Loading in 2 Seconds...

Sensitivity Analysis for Ungapped Markov Models of Evolution

Loading in 2 Seconds...

- By
**paul2** - Follow User

- 286 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Sensitivity Analysis for Ungapped Markov Models of Evolution' - paul2

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Sensitivity Analysis for Ungapped Markov Models of Evolution

David Fernández-Baca

Department of Computer Science

Iowa State University

(Joint work with Balaji Venkatachalam)

CPM '05

Motivation

- Alignment scoring schemes are often based on Markov models of evolution
- Optimum alignment depends on evolutionary distance
- Our goal: Understand how optimum alignments are affected by choice of evolutionary distance

CPM '05

X

Y

Ungapped local alignmentsAn ungapped local alignmentof sequences X and Y is a pair of equal-length substrings of X and Y

Only matches and mismatches — no gaps

CPM '05

23 matches

2 mismatches

34 matches

11 mismatches

Ungapped local alignmentsA:

B:

P. Agarwal and D.J. States. Bayesian evolutionary distance. Journal of Computational Biology 3(1):1—17, 1996

CPM '05

> 0

< 0

score(B)

score(A)

/

-11/9

Which alignment is better?Score = ∙ #matches + ∙ #mismatches

In practice, scoring schemes depend on evolutionary distance

CPM '05

Log-odds scoring

Let qX =base frequency of nucleotide XmXY(t) =Prob(XY mutation in t time units) A be an alignment X1X2X3 XnY1Y2Y3 Yn

Then, Log odds score ofA =

CPM '05

Log-odds scoring

- Simplest model:
- mXX(t) = r(t) for all X
- mXY(t) = s(t) for all XY
- qX = ¼ for all X
- Log-odds score of alignment:(t)∙ #matches + (t) ∙ #mismatcheswhere(t) = 4 + log r(t) (t) = 4 + log s(t)

CPM '05

This talk

- An efficient algorithm to compute optimum alignments for all evolutionary distances
- Techniques
- Linearization
- Geometry
- Divide-and-conquer

CPM '05

Related Work

- Combinatorial/linear scoring schemes:
- Waterman, Eggert, and Lander 1992: Problem definition
- Gusfield, Balasubramanian, and Naor1994: Bounds on number of optimality regions for pairwise alignment
- F-B, Seppäläinen, and Slutzki 2004: Generalization to multiple and phylogenetic alignment
- Sensitivity analysis for statistical models:
- P. Agarwal and D.J. States 1996
- L. Pachter and B. Sturmfels2004a & b: connections between linear scoring and Markov models

CPM '05

A simple Markov model of evolution

- Sites evolve independently through mutation according to a Markov process
- For each site:
- Transition probability matrix:M = [mij], i, j {A, C, T, G}where mij = Prob(i j mutation in 1 time unit)
- Transition matrix for t time units is M(t)

CPM '05

Recall:

Score(A) = ∙ #matches + ∙ #mismatches

Linearization- Allow and to vary arbitrarily, ignoring that they
- are functions of t and
- must satisfy laws of probability
- Result is a linear parametric problem

CPM '05

Theorem

Letn be the length of the shorter sequence. Then,

(ii) The parameter space decomposition looks like this:

(i) The number of distinct optimal solutions over all values of and is O(n2/3).

CPM '05

The optimum solutions for t = 0 to +are found by varying / from - to 1

Non-linear problem in t reduces to a linear one-parameter problem in /

Re-introducing distanceThe vs. curve intersects every boundary line with slope (-∞, +1]

CPM '05

An algorithm

- Start with a simple, but highly parallel, algorithm for fixed-parameter problem
- Lift the fixed-parameter algorithm
- Lifted algorithm runs simultaneously for all parameter values in linearized problem
- Output: A decomposition of parameter space into optimality regions
- Construct solution to original problem by finding the optimality regions intersected by the (t), (t) curve

CPM '05

A naïve dynamic programming algorithm

Y

- Let C be the matrix whereCij = score of opt alignment ending at Xi and Yj
- Subdiagonals correspond to alignments
- Diagonals are independent of each other
- Process each diagonal separately
- Pick best answer over all diagonals
- Total time: O(nm)

aattcaattcaatc . . .

caatttgtcacttttt . . .

X

C

CPM '05

X(1)

X(2)

X(1)

X(2)

Y(1)

Y(2)

Y(1)

Y(2)

X(1)

X(2)

Y(1)

Y(2)

length of diagonal

Divide and conquer for diagonalsSplit diagonal in half, solve each side recursively, and combine answers. E.g.:

X

Y

T(N) = 2 T(N/2) + O(1)

T(N) = O(N)

#subproblems

CPM '05

Lifting

- Run naïve DP algorithm for all parameter values by manipulating piecewise linear functions instead of numbers:
- “+” “+” for piecewise linear functions
- “max” “max” of piecewise linear functions

CPM '05

f

max (f,g)

g

Computing the maximum of piecewise linear functionsTime = O(total number of segments)

CPM '05

#(optimum solutions for diagonal)Analysis

- Processing a diagonal:
- T(n) = 2 T(n/2) + O(n2/3)T(n) = O(n)
- Merging score functions for diagonals:
- O(n2/3) line segments per function, m+n-1 diagonals
- Total time: O(mn + mn2/3 lg m)

CPM '05

ACT

AAT

AGC

Further Results (1): Parametric ancestral reconstruction- Given a phylogeny, find most likely ancestors

AAC

- Sensitive to edge lengths

AAT

- Result: O(n) algorithm for uniform model (all edge lengths equal)

CPM '05

Further Results (2)

- Bounds on number of regions for gapped alignment (indels are allowed)
- Lead to algorithms, but not as efficient as ungapped case

CPM '05

Open Problems

- Tight bounds on size of parameter space decomposition
- Evolutionary trees with different branch lengths
- Efficient sensitivity analysis for gapped models
- Evaluation of sensitivity to changes in structure and parameters
- Useful in branch-swapping

CPM '05

Download Presentation

Connecting to Server..