evolution of plasmodium falciparum var antigen genes

Evolution of Plasmodium falciparum var antigen genes

Evolution of Plasmodium falciparum var antigen genes

Przeworski lab meeting Przeworski lab meeting

Evolution of Plasmodium falciparum var genesEvolution of Plasmodium falciparum var genes

Malaria: a protozoan blood parasite carried by mosquito vectors principally of the genus Anopheles Plasmodium falciparum and Plasmodium vivax are the main species

infectious to humans Related species infect primates, rodents, other mammals, birds and

reptiles

P. falciparum is the most deadly: in humans it is responsible for 80% of infections (250 million annually) 90% of deaths (1 million annually)

Endemic in tropical and sub-tropical regions

Member of the Apicomplexa, a phylum characterized by a unique organelle called the apical complex/apicoplast responsible for lipid synthesis

Malaria: a protozoan blood parasite carried by mosquito vectors principally of the genus Anopheles Plasmodium falciparum and Plasmodium vivax are the main species

infectious to humans Related species infect primates, rodents, other mammals, birds and

reptiles

P. falciparum is the most deadly: in humans it is responsible for 80% of infections (250 million annually) 90% of deaths (1 million annually)

Endemic in tropical and sub-tropical regions

Member of the Apicomplexa, a phylum characterized by a unique organelle called the apical complex/apicoplast responsible for lipid synthesis

QuickTime™ and a decompressorare needed to see this picture.


Key elements of life cycle: Infection with diploid cells from mosquito bite Asexual replication in liver and then red blood cells Haploid cells produced in red blood cells, ingested by mosquito Sex in mosquito produces diploid cells that migrate to saliavary gland

Cycles of replication and aggregation in blood vessels (sequestration) causes the symptoms of malaria: Anaemia Fever, chills, malaise Coma (especially in cerebral malaria)

Sequestration prevents clearing of infected redblood cells by the spleen

Key elements of life cycle: Infection with diploid cells from mosquito bite Asexual replication in liver and then red blood cells Haploid cells produced in red blood cells, ingested by mosquito Sex in mosquito produces diploid cells that migrate to saliavary gland

Cycles of replication and aggregation in blood vessels (sequestration) causes the symptoms of malaria: Anaemia Fever, chills, malaise Coma (especially in cerebral malaria)

Sequestration prevents clearing of infected redblood cells by the spleen

QuickTime™ and a decompressor

are needed to see this picture.

Evolution of Plasmodium falciparum var genesEvolution of Plasmodium falciparum var genes The var genes are responsible for antigenic

variation and sequestering by cytoadhesion

They encode PfEMP-1 (P. falciparum erythrocyte membrane protein), a parasite protein exported to the outside of the host red blood cells

Typically 60 var genes per genome, clustered telomerically. Differential expression allows immune evasion

Protein consists of Duffy binding-like domain (DBL) and cysteine-rich interdomain region (CIDR)

All proteins begin with DBL1, hence its use in population biology

The var genes are responsible for antigenic variation and sequestering by cytoadhesion

They encode PfEMP-1 (P. falciparum erythrocyte membrane protein), a parasite protein exported to the outside of the host red blood cells

Typically 60 var genes per genome, clustered telomerically. Differential expression allows immune evasion

Protein consists of Duffy binding-like domain (DBL) and cysteine-rich interdomain region (CIDR)

All proteins begin with DBL1, hence its use in population biology



Kraemer et al. 2007 BMC Genomics 8:45


Collaboration with Caroline Buckee (Oxford, Santa Fe and Kilifi) and Pete Bull (Kilifi)

Looking at 1000 DBL sequences

Aims Investigate genetic structuring imposed by geography,

selection and genomic position Understand population biology: evidence for strain structure? Reconstruct evolutionary history: quantify the roles of gene

duplication, homologous and non-homologous recombination

Current hypotheses: Categorization based on conserved 5’ regions and double-W Recombination hierarchy Some genotypes are over-represented in severe malaria

Collaboration with Caroline Buckee (Oxford, Santa Fe and Kilifi) and Pete Bull (Kilifi)

