A Theoretical Understanding of 2 Base Color Codes and Its Application to Annotation, Error Detection, and Error Correction


 Brett Daniel
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1 Whte Paper SOLD System heoretcal Understandng of 2 Base olor odes and Its pplcaton to nnotaton, Error Detecton, and Error orrecton Methods for nnotatng 2 Base olor Encoded Reads n the SOLD System Henz Breu Introducton he SOLD System enables massvely parallel sequencng of clonally amplfed DN fragments lned to beads. hs unque sequencng methodology s based on sequental lgaton of dyelabeled olgonucleotde probes whereby each probe assays two base postons at a tme. he system uses four fluorescent dyes to encode for the sxteen possble twobase combnatons. hs unque approach employs a scheme that represents a fragment of DN as an ntal base followed by a sequence of overlappng dmers (adacent pars of bases). he system encodes each dmer wth one of four colors usng a degenerate codng scheme that satsfes a number of rules. sngle color n the read can represent any of four dmers, but the overlappng propertes of the dmers and the nature of the color code allow for errorcorrectng propertes. In ths document, we dscuss the theory that explans these errorcorrectng propertes, show how to correct the msapplcatons of these propertes, and descrbe software algorthms to utlze and verfy the 2 base encodng scheme. For example, we can dentfy and annotate solated erroneous color calls, as well as colorreads that correspond to solated blocs of adacent nucleotde varants from a reference, most realstcally one, two, or three, but as many as the applcaton mght requre. onstructng the 2 Base olor ode he SOLD System s 2 base color codng scheme s shown n Fgure 1. code dye FM y3 XR y5 Use the followng steps to encode a DN sequence * : 1. start at the 5' end, Fgure 1: SOLD System s 2 base odng Scheme. he column under code lsts the correspondng dye and the dbases (adacent nucleotdes) encoded by color. For example, s labeled wth y3 and coded as replace the dbase at ths poston wth ts correspondng code 3 from the table, 3. advance by one base, whch exposes the dbase, and 4. contnue, as shown below. Base Sequence: olor Strng: hs process encodes a mer of bases as a (1)mer of colors. lthough ths color strng codes for four dfferent mers, nowledge of the type and poston of any of ts bases helps to encode the sequence. For SOLD sequencng applcatons, prepend the leadng base to result n a mer (from the example above) from whch the base sequence can be reconstructed. * he example depcted here uses an as the frst base. In practce, the current chemstry on the SOLD System uses a as the frst base.
2 SOLD Substrate Dbase Probes 1 μm bead 5 P1 dapter emplate Sequence n n n z z z 5 n n n z z z 5 n n n z z z 5 EMPLE 2nd Base 1st Base n n n z z z 5 lass Slde leavage Ste 1. Prme and Lgate 5. Repeat steps 14 to Extend Sequence P OH PRIMER ROUND 1 + Lgase Lgaton cycle (n cycles) 1 μm bead Unversal seq prmer (n) P1 dapter emplate Sequence 2. Image Excte Fluorescence 6. Prmer Reset Unversal seq prmer (n1) 1 μm bead 2. Prmer reset 1. Melt off extended sequence 3. ap Unextended Strands Phosphatase PO 4 7. Repeat steps 15 wth new prmer 1 base shft PRIMER ROUND leave off Fluor leavage gent HO Unversal seq prmer (n1) 1 μm bead P 8. Repeat Reset wth, n2, n3, n4 prmers Read Poston Unversal seq prmer (n) Prmer Round Unversal seq prmer (n1) Unversal seq prmer (n2) Unversal seq prmer (n3) Brdge Probe Brdge Probe 5 Unversal seq prmer (n4) Brdge Probe Indcates postons of nterogaton Lgaton ycle Fgure 2. Lgaton based sequencng wth dbase probes usng the SOLD System. hs schematc shows bases nterrogated by the dbase probes at postons 1 and 2.
