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In general, we have Bayes theorem:

P(X|Y) = P(Y|X)P(X)/P(Y)

- Event X: the die is loaded, Event Y: 3 sixes.
- Example: Assume we know that on average extracellular proteins have a slightly different a.a. composition than intracellular ones. Eg. More cysteines. How do we use this information to predict a new protein sequence x=x1x2…xn whether it is intracellular or extracellular.
- We first split the training examples from Swiss-Prot into intracellular and extracellular proteins, leaving aside those unclassifiable.
- We then estimate a set of frequencies for intraceullar proteins and a set of extracellular frequencies.
- Also estimate the probability that any new sequence is extracelluar, pext and intracellular pint , called prior probabilites, because they are best guesses about a sequence before we actually see the sequence itself.

We now have:

- Because we assume that every sequence must be either extracellular or intracelluar, we have:
- By Bayes’ theorem,
- This is the number we want: the posterior probability that a sequence is extracellular.
- It is our best guess after we have seen the data.
- More complicated: transmembrane proteins have both intra and extra cellular components.

Random Model R: For two sequences x and y, of lengths n and m. If xi is the i th symbol in x, and yi the i th symbol in y. Assume that letter a occurs independently with some frequency qa.

- The probability of the two sequences x and y is just the product of the probabilities of each amino acid:

P(x,y|R) = P qxiP qyi

- An alternative model: Match Model M: Aligned pairs of residues occur with a joint probability Pab. Its value can be thought of as the probability that the resdiues a and b have each independently been derived from some unknown original residue c in their common ancester.
- c might be the same as a and/or b.
- The probability of the whole alignment is:

P(x,y|M) = P pxiyi

- The ratio of these two likelihoods is the odds ratio:

P(x,y|M) / P(x,y|R) = P pxiyi / (P qxiP qyi )= P pxiyi / qxi qyi

- To make this additive, we take the logarithm of this ratio, the log-odd ratio.

S = S s(xi, yi), where s(a, b) = log (pab / qa qb ),

Here s(a, b) is the log likelihood ratio of the residue pair (a,b) occurring as an aligned pair, as opposed to an unaligned pair.

A biologist may write down and ad hoc substitution matrix based on intuition, but it actually implies the “target frequencies” pab . Any substitution matrix is making a statement about the probability of observing ab pairs in real alignment.

- How to develop an evolutionary model?
- Parameterized by probability of residue A mutated to residue B: PAB
- Statistical modeling: these parameters cannot be assigned, rather, they have to be estimated from data.
- Suppose we know sequences s and s’ are related: find PAB that maximizes:
- Maximum likelihood: maximize data likelihood under model.
- Results:

Substitution matrix can be obtained when alignment of sequences are compiled.

- Different matrix for different evolutionary time t :
- How do we estimate it?
- The probability of given a residue A and it is substituted by B within evolutionary distance t :
- Ignore directionality of time:
- Assume that the distribution of amino acid (a.a.) does not change during evolution:

Can be estimated from:

- relative frequency of pair (A,B) in the known alignment of s and s’, and
- relative frequency qA of residue A.
- Substitution matrix over a longer time scale:

Regular Expression

- Widely used in many programs, especially those on Unix: awk, grep, sed, and perl.
- Used for searching text files for a pattern. Eg. Search for all files that containing “C.elegans” or “Caenorhabditis elegans” with the regular expression:

% grep “C[\.a-z]* elegans” *

- This matches any line containing a C followed by any number of lower-case leters or “.”, then a space, and then “elegans”.
- Another example, PROSITE.
- Difficulty: need to be very broad and complex, because protein spelling is much more free than English spelling.

Example: ( use DNA because of the smaller number of letters than a.a.)

ACA---ATG

TCAACTATC

ACAC--AGC

AGA---ATC

ACCG--ATC

- A regular expression for this is:

[AT][CG][AC][ACGT]*A[TG][GC]

Problem: It does not distinguish:

TGCT--AGG Highly implausible, exceptional character in each position

ACAC--ATC Consensus sequence

Alternative: score sequences by how well they fit the alignment.

- Eg. A proabability of 4/5=0.8 for A in the first position, 1/5=0.2 for a T; etc.
- After the third position in the alignment, 3 out of 5 sequences have “insertions” of varying lengths, so we say the probability of making an insertion is 3/5 and thus 2/5 for not makng one.
- A diagram: This is a hidden Markov model!

ACA---ATG

TCAACTATC

ACAC--AGC

AGA---ATC

ACCG--ATC

?

A 0.8

C 0.0

G 0.0

T 0.2

A 0.0

C 0.8

G 0.2

T 0.0

A ?

C ?

