\sum _{H_{i-1}}^{}P\left(M_{\left(1,i-1\right)}❘H_{\left(1,i-1\right)}\right)*1/4*T\left(H_{\left(1,i-1\right)}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)\right)*\left(\sum _{H_{i+1}}^{}P\left(M_{\left(i+1,k\right)}❘H_{\left(i+1,k\right)}\right)*1/4*T\left(H_{\left(i+1,k\right)}\right)*\mathrm{PF}\left(H_i^{*},H_{i+1}\right)\right)=4*P\left(M_i❘H_i^{*}\right)*\left(\sum _{H_{i-1}}^{}W\left(M_{\left(1,i-1\right)}❘H_{i-1}\right)*\mathrm{PF}\left(H_{i-1},H_i^{*}\right)\right)*\left(\sum _{H_{i+1}}^{}W\left(M_{\left(i+,k\right)}❘H_{i+1}\right)*\mathrm{PF}\left(H_i^{*},H_{i+1}\right)\right)$

[0125] Again a case of size k has been reduced to two pieces of smaller size, albeit a bit more complicated than before. Each of the pieces can be calculated as

[0126] $W\left(M_{\left(1,n\right)},H_n\right)=P\left(M_n❘H_n\right)*\left(\sum _{H_{n-1}}^{}W\left(M_{\left(1,n-1\right)},H_{n-1}\right)*\mathrm{PF}\left(H_{n-1},H_n\right)\right)$ $W\left(M_{\left(m,k\right)},H_m\right)=P\left(M_m❘H_m\right)*\left(\sum _{H_{m+1}}^{}W\left(M_{\left(m+1,k\right)},H_{m+1}\right)*\mathrm{PF}\left(H_m,H_{m+1}\right)\right)$
So the algorithm will, for n=1, . . . , k, m=k, . . . 1, for each of 4 different H_n, H_m calculate W(M_(1,n),H_n),W(M_(m,k),H_m) and then combine them as needed to calculate W(M_(1,k),H_i*), for i=1, . . . , k. The number of operations is still linear in k.
Application of the Disclosed Method to Embryonic Data when a Smaller or Different Set of Data is Available

[0127] In one embodiment of the system it is only necessary to make use of diploid data from one parent (presumably the mother), with or without haploid data from either or both of the parents, and when that data is known to a greater or lesser degree of certainty. For example it is expected that, given the grueling nature of egg donation, there will be occasions when maternal haploid data is not readily available. It will be clear to one skilled in the art, after reading this description, how the statistical methods for computing the likelihood of a particular SNP can be modified given a limited data set.

[0128] An alternative approach uses data from more distant relatives to make up for missing diploid or haploid data of one or both parents. For example, since it is known that one set of an individual's chromosomes come from each of his or her parents, diploid data from the maternal grandparents could be used to partially reconstruct missing or poorly measured maternal haploid data.

[0129] Note the recursive nature of this method: given the naturally noisy measurement of single cell parental haploid data, along with the diploid and/or haploid data of the appropriate grandparents, the disclosed method could be used to clean the parental haploid data, which in turn will provide more accurate genotyping of the embryo. It should be obvious to one skilled in the arts how to modify the method for use in these cases.

[0130] It is preferable to use more information rather than less, as this can increase the chances of making the right call at a given SNP, and can increase the confidence in those calls. This must be balanced with the increasing complexity of the system as additional techniques and sources of data are used. There are many sources of additional information, as well as techniques available to use the information to augment the data. For example, there are informatics based approaches which take advantage of correlations which can be found in Hapmap data, or other repositories of genomic data. In addition there are biological approaches which can allow for the direct measurement of genetic data that otherwise would need to be recreated in silico. For example, haploid data otherwise unavailable may be measureable by extracting individual chromosomes from diploid cells using flow cytometry techniques to isolate fluorescently tagged chromosomes. Alternately, one may use cell fusion to create monoallelic hybrid cells to effect diploid to haploid conversion.

Application of the Disclosed Method to Selecting which Embryo is Likely to Implant

[0131] In one embodiment, the system can be used to determine the likelihood of an embryo to implant in the mother and develop into a baby. To the extent that the likelihood of the embryo implanting is determined by SNPs of the embryo, and/or their relation to SNPs of the mother, the disclosed method will be important in helping the selection of embryos, based on making a reliable prediction of which will successfully implant based on the clean SNP data. To best predict the likelihood it will be necessary to take into account the determined genotype of the embryo possibly combined with the levels of gene expression in the embryo, the levels of gene expression in the mother, and/or the determined genotype of the mother.

[0132] In addition, it is well known that aneuploid embryos are less likely to implant, less likely to result in a successful pregnancy, and less likely to result in a healthy child. Consequently, screening for aneuploides is an important facet to selecting the embryo that is most likely to result in a successful outcome. More detail on this approach is given below.

Deducing Parental Haploid Data

[0133] In one embodiment of the method, it may be necessary to deduce parental haplotypes, given detailed knowledge of the diploid data of a parent. There are multiple ways this can be done. In the simplest case, haplotypes have already been inferred by molecular assay of single haploid cells of a direct relation (mother, father, son or daughter). In this case, it is a trivial matter to one skilled in the art to deduce the sister haplotype by subtracting the known haplotype from the diploid genotype measured by molecular assay. For example, if a particular locus is heterozygous, an unknown parental haplotype is the opposite allele from the known parental haplotype.

[0134] In another case, the noisy haploid data of the parent may be known from molecular biological haplotyping of individual parental haploid cells, such as a sperm cell, or from individual chromosomes, which may be isolated by various methods including magnetic beads and flow cytometry. In this case, the same procedure can be used as above, except that the determined haplotype will be as noisy as the measured haplotype.

[0135] There are also methods for deducing haploid data sets directly from diploid data, using statistical methods that utilize known haplotype blocks in the general population (such as those created for the public Hapmap project). A haplotype block is essentially a series of correlated alleles that occur repeatedly in a variety of populations. Since these haplotype blocks are often ancient and common, they may be used to predict haplotypes from diploid genotypes. The parents' inferred haplotype blocks can then be used as input for the method described herein to clean the noisy data from the embryos. Publicly available algorithms that would accomplish this task include an imperfect phylogeny approach, Bayesian approaches based on conjugate priors, and priors from population genetics. Some of these algorithms use hidden Markov models. One study used public trio and unrelated individual data to demonstrate that these algorithms perform with error rates as low as 0.05% across 1 MB of sequence. However, as expected, accuracy is lower for individuals with rare haplotype blocks. In one estimate, computational methods failed to phase as many as 5.1% of loci with minor allele frequency of 20%.

