prologue: pitfalls of standard alignments

PROLOGUE:

Pitfalls of standard alignments

A:ALAEVLIRLITKLYP B:ASAKHLNRLITELYP

Blosum62Scoring a pairwise alignment

Alignment of a family (globins).Different positions are not equivalent

http://weblogo.berkeley.edu/cache/file5h2DWc.pngThe substitution score IN A FAMILY should depend on the position (the same for gaps)Sequence logosFor modelling families we need more flexible tools

Probabilistic Models for Biological SequencesWhat are they?

Generative definition:

Objects producing different outcomes (sequences) with different probabilities

The probability distribution over the sequences space determines the model specificity

Probabilistic models for sequencesMSequence spaceProbabilityGenerates si with probability P(si | M)e.g.: M is the representation of the family of globins

Associative definition:

Objects that, given an outcome (sequence), compute a probability value

Probabilistic models for sequencesMSequence spaceProbabilityAssociates probability P(si | M) to sie.g.: M is the representation of the family of globins We dont need a generator of new biological sequences

the generative definition is useful as operative definition

Probabilistic models for sequencesSequence spaceProbabilityMost useful probabilistic models are Trainable systems

The probability density function over the sequence space is estimated from known examples by means of a learning algorithmSequence spaceProbabilityKnown examplesPdf estimate (generalization)e.g.: Writing a generic representation of the sequences of globins starting from a set of known globins

Probabilistic Models for Biological SequencesWhat are they?Why to use them?

Modelling a protein familyProbabilistic modelGiven a protein class (e.g. Globins), a probabilistic model trained on this family can compute a probability value for each new sequenceThis value measures the similarity between the new sequence and the family described by the model

Probabilistic Models for Biological SequencesWhat are they?Why to use them?Which probabilities do they compute?

A model M associates to a sequence s the probability P( s | M )

This probability answers the question:

Which is the probability for a model describing the Globins to generate the sequence s ?

The question we want to answer is:

Given a sequence s, is it a Globin?

We need to compute P( M | s ) !!P( s | M ) or P( M | s ) ?

P(M | s) = P(s | M) P(M)P(s)A priori probabilitiesThe A priori probabilitiesP(M) is the probability of the model (i.e. of the class described by the model) BEFORE we know the sequence:

can be estimated as the abundance of the classP(s) is the probability of the sequence in the sequence space.

Cannot be reliably estimated!!

Comparison between models===We can overcome the problem comparing the probability of generating s from different models Ratio between the abundance of the classes

Null modelOtherwise we can score a sequence for a model M comparing it to a Null Model: a model that generates ALL the possible sequences with probabilities depending ONLY on the statistical amino acid abundance

S(M, s) = log P(s | M) P(s | N)In this case we need a threshold and a statistic for evaluating the significance (E-value, P-value)S(M, s)Sequences belonging to model MSequences NOT belonging to model M

The simplest probabilistic models:Markov ModelsDefinition

CRSFC: CloudsR: RainF: FogS: SunMarkov Models Example: WeatherRegister the weather conditions day by day:

as a first hypothesis the weather condition in a day depends ONLY on the weather conditions in the day before.

Define the conditional probabilities

P(C|C), P(C|R),. P(R|C)..

The probability for the 5-days registration CRRCS

P(CRRCS) = P(C)P(R|C) P(R|R) P(C|R) P(S|C)

Stochastic generator of sequences in which the probability of state in position i depends ONLY on the state in position i-1

Given an alphabet C = {c1; c2; c3; cN }

a Markov model is described with N(N+2) parameters {art , aBEGIN t , ar END; r, t C}

arq = P( s i = q| s i-1 = r ) aBEGIN q = P( s 1 = q ) ar END = P( s T = END | s T-1 = r )Markov Model t art + ar END = 1 r t aBEGIN t = 1c3c1c2c4cNENDBEGIN

Given the sequence: s = s1s2s3s4s6 sT with si C = {c1; c2; c3; cN }

P( s | M ) = P( s1 ) i=2 P( s i | s i-1 ) =

Markov ModelsP(ALKALI)= aBEGIN A aA L aL K aK A aA L aL I aI END

Markov Models: Exercise1) Fill the non defined values for the transition probabilities

Markov Models: Exercise2) Which model better describes the weather in summer? Which one describes the weather in winter?

Markov Models: Exercise3) Given the sequenceCSSSCFS

which model gives the higher probability?[Consider the starting probabilities: P(X|BEGIN)=0.25]WinterSummer

Markov Models: ExerciseP (CSSSCFS | Winter) ==0.25x0.1x0.2x0.2x0.3x0.2x0.2==1.2 x 10-5

P (CSSSCFS | Summer) ==0.25x0.4x0.8x0.8x0.1x0.1x1.0==6.4 x 10-4

4) Can we conclude that the observation sequence refers to a summer week?WinterSummer

Markov Models: ExerciseP (Seq | Winter) =1.2 x 10-5

P (Seq | Summer) =6.4 x 10-4

WinterSummerP(Seq |Summer) P(Summer)P(Seq| Winter) P(Winter)==

DNA:C = {Adenine, Citosine, Guanine, Timine }

16 transition probabilities (12 of which independent) +4 Begin probabilities +4 End probabilities.Simple Markov Model for DNA sequencesThe parameters of the model are different in different zones of DNA

They describe the overall composition and the couple recurrences

Example of Markov Models: GpC IslandIn the Markov Model of GpC Islands aGC is higher than in Markov Model Non-GpC IslandsGpC IslandsNon-GpC IslandsGiven a sequence s we can evaluate

GATGCGTCGC

CTACGCAGCG

The simplest probabilistic models:Markov ModelsDefinitionTraining

Probabilistic training of a parametric methodGenerally speaking, a parametric model M aims to reproduce a set of known dataModel MParameters TModelled dataReal data (D)How to compare them?

Let M be the set of parameters of model M.

During the training phase, M parameters are estimated from the set of known data D

Maximum Likelihood Extimation (ML)

ML = argmax P( D | M, )

It can be proved that:Training of Markov ModelsMaximum A Posteriori Extimation (MAP)

MAP = argmax P( | M, D ) = argmax [P( D | M, ) P( ) ]

Frequency of occurrence as counted in the data set D

Example (coin-tossing)Given N tossing of a coin (our data D), the outcomes are h heads and t tails (N=t+h)ASSUME the modelP(D|M)= ph (1- p)t Computing the maximum likelihood of P(D|M) We obtain that our estimate of p is p = h / (h+t) = h / N

Example (Error measure)Suppose you think that your data are affected by a Gaussian errorSo that they are distributed according toF(xi)=A*exp-[(xi m)2 /2s 2] With A=1/sqrt(2 s)If your measures are independent the data likelihood is P(Data| model) = Pi F(xi)Find m and s that maximize the P(Data| model)

Maximum Likelihood training: Proof

Given a sequence s contained in D:s = s1s2s3s4s6 sT We can count the number of transitions between any to states j and k: njkWhere states 0 and N+1 are BEGIN and ENDNormalisation contstraints are taken into account using the Lagrange multipliers lk

Hidden Markov ModelsPreliminary examples

Given a sequence:

4156266656321636543662152611536264162364261664616263

We dont know the sequence of dice that generated it.

RRRRRLRLRRRRRRRLRRRRRRRRRRRRLRLRRRRRRRRLRRRRLRRRRLRRLoaded dice

We have 99 regular dice (R) and 1 loaded die (L).

P(1)P(2)P(3)P(4)P(5)P(6)R1/61/61/61/61/61/6L1/101/101/101/101/101/2

Hypothesis:

We chose a different die for each roll

Two stochastic processes give origin to the sequence of observations.

1) Choosing the die ( R o L ). 2) Rolling the die

The sequence of dice is hidden

The first process is assumed to be Markovian (in this case a 0-order MM)

The outcome of the second process depends only on the state reached in the first process (that is the chosen die)Loaded dice

Model

Each state (R and L) generates a character of the alphabet C = {1, 2, 3, 4, 5, 6 }

The emission probabilities depend only on the state.

The transition probabilities describe a Markov model that generates a state path: the hidden sequence (p)

The observations sequence (s) is generated by two concomitant stochastic processesRL0.010.01

0.99

0.99

Casin

The observations sequence (s) is generated by two concomitant stochastic processes

RL0.010.01

0.99

0.99

Choose the State : RProbability= 0.99

Chose the Symbol: 1 Probability= 1/6 (given R)4156266656321636543662152611RRRRRLRLRRRRRRRLRRRRRRRRRRRR4156266656321636543662152611RRRRRLRLRRRRRRRLRRRRRRRRRRRR

The observations sequence (s) is generated by two concomitant stochastic processes

RL0.010.01

0.99

0.99

Choose the State : LProbability= 0.99

Chose the Symbol: 5 Probability= 1/10 (given L)415626665632163654366215261RRRRRLRLRRRRRRRLRRRRRRRRRRR41562666563216365436621526115RRRRRLRLRRRRRRRLRRRRRRRRRRRRL

Model

Each state (R and L) generates a character of the alphabet C = {1, 2, 3, 4, 5, 6 }

The emission probabilities depend only on the state.

The transition probabilities describe a Markov model that generates a state path: the hidden sequence (p)

The observations sequence (s) is generated by two concomitant stochastic processesLoaded dice

Some not so serious example1) DEMOGRAPHY

Observable: Number of births and deaths in a year in a village. Hidden variable: Economic conditions (as a first approximation we can consider the success in business as a random variable, and by consequence, the wealth as a Markov variable

---> can we deduce the economic conditions of a village during a century by means of the register of births and deaths?

