Hidden Markov model BioE 480 Sept 16, 2004

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=x 1 x 2 …x n 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, p ext and intracellular p int, 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 x i is the i th symbol in x, and y i the i th symbol in y. Assume that letter a occurs independently with some frequency q a. –The probability of the two sequences x and y is just the product of the probabilities of each amino acid: P(x,y|R) =  q xi  q yi An alternative model: Match Model M: Aligned pairs of residues occur with a joint probability P ab. 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 xiyi The ratio of these two likelihoods is the odds ratio: P(x,y|M) / P(x,y|R) =  p xiyi / (  q xi  q yi )=  p xiyi / q xi q yi To make this additive, we take the logarithm of this ratio, the log-odd ratio. S =  s(x i, y i ), where s(a, b) = log (p ab / q a q b ),

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” p ab. 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 q A 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 ? ? ? ? ?

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.