This application is a continuation of U.S. patent application Ser. No. 10/272,461, entitled METHODS AND SYSTEMS FOR POLYNUCLEOTIDE DETECTION, filed Oct. 16, 2002, now pending; which is a continuation-in-part of U.S. patent application Ser. No. 10/146,720, entitled A METHOD TO ASSEMBLE SPLICE VARIANTS FOR BOTH KNOWN AND PREDICTED GENES, A METHOD FOR VALIDATING THEIR EXPRESSION IN CELLS AND A METHOD OF DISCOVERING NOVEL EXON/EXON EXTENSIONS/TRIMS, filed May 14, 2002, now pending, which claims the benefit of U.S. Provisional Patent Application No. 60/307,911, entitled A METHOD TO ASSEMBLE SPLICE VARIANTS FOR BOTH KNOWN AND PREDICTED GENES, A METHOD FOR VALIDATING THEIR EXPRESSION IN CELLS AND A METHOD OF DISCOVERING NOVEL EXON/EXON EXTENSIONS/TRIMS, filed Jul. 25, 2001, now abandoned; the Ser. No. 10/146,720 application also claim the benefit of U.S. Provisional Patent Application No. 60/343,298, entitled METHODS OF OLIGO SELECTION AND OPTIMIZATION, filed Dec. 21, 2001, now abandoned; and U.S. Provisional Patent Application No. 60/329,914, entitled A METHOD TO ASSEMBLE SPLICE VARIANTS FOR BOTH KNOWN AND PREDICTED GENES, A METHOD FOR VALIDATING THEIR EXPRESSION IN CELLS AND A METHOD OF DISCOVERING NOVEL EXON/EXON EXTENSIONS/TRIMS, filed Oct. 17, 2001, now abandoned, all of which are hereby incorporated by reference in their entirety.
The described technology relates generally to selecting indicator polynucleotides and to detecting polynucleotides.
Polymerase chain reaction (PCR) analyses, nucleotide array experiments, in situ hybridizations, and Southern, Northern and Dot blot experiments attempt to form DNA-DNA, RNA-RNA, or DNA-RNA hybrids. In such experiments, an “indicator polynucleotide,” such as an oligonucleotide probe, hybridizes to a polynucleotide that includes a polynucleotide subsequence complementary to the indicator polynucleotide. The “melting temperature,” which depends in part upon the nucleotide sequence of the indicator polynucleotide, characterizes the stability of the hybridization product given a set of experimental conditions. The melting temperature is the temperature at which 50% of a given indicator polynucleotide hybridizes to complementary polynucleotides of sufficient abundance. The melting temperature is critical for determining the selectivity and sensitivity of indicator polynucleotides when used as primers in polymerase chain reaction (PCR) experiments, as probes for in situ hybridizations, as probes for nucleotide array experiments, and in Southern, Northern, or Dot blot experiments. If the melting temperature is too low, few indicator polynucleotides will hybridize to their complementary polynucleotides. If the melting temperature is too high, indicator polynucleotides may hybridize to polynucleotides weakly homologous to their complementary polynucleotides. Even with an optimal melting temperature, the formation of hybridization products may be influenced by experimental conditions and the nucleotide sequences of the indicator polynucleotide and the polynucleotides present in the biological sample or environment. It would be desirable to select indicator polynucleotides to maximize hybridization to complementary polynucleotides while minimizing hybridization to other polynucleotides. Also, it would be desirable if the post-hybridization analysis would factor in expected variations due to differing melting temperatures of indicator polynucleotides as well as expected variations due to homologous polynucleotides and other polynucleotides present in the sample or environment.
FIG. 1 is a flow diagram illustrating the overall process of selecting indicator polynucleotides to minimize the errors between the actual melting temperatures and the desired melting temperature in one embodiment.
FIG. 2 is a flow diagram illustrating the overall process of analytically correcting for variations in melting temperatures in one embodiment.
FIG. 3 is a flow diagram illustrating the overall process of correcting for hybridization of an indicator polynucleotide with a homologue of the target polynucleotide in one embodiment.
Optimization techniques for selecting indicator polynucleotides for an experiment and for determining expression levels resulting from the experiment are provided. In one embodiment, the optimization technique corrects for variations in polynucleotide melting temperatures during analysis of the experimental results. The optimization technique selects set of indicator polynucleotides for the experiment. The optimization technique then performs the experiment with the indicator polynucleotides and a sample and identifies the relative amounts of the indicated polynucleotides. The optimization technique then adjusts the relative amounts of the indicated polynucleotides based on melting temperatures associated with the indicator polynucleotides. For example, if one of the indicator polynucleotides has a high melting temperature relative to the hybridization and wash temperatures, then the optimization technique may increase the relative amount of the corresponding indicated polynucleotide to account for the high melting temperature. In an alternate embodiment, the optimization technique includes a control indicator polynucleotide to identify relative amounts of a homologue polynucleotide whose presence is incidentally detected by an indicator polynucleotide designed to detect the presence of a target polynucleotide.
In another embodiment, the optimization technique attempts to minimize the difference between the actual melting temperature of an indicator polynucleotide and a desired melting temperature. When multiple indicator polynucleotides are used in a single experiment, the optimization technique attempts to minimize the overall error resulting from differences in the melting temperatures of the indicator polynucleotides and the desired melting temperature. The optimization technique may modify indicator polynucleotides prior to performing an experiment to reduce the error. The optimization technique may modify an indicator polynucleotide by shifting the location of an indicator polynucleotide to hybridize with upstream or downstream portions of the target polynucleotide. The optimization technique may also vary the lengths of the indicator polynucleotides to minimize the error. When detecting exon-exon junctions, the optimization technique may attempt to balance the melting temperature for each exon portion of the indicator polynucleotide so that the indicator polynucleotide will hybridize to both exons equally. These and another optimization techniques are described more fully in the following.