3 he SOLD System generates ts reads n precsely ths encoded form. One way of accomplshng ths s shown n Fgure 2. Requrements/Propertes for a 2 Base olor ode Scheme he 2 base color codng scheme possesses certan propertes that wll be dscussed later n the document. Interestngly, ths scheme s essentally the only code that satsfes these propertes. hs can be observed by treatng the propertes as requrements and constructng the color code from them. hs document deals only wth bases, not other IUB (Internatonal Unon of Bochemstry) codes. So let B = {,,, }. he color code should satsfy the followng requrements: For all bases b, d, e n B: 1. he avalable colors are 0, 1, 2, and 3: color (bd) {0, 1, 2, 3}. 2 wo dfferent dbases that have the same frst base get dfferent colors: color (bd) color (be) f d e. For example, color () color (). 3. dbase and ts reverse get the same color: color (bd) = color (db). For example, color () = color (). 4. Monodbases get the same color: color (bb) = color (dd). hat s, color () = color () = color () = color (). he followng are not requrements, but nterestng propertes that follow from these four. Property 5 follows from requrements 2 and 3, and wll mae our constructon easer. 5. wo dfferent dbases that nevertheless have the same second base get dfferent colors: color (bd) color (ed), f b e. For example, color () color (). Property 6 also follows from requrements 14, but t s most easly verfed aganst the completed code (Fgure 3, Panel E). 6. dbase and ts complement get the same color: color (b c d c ) = color (d c b c ). For example, color () = color (). Satsfyng the Requrements for a 2 Base odng System he remander of ths document uses a notaton dfferent from Fgure 1. Fgure 3 lsts the colors for each dbase. For example, the value n row and column wll be the color (2) for dbase (Fgure 3, Panel ). Requrements 1 and 2 requre that all colors are present n the frst row. Because the system can use any onetoone mappng between the actual dyes and the labels 0, 1, 2, and 3 (provded that requrements 1 and 2 reman satsfed) then the frst row (row ) can be Frst Base 2 Frst Base Frst Base Panel Panel B Panel Frst Base Frst Base Panel D Panel E Fgure 3. Requrements that ssgn the olor for a 2 Base ode.
4 labeled as shown (Fgure 3, Panel B.) Requrement 3, that color (bd) = color (db), gves a unque labelng for column (Fgure 3, Panel ). Requrement 4, that color (bb) = color (), gves a unque labelng for the dagonal (Fgure 3, Panel D). Fnally, requrements 1, 2, and 5, state that every color must appear n every row and every column exactly once (Fgure 3, Panel E). he table (Fgure 3, Panel E) s easy to memorze and wor wth because, by vrtue of Property 3, one can thn of dbases as twoelement sets for assgnng colors. he dbases startng wth get colors 0, 1, 2, and 3 respectvely. herefore: 0.,,, all get color 0, 1. and get color 1, and so must and, 2. and get color 2, and so must and, 3. and get color 3, and so must and. Determnng the olor Strng of the Sequence Wthout the leadng base, t s not possble to determne f a partcular DN sequence s rch from ts color strng alone. Here, we wll show that for any rch sequence S, there exsts another sequence S', wth exactly the same color strng. Mae the new sequence S' from S by replacng wth, wth, wth, and wth (.e., replace the base wth the other purne or pyrmdne). he result s an rch sequence that has the same color strng. Example: S = S' = hs s because, f a dbase bd has a color, then so does ts replacement b'd'. For example: 0:, 3:, 2:, 1:. If the dbase n S has color 0, then ts bases are equal and reman so n S'. If t has color 3, ts bases are complementary and reman so n S'. If t has color 2 and ts bases are both purnes, they wll both be pyrmdnes n S', and vce versa. Fnally, f t has color 1 then t s,,, or, and so replaced by,,, or respectvely, all of whch agan have color 1. olors as ransformatons of Bases Up untl ths pont, colors are assgned as a result of an encodng process, ether a specfc chemcal one, le the SOLD sequencng process, or a purely mathematcal one. Here, there s ust one functon, color: B B {0, 1, 2, 3}, that maps dbases to colors, e.g., color () = 3. For transformaton of bases, color