G ?

T ?

A ?

C ?

G ?

T ?

A 1.0

C 0.0

G 0.0

T 0.0

A 0.0

C 0.0

G 0.2

T 0.8

A 0.0

C 0.8

G 0.2

T 0.2

?

?

?

?

1.0

1.0

1.0

Hidden Markov Model

- A box is called a “(Match) state”:
- one state for each term in the regular expression.
- Probabilities: counting in the multiple alignment how many times each event occurs.
- “Insertion”: A state above the other states.
- Probabilities of NTs: counting all occurrences of the four NTs in this region of the alignment: A 1/5; C 2/5; G 1/5, and T 1/5.
- Probabilities of transitions:
- After sequences 2, 3 and 5 have made one insertion each, there are two more (from sequence 2)
- Total number of transitions back to the main line is 3: there are three sequences that have insertions. All will come back to the main states.
- Therefore, probability of making a transition to the next state: 3/5
- Probability of making a transition to itself: 2/5 --- keep inserting.

Scoring Sequences

- Consensus sequences: ACACATC.
- Probability of the 1st A: 4/5.
- This is multiplied by the probability of the transition from the first state to the second, which is 1.
- ….
- How do we score the exceptional sequence TGCT--AGG?
- This is 2000 times smaller.
- We can now get a score for each sequence to measure how well it fits the motif.

For the other four original sequences:

- The probability depends very strongly on the length of the sequence.
- Probability: Not a good number as score.
- Use log-odds ratio: ln( observed/random), here the random model (null model) is that the sequences are random strings of NTs:
- the probability of a sequence of length L is: 0.25 L

The log-odds score is:

- Other null model: not 0.25, but background NT compositions.
- When a sequence fits a HMM very well: high log-odds score
- When it fits a null model better: negative score.

The second sequence has raw score as low as the exceptional score

- because it has three inserts.
- But the log-odds score is much higher than the exceptional seq.
- Excellent discrimination.
- But, high log-odds may not be a “hit”: there will always be random hits when searching a database. Need to look at E-value and P-value.
- If the alignment had no gaps or insertions:
- No insert state.
- All probabilities associated with the arrows (the transition probabilities) = 1. Can all be ignored.
- HMM works then exactly as a weight matrix of log-odds scores.

What is hidden

- Come back to the occasional dishonest casino: they use a fair die most of the time, with a probability of 0.05 it switching to loaded die, and with a probability of 0.1 of switch back.
- The switch between die is a Markov process (it only depends on the previous state).
- The observation of the sequence of rolls is hidden Markov process because the casino wouldn’t tell you in which role they were using loaded die.

Profile HMMs

- Profile HMMs: allows position-dependent gap penalties.
- Obtained from a multiple alignment.
- Can be used to search a database for other members of the family just like a standard profile.
- Structure of the Model:
- Main states (bottom): model the columns of the alignment, are the main states.
- Probabilities are calculated by the frequency of the a.a. or NT.
- Insert states (diamond): model highly variable regions in the alignment
- Often the probabilities are a fixed distribution, eg, by composition

Delete states (circle): silent or null state. Do not match any residues, they are there so it is possible to jump over one or more columns:

- For modeling when just a few of the sequences have a “-” at a position.
- Example:

Insertion region (white): an alignment of this region is highly uncertain.

- Shaded region: columns that correspond to main states in the HMM model.
- Probabilities: For each non-insert column, we make a main state and set the probabilities equal to the amino acid frequencies.
- Transition probabilities: count how many sequences use the various transitions, like before.
- Delete states: Two transitions from a main state to a delete state, shown with dashed lines:
- from “begin” to the first delete state
- from main state 12 to delete state 13.
- Both correspond to dashes in the alignment:
- Only one sequence has gaps, the probability of these delete transitions is 1/30.
- The 4th sequence continues deletion to the end:
- Probability from delete 13 to 14 is 1, and from delete 14 to the end is also 1.

Pseudo-counts

- Dangerous to estimate a probability distribution from just a few observed amino acids.
- If there are two sequences, with Leu at a position:
- P for Leu =1, but P = 0 for all other residues at this position
- But we know that often Val substitutes Leu.
- The probability of the whole sequence are easily become 0 if a single Leu is substituted by a Val.
- Or , the log-odds is minus infinity.
- How to avoid “over-fitting” (strong conclusions drawn from very little evidence)?
- Use pseudocounts:
- Pretend to have more counts than those from the data.
- A. Add 1 to all the counts:
- Leu: 3/23, other a.a.: 1/23

Adding 1 to all counts is as assuming a priori all a.a. are equally likely.

- Another approach: use background composition as pseudocounts.

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