[0136] In one embodiment of the invention, genetic data from multiple blastomeres taken from different embryos during an IVF cycle is used to infer the haplotype blocks of the parents with greater reliability.

Techniques for Screening for Aneuploidy Using High and Medium Throughput Genotyping

[0137] In one embodiment of the system the measured genetic data can be used to detect for the presence of aneuploides and/or mosaicism in an individual. Disclosed herein are several methods of using medium or high-throughput genotyping to detect the number of chromosomes or DNA segment copy number from amplified or unamplified DNA from tissue samples. The goal is to estimate the reliability that can be achieved in detecting certain types of aneuploidy and levels of mosaicism using different quantitative and/or qualitative genotyping platforms such as ABI TAQMAN, MIPS, or Microarrays from ILLUMINA, AGILENT and AFFMETRIX. In many of these cases, the genetic material is amplified by PCR before hybridization to probes on the genotyping array to detect the presence of particular alleles. How these assays are used for genotyping is described elsewhere in this disclosure.

[0138] Described below are several methods for screening for abnormal numbers of DNA segments, whether arising from deletions, aneuploides and/or mosaicism. The methods are grouped as follows: (i) quantitative techniques without making allele calls; (ii) qualitative techniques that leverage allele calls; (iii) quantitative techniques that leverage allele calls; (iv) techniques that use a probability distribution function for the amplification of genetic data at each locus. All methods involve the measurement of multiple loci on a given segment of a given chromosome to determine the number of instances of the given segment in the genome of the target individual. In addition, the methods involve creating a set of one or more hypotheses about the number of instances of the given segment; measuring the amount of genetic data at multiple loci on the given segment; determining the relative probability of each of the hypotheses given the measurements of the target individual's genetic data; and using the relative probabilities associated with each hypothesis to determine the number of instances of the given segment. Furthermore, the methods all involve creating a combined measurement M that is a computed function of the measurements of the amounts of genetic data at multiple loci. In all the methods, thresholds are determined for the selection of each hypothesis H_i based on the measurement M, and the number of loci to be measured is estimated, in order to have a particular level of false detections of each of the hypotheses.

[0139] The probability of each hypothesis given the measurement M is P(H_i|M)=P(M|H_i)P(H_i)/P(M). Since P(M) is independent of H_i, we can determine the relative probability of the hypothesis given M by considering only P(M|H_i)P(H_i). In what follows, in order to simplify the analysis and the comparison of different techniques, we assume that P(H_i) is the same for all {H_i}, so that we can compute the relative probability of all the P(H_i|M) by considering only P(M|H_i). Consequently, our determination of thresholds and the number of loci to be measured is based on having particular probabilities of selecting false hypotheses under the assumption that P(H_i) is the same for all {H_i}. It will be clear to one skilled in the art after reading this disclosure how the approach would be modified to accommodate the fact that P(H_i) varies for different hypotheses in the set {H_i}. In some embodiments, the thresholds are set so that hypothesis H_i, is selected which maximizes P(H_i|M) over all i. However, thresholds need not necessarily be set to maximize P(H_i|M), but rather to achieve a particular ratio of the probability of false detections between the different hypotheses in the set {H_i}.

[0140] It is important to note that the techniques referred to herein for detecting aneuploides can be equally well used to detect for uniparental disomy, unbalanced translocations, and for the sexing of the chromosome (male or female; XY or XX). All of the concepts concern detecting the identity and number of chromosomes (or segments of chromosomes) present in a given sample, and thus are all addressed by the methods described in this document. It should be obvious to one skilled in the art how to extend any of the methods described herein to detect for any of these abnormalities.

The Concept of Matched Filtering

[0141] The methods applied here are similar to those applied in optimal detection of digital signals. It can be shown using the Schwartz inequality that the optimal approach to maximizing Signal to Noise Ratio (SNR) in the presence of normally distributed noise is to build an idealized matching signal, or matched filter, corresponding to each of the possible noise-free signals, and to correlate this matched signal with the received noisy signal. This approach requires that the set of possible signals are known as well as the statistical distribution—mean and Standard Deviation (SD)—of the noise. Herein is described the general approach to detecting whether chromosomes, or segments of DNA, are present or absent in a sample. No differentiation will be made between looking for whole chromosomes or looking for chromosome segments that have been inserted or deleted. Both will be referred to as DNA segments. It should be clear after reading this description how the techniques may be extended to many scenarios of aneuploidy and sex determination, or detecting insertions and deletions in the chromosomes of embryos, fetuses or born children. This approach can be applied to a wide range of quantitative and qualitative genotyping platforms including TAQMAN, qPCR, ILLUMINA Arrays, AFFMETRIX Arrays, AGILENT Arrays, the MIPS kit etc.

Formulation of the General Problem

[0142] Assume that there are probes at SNPs where two allelic variations occur, x and y. At each locus i, i=1 . . . N, data is collected corresponding to the amount of genetic material from the two alleles. In the TAQMAN assay, these measures would be, for example, the cycle time, Ct, at which the level of each allele-specific dye crosses a threshold. It will be clear how this approach can be extended to different measurements of the amount of genetic material at each locus or corresponding to each allele at a locus. Quantitative measurements of the amount of genetic material may be nonlinear, in which case the change in the measurement of a particular locus caused by the presence of the segment of interest will depend on how many other copies of that locus exist in the sample from other DNA segments. In some cases, a technique may require linear measurements, such that the change in the measurement of a particular locus caused by the presence of the segment of interest will not depend on how many other copies of that locus exist in the sample from other DNA segments. An approach is described for how the measurements from the TAQMAN or qPCR assays may be linearized, but there are many other techniques for linearizing nonlinear measurements that may be applied for different assays.

[0143] The measurements of the amount of genetic material of allele x at loci 1 . . . N is given by data d_x=[d_x1 . . . d_xN]. Similarly for allele y, d_y=[d_y1 . . . d_yN]. Assume that each segment j has alleles a_j=[a_j1 . . . a_jN] where each element a_ji is either x or y. Describe the measurement data of the amount of genetic material of allele x as d_x=s_x+υ_x where s_x is the signal and υ_x is a disturbance. The signal s_x=[f_x(a₁₁, . . . , a_J1) . . . f_x(a_JN, . . . , a_JN)] where f_x is the mapping from the set of alleles to the measurement, and J is the number of DNA segment copies. The disturbance vector υ_x is caused by measurement error and, in the case of nonlinear measurements, the presence of other genetic material besides the DNA segment of interest. Assume that measurement errors are normally distributed and that they are large relative to disturbances caused by nonlinearity (see section on linearizing measurements) so that υ_xi≈n_xi where n_xi has variance σ_xi² and vector n_x is normally distributed ˜N(0,R), R=E(n_xn_x^T). Now, assume some filter h is applied to this data to perform the measurement m_x=h^Td_x=h^Ts_x+h^Tυ_x. In order to maximize the ratio of signal to noise (h^Ts_x/h^Tn_x) it can be shown that h is given by the matched filter h=μR⁻¹s_x where μ is a scaling constant. The discussion for allele x can be repeated for allele y.