can be represented n a dfferent way. Each color d s a functon f d : B B that transforms base u to base v. here can be any number of colors, but n ths case there are four. he transformatons are also specfed n Fgure 3, Panel E. o transform base u wth color d, loo up color d n row u and report the unque column (by Requrement 2) n whch t resdes. For example, color 3 transforms base to base, whch can be wrtten n dfferent ways: f 3 () =, 3 =, 3, or even 3. Swappng Rules he followng rules can be appled to verfy the observatons from Fgure 3: 1. olor 0 s the dentty functon. hat s f 0 (b) = b for every base b. 2. olor 1 swaps wth, and t swaps wth. For example, f 1 () =. 3. olor 2 swaps wth, and wth. 4. olor 3 swaps wth, and wth. Functon omposton on olors Strngs of colors can also be treated as transformatons by smply applyng one color transformaton after another. hs s how to decode a color read. For example, to decode apply color 3 to to get f 3 () =. hen apply the next color, 2, to ths result, to get f 2 () =. ontnung n ths way, decode all bases, ncludng the last: hat s, decodes to. o compose all color functons n an entre strng, the next step s to gnore ntermedate bases. Just as a sngle color transforms one base nto another, so does a strng of colors. he example above transforms the frst base nto the last base of the sequence. hs s true of the whole strng and also of substrngs. In the example above, we can thn of the substrng 102 as transformng nto : v h r v h r v v r h h h r v r r h v Panel Panel B Fgure 4. ddton able to Obtan the ode for Strngs of olors as ransformatons of Bases. () he orgnal Klen fourgroup addton table. hs addton table has been obtaned by the Klen fourgroup en.wpeda.org/w/klen_fourgroup, whch s the symmetry group of a rectangle. (B) he correspondng addton table for strngs of colors.
5 For nput color strng 102 acts ust le color 3, and ths also transforms to. lso, 102 acts ust le color 3, for all nputs,,, and. o understand ths concept, begn by composng ust two adacent colors as follows: 10 = f 0 o f 1 () = f 0 (f 1 ()) = f 0 () = = f 1 () In ths example, color strng 10 behaves ust le the sngle color 1, whch also maps to. Usng the swappng rules, color strng 10 behaves le color 1 for all nput bases. olor 1 swaps wth, and wth. olor 0 does not swap any of the bases. he rules for all parwse combnatons of colors, showng that each twocolor strng behaves le a partcular sngle color, have already been determned n the Klen fourgroup. he Klen fourgroup (http://en.wpeda.org/ w/klen_fourgroup) s the symmetry group of a rectangle, whch has four elements, the dentty, the vertcal reflecton, the horzontal reflecton, and a 180 degree rotaton, as shown n Fgure 4. he rectangle symmetry group has the addton table shown n Fgure 4, Panel. he symbol means followed by. For example, v h = r means that a vertcal reflecton followed by a horzontal reflecton s the same, as f t had rotated the rectangle by 180 degrees. he set of color operatons wth functonal composton that were dscussed above s somorphc to the Klen fourgroup. hs can be observed by labelng the corners of a rectangle wth the bases and wtnessng ther rearrangements (transformatons), as shown n Fgure 5. he dentty leaves the bases unchanged, exactly le color 0. he vertcal reflecton swaps wth, and wth, exactly le color 1. he horzontal reflecton swaps wth, and wth, exactly le color 2; and the 180 degree rotaton swaps wth, and wth, exactly le color 3. n addton table can be created for colors from the addton table for rectangle symmetres smply by substtutng, v, h, and r wth 0, 1, 2, and 3 respectvely, as n Fgure 4, Panel B. he symbol means followed by n ths context too, but t s usually more convenent to leave t out n the wrtten expressons. For example, nstead of , ust wrte , and = means the same as f2(f2(f0(f3(f2(f0(f1(f2(f3 ())))))))) =, the former beng somewhat easer to read, wrte, and parse. Propertes of roups group (http://en.wpeda.org/w/roup_theory) s a set of elements (colors, n our case) wth an operator that satsfes the followng four propertes: 1) losure: If a and b are elements, then a b s also an element. 