Measuring Error in Melting Temperatures
In one embodiment, the optimization technique selects an indicator polynucleotide from among multiple possibilities that will minimize an error equation, such as the following distance equation:
E^{2}=(T_{d}−T_{m}) (1)
where E is the error, T_{d }is the desired melting temperature, and T_{m }is the theoretical or empirical melting temperature of the polynucleotide. Equation 1 is generally referred to as an Euclidean distance measure. The indicator polynucleotide with the smallest error may be the best choice.
When multiple indicator polynucleotides are to be used in the same experiment, the optimization technique selects indicator polynucleotides that tend to minimize the overall error in melting temperatures. In one embodiment, the optimization technique calculates the error according to the following equation.
E_{t}^{2}=E_{1}^{2}+E_{2}^{2}+ . . . +E_{n}^{2} (2)
where E_{t }is the total error, E_{i }is the error from the ith polynucleotide, and n is the number of indicator polynucleotides.
A more general formulation of error equation 1 that applies to a single polynucleotide is the following equation:
E=f(T_{d}, T_{m}) (3)
where f is any arbitrary error function. The substitution of equation 3 into equation 2 for multiple polynucleotides results in the following equation:
E_{t}=g(f(T_{d}, T_{m1}), f(T_{d}, T_{m2}), . . . , f(T_{d}, T_{mn})) (4)
where g is a function that combines the individual error measures of the n indicator polynucleotides. When an experimental design includes constraints, such as which genes, exons, or exon-exon junctions the indicator polynucleotides identify, the optimization technique selects those indicator polynucleotides that minimize the value of E_{t }given the desired melting temperature T_{d}.
To illustrate the error calculation, an example nucleotide array experiment involving indicator polynucleotides for each exon and exon-exon junction of a transcript of the gene CD44 with the GenBank locus name XM_{—}030326, which contains 18 exons and 17 exon-exon junctions, is used. One indicator polynucleotide selection technique might select indicator polynucleotides with a length of 20 bases. Each indicator polynucleotide for an exon is selected to hybridize to the center of the exon, and each indicator polynucleotide for an exon-exon junction is selected to hybridize to 10 bases on each side of the exon-exon junction. The selected indicator polynucleotides are presented in the following table:
TABLE 1 | ||||
SEQ ID NO | Junction/Exon | Melting Temp ° C. | Sequence | |
SEQ ID NO:1 | E1 | 62.030003 | cgcgcccagggatcctccag | |
SEQ ID NO:2 | J1-2 | 22.0 | gcgcagatcgatttgaatat | |
SEQ ID NO:3 | E2 | 59.980003 | ggaggccgctgacctctgca | |
SEQ ID NO:4 | J2-3 | 30.0 | agacctgcaggtatgggttc | |
SEQ ID NO:5 | E3 | 53.83 | tgcagcaaacaacacagggg | |
SEQ ID NO:6 | J3-4 | 32.0 | aatgcttcagctccacctga | |
SEQ ID NO:7 | E4 | 53.83 | agtcacagacctgcccaatg | |
SEQ ID NO:8 | J4-5 | 24.0 | attaccataactattgttaa | |
SEQ ID NO:9 | E5 | 55.880005 | gctcctccagtgaaaggagc | |
SEQ ID NO:10 | J5-6 | 28.0 | cctgctaccactttgatgag | |
SEQ ID NO:11 | E6 | 51.780003 | cctgggattggttttcatgg | |
SEQ ID NO:12 | J6-7 | 30.0 | caaatggctggtacgtcttc | |
SEQ ID NO:13 | E7 | 49.730003 | gatgaaagagacagacacct | |
SEQ ID NO:14 | J7-8 | 28.0 | tccagcaccatttcaaccac | |
SEQ ID NO:15 | E8 | 55.880005 | tggacccagtggaacccaag | |
SEQ ID NO:16 | J8-9 | 28.0 | aggatgactgatgtagacag | |
SEQ ID NO:17 | E9 | 53.83 | aagcacaccctcccctcatt | |
SEQ ID NO:18 | J9-10 | 32.0 | acaagcacaatccaggcaac | |
SEQ ID NO:19 | E10 | 51.780003 | ggtttggcaacagatggcat | |
SEQ ID NO:20 | J10-11 | 34.0 | gggacagctgcagcctcagc | |
SEQ ID NO:21 | E11 | 49.730003 | gacagttcctggactgattt | |
SEQ ID NO:22 | J11-12 | 28.0 | agaaggatggatatggactc | |
SEQ ID NO:23 | E12 | 47.68 | aatccaaacacaggtttggt | |
SEQ ID NO:24 | J12-13 | 28.0 | atgacaacgcagcagagtaa | |
SEQ ID NO:25 | E13 | 47.68 | gaaggcttggaagaagataa | |
SEQ ID NO:26 | J13-14 | 26.0 | acatcaagcaataggaatga | |
SEQ ID NO:27 | E14 | 51.780003 | acgaaggaaagcaggacctt | |
SEQ ID NO:28 | J14-15 | 30.0 | tccttatcaggagaccaaga | |
SEQ ID NO:29 | E15 | 57.930004 | cagtggggggtcccatacca | |
SEQ ID NO:30 | J15-16 | 30.0 | gaatcagatggacactcaca | |
SEQ ID NO:31 | E16 | 53.83 | ggagcaaacacaacctctgg | |
SEQ ID NO:32 | J16-17 | 28.0 | caaattccagaatggctgat | |
SEQ ID NO:33 | E17 | 49.730003 | ccttggctttgattcttgca | |
SEQ ID NO:34 | J17-18 | 34.0 | gtcgaagaaggtgtgggcag | |
SEQ ID NO:35 | E18 | 53.83 | tcgttccagttcccacttgg | |
The NCBI locus name and version for the genomic sequence containing the CD44 gene is NT_{—}024229.8. The NCBI locus name and version for the mRNA sequence of the CD44 transcript is XM_{—}030326.3. The identifiers E1, E2, E3, and etc. identify indicator polynucleotides for exon 1, exon 2, exon 3, and etc. The identifiers J1-2, J2-3, and etc. identify indicator polynucleotides for the exon-exon junction between exon 1 and exon 2, the exon-exon junction between exon 2 and exon 3, and etc. The melting temperatures are theoretical melting temperatures of one exon portion of the indicator polynucleotide, calculated according to the following equation:
T_{m}=64.9° C.+41° C.*(GC−16.4)/L (5)
where L is the length of the target polynucleotide and GC is the GC content. One skilled in the art will appreciate the theoretical melting temperature can be calculated using various well-known equations. The calculated melting temperature varies from a low of 22° C. to a high of 62.03° C. in this example. This wide range of melting temperatures will lead to substantial variation in the number of polynucleotides that hybridize to instances of a given indicator polynucleotides in an experiment with fixed hybridization and washing temperatures. For example, in an experiment with a sample containing CD44 using a standard protocol with a hybridization temperature of 52° C. and a washing temperature of 52° C., the indicator polynucleotides with melting temperatures under 52° C. will form and retain hybridization products less frequently than those with melting temperatures above 52° C. In general, indicator polynucleotides with low calculated melting temperatures may not bind to their target exons or exon-exon junctions with any measurable strength above a background level. The error with a target temperature of 52° C. is calculated using equation 2 in the following:
E_{t}^{2}=(52−62.03)^{2}+(52−22)^{2}+ . . . +(52−53.83)^{2}=(97.46)^{2}
Modifying Indicator Polynucleotides to Reduce Errors in Melting Temperatures
The optimization technique can optimize an indicator polynucleotide prior to running the experiment. The optimization technique may optimize indicator polynucleotides by shifting the location of the indicator polynucleotides slightly upstream or downstream within their respective RNAs. For example, the optimization technique can select indicator polynucleotides for exons from any base range within the exon that the indicator polynucleotide identifies, using any selection criteria, such as GC content. Alternatively, the optimization technique can optimize indicator polynucleotides by varying their lengths. For example, rather than selecting only indicator polynucleotides with a length of 20 bases, the optimization technique may select indicator polynucleotides of varying lengths to reduce the error measure for each indicator polynucleotide. These various optimizations may be used alone or in conjunction with other indicator polynucleotide selection, analytical correction, or other optimizations. When varying the length of the indicator polynucleotides for the same CD44 transcript, without using other indicator polynucleotide selection or optimizations, the resulting indicator polynucleotide are shown in the following:
TABLE 2 | ||||
SEQ ID NO | Junction/Exon | Melting Temp ° C. | Sequence | |
SEQ ID NO:36 | E1 | 51.062504 | ccca atcctcca | |
SEQ ID NO:37 | J1-2 | 52.258335 | cgcagatcgatttgaatataacct | |
SEQ ID NO:38 | E2 | 51.062504 | aggccgctgacctctg | |
SEQ ID NO:39 | J2-3 | 53.24737 | gacctgcaggtatgggttc | |
SEQ ID NO:40 | E3 | 51.089478 | tgcagcaaacaacacaggg | |
SEQ ID NO:41 | J3-4 | 51.089478 | aatgcttcagctccacctg | |
SEQ ID NO:42 | E4 | 51.089478 | agtcacagacctgcccaat | |
SEQ ID NO:43 | J4-5 | 51.653847 | caattaccataactattgttaaccgt | |
SEQ ID NO:44 | E5 | 53.24737 | ctcctccagtgaaaggagc | |
SEQ ID NO:45 | J5-6 | 51.780003 | cctgctaccactttgatgag | |
SEQ ID NO:46 | E6 | 52.97273 | cctgggattggttttcatggtt | |
SEQ ID NO:47 | J6-7 | 51.780003 | caaatggctggtacgtcttc | |
SEQ ID NO:48 | E7 | 51.109093 | aagatgaaagagacagacacct | |
SEQ ID NO:49 | J7-8 | 51.780003 | ccagcaccatttcaaccaca | |
SEQ ID NO:50 | E8 | 52.600002 | ggacccagtggaacccaa | |
SEQ ID NO:51 | J8-9 | 51.7087 | aaggatgactgatgtagacagaa | |
SEQ ID NO:52 | E9 | 51.876472 | gcacaccctcccctcat | |
SEQ ID NO:53 | J9-10 | 52.404762 | tacaagcacaatccaggcaac | |
SEQ ID NO:54 | E10 | 51.780003 | ggtttggcaacagatggcat | |
SEQ ID NO:55 | J10-11 | 51.876472 | ggacagctgcagcctca | |
SEQ ID NO:56 | E11 | 52.404762 | gacagttcctggactgatttc | |
SEQ ID NO:57 | J11-12 | 52.97273 | aagaaggatggatatggactcc | |
SEQ ID NO:58 | E12 | 51.109093 | caaatccaaacacaggtttggt | |
SEQ ID NO:59 | J12-13 | 51.7087 | aatgacaacgcagcagagtaatt | |
SEQ ID NO:60 | E13 | 51.7087 | gaaggcttggaagaagataaaga | |
SEQ ID NO:61 | J13-14 | 51.7087 | gacatcaagcaataggaatgatg | |
SEQ ID NO:62 | E14 | 51.089478 | acgaaggaaagcaggacct | |
SEQ ID NO:63 | J14-15 | 50.452385 | ttccttatcaggagaccaaga | |
SEQ ID NO:64 | E15 | 51.062504 | tggggggtcccatacc | |
SEQ ID NO:65 | J15-16 | 51.109093 | tgaatcagatggacactcacat | |
SEQ ID NO:66 | E16 | 51.089478 | ggagcaaacacaacctctg | |
SEQ ID NO:67 | J16-17 | 52.97273 | ccaaattccagaatggctgatc | |
SEQ ID NO:68 | E17 | 52.404762 | ccttggctttgattcttgcag | |
SEQ ID NO:69 | J17-18 | 52.600002 | gtcgaagaaggtgtgggc | |
SEQ ID NO:70 | E18 | 51.089478 | tcgttccagttcccacttg | |
Since the calculated polynucleotide melting temperatures have a small range, from 50.45° C. to 52.97° C. (rather than 22° C. to 62.03° C.), the error measure E_{t }of 4.51 (rather than 97.46) is also small. This varying of the lengths of the indicator polynucleotides has eliminated over 95% of the error encountered when the indicator polynucleotides were selected as shown in Table 1.
