. פרויקט בתכנות מתקדם – פונקציות מרחק אופטימליות לשיחזור עצי אבולוציה סמסטר אביב דואר אלקטרוני חדרטלפון שלמה מורן דניאל דור

. ההשפעה של פונקציות מרחק על שיחזור עצי אבולוציה לאחר שלב ההודעות, נעביר היום קורס בזק מקוצר על: 1.עצי אבולוציה: הגדרות ומודלים מבוססי DNA 2.שיטות מבוססות מרחקים לבניית עצי אבולוציה 3.פונקציות מרחק למודלים אבולוציוניים 4.הערכת מרחקים בין זנים על סמך השוני בין סדרות הDNA לאחר ה"קורס המזורז" עדיין תזדקקו להשלמות מסוימות בהמשך הסמסטר. במהלך "קורס הבזק" יוצגו הפרויקטים. נושא הפרויקט

. דרישות קדם: אלגוריתמים 1, הסתברות רצוי (אך לא הכרחי): אלגוריתמים בביולוגיה חישובית ככלל, הפרויקטים יעשו בזוגות. תוך שבוע הודיעונו על החלוקה לזוגות (בדוא"ל) בחירת פרוייקט: יהיו שני כיוונים עיקריים. השלב הראשוני דומה בשני הכיוונים. התמקדות בכיוון מסוים תעשה בהמשך (תוך כחודש). (מכאן והלאה שקפים באנגלית) אדמיניסטרציה

4 Crash course on evolutionary distances

5 The Phylogenetic Reconstrutction Problem

6 AATCCTG ATAGCTG AATGGGC GAACGTA AAACCGA ACGGTCA ACGGATA ACGGGTA ACCCGTG ACCGTTG TCTGGTA TCTGGGA TCCGGAAAGCCGTG GGGGATT AAAGTCA AAAGGCG AAACACA AAAGCTG Evolution is modeled by DNA sequences which evolve along an Evolution Tree (Phylogeny) (All our sequences are DNA sequences, consisting of {A,G,C,T})

7 AATCCTG ATAGCTG AATGGGC GAACGTA AAACCGA ACCGTTG TCTGGGA TCCGGAAAGCCGTG GGGGATT Phylogenetic Reconstruction

8 B : AATCCTG C : ATAGCTG A : AATGGGC D : GAACGTA E : AAACCGA J : ACCGTTG G : TCTGGGA H : TCCGGAA I : AGCCGTG F : GGGGATT Goal: reconstruct the ‘true’ tree as accurately as possible reconstruct A B C F G IHJ D E A B C F G I H J D E (root) Phylogenetic Reconstruction

9 Three Methods of Tree Construction u Parsimony – A tree with minimum number of mutations. u Maximum likelihood - Finding the “most probable” tree. u Distance- A weighted tree that realizes the distances between the species.

10 A C B D F G E edge-weighted ‘true’ tree reconstructed tree reconstruction B C A D F G E noise α Major problem: sensitivity to noise reconstruction in O(n 2 ) Distance Based Reconstruction: Exact vs. approximate distances Exact distances

11 A C B D F G E edge-weighted ‘true’ tree reconstruction in O(n 2 ) The Algorithmic Aspect Exact distances Many algorithms can reconstruct a weighted tree from the exact distances. In this project we will use the “Saitou&Nei Neighbor Joining algorithm”, or simply the “NJ algorithm”.

12 Evolutionary Distances: - How are they defined? - How are they extracted from the DNA sequences? We’ll show this on a specific model the Kimura 2 Parameters (K2P) model The Distance Estimation Aspect noise α

13 The Kimura 2 Parameter (K2P) model [Kimura80]: each edge corresponds to a “Rate Matrix” Transitions Transversions Transitions K2P generic rate matrix u v

14 K2P standard distance: Δ total = Total substitution rate u vw The total substitution rate of a K2P rate matrix R is This is the expected number of mutations per site. It is an additive distance. + α + 2βα’ + 2β’ (α+α’) + 2(β+ β’)

15 The distance Δ total (R uv ) = d K2P (u,v) is estimated from the aligned sequences u AACA…GTCTTCGAGGCCC v AGCA…GCCTATGCGACCT K2P total rate “distance correction” procedure since mutations may overwrite each other, this is a “noisy” process

A basic question: How good is a reconstruction method which uses K2P distances? AC BD w sep The performance of tree reconstructions method is often tested on quartets, which are trees with 4 taxa. A quartet contains a single internal edge, which defines the quartet-split.