Looking at 1000 DBL sequences

Aims Investigate genetic structuring imposed by geography,

selection and genomic position Understand population biology: evidence for strain structure? Reconstruct evolutionary history: quantify the roles of gene

duplication, homologous and non-homologous recombination

Current hypotheses: Categorization based on conserved 5’ regions and double-W Recombination hierarchy Some genotypes are over-represented in severe malaria





var genes: raw datavar genes: raw data




Alignment is a major problem even in the conserved DBL domain

Multiple alignment algorithms Clustal (Larkin et al. 2007 Bioinformatics 23: 2947-8) Muscle (Edgar 2004 Nucleic Acids Research 32: 1792-7) T-Coffee (Notredame et al. 2000 JMB 302: 205-17)

Pairwise alignment BLAST (Tatiana et al. 1999 FEMS Microbiol Lett 174: 247-250) Hidden Markov models (e.g. Lunter 2007 Bioinformatics 23: 2485-7)

Multiple statistical alignment Phylogenetic (Holmes 2003 Bioinformatics S1: i147-57)

Alignment is a major problem even in the conserved DBL domain

Multiple alignment algorithms Clustal (Larkin et al. 2007 Bioinformatics 23: 2947-8) Muscle (Edgar 2004 Nucleic Acids Research 32: 1792-7) T-Coffee (Notredame et al. 2000 JMB 302: 205-17)

Pairwise alignment BLAST (Tatiana et al. 1999 FEMS Microbiol Lett 174: 247-250) Hidden Markov models (e.g. Lunter 2007 Bioinformatics 23: 2485-7)

Multiple statistical alignment Phylogenetic (Holmes 2003 Bioinformatics S1: i147-57)

Malign: probabilistic sequence alignment in malariaas a pre-requisite for evolutionary inference

Malign: probabilistic sequence alignment in malariaas a pre-requisite for evolutionary inference

Simulating a sequence alignment

The likelihood of a sequence alignment

Bayesian inference

Testing, testing

Inference proper

Simulating a sequence alignment

The likelihood of a sequence alignment

Bayesian inference

Testing, testing

Inference proper

Simulating an alignment: beta-binomial distributionSimulating an alignment: beta-binomial distribution

N=5 sequences, T=1 site Simulate the frequency of indels from a

symmetric beta distribution with parameters (,)

Given the frequency f, draw the number of indels from a binomial distribution with parameters (N,f)

N=5 sequences, T=1 site Simulate the frequency of indels from a

symmetric beta distribution with parameters (,)

Given the frequency f, draw the number of indels from a binomial distribution with parameters (N,f)




For example, the beta-binomial distribution with N=15 and =0.1 (red) =1 (yellow) =10 (green)

The binomial distribution with p=0.5 (blue)

For example, the beta-binomial distribution with N=15 and =0.1 (red) =1 (yellow) =10 (green)

The binomial distribution with p=0.5 (blue)



Simulating nucleotides: multinomial-Dirichlet distributionSimulating nucleotides: multinomial-Dirichlet distribution

Simulate the nucleotide frequencies from a symmetric Dirichlet distribution with parameter =(,,,)

Given the frequencies f=(fA, fG, fC, fT), simulate the number of As Gs Cs and Ts from a multinomial distribution with parameters (N,f)

Simulate the nucleotide frequencies from a symmetric Dirichlet distribution with parameter =(,,,)

Given the frequencies f=(fA, fG, fC, fT), simulate the number of As Gs Cs and Ts from a multinomial distribution with parameters (N,f)

= 0.1 = 1 = 10

Simulating nucleotides: multinomial-Dirichlet distributionSimulating nucleotides: multinomial-Dirichlet distribution





Simulating a nucleotide alignmentSimulating a nucleotide alignment

T = 100 L = 77

Simulating a nucleotide alignmentSimulating a nucleotide alignment



True alignment

Raw data



Representing an alignmentRepresenting an alignment

Matrix of size N by L 0 indicates an indel Other integers correspond to the position in

the raw sequence

Matrix of size N by L 0 indicates an indel Other integers correspond to the position in

the raw sequence

The likelihood of an alignment: indel patternThe likelihood of an alignment: indel pattern

Let I be the number of indels, and be the parameter of the beta distribution. N is the number of sequences.

