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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 dye-labeled olgonucleotde probes whereby each probe assays two base postons at a tme. he system uses four fluorescent dyes to encode for the sxteen possble two-base 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 error-correctng 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 color-reads 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 d-bases (adacent nucleotdes) encoded by color. For example, s labeled wth y3 and coded as replace the d-base at ths poston wth ts correspondng code 3 from the table, 3. advance by one base, whch exposes the d-base, 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 D-base 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 1-4 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 (n-1) 1 μm bead 2. Prmer reset 1. Melt off extended sequence 3. ap Unextended Strands Phosphatase PO 4 7. Repeat steps 1-5 wth new prmer 1 base shft PRIMER ROUND leave off Fluor leavage gent HO Unversal seq prmer (n-1) 1 μm bead P 8. Repeat Reset wth, n-2, n-3, n-4 prmers Read Poston Unversal seq prmer (n) Prmer Round Unversal seq prmer (n-1) Unversal seq prmer (n-2) Unversal seq prmer (n-3) Brdge Probe Brdge Probe 5 Unversal seq prmer (n-4) Brdge Probe Indcates postons of nterogaton Lgaton ycle Fgure 2. Lgaton based sequencng wth d-base probes usng the SOLD System. hs schematc shows bases nterrogated by the d-base 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 d-bases that have the same frst base get dfferent colors: color (bd) color (be) f d e. For example, color () color (). 3. d-base 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 d-bases 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 1-4, but t s most easly verfed aganst the completed code (Fgure 3, Panel E). 6. d-base 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 d-base. For example, the value n row and column wll be the color (2) for d-base (Fgure 3, Panel ). Requrements 1 and 2 requre that all colors are present n the frst row. Because the system can use any one-to-one 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 d-bases as two-element sets for assgnng colors. he d-bases 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 d-base bd has a color, then so does ts replacement b'd'. For example: 0:, 3:, 2:, 1:. If the d-base 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 d-bases 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 four-group addton table. hs addton table has been obtaned by the Klen four-group en.wpeda.org/w/klen_four-group, 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 two-color strng behaves le a partcular sngle color, have already been determned n the Klen four-group. he Klen four-group (http://en.wpeda.org/ w/klen_four-group) 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 a-1, such that a a-1 = a-1 a =. v In addton, the Klen four-group 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 four-group 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 Sngle-Base Varant sngle-base 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 two-base 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 sngle-color 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 sngle-base 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 two-color 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 sngle-color 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 two-base 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 sngle-color change (3 2). Both mpossble color changes are nevertheless perfectly consstent wth a three-base varant, n partcular, the one shown above. gan, t s easy to verfy that the colors satsfy the requrements for a three-base 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 comma-separated 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 1-based 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 color-call on ts left nor the color-call 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 two-base 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 three-base 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 m-base 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 m-base 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 m-base 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 m-base msmatches he frst base of the m-base 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 pseudo-code 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 pseudo-code 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 well-nown Klen four-group. 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 color-calls 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 ycled-lgaton, 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 d-bases that nevertheless have the same frst base get dfferent colors: color (bd ) color (be) f d e. For example, color () color (). 3. d-base 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 pseudo-code 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 139WP01-02 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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Purpose & Goals Volunteers in the Sheriff's Office offer their talents, skills, abilities and dedication to provide services to their community. Our Volunteers epitomise our Community Orientated Policing philosophy. They are involved in many aspects of our day to day operations, and continue to be an invaluable resource to the department and our community.
For information about becoming a volunteer call the Sheriff's Office at (530) 265-1471

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MCB YLD - Making the Holidays a Little Brighter The Salvation Army Angel Tree Program
The MCB YLD is working with The Salvation Army Angel Tree Program to provide gifts to children in our community. This is an excellent opportunity to make the holidays brighter for children who come from low-income families and/or live in foster care or group homes.
