Lecture 9 February 21

Transcription

1 Math 239: Dscrete Mathematcs for the Lfe Sceces Sprg 2008 Lecture 9 February 21 Lecturer: Lor Pachter Scrbe/ Edtor: Sudeep Juvekar/ Alle Che 9.1 What s a Algmet? I ths lecture, we wll defe dfferet types of algmets ad expla some of ther propertes. We beg wth a sutable alphabet Σ. Throughout the dscusso, we wll choose Σ = {A, C, G, T }, the four ucleotdes, although ote that the deftos hold for all fte alphabets. Examples of other alphabets clude Σ = {A, C, G, T, N}, where N deotes a ukow base. The alphabet Σ s wdely used computatoal bology lterature. Notato. Cosder k sequeces over Σ, σ 1, σ 2, σ k. The legth of the th sequece s deoted as σ =. The sequece legths may be dfferet. We use σj to deote j th character of th sequece. We also defe the postos of the sequeces, depedet of the characters occupyg them. Thus, S σ = {(, j j {1, 2,, }} deotes the set of postos the sequece σ. Therefore, S σ1 σ k = k deotes the uo of postos of all sequeces σ 1, σ 2,, σ k. Sequece algmet s complcated by the fact that DNA has a double helcal structure, whch the two strads are reverse complemets of each other as show the Fgure 9.1. Further, the two strads of the helx have opposte drectoalty, deoted by (5 3. Ths drectoalty ad reverse complemetarty must be corporated may computatoal bology applcatos, cludg sequece algmet. =1 S σ Fgure 9.1. Reverse complemetarty of DNA strads. 9-1

2 Math 239 Lecture 9 February 21 Sprg 2008 Throughout ths dscusso, we use the followg coveto to corporate the reverse complemetarty of DNA: gve ay k DNA sequeces, we add k more sequeces to produce 2k sequeces, σ 1, σ 2,, σ 2k, where the odd sequeces {σ 2 1 {1, 2,, }} deote the orgal set of sequeces, ad the eve sequeces σ 2 are the reverse complemets of σ 2 1. Defto 9.1. Gve a set of sequeces σ 1, σ 2,, σ k, a homology forest s a lamar famly for S σ1 σ k. 9.2 Local Algmet We ow defe a algmet of sequeces. Defto 9.2. A algmet of sequeces σ 1, σ 2,, σ k s a equvalece relato S S σ1 σ k σ1 σk. We say a character σj s alged to the character σl k f σj σl k. There s a atural algmet assocated wth a homology forest, amely the equvalet relato gve by the compoets. Thus, two characters are -related f they belog to same compoet (tree of the homology forest. More formally, (, j s -related to (k, l ff there exst edges e 1, e 2,, e k ad vertces Z 1, Z 2,, Z k+1 such that, (, j = Z 1, (k, l = Z k+1 ad {Z r, Z r+1 } e r for every r {1, 2,, k}. It s easy to see that defes a equvalece relato. Fgure 9.2. Algmet of four sequeces. Fgure 9.2 shows the equvalet classes formed by homology forests o a set of four sequeces. Equvalet postos of the sequeces are coected by a le. Note that by ths defto, two postos the same sequeces ca belog to same equvalet class ad hece ca be alged. We ow defe dfferet classes of algmet that stregthe the defto

3 Math 239 Lecture 9 February 21 Sprg 2008 Defto 9.3. A mootopoorthologous (MTO homology forest s a lamar famly where o edge cotas (, a ad (, b for ay, a, b. It s easy to see that the homology forest Fgure 9.2 s ot a MTO forest because a le coects dfferet postos of sequece σ 1. Elmatg all such les from the fgure, however, wll result a MTO homology forest. We ow defe a local algmet based o defto of MTO forest above. Defto 9.4. A local algmet of σ 1, σ 2,, σ k s the equvalece relato duced by the set of compoets of a MTO homology forest H wth a partal order H o compoets C(H of H, such that f (, j 1 C r ad (, j 2 C s for some C r H C s, the j 1 j 2. The ame local algmet s used because the order o the compoets s defed usg local formato of postos dvdual sequeces. The local algmet s represeted as a tuple (H, H. I further dscusso, we wll refer to the poset (C(H, H, as a algmet poset. For two sequeces, the local algmet ca also be represeted as a dotplot show Fgure 9.3. I the fgure, x-axs represets the postos of sequece σ 1 ad y-axs represets the postos of sequece σ 2. A dot at posto (, j meas that (1, ad (2, j are alged. Fgure 9.3. A dotplot for two sequeces. 9.3 Partal Global Algmet We ow defe a partal global algmet of a set of sequeces. 9-3