FIG. 1 is a flow diagram illustrating the overall process of selecting indicator polynucleotides to minimize the errors between the actual melting temperatures and the desired melting temperature in one embodiment. In block 101, the technique selects an initial set of indicator polynucleotides. In block 102, the technique inputs the desired melting temperature of the indicator polynucleotides. In blocks 103-105, the technique modifies each indicator polynucleotide to minimize its error. In block 103, the technique selects the next indicator polynucleotide. In decision block 104, if all the indicator polynucleotides have already been selected, then the technique completes, else the technique continues at block 105. In decision block 105, if the error between actual melting temperature of the selected indicator polynucleotide and the desired melting temperature is acceptable, then the technique selects the next indicator polynucleotide, else the technique continues at block 106. In block 106, the technique modifies the selected indicator polynucleotide using one or more of the described techniques (e.g., adjusting the length of the indicator polynucleotide or moving the indicator polynucleotide upstream or downstream) and then continues at a block 105.
One skilled in the art will recognize that the error E_{t }may be further reduced in a variety of ways, such as by introducing molecules other than the bases A, C, G, or T with different binding characteristics. Such bases may be appended, prepended, inserted, or selectively substituted within the indicator polynucleotides.
CD44 is known to have several splice variants. For example, the transcript used above contains several exons not present in other splice forms of CD44. In particular, J6-7 does not exist in some species of CD44. Instead, E6 joins with a later exon E7′, yielding J6-7′. The RNA transcript containing the J6-7 is referred to as R^{1}, and the RNA transcript containing J6-7′ is referred to as R^{2}.
When using indicator polynucleotides to identify exon-exon junctions, binding may be less specific than desired if polynucleotides bind to one or the other half of an indicator polynucleotide. For example, if J6-7 is not present in a given sample, but E6 is present (because a different splice form of the gene is present in that sample), then an indicator polynucleotide for J6-7 may hybridize to E6 even though E6 is not joined to E7 in the splice variant in the sample. The hybridization, however, will likely be weaker than if a splice variant containing J6-7 was present. The expression levels may be measured as
H(E6)=1069
H(J6-7)=388
where H(E6) is the measured expression level of E6 in the experiment and H(J6-7) is the measured expression level of J6-7. These measured expression may represent the following scenarios:
Techniques described in U.S. patent application Ser. No. 10/146,720, entitled “Method and System for Identifying Splice Variants of a Gene,” can be used to differentiate these scenarios. Those techniques assign an expected expression level to J6-7 in the presence of R_{1}, in the presence of R^{2}, and in the presence of R^{3}. For example, if there are indicator polynucleotides for E6, E7, and J6-7, then a matrix M with a column for each expected splice variant (e.g., R^{1}, R^{2 }and R^{3}) and a row for each indicator polynucleotide (e.g., E6, E7, and J6-7) is created. The values in the matrix correspond to the expected expression level for the target polynucleotides. When the partial expression levels are not expected, the matrix might look like the following:
If the indicator polynucleotide for J6-7, however, weakly binds in the presence of a different splice site, J6-7′, a correction can be applied. The values in the correction matrix can be calculated or empirically derived. The values can be empirically derived by performing a hybridization experiment containing J6-7′ but not J6-7. The correction matrix may be derived using one or more samples containing antisense polynucleotides. A sample could include antisense polynucleotides for J6 or J7 or both. (“J6” refers to the portion of an indicator polynucleotide for a J6-X junction that is used to identify the E6 portion of the junction.) For example, an antisense polynucleotide for J6 might contain the complementary polynucleotide for J6. Alternatively, the antisense polynucleotide might contain the complementary polynucleotide for J6 appended to a sequence of some additional number of bases, perhaps chosen randomly. In yet another scenario, the antisense polynucleotide might contain J6 with J7 prepended. After the hybridization experiments, the following expression values may result:
H(J6-7|J6)=556
H(J6-7|J7)=310
H(J6-7|J6-7)=1544
H(J6-7|J6, J7)=756
where H(J6-7|J6) is the empirically derived expression level of the indicator polynucleotide for J6-7 in the presence of a sample containing antisense polynucleotides for J6, H(J6-7|J7) is the expression level of J6-7 in the presence of a sample containing antisense polynucleotides for J7, H(J6-7|J6-7) is the ordinary expression level of J6-7 in the presence of antisense polynucleotides for J6-7, and H(J6-7|J6, J7) is the expression level of J6-7 in the presence of separate antisense polynucleotides for J6 and J7. The expression values are not independent and are preferably measured in separate hybridization experiments. If more than one expression level is measured in the same experiment, the individual values can be solved using a system of linear equations, a least squares equation, or another deconvolution method.
Once the expression levels have been determined either empirically or theoretically, the values in the matrix M could be determined using an equation such as:
M_{i,j}=H(P_{i}|R_{j})/H(P_{i}) (6)
where M_{i,j }is a coefficient matrix, P_{i }is an indicator polynucleotide that identifies a subsequence of RNA transcript R_{j}, H(P_{i}|R_{j}) is the expression level of indicator polynucleotide P_{i }given that RNA R_{j }is expressed, and H(P_{i}) is the expression level of the hybridization product of indicator polynucleotide i. The solution to this equation is:
M_{2,1}=H(J6-7|J6-7)/H(J6-7|J6-7)=1544/1544−1=1
M_{2,2}=H(J6-7|J6)/H(J6-7|J6-7)=556/1544=0.36
M_{2,3}=H(J6-7|J6,J7)/H(J6-7|J6-7)=756/1544=0.49
In this example, H(J6-7|J6-7) is the expression of the indicator polynucleotide for J6-7 given RNA containing J6-7 is present. The remaining matrix elements all have a value of 1, for the same reason as M_{2,1}. The resulting matrix M is:
The value 0.36 indicates that the indicator polynucleotide for J6-7 will yield a relative expression level of 0.36 times the full value if R^{2 }is present, and the other two splice variants are not present. The expression level is non-zero in this case because the indicator polynucleotide for J6-7 will hybridize weakly in the presence of E6 even if J6-7 is not present. The value 0.49 indicates that the indicator polynucleotide for J6-7 will yield a relative expression level of 0.49 times the full value if R^{3 }is present and the other two splice variants are not present. The value for J6-7 in R^{3 }is larger than the corresponding value in R^{2 }because the indicator polynucleotide will hybridize weakly to both E6 and E7 rather than only to E6.