2) THE METEREOPATHIC TEACHER

Observable: Average of the marks that a metereopathic teacher gives to their students during a day. Hidden variable: Weather conditions

---> can we deduce the weather conditions during a years by means of the class register?

To be more serious1) SECONDARY STRUCTURE Observable: protein sequence Hidden variable: secondary structure

---> can we deduce (predict) the secondary structure of a protein given its amino acid sequence?

2) ALIGNMENT Observable: protein sequence Hidden variable: position of each residue along the alignment of a protein family

---> can we align a protein to a family, starting from its amino acid sequence?

Hidden Markov ModelsPreliminary examplesFormal definition

A HMM is a stochastic generator of sequences characterised by: N states A set of transition probabilities between two states {akj}akj = P( (i) = j | (i-1) = k ) A set of starting probabilities {a0k}a0k = P( (1) = k ) A set of ending probabilities {ak0}ak0 = P( p (i) = END | (i-1) = k ) An alphabet C with M characters. A set of emission probabilities for each state {ek (c)}ek (c) = P( s i = c | (i) = k )Constraints:k a0 k = 1ak0 + j ak j = 1 kc C ek (c) = 1 k

Formal definition of Hidden Markov Models s: sequencep: path through the states

Generating a sequence with a HMM

s :AGCGCGTAATCTGp :YYYYYYYNNNNNN

P( s, p | M ) can be easily computed

GpC Island, simple model

P( s, p | M ) can be easily computed s : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y N N N N N NEmission: 0.1 0.4 0.4 0.4 0.4 0.4 0.10.250.250.250.250.250.25Transition: 0.2 0.7 0.7 0.7 0.7 0.7 0.7 0.20.8 0.8 0.80.8 0.8 0.1 Multiplying all the probabilities gives the probability of having the sequence AND the path through the states

Evaluation of the joint probability of the sequence ad the path

Hidden Markov ModelsPreliminary examplesFormal definitionThree questions

s :AGCGCGTAATCTGp:?????????????

P( s, p | M ) can be easily computed How to evaluate P ( s | M )?


How to evaluate P ( s | M )?s : A G C G C G T A A T C T Gp1: Y Y Y Y Y Y Y Y Y Y Y Y Yp2: Y Y Y Y Y Y Y Y Y Y Y Y Np3: Y Y Y Y Y Y Y Y Y Y Y N Yp4: Y Y Y Y Y Y Y Y Y Y Y N Np5: Y Y Y Y Y Y Y Y Y Y N Y Y213 different pathsSumming over all the path will give the probability of having the sequenceP ( s | M ) = p P( s, p | M )

s :AGCGCGTAATCTGp :?????????????

P( s, p | M ) can be easily computed How to evaluate P ( s | M )?Can we show the hidden path?

Resum: GpC Island, simple model

Can we show the hidden path? s : A G C G C G T A A T C T Gp1: Y Y Y Y Y Y Y Y Y Y Y Y Yp2: Y Y Y Y Y Y Y Y Y Y Y Y Np3: Y Y Y Y Y Y Y Y Y Y Y N Yp4: Y Y Y Y Y Y Y Y Y Y Y N Np5: Y Y Y Y Y Y Y Y Y Y N Y Y213 different pathsViterbi path: path that gives the best joint probabilityp* = argmax p [ P( p | s, M ) ] = argmax p [ P( p , s | M ) ]

A Posteriori decoding

For each position choose the state p (t) : p (i) = argmax k [ P( p (i) = k| s, M ) ]

The contribution to this probability derives from all the paths that go through the state k at position i.

The A posteriori path can be a non-sense path (it may not be a legitimate path if some transitions are not permitted in the model)

Can we show the hidden path?

s :AGCGCGTAATCTGp :YYYYYYYNNNNNN

P( s, p | M ) can be easily computed How to evaluate P ( s | M )?Can we show the hidden path? Can we evaluate the parameters starting from known examples?


Can we evaluate the parameters starting from known examples? s : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y N N N N N NEmission: eY (A)eY (G)eY (C)e Y(G)eY (C)eY (G)eY (T)eN (A)eN (A)eN (T)eN (C)eN (T)eN (G) Transition: a0Y aYY aYY aYY aYY aYY aYY aYN aNN aNN aNN aNN aNN aN0How to find the parameters e and a that maximises this probability?How if we dont know the path?

Hidden Markov Models:Algorithms ResumEvaluating P(s | M): Forward Algorithm

Computing P( s,p | M ) for each path is a redundant operations : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y Y Y Y Y Y YEmission: 0.1 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.4 0.1 0.4Transition: 0.2 0.7 0.7 0.7 0.7 0.7 0.7 0.70.7 0.7 0.70.7 0.7 0.1 s : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y Y Y Y Y Y NEmission: 0.1 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.4 0.1 0.25Transition: 0.2 0.7 0.7 0.7 0.7 0.7 0.7 0.70.7 0.7 0.70.7 0.2 0.1

Computing P( s,p | M ) for each path is a redundant operationIf we compute the common part only once we gain 2(T-1) operations

Summing over all the possible pathss : A G p : Y Y Emission: 0.1 0.4 Transition: 0.2 0.7 s : A G p : Y N Emission: 0.1 0.25 Transition: 0.2 0.2 s : A G p : N Y Emission: 0.250.4 Transition: 0.8 0.1 s : A G p : N N Emission: 0.250.25 Transition: 0.8 0.8 0.00560.0010.040.008

s : A G C p : X Y Y0.0136Summing over all the possible paths 0.4 0.7 s : A G C p : X Y N0.0136 0.25 0.2 s : A G C p : X N Y0.041 0.4 0.1 s : A G C p : X N N0.041 0.25 0.8 ++

Summing over all the possible pathsA G C G C G T A A T C T GX X X X X X X X X X X X YA G C G C G T A A T C T GX X X X X X X X X X X X N 0.1 (aY0) 0.1 (aN0) +P(s|M)Iterating until the last position of the sequence:

If we know the probabilities of emitting the two first characters of the sequence ending the path in states L and R respectively:

FR(2) P(s1,s2,p(2) = R | M) and FL(2) P(s1,s2,p(2) = L | M)

then we can compute:

P(s1,s2,s3,p(3) = R | M) = FR(2) aRR eR(s3) + FL(2) aLR eR(s3) Summing over all the possible paths

On the basis of preceding observations the computation of P(s | M) can be decomposed in simplest problems

For each state k and each position i in the sequence, we compute:Fk(i) = P( s1s2s3s i, p (i) = k | M)

Initialisation: FBEGIN (0) = 1 Fi (0) = 0 i BEGINRecurrence: Fl ( i+1) = P( s1s2s is i+1, p (i + 1) = l ) = = k P( s1s2 s i, p (i) = k ) a k l e l ( s i+1 ) = = e l ( s i+1 ) k Fk ( i ) a k lTermination: P( s ) = P( s1s2s3s T, p (T + 1) = END ) = =k P( s1s2 s T , p (T) = k ) a k0 = k Fk ( T ) a k 0

Forward AlgorithmWill be understood

STATEIterationBEGINENDAB01FB (2)SeB (s2)TT + 1Fi (1) aiBP(s | M)Forward Algorithm

Naf method P ( s | M ) = p P( s, p | M )There are N T possible paths.Each path requires about 2T operations.The time for the computation is O( T N T )Forward algorithm: computational complexitys : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y Y Y Y Y Y YEmission: 0.1 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.4 0.1 0.4Transition: 0.2 0.7 0.7 0.7 0.7 0.7 0.7 0.70.7 0.7 0.70.7 0.7 0.1 s : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y Y Y Y Y Y NEmission: 0.1 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.4 0.1 0.25Transition: 0.2 0.7 0.7 0.7 0.7 0.7 0.7 0.70.7 0.7 0.70.7 0.2 0.1

s : A G C p : X Y Y0.0136 0.4 0.7 s : A G C p : X Y N0.0136 0.25 0.2 s : A G C p : X N Y0.041 0.4 0.1 s : A G C p : X N N0.041 0.25 0.8 s : A G C p : X X Y s : A G C p : X X N0.0054480.00888++SumForward algorithmT positions, N values for each positionEach element requires about 2N product and 1 sum The time for the computation is O(T N2)

Forward algorithm: computational complexity

Forward algorithm: computational complexityNaf method Forward algorithm

Grafico1

88

2412

6416

16020

38424

89628

204832

T

No. of operations

Foglio1

T

124

288

32412

46416

516020

638424

789628

8204832

9460836

101024040

Foglio2

Foglio3

Hidden Markov Models:Algorithms ResumEvaluating P(s | M): Forward AlgorithmEvaluating P(s | M): Backward Algorithm

Backward AlgorithmSimilar to the Forward algorithm: it computes P( s | M ), reconstructing the sequence from the end

For each state k and each position i in the sequence, we compute:Bk(i) = P( s i+1s i+2s i+3s T | p (i) = k )

Initialisation: Bk (T) = P(p (T+1) = END | p (T) = k ) = ak0Recurrence: Bl ( i-1) = P(s is i+1s T | p (i - 1) = l ) = = k P(s i+1s i+2s T | p (i) = k) a l k e k (s i )= = k Bk ( i ) e k ( s i ) a l kTermination: P( s ) = P( s1s2s3s T | p (0) = BEGIN ) = = k P( s2 s T | p (1) = k ) a 0 k e k ( s 1 ) = = k Bk ( 1 ) a 0k e k ( s 1 )

STATEIterationBEGINENDAB012TT + 1Backward AlgorithmBB (T-1)Bk(T) aB T e k (s T-1 )T-1SP(s | M)

