Course Introduction

• What these courses are about

• What I expect

• What you can expect

What these courses are about

• overview of ways in which computers are used to solve problems in biology

• supervised learning of illustrative or frequently-used algorithms and programs

• supervised learning of programming techniques and algorithms selected from these uses

I Expect

• students will have basic knowledge of biology and chemistry (at the level of Modern Biology/Chemistry)

• students will have basic familiarity with statistics

• students have some programming experience and willingness to work to improve

You can expect

• Homework assignments– 80% of grade

• Final (20% of grade)

• Grades totally determined by points system

Textbook

• Required textbook: Biological Sequence Analysis: Probabilistic models of proteins and nucleic acids by Durbin et al.

• Recommended additional textbook:

Introduction to Computational Biology by Waterman

Chapter 1

Introduction

Purpose

• A great acceleration in the accumulation of biological knowledge started in our era

• Part of the challenge is to organize, classify and parse the immense richness of sequence data

• This is not just a task of string parsing, for behind the string of base or amino acids is the whole complexity of molecular biology

• A major task in computational molecular biology is to “decipher” information contained in biological sequences

• Since the nucleotide sequence of a genome contains all information necessary to produce a functional organism, we should in theory be able to duplicate this decoding using computers

Information Flow

Review of basic biochemistry

• Central Dogma: DNA makes RNA makes protein

• Sequence determines structure determines function

Structure

• DNA composed of four nucleotides or "bases": A,C,G,T

• RNA composed of four also: A,C,G,U (T transcribed as U)

• proteins are composed of amino acids

Purpose

This class is about methods which are in

principle capable of capturing some of the

complexity of biology, by integrating diverse

sources of biological information into clean,

general, and tractable probabilistic models

for sequence analysis.

However,

The most reliable way to determine a biological molecule’s structure or function is by direct experimentation.

It is far easier to obtain the DNA sequence of the gene corresponding to an RNA or protein than it is to experimentally determine its function or its structure.

The Human Genome Project

Gives us the raw sequence of an estimated 20,000-25,000 human genes, only a small fraction of which have been studied experimentally.

The development of computational methods have become more important (computer science, statisticians, and etc….)

Basic Information

• New sequences are adapted from pre-existing sequences

• We compare a new sequence with an old sequence with known structure or function

• Two related sequences are called homologous and we are transferring information by homology

• It is somewhat similar to determine the similarity between two text strings

• In fact, we will be trying to find a plausible alignment between sequences

Definition

• A sequence is a linear set of characters (sequence elements) representing nucleotides or amino acids– DNA composed of four nucleotides or

"bases": A,C,G,T– RNA composed of four also: A,C,G,U (T

transcribed as U)– proteins are composed of amino acids (20)

Character representation of sequences• DNA or RNA

– use 1-letter codes (e.g., A,C,G,T)

• protein– use 1-letter codes

• can convert to/from 3-letter codes

(e.g., A = Ala = Alanine

C = Cys = Cysteine)

Alignment

• Find the best alignment between two strings under some scoring system

• “+1” for a match; “-1” for a mismatch• Most important, we want a scoring system to give the

biologically most likely alignment the highest score• Note that biological molecules have evolutionary

histories, 3D folded structures, and other features• This is more the realm of statistics than computer

science• Probabilistic modeling approach might be used and

extend

Probabilities & Probabilistic Models• A model means a system that simulates the

object under consideration• A probabilistic model is to produce different

outcomes with different probabilities• That is, it stimulates a whole class of objects,

and assign each object an associated probability• The objects will be sequences, and a model

might describe a family of related sequences

Example: Rolling a six-sided die

• A probability model of rolling a 6-sided die involves 6 parameters p1, p2, p3, p4, p5, and p6

• The probability of rolling i is pi

• pi 0 and Σ≧ pi=1

• Rolling the die 3 times independently, P([1,6,3])= p1 p6 p3

Example: Biological Sequence

• Biological sequences are strings from finite alphabet of residues (4 nucleotides or 20 amino acids)

• A residue a occurs at random with probability qa, independent of all other residues in the sequence

• If the sequence is denoted by x1… xn, the probability of the whole sequence is

qx1 qx2… qxn

Maximum Likelihood Estimation

• The parameters of a probability model is estimated from a training set (sample)

• The probability qa for amino acid a can be estimated as the observed frequency of residues in a database of known protein sequences (SWISS-PROT)

• The training sequences are not systematically biased towards a peculiar residue composition

http://au.expasy.org/sprot/

MLE (continued)

• This way of estimating models is called maximum likelihood estimation (MLE)

• The MLE maximizes the total probability of all sequences given the model (the likelihood)

• Given a model with parameters θ and a set of data D, the maximum likelihood estimate for θ is that value which maximizes P(D|θ)

Estimation

• If estimating parameters from a limited amount of data, there is a danger of overfitting

• Overfitting: The model becomes very well adapted to the training data, but it will not generalize well to new data

• For example, observing the three flips of a coin [tail, tail, tail] would lead to the maximum likelihood estimate that the probability of head is 0 and that of tail is 1

Conditional, Joint, and Marginal

• We have two dies, D1 and D2

• The conditional probability of rolling i given die D1 is called P(i|D1)

• We pick a die with probability P(Dj), j=1, 2• The probability for picking die Dj and rolling an i is

the product of the two probabilities, P(i, Dj)=P(Dj)P(i|Dj), the joint probability

• P(X, Y)=P(X|Y)P(Y)• P(X)=ΣYP(X, Y)=ΣY P(X|Y)P(Y), the marginal prob

ability

Bayes Theorem

• Bayes’ theorem

• The denominator is the marginal• The numerator is the joint

)(

)()|()|(

YP

XPXYPYXP

Example 1

Consider an occasionally dishonest casino

that uses two kinds of dice. Of the dice 99%

are fair but 1% are loaded so that a six

comes up 50% of the time. Suppose we pick

a die at random and roll it three times,

getting three consecutive sixes. What is

P(Dloaded|3 sixes)?