Modifying a Junction Indicator Polynucleotide to Balance Its Melting Temperature
The indicator polynucleotide for J6-7 consists of a portion J6 that identifies the 3′ end of E6 as well as a portion J7 that identifies the 5′ end of E7. Each of these portions has its own melting temperature. In other words, the indicator polynucleotide for J6-7 will hybridize to E6 based on the melting temperature of J6 even if J6-7 is not present in the sample. Likewise, the indicator polynucleotide will hybridize to E7 based on the melting temperature of J7 even if J6-7 is not present in the sample. In one embodiment, the optimization technique balances the melting temperature of each exon portion of a junction indicator polynucleotide.
A full set of exon-exon junction indicator polynucleotides with a length of 30 bases for CD44 selected to have 15 bases for each exon is shown in the following:
TABLE 3 | ||||||
Junction/ | Melting | Left Melting | Right Melting | |||
SEQ ID No | Exon | Temp ° C. | Temp ° C. | Temp ° C. | Sequence | |
SEQ ID | J1-2 | 61.620003 | 50.140003 | 28.273338 | gcctggcgcagatcgatttgaatataacct | |
NO: 71 | ||||||
SEQ ID | J2-3 | 60.253334 | 39.20667 | 36.473335 | atttgagacctgcaggtatgggttcataga | |
NO: 72 | ||||||
SEQ ID | J3-4 | 62.986668 | 39.20667 | 41.940002 | gcttcaatgcttcagctccacctgaagaag | |
NO: 73 | ||||||
SEQ ID | J4-5 | 57.520004 | 33.74 | 36.473335 | gaccaattaccataactattgttaaccgtg | |
NO: 74 | ||||||
SEQ ID | J5-6 | 61.620003 | 41.940002 | 36.473335 | gaatccctgctaccactttgatgagcacta | |
NO: 75 | ||||||
SEQ ID | J6-7 | 58.88667 | 39.20667 | 33.74 | caacacaaatggctggtacgtcttcaaata | |
NO: 76 | ||||||
SEQ ID | J7-8 | 61.620003 | 39.20667 | 39.20667 | ttatctccagcaccatttcaaccacaccac | |
NO: 77 | ||||||
SEQ ID | J8-9 | 60.253334 | 41.940002 | 33.74 | ccacaaggatgactgatgtagacagaaatg | |
NO: 78 | ||||||
SEQ ID | J9-10 | 60.253334 | 33.74 | 41.940002 | attctacaagcacaatccaggcaactccta | |
NO: 79 | ||||||
SEQ ID | J10-11 | 65.72 | 44.673336 | 41.940002 | caacagggacagctgcagcctcagctcata | |
NO: 80 | ||||||
SEQ ID | J11-12 | 62.986668 | 41.940002 | 39.20667 | caggaagaaggatggatatggactccagtc | |
NO: 81 | ||||||
SEQ ID | J12-13 | 58.88667 | 36.473335 | 36.473335 | tttcaatgacaacgcagcagagtaattctc | |
NO: 82 | ||||||
SEQ ID | J13-14 | 58.88667 | 39.20667 | 33.74 | ctctgacatcaagcaataggaatgatgtca | |
NO: 83 | ||||||
SEQ ID | J14-15 | 60.253334 | 36.473335 | 39.20667 | atcgttccttatcaggagaccaagacacat | |
NO: 84 | ||||||
SEQ ID | J15-16 | 61.620003 | 36.473335 | 41.940002 | gatctgaatcagatggacactcacatggga | |
NO: 85 | ||||||
SEQ ID | J16-17 | 61.620003 | 41.940002 | 36.473335 | caccccaaattccagaatggctgatcatct | |
NO: 86 | ||||||
SEQ ID | J17-18 | 62.986668 | 39.20667 | 41.940002 | caacagtcgaagaaggtgtgggcagaagaa | |
NO: 87 | ||||||
The temperature difference in J6-7 is approximately 5.5° C. The expected result of the difference is that the indicator polynucleotide will yield a larger expression value if E6 is present without J6-7 than if E7 is present without J6-7. A similar effect will be observed to varying degrees for all of the indicator polynucleotides. In one embodiment, the optimization technique corrects for these effects using the linear equations presented above. For example, different values for a transcript containing E6 but not E7 than for a transcript containing E7 but not E6 may be used. The corrected matrix might look like this:
where R^{2 }is a transcript containing E6 but not E7 and R^{3 }is a transcript containing E7 but not E6. However, it would be desirable to minimize the need to correct experimental results in this way, since it is generally desirable to minimize the number of experimental parameters that vary in the same experiment. The optimization technique minimizes or eliminates the need for this correction by balancing the melting temperature on each side of the exon-exon junction, so that the melting temperatures of indicator polynucleotide portions J6 and J7 are close to equal.
An optimization equation for error from imbalanced melting temperature using Euclidean distance can be written as:
E_{x}^{2}=(T_{ia}−T_{ib})^{2 } (7)
where E_{x }is the error from imbalance in melting temperature, T_{ia }is the calculated or empirical melting temperature of the portion of the indicator polynucleotide which identifies the 3′ end of the first exon, and T_{ib }is the calculated melting temperature of the portion of the indicator polynucleotide which identifies the 5′ end of the second exon. If the indicator polynucleotide does not identify an exon-exon junction, the error is zero. The total error in the experiment then is given by:
E_{xt}^{2}=E_{x1}^{2}+E_{x2}^{2}+ . . . +E_{xn}^{2 } (8)
where E_{xt }is the total error from multiple indicator polynucleotides as a result of temperature imbalance in indicator polynucleotides that identify exon-exon junctions. One skilled in the art will appreciate that error metrics other than Euclidean distance may be used.