Hidden Markov Models:Algorithms ResumEvaluating P(s | M): Forward AlgorithmEvaluating P(s | M): Backward AlgorithmShowing the path: Viterbi decoding

Finding the best paths : A G p : Y Y Emission: 0.1 0.4 Transition: 0.2 0.7 s : A G p : Y N Emission: 0.1 0.25 Transition: 0.2 0.2 s : A G p : N Y Emission: 0.250.4 Transition: 0.8 0.1 s : A G p : N N Emission: 0.250.25 Transition: 0.8 0.8 0.00560.0010.040.008

s : A G C p : N Y Y0.008 0.4 0.7 s : A G C p : N Y N0.008 0.25 0.2 s : A G C p : N N Y0.04 0.4 0.1 s : A G C p : N N N0.04 0.25 0.8 ;=0.00224=0.0016=0.0004=0.008Finding the best path

A G C G C G T A A T C T GN Y Y Y Y Y Y N N N Y Y YA G C G C G T A A T C T GN N N N N N N N N N N N N 0.1 (aY0) 0.1 (aN0) Choose the MaximumIterating until the last position of the sequence:Finding the best path

Viterbi Algorithmp* = argmax p [ P( p , s | M ) ]The computation of P(s,p*| M) can be decomposed in simplest problems

Let Vk(i) be the probability of the most probable path for generating the subsequence s1s2s3s i ending in the state k at iteration iInitialisation: VBEGIN (0) = 1 Vi (0) = 0 i BEGINRecurrence: Vl ( i+1) = e l ( s i+1 ) Max k ( Vk ( i ) a k l )ptr i ( l ) = argmax k ( Vk ( i ) a k l )Termination: P( s, p* ) =Maxk (Vk ( T ) a k 0 )p* ( T ) = argmax k (Vk ( T ) a k 0 )Traceback:p* ( i-1 ) = ptr i (p* ( i ))

Viterbi AlgorithmSTATEIterationBEGINENDAB01VB (2)MAXeB (s2)TT + 1Vi (1) aiBP(s, *| M)ptr2 (B)

Viterbi AlgorithmViterbi pathDifferent paths can have the same probability

Hidden Markov Models:Algorithms ResumEvaluating P(s | M): Forward AlgorithmEvaluating P(s | M): Backward AlgorithmShowing the path: Viterbi decodingShowing the path: A posteriori decodingTraining a model: EM algorithm

If we know the path generating the training sequences : A G C G C G T A A T C T Gp : Y Y Y Y Y Y Y N N N N N NEmission: eY (A)eY (G)eY (A)e Y(G)eY (C)eY (G)eY (T)eN (A)eN (A)eN (T)eN (C)eN (T)eN (G) Transition: a0Y aYY aYY aYY aYY aYY aYY aYN aNN aNN aNN aNN aNN aN0Just count!Example: aYY= nYY /(nYY+ nYN)= 6/7eY(A) = nY(A) /[nY(A)+nY(C) +nY(G) +nY(T)]= 1/7

Expectation-Maximisation algorithmWe need to estimate the Maximum Likelihood parameters when the paths generating the training sequences are unknown

qML = argmaxq [P ( s | q, M)]

Given a model with parameters q0 the EM algorithm finds new parameters q that increase the likelihood of the model:

P( s | q ) > P( s| q0 )

Ak,l = Sp P(p | s,q0) Ak,l(p)Ek (c) = Sp P(p | s,q0) Ek (c,p)We can compute the expected values over all the pathsak,l = ek(c) = Expectation-Maximisation algorithmGiven a path p we can countthe number of transition between states k and l: Ak,l(p)the number on emissions of character c from state k: Ek (c,p) s : A G C G C G T A A T C T Gp1: Y Y Y Y Y Y Y Y Y Y Y Y Yp2: Y Y Y Y Y Y Y Y Y Y Y Y Np3: Y Y Y Y Y Y Y Y Y Y Y N Yp4: Y Y Y Y Y Y Y Y Y Y Y N N...The updated parameters are:

Expectation-Maximisation algorithmWe need to estimate the Maximum Likelihood parameters when the paths generating the training sequences are unknown

qML = argmaxq [P ( s | q, M)]

Given a model with parameters q0 the EM algorithm finds new parameters q that increase the likelihood of the model:

P( s | q ) > P( s| q0 )

or equivalentely

log P( s | q ) > log P( s| q0 )

Expectation-Maximisation algorithmThe EM algorithm is an iterative process

Each iteration performs two steps:E-step: evaluation of Q(q |q0) = Sp P(p | s,q0) log P(s,p | q ) M-step: Maximisation of Q(q |q0) over all q

It does NOT assure to converge to the GLOBAL Maximum Likelihood

E-step:Q( q |q0) = Sp P(p | s,q0) log P(s,p |q ) P(s,p |q ) = a0,p(1) Pi = 1 ap(i),p(i+1) ep(i)(si) = = Pk = 0 Pl = 1 ak,l Pk = 1 Pc C ek (c) TNNNAk.l (p)Ek (c,p)Ak,l = Sp P(p | s,q0) Ak,l(p)Ek (c) = Sp P(p | s,q0) Ek (c,p)Baum-Welch implementation of the EM algorithmExpected values over all the actual paths

ak,l =

ek(c) = Baum-Welch implementation of the EM algorithmM-step:For any state k and character c, with Sc ek(c) = 1By means of Lagranges multipliers techniques, we can solve the system

Fk(i) = P( s1s2s3s i, p (i) = k )

Bk(i) = P( s i+1s i+2s i+3s T | p (i) = k )

Ak,l= P(p (i ) = k , p ( i +1) = l | s,q) =

Ek (c) = P( s i = c , p (i ) = k | s,q) =

Baum-Welch implementation of the EM algorithmHow to compute the expected number of transitions and emissions over all the paths

Baum-Welch implementation of the EM algorithmAlgorithmStart with random parametersCompute Forward and Backward matrices on the known sequencesCompute Ak,l and Ek (c) expected numbers of transitions and emissions Update a k,l Ak,l ek (c) Ek (c)Has P(s|M) incremented ?YesNoEnd

Profile HMMsHMMs for alignments

M0M1M2M3M4How to align?Each state represent a position in the alignment.ACGGTAM0M1M2M3M4M5

ACGATCM0M1M2M3M4M5

ATGTTCM0M1M2M3M4M5

M5Each position has a peculiar composition

M0M1M2M3M4ACGGTAACGATCATGTTCM5Given a set of sequences....we can train a model..A1000.3300.33C00.660000.66G0010.3300T00.3300.3310..estimating the emission probabilities.

M0M1M2M3M4M5Given a trained model....we can align a new sequence..A1000.3300.33C00.660000.66G0010.3300T00.3300.3310ACGATC..computing the emission probabilityP(s|M) = 1 0.66 1 0.33 1 0.66

M0M1M2M3M4And for the sequence AGATC ?

AGATCM0M2M3M4M5M5M5A1000.3300.33C00.660000.66G0010.3300T00.3300.3310We need a way to introduce gaps

Silent statesRed transitions allow gaps(N-1) ! transitionsTo reduce the number of parameters we can use states that doesnt emit any character4N-8 transitions

Profile HMMsDelete statesInsert statesMatch statesA C G G T AM0 M1 M2 M3 M4 M5

A C G C A G T CM0 I0 I0 M1 M2 M3 M4 M5

A G A T CM0 D1 M2 M3 M4 M5

Example of alignmentSequence 1A S T R A LViterbi pathM0 M1 M2 M3 M4 M5A S T R A L

Sequence 2A S T A I LViterbi pathM0 M1 M2 D3 M4 I4 M5A S T A I L

Sequence 3A R T IViterbi pathM0 M1 M2 D3 D4 M5A R T I

Example of alignment-Log P(s | M) Is an alignment score

Grouping by vertical layers

0

1

2

3

4

5

s1

A

S

T

R

A

L

s2

A

S

T

AI

L

s3

A

R

T

I

Alignment

ASTRA-L

AST-AIL

ART---I

Searching for a structural/functional pattern in protein sequenceZn binding loop: C H C I C R I C C H C L C K I C C H C I C S L C D H C L C T I C C H C I D S I C C H C L C K I C

Cysteines can be replaced by an Aspartic Acid, but only ONCE for each sequence

Searching for a structural/functional pattern in protein sequences..ALCPCHCLCRICPLIY.. ..WERWDHCIDSICLKDE..obtains higher probability than.. because M0 and M4 have low emission probability for Aspartic Acid and we multiply them twice.