2) ssocatve: (a b) c = a (b c) 3) Identty: here s an element such that a = a = a. 4) Inverse: For every element a, there s an element a1, such that a a1 = a1 a =. v In addton, the Klen fourgroup s also belan, whch s to say that t s 5) ommutatve: a b = b a. h r In ths group of color calls, 0 s the dentty element and every element s ts own nverse (the dagonal of Fgure 4, Panel B are all 0 s). It should be noted that except for the dentty color 0, any color composed wth any other color s the thrd color. hese propertes can be used when desgnng protocols, wrtng programs, and provng correctness. Fgure 5: Usng the Klen fourgroup to Obtan the ode for Strngs of olors as ransformatons of Bases. Rectangle symmetres, v, h, and r transform bases exactly le color operators 0, 1, 2, and 3 respectvely. pplcatons to SOLD Sequencng Events Usng the theoretcal methods dscussed above, here are a few examples on how to analyze typcal SOLD resequencng events. Each of the next few sectons analyzes common resequencng events by usng an example of a nucleotde sequence read, the correspondng reference, and ther colorstrng representatons. In general, a color strng that does NO satsfy these requrements s sad to be nvald.
6 Msmatched Leadng Base In ths applcaton, a gven color read and ts algnment wth a reference s shown below. If all colors match, but the leadng base does not, then LL decoded bases wll be msmatches. hs s observed n the example below he color code group propertes can be used to prove ths wll always be the case. Begn wth ust the algnment of color reads: For all postons, the reference base at s not equal to the read base at. In consderng an arbtrary poston, t has already been shown that the base n reference poston s b, where b s the sum of the reference colors n postons 1 through, usng the addton table n Fgure 4, Panel B. Smlarly the base n read poston s d, where d s the sum of read colors n postons 1 through. However, b must be equal to d because the read colors are the same as the reference colors. herefore, by consderng color code requrement 2, b b, whch shows all bases must be msmatches. n Isolated SngleBase Varant snglebase varant s a read base that dffers from ts algned reference base. he varant s sad to be solated f t s not adacent to a varant on ether sde. For example, the dot mars the solated sngle base varant n the algnment below Reference: Read: o analyze ths case, let s focus on the varant and ntroduce varables, as shown n the algnment below. u v Reference: B D F Read: B E F w x Here D E are msmatched bases, and they are solated because they are flaned by matched bases B and F. In order for colors u, v, w, and x to be consstent wth ths varant, they must satsfy the followng propertes: 1. u w because color (BD) color (BE) by ode Requrement v x follows from the frst two requrements. hs does not need to be tested snce the frst two are true. wo dacent Varants In the prevous applcaton, both relevant color postons were msmatches. he two examples n ths secton show that some colors n other nds of resequencng events, although otherwse constraned, may result n matches or msmatches. herefore, when annotatng color reads, t s not suffcent to annotate only colors that are msmatches. Here s an example where all relevant postons are color msmatches: Reference: Read: nalyss: u v w Reference: B D E F Read: B H F x y z Propertes: 1. u x by ode Requrement uv xy because they must transform B to dfferent bases E and H. 3. uvw = xyz because both uvw and xyz must transform B to F. 4. v and y could be equal or they could be dfferent. here are two addtonal nterestng propertes that follow from the propertes above, therefore, there s no reason to test for them when dentfyng twobase varants above. 5. vw yz because they must transform dfferent bases D and to F. 6. w z because they must transform dfferent bases E and H to F. Here s another example of two adacent varants resultng n three color postons n whch only the outer two are msmatches Reference: Read: Even though the center two color postons, both wth color 1, are not msmatches, the analyss above stll apples, and ths new example satsfes all of the above propertes. 2. uv = wx because both colors uv and wx transform base B to base F, so that uv = color (BF) = wx.