For a single column in the alignment, the beta-binomial distribution gives the probability of I

However, the sequences are labelled, so the ordering matters. Therefore the likelihood is

Let I be the number of indels, and be the parameter of the beta distribution. N is the number of sequences.

For a single column in the alignment, the beta-binomial distribution gives the probability of I

However, the sequences are labelled, so the ordering matters. Therefore the likelihood is

Pr I | N ,( ) = Pr I |N, f( ) p f |( )df0

1

∫

=NI

⎛⎝⎜

⎞⎠⎟

f I 1− f( )N−I⎡

⎣⎢

⎤

⎦⎥

f −1 1− f( )−1

Β ,( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥df

0

1

∫

=NI

⎛⎝⎜

⎞⎠⎟Β I + ,N −I + ( )

Β ,( )

Pr I | N ,( ) =Β I + ,N −I + ( )

Β ,( )

The likelihood of an alignment: nucleotide patternThe likelihood of an alignment: nucleotide pattern

Let Yi, i={A,G,C,T} be the number of each type of nucleotide in the remaining (N-I) non-indels. Let be the parameter of the Dirichlet distribution.

For a single column, conditional on the indel pattern, the multinomial-Dirichlet distribution gives the likelihood of Y

Again the ordering matters, so the likelihood is

Let Yi, i={A,G,C,T} be the number of each type of nucleotide in the remaining (N-I) non-indels. Let be the parameter of the Dirichlet distribution.

For a single column, conditional on the indel pattern, the multinomial-Dirichlet distribution gives the likelihood of Y

Again the ordering matters, so the likelihood is

Pr Y | N −I ,( ) = Pr Y |N −I , f( ) p f |( )f∫

=N −I

Y⎛⎝⎜

⎞⎠⎟

fiYi

i∏

⎡

⎣⎢

⎤

⎦⎥

fi−1

i∏Β ( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥f

∫

=N −I

Y⎛⎝⎜

⎞⎠⎟Β Y +( )

Β ( )

Pr Y | N −I ,( ) =Β Y +( )

Β ( )

The likelihood of an alignment: missing columnsThe likelihood of an alignment: missing columns

Because our representation of the alignment does not specify the exact position of columns fixed for indels, we need to take that into account in the likelihood.

When there are (T-L) columns fixed for indels, their likelihood is

Because our representation of the alignment does not specify the exact position of columns fixed for indels, we need to take that into account in the likelihood.

When there are (T-L) columns fixed for indels, their likelihood is

Β N + β ,β( )

Β β ,β( )

⎛

⎝⎜⎞

⎠⎟

T − L

×T

L

⎛⎝⎜

⎞⎠⎟

The full likelihood of an alignmentThe full likelihood of an alignment

Pr I,Y |,( ) =TL

⎛⎝⎜

⎞⎠⎟

Β N + ,( )Β ,( )

⎛

⎝⎜⎞

⎠⎟

T −L Β I j + ,N −I j + ( )Β ,( )

Β Y j +( )Β ( )j=1

L

∏



Review of assumptionsReview of assumptions

Independence between sites = free recombination

Exchangeability between sequences = panmictic population structure

Equal base usage and stationary frequency distribution

Simplistic indel model

Independence between sites = free recombination

Exchangeability between sequences = panmictic population structure

Equal base usage and stationary frequency distribution

Simplistic indel model

Bayesian inferenceBayesian inference

Fix the auxiliary variables, e.g. at N=15 and T=100

Priors on and , e.g. exponential distribution with mean 0.1

Markov chain Monte Carlo (MCMC) for sampling from the joint posterior distribution of , and the alignment

Fix the auxiliary variables, e.g. at N=15 and T=100

Priors on and , e.g. exponential distribution with mean 0.1

Markov chain Monte Carlo (MCMC) for sampling from the joint posterior distribution of , and the alignment



MCMC: Multiplicative updates to and MCMC: Multiplicative updates to and

E.g. propose an update from to ´: Let U ~ Uniform(-1,1) Let ´ = exp(U)