Profile HMMsHMMs for alignmentsExample on globins

Structural alignment of globins

Structural alignment of globinsBashdorf D, Chothia C & Lesk AM, (1987) Determinants of a protein fold: unique features of the globin amino sequence. J.Mol.Biol. 196, 199-216

Alignment of globins reconstructed with profile HMMsKrogh A, Brown M, Mian IS, Sjolander K & Haussler D (1994) Hidden Markov Models in computational biology: applications to protein modelling. J.Mol.Biol. 235, 1501-1531

Discrimination power of profile HMMsKrogh A, Brown M, Mian IS, Sjolander K & Haussler D (1994) Hidden Markov Models in computational biology: applications to protein modelling. J.Mol.Biol. 235, 1501-1531

Profile HMMsHMMs for alignmentsExample on globinsOther applications

BeginProfile HMM specific for the considered domain

EndFinding a domain

BEGINHMM 1HMM 2HMM 3HMM n.END.Clustering subfamiliesEach sequence s contributes to update HMM i with a weight equal to P ( s | Mi )

Profile HMMsHMMs for alignmentsExample on globinsOther applicationsAvailable codes and servers

HMMER at WUSTL: http://hmmer.wustl.edu/Eddy SR (1998) Profile hidden Markov models. Bioinformatics 14:755-763

HMMERAlignment of a protein familyhmmbuildTrained profile-HMM hmmcalibateHMM calibrated with the accurate E-value statistics Takes the aligned sequences, checks for redundancy and sets the emission and the transitions probabilities of a HMMTakes a trained HMM, generates a great number of random sequences, score them and fits the Extreme Value Distribution to the computed scores

HMMERSet of sequencesAlignment of all the sequences to the modelhmmalignList of sequences that match the HMM (sorted by E-value)hmmsearchHMMSet of HMMsSequencehmmpfamList of HMMs that match the sequence

MSF Format: globins50.msf

!!AA_MULTIPLE_ALIGNMENT 1.0

PileUp of: *.pep

Symbol comparison table: GenRunData:blosum62.cmp CompCheck: 6430

GapWeight: 12

GapLengthWeight: 4

pileup.msf MSF: 308 Type: P August 16, 1999 09:09 Check: 9858 ..

Name: lgb1_pea Len: 308 Check: 2200 Weight: 1.00

Name: lgb1_vicfa Len: 308 Check: 214 Weight: 1.00

Name: myg_escgi Len: 308 Check: 3961 Weight: 1.00

Name: myg_horse Len: 308 Check: 5619 Weight: 1.00

Name: myg_progu Len: 308 Check: 6401 Weight: 1.00

Name: myg_saisc Len: 308 Check: 6606 Weight: 1.00

//

1 50

lgb1_pea ~~~~~~~~~G FTDKQEALVN SSSE.FKQNL PGYSILFYTI VLEKAPAAKG

lgb1_vicfa ~~~~~~~~~G FTEKQEALVN SSSQLFKQNP SNYSVLFYTI ILQKAPTAKA

myg_escgi ~~~~~~~~~V LSDAEWQLVL NIWAKVEADV AGHGQDILIR LFKGHPETLE

myg_horse ~~~~~~~~~G LSDGEWQQVL NVWGKVEADI AGHGQEVLIR LFTGHPETLE

myg_progu ~~~~~~~~~G LSDGEWQLVL NVWGKVEGDL SGHGQEVLIR LFKGHPETLE

myg_saisc ~~~~~~~~~G LSDGEWQLVL NIWGKVEADI PSHGQEVLIS LFKGHPETLE

Alignment of a protein familyhmmbuildTrained profile-HMM hmmbuild globin.hmm globins50.msfAll the transition and emission parameters are estimated by means of the Expectation Maximisation algorithm on the aligned sequences.In principle we could use also NON aligned sequences to train the model. Nevertheless it is more efficient to build the starting alignment using, for example, CLUSTALW

hmmcalibrate [-num N] -histfile globin.histo globin.hmmTrained profile-HMM hmmcalibateHMM calibrated with the accurate E-value statistics A number of N (default 5000) random sequences are generated and scored with the model. Random sequencesLog P(s|M)/P(s|N)Range for globin sequencesE-value(S): expected number of random sequences with a score > S

Trained model (globin.hmm)HMMER2.0 [2.3.2]NAME globins50LENG 143ALPH AminoRF noCS noMAP yesCOM /home/gigi/bin/hmmbuild globin.hmm globins50.msfCOM /home/gigi/bin/hmmcalibrate --histfile globin.histo globin.hmmNSEQ 50DATE Sun May 29 19:03:18 2005CKSUM 9858XT -8455 -4 -1000 -1000 -8455 -4 -8455 -4 NULT -4 -8455NULE 595 -1558 85 338 -294 453 -1158 197 249 902 -1085 -142 -21 -313 45 531 201 384 -1998 -644 EVD -38.893742 0.243153HMM A C D E F G H I K L M N P Q R S T V W Y m->m m->i m->d i->m i->i d->m d->d b->m m->e -450 * -1900 1 591 -1587 159 1351 -1874 -201 151 -1600 998 -1591 -693 389 -1272 595 42 -31 27 -693 -1797 -1134 14 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 -450 * 2 -926 -2616 2221 2269 -2845 -1178 -325 -2678 -300 -2596 -1810 220 -1592 939 -974 -671 -939 -2204 -2785 -1925 15 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * * 3 -638 -1715 -680 497 -2043 -1540 23 -1671 2380 -1641 -840 -222 -1595 437 1040 -564 -523 -1363 2124 -1313 16 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * *

Trained model (globin.hmm)null modelHMMER2.0 [2.3.2]NAME globins50LENG 143ALPH AminoRF noCS noMAP yesCOM /home/gigi/bin/hmmbuild globin.hmm globins50.msfCOM /home/gigi/bin/hmmcalibrate --histfile globin.histo globin.hmmNSEQ 50DATE Sun May 29 19:03:18 2005CKSUM 9858XT -8455 -4 -1000 -1000 -8455 -4 -8455 -4 NULT -4 -8455NULE 595 -1558 85 338 -294 453 -1158 197 249 902 -1085 -142 -21 -313 45 531 201 384 -1998 -644 EVD -38.893742 0.243153HMM A C D E F G H I K L M N P Q R S T V W Y m->m m->i m->d i->m i->i d->m d->d b->m m->e -450 * -1900 1 591 -1587 159 1351 -1874 -201 151 -1600 998 -1591 -693 389 -1272 595 42 -31 27 -693 -1797 -1134 14 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 -450 * 2 -926 -2616 2221 2269 -2845 -1178 -325 -2678 -300 -2596 -1810 220 -1592 939 -974 -671 -939 -2204 -2785 -1925 15 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * * 3 -638 -1715 -680 497 -2043 -1540 23 -1671 2380 -1641 -840 -222 -1595 437 1040 -564 -523 -1363 2124 -1313 16 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * *

Score = INT [1000 log2(prob/null_prob)]null_prob = 1 for transitions

Trained model (globin.hmm)null modelHMMER2.0 [2.3.2]NAME globins50LENG 143ALPH AminoRF noCS noMAP yesCOM /home/gigi/bin/hmmbuild globin.hmm globins50.msfCOM /home/gigi/bin/hmmcalibrate --histfile globin.histo globin.hmmNSEQ 50DATE Sun May 29 19:03:18 2005CKSUM 9858XT -8455 -4 -1000 -1000 -8455 -4 -8455 -4 NULT -4 -8455NULE 595 -1558 85 338 -294 453 -1158 197 249 902 -1085 -142 -21 -313 45 531 201 384 -1998 -644 EVD -38.893742 0.243153HMM A C D E F G H I K L M N P Q R S T V W Y m->m m->i m->d i->m i->i d->m d->d b->m m->e -450 * -1900 1 591 -1587 159 1351 -1874 -201 151 -1600 998 -1591 -693 389 -1272 595 42 -31 27 -693 -1797 -1134 14 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 -450 * 2 -926 -2616 2221 2269 -2845 -1178 -325 -2678 -300 -2596 -1810 220 -1592 939 -974 -671 -939 -2204 -2785 -1925 15 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * * 3 -638 -1715 -680 497 -2043 -1540 23 -1671 2380 -1641 -840 -222 -1595 437 1040 -564 -523 -1363 2124 -1313 16 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * *

Score = INT [1000 log2(prob/null_prob)]

= natural abundance for emissions

HMMER2.0 [2.3.2]NAME globins50LENG 143ALPH AminoRF noCS noMAP yesCOM /home/gigi/bin/hmmbuild globin.hmm globins50.msfCOM /home/gigi/bin/hmmcalibrate --histfile globin.histo globin.hmmNSEQ 50DATE Sun May 29 19:03:18 2005CKSUM 9858XT -8455 -4 -1000 -1000 -8455 -4 -8455 -4 NULT -4 -8455NULE 595 -1558 85 338 -294 453 -1158 197 249 902 -1085 -142 -21 -313 45 531 201 384 -1998 -644 EVD -38.893742 0.243153HMM A C D E F G H I K L M N P Q R S T V W Y m->m m->i m->d i->m i->i d->m d->d b->m m->e -450 * -1900 1 591 -1587 159 1351 -1874 -201 151 -1600 998 -1591 -693 389 -1272 595 42 -31 27 -693 -1797 -1134 14 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 -450 * 2 -926 -2616 2221 2269 -2845 -1178 -325 -2678 -300 -2596 -1810 220 -1592 939 -974 -671 -939 -2204 -2785 -1925 15 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * * 3 -638 -1715 -680 497 -2043 -1540 23 -1671 2380 -1641 -840 -222 -1595 437 1040 -564 -523 -1363 2124 -1313 16 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * *

Trained model (globin.hmm)Transitions

HMMER2.0 [2.3.2]NAME globins50LENG 143ALPH AminoRF noCS noMAP yesCOM /home/gigi/bin/hmmbuild globin.hmm globins50.msfCOM /home/gigi/bin/hmmcalibrate --histfile globin.histo globin.hmmNSEQ 50DATE Sun May 29 19:03:18 2005CKSUM 9858XT -8455 -4 -1000 -1000 -8455 -4 -8455 -4 NULT -4 -8455NULE 595 -1558 85 338 -294 453 -1158 197 249 902 -1085 -142 -21 -313 45 531 201 384 -1998 -644 EVD -38.893742 0.243153HMM A C D E F G H I K L M N P Q R S T V W Y m->m m->i m->d i->m i->i d->m d->d b->m m->e -450 * -1900 1 591 -1587 159 1351 -1874 -201 151 -1600 998 -1591 -693 389 -1272 595 42 -31 27 -693 -1797 -1134 14 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 -450 * 2 -926 -2616 2221 2269 -2845 -1178 -325 -2678 -300 -2596 -1810 220 -1592 939 -974 -671 -939 -2204 -2785 -1925 15 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * * 3 -638 -1715 -680 497 -2043 -1540 23 -1671 2380 -1641 -840 -222 -1595 437 1040 -564 -523 -1363 2124 -1313 16 - -149 -500 233 43 -381 399 106 -626 210 -466 -720 275 394 45 96 359 117 -369 -294 -249 - -23 -6528 -7571 -894 -1115 -701 -1378 * *