Method 1a: Measuring Aneuploidy or Sex by Quantitative Techniques that do not Make Allele Calls when the Mean and Standard Deviation for Each Locus is Known

[0144] Assume for this section that the data relates to the amount of genetic material at a locus irrespective of allele value (e.g. using qPCR), or the data is only for alleles that have 100% penetrance in the population, or that data is combined on multiple alleles at each locus (see section on linearizing measurements) to measure the amount of genetic material at that locus.

[0145] Consequently, in this section one may refer to data d_x and ignore d_y. Assume also that there are two hypotheses: h₀ that there are two copies of the DNA segment (these are typically not identical copies), and h₁ that there is only 1 copy. For each hypothesis, the data may be described as d_xi(h₀)=s_xi(h₀)+n_xi and d_xi(h₁)=s_xi(h₁)+n_xi respectively, where s_xi(h₀) is the expected measurement of the genetic material at locus i (the expected signal) when two DNA segments are present and s_xi(h₁) is the expected data for one segment. Construct the measurement for each locus by differencing out the expected signal for hypothesis h₀: m_xi=d_xi−s_xi(h₀). If h₁ is true, then the expected value of the measurement is E(m_xi)=s_xi(h₁)−s_xi(h₀). Using the matched filter concept discussed above, set h=(1/N)R⁻¹(s_xi(h₁)−s_xi(h₀)). The measurement is described as m=h^Td_x=(1/N)Σ_{i=1 . . . N}((s_xi(h₁)−s_xi(h₀))/σ_xi²)m_xi.

[0146] If h₁ is true, the expected value of E(m|h₁)=m₁=(1/N)Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi² and the standard deviation of m is σ_m|h1²=(1/N²)Σ_{i=1 . . . N}((s_xi(h₁)−s_xi(h₀))²/σ_xi⁴)σ_xi²=(1/N²)Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi².

[0147] If h₀ is true, the expected value of m is E(m|h₀)=m₀=0 and the standard deviation of m is again σm|h₀²=(1/N²)Σ_{i=1 . . . N}(s_xi(h)−s_xi(h₀))²/σ_xi².

[0148] FIG. 3 illustrates how to determine the probability of false negatives and false positive detections. Assume that a threshold t is set half-way between m₁ and m₀ in order to make the probability of false negatives and false positives equal (this need not be the case as is described below). The probability of a false negative is determined by the ratio of (m₁−t)/σ_m|h1=(m₁−m₀)/(2σ_m|h1). “5-Sigma” statistics may be used so that the probability of false negatives is 1−normcdf(5,0,1)=2.87e-7. In this case, the goal is for (m₁−m₀)/(2σ_m|h0)>5 or 10sqrt((1/N²)Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi²)<(1/N)Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi² or sqrt(Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi²)>10. In order to compute the size of N, Mean Signal to Noise Ratio can be computed from aggregated data: MSNR=(1/N)Σ_{i=1 . . . N}(s_xi(h₁)−s_xi(h₀))²/σ_xi². N can then be found from the inequality above: sqrt(N).sqrt(MSNR)>10 or N>100/MSNR.

[0149] This approach was applied to data measured with the TAQMAN Assay from Applied BioSystems using 48 SNPs on the X chromosome. The measurement for each locus is the time, Ct, that it takes the die released in the well corresponding to this locus to exceed a threshold. Sample 0 consists of roughly 0.3 ng (50 cells) of total DNA per well of mixed female origin where subjects had two X chromosomes; sample 1 consisted of roughly 0.3 ng of DNA per well of mixed male origin where subject had one X chromosome. FIG. 4 and FIG. 5 show the histograms of measurements for samples 1 and 0. The distributions for these samples are characterized by m₀=29.97; SD₀=1.32, m₁=31.44, SD₁=1.592. Since this data is derived from mixed male and female samples, some of the observed SD is due to the different allele frequencies at each SNP in the mixed samples. In addition, some of the observed SD will be due to the varying efficiency of the different assays at each SNP, and the differing amount of dye pipetted into each well. FIG. 6 provides a histogram of the difference in the measurements at each locus for the male and female sample. The mean difference between the male and female samples is 1.47 and the SD of the difference is 0.99. While this SD will still be subject to the different allele frequencies in the mixed male and female samples, it will no longer be affected the different efficiencies of each assay at each locus. Since the goal is to differentiate two measurements each with a roughly similar SD, the adjusted SD may be approximated for each measurement for all loci as 0.99/sqrt(2)=0.70. Two runs were conducted for every locus in order to estimate σ_xi for the assay at that locus so that a matched filter could be applied. A lower limit of σ_xi was set at 0.2 in order to avoid statistical anomalies resulting from only two runs to compute σ_xi. Only those loci (numbering 37) for which there were no allele dropouts over both alleles, over both experiment runs and over both male and female samples were used in the plots and calculations. Applying the approach above to this data, it was found that MSNR=2.26, hence N=2²5²/2.26{circumflex over ( )}=17 loci.

Method 1b: Measuring Aneuploidy or Sex by Quantitative Techniques that do not Make Allele Calls when the Mean and Std. Deviation is not Known or is Uniform

[0150] When the characteristics of each locus are not known well, the simplifying assumptions that all the assays at each locus will behave similarly can be made, namely that E(m_xi) and σ_xi are constant across all loci i, so that it is possible to refer instead only to E(m_x) and σ_x. In this case, the matched filtering approach m=h^Td_x reduces to finding the mean of the distribution of d_x. This approach will be referred to as comparison of means, and it will be used to estimate the number of loci required for different kinds of detection using real data.