If you or your firm are interested in buying gifts for children in our community, please e-mail Stacey Vandiford at firstname.lastname@example.com by Friday, December 6 with the following information: 1) How many "Angels" you would like to "adopt" 2) A contact name 3) The e-mail address to deliver your "Angel" wish list(s)You will be provided an "Angel" tag that indicates the child's clothing size and his/her wish list -- making shopping a breeze! The Salvation Army encourages sponsors to buy appropriate gifts at prices they feel comfortable spending. As a rule of thumb, gifts typically average $50 to $150 per "angel". If you plan to spend more than this amount, please consider adopting more than one "angel" so that we can brighten the holidays for as many children as possible.
Gifts should be dropped off at one of the Salvation Army drop-off locations listed here by Monday, December 9. If you have any questions, please contact Stacey Vandiford at firstname.lastname@example.com or 704/331-3656 or Debra Lofano at 704/444-1094.
Please consider participating in this very worthy cause, which is sure to get your holidays off to a wonderful start!
In conjunction with the YLD's Angel Tree Project to benefit the Salvation Army, the YLD Legal Assistance Committee will be holding a small legal clinic at the Salvation Army's Center for Hope on December 16, 2013 from 7 p.m. - 8:30 p.m. The Center for Hope provides shelter to women and children in our community. Volunteers will be made available at the Center for Hope to answer general legal questions and provide guidance to the shelter's residents. The Committee would like to have a large number of volunteers to ensure all residents will be afforded ample time to address their concerns. If you would like to volunteer, please contact Jeremy Sugg at firstname.lastname@example.com.


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PBLS - Amsterdam
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Navigate the conference and plan your daily agenda straight from your mobile device. This application includes an agenda with descriptions of each presentation, maps of the venue, a personal schedule and much more information about The Premier Business Leadership Series in Amsterdam.
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Imonni the ahn pant is the most comfortable pant you will wear this season, featuring an elasticated waist and drop crotch. The loose fit pull on pant is constructed of amazingly soft fabric making this pant a must have for relaxing. Team with slides and a tee for a casual look. Made of 60% cupro / 40% viscose sandwash taffeta.


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Cape Tech students at NJ SkillsUSA
On April 24, 45 students from Cape May County Technical High School attended the 2013 NJ SkillsUSA Championship held at the Garden State Exhibit Center in Somerset County and the Somerset County Technical School in Bridgewater. More than 1,600 students competed in more than 90 real-world skill and leadership contests.
Students include: Audio/ Radio Production, first place Austin DiCola and Patrick Haigh; Advertising Design, second place Matt Powers; Career Pathways Showcase: Law and Public Safety, first place Farah Almadani, Joslyn Nelson, Chris Russ; Crime Scene Investigation, third place Jeanette Champion, Harry Wozunk, Brian DeVico; Extemporaneous Speaking, second place Mickayla Ritzler; Quiz Bowl, second place Mike Hawthorne, Josh Greenberg, Liam Munz, Sean Popovick, Connor Kelly and alternates Hannah Dietrich, Stephanie Miller; SkillsUSA Safety Display, second place Christina Gibboney; Wedding Cake Decorating, second place Elizabeth Farmer, Laura McKeon; Esthetics second place post-secondary division Elizabeth Hayoun, Model Juliette King. Nancy Fissel was the adviser.
SkillsUSA provides education experiences for students in leadership, teamwork, citizenship and character development. See NJ-SkillsUSA.org and capemaytech.com.
Tyndall inducted into honor society
Ryan W. Tyndall, of Cape May Court House, was one of 24 New Jersey residents and undergraduate students in the Rutgers-Camden School of Business to be inducted recently into the Athenaeum Honor Society.
Athenaeum is the undergraduate honor society at the Camden Campus of Rutgers University. Membership is limited to juniors maintaining a cumulative grade-point average of no less than 3.7 (out of 4.0) and to seniors maintaining a cumulative average of no less than 3.5.


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View a photo collection of the best cars on a desktop screensaver.Free to try
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Former RSL defender Robbie Russell retiring from soccer to pursue medical career
05/16/2013 10:29amD.C. United announced Wednesday that former RSL defender Robbie Russell is retiring from the game of soccer to attend Georgetown University and pursue a medical career.
Real Salt Lake GK Nick Rimando Called Up to U.S. National Team for World Cup Qualifiers
05/16/2013 3:04pmNick Rimando will play for the U.S. Men's National Team starting in June.