Trained model (globin.hmm)Emissions

hmmemit [-n N] globin.hmmTrained profile-HMM hmmemitSequences generated by the model The parameters of the model are used to generate new sequences

hmmsearch globin.hmm Artemia.fa > Artemia.globinSet of sequencesList of sequences that match the HMM (sorted by E-value)hmmsearchTrained profile-HMM

Search results (Artemia.globin)Sequence Description Score E-value N -------- ----------- ----- ------- ---S13421 S13421 474.3 1.7e-143 9

Parsed for domains:Sequence Domain seq-f seq-t hmm-f hmm-t score E-value-------- ------- ----- ----- ----- ----- ----- -------S13421 7/9 932 1075 .. 1 143 [] 76.9 7.3e-24S13421 2/9 153 293 .. 1 143 [] 63.7 6.8e-20S13421 3/9 307 450 .. 1 143 [] 59.8 9.8e-19S13421 8/9 1089 1234 .. 1 143 [] 57.6 4.5e-18S13421 9/9 1248 1390 .. 1 143 [] 52.3 1.8e-16S13421 1/9 1 143 [. 1 143 [] 51.2 4e-16S13421 4/9 464 607 .. 1 143 [] 46.7 8.6e-15S13421 6/9 775 918 .. 1 143 [] 42.2 2e-13S13421 5/9 623 762 .. 1 143 [] 23.9 6.6e-08

Alignments of top-scoring domains:S13421: domain 7 of 9, from 932 to 1075: score 76.9, E = 7.3e-24 *->eekalvksvwgkveknveevGaeaLerllvvyPetkryFpkFkdLss +e a vk+ w+ v+ ++ vG +++ l++ +P+ +++FpkF d+ S13421 932 REVAVVKQTWNLVKPDLMGVGMRIFKSLFEAFPAYQAVFPKFSDVPL 978

adavkgsakvkahgkkVltalgdavkkldd...lkgalakLselHaqklr d++++++ v +h V t+l++ ++ ld++ +l+ ++L+e H+ lr S13421 979 -DKLEDTPAVGKHSISVTTKLDELIQTLDEpanLALLARQLGEDHIV-LR 1026

vdpenfkllsevllvvlaeklgkeftpevqaalekllaavataLaakYk< v+ fk +++vl+ l++ lg+ f+ ++ +++k+++++++ +++ + S13421 1027 VNKPMFKSFGKVLVRLLENDLGQRFSSFASRSWHKAYDVIVEYIEEGLQ 1075

Number of domainsDomains sorted byE-valueStartEndConsensus sequenceSequence

hmmalign globin.hmm globins630.faSet of sequenceshmmalignAlignment of all sequences to the modelTrained profile-HMM InsertionsBAHG_VITSP QAG-..VAAAHYPIV.GQELLGAIKEV.L.G.D.AATDDILDAWGKAYGVGLB1_ANABR TR-K..ISAAEFGKI.NGPIKKVLAS-.-.-.K.NFGDKYANAWAKLVAVGLB1_ARTSX NRGT..-DRSFVEYL.KESL-----GD.S.V.D.EFT------VQSFGEVGLB1_CALSO TRGI..TNMELFAFA.LADLVAYMGTT.I.S.-.-FTAAQKASWTAVNDVGLB1_CHITH -KSR..ASPAQLDNF.RKSLVVYLKGA.-.-.T.KWDSAVESSWAPVLDFGLB1_GLYDI GNKH..IKAQYFEPL.GASLLSAMEHR.I.G.G.KMNAAAKDAWAAAYADGLB1_LUMTE ER-N..LKPEFFDIF.LKHLLHVLGDR.L.G.T.HFDF---GAWHDCVDQGLB1_MORMR QSFY..VDRQYFKVL.AGII-------.-.-.A.DTTAPGDAGFEKLMSMGLB1_PARCH DLNK..VGPAHYDLF.AKVLMEALQAE.L.G.S.DFNQKTRDSWAKAFSIGLB1_PETMA KSFQ..VDPQYFKVL.AAVI-------.-.-.V.DTVLPGDAGLEKLMSMGLB1_PHESE QHTErgTKPEYFDLFrGTQLFDILGDKnLiGlTmHFD---QAAWRDCYAV

Gaps

HMMER applications:PFAMhttp://www.sanger.ac.uk/Software/Pfam/

PFAM ExerciseGenerate with hmmemit a sequence from the globin model and search it in PFAM database

Search in the SwissProt database the sequencesCG301_HUMANQ9H5F4_HUMAN

1) search them in the PFAM data base.2)launch PSI-BLAST searches. Is it possible to annotate the sequences by means of the BLAST results?PFAM Exercise

SAM at UCSD:http://www.soe.ucsc.edu/research/compbio/sam.htmlKrogh A, Brown M, Mian IS, Sjolander K & Haussler D (1994) Hidden Markov Models in computational biology: applications to protein modelling. J.Mol.Biol. 235, 1501-1531

SAM applications:http://www.cse.ucsc.edu/research/compbio/HMM-apps/T02-query.html

HMMPRO: http://www.netid.com/html/hmmpro.htmlPierre Baldi, Net-ID

HMMs for Mapping problemsMapping problems in protein prediction

Covalent structureTTCCPSIVARSNFNVCRLPGTPEAICATYTGCIIIPGATCPGDYANSecondary structure

position of Trans Membrane Segments along the sequenceTopographyTopology of membrane proteins

ALALMLCMLTYRHKELKLKLKK ALALMLCMLTYRHKELKLKLKK ALALMLCMLTYRHKELKLKLKK

HMMs for Mapping problemsMapping problems in protein predictionLabelled HMMs

HMM for secondary structure prediction

Simplest modelIntroducing a grammara1b1ca2a3b2

HMM for secondary structure predictionLabels

The states a1, a2 and a3 share the same label, so states b1 and b2 do.Decoding the Viterbi path for emitting a sequence s, makes a mapping between the sequence s and a sequence of labels y

S A L K M N Y T R E I M V A S N Q s: Sequencec a1 a2 a3 a4 a4 a4 c c c c b1 b2 b2 b2 c cp: Pathc a a a a a a c c c c b b b b c cY(p): Labelsa1b1ca2a3b2

Computing P(s, y | M)Only the path whose labelling is y have to be considered in the sumIn Forward and Backward algorithms it means to set

Fk(i) = 0, Bk(i) = 0 if Y(k) yi

Baum-Welch training algorithm for labelled HMMsGiven a set of known labelled sequences (e.g. amino acid sequences and their native secondary structure) we want to find the parameters of the model, without knowing the generating paths:

qML = argmaxq [P ( s, y | q, M)]

The algorithm is the same as in the non-labelled case if we use the Forward and Backward matrices defined in the last slide.

Supervised learning of the mapping

HMMs for Mapping problemsMapping problems in protein predictionLabelled HMMsDuration modelling

Self loops and geometric decayP(l) = p l-1 ( 1-p )The length distribution of the generated segments is always exp-like

How can we model other length distributions?Limited caseThis topology can model any length distribution between 1 and N

How can we model other length distributions?Non limited caseThis topology can model any length distribution between 1 and N-1 and a geometrical decay from N and

Secondary structure: length statistic

Grafico1

000

000.11612

00.1585330.154294

0.1972340.1359660.130081

0.1081250.1598030.131099

0.0513910.1386460.111321

0.0526480.1238360.073875

0.055320.0935120.057951

0.0523340.0671370.046099

0.0520190.0407620.035192

0.0609780.027080.027848

0.0605060.0187590.021377

0.0474620.0124120.017596

0.0474620.0088860.013742

0.0411760.0056420.010325

0.0326890.0029620.00938

0.0227880.0012690.006762

0.0235740.001410.006399

0.0176020.0005640.004508

0.012730.0007050.005017

0.0099010.0005640.002327

0.0102150.0007050.003127

0.0086440.0002820.002327

0.0040860.0001410.001818

0.00628600.001382

0.0034570.0002820.001527

0.00282900.000945

0.00314300.000873

0.003300.000582

0.00267200.000436

0.00204300.000727

0.00157200.000364

0.002200.000727

0.0003140.0001410.000291

0.00062900.000291

0.00031400.000291

&A

Pagina &P

Helix

Strand

Coil

Length (residues)

Frequency

Database

8rxna.odc.txt

6363709013753

HelixStrandCoil

0000

1000.11612

200.1585330.154294

30.1972340.1359660.130081

40.1081250.1598030.131099

50.0513910.1386460.111321

60.0526480.1238360.073875

70.055320.0935120.057951

80.0523340.0671370.046099

90.0520190.0407620.035192

100.0609780.027080.027848

110.0605060.0187590.021377

120.0474620.0124120.017596

130.0474620.0088860.013742

140.0411760.0056420.010325

150.0326890.0029620.00938

160.0227880.0012690.006762

170.0235740.001410.006399

180.0176020.0005640.004508

190.012730.0007050.005017

200.0099010.0005640.002327

210.0102150.0007050.003127

220.0086440.0002820.002327

230.0040860.0001410.001818

240.00628600.001382

250.0034570.0002820.001527

260.00282900.000945

270.00314300.000873

280.003300.000582

290.00267200.000436

300.00204300.000727

310.00157200.000364

320.002200.000727

330.0003140.0001410.000291

340.00062900.000291

350.00031400.000291

360.00015700.000218

370.00062900.000145

380.00015700.000291

390.00015700.000073

400.00015700.000218

41000.000145

42000.000145

430.00015700.000073

44000.000145

45000

46000

470.00031400.000145

48000.000145

490.00015700.000073

50000.000073

51000.000073

52000.000145

53000

54000.000073

55000.000073

56000.000073

57000

58000.000073

59000

60000

61000.000073

62000.000073

63000

640.00015700

65000.000145

660.00015700

670.00015700

68000

69000

70000

71000

72000

73000

74000

75000

76000

77000.000073

78000

79000

80000

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86000.000073

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99000.000073

100000.000073

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&A

Pagina &P

a3a4a5a6a7a2a1a8a9a10a13a14a12a11c3c4c2c1b3b4b5b2b1Secondary structure: modelDo we use the same emission probabilities for states sharing the same label?