The acceptanceprobability is

E.g. propose an update from to ´: Let U ~ Uniform(-1,1) Let ´ = exp(U)

The acceptanceprobability is Pr ′ ≤x( ) =

12

logx

−1⎛⎝⎜

⎞⎠⎟

q → ′( ) =1

2 ′min 1,π ′( )π ( )

q ′ → ( )q → ′( )

⎧⎨⎩

⎫⎬⎭

=min 1,π ′( )π ( )

′

⎧⎨⎩

⎫⎬⎭

MCMC: Updating the alignmentMCMC: Updating the alignment

General strategy is to partition the current alignment at random

General strategy is to partition the current alignment at random


And re-align one partition relative to the other And re-align one partition relative to the other


Gaps are stripped from both Gaps are stripped from both


And the reference partition (top) is labelled Odd-numbered sites correspond to gaps, even-numbered

sites to matches

And the reference partition (top) is labelled Odd-numbered sites correspond to gaps, even-numbered

sites to matches

10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22


Each site in the non-reference partition is aligned to a numbered site in the reference partition

Each site in the non-reference partition is aligned to a numbered site in the reference partition

10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22


Gaps may have multiple hits, but matches cannot Gaps may have multiple hits, but matches cannot

10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22

0 0 1 7 11 15 17 18 21


Finally the alignment is reconstructed Finally the alignment is reconstructed


In this example, it has grown by two columns In this example, it has grown by two columns


10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22

0 0 1 7 11 15 17 18 21

Partitioning the alignmentprobabilistic

Stripping indelsbook-keeping

Realigning the partitionsprobabilistic

Reconstructing the alignmentbook-keeping

MCMC: Partitioning the alignmentMCMC: Partitioning the alignment

The partition size is drawn from some discrete distribution between 1 and N-1, e.g. uniform, or inverse.

Sequences are then assigned to one partition or the other uniformly at random (formally, the hypergeometric distribution describes this method of sampling without replacement).

The proposal probability is independent of the alignment, so the ratio of reverse and forward proposal probabilities (the Hastings ratio) is 1.

The partition size is drawn from some discrete distribution between 1 and N-1, e.g. uniform, or inverse.

Sequences are then assigned to one partition or the other uniformly at random (formally, the hypergeometric distribution describes this method of sampling without replacement).

The proposal probability is independent of the alignment, so the ratio of reverse and forward proposal probabilities (the Hastings ratio) is 1.

MCMC: Realigning the partitionsMCMC: Realigning the partitions

In principal, you could enumerate over all possible alignments of the non-reference to the reference partition. (Very approximately (2L)L combinations).

For each combination you could calculate the posterior probability and Gibbs sample.

In practice, not computationally tractable so I impose a window of, e.g. ±40 matches, which constrains the maximum distance a site in the non-reference partition can move from its current position.

The likelihood can be calculated it an efficient manner (computer scientists would call it dynamic programming).

In principal, you could enumerate over all possible alignments of the non-reference to the reference partition. (Very approximately (2L)L combinations).

For each combination you could calculate the posterior probability and Gibbs sample.

In practice, not computationally tractable so I impose a window of, e.g. ±40 matches, which constrains the maximum distance a site in the non-reference partition can move from its current position.

The likelihood can be calculated it an efficient manner (computer scientists would call it dynamic programming).

10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22

0 0 1 7 11 15 17 18 21

MCMC: Realigning the partitionsMCMC: Realigning the partitions

Since not all possible moves are allowed, the proposal is a Metropolis-Hastings move.

To calculate the reverse proposal probability for the Hastings ratio, it is necessary to implement the move and carry out the full computations a second time.

The proposed change in the alignment, A´, is accepted with the usual probability

Since not all possible moves are allowed, the proposal is a Metropolis-Hastings move.

To calculate the reverse proposal probability for the Hastings ratio, it is necessary to implement the move and carry out the full computations a second time.