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The event is free and open to all who want to learn more about screening, detection, and treatment options for this second leading cause of cancer deaths in men.
When detected early, prostate cancer is curable. But to detect prostate cancer early, men have to get screened. And too few of them do.
"It's a shame, because this is one of the few cancers that we have a marker for, and the test is very simple," said Peter Doherty, a prostate cancer survivor and senior advisor to the hospital's Prostate Cancer Support Group.
"But men do not practice enough preventive health. The prostate is something they don't want to talk about. They don't even want to know about it."
Prostate cancer often strikes without any warning symptoms.
The National Prostate Cancer Coalition estimates that almost 200,000 men will be diagnosed this year, and there will be almost 32,000 deaths from prostate cancer.
Regular screening is recommended for all men over age 50, and even earlier for men who have a family history of prostate cancer.
African-American men, who get prostate cancer at a rate 60 percent higher than caucasian men, should also seek out early screening, starting at age 40.
According to Doherty, who will speak at the Healthy Prostate 2001 program, many people assume that prostate cancer is an old man's disease.
In fact, it can strike men in the prime of their lives. About one-third of the diagnoses occur in men under age 65. The early death of a father or uncle is a strong warning to have a screening test.
Also featured in the program are Dr. David Taylor and Dr. Arthur Israel, both attending urologists at Morristown Memorial Hospital.
"We can detect prostate cancer at very, very early stages," said Dr. Taylor, "And we have very high success rates for cure at early stages."
As they age, most men experience voiding symptoms that are associated with prostate cancer, like restricted urine flow, and frequent urination that interrupts sleep. Since prostate cancer is the most common cancer, it makes sense to get screened as soon as the symptoms appear, if not before.
Dr. Taylor and Dr. Israel will also discuss the symptoms and treatment of other non-cancerous conditions, like infection and enlargement of the prostate.
"Healthy Prostate 2001" will also feature Mary-Giselle Ulbrich, dietitian for Oncology Outpatient Services at the hospital, who will outline the elements of a healthy diet. Avoiding fatty foods has been linked to a lower risk of prostate cancer.
A question and answer session will follow the speakers, and free literature will be available to take home.
"Healthy Prostate 2001" is a program of the Carol G. Simon Cancer Center, an Atlantic Cancer Center at Morristown Memorial Hospital, and will be held in the Forbes Amphitheater.
To register and get directions, call (800) 247- 9580 or register on-line and get more information about the Peter B. Doherty Men's Cancer Program at www .AtlanticHealth.org.


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Using the "offers from" options you can filter merchants from regions which are most common in our database: the UK, Germany, Austria and Poland.
If a checkbox near a flag is selected, offers from merchants from that particular region will be displayed, otherwise they will not be visible.
If you select offers from "all regions", all offers from all merchants will be displayed. These incude offers from merchants in countries not listed here.
Type: Board game Genre: Action Players: 2 to 4 Playing time: 20 minutes Age: from 6 years Author: Inka fire, Markus fire Publisher: KOSMOS ...


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* Prices may differ from that shown
Having a large family I was very lucky that when I found out I was expecting my first child I had plenty of people to buy alot of the things I needed. My sister kindly bought me this morse basket and at fittest I loved it. I kept it covered away until the day we bought our son home from the hospital. Immediately I noticed there was a considerable gap between the mattress and cot ( this was the mattress it came with!! ) this had me very worried as I has read about heads getting stuck in holes. I used sheets to fill the holes and although my son did not move an inch while sleeping I found it hard to sleep when he was in it. I read online and many people were saying the gap was normal but it still worried me alot. The design of the Moses basket is very cute and nice to look at I however could not sleep at night with my son in their so he ended up sharing a bed with me. For me it was good during the day when I could watch him in it but at night a big no no ( maybe I'm too much of a worrier ) if you are like me I would suggest at skipping the Moses basket and put your babie straight in a cot. There are no gaps I between the mattress and cot and, to me anyway, seem alot safer than a Moses basket.
When I found I was expecting my first child, I was very lucky to be given an almost new Mamas & Papas Moses Basket in the Elfie and Mop style, by my cousin who bought it but had a baby who took to his cot so well they barely used it. As such, it was brand new when we got it and we really appreciated being given it as the 'start up' costs of a baby are pretty high!
If you wish to buy a Mamas & Papas Moses Basket however you can do so for £55 which includes the soft foam mattress but not the stand, which needs to be purchased seperately. The stand is white, if you want one in a natural colour to match the wicker, I recommend the Toys R Us moses stand.
The Mamas & Papas Moses Basket is woven from a light wicker and is very sturdy, with two woven handles midway down each side, and a pull up hood. The soft foam mattress fits a standard size moses cover well and all standard size moses blankets, though of course you should never use a pillow in any moses basket or place anything into it to prop baby up or lay them into, as they may roll over and suffocate.
The Mamas & Papas Moses Basket feels well made initially and we did get plenty of use out of it - our little girl was sleeping in it next to our bed up to six months old as she was breast fed and it made for easier night time feeds. However, when she came out of it at six months and went to her cot we noticed that just under the handles were ripped, which means we have had to throw it away because I am too scared to use it with another baby. I know that we were given it second hand but it had been used less than five times and was so new it still had the tags underneath it - surely such damage isn't good for a moses basket.
I've since asked a few friends who have Mamas & Papas Moses Baskets and all of them agree that the handles are pretty much for decoration only. I didn't know this and I'm sure they are intended to be used - and I only used the handles myself to lift my very light baby from the lounge to her nursery - literally, five or six steps as we only lived in a flat at the time. I am a bit dubious now of replacing it with another Mamas & Papas Moses Basket and would probably consider buying our next moses from another retailer, so really I can't recommend it for the money as you can get cheaper moses baskets which I've read good reviews on. The Elfie and Mop brand is lovely, and it would be nice to have a fully matching set (we had the cot blankets, curtains and rug), but I don't think I'd risk re-using this moses basket just to have that matching style, though I could use the bedding minus the hood in another moses if I wanted to, I suppose.
Overall then, appreciating that we didn't buy this from new I will give the Mamas & Papas Moses Basket a generous 3 stars out of 5 (a low 3 at that) because though we did get good use from it and it looks nice and is very comfortable for baby the ripping under the handles that myself and members of my family and friends have experienced is a worrying possible design issue which would put me off recommending it or re buying for any future children.
The "Elfie and Mop" Moses Basket from Mamas and Papas was recently featured in an online sale available at half price costing £49.99 instead of £99.99 and having been previously impressed with the quality of M&P's items I thought this made it a great buy and snapped one up a couple of weeks ago. It was available with free next day delivery and came packaged in a large cardboard box and other than a protective plastic sleeve that covered the basket it was good to go from the moment it was unboxed.
It's a large Moses basket measuring 85cm in length, 45cm in width and 26cm in depth and comes complete with basket hood, foam mattress and bedding. A 2.5 tog quilt is included for the money and the basket is constructed from Maize and is suitable for babies weighing up to 9Kgs. The characters of Elfie and Mop (an Elephant and Mouse) feature on the white fabric dressing and are subtly detailed in embroidery. The hood of the basket is of sufficient rigidity that makes it fully adjustable and you can position it either as a full or part cover that should make your little one feel safe and secure. It's a very well made Moses Basket with a tightly woven construction and the two carry handles feel strong and sturdy making it easy to move around and place in position on a stand. It's designed to fit on any Mamas and Papas stand although I have mine on a Mothercare one and it sits in place without any problems or overlaps on mine and overall I have been very impressed with my purchase and my young son settles and sleeps well in it.
I'd say it was a unisex item, the quilt and hood cover are predominately white with green and orange detailing and in my opinion it always looks fresh and clean. The fabric used in 100% cotton which gives it a cool and smooth feel and its easy to wash and dry. The supplied mattress is foam constructed which feels soft has a slight bounce to it, it's not terribly thick but I didn't expect it to be given it's a Moses basket mattress and generally they are quite thin, you will need a mattress cover though as one isn't supplied but they are readily and cheaply available so finding one shouldn't be a problem.
My Opinion
This is my son's 'other' Moses basket which I bought for my home for him, he already has one at his mummy's house and I got this for him for when he stays at my place rather than having to carry his other one backwards and forwards all the time. Compared to the one he has (a Mothercare basket from a different character range) both me and my partner have commented that this one from Mamas and Papas is a lot nicer and better made, the hood on this one stays in place whereas his other one slips out of position and I think the quality of the fabrics used is a lot better too.
For something that should have cost £100 I'd expect quality though and this basket lived up to my expectations from the moment my son was laid in it, he looks comfortable when he's asleep and seems to settle well on a night time. It looks expensive I have to admit, the fabrics used are high quality and whilst it's only going to be used for a relatively short time I am happy to let my son sleep in this basket until he becomes to big for it.
Being brand new parents we have made some expensive mistakes with some of the items we have bought for our son but I'm happy to say that this Moses basket hasn't been one of them, for a shade under £50.00 as it's priced at now I would say it's a steal and would definitely recommend this one to other parents-to-be. Suitable for either a boy or a girl and beautifully made with some fine detailing the Mamas and Papas "Elfie and Mop" Moses basket gets a perfect five star rating from me. Still available online on the M&P website I'd recommend buying one before they either sell out or discontinue the range, thanks for reading my review.


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- Description
Why wear synthetic when you can wear the real thing? Made from Icebreaker's all-natural Realfleece brushed merino fabric, the Camper Hoody will keep kids warm and comfortable during even the coldest cold snap—and protects them with a UPF rating of 30.
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Hiya, how much should I charge for renting my 2 double bedroom flat in W9/ Maida Hill?
I've been living in one room & renting the other but thinking of moving out & renting whole place to couple/2 singles. Good condition (redecorated 4.5 years ago). Period raised ground floor flat, very tall ceilings, 1 bathroom, large sitting room, 550 sq ft approx.
Asked on Mar 12 2012, Renting in London | Report content
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It would be best to get a couple of letting agents to value. You will be able to get a good idea of what you might achieve for your property by doing a Zoopla search for your area.Answered on Mar 13 2012, Report content
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Originally published Monday, February 9, 2009 at 6:37 PM
Players won't be suspended for leaving bench
No Garfield or Redmond player who left the bench Friday, after a Redmond fan ran out of the stands and punched a Garfield player in the face, will face a suspension for leaving the bench. But both high schools could be handing down their own sanctions to players for their behavior during the incident, the KingCo Conference determined.
Seattle Times staff reporter
Scores & stats
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No Garfield or Redmond player who left the bench Friday, after a Redmond fan ran out of the stands and punched a Garfield player in the face, will face a suspension for leaving the bench. But both high schools could be handing down their own sanctions to players for their behavior during the incident.
In a league meeting Monday, the KingCo Conference overturned the ejections of Garfield and Redmond players who left the bench Friday, when Redmond senior Sean Shewey punched Garfield guard DeAndre Taylor in the face during a break in play. The ejections would have carried a mandatory one-game suspension.
But with witness accounts circulating of players fighting after Shewey's punch, — both Garfield and Redmond are investigating the incident.
"Any additional sanctions that come out of this are going to come from the two schools involved, pending league approval," KingCo president Jeff Lowell said.
Those sanctions will likely be decided Tuesday before each team plays their second-to-last game of the regular season. Garfield hosts Eastlake in the Bulldogs' final home game, and Redmond plays at Skyline.
Neither athletic director from Garfield or Redmond could speak on the record Monday evening. The Lake Washington School District is also investigating the incident, according to a statement.
A high-school rule forbids players from leaving the bench during a fight, but after seeking advice from the National Federation of State High School Associations, the conference decided that a fan attacking a player did not constitute a fight.
The conference was able to view a video clip of the incident, which has since been passed on to the Redmond police. The police have arrested Shewey, 18, on suspicion of assault and disorderly conduct.
The league also finalized Redmond's decision to have Garfield named the winner. The score when the game was called with about five minutes left — Garfield 50, Redmond 33 — will be the final score.
Tom Wyrwich: 206-515-5653 or email@example.com
Copyright © 2009 The Seattle Times Company
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A fact without the context can mean anything. When you are trying to reach any sort of "truth", you are actually looking for the context. Every issue, whether it's going to be economic, political, social or personal can only be understood by thinking of it as part of a bigger picture.
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