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Announcements: Start chapter 12 Problem sets to hand back

Review of the motif finding problem

A motif logo

Motifs are captured by profiles

Scoring function to reflect nearness to consensus

Brute force enumeration can take a great deal of time

The basic problem

This is a ridiculously general version of the problem:

• Given a multivariable scoring function f(y{1}...y{n}) >= 0
• Find max f

This is a ridiculous sounding solution:

• Set up a probabilistic system where f(opt) has a good chance to appear
• Run the system for a while and take the best f seen

Candidate probabilistic systems

• prob(y{1}...y{n}) =~ f(y)
• prob(y{1}...y{n}) =~ f(y)^(1/T)
• prob(y{1}...y{n}) =~ exp(f(y)/T) (f as a tropical quantity)

Tropics: sampling from f^(1/T) as T->0 should give f(opt)

Gibbs sampling

Solves the following problem:

• Input: prob(y{1}...y{n}), where |y|^n is big, but |y| is managable
• Output: a random y chosen with prob, or a series of such y's

How do we do it?

• Setup Markov chain with states y
• Transitions: (y{1}...y{m}...y{n})-->(y{1}...y'{m}...y{n})
• Transition Probs: pick m uniformly, then proportional to prob(y{1}...y'{m}...y{n})

Specialized instance of more general idea Monte Carlo Markov Chains

Recasting the motif problem as a probability problem

Given letter frequencies p(x) and profile positions s{1}...s{n}

• prob(C) =~ PROD p(x)^C{x,i} / C{x,i}!
• 1/prob(C) = score of candidate s{1}...s{n}
• logarithmic version: SUM C{x,i} log(C{x,i}/p(x))

This is incidental motivation for this scoring function -- it will have nice properties

Gibbs sampling of motifs

1. generate start state: s{1}...s{n}
2. pick uniformly m from 1...n
3. replace s{m} <-- s picked by 1/prob(C(s{1}...s...s{n})) weights
4. <do whatever with the sample>
5. goto 2

At this point, the algorithm is essentially done.

For the chosen objective function, something cute happens

prob(s) =~ C(s{1}...s...s{n}){T[s+i],i} / p(T[s+i])

Typical hack:

• C(s{1}...s...s{n}){T[s+i],i} = C(s{1}...()...s{n}){T[s+i],i}

What can we do with this?

Other applications

• sampling of phylogenetic histories
• haplotype phasing problem (maybe I'll talk about it next time
• exploring hidden state sequences in a random model (e.g. HMM)
• training an random model (e.g. HMM)

1. measure relative uncertainties in the profile quantities
2. in general: establish sensitivity coefficients or find moments
3. what structures are stable over time?

Simulated Annealing

We found a way to sample from prob =~ f(y{1}...y{k})

Annealing is a process by which glass is put into a highly durable state via a process of slow cooling

Same idea here: prob(T) =~ f^(1/T) and T-->0

What can go wrong?

• How long do we run?
• How often can we sample?