7 hree dacent Varants Here s an example of three adacent varants Reference: Read: hs example has two snglecolor msmatches solated by two color matches. In the analyss below, the B are bases and the c are colors: c 1 c 2 c 3 c 4 Reference: B 1 B 2 B 3 B 4 B 5 Read: B 1 B 6 B 7 B 8 B 5 c 5 c 6 c 7 c 8 Propertes: 1. c 1 c 5 2. c 1 c 2 c 5 c 6 3. c 1 c 2 c 3 c 5 c 6 c 7 4. c 1 c 2 c 3 c 4 = c 5 c 6 c 7 c 8 Inserton and Deletons (Indels) In these examples, a base deleted from the read s depcted as a n the read. Smlarly, a base nserted nto the read s depcted as a n the reference. It should be noted that the reference would never be stored wth a n those places, nor does the SOLD System ever call a for any of ts colors. he would be generated only by algnng the read wth the reference. For example, here s a snglebase deleton: Reference: Read: he analyss must explan two stuatons, whch t does: u v u v Reference: B D E B D E and Read: B  E B   w w  Propertes: 1. uv = w because both colors uv and w must transform base B to base E. sngle base deleton causes two elementary colors, u and v, to be replaced by one, w. In the above deleton, for example, 31 = 2, as s easly confrmed by the addton table. 2. Property 1 s a suffcent test, but t may be of nterest that the addton table mples that ether u, v, and w are all dfferent, or one s 0 and the other two are dentcal. Property 1 apples also for multple bases. For example, here s a three base nserton: Reference: Read: hs algnment shows the 3 n the nserton algnng wth a 3 n the reference, even though these two have nothng to do wth one another; the 3 n the reference encodes, whose bases bracet the nserton, and the nserton has a, whch s concdently also encoded wth color 3. nalyss: u Reference: B H Read: B D E F H v w x y he algnment above shows the reference color u algned wth read color v, but n fact mght algn wth color w, x, or y nstead. he followng property holds n all cases: u = vwxy. One elementary color, u, n the reference s replaced by four, c 1, c 2, c 3, and c 4, n the read. orrectng wo ommon Msunderstandngs cursory overvew of the SOLD 2 base color codng system may lead to some common msunderstandngs. he frst s that an solated color msmatch must always correspond to a sequencng error. he second s that certan (3/4 of the total) solated twocolor changes always correspond to errors. hese msunderstandngs do not actually appear n prnt, but there s publc nformaton that can be easly msread that way f taen out of context. In ths secton, two examples are presented that dscuss these common msunderstandngs. he frst nvolves an solated par of adacent base varants and addresses only the solated color change problem. he second example nvolves an solated cluster of three adacent base changes. he second example has both an solated color change and an mpossble two color change. ounter Example to an solated color change always corresponds to a sequencng error Reference: Read: hs counter example shows two snglecolor msmatches, both 1 2, solated from one another by a color match. It s easy to verfy that ths counter example does ndeed satsfy the requrements for a twobase varant: 1 2, 13 = 2 1 = 23, and 131 = 3 = 232.