[0151] As above, consider the scenario when there are two chromosomes present in the sample (hypothesis h₀) or one chromosome present (h₁). For h₀, the distribution is N(μ₀,σ₀²) and for h₁ the distribution is N(μ₁,σ₁²). Measure each of the distributions using N₀ and N₁ samples respectively, with measured sample means and SDs m₁, m₀, s₁, and s₀. The means can be modeled as random variables M₀, M₁ that are normally distributed as M₀˜N(μ₀, σ₀²/N₀) and M₁˜N(μ₁, σ₁²/N₁). Assume N₁ and N₀ are large enough (>30) so that one can assume that M₁˜N(m₁, s₁²/N₁) and M₀˜N(m₀, s₀²/N₀). In order to test whether the distributions are different, the difference of the means test may be used, where d=m₁−m₀. The variance of the random variable D is σ_d²=σ₁²/N₁+σ₀²/N₀ which may be approximated as σ_d²=s₁²/N₁+s₀²/N₀. Given h₀, E(d)=0; given h₁, E(d)=μ₁−μ₀. Different techniques for making the call between h₁ for h₀ will now be discussed.

[0152] Data measured with a different run of the TAQMAN Assay using 48 SNPs on the X chromosome was used to calibrate performance. Sample 1 consists of roughly 0.3 ng of DNA per well of mixed male origin containing one X chromosome; sample 0 consisted of roughly 0.3 ng of DNA per well of mixed female origin containing two X chromosomes. N₁=42 and N₀=45. FIG. 7 and FIG. 8 show the histograms for samples 1 and 0. The distributions for these samples are characterized by m₁=32.259, s₁=1.460, σ_m1=s₁/sqrt(N₁)=0.225; m₀=30.75; s₀=1.202, σ_m0=s₀/sqrt(N₀)=0.179. For these samples d=1.509 and σ_d=0.2879.

[0153] Since this data is derived from mixed male and female samples, much of the standard deviation is due to the different allele frequencies at each SNP in the mixed samples. SD is estimated by considering the variations in C_t for one SNP at a time, over multiple runs. This data is shown in FIG. 9. The histogram is symmetric around 0 since C_t for each SNP is measured in two runs or experiments and the mean value of C_t for each SNP is subtracted out. The average std. dev. across 20 SNPs in the mixed male sample using two runs is s=0.597. This SD will be conservatively used for both male and female samples, since SD for the female sample will be smaller than for the male sample. In addition, note that the measurement from only one dye is being used, since the mixed samples are assumed to be heterozygous for all SNPs. The use of both dyes requires the measurements of each allele at a locus to be combined, which is more complicated (see section on linearizing measurements). Combining measurements on both dyes would double signal amplitude and increase noise amplitude by roughly sqrt(2), resulting in an SNR improvement of roughly sqrt(2) or 3 dB.

Detection Assuming No Mosaicism and No Reference Sample

[0154] Assume that m₀ is known perfectly from many experiments, and every experiment runs only one sample to compute m₁ to compare with m₀. N₁ is the number of assays and assume that each assay is a different SNP locus. A threshold t can be set half way between m₀ and m₁ to make the likelihood of false positives equal the number of false negatives, and a sample is labeled abnormal if it is above the threshold. Assume s₁=s₂=s=0.597 and use the 5-sigma approach so that the probability of false negatives or positives is 1−normcdf(5,0,1)=2.87e-7. The goal is for 5s₁/sqrt(N₁)<(m₁−m₀)/2, hence N₁=100 s₁²/(m₁−m₀)²=16. Now, an approach where the probability of a false positive is allowed to be higher than the probability of a false negatives, which is the harmful scenario, may also be used. If a positive is measured, the experiment may be rerun. Consequently, it is possible to say that the probability of a false negative should be equal to the square of the probability of a false positive. Consider FIG. 3, let t=threshold, and assume Sigma_0=Sigma_l=s. Thus (1−normcdf((t−m₀)/s,0,1))²=1−normcdf((m₁−t)/s,0,1). Solving this, it can be shown that t=m₀+0.32(m₁−m₀). Hence the goal is for 5s/sqrt(N₁)<m₁−m₀−0.32(m₁−m₀)=(m₁−m₀)/1.47, hence N₁=(5²)(1.47²)s²/(m₁−m₀)²=9.

Detection with Mosaicism without Running a Reference Sample

[0155] Assume the same situation as above, except that the goal is to detect mosaicism with a probability of 97.7% (i.e. 2-sigma approach). This is better than the standard approach to amniocentesis which extracts roughly 20 cells and photographs them. If one assumes that 1 in 20 cells is aneuploid and this is detected with 100% reliability, the probability of having at least one of the group being aneuploid using the standard approach is 1−0.95²⁰=64%. If 0.05% of the cells are aneuploid (call this sample 3) then m₃=0.95m₀+0.05m₁ and var(m₃)=(0.95s₀²+0.05s₁²)/N₁. Thus std(m₃)2<(m₃−m₀)/2=>sqrt(0.95s₀²+0.05s²)/sqrt(N₁)<0.05(m₁−m₂)/4=>N₁=16(0.95s₂²+0.05s₁²)/(0.05²(m₁−m₂)²)=1001. Note that using the goal of 1−sigma statistics, which is still better than can be achieved using the conventional approach (i.e. detection with 84.1% probability), it can be shown in a similar manner that N₁=250.

Detection with No Mosaicism and Using a Reference Sample

[0156] Although this approach may not be necessary, assume that every experiment runs two samples in order to compare m₁ with truth sample m₂. Assume that N=N₁=N₀. Compute d=m₁−m₀ and, assuming σ₁=σ₀, set a threshold t=(m₀+m₁)/2 so that the probability of false positives and false negatives is equal. To make the probability of false negatives 2.87e-7, it must be the case that (m₁−m₂)/2>5sqrt(s²/N+s₂²/N)=>N=100(s₁²+s₂²)/(m₁−m₂)²=32.

Detection with Mosaicism and Running a Reference Sample

[0157] As above, assume the probability of false negatives is 2.3% (i.e. 2-sigma approach). If 0.05% of the cells are aneuploid (call this sample 3) then m₃=0.95m₀+0.05m₁ and var(m₃)=(0.95s₀²+0.05s₁²)/N₁. d=m₃−m₂ and σ_d²=(1.95s₀²+0.05s₁²)/N. It must be that std(m₃)2<(m₀−m₂)/2=>sqrt(1.95s₂²+0.05s₁²)/sqrt(N)<0.05(m₁−m₂)/4=>N=16(1.95s₂²+0.05s₁²)/(0.05²(m₁−m₂)²)=2002. Again using 1-sigma approach, it can be shown in a similar manner that N=500.

