A retrospective look at our models
First we saw the finite state automaton
a
S
g
1
c
2
g
3
c
4
g
5
g
6
c
7
t
8
g
e
c
The finite state automaton above is nondeterministic, but is NOT probabilistic
Sequences under test are either elements of the set of
accepted sequences, or are rejected
The rigid non-stochastic nature of these structures
ultimately limited their usefulness to us as models of DNA
Markov Model
What do we need to probabilistically model DNA sequences?
States
Transition
probabilities
A
C
G
T
Here each observation necessarily corresponded to the underlying
biological state…which also limited the usefulness and generality of
the concept as a model for DNA or amino acid sequences
HMM Topology
Many different HMM topologies are possible…
a
S
R
e
b
Careful selection of alternative HMM topologies will allow us to address a
variety of different problems using essentially the same algorithmic toolkit
HMM Topology
Duration modeling with HMMs
C pG +
S
e
p
CpG Consider our CpG island model with two underlying states….
How long does our model dwell in a particular state?
P(L residues) = (1-p)·pL-1
This should be familiar as an exponentially decaying value
HMM Topology
Duration modeling with HMMs
C pG +
S
e
p
CpG P
P(L residues) =
(1-p)·pL-1
L
What if this doesn’t accurately reflect the real distribution of lengths?
HMM Topology
Duration modeling with HMMs
etc.
etc.
Assuming that each state shown here has the same emission
frequencies, this forms an example of a submodel topology that will
guarantee a minimum of four symbols drawn from the same
underlying emission distribution, but with a length distribution that
is still geometric. The duration of the model has been tweaked.
There are many more complex possibilities depending on the
properties of the length distribution you seek to model
Note: this and numerous other figures that follow in this presentation are after Durbin et al.
etc.
HMM Topology
Duration modeling with HMMs
etc.
The topology shown in this variation can model any
desired distribution of lengths varying between 2 and 6
HMM Topology
Duration modeling with HMMs
p
etc.
p
1-p
𝑷 𝒍 =
p
1-p
p
1-p
1-p
etc.
𝒍−𝟏
𝒍−𝒏
𝒏
𝒑 (𝟏 − 𝒑)
𝒏−𝟏
Where n = number of repetitive states, so here n = 4
Without belaboring the derivation, we’ll just mention that this negative binomial
distribution offers a great deal of flexibility in modeling length distributions
HMM Topology
Duration modeling with HMMs
http://www.boost.org/doc/libs/1_40_0/libs/math/doc/sf_and_dist/html/
math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html
By selecting appropriate values for the number of repeating states and the
probability of re-entering each state, you can flexibly model the length distribution
HMM Topology
Duration modeling with HMMs
http://www.boost.org/doc/libs/1_40_0/libs/math/doc/sf_and_dist/html/
http://www.boost.org/doc/libs/1_40_0/libs/math/doc/sf_and_dist/html/
math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html
math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html
In practical HMMs we would probably use n much smaller than in these
examples, but with probability of state re-entry much closer to 1
HMM Topology
A trivial HMM is equivalent to a Position Specific Scoring Matrix
S
M1
M2
...
ML
e
The transitions are deterministic and Pr{aMi→Mi+1}= 1 but
the emissions correspond to the estimated amino acid
or nucleotide frequencies in the columns of a PSSM
We refer collectively to M1…Mj as match states
This simple model cannot account for indels and does not
capture all of the information present in a multiple alignment
HMM Topology
Insertions are handled with special states with background-like emission
Ij
S
M1
...
Mj
Mj+1
e
Insertion states correspond to states that do not match
anything in the model. They almost always have emission
probabilities matching the background distribution
We still need a way to deal with deletions…
HMM Topology
How do insertions affect the overall probability of a sequence?
For an insertion of
length k:
S
M1
...
Ij
Mj
Mj+1
e
log aMj→Ij + (k-1)·log aIj→Ij + log aIj→Mj+1
Assuming log-odds scoring relative to some background distribution, emissions
from Ij will not contribute to the overall score, and only the transitions will matter
HMM Topology
How do insertions affect the overall probability of a sequence?
For an insertion of
length k:
S
M1
...
Ij
Mj
Mj+1
e
log aMj→Ij + (k-1)·log aIj→Ij + log aIj→Mj+1
This is equivalent to the familiar gap opening + gap extension penalties used in
many sequence alignment methods. This is therefore a form of affine gap scoring
HMM Topology
How best to handle deletions?
e
S
The problem with allowing arbitrarily long gaps in any
practically sized model is that we would need to
estimate far too many transition probabilities
Estimation is the hardest HMM problem, so we should avoid an
unnecessary proliferation of transitions wherever possible!
HMM Topology
In general we handle deletions by adding silent states…
Dj
S
Mj
e
We can use a sequence of silent-state transitions to
move from any match state to any other match state!
Note that not all Dj→Dj+1 transitions need have the same probability
The Profile HMM
If we combine all these features, we have the famous profile HMM
Dj
Ij
S
Mj
We can use a sequence of silent-state transitions to
move from any match state to any other match state!