In one embodiment, the optimization technique considers both temperature balancing in exon-exon junction indicator polynucleotides and total indicator polynucleotide melting temperature, which can be represented by the following equation:
E_{jt}=k_{1}E+k_{2}E_{x } (9)
where E_{jt }is the joint error measure, E is given by equation 1, E_{x }is given by equation 7, and k_{1 }and k_{2 }are constants. The total joint error measure can be represented by the following equation:
E_{jt}=E_{j1}+E_{j2}+ . . . +E_{jn } (10)
where E_{jt }is the total error from both temperature imbalance in exon-exon junction indicator polynucleotides and each E_{ji }is the error for each individual indicator polynucleotide i calculated using equation 9. One skilled in the art will appreciate that error metrics other than Euclidea distance can be used and that equations 7-10 can be generalized as equations 3 and 4 were generalized.
The exon-exon junction indicator polynucleotides selected from CD44 according to equations 7-10, using a desired total melting temperature T_{d}=60° C. and k_{1}=k_{2}=1.0 are shown in the following:
TABLE 4 | ||||||
Junction/ | Melting | Left Melting | Right Melting | |||
SEQ ID No | Exon | Temp ° C. | Temp ° C. | Temp ° C. | Sequence | |
SEQ ID | J1-2 | 59.921432 | 38.0 | 37.405884 | ggcgcagatcgatttgaatataacctgc | |
NO:88 | ||||||
SEQ ID | J2-3 | 59.921432 | 38.0 | 38.25 | tgagacctgcaggtatgggttcatagaa | |
NO:89 | ||||||
SEQ ID | J3-4 | 61.255558 | 39.20667 | 38.0 | gcttcaatgcttcagctccacctgaag | |
NO:90 | ||||||
SEQ ID | J4-5 | 59.59412 | 39.81765 | 39.81765 | tggaccaattaccataactattgttaaccgtgat | |
NO:91 | ||||||
SEQ ID | J5-6 | 59.737038 | 37.37143 | 38.0 | aatccctgctaccactttgatgagcac | |
NO:92 | ||||||
SEQ ID | J6-7 | 60.40323 | 39.20667 | 38.25 | caacacaaatggctggtacgtcttcaaatac | |
NO:93 | ||||||
SEQ ID | J7-8 | 59.737038 | 37.37143 | 38.0 | tatctccagcaccatttcaaccacacc | |
NO:94 | ||||||
SEQ ID | J8-9 | 60.253334 | 37.37143 | 38.25 | cacaaggatgactgatgtagacagaaatgg | |
NO:95 | ||||||
SEQ ID | J9-10 | 59.921432 | 38.25 | 38.0 | cattctacaagcacaatccaggcaactc | |
NO:96 | ||||||
SEQ ID | J10-11 | 59.980003 | 34.0 | 34.0 | gggacagctgcagcctcagc | |
NO:97 | ||||||
SEQ ID | J11-12 | 59.737038 | 37.37143 | 38.0 | aggaagaaggatggatatggactccag | |
NO:98 | ||||||
SEQ ID | J12-13 | 58.88667 | 36.473335 | 36.473335 | tttcaatgacaacgcagcagagtaattctc | |
NO:99 | ||||||
SEQ ID | J13-14 | 60.54375 | 39.20667 | 39.81765 | ctctgacatcaagcaataggaatgatgtcaca | |
NO:100 | ||||||
SEQ ID | J14-15 | 59.737038 | 38.0 | 37.37143 | cgttccttatcaggagaccaagacaca | |
NO:101 | ||||||
SEQ ID | J15-16 | 61.506897 | 40.8125 | 40.0 | ggatctgaatcagatggacactcacatgg | |
NO:102 | ||||||
SEQ ID | J16-17 | 60.093105 | 37.37143 | 36.473335 | accccaaattccagaatggctgatcatct | |
NO:103 | ||||||
SEQ ID | J17-18 | 59.324 | 38.0 | 38.0 | acagtcgaagaaggtgtgggcagaa | |
NO:104 | ||||||
Variations between the left melting temperature and right melting temperature values has decreased significantly; the maximum difference between left melting temperature and right melting temperature is now only approximately 0.95° C. The temperature balancing technique has eliminated over 95% of the variability in melting temperature between the two portions of each exon-exon junction indicator polynucleotide as compared to the indicator polynucleotide selection technique. The melting temperature of a fixed-length indicator polynucleotide can be balanced by decreasing the length of one exon's portion and increasing the length of the other exon's portion. For example, if the length of the indicator polynucleotide for a junction is 20 and the length of each portion is 10, then the length of one portion may be decreased to 8 and the length of the other portion might be increased to 12 to balance the melting temperature, keeping the overall length at 20. Corrections can be applied to experimental results, as described above, to further reduce the variability.
Correcting Analytically for Variations in Melting Temperature
In one embodiment, the optimization technique corrects for variations in polynucleotide melting temperatures during analysis of the experimental results.
This correction can be used whether or not an optimal set of indicator polynucleotides is selected as described above. The optimization technique adjusts the detected expression levels of the polynucleotides based on theoretical and desired melting temperatures. FIG. 2 is a flow diagram illustrating the overall process of analytically correcting for variations in melting temperatures in one embodiment. The steps of the processes illustrated by the flow diagrams may be controlled by, performed with the assistance of, or performed by a computer system. One skilled in the art will appreciate that various steps may be performed manually. In block 201, the technique selects the indicator polynucleotides that are to be used to identify target polynucleotides in a sample. In block 202, the technique performs the hybridization experiment, for example, using nucleotide array technology. In block 203, the technique receives the resulting expression levels from the experiment. In block 204, the technique calculates or inputs the melting temperature for each of the indicator polynucleotides. In block 205, the technique adjusts the expression levels based on the melting temperature for each of the indicator polynucleotides and temperature (e.g., hybridization temperature or wash temperature) at which the experiment was performed to give the final calculated expression levels for the experiment. One example analysis technique is described in U.S. patent application Ser. No. 10,146,720, filed on May 14, 2002 and entitled “Method and System for Identifying Splice Variants of a Gene,” which is hereby incorporated by reference. That splice variants analysis calculates the expression levels according to the following equation:
S=MH (11)
where S is a solution matrix of expression levels with a row for each expected RNA transcript and a column for each experiment, M is a matrix of coefficients in which each column corresponds to an expected RNA transcript and each row corresponds to an indicator polynucleotide in the experiment (each coefficient indicates the relative expected expression level of the indicator polynucleotide for the expected RNA transcript), and H is a matrix in which each of the columns corresponds to expression values derived from an expression array experiment. Because of experimental noise, an exact solution to this equation may not exist. The splice variants analysis can find an approximation using a variety of techniques such as a least squares regression using the following equation:
S=(MM^{T})^{−1}M^{T}H (12)
where M^{T }transpose of M. The values in the coefficient matrix M can be represented by the following equation:
M_{i,j}=L(P_{i}|R_{j}) (13)
where M_{i,j }is the matrix element in the ith row and jth column and L(P_{i}|R_{j}) defines a coefficient for indicator polynucleotide P (in the ith row of M) given that the expected RNA transcript R (in the jth column of M) is present in a sample.
If all indicator polynucleotides that identify subsequences of R are expected to be expressed at an identical level when R is present in the sample, and all indicator polynucleotides that do not identify subsequences of R are expected not to be expressed at all, matrix M might consist entirely of zeros and ones. The matrix M for two splice variants of CD44, R^{1 }and R^{2}, is represented by the following:
The value of 0 indicates the expected expression level will be zero if the target polynucleotide (e.g., J6-10) is not present in the sample. The uniform value of 1 indicates the expected expression levels of all other indicator polynucleotides will be equal if the expected RNA transcript is present in the sample. The indicator polynucleotides, however, may identify their target polynucleotides at varying expression levels because of variations in the melting temperature of the indicator polynucleotides.
In one embodiment, the optimization technique adjusts the detected expression levels to account for the variations in melting temperatures. Continuing with the CD44 gene example, the calculated melting temperatures are used to scale the expected expression levels in matrix M. If the expected expression levels varied linearly with respect to temperature, then the expected expression levels can be adjusted according to the following equation:
M_{i,j}=T_{i}/T_{d } (14)
where M_{i,j }element in the ith row and jth column of the matrix M, T_{i }is the expected melting temperature of the indicator polynucleotide corresponding to row i, and T_{d }is the desired melting temperature for use in the experiment. If the melting temperature is 62.03 and the desired melting temperature is 52, then the adjusted expression level would be 1.19 (i.e., 62.02/52).
Alternatively, the optimization technique performs the correction by adjusting the detected expression levels of matrix H, rather than adjusting the coefficients of matrix M. If the detected expression level varied linearly with respect to temperature, the detected expression levels can be adjusted according to the following equation:
H′_{i,j}=H_{i,j}T_{d}/T_{i } (15)
where H_{i,j }is the detected expression level for indicator polynucleotide i in experiment j, H′_{i,j }is the corrected expression level, T_{i }is the expected melting temperature of the indicator polynucleotide corresponding to row i, and T_{d }is the desired melting temperature. If the melting temperature is 62.03, the desired melting temperature is 52, and the detected expression level is 11,869, then the adjusted expression level is 9950 (i.e., 11,869*52/62.03).
Since expected expression levels will likely vary with respect to temperature according to an equation that is more complex than a linear equation, the optimization technique in one embodiment may apply a correction based on one or more empirical hybridization experiments. One such experiment creates a sample consisting of synthesized polynucleotides to bind to indicator polynucleotides on a nucleotide array. The optimization technique then calculates the expected expression levels of matrix M according to the following equation:
M_{i,j}=k H_{i } (16)
where M_{i,j }is the element in the ith row and jth column of the matrix M, H_{i }is an empirically derived expression level for the indicator polynucleotide i from a hybridization experiment, and k is a constant. For example, if the expression level of an indicator polynucleotide is 1481 and the average expression level from the experiment is 1208, then k is 1/1208 resulting in a corrected expected expression level of 1.23 (e.g., (1/1208)*1481).
The optimization technique can also correct for 3′ bias, which occurs in some methods of RNA amplification, such as T7 amplification. In one embodiment, the optimization technique calculates the corrected expression level according to the following equation:
M_{i,j}=f(H_{i,b}) (17)
where f is a function such as a linear or nonlinear equation and H_{i,b }is the expected expression level for an indicator polynucleotide i which identifies a target polynucleotide located b bases from the 3′ end of the RNA transcript in column j. For example, if the expression level for polynucleotides drops off linearly measured by the number of bases from the 3′ end of the RNA transcript, function f can be represented by the following equation:
f(H_{i,b})=k_{1}−k_{2}b (18)
where k_{1 }is the maximum value at the 3′ end of the polynucleotide and k_{2 }is the rate at which the expression level drops for each nucleotide base from the 3′ end. If k_{1 }is 1.00, k_{2 }is 1.2E-2, and the target polynucleotide is located 1,303 bases from the 3′ end of the RNA transcript, then the value of function f is represented by the following equation:
M_{i,j}=f(H_{i, 1303})=1.00−1.2E-4*1303=0.84364
Alternatively, the optimization technique can apply the 3′ bias correction to the empirically derived expression levels of matrix H, rather than matrix M, resulting in the following equation:
H′_{1,1}=H_{i,j}f(H_{i,b})^{−1}=11,869*(0.84364)^{−1}=14,069
The optimization technique can use more complex functions when the expression level varies nonlinearly with respect to the distance from the 3′ end. For example, the expression level might drop precipitously when b exceeds some value. The optimization technique can use a nonlinear equation to represent such a drop.
The optimization technique can apply multiple corrections to the same matrix as indicated by the following equation:
M′_{i,j}=M_{i,j}*C^{1}_{i,j}*C^{2}_{i,j}* . . . *C^{n}_{i,j } (19)
where M′ is the corrected matrix, M_{i,j }is a starting matrix element value, and the various C_{i,j }are element-by-element multiplicative corrections such as the 3′ bias correction described above. The optimization technique can represent the correction more generally by the following equation:
M′_{i,j}=f(M_{i,j}, C^{1}_{i,j}, C^{2}_{i,j}, . . . , C^{n}_{i,j}) (20)
where the C_{i,j}s are correction factors that may be applied using element-by-element corrections, such as multiplication, addition, or subtraction. The optimization technique can alternatively apply the multiple correction factors to matrix H, rather than matrix M according to the following equations:
H′_{i,j}=H_{i,j}*(C^{1}_{i,j})^{−1}*(C^{2}_{i,j})^{−1}* . . . *(C^{n}_{i,j})^{−1} (21)
H′_{i,j}=f(H_{i,j}, C^{1}_{i,j}, C^{2}_{i,j}, . . . , C^{n}_{i,j}) (22)
where f is a function that applies each individual correction to the empirically derived expression level.
Correcting for Presence of Homologous Polynucleotides
In one embodiment, the optimization technique corrects for the presence of homologous polynucleotides in a sample. In particular, an indicator polynucleotide identifies a target polynucleotide in an RNA transcript, but it may also identify, albeit less strongly, a homologue of the target polynucleotide in a different RNA transcript. The RNA transcripts may differ by only a few bases. The optimization technique can correct for a homologue in the sample by adding a column to matrix M for the homologue. The column might have a value only in row i if no other indicator polynucleotide in the experiment identifies a subsequence of the different RNA transcript. The solution can be found in various ways, such as using a system of linear equations or a least squares algorithm as described above.
Alternatively, the optimization technique may determine the contribution to the expression level of the indicator polynucleotide resulting from the homologue if the expression level of the homologue by itself is known. Such determination would not have to rely on a single indicator polynucleotide to identify the expression level of both the target polynucleotide and the homologue. The optimization technique can determine the expression level of the homologue by selecting one or more “control indicator polynucleotides” specific to the homologue. FIG. 3 is a flow diagram illustrating the overall process of correcting for hybridization of an indicator polynucleotide with a homologue of the target polynucleotide in one embodiment. In block 301, the technique selects the indicator polynucleotide. In block 302, the technique selects a control polynucleotide. In block 303, the technique performs the experiment using the indicator polynucleotide and a control indicator polynucleotide. In block 304, the technique retrieves the expression levels from the experiment. In block 305, the technique adjusts the expression levels based on the characteristics of the control indicator polynucleotide. The optimization techniques can identify control indicator polynucleotides by selecting a region of the 3′ untranslated of the homologue, confirming that it has no homologues by performing a similarity search (such as a BLAST search) against a database of known sequences, and designing control indicator polynucleotide to identify that region.
The optimization technique includes this control indicator polynucleotide in the experiment to correct the original indicator polynucleotide that may be weakly identifying the homologue instead of or in addition to the target polynucleotide.
The correction may be performed according to the following equation:
H(P_{i})=H(N_{i})+kH(N′_{i}) (23)
where H(P_{i}) is the expression level of indicator polynucleotide P_{i}, H(N_{i}) is the detected expression level of the target polynucleotide N_{i}, k is a constant, and H(N′_{i}) is the detected expression level of the homologous polynucleotide N′_{i}. The constant k accounts for differences between the target polynucleotide and its homologue. In such a case, the contribution of the homologous polynucleotide to the expression level measured for an indicator polynucleotide P_{i }will be less than its independent expression level as measured by the control indicator polynucleotide. For example, if it is expected that the homologue will bind at ⅔ the proportion of its actual expression level, the value of k would be ⅔. The value of k may be determined empirically by performing hybridizations with the homologous polynucleotide and the indicator polynucleotide whose expression level is the numerator of k and with the homologous polynucleotide and the control indicator polynucleotide whose expression level is the denominator of k. Alternatively it may be calculated theoretically using an equation such as the following equation:
H(P_{i}|N′_{i})=f(P_{i}, N′_{i}) (24)
where H(P_{i}|N′_{i}) is the empirically derived expression level of polynucleotide P_{i }when hybridized to a sample containing polynucleotide N′_{i}, and f(P_{i}, N′_{i}) is a function of the two polynucleotides that are binding. The function f may take into consideration various factors such as the number of matches, the individual melting temperatures of the polynucleotides assuming perfect matches, base stacking, salt concentration, GC content, the precise pairing of mismatched bases, number of hydrogen bonds, magnesium concentration, primer concentration, length of perfectly matching regions, melting temperature of perfectly matching regions, distance of mismatches from the ends of the polynucleotides, and so on.
In the following example, the optimization technique corrects for an indicator polynucleotide that identifies a target polynucleotide and its homologue. The control indicator polynucleotide is used to determine the independent expression level of the RNA containing the homologue. The optimization technique calculates the value of the constant k as 0.723. When the experiment is run with the indicator polynucleotide and the control indicator polynucleotide, the resulting expression levels are 36,366 and 15,695, respectively. The optimization technique calculates the independent expression level of the target polynucleotide as follows:
36366=H(N_{i})+0.723*15695
H(N_{i})=36366−0.723*15695=25018
Alternatively, the optimization technique can apply the homologue correction as a correction matrix to matrix H, rather than matrix M.
From the foregoing, it will be appreciated that specific embodiments of the invention have been described herein for purposes of illustration, but that various modifications may be made without deviating from the spirit and scope of the invention. For example, one skilled in the art will recognize the various ways in which the temperature balancing method and other indicator polynucleotide selection, optimization and correction techniques described here can be combined with existing methods. For example, the indicator polynucleotide selection techniques may be implemented or executed by means of a computer. Accordingly, the invention is not limited except as by the appended claims.