MAT 4830 Mathematical Modeling 4.5 Phylogenetic Distances I http://myhome.spu.edu/lauw

Preview Phylogenetic: of or relating to the evolutionary development of organisms Estimate the amount of total mutations (observed and hidden mutations).

Example from 4.1 S0 : Ancestral sequence S1 : Descendant of S0

Example from 4.1 S0 : Ancestral sequence S1 : Descendant of S0 Observed mutations: 2

Example from 4.1 S0 : Ancestral sequence S1 : Descendant of S0 Actual mutations: 5

Example from 4.1 S0 : Ancestral sequence S1 : Descendant of S0 Actual mutations: 5, (some are hidden mutations)

Distance of Two Sequences We want to define the “distance” between two sequences. It measures the average no. of mutations per site that occurred, including the hidden ones.

Distance of Two Sequences Let d(S0,S) be the distance between sequences S0 and S. What properties it “should” have? 1. 2. 3.

Jukes-Cantor Model Assume α is small. Mutations per time step are “rare”.

Jukes-Cantor Model q(t)=conditional prob. that the base at time t is the same as the base at time 0 A

Jukes-Cantor Model q(t)=fraction of sites with no observed mutations A

Jukes-Cantor Model p(t)=1-q(t)=fractions of sites with observed mutations A

Jukes-Cantor Model p(t)=1-q(t)=fractions of sites with observed mutations A

Jukes-Cantor Model p can be estimated from the two sequences A

Example from 4.1 Observed mutations: 2

Jukes-Cantor Distance Given p (and t), the J-C distance between two sequences S0 and S1 is defined as

Jukes-Cantor Distance Given p (and t), the J-C distance between two sequences S0 and S1 is defined as Why?

Jukes-Cantor Distance

Jukes-Cantor Distance

Jukes-Cantor Distance

Example from 4.3 Suppose a 40-base ancestral and descendent DNA sequences are

Example from 4.3 Suppose a 40-base ancestral and descendent DNA sequences are

Example from 4.3 0.275 observed sub. per site. 0.3426 sub. estimated per site.

Example from 4.3 11 observed sub. 13.7 sub. estimated.

Performance of JC distance (Homework Problem 4) Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α.

Performance of JC distance (Homework Problem 4) Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α. Count the number of base substitutions occurred.

Performance of JC distance (Homework Problem 4) Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α. Count the number of base substitutions occurred. Compute the Jukes-Cantor distance of the initial and finial sequence.

Performance of JC distance (Homework Problem 4) Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α. Count the number of base substitutions occurred. Compute the Jukes-Cantor distance of the initial and finial sequence. Compare the actual number of base substitutions and the estimation from the Jukes-Cantor distance.

Performance of JC distance (Homework Problem 4)

Maple: Strings Handling II Concatenating two strings

Maple: Strings Handling II However, no “re-assignment”.

Classwork Work on HW #1, 2