17 A correct reconstruction of the quartet requires finding of the true quartet-split AC BD AB C D AC DB Distance methods reconstruct the true split by the 4-point condition: There are 3 possible splits: w sep The 4-point condition for noisy distances is:

18 We evaluate the accuracy of the K2P distance estimation by Split Resolution Test: root D C A B t is “evolutionary time” The diameter of the quartet is 22t

19 Phase A: simulate evolution D C A B

20 Phase B: reconstruct the split by the 4p condition DCBA                   Apply the 4p condition. Was the correct split found? compute distances between sequences, Repeat this process 10,000 times, count number of failures

21 the split resolution test was applied on the model quartet with various diameters  For each diameter, mark the fraction (percentage) of the simulations in which the 4p condition failed (next slide) ……

22 Performance of K2P distances in resolving quartets, small diameters: Template quartet

23 Performance for larger diameters “site saturation”

24 Transitions Transversions Transitions When β < α, we can postpone the “site saturation” effect. For this, use another distance function for the same model, Δ tv, which counts only transversions: This is the CFN model [Cavendar78, Farris73, Neymann71] α α β

25 Apply the same split resolution test on the transversions only distance: u AACA…GTCTTCGAGGCCC v AGCA…GCCTATGCGACCT Transversions only Distance correction procedure

26 transversions only performs better on large, worse on small rates Transversions only total K2P rate

Conclusion: Distance based reconstruction methods should be adaptive: Find a distance function d which is good for the input                      Projects goal: Evaluate the performance of distance functions in reconstructing phylogenies

28 1 st step in finding good distance functions ( for the K2P model): Characterize the available distance functions. Ideally, we would like to use the K2P distance associated with the rate matrix of each edge, but...

29 Rate matrices are hard to observe, hence we use Substitution matrices AACA…GTCTTCGAGGCCC u v AGCA…GCCTATGCGACCT Evolution of a finite sequence by unknown model parameters α, β A stochastic substitution matrix P uv

30 Subtitution matrices are extended to paths: u v w

31 Substitution matrices are converted to distances by a Substitution Rate function u v w SR function need to satisfy the following for all substitution matrices P,Q in K2P : 1.Δ(PQ) = Δ(P)+ Δ(Q) (additivity) 2.Δ(P)>0 (positivity)

32 To define SR functions which are additive: Δ(PQ) = Δ(P)+ Δ(Q) We use some linear algebra

33 Lemma: There is a matrix U which diagonalizes each K2P Substitution Matrix P: λPλP 00 00μPμP 0 000μPμP Where: U -1 PU = P = 0 < λ P <1 0 < μ P < 1

34 μPμP 000 0μPμP 00 00λPλP U -1 P U = μQμQ 000 0μQμQ 00 00λQλQ U -1 Q U = U -1 PQ U = Let P,Q be two matrices in K2P. Then: μ P μ Q λ P λ Q U -1 PQ U =

35 Proof: D λ (PQ) = -ln(λ P λ Q ) = -ln(λ P ) -ln(λ Q ) = D λ (P)+ D λ (Q) And the same for D μ (P )= -ln(μ P ) Hence, the functions: D λ (P)= -ln(λ P ), D μ (P)=-ln(μ P ) are additive distance functions For the K2P model

36 Moreover, Each positive linear combination of D λ and D μ is an additive distance function u v w Our goal: given set of input sequences, select D which guarantees best reconstruction of the true tree.

37 ACGGTCA ACGGATA GGGGATT The approximate distance function is defined by the observable noisy version of the substitution matrices w v u We would like to use functions which minimize the influence of the “noise” on the reconstruction. Such a function can be defined&computed analytically for a single distance. Computing it for even small trees looks hard.

38 Summary We have infinitely many additive distance functions for the K2P model. Which one should we use for the given input DNA sequences? If we have the exact substitution matrices for all pairs of taxa, then all functions are equally good. But we have only finite sequences, whose alignments provide only estimations of the true substitution matrices

39 3 phases of the project Phase 1: Distance functions on simulated quartets :1 month Phase 2: Distance functions on larger simulated trees: (1+) month Phase 3: Extensions to real data and/or different models: 1 month Phase 2 and 3 are flexible

40 Phase I: Quartets (~one month) Study the relevant info in “Towards Optimal....” Write a program (in MATLAB or C..) which compute optimal distance functions as in the above paper Repeat the “quartet resolution test” given in this presentation, and extend it to include optimal distance functions. Feel free modify the simulation by your judgment.

41 Phase II: Reconstructing Larger Trees using the Neighbor Joining Algorithm 1.Study the Neighbor Joining algorithm 2.Newick trees representations, and Robinson Fould measure. 3.Make similar tests, but this time on larger trees. 4.Implementation of NJ, and “Tree Templates” can be downloaded from the www. More information will be given later, either via the course site or in a meeting.

42 Phase III: Trees from Real Data 1.Get Homologeous DNA sequences from existing databases. 2.Align the sequences using public domain software. 3.Select appropriate distance functions, and estimate distances between the aligned sequences, using appropriate distance functions 4.Use the various distance functions to reconstruct the trees, and compare their perfomance.