8 ounter Example for mpossble two poston changes always correspond to measurement errors Reference: Read: he example shows a forbdden (.e., mpossble adacent msmatches when only a sngle base change s present) twocolor change, 12 31, solated by a color match (0 0) from a snglecolor change (3 2). Both mpossble color changes are nevertheless perfectly consstent wth a threebase varant, n partcular, the one shown above. gan, t s easy to verfy that the colors satsfy the requrements for a threebase varant: 1 3, 12 31, , and 1203 = clarfcaton of the msunderstandng follows easly by understandng that they are equvalent to assumng that the only base varants permtted n sequences are solated snglebase substtutons. nnotatng lgnment Msmatches n the SOLD System From the prevous dscusson, t s clear that when a SOLD read s algned to a reference sequence, only a subset of possble msmatches represent underlyng sequence varatons. It s therefore mportant to dentfy msmatches that are canddate sequence varatons and to dstngush them from other patterns that are the result of sequencng error. hs nformaton can then be used to correct errors from algned reads, resultng n a sgnfcant mprovement n the accuracy of the system, and to pass putatve sequence varaton to downstream algorthms for the assessment of genotype calls and dentfcaton of new alleles n the sample. hese annotatons are typcally conveyed n fles that descrbe the attrbutes of sequence reads, such as the SOLD FF fle. he SOLD FF s an mportant fle format for the SOLD System. In essence, a fle of ths type s a lst of reads, each wth several attrbutes. One of those attrbutes s the s, or annotatons attrbute. Here s the defnton of the annotatons attrbute of the SOLD FF v 0.2 fle: hs s a commaseparated strng representng annotatons on the sequence. he format s {char}{poston} where {char} s a character representng the type of annotaton (typcally a formattng request to vsualzaton software) and {poston} s the poston of ths annotaton n the read. he poston s 1based on the strng recorded n the g attrbute. hat s, the prepended base has poston 1, and the frst color has poston 2. For example, a5, g7, g8 means format the color call at poston 5 gray, format call 7 green, and format call 8 green. he SOLD System follows ths conventon: a (ray) s an solated msmatch; t s a msmatch and nether the colorcall on ts left nor the colorcall on ts rght s a msmatch, g (reen) s a vald adacent msmatch; t s a msmatch and ths msmatch, together wth the adacent msmatch on ts left or rght, could correspond to an solated SNP, y (Yellow) s a color call that s consstent wth an solated twobase change. In general, these wll be msmatches, but a conserved color between two msmatches s also a possblty. r (Red) s a color call that s consstent wth an solated threebase change. b (Blue) s an nvald adacent msmatch; t s any other msmatch. n effcent algorthm for calculatng these annotatons, gven an algnment of the color reads wth a correspondng color reference, can be derved from the theory descrbed prevously. o begn, the followng theorem correlates the applcatons n ths secton to SOLD sequencng events. heorem 1: Let c = c 1 c 2 c 3 c be a color substrng of a read algned wth the correspondng color reference r = r 1 r 2 r 3 r. hen c encodes an solated (1)base change f and only f the base poston precedng c s not a varant, and the followng two equatons hold under the olor ddton able (Fgure 4, Panel B): Equaton 1: c = r Equaton 2: c = r For all from 1 to 1 c r Proof: For the forward drecton, suppose that c encodes an solated 1 base change. r 1 r 2 r 1 r Reference: B 0 B 1 B 2 B 1 B Read: B 0 D 1 D 2 D 1 B c 1 c 2 c 1 c hen the base B 0 precedng c s not a varant because t must serve to solate the 1 base change. Smlarly, the base B followng c s also not a varant because t too must serve to solate the base change. Equaton c =color 1 and ( B02 Bare ) derved by usng the color addton propertes. c =color ( B0B ) and r =color ( B0B ) Equaton 2 stays the same, because all bases between B 0 to B r are =color varants. ( B0B )