4 Math 239 Lecture 9 February 21 Sprg 2008 Defto 9.5. A partal global algmet of a set of sequeces σ 1, σ 2, σ k s the equvalece relato duced by the compoets of a MTO homology forest H for whch there exsts a partal order H o ts compoets C(H, such that f (, j 1 C r ad (, j 2 C s ad j 1 j 2, the C r H C s. Note that there s a subtle dfferece betwee the deftos of local algmet ad partal global algmet. Cosder two compoets C = {(1, 2, (2, 2} ad D = {(1, 3, (3, 3}. By the defto of partal global algmet, C H D. However, local algmet asserts oly that D H C. Example. A mportat example of partal global algmet s a ull algmet show Fgure 9.4. The homology forest a ull algmet s a set of dsjot vertces. The oly order relatos are those mposed by the defto of global algmet, amely the sgleto compoet {(, j 1 } s H the sgleto compoet {(, j 2 } for all ad j 1 j 2. For smplcty, reverse complemets are omtted from Fgure 9.4. Fgure 9.4. Null algmet of three sequeces. Proposto ( 9.6. For two sequeces of legth ad m, the umber of partal global algmets s. m + Proof: Cosder ay partal global algmet of the two sequeces, where exactly compoets are of sze 2 ad all remag compoets are sgleto. Ths ( ca be doe as follows: select ay postos the frst sequece, whch ca by doe ways. Smlarly, the 9-4

5 Math 239 Lecture 9 February 21 Sprg 2008 ( m postos of secod sequece ca be chose ways. Because of the codto o MTO homology forest, these postos of the two sequeces ca be alged( wth each ( other m a uque way, amely, by algg the pars order. Thus, there are ways of algg exactly characters. Thus, the total umber of ways of algmets of the two sequeces s m(m, ( ( m, (9.1 whch s equal to ( m + =0. There s a alteratve way to compute the sum ( 9.1. Cosder a m rectagular grd as show Fgure 9.5. Ay path the grd from the lower left corer to the upper rght corer volves m + moves, out of whch ( are horzotal ad remag m are m + vertcal. Thus, the total umber of moves s. Now, cosder a move lke the oe show the fgure. Every such move has corers, where the path chages drecto from horzotal to vertcal or vce-versa. The total umber of paths cotag exactly horzotal-to-vertcal corers ca be foud by detfyg x-coordates ad y-coordates ( ( m ( 9.1, whch s equal to ways. The total umber of paths s thus equal to the summato gve ( m +. Fgure 9.5. Partal global algmets of two sequeces. 9-5

6 Math 239 Lecture 9 February 21 Sprg Global Algmet We ow tur our atteto to global algmets. Defto 9.7. A lear exteso of a partally ordered set (poset (C 1, C 2,, C m, s a bjecto π : {1, 2,, m} {1, 2,, m} such that f C C j, the π( π(j. Fgure 9.4 shows oe lear exteso of the ull algmet where a teger ext to each posto deotes the value of π at that posto. Defto 9.8. A global algmet s a partal global algmet together wth a lear exteso o the algmet poset. A global algmet s ofte represeted usg a k matrx T, where k = umber of sequeces ad = umber of compoets of the algmet. The matrx T s defed as: { σ T,j = k f k σk C π(; otherwse. Example. The global algmet show Fgure 9.4 has followg matrx represetato: σ1 1 σ2 1 σ3 1 σ1 3 σ2 3 σ3 3. σ1 5 σ2 5 σ3 5 σ4 5 There s a graph represetato of a global algmet, whch s smlar to the oe show Fgure 9.5. The umber of global algmets betwee two sequeces of legth m ad s gve by the umber of paths betwee the lower left ad the upper rght corers of Fgure 9.6. Fgure 9.6. Global algmets of two sequeces. 9-6

7 Math 239 Lecture 9 February 21 Sprg 2008 Defto 9.9. The algmet graph G,m s the drected graph o the set of odes {0, 1,, } {0, 1,, m} ad three classes of drected edges as follows: there are edges labeled by I (serto, move up betwee pars of odes (, j (, j + 1; there are edges labeled by D (deleto, move rght betwee pars of odes (, j ( + 1, j, ad; there are edges labeled by H (homology, move dagoal betwee pars of odes (, j ( + 1, j + 1. Example. The followg table shows a algmet of sequeces σ 1 ad σ 2 ad the edges of ts drected graph. G G A T T A C A G C T T A G Example. We gve a typcal example of global algmet of multple sequeces: G A T C A T C C A G G G G G A C T A C C T 9.5 Homework Partal global algmets of three sequeces Let N( 1, 2, 3 deote the umber of partal global algmets of three sequeces of legths 1, 2, 3 respectvely. Prove that N( 1, 2, 3 = N( 1 1, 2, 3 +N( 1, 2 1, 3 +N( 1, 2, 3 1 N( 1 1, 2 1, 3 1. Number of global algmets Prove that the umber of global algmets betwee two sequeces of legths ad m respectvely, s gve by 9.6 Refereces m(m, =1 ( ( m 2. Secto 2.2, Sequece Algmet. Pachter ad Sturmfels (2005: