Math 591 Homework 1 Due: Thursday, January 26, 2012
Problem 1. Let P be a 3 dimensional polytope with 9 vertices. Give upper and lower bounds
on the number of edges of P . Show that your bounds are tight by constructing examples of
3 dimensional polytopes
Math 591 Class Notes Tuesday, February 7, 2012
Introduction
Previously, we had been examining a hidden Markov Model where
A = the transition matrix
B = the emission matrix
We had an observed sequence Y0 = 0 , . . . , Yn = n and we were interested in deter
Discrete Mathematical Biology
Lecture 10: Sequence Alignment
Ismail Demir
February 09, 2012
1
Sequence Alignment
Evolutionary theory tells us that DNA of extant species evolved to its present form from common ancestral DNA sequences. There are a number of
Discrete Mathematical Biology
Seth Sullivant
February 14, 2012
Lecture 11: Graphs related to sequence alignment
The computational time of the Needleman-Wunsch algorithm for calculating the alignment
of two sequences S1 and S2 (of length m and n) with the
Discrete Mathematical Biology
Lecture 12
Tulay Ayyildiz Akoglu
February 15, 2012 (Make-Up Class)
Pair Hidden Markov Model (contd)
Graph G(n, ) and hidden states i for i = 1, . . . , n.
P r(1 , . . . , n , S2 )
S1
How to compute P r(S2 )?
S1
P r(1 , . . .
Lecture Notes - Thursday, February 16, 2012
We return to the consideration of the sequence alignment problem, specically, the 2-parameter
model with gap penalty g and mismatch penalty m.
Question: What are the scores of the two alignments given in the gra
MATH 591 - 21 February 2012 Notes
1. (From Last Time)
If F (a, b) = P r (Y1 = 1 , . . . , Yn = n ) then N ewt(f0 (a, b) = V P ( ),
where Newt denotes the Newton polytope and VP denotes the Viterbi Polytope. How do we exploit this relationship?
Similarly,
MATH 591 - 23 February 2012 Notes
1. More on optimal alignments for polytopes
Theorem 1 (Andrews, 63). Let C Rd be a strictly convex set with N
integer points on the boundary, not all contained in an ane hyperplane.
d+1
Then there exists (d) such that V o
Discrete Mathematical Biology
Seth Sullivant
February 28, 2012
Lecture 16: Splits
Some terminology
Denition 1. An X -split A|B is a partition of X , a set of labels, into two non-empty sets A, B .
Question 2. If #X = n, how many X -splits are there?
Answe
Math 591 March 1, 2012
Denition: Let X be a set of labels. A character on X is a map : X C (another
set of labels).
Let T be an X -tree. A character is convex on T if for all c1 = c2 C , the induced trees
T |1 (c1 ) T |1 (c2 ) = .
Example: Which 1 , 2 is
MA 591 Notes: Parsimony
Carl Giure
March 12, 2012
Thm 1. Let C be a collection of characters on X , cfw_X1 ,.,Xr . Then C is compatible ( an
X -tree displaying all X C convexly) int(C ) has a restricted chordal completion.
Proof.
( = )
Suppose C is compat
Math 591 Class Notes Tuesday, March 13, 2012
Question: Let X =[4] and C be a list of binary characters. There are only seven ways to partition [4] into
two blocks, and so there are only seven possible binary characters.
1
2
3
4
12
13
14
1|234 2|134 3|124
Math 591 Lecture Notes March 20th, 2012
The goal of today is to develop the denitions and theorems needed to prove the following
theorem. Following this we examine the set/space of all tree metrics and its properties.
Theorem: A metric d is a tree metric
Discrete Mathematical Biology
Lecture 21 Building a tree metric from data
Tulay Ayyildiz Akoglu
March 22, 2012
Proposition 1. Let T be an X tree. The set of all tree matrics on T is
CT = cfw_
A|B A|B : A|B 0
A|B (T )
Proof:
d(x, y ) =
(e)
e in path
from
Math 591 Notes Thursday, February 2, 2012
Hidden Markov Models: Consider two alphabets, and , a collection of random variables
X0 , X1 , X2 , . with state space , and another collection of random variables Y0 , Y1 , Y2 , . with
state space . Let X0 , X1 ,
MATH 591 - DISCRETE MATHEMATICAL BIOLOGY
LECTURE NOTES
JANUARY 31ST, 2012
1. The Annotation Problem
1.1. Probability. Before we dive into the annotation problem, we will review some of the
basic probability we will need. Let X denote a random variable wit
Discrete Mathematical Biology: Lecture 6, January 26, 2012
Ruth Davidson
January 31, 2012
De Bruijn Graphs and Other Strategies
Recall that the Shortest Common Superstring Problem (SSP) is NP-Hard. In particular, the greedy
algorithm solution from Lecture
Math 591 Homework 2 Due: Thursday, February 2, 2012
Problem 1. It is conjectured that the greedy algorithm discussed in class gives a 2-approximation
algorithm for the shortest common superstring problem. (That is, the output of the greedy algorithm is, a
Math 591 Homework 3 Due: Thursday, February 9, 2012
Problem 1. Let A be the matrix
1/3 1/3 1/3
A = 0 1/4 3/4
0 1 /5 4 /5
which we will use as the transition matrix of a Markov chain with state space = cfw_1, 2, 3.
(1) Show that A violates the conditions o
Math 591 Homework 4 Due: Thursday, February 16, 2012
Problem 1.
(1) Write a computer program that implements the Viterbi algorithm (use any
software of your choice).
(2) Test your Viterbi algorithm code on the HMM with the following parameters. Transition
Math 591 Homework 5 Due: Thursday, February 23, 2012
Problem 1.
(1) Write a computer program that implements the Needleman-Wunsch algorithm for sequence alignment (use any software of your choice). You may implement
the simplest version of the algorithm;
Math 591 Homework 6 Due: Thursday, March 15, 2012
Problem 1 (From Semple and Steel). A subforest of a phylogenetic tree T is a collection of
edge-disjoint subtrees of T such that all subtrees have at least one edge and each degree one
vertex of every subt
Math 591 Homework 7 Due: Thursday, March 22, 2012
Problem 1 (From Semple and Steel). Let d be a metric on X and let > 0. Let G denote the
graph that has vertex set X , and an edge joining two vertices x and y precisely when d(x, y ) < .
(1) Show that G is
Math 591 Homework 8 Due: Thursday, March 29, 2012
Problem 1. Let be the dissimilarity map on X = cfw_a, b, c, d, e given by the array:
a
b
c
d
e
abc
3 10
8
(1)
(2)
(3)
(4)
de
75
77
13 12
10
Is a metric?
Is an ultrametric?
Is a tree metric?
Apply the UPGMA
Math 591 Homework 9 Due: Thursday, April 12, 2012
Problem 1. Let be the dissimilarity map on X = cfw_a, b, c, d, e given by the array:
a
b
c
d
e
abc
3 10
8
de
75
77
13 12
10
(1) Find the subdominant ultrametric U .
(2) Find the ultrametric that minimizes
Discrete Mathematical Biology
Seth Sullivant
January 10, 2012
1
Lecture 1: Introduction
Molecular biology is fundamentally about the analysis of sequences: DNA sequences, RNA
sequences, and protein (amino acid) sequences. This course will be primarily con
Discrete Mathematical Biology
Seth Sullivant
January 12, 2012
Lecture 2: Some background on molecular biology
Note: All gures in this lecture were taken from the internet.
Organisms are made up of cells. Each cell consists of many dierent organelles, whic
DISCRETE MATHEMATICAL BIOLOGY
SETH SULLIVANT
Lecture 3: Graph Theory
1
1
5
5
1
1
2
2
5
5
4
4
33
22
44
33
Figure 1. Same Graph
Denition. An undirected graph G = (V, E ) is a set V and a set E of unordered pairs of
elements of V . The elements of V are call
Discrete Mathematical Biology
Lecture 4: Graph Theory and Optimization
Tulay Ayyildiz Akoglu
January 19, 2012
1
Shortest Path Problem
Denition 1. Let D be a directed graph. For each directed edge from (i, j ) D, let wij R a weight
k
for a directed path P
Discrete Mathematical Biology: Lecture 5, January 24, 2012
Ruth Davidson
January 27, 2012
Introduction to Lecture 5: The Sequence Assembly Problem
Recall from Lecture 1 (January 10) that for practical reasons, DNA sequence data is acquired by reading
shor