HMMs for Mapping problemsMapping problems in protein predictionLabelled HMMsDuration modellingModels for membrane proteins

Porin (Rhodobacter capsulatus)Bacteriorhodopsin(Halobacterium salinarum)Bilayer-barrel-helicesOuter MembraneInner Membrane

position of Trans Membrane Segments along the sequenceTopographyTopology of membrane proteins

ALALMLCMLTYRHKELKLKLKK ALALMLCMLTYRHKELKLKLKK ALALMLCMLTYRHKELKLKLKK

A generic model for membrane proteins (TMHMM)TransmembraneInner SideOuter Side End Begin

Model of b-barrel membrane proteins

Labels:

Transmembrane states

Loop statesModel of b-barrel membrane proteinsTransmembraneInner SideOuter Side End Begin

Length of transmembrane b-strands:

Minimum: 6 residues

Maximum: unboundedModel of b-barrel membrane proteins

Six different sets of emission parameters:

Outer loop Inner loop Long globular domains

TM strands edgesTM strands coreModel of b-barrel membrane proteins

Model of a-helix membrane proteins (HMM1)TransmembraneInner SideOuter Side

Model of a-helix membrane proteins (HMM2)TransmembraneInner SideOuter Side....x 10....x 10......

Dynamic programming filtering procedure

Grafico1

0.0000174787

0.0000235731

0.0000415042

0.0000553066

0.000153429

0.000167835

0.000172728

0.000172356

0.000171323

0.000170204

0.000171257

0.000166853

0.000158308

0.000148597

0.000141162

0.000132089

0.0000570664

0.0000617807

0.0000673372

0.0000777819

0.0000753772

0.000097771

0.00011462

0.000126552

0.000133944

0.000147535

0.000149896

0.000139449

0.000127235

0.00012101

0.00011239

0.0001101

0.000120898

0.000122121

0.000121798

0.000138214

0.000231619

0.0003353

0.000374969

0.000451195

0.00048892

0.00055561

0.000577447

0.000596539

0.000642954

0.000699476

0.00073945

0.000710173

0.000763287

0.000952486

0.00145005

0.00344736

0.00571033

0.00788937

0.0139086

0.0162845

0.0182426

0.019068

0.0226285

0.02366

0.0258444

0.0261305

0.0284355

0.0295408

0.0324913

0.0318368

0.0285251

0.0268962

0.0371214

0.0521017

0.0770203

0.100266

0.166072

0.251925

0.577052

0.720923

0.912751

0.942341

0.945026

0.942038

0.930935

0.912393

0.896836

0.840698

0.825128

0.721808

0.520763

0.401477

0.221122

0.193517

0.166473

0.188099

0.180852

0.18561

0.196164

0.22403

0.251177

0.268332

0.264953

0.264259

0.278232

0.3047

0.361445

0.407271

0.418106

0.417674

0.435532

0.446926

0.453063

0.425028

0.40118

0.318551

0.286291

0.306349

0.317463

0.450198

0.815417

0.895298

0.905273

0.9123

0.926058

0.926184

0.925919

0.924039

0.927282

0.912855

0.836395

0.779566

0.581614

0.406334

0.215819

0.22083

0.401683

0.499017

0.669711

0.704546

0.714602

0.723647

0.744263

0.737763

0.740447

0.735937

0.731559

0.691716

0.627088

0.576156

0.443201

0.404786

0.270514

0.208538

0.245984

0.242904

0.238868

0.255671

0.266423

0.318376

0.31008

0.358739

0.43441

0.558658

0.647963

0.684144

0.719301

0.722901

0.7581

0.743032

0.736886

0.727311

0.685591

0.660593

0.560776

0.539008

0.524622

0.500756

0.396593

0.496948

0.537717

0.54623

0.58491

0.74687

0.782884

0.833712

0.838098

0.799548

0.765294

0.764944

0.713315

0.619098

0.570536

0.499848

0.425812

0.27176

0.283301

0.263614

0.370863

0.418584

0.439832

0.450443

0.460137

0.524176

0.599266

0.655459

0.693898

0.672716

0.679209

0.65671

0.629125

0.622578

0.585368

0.576914

0.615416

0.666336

0.726034

0.6798

0.713449

0.604731

0.593474

0.549547

0.575317

0.622605

0.677113

0.701663

0.677405

0.637634

0.601581

0.58019

0.542949

0.514777

0.472724

0.435447

0.30936

0.25235

0.186252

0.168481

0.189475

0.219068

0.276261

0.352376

0.416928

0.534413

0.626717

0.709252

0.753832

0.787217

0.803077

0.802876

0.790199

0.757288

0.649386

0.552361

0.407434

0.414973

0.38495

0.631403

0.737471

0.885202

0.909909

0.932449

0.933491

0.936373

0.933199

0.908904

0.845921

0.760906

0.686365

0.405323

0.348651

0.242403

0.222158

0.21915

0.248145

0.23981

0.223253

0.218881

0.208017

0.207941

0.205119

0.216356

0.23604

0.240296

0.246139

0.288373

0.317333

0.373995

0.477986

0.568314

0.620188

0.711002

0.76414

0.784637

0.783317

0.757715

0.688726

0.669211

0.676452

0.647544

0.669277

0.757138

0.791851

0.787456

0.75278

0.733017

0.706812

0.591633

0.585345

0.592416

0.633283

0.668552

0.734997

0.690004

0.726673

0.724654

0.728118

0.694509

0.662954

0.629524

0.581039

0.562001

0.521694

0.479281

0.377203

0.327218

0.212958

0.16657

0.119169

0.113159

0.108245

0.114516

0.112822

0.127794

0.145578

0.176129

0.210041

0.33746

0.508619

0.659039

0.836691

0.878103

0.898896

0.90666

0.9137

0.921421

0.891898

0.878043

0.868452

0.74794

0.567508

0.326562

0.244662

0.302795

0.60396

0.719897

0.843056

0.90701

0.935406

0.942954

0.938652

0.934715

0.917907

0.868289

0.770857

0.642134

0.412903

0.284398

0.132874

0.0686025

0.0508897

0.0572534

0.107611

0.194576

0.387605

0.649729

0.827149

0.935307

0.971441

0.984669

0.988886

0.988197

0.988422

0.986796

0.979964

0.972433

0.920744

0.786618

0.523556

0.387433

0.166555

0.254606

0.544102

0.659614

0.919984

0.953069

0.974823

0.976202

0.970414

0.963163

0.95289

0.896097

0.777576

0.662156

0.47229

0.38383

0.205591

0.178812

0.128526

0.103349

0.0888854

0.0940384

0.106731

0.15551

0.205257

0.24824

0.302063

0.351648

0.411679

0.529721

0.652949

0.82179

0.834566

0.851302

0.860777

0.862223

0.862947

0.864675

0.877207

0.868468

0.838861

0.721514

0.655336

0.554857

0.529138

0.328663

0.371278

0.598594

0.638451

0.727219

0.72837

0.755185

0.803927

0.844441

0.871994

0.893023

0.89633

0.886404

0.871464

0.84975

0.794927

0.52352

0.490371

0.294861

0.267727

0.180829

0.165128

0.156195

0.154401

0.150078

0.125137

0.11442

0.105883

0.0969399

0.0768029

0.0684502

0.0650629

0.0684616

0.0723199

0.0740659

0.140617

0.206073

0.347735

0.467618

0.634927

0.866976

0.967974

0.989465

0.992942

0.994601

0.994744

0.994946

0.987479

0.956531

0.871111

0.758467

0.344668

TMS probability

Sequence (1A0S)