9 o prove the reverse drecton, suppose that the base B 0 precedng color c s not a varant, as shown below, and that Equatons 1 and 2 hold. r 1 r 2 r 1 r Reference: B 0 B 1 B 2 B 1 B Read: B 0 D 1 D 2 D 1 D c 1 c 2 c 1 c hen Equaton 1 mples that D = B, as follows: D = B0 c = B0 r = B Bases D 1 through D 1 are therefore solated by matched bases B 0 and B. Furthermore, all solated bases are varants D = B 0 c c = Br 0 = Br = B that are 1D B for 1 all between 1 and 1, whch follows wth the help of Equaton 2: D = c r = B he equaton above completes ths proof. If a substrng of colors s consstent wth a base change, assume the leadng base s not a varant and apply heorem 1. o annotate color calls, note that a color call s consstent wth a base change f t s part of a color strng that s consstent wth a base change. heorem 2: color call that s consstent wth a base change s not also consstent wth an mbase change for m. Proof: ssume the opposte, so that there s a color call c that s part of a color strng c that s consstent wth a base change, and also part of a color strng d that s consstent wth an mbase change. here are two substrngs of bases, of dfferng lengths +2 and m+2, where the outer two bases are matches and the nner and m form the solated varants. ssume wthout loss of generalty that the base sequence overlaps the mbase sequence, at color c, on the left. he extreme case s setched below, n whch a matched base s a 1 and a msmatched base s an x. Recall that color calls appear only between bases. By heorem 1, poston s compatble wth m solated varants at lne brea. By heorem 2, there s no need to contnue searchng, because wll not be compatble wth any other number. Note the effcency of ths algorthm: f s compatble wth m varants, the algorthm maes the least number of tests possble before breang from the loop. On the other hand, all the tests that t does mae are requred by heorem 1 because f t sps any test, an adversary could devse an nput on whch the algorthm would gve the wrong answer. See ppendx 2 for an mplementaton n Java. Fgure 6 shows a small porton of a vsualzaton of a FF fle annotated wth ths method. lgorthm: getompatablty Input: Read colors, Reference colors R, Query poston, Maxmum acceptable varants M Output: Number of compatble solated base varants m. Method: //summ s the group theoretc sum of m read colors c startng at =. //sumrm s the group theoretc sum of m read colors r startng at =. Intalze: m = summ = sumdm = 0; n = sze(c) whle m M, do { summ = addtonable[summ][c]; sumrm = addtonable[sumrm][d]; f (summ == sumrm) brea; // Substrng s = c c +m s compatble wth an solated group of m adacent varants. m = m + 1 } // for m // If the loop termnates wth m = M+1, then s not compatble wth M or less solated varants. f (m > M) return Incompatble else return m 1xxx1 1xxxx1 base msmatches on the left mbase msmatches he frst base of the mbase sequence, a match, concdes wth a msmatch n the base sequence, whch contradcts the fact that a base cannot smultaneously match and msmatch the reference. hs contradcton proves the theorem. he lgorthm he pseudocode functon getompatblty, shown here, returns m f the color call at poston s compatble wth an solated group of m msmatched bases (varants), where m s between 0 and a preset maxmum acceptable number M. For readablty, ths pseudocode gnores read edge effects.
10 oncluson he exstng 2 base color code for SOLD sequencng follows from a small set of requrements. hs color code s essentally the only one that satsfes these requrements. If colors are treated as transformers of bases, these requrements lead to a formulaton of color combnatons as the wellnown Klen fourgroup. he group theoretc propertes lead to effcent and systematc methods for annotatng color reads from the SOLD System. In partcular, they allow for a smple Java mplementaton that annotates colorcalls wth any gven number of DN varants. Frst Base Fgure 7. permuted color addton table. ppendx 1: lternatve 2 Base olor odng Schemes hs appendx explores alternatve color codng schemes that exst, and ther effects on subsequent group propertes. Mantanng the Requrements here s essentally only one codng scheme that satsfes all the requrements. Once the frst row has been decded, the requrements determne the rest of the