The proposed change in the alignment, A´, is accepted with the usual probability

10 2 3 4 65 7 8 9 1110 12 13 14 1615 17 18 19 20 21 22

0 0 1 7 11 15 17 18 21

min 1,π ′A( )π A( )

q ′A → A( )q A → ′A( )

⎧⎨⎩

⎫⎬⎭

R/C implementationR/C implementation

The MCMC code is written in C for speed

Linked as a static library and called from R for graphical monitoring and post-processing

Can watch the chain and the current alignment in real time

The MCMC code is written in C for speed

Linked as a static library and called from R for graphical monitoring and post-processing

Can watch the chain and the current alignment in real time



Testing, testingTesting, testing

It is criminally insane to assume your MCMC will work

(straight away) Two methods of testing are highly

recommended Do you retrieve the prior when there is no data? Is the posterior well-calibrated?

It is criminally insane to assume your MCMC will work

(straight away) Two methods of testing are highly

recommended Do you retrieve the prior when there is no data? Is the posterior well-calibrated?

Do you retrieve the prior when there is no data?Do you retrieve the prior when there is no data?

Testing the prior for alpha and beta is straightforward: you feed the program no data and compare to the specified prior distribution.

How do you test the “prior” on the alignment? The “prior” on the alignment is conditional on the sequence

lengths but not on the nucleotide sequences themselves

Calculate it theoretically for sufficiently simple examples

Use an alternative method for sampling from the prior, such as importance sampling, and compare.

Testing the prior for alpha and beta is straightforward: you feed the program no data and compare to the specified prior distribution.

How do you test the “prior” on the alignment? The “prior” on the alignment is conditional on the sequence

lengths but not on the nucleotide sequences themselves

Calculate it theoretically for sufficiently simple examples

Use an alternative method for sampling from the prior, such as importance sampling, and compare.

Testing the alignment prior: theoretical approachTesting the alignment prior: theoretical approach

For the 2-sequence case, this is theoretically tractable. Suppose both sequences have length S, and the maximum alignment length is T. Calculate the alignment length Pr(L|S).

For the 2-sequence case, this is theoretically tractable. Suppose both sequences have length S, and the maximum alignment length is T. Calculate the alignment length Pr(L|S).

Configuration Count

T-L

L-S

2S-L

Prior probability

Β 2 + β ,β( )

Β β ,β( )

Β ,2 + β( )

Β β ,β( )

Β 1+ β ,1+ β( )

Β β ,β( )

L-SΒ 1+ β ,1+ β( )

Β β ,β( )

Testing the alignment prior: theoretical approachTesting the alignment prior: theoretical approach

E.g. T=100, S=50, =0.1 Key

MCMC: black lines Theory: red lines

E.g. T=100, S=50, =0.1 Key

MCMC: black lines Theory: red lines

Pr L | S( ) ∝T !

2S−L( )! L −S( )! L −S( )! T −L( )!Β 2 + ,( )

Β ,( )

⎛

⎝⎜⎞

⎠⎟

T −2 L−S( )Β 1+ ,1+ ( )

Β ,( )

⎛

⎝⎜⎞

⎠⎟

2 L−S( )


are needed to see this picture.QuickTime™ and a

decompressorare needed to see this picture.

Testing the alignment prior: importance samplingTesting the alignment prior: importance sampling

Importance sampling is an alternative to MCMC.

Instead of proposing local moves to the parameters and missing data, they are independently proposed de novo each iteration.

Each simulation is given a weighting equal to the ratio of the posterior and proposal probabilities.

The proposal distribution is chosen to approximate the posterior distribution, and thus minimize the variance in the weightings.

I used a sequential importance sampler (SIS) that simulates the first sequence, then the second given the first, and so on. Each sequence is simulated to ensure agreement with the observed sequence

length. The importance sampler represents a better choice for simulating from the prior,

but not necessarily the posterior.

Importance sampling is an alternative to MCMC.

Instead of proposing local moves to the parameters and missing data, they are independently proposed de novo each iteration.

Each simulation is given a weighting equal to the ratio of the posterior and proposal probabilities.

The proposal distribution is chosen to approximate the posterior distribution, and thus minimize the variance in the weightings.

I used a sequential importance sampler (SIS) that simulates the first sequence, then the second given the first, and so on. Each sequence is simulated to ensure agreement with the observed sequence

length. The importance sampler represents a better choice for simulating from the prior,

but not necessarily the posterior.