[0158] Consider the case if the goal is only to detect 5% mosaicism with a probability of 64% as is the current state of the art. Then, the probability of false negative would be 36%. In other words, it would be necessary to find x such that 1-normcdf(x,0,1)=36%. Thus N=4(0.36{circumflex over ( )}2)(1.95s₂²+0.05s₁²)/(0.05²(m₁−m₂)²)=65 for the 2-sigma approach, or N=33 for the 1-sigma approach. Note that this would result in a very high level of false positives, which needs to be addressed, since such a level of false positives is not currently a viable alternative.

[0159] Also note that if N is limited to 384 (i.e. one 384 well TAQMAN plate per chromosome), and the goal is to detect mosaicism with a probability of 97.72%, then it will be possible to detect mosaicism of 8.1% using the 1-sigma approach. In order to detect mosaicism with a probability of 84.1% (or with a 15.9% false negative rate), then it will be possible to detect mosaicism of 5.8% using the 1-sigma approach. To detect mosaicism of 19% with a confidence of 97.72% it would require roughly 70 loci. Thus one could screen for 5 chromosomes on a single plate.

[0160] The summary of each of these different scenarios is provided in FIG. 20. Also included in FIG. 20 are the results generated from qPCR and the SYBR assays. The methods described above were used and the simplifying assumption was made that the performance of the qPCR assay for each locus is the same. FIG. 10 and FIG. 11 show the histograms for samples 1 and 0, as described above. N₀=N₁=47. The distributions of the measurements for these samples are characterized by m₁=27.65, s₁=1.40, σ_m1=s₁/sqrt(N₁)=0.204; m₀=26.64; s₀=1.146, σ_m0=s₀/sqrt(N₀)=0.167. For these samples d=1.01 and σ_d=0.2636. FIG. 12 shows the difference between C_t for the male and female samples for each locus, with a standard deviation of the difference over all loci of 0.75. The SD was approximated for each measurement of each locus on the male or female sample as 0.75/sqrt(2)=0.53.

Method 2: Qualitative Techniques that Use Allele Calls

[0161] In this section, no assumption is made that the assay is quantitative. Instead, the assumption is that the allele calls are qualitative, and that there is no meaningful quantitative data coming from the assays. This approach is suitable for any assay that makes an allele call. FIG. 13 describes how different haploid gametes form during meiosis, and will be used to describe the different kinds of aneuploidy that are relevant for this section. The best algorithm depends on the type of aneuploidy that is being detected.

[0162] Consider a situation where aneuploidy is caused by a third segment that has no section that is a copy of either of the other two segments. From FIG. 13, the situation would arise, for example, if p₁ and p₄, or p₂ and p₃, both arose in the child cell in addition to one segment from the other parent. This is very common, given the mechanism which causes aneuploidy. One approach is to start off with a hypothesis h₀ that there are two segments in the cell and what these two segments are. Assume, for the purpose of illustration, that h₀ is for p₃ and m₄ from FIG. 13. In a preferred embodiment this hypothesis comes from algorithms described elsewhere in this document. Hypothesis h₁ is that there is an additional segment that has no sections that are a copy of the other segments. This would arise, for example, if p₂ or m₁ was also present. It is possible to identify all loci that are homozygous in p₃ and m₄. Aneuploidy can be detected by searching for heterozygous genotype calls at loci that are expected to be homozygous.

[0163] Assume every locus has two possible alleles, x and y. Let the probability of alleles x and y in general be p_x and p_y respectively, and p_x+p_y=1. If h₁ is true, then for each locus i for which p₃ and m₄ are homozygous, then the probability of a non-homozygous call is p_y or p_x, depending on whether the locus is homozygous in x or y respectively. Note: based on knowledge of the parent data, i.e. p₁, p₂, p₄ and m₁, m₂, m₃, it is possible to further refine the probabilities for having non-homozygous alleles x or y at each locus. This will enable more reliable measurements for each hypothesis with the same number of SNPs, but complicates notation, so this extension will not be explicitly dealt with. It should be clear to someone skilled in the art how to use this information to increase the reliability of the hypothesis.

[0164] The probability of allele dropouts is pd. The probability of finding a heterozygous genotype at locus i is p_0i given hypothesis h₀ and p_1i given hypothesis h₁.

[0165] Given h₀: p_0i=0

[0166] Given h₁: p₁=p_x(1−p_d) or p_1i=p_y(1−p_d) depending on whether the locus is homozygous for x or y.

[0167] Create a measurement m=1/N_h Σ_{i=1 . . . Nh} I_i where I_i is an indicator variable, and is 1 if a heterozygous call is made and 0 otherwise. N_h is the number of homozygous loci. One can simplify the explanation by assuming that p_x=p_y and p_0i, p_1i for all loci are the same two values p₀ and p₁. Given h₀, E(m)=p₀=0 and σ²_m|h0=p₀(1−p₀)/N_h. Given h₁, E(m)=p₁ and σ²_m|h1=p₁(1−p₁)/N_h. Using 5 sigma-statistics, and making the probability of false positives equal the probability of false negatives, it can be shown that (p₁−p₀)/2>5σ_m|h1 hence N_h=100(p₀(1−p₀)+p₁(1−p₁))/(p₁−p₀)². For 2-sigma confidence instead of 5-sigma confidence, it can be shown that N_h=4.2²(p₀(1−p₀)+p₁(1−p₁))/(p₁−p₀)².

[0168] It is necessary to sample enough loci N that there will be sufficient available homozygous loci N_h-avail such that the confidence is at least 97.7% (2-sigma). Characterize N_h-avail=Σ_{i=1 . . . N} J_i where J_i is an indicator variable of value 1 if the locus is homozygous and 0 otherwise. The probability of the locus being homozygous is p_x²+p_y². Consequently, E(N_h-avail)=N(p_x²+p_y²) and σ_Nh-avail²=N(p_x²+p_y²)(1−p_x²−p_y²). To guarantee N is large enough with 97.7% confidence, it must be that E(N_h-avail)−2σ_Nh-avail=N_h where N_h is found from above.

[0169] For example, if one assumes p_d=0.3, p_x=p_y=0.5, one can find N_h=186 and N=391 for 5-sigma confidence. Similarly, it is possible to show that N_h=30 and N=68 for 2-sigma confidence i.e. 97.7% confidence in false negatives and false positives.

[0170] Note that a similar approach can be applied to looking for deletions of a segment when h₀ is the hypothesis that two known chromosome segment are present, and h₁ is the hypothesis that one of the chromosome segments is missing. For example, it is possible to look for all of those loci that should be heterozygous but are homozygous, factoring in the effects of allele dropouts as has been done above.

[0171] Also note that even though the assay is qualitative, allele dropout rates may be used to provide a type of quantitative measure on the number of DNA segments present.