table, as shown n Fgure 3. Each of the 4! = 24 ways to lay down the frst row (gven requrements 1 and 2) corresponds to a dfferent table. But these are all somorphc to one another. Even though there can only be one table, the colors can be permuted to obtan 23 other tables. For example, the permutaton: Fgure 6. Vsualzaton of an nnotated FF fle Usng the getompatblty lgorthm. References 1. osta, na, et al. pplcatons of Next eneraton Sequencng Usng Stepwse ycledlgaton, Product/Sold_Knowledge/PDF/SH_2007_2791.pdf generates the table n Fgure Pecham, Heather E. et al. SOLD Sequencng and 2 Base Encodng, Product/Sold_Knowledge/PDF/SH_2007_2624.pdf 3. Rhodes M. SOLD System Data 2 Base Encodng, Webnar, 2be/ /
11 Relaxng the onstrants Dfferent codng schemes can be generated by relaxng the constrants. Unfortunately, these alternatve schemes are not guaranteed to satsfy the group theoretc propertes that are needed for annotaton, error detecton, and correcton. Here are four of the requrements: For all bases b, d, e n B: 1. he avalable colors are 0, 1, 2, and 3: color (bd ) {0,1,2,3}. 2. wo dfferent dbases that nevertheless have the same frst base get dfferent colors: color (bd ) color (be) f d e. For example, color () color (). 3. dbase and ts reverse get the same color: color (bd ) = color (db) For example, color() = color(). 4. Monodbases get the same color: color (bb) = color (dd) hat s, color () = color () = color () = color (). prvate nt getompatablty( Strng readolors, Strng refolors, nt ) { //summ s the group theoretc sum of ref color c from = to +m. nt m = 0; // Number of color msmatches. nt summ = 0; nt sumdm = 0; nt n = Math.mn(readolors.length(), refolors.length()); nt numremanng = n   1; // he number of postons past. // maxm s the largest bloc sze m to chec. nt maxm = (MX_DJEN_VRINS < numremanng)? MX_DJEN_VRINS : numremanng; for (m=0; m<=maxm; m++) { nt c = haracter.getnumercvalue(refolors.chart(+m)); nt d = haracter.getnumercvalue(readolors.chart(+m)); // dd n the new colors. // ssume mssng data f ether s out of range. f (0 <= c && c <=3 && 0 <= d && d <= 3) // n range. { summ = addolors[summ][c]; sumdm = addolors[sumdm][d]; f (summ == sumdm) brea; // ompatble wth solated group of m adacent varants. } else // out of range. ermnate ths test as f we ran over maxn. { m = maxm + 1; } } // for m Frst Base Frst Base // Return m f t s n range. Otherwse, 0 means ncompatble and solated, // and 1 means ncompatble but not solated. f (m < 1 m > maxm) { m = sisolated(, readolors, refolors)? 0 : 1; } return m; } // getompatablty() Panel Panel B Fgure 8. Droppng olor ode Requrements Wll Not Satsfy ll roup Propertes. () color code that satsfes only requrements 1, 2, and 3, now does not allow 01 to behave le a sngle color. (B) color code that satsfes only requrements 1, 2, and 4, now does not allow 23 to behave le a sngle color. an requrement 4 be dropped? One such table s shown n Fgure 8, Panel, but t no longer satsfes the group propertes. For example 01 does not behave le a sngle color anymore: 01 = and 1 =, but 01 = and 2 =. So color 0 1 behaves le color 1 for, but t behaves le color 2 for. an we drop ust requrement 3? Such a table s shown n Fgure 8, Panel B, but t no longer satsfes the group propertes. For example 23 does not behave le a sngle color anymore: 23 = and 1 =, but 23 = and 0 =. So color 2 3 behaves le color 1 for, but t behaves le color 0 for. ppendx 2: Java Implementaton of getompatblty he Java functon below, getompatblty, mplements the pseudocode n the Secton he lgorthm.
12 For Research Use Only. Not for use n dagnostc procedures. Notce to Purchaser: Lcense Dsclamer ppled Bosystems. ll rghts reserved. ll other trademars are the property of ther respectve owners. pplera, ppled Bosystems, and B (Desgn) are regstered trademars and SOLD s a trademar of pplera orporaton or ts subsdares n the U.S. and/or certan other countres. Prnted n the US. 07/2010 Publcaton 139WP0102 O13982 Headquarters 850 Lncoln entre Drve Foster ty, US Phone oll Free Internatonal Sales For our offce locatons please call the dvson headquarters or refer to our Web ste at
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