Testing the alignment prior: importance samplingTesting the alignment prior: importance sampling

E.g. N=15, T=100, =0.3S=[51, 52, 47, 47, 57, 47, 50, 53, 48, 49, 44, 47, 52, 50,

50]

Key MCMC: white bars

3000 iterations

SIS: grey bars

1000 iterations

E.g. N=15, T=100, =0.3S=[51, 52, 47, 47, 57, 47, 50, 53, 48, 49, 44, 47, 52, 50,

50]

Key MCMC: white bars

3000 iterations

SIS: grey bars

1000 iterationsQuickTime™ and a

decompressorare needed to see this picture.



Is the posterior well-calibrated?Is the posterior well-calibrated?

The well-calibrated Bayesian is a person whose prior beliefs reflect the long-run probabilities of an event.

Dawid, A.P. (1982) JASA 77: 605-10

For example, if a weather forecaster predicts the chance of rain each day, s/he is well calibrated if, in the long-run, it rained on at least 80% of the days when the predicted chance of rain was 80% or more.

This is a minimum requirement for good inference: a lazy weather forecaster would be well-calibrated if s/he used the annual precipitation rate as his/her estimated chance of rain every day.

The well-calibrated Bayesian is a person whose prior beliefs reflect the long-run probabilities of an event.

Dawid, A.P. (1982) JASA 77: 605-10

For example, if a weather forecaster predicts the chance of rain each day, s/he is well calibrated if, in the long-run, it rained on at least 80% of the days when the predicted chance of rain was 80% or more.

This is a minimum requirement for good inference: a lazy weather forecaster would be well-calibrated if s/he used the annual precipitation rate as his/her estimated chance of rain every day.


The idea of calibration can be used to test a Bayesian inference procedure, using the following simulation scheme Simulate a parameter from a prior p() Simulate data X from a model p(X|) Perform inference using the same prior and likelihood to obtain a

posterior distribution phat(|X)

Calculate a 95% credible interval based on phat(|X)

Repeat many times

If the method of inference is working correctly, the credible intervals must be well-calibrated.

A calibration curve shows the proportion of simulations for which the true value of fell within the 100(1-)% credible interval. To be well-calibrated means that proportion equals 1-

The idea of calibration can be used to test a Bayesian inference procedure, using the following simulation scheme Simulate a parameter from a prior p() Simulate data X from a model p(X|) Perform inference using the same prior and likelihood to obtain a

posterior distribution phat(|X)

Calculate a 95% credible interval based on phat(|X)

Repeat many times

If the method of inference is working correctly, the credible intervals must be well-calibrated.

A calibration curve shows the proportion of simulations for which the true value of fell within the 100(1-)% credible interval. To be well-calibrated means that proportion equals 1-

DebuggingDebugging

Check the likelihood calculations are consistent

For accepted moves, check the calculated reverse move probability was correct

Go through the calculations for simple examples by hand

Theory - go through again by hand

Check the likelihood calculations are consistent

For accepted moves, check the calculated reverse move probability was correct

Go through the calculations for simple examples by hand

Theory - go through again by hand

Inference properInference proper

Screw debugging, let’s analyse the data

What will we do with a posterior distribution of sequence alignments?

For inference Improper inverse priors on and Improper uniform distribution on the maximum

alignment length T

Screw debugging, let’s analyse the data

What will we do with a posterior distribution of sequence alignments?

For inference Improper inverse priors on and Improper uniform distribution on the maximum

alignment length T

ConclusionsConclusions

There was a bug in my MCMC

Multiple sequence alignment allows Exploratory analysis of quantities of interest

pertaining to selection, recombination and demography

Formal model fitting using standard tools such as omegaMap, LDhat and Structure-like analyses

There was a bug in my MCMC

Multiple sequence alignment allows Exploratory analysis of quantities of interest

pertaining to selection, recombination and demography

Formal model fitting using standard tools such as omegaMap, LDhat and Structure-like analyses

evolution of plasmodium falciparum var antigen genes

Documents

plasmodium vivax

red blood cellshaploid

diploid cells

protozoan blood parasite

host red blood cellstypically

blood vessels sequestration

domain dbl

bioinformatics s1