TMS probability

1a0spTOT.SumPostHmm

1.75E-050000

2.36E-050000

4.15E-050000

5.53E-050000

0.0001534290000

0.0001678350000

0.0001727280000

0.0001723560000

0.0001713230000

0.0001702040000

0.0001712570000

0.0001668530000

0.0001583080000

0.0001485970000

0.0001411620000

0.0001320890000

5.71E-050000

6.18E-050000

6.73E-050000

7.78E-050000

7.54E-050000

9.78E-050000

0.000114620000

0.0001265520000

0.0001339440000

0.0001475350000

0.0001498960000

0.0001394490000

0.0001272350000

0.000121010000

0.000112390000

0.00011010000

0.0001208980000

0.0001221210000

0.0001217980000

0.0001382140000

0.0002316190000

0.00033530000

0.0003749690000

0.0004511950000

0.000488920000

0.000555610000

0.0005774470000

0.0005965390000

0.0006429540000

0.0006994760000

0.000739450000

0.0007101730000

0.0007632870000

0.0009524860000

0.001450050000

0.003447360000

0.005710330000

0.007889370000

0.01390860000

0.01628450000

0.01824260000

0.0190680000

0.02262850000

0.023660000

0.02584440000

0.02613050000

0.02843550000

0.02954080000

0.03249130000

0.03183680000

0.02852510000

0.02689620000

0.03712140000

0.05210170000

0.07702030000

0.1002660000

0.1660720.50.5010

0.2519250.50.5010

0.5770520.50.40.50.411

0.7209230.50.40.50.411

0.9127510.50.40.50.411

0.9423410.50.40.50.411

0.9450260.50.40.50.411

0.9420380.50.40.50.411

0.9309350.50.40.50.411

0.9123930.50.40.50.411

0.8968360.50.40.50.411

0.8406980.50.40.50.411

0.8251280.50.40.50.411

0.7218080.50.40.50.411

0.5207630000

0.4014770000

0.2211220000

0.1935170000

0.1664730000

0.1880990000

0.1808520000

0.185610000

0.1961640000

0.224030000

0.2511770000

0.2683320000

0.2649530000

0.2642590000

0.2782320000

0.30470000

0.3614450000

0.4072710000

0.4181060000

0.4176740000

0.4355320000

0.4469260000

0.4530630000

0.4250280000

0.401180000

0.3185510000

0.2862910000

0.3063490000

0.3174630000

0.4501980000

0.8154170.50.40.50.411

0.8952980.50.40.50.411

0.9052730.50.40.50.411

0.91230.50.40.50.411

0.9260580.50.40.50.411

0.9261840.50.40.50.411

0.9259190.50.40.50.411

0.9240390.50.40.50.411

0.9272820.50.40.50.411

0.9128550.50.40.50.411

0.8363950.50.40.50.411

0.7795660.50.40.50.411

0.5816140000

0.4063340000

0.2158190000

0.220830000

0.4016830000

0.4990170.50.5010

0.6697110.50.40.50.411

0.7045460.50.40.50.411

0.7146020.50.40.50.411

0.7236470.50.40.50.411

0.7442630.50.40.50.411

0.7377630.50.40.50.411

0.7404470.50.40.50.411

0.7359370.50.40.50.411

0.7315590.50.40.50.411

0.6917160.50.40.50.411

0.6270880.50.40.50.411

0.5761560000

0.4432010000

0.4047860000

0.2705140000

0.2085380000

0.2459840000

0.2429040000

0.2388680000

0.2556710000

0.2664230000

0.3183760000

0.310080000

0.3587390000

0.434410.50.5010

0.5586580.50.5010

0.6479630.50.40.50.411

0.6841440.50.40.50.411

0.7193010.50.40.50.411

0.7229010.50.40.50.411

0.75810.50.40.50.411

0.7430320.50.40.50.411

0.7368860.50.40.50.411

0.7273110.50.40.50.411

0.6855910.400.401

0.6605930.400.401

0.5607760000

0.5390080000

0.5246220000

0.5007560000

0.3965930000

0.4969480000

0.5377170000

0.546230000

0.584910000

0.746870.400.401

0.7828840.50.40.50.411

0.8337120.50.40.50.411

0.8380980.50.40.50.411

0.7995480.50.40.50.411

0.7652940.50.40.50.411

0.7649440.50.40.50.411

0.7133150.50.40.50.411

0.6190980.50.5010

0.5705360000

0.4998480000

0.4258120000

0.271760000

0.2833010000

0.2636140000

0.3708630000

0.4185840000

0.4398320000

0.4504430000

0.4601370000

0.5241760000

0.5992660000

0.6554590.400.401

0.6938980.400.401

0.6727160.400.401

0.6792090.50.40.50.411

0.656710.50.40.50.411

0.6291250.50.40.50.411

0.6225780.50.40.50.411

0.5853680.50.40.50.411

0.5769140.50.40.50.411

0.6154160.50.40.50.411

0.6663360.50.40.50.411

0.7260340.50.40.50.411

0.67980.400.401

0.7134490.400.401

0.6047310000

0.5934740000

0.5495470000

0.5753170000

0.6226050.400.401

0.6771130.400.401

0.7016630.50.40.50.411

0.6774050.50.40.50.411

0.6376340.50.40.50.411

0.6015810.50.40.50.411

0.580190.50.40.50.411

0.5429490.50.5010

0.5147770.50.5010

0.4727240.50.5010

0.4354470.50.5010

0.309360.50.5010

0.252350.50.5010

0.1862520.50.5010

0.1684810000

0.1894750000

0.2190680000

0.2762610000

0.3523760000

0.4169280000

0.5344130000

0.6267170.50.40.50.411

0.7092520.50.40.50.411

0.7538320.50.40.50.411

0.7872170.50.40.50.411

0.8030770.50.40.50.411

0.8028760.50.40.50.411

0.7901990.50.40.50.411

0.7572880.50.40.50.411

0.6493860.50.40.50.411

0.5523610.50.5010

0.4074340.50.5010

0.4149730.5010

0.384950000

0.6314030000

0.7374710.400.401

0.8852020.50.40.50.411

0.9099090.50.40.50.411

0.9324490.50.40.50.411

0.9334910.50.40.50.411

0.9363730.50.40.50.411

0.9331990.50.40.50.411

0.9089040.50.40.50.411

0.8459210.50.40.50.411

0.7609060.50.40.50.411

0.6863650.400.401

0.4053230000

0.3486510000

0.2424030000

0.2221580000

0.219150000

0.2481450000

0.239810000

0.2232530000

0.2188810000

0.2080170000

0.2079410000

0.2051190000

0.2163560000

0.236040000

0.2402960000

0.2461390000

0.2883730000

0.3173330000

0.3739950000

0.4779860000

0.5683140.50.5010

0.6201880.50.5010

0.7110020.50.40.50.411

0.764140.50.40.50.411

0.7846370.50.40.50.411

0.7833170.50.40.50.411

0.7577150.50.40.50.411

0.6887260.50.40.50.411

0.6692110.50.40.50.411

0.6764520.50.40.50.411

0.6475440.50.40.50.411

0.6692770.400.401

0.7571380.400.401

0.7918510.400.401

0.7874560.400.401

0.752780.400.401

0.7330170.400.401

0.7068120.400.401

0.5916330000

0.5853450000

0.5924160.50.5010

0.6332830.50.5010

0.6685520.50.40.50.411

0.7349970.50.40.50.411

0.6900040.50.40.50.411

0.7266730.50.40.50.411

0.7246540.50.40.50.411

0.7281180.50.40.50.411

0.6945090.50.40.50.411

0.6629540.50.40.50.411

0.6295240.50.5010

0.5810390000

0.5620010000

0.5216940000

0.4792810000

0.3772030000

0.3272180000

0.2129580000

0.166570000

0.1191690000

0.1131590000

0.1082450000

0.1145160000

0.1128220000

0.1277940000

0.1455780000

0.1761290000

0.2100410000

0.337460000

0.5086190.50.5010

0.6590390.50.40.50.411

0.8366910.50.40.50.411

0.8781030.50.40.50.411

0.8988960.50.40.50.411

0.906660.50.40.50.411

0.91370.50.40.50.411

0.9214210.50.40.50.411

0.8918980.50.40.50.411

0.8780430.50.40.50.411

0.8684520.400.401

0.747940.400.401

0.5675080000

0.3265620000

0.2446620000

0.3027950000

0.603960.50.5010

0.7198970.50.40.50.411

0.8430560.50.40.50.411

0.907010.50.40.50.411

0.9354060.50.40.50.411

0.9429540.50.40.50.411

0.9386520.50.40.50.411

0.9347150.50.40.50.411

0.9179070.50.40.50.411

0.8682890.50.40.50.411

0.7708570.50.40.50.411

0.6421340.50.5010

0.4129030.50.5010

0.2843980000

0.1328740000

0.06860250000

0.05088970000

0.05725340000

0.1076110000

0.1945760000

0.3876050.50.5010

0.6497290.50.5010

0.8271490.50.40.50.411

0.9353070.50.40.50.411

0.9714410.50.40.50.411

0.9846690.50.40.50.411

0.9888860.50.40.50.411

0.9881970.50.40.50.411

0.9884220.50.40.50.411

0.9867960.50.40.50.411

0.9799640.50.40.50.411

0.9724330.50.40.50.411

0.9207440.50.40.50.411

0.7866180.50.40.50.411

0.5235560000

0.3874330000

0.1665550000

0.2546060000

0.5441020.50.5010

0.6596140.50.40.50.411

0.9199840.50.40.50.411

0.9530690.50.40.50.411

0.9748230.50.40.50.411

0.9762020.50.40.50.411

0.9704140.50.40.50.411

0.9631630.50.40.50.411

0.952890.50.40.50.411

0.8960970.50.40.50.411

0.7775760.50.40.50.411

0.6621560.50.40.50.411

0.472290.50.5010