Profile HMMs fully generalise the concept of pairwise alignment!
e
The Profile HMM
If we combine all these features, we have the famous profile HMM
Dj
Ij
S
Mj
e
Profile HMMs are extensively used for the identification
of new members of conserved sequence families
This relies on the ability to estimate parameters for the profile
HMM on the basis of multiple sequence alignments
Basic estimation for profile HMMs
Consider a multiple sequence alignment
#pos
01234567890
>glob1 VGA--HAGEY
>glob2 V----NVDEV
>glob3 VEA--DVAGH
>glob4 VKG------D
>glob5 VYS--TYETS
>glob6 FNA--NIPKH
>glob7 IAGADNGAGV
*** *****
First heuristic: positions with more than 50% gaps will be modeled as
inserts, the remainder as matches. In this example only the starred
columns will correspond to matches
We can now count simply count the transitions and emissions to calculate our maximum
likelihood estimators from frequencies. But what about missing observations?
Basic estimation for profile HMMs
Parameter estimation from multiple alignments
Second heuristic: use pseudocounts to fill in missing observations..
𝒂𝒌→𝒍 =
𝒆𝒌 (𝒃) =
𝒌→𝒍 + 𝑩 ·𝒒𝒌→𝒍
𝑩 + 𝒍′ |𝒌→𝒍′ |
𝒌 𝒂 +|𝑩|·𝒒𝒌(𝒃)
𝑩 + 𝒃′ |𝒌(𝒃′ )|
Here B is the total number of pseudocounts, and q represents the
fraction of the total number that have been allocated to that
particular transition or emission
Let’s start by simply applying Laplace’s rule: add one pseudocount to every frequency calculation
Basic estimation for profile HMMs
Consider a multiple sequence alignment
#pos
01234567890
>glob1 VGA--HAGEY
>glob2 V----NVDEV
>glob3 VEA--DVAGH
>glob4 VKG------D
>glob5 VYS--TYETS
>glob6 FNA--NIPKH
>glob7 IAGADNGAGV
*** *****
eM1(V) = 6/27, eM1(I) = eM1(F) = 2/27, eM1(all other aa) = 1/27
aM1→M2 = 7/10, aM1→D2 = 2/10, aM1→I1 = 1/10, etc.
These are the estimates when we apply Laplace’s rule for adding pseudocounts..
Scoring with profile HMMs
The state path of sequences under test are always unknown
Since the state path is unknown we have two basic options:
• Employ the Viterbi algorithm to determine the joint probability
of the observed sequence and the most probable state path
• Employ the forward (or backward) algorithm to determine the
full probability summed over all possible state paths
Either way, one fly in the ointment is that we are usually interested in a
score expressed as a log-odds ratio relative to our random model:
𝑷 𝒙 𝒓𝒂𝒏𝒅𝒐𝒎 =
𝒒𝒙𝒊
𝒊
We could calculate separate scores for the models, then combine, but this is inefficient
Scoring with profile HMMs
Modifying Viterbi and Forward for profile HMMs
We could rewrite the recurrences to accommodate log-odds scoring.
For example, Viterbi might look like this:
𝒗𝒋
𝒌
𝒊 = 𝒍𝒐𝒈
𝒆𝒌𝒋 (𝒙𝒊 )
𝒒𝒙 𝒊
𝒗𝑴 𝒋−𝟏 𝒊 − 𝟏 + 𝒍𝒐𝒈 𝒂𝑴𝒋−𝟏→𝑴𝒋
+𝒎𝒂𝒙
𝒗𝑰 𝒋−𝟏 𝒊 − 𝟏 + 𝒍𝒐𝒈𝒂𝑰𝒋−𝟏→𝑰𝒋
𝒗𝑫 𝒋−𝟏 𝒊 − 𝟏 + 𝒍𝒐𝒈𝒂𝑫𝒋−𝟏→𝑫𝒋
This would be a serious nuisance except we already have a log_float
class!!! So, we can instead just alter our emission terms to the form below:
emission = log_float(
𝒆𝒌𝒋 (𝒙𝒊 )
𝒒𝒙 𝒊
)
In other words, we need only adjust and “log_floatize” our emissions, which is
something we can do up front in the self.emissions dict rather than in the algorithms
Handling multiple sequences in Python
Inheritance is a powerful tool for deriving new classes
class FastA(list):
# note derivation from list, not object
def __init__(self):
#stuff
def read_file(self, filepath):
# stuff
def calculate_MLE(self, pseudocounts)
# stuff
return (transitions, emissions)
Our goal is to develop a class, FastA, that behaves like a python list that serves as a
container for sequences read from a file. Each element will contain a tuple of the
annotation and the sequence stored as a string. It should also be able to use these
sequences to then calculate “dict-of-dict” transition and emission distributions suitable
for use in our increasingly sophisticated HMM class
I’ll provide more information about the key methods in the form of a Python docstring