Example 1 (Continued)

)()|sixes 3(

)()|sixes 3(sixes) 3(

125.05.0)|sixes 3(

0.01)(

)sixes 3(

)()|sixes 3()sixes 3|(

fairfair

loadedloaded

3loaded

loaded

loadedloaded

loaded

DPDP

DPDPP

DP

DP

P

DPDPDP

Example 1 (Continued)

We will still more likely pick up a fair die, despite

seeing three successive sixes.

0.21

)99.0()6/1()01.0)(5.0(

)01.0)(5.0()sixes 3|(

33

3

loaded

DP

Example 2

• Assume that, on average, extracellular protein have a slightly different amio acid composition than intracellular proteins

• For example, cysteine is more common in extracellular than intracellular proteins

• Question: whether a new protein sequence x=x1…xn is intracellular or extracellular?

Example 2 (continued)

• We first split our training examples from SWISS-PROT into extracellular and intracellular proteins

• Estimate a set of frequencies for intracellular proteins, and a corresponding set of extracellular frequencies

• The probability that any new sequence is extracellular is pext, and the corresponding probability of being intracellular is pint. Note that pint=1- pext

intaq

extaq

Example 2 (continued)

intintextext

extext

intext

intext

)|ext(

)int|()ext|()(

)int|( and )ext|(

ii

i

ii

xixi

xi

xixi

qpqp

qpxP

xppxppxP

qxPqxP

Bayesian Model

• θ is the parameter of interest• Before collecting data, the information regarding

θ is called the prior information, P(θ)• After collected the data, the information regardin

g θ is called the posterior information, P(θ|D)• If we do not have enough data to reliably estimat

e the parameters, we can use prior knowledge to constrain the estimates

Bayesian and Frequentist

D ~ N(θ,1)

• To frequentists, θ is fixed (unknown)

• To Bayesians, θ is random

• If θ is random, what should its distribution

be?

• Frequentists argue that the determination of the prior distribution of θ is very subjective

Prior and Posterior

• Suppose that θ has a probability distribution

P(θ) (prior)• Assume that θ and D|θ are independent • P(D, θ) is the joint distribution of D and θ• P(D | θ) is the conditional distribution of D given

θ• P(θ | D) is the conditional distribution of θ

given D (the posterior)

Prior and Posterior

• P(D| θ)P(θ)=P(D, θ)=P(θ | D)P(D)• Bayes’ theorem:

DPPDP

DP)(|

|

Posterior Distribution

Given D’s density p(D|θ) and a prior

probability density P(θ), the posterior

density for θ is given as

p(θ|D)=cp(θ) p(D|θ) ,

where

c-1=∫ p(θ) p(D|θ) dθ

(the marginal of D).

Example

D ~ N(θ,2), is known.

P()=N( 0, 02)

Then the posterior density is normal with

and

220

2

220

20

0 /1/1

/1

/1/1

/1mean

d

1220 /1/1variance

Conjugate Prior

• D ~ N(θ,2) is a normal distribution

• Prior distribution, P()=N( 0, 02), is

also a normal distribution

• Posterior distribution, P(|D), is also a normal distribution

• The normal distribution is conjugate to the normal

Specification of the Prior

• Conjugate priors:

–The beta distribution is conjugate to the

binomial

–The normal distribution is conjugate to the

normal

–The gamma distribution is conjugate to the

Poisson

Specification of the Prior

• Noninformative (uninformative) priors: P(θ) constant• When we don’t have a strong belief or in

public policy situations strongly differ

Specification of the Prior

Sometimes, we will wish to use an

informative P(θ). We know a priori that the

amino acids phenylalanine (Phe, F), tyrosine (Tyr,

Y), and tryptophan (Trp, W) are structurally similar

and often evolutionarily interchangeable. We

would want to use a P(θ) that tends to favor

parameter sets that assign them very similar

probabilities.

Parameter Estimation

• Choose the parameter value for θ that maximize P(|D)

• This is called maximum a posteriori or MAP estimation

• MAP estimation maximizes the likelihood times the prior

• If the prior is flat (uninformative), then MAP is equal to the MLE

• Another parameter estimation is to choose the mean of the posterior

Maximum A Posteriori (MAP) Estimation

• Ex: estimating probabilities for a die

We roll 1, 3, 4, 2, 4, 6, 2, 1, 2, 2

MLE: p5=0 ~ overfitting

add 1 to each observed number of counts (pseudocount):

MAP: p5=1/16

When estimating large parameter sets from

small amounts of data, we believe that

Bayesian methods provide a consistent

formalism for bringing in additional

information from previous experience with

the same type of data.