0.383830.50.5010

0.2055910.50.5010

0.1788120000

0.1285260000

0.1033490000

0.08888540000

0.09403840000

0.1067310000

0.155510000

0.2052570000

0.248240000

0.3020630.50.5010

0.3516480.50.5010

0.4116790.50.5010

0.5297210.50.5010

0.6529490.50.40.50.411

0.821790.50.40.50.411

0.8345660.50.40.50.411

0.8513020.50.40.50.411

0.8607770.50.40.50.411

0.8622230.50.40.50.411

0.8629470.50.40.50.411

0.8646750.50.40.50.411

0.8772070.50.40.50.411

0.8684680.50.40.50.411

0.8388610.400.401

0.7215140.400.401

0.6553360000

0.5548570000

0.5291380000

0.3286630000

0.3712780000

0.5985940000

0.6384510.400.401

0.7272190.400.401

0.728370.400.401

0.7551850.400.401

0.8039270.50.40.50.411

0.8444410.50.40.50.411

0.8719940.50.40.50.411

0.8930230.50.40.50.411

0.896330.50.40.50.411

0.8864040.50.40.50.411

0.8714640.50.40.50.411

0.849750.50.40.50.411

0.7949270.50.40.50.411

0.523520.50.5010

0.4903710.50.5010

0.2948610000

0.2677270000

0.1808290000

0.1651280000

0.1561950000

0.1544010000

0.1500780000

0.1251370000

0.114420000

0.1058830000

0.09693990000

0.07680290000

0.06845020000

0.06506290000

0.06846160000

0.07231990000

0.07406590000

0.1406170000

0.2060730000

0.3477350000

0.4676180000

0.6349270000

0.8669760.50.40.50.411

0.9679740.50.40.50.411

0.9894650.50.40.50.411

0.9929420.50.40.50.411

0.9946010.50.40.50.411

0.9947440.50.40.50.411

0.9949460.50.40.50.411

0.9874790.50.40.50.411

0.9565310.50.40.50.411

0.8711110.50.40.50.411

0.7584670.50.40.50.411

0.3446680000

Dynamic programming filtering procedureMaximum-scoring subsequences with constrained segment length and number

Grafico1

0.0000174787

0.0000235731

0.0000415042

0.0000553066

0.000153429

0.000167835

0.000172728

0.000172356

0.000171323

0.000170204

0.000171257

0.000166853

0.000158308

0.000148597

0.000141162

0.000132089

0.0000570664

0.0000617807

0.0000673372

0.0000777819

0.0000753772

0.000097771

0.00011462

0.000126552

0.000133944

0.000147535

0.000149896

0.000139449

0.000127235

0.00012101

0.00011239

0.0001101

0.000120898

0.000122121

0.000121798

0.000138214

0.000231619

0.0003353

0.000374969

0.000451195

0.00048892

0.00055561

0.000577447

0.000596539

0.000642954

0.000699476

0.00073945

0.000710173

0.000763287

0.000952486

0.00145005

0.00344736

0.00571033

0.00788937

0.0139086

0.0162845

0.0182426

0.019068

0.0226285

0.02366

0.0258444

0.0261305

0.0284355

0.0295408

0.0324913

0.0318368

0.0285251

0.0268962

0.0371214

0.0521017

0.0770203

0.100266

0.166072

0.251925

0.5770520.4

0.7209230.4

0.9127510.4

0.9423410.4

0.9450260.4

0.9420380.4

0.9309350.4

0.9123930.4

0.8968360.4

0.8406980.4

0.8251280.4

0.7218080.4

0.520763

0.401477

0.221122

0.193517

0.166473

0.188099

0.180852

0.18561

0.196164

0.22403

0.251177

0.268332

0.264953

0.264259

0.278232

0.3047

0.361445

0.407271

0.418106

0.417674

0.435532

0.446926

0.453063

0.425028

0.40118

0.318551

0.286291

0.306349

0.317463

0.450198

0.8154170.4

0.8952980.4

0.9052730.4

0.91230.4

0.9260580.4

0.9261840.4

0.9259190.4

0.9240390.4

0.9272820.4

0.9128550.4

0.8363950.4

0.7795660.4

0.581614

0.406334

0.215819

0.22083

0.401683

0.499017

0.6697110.4

0.7045460.4

0.7146020.4

0.7236470.4

0.7442630.4

0.7377630.4

0.7404470.4

0.7359370.4

0.7315590.4

0.6917160.4

0.6270880.4

0.576156

0.443201

0.404786

0.270514

0.208538

0.245984

0.242904

0.238868

0.255671

0.266423

0.318376

0.31008

0.358739

0.43441

0.558658

0.6479630.4

0.6841440.4

0.7193010.4

0.7229010.4

0.75810.4

0.7430320.4

0.7368860.4

0.7273110.4

0.6855910.4

0.6605930.4

0.560776

0.539008

0.524622

0.500756

0.396593

0.496948

0.537717

0.54623

0.58491

0.746870.4

0.7828840.4

0.8337120.4

0.8380980.4

0.7995480.4

0.7652940.4

0.7649440.4

0.7133150.4

0.619098

0.570536

0.499848

0.425812

0.27176

0.283301

0.263614

0.370863

0.418584

0.439832

0.450443

0.460137

0.524176

0.599266

0.6554590.4

0.6938980.4

0.6727160.4

0.6792090.4

0.656710.4

0.6291250.4

0.6225780.4

0.5853680.4

0.5769140.4

0.6154160.4

0.6663360.4

0.7260340.4

0.67980.4

0.7134490.4

0.604731

0.593474

0.549547

0.575317

0.6226050.4

0.6771130.4

0.7016630.4

0.6774050.4

0.6376340.4

0.6015810.4

0.580190.4

0.542949

0.514777

0.472724

0.435447

0.30936

0.25235

0.186252

0.168481

0.189475

0.219068

0.276261

0.352376

0.416928

0.534413

0.6267170.4

0.7092520.4

0.7538320.4

0.7872170.4

0.8030770.4

0.8028760.4

0.7901990.4

0.7572880.4

0.6493860.4

0.552361

0.407434

0.414973

0.38495

0.631403

0.7374710.4

0.8852020.4

0.9099090.4

0.9324490.4

0.9334910.4

0.9363730.4

0.9331990.4

0.9089040.4

0.8459210.4

0.7609060.4

0.6863650.4

0.405323

0.348651

0.242403

0.222158

0.21915

0.248145

0.23981

0.223253

0.218881

0.208017

0.207941

0.205119

0.216356

0.23604

0.240296

0.246139

0.288373

0.317333

0.373995

0.477986

0.568314

0.620188

0.7110020.4

0.764140.4

0.7846370.4

0.7833170.4

0.7577150.4

0.6887260.4

0.6692110.4

0.6764520.4

0.6475440.4

0.6692770.4

0.7571380.4

0.7918510.4

0.7874560.4

0.752780.4

0.7330170.4

0.7068120.4

0.591633

0.585345

0.592416

0.633283

0.6685520.4

0.7349970.4

0.6900040.4

0.7266730.4

0.7246540.4

0.7281180.4

0.6945090.4

0.6629540.4

0.629524

0.581039

0.562001

0.521694

0.479281

0.377203

0.327218

0.212958

0.16657

0.119169

0.113159

0.108245

0.114516

0.112822

0.127794

0.145578

0.176129

0.210041

0.33746

0.508619

0.6590390.4

0.8366910.4

0.8781030.4

0.8988960.4

0.906660.4

0.91370.4

0.9214210.4

0.8918980.4

0.8780430.4

0.8684520.4

0.747940.4

0.567508

0.326562

0.244662

0.302795

0.60396

0.7198970.4

0.8430560.4

0.907010.4

0.9354060.4

0.9429540.4

0.9386520.4

0.9347150.4

0.9179070.4

0.8682890.4

0.7708570.4

0.642134

0.412903

0.284398

0.132874

0.0686025

0.0508897

0.0572534

0.107611

0.194576

0.387605

0.649729

0.8271490.4

0.9353070.4

0.9714410.4

0.9846690.4

0.9888860.4

0.9881970.4

0.9884220.4

0.9867960.4

0.9799640.4

0.9724330.4

0.9207440.4

0.7866180.4

0.523556

0.387433

0.166555

0.254606

0.544102

0.6596140.4

0.9199840.4

0.9530690.4

0.9748230.4

0.9762020.4

0.9704140.4

0.9631630.4

0.952890.4

0.8960970.4

0.7775760.4

0.6621560.4

0.47229

0.38383

0.205591

0.178812

0.128526

0.103349

0.0888854

0.0940384

0.106731

0.15551

0.205257

0.24824

0.302063

0.351648

0.411679

0.529721

0.6529490.4

0.821790.4

0.8345660.4

0.8513020.4

0.8607770.4

0.8622230.4

0.8629470.4

0.8646750.4

0.8772070.4

0.8684680.4

0.8388610.4

0.7215140.4

0.655336

0.554857

0.529138

0.328663

0.371278

0.598594

0.6384510.4

0.7272190.4

0.728370.4

0.7551850.4

0.8039270.4

0.8444410.4

0.8719940.4

0.8930230.4

0.896330.4

0.8864040.4

0.8714640.4

0.849750.4

0.7949270.4

0.52352

0.490371

0.294861

0.267727

0.180829

0.165128

0.156195

0.154401

0.150078

0.125137

0.11442

0.105883

0.0969399

0.0768029

0.0684502

0.0650629

0.0684616

0.0723199

0.0740659

0.140617

0.206073

0.347735

0.467618

0.634927

0.8669760.4

0.9679740.4

0.9894650.4

0.9929420.4

0.9946010.4

0.9947440.4

0.9949460.4

0.9874790.4

0.9565310.4

0.8711110.4

0.7584670.4

0.344668

TMS probability

Predicted TMS

Sequence (1A0S)

TMS probability

1a0spTOT.SumPostHmm

1.75E-050000

2.36E-050000

4.15E-050000

5.53E-050000

prologue: pitfalls of standard alignments

Documents