Method 3: Making Use of Known Alleles of Reference Sequences, and Quantitative Allele Measurements

[0172] Here, it is assumed that the alleles of the normal or expected set of segments are known. In order to check for three chromosomes, the first step is to clean the data, assuming two of each chromosome. In a preferred embodiment of the invention, the data cleaning in the first step is done using methods described elsewhere in this document. Then the signal associated with the expected two segments is subtracted from the measured data. One can then look for an additional segment in the remaining signal. A matched filtering approach is used, and the signal characterizing the additional segment is based on each of the segments that are believed to be present, as well as their complementary chromosomes. For example, considering FIG. 13, if the results of PS indicate that segments p2 and m1 are present, the technique described here may be used to check for the presence of p2, p3, m1 and m4 on the additional chromosome. If there is an additional segment present, it is guaranteed to have more than 50% of the alleles in common with at least one of these test signals. Note that another approach, not described in detail here, is to use an algorithm described elsewhere in the document to clean the data, assuming an abnormal number of chromosomes, namely 1, 3, 4 and 5 chromosomes, and then to apply the method discussed here. The details of this approach should be clear to someone skilled in the art after having read this document.

[0173] Hypothesis h₀ is that there are two chromosomes with allele vectors a₁, a₂. Hypothesis h₁ is that there is a third chromosome with allele vector a₃. Using a method described in this document to clean the genetic data, or another technique, it is possible to determine the alleles of the two segments expected by h₀: a₁=[a₁₁ . . . a_1N] and a₂=[a₂₁ . . . a_2N] where each element a_ji is either x or y. The expected signal is created for hypothesis h₀: s_0x=[f_0x(a₁₁, a₂₁) . . . f_x0(a_1N, a_2N)], S_0y=[f_y(a₁₁, a₂₁) . . . f_y(a_1N, a_2N)] where f_x, f_y describe the mapping from the set of alleles to the measurements of each allele. Given h₀, the data may be described as d_xi=s_0xi+n_xi, n_xi˜N(0,σ_xi²); d_yi=s_0yi+n_yi, n_yi˜N(0,σ_yi²). Create a measurement by differencing the data and the reference signal: m_xi=d_xi−s_xi; m_yi=d_yi−s_yi. The full measurement vector is m=[m_x^T m_y^T]^T.

[0174] Now, create the signal for the segment of interest—the segment whose presence is suspected, and will be sought in the residual—based on the assumed alleles of this segment: a₃=[a₃₁ . . . a_3N]. Describe the signal for the residual as: s_r=[s_rx^T s_ry^T]^T where s_rx=[f_rx(a₃₁) . . . f_rx(a_3N)], s_ry=[f_ry(a₃₁) . . . f_ry(a_3N)] where f_rx(a_3i)=δ_xi if a_3i=x and 0 otherwise, f_ry(a_3i)=δ_yi if a_3i=y and 0 otherwise. This analysis assumes that the measurements have been linearized (see section below) so that the presence of one copy of allele x at locus i generates data δ_xi+n_xi and the presence of κ_x copies of the allele x at locus i generates data κ_xδ_xi+n_xi. Note however that this assumption is not necessary for the general approach described here. Given h₁, if allele a_3i=x then m_xi=δ_xi+n_xi, m_yi=n_yi and if a_3i=y then m_xi=n_xi, m_yi=δ_yi+n_yi. Consequently, a matched filter h=(1/N)R⁻¹s_r can be created where R=diag([σ_xi² . . . σ_xN² σ_yi² . . . σ_yN²]). The measurement is m=h^Td.
m=(1/N)Σ_{i=1 . . . N}s_rxin_xi/σ_xi²+s_ryin_yi/σ_yi²  h₀:
m=(1/N)Σ_{i=1 . . . N}s_rxi(δ_xi+n_xi)/σ_xi²+s_ryi(δ_yi+n_yi)/σ_yi²  h₁:
In order to estimate the number of SNPs required, make the simplifying assumptions that all assays for all alleles and all loci have similar characteristics, namely that δ_xi=δ_yi=δ and σ_xi=σ_yi=σ for i=1 . . . N. Then, the mean and standard deviation may be found as follows:
E(m)=m₀=0;σ_m|h0²=(1/N²σ⁴)(N/2)(σ²δ²+σ²δ²)=δ²/(Nσ²)  h₀:
E(m)=m₁=(1/N)(N/2σ²)(δ²+δ²)=δ²/σ²;σ_m|h1²=(1/N²σ⁴)(N)(σ²δ²)=δ²/(Nσ²)  h₁:
Now compute a signal-to-noise ratio (SNR) for this test of h₁ versus h₀. The signal is m₁−m₀=δ²/σ², and the noise variance of this measurement is σ_m|h0²+σ_m|h1²=2δ²/(Nσ²). Consequently, the SNR for this test is (δ⁴/σ⁴)/(2δ²/(Nσ²))=Nδ²/(2σ²).

[0175] Compare this SNR to the scenario where the genetic information is simply summed at each locus without performing a matched filtering based on the allele calls. Assume that h=(1/N)i where i is the vector of N ones, and make the simplifying assumptions as above that δ_xi=δ_yi=δ and σ_xi=σ_yi=σ for i=1 . . . N. For this scenario, it is straightforward to show that if m=h^Td:
E(m)=m₀=0;σ_m|h0²=Nσ²/N²+Nσ²/N²=2σ²/N  h₀:
E(m)=m₁=(1/N)(Nδ/2+Nδ/2)=δ;σ_m|h1²=(1/N²)(Nσ²+Nσ²)=2σ²/N  h₁:
Consequently, the SNR for this test is Nδ²/(4σ²). In other words, by using a matched filter that only sums the allele measurements that are expected for segment a₃, the number of SNPs required is reduced by a factor of 2. This ignores the SNR gain achieved by using matched filtering to account for the different efficiencies of the assays at each locus.

[0176] Note that if we do not correctly characterize the reference signals s_xi and s_yi then the SD of the noise or disturbance on the resulting measurement signals m_xi and m_yi will be increased. This will be insignificant if δ<<σ, but otherwise it will increase the probability of false detections. Consequently, this technique is well suited to test the hypothesis where three segments are present and two segments are assumed to be exact copies of each other. In this case, s_xi and s_yi will be reliably known using techniques of data cleaning based on qualitative allele calls described elsewhere. In one embodiment method 3 is used in combination with method 2 which uses qualitative genotyping and, aside from the quantitative measurements from allele dropouts, is not able to detect the presence of a second exact copy of a segment.

[0177] We now describe another quantitative technique that makes use of allele calls. The method involves comparing the relative amount of signal at each of the four registers for a given allele. One can imagine that in the idealized case involving a single, normal cell, where homogenous amplification occurs, (or the relative amounts of amplification are normalized), four possible situations can occur: (i) in the case of a heterozygous allele, the relative intensities of the four registers will be approximately 1:1:0:0, and the absolute intensity of the signal will correspond to one base pair; (ii) in the case of a homozygous allele, the relative intensities will be approximately 1:0:0:0, and the absolute intensity of the signal will correspond to two base pairs; (iii) in the case of an allele where ADO occurs for one of the alleles, the relative intensities will be approximately 1:0:0:0, and the absolute intensity of the signal will correspond to one base pair; and (iv) in the case of an allele where ADO occurs for both of the alleles, the relative intensities will be approximately 0:0:0:0, and the absolute intensity of the signal will correspond to no base pairs.

[0178] In the case of aneuploides, however, different situations will be observed. For example, in the case of trisomy, and there is no ADO, one of three situations will occur: (i) in the case of a triply heterozygous allele, the relative intensities of the four registers will be approximately 1:1:1:0, and the absolute intensity of the signal will correspond to one base pair; (ii) in the case where two of the alleles are homozygous, the relative intensities will be approximately 2:1:0:0, and the absolute intensity of the signal will correspond to two and one base pairs, respectively; (iii) in the case where are alleles are homozygous, the relative intensities will be approximately 1:0:0:0, and the absolute intensity of the signal will correspond to three base pairs. If allele dropout occurs in the case of an allele in a cell with trisomy, one of the situations expected for a normal cell will be observed. In the case of monosomy, the relative intensities of the four registers will be approximately 1:0:0:0, and the absolute intensity of the signal will correspond to one base pair. This situation corresponds to the case of a normal cell where ADO of one of the alleles has occurred, however in the case of the normal cell, this will only be observed at a small percentage of the alleles. In the case of uniparental disomy, where two identical chromosomes are present, the relative intensities of the four registers will be approximately 1:0:0:0, and the absolute intensity of the signal will correspond to two base pairs. In the case of UPD where two different chromosomes from one parent are present, this method will indicate that the cell is normal, although further analysis of the data using other methods described in this patent will uncover this.

[0179] In all of these cases, either in cells that are normal, have aneuploides or UPD, the data from one SNP will not be adequate to make a decision about the state of the cell. However, if the probabilities of each of the above hypothesis are calculated, and those probabilities are combined for a sufficient number of SNPs on a given chromosome, one hypothesis will predominate, it will be possible to determine the state of the chromosome with high confidence.

Methods for Linearizing Quantitative Measurements

[0180] Many approaches may be taken to linearize measurements of the amount of genetic material at a specific locus so that data from different alleles can be easily summed or differenced. We first discuss a generic approach and then discuss an approach that is designed for a particular type of assay.

[0181] Assume data d_xi refers to a nonlinear measurement of the amount of genetic material of allele x at locus i. Create a training set of data using N measurements, where for each measurement, it is estimated or known that the amount of genetic material corresponding to data d_xi is β_xi. The training set β_xi, i=1 . . . N, is chosen to span all the different amounts of genetic material that might be encountered in practice. Standard regression techniques can be used to train a function that maps from the nonlinear measurement, d_xi, to the expectation of the linear measurement, E(β_xi). For example, a linear regression can be used to train a polynomial function of order P, such that E(β_xi)=[1 d_xi d_xi² . . . d_xi^P]c where c is the vector of coefficients c=[c₀ c₁ . . . c_P]^T. To train this linearizing function, we create a vector of the amount of genetic material for N measurements β_x=[β_x . . . β_xN]^T and a matrix of the measured data raised to powers 0 . . . P: D=[[1 d_x1 d_x1² . . . d_x1^P]^T [1 d_x2 d_x2² . . . d_x2^P]^T . . . [1 d_xN d_xN² . . . d_xN^P]^T]^T. The coefficients can then be found using a least squares fit c=(D^TD)⁻¹D^Tβ_x.

[0182] Rather than depend on generic functions such as fitted polynomials, we may also create specialized functions for the characteristics of a particular assay. We consider, for example, the TAQMAN assay or a qPCR assay. The amount of die for allele x and some locus i, as a function of time up to the point where it crosses some threshold, may be described as an exponential curve with a bias offset: g_xi(t)=α_xi+β_xiexp(γ_xit) where α_xi is the bias offset, γ_xi is the exponential growth rate, and β_xi corresponds to the amount of genetic material. To cast the measurements in terms of β_xi, compute the parameter α_xi by looking at the asymptotic limit of the curve g_xi(−∞) and then may find β_xi and γ_xi by taking the log of the curve to obtain log(g_xi(t)−α_xi)=log(β_xi)+γ_xit and performing a standard linear regression. Once we have values for α_xi and γ_xi, another approach is to compute β_xi from the time, t_x, at which the threshold g_x is exceeded. β_xi=(g_x−α_xi)exp(−γ_xit_x). This will be a noisy measurement of the true amount of genetic data of a particular allele.

[0183] Whatever techniques is used, we may model the linearized measurement as β_xi=κ_xδ_xi+n_xi where x is the number of copies of allele x, δ_xi is a constant for allele x and locus i, and n_xi˜N(0, σ_x²) where σ_x² can be measured empirically.

Method 4: Using a Probability Distribution Function for the Amplification of Genetic Data at Each Locus

[0184] The quantity of material for a particular SNP will depend on the number of initial segments in the cell on which that SNP is present. However, due to the random nature of the amplification and hybridization process, the quantity of genetic material from a particular SNP will not be directly proportional to the starting number of segments. Let q_s,A, q_s,G, q_s,T, q_s,C represent the amplified quantity of genetic material for a particular SNP s for each of the four nucleic acids (A,C,T,G) constituting the alleles. Note that these quantities may be exactly zero, depending on the technique used for amplification. Also note that these quantities are typically measured from the intensity of signals from particular hybridization probes. This intensity measurement can be used instead of a measurement of quantity, or can be converted into a quantity estimate using standard techniques without changing the nature of the invention. Let q_S be the sum of all the genetic material generated from all alleles of a particular SNP: q_s=q_s,A+q_s,G+q_s,T+q_s,C. Let N be the number of segments in a cell containing the SNP s. N is typically 2, but may be 0, 1 or 3 or more. For any high or medium throughput genotyping method discussed, the resulting quantity of genetic material can be represented as q_s=(A+A_θ,s)N+θ_s where A is the total amplification that is either estimated a-priori or easily measured empirically, A_θ,s is the error in the estimate of A for the SNP s, and θ_s is additive noise introduced in the amplification, hybridization and other process for that SNP. The noise terms A_θ,s and θ_s are typically large enough that q_s will not be a reliable measurement of N. However, the effects of these noise terms can be mitigated by measuring multiple SNPs on the chromosome. Let S be the number of SNPs that are measured on a particular chromosome, such as chromosome 21. It is possible to generate the average quantity of genetic material over all SNPs on a particular chromosome as follows:

[0185] $\begin{array}{cc}q=\frac{1}{S}\sum _{s=1}^S{q}_s=\mathrm{NA}+\frac{1}{S}\sum _{s=1}^S{A}_{\theta ,s}N+{\theta }_s& \left(16\right)\end{array}$
Assuming that A_θ,s and θ_s are normally distributed random variables with 0 means and variances σ²_Aθ,s and σ²_θs, one can model q=NA+φ where φ is a normally distributed random variable with 0 mean and variance

[0186] $\frac{1}{S}\left(N^2{\sigma }_{A_{\theta ,s}}^2+{\sigma }_{\theta }^2\right).$
Consequently, if sufficient number of SNPs are measured on the chromosome such that S>>(N²σ²_Aθ,S+σ²_θ), then N=q/A can be accurately estimated.

[0187] In another embodiment, assume that the amplification is according to a model where the signal level from one SNP is s=a+α where (a+α) has a distribution that looks like the picture in FIG. 14A, left. The delta function at 0 models the rates of allele dropouts of roughly 30%, the mean is a, and if there is no allele dropout, the amplification has uniform distribution from 0 to a₀. In terms of the mean of this distribution a₀ is found to be a₀=2.86a. Now model the probability density function of a using the picture in FIG. 14B, right. Let s_c be the signal arising from c loci; let n be the number of segments; let α_i be a random variable distributed according to FIGS. 14A and 14B that contributes to the signal from locus i; and let σ be the standard deviation for all {α_i}. s_c=anc+Σ_{i=1 . . . nc} α_i; mean(s_c)=anc; std(s_c)=sqrt(nc)σ. If σ is computed according to the distribution in FIG. 14A, right, it is found to be σ=0.907a². We can find the number of segments from n=s_c/(ac) and for “5-sigma statistics” we require std(n)<0.1 so std(s_c)/(ac)=0.1=>0.95a.sqrt(nc)/(ac)=0.1 so c=0.95² n/0.1²=181.

[0188] Another model to estimate the confidence in the call, and how many loci or SNPs must be measured to ensure a given degree of confidence, incorporates the random variable as a multiplier of amplification instead of as an additive noise source, namely s=a(1+α). Taking logs, log(s)=log(a)+log(1+α). Now, create a new random variable γ=log(1+α) and this variable may be assumed to be normally distributed ˜N(0,σ). In this model, amplification can range from very small to very large, depending on σ, but never negative. Therefore α=e^γ−1; and s_c=Σ_{i=1 . . . cn}a(1+α_i). For notation, mean(s_c) and expectation value E(s_c) are used interchangeably
E(s_c)=acn+aE(Σ_{i=1 . . . cn}α_i)=acn+aE(Σ_{i=1 . . . cn}α_i)=acn(1+E(α))
To find E(α) the probability density function (pdf) must be found for a which is possible since α is a function of γ which has a known Gaussian pdf. p_α(α)=p_γ(γ)(dγ/dα). So:

[0189] $p_{\gamma }\left(\gamma \right)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{\frac{-{\gamma }^2}{2{\sigma }^2}}\mathrm{and}$ $\frac{d\gamma }{d\alpha }=\frac{d}{d\alpha }\left(\log \left(1+\alpha \right)\right)=\frac{1}{1+\alpha }=e^{-\gamma }\mathrm{and}\text{:}$ $p_{\alpha }\left(\alpha \right)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{\frac{-{\gamma }^2}{2{\sigma }^2}}{e}^{-\gamma }=\frac{1}{\sqrt{2\pi }\sigma }{e}^{\frac{-{\left(\log \left(1+\alpha \right)\right)}^2}{2{\sigma }^2}}\frac{1}{1+\alpha }$
This has the form shown in FIG. 15 for σ=1. Now, E(α) can be found by integrating over this pdf E(α)=∫_−∞^∞αp_a(α)d_α which can be done numerically for multiple different σ. This gives E(s_c) or mean(s_c) as a function of σ. Now, this pdf can also be used to find var(s_c):

[0190] $\begin{array}{c}\mathrm{var}\left(s_c\right)={E\left(s_c-E\left(s_c\right)\right)}^2\\ ={E\left(\sum _{i=1\dots \mathrm{cn}}^{}a\left(1+{\alpha }_i\right)-\mathrm{acn}-\mathrm{aE}\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i\right)\right)}^2\\ ={E\left(\sum _{i=1\dots \mathrm{cn}}a{\alpha }_i-\mathrm{aE}\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i\right)\right)}^2\\ =a^2{E\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i-\mathrm{cnE}\left(\alpha \right)\right)}^2\\ =a^{2}E\left({\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i\right)}^2-2\mathrm{cnE}\left(\alpha \right)\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i\right)+c^2{n}^2{E\left(\alpha \right)}^2\right)\\ =a^{2}E\left(\mathrm{cn}{\alpha }^2+\mathrm{cn}\left(\mathrm{cn}-1\right){\alpha }_i{\alpha }_j-2\mathrm{cnE}\left(\alpha \right)\left(\sum _{i=1\dots \mathrm{cn}}{\alpha }_i\right)+c^2{n}^2{E\left(\alpha \right)}^2\right)\\ =a^2{c}^2{n}^2\left(E\left({\alpha }^2\right)+\left(\mathrm{cn}-1\right)E\left({\alpha }_i{\alpha }_j\right)-2\mathrm{cnE}\left({\alpha }^2\right)+{\mathrm{cnE}\left(\alpha \right)}^2\right)\\ =a^2{c}^2{n}^2\left(E\left({\alpha }^2\right)+\left(\mathrm{cn}-1\right)E\left({\alpha }_i{\alpha }_j\right)-{\mathrm{cnE}\left(\alpha \right)}^2\right)\end{array}$
which can also be solved numerically using p_α(α) for multiple different σ to get var(s_c) as a function of σ. Then, we may take a series of measurements from a sample with a known number of loci c and a known number of segments n and find std(s_c)/E(s_c) from this data. That will enable us to compute a value for σ. In order to estimate n, E(s_c)=nac(1+E(α)) so