Evolutionary Changes In Nucleotide Sequences

Workshop on Computational Molecular Evolution Chau-Ti Ting 12/22/2007

Substitution Models

Readings: Chapter 3 of Fundamentals of Molecular Evolution by Graur and Li (2000) Chapter 1 & 2 of Computational Molecular Evolution by Yang (2006)

Models of nucleotide substitution Introduction Calculate of the distance between two sequences is the simplest phylogenetic analysis

Important because The first step in distance methods for phylogeny reconstruction Markov-process models of nucleotide substitution used in distance calculation form the basis of likelihood and Bayesian analysis

The distance between two nucleotide sequences is defined as the expected number of nucleotide substitutions per site.

A simplest distance measure is the proportion of different sites, sometimes called the p- distance. If 10 sites are different between two sequences, each 100 bp long, then p= 10% = 0.1

However, a variable site may result from more than one substitutions that have occurred, and even a constant site may harbor back or parallel substitutions.

Multiple hits: multiple substitutions at the same site (i.e., some changes are hidden) Note: p is usable only for high similar sequences, with p < 5%.

Nucleotide substitution in a DNA sequence Jukes and Cantor’s one-parameter model (JC69) This simple model assumes that substitutions occur with equal probability among the four nucleotide types. The rate of substitution for each nucleotide is 3 per unit time, and the rate of substitution is in each of the three possible directions of change is . It is called the one-parameter model because the model involves a single parameter, .

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Since we start with A, the probability hat this site is occupied by A at time 0 is PA(0) = 1. At time 1, the probability of still having A at this site is given by

PA(1) = 1 – 3

In which 3 is the probability of A changing to T, C or G, and 1 – 3 is the probability that A has remained unchanged.

The probability of having A at time 2 is

PA(2) = (1 – 3 PA(1) + ( PA(1))

To derive this equation, we consider two possible scenarios: 1. the nucleotide has remained unchanged from time 0 to time 2, and 2. the nucleotide has changed to T, C, or G at time 1, but has subsequently reverted to A at time 2.

Using the above formulation, we can show that the following recurrence equation applies to any t:

PA(t+1) = (1 – 3 PA(t) +  PA(t))

We can rewrite this equation in terms of the amount of change in PA(t) per unit time as

PA(t) = PA(t+1)  PA(t) = [(1 – 3 PA(t) + ( PA(t))]  PA(t) = – 3 PA(t) +  PA(t) = – 4 PA(t) + 

We can approximate this process by a continuous-time model, by regarding PA(t) as the rate of change at time t. With this approximation,

dP A(t ) = - 4l P + l dt A(t )

1 1 P = + (P - )e- 4l t A(t ) 4 A(0) 4

PA(0) = 1,

1 3 P = + ( )e- 4 l t A(t ) 4 4

This also can be rewritten in a more explicit form to take into account the facts that the initial nucleotide is A and the nucleotide at time t is also A.

2 Workshop on Computational Molecular Evolution Chau-Ti Ting 12/22/2007

1 3 P = + ( )e- 4 l t AA(t) 4 4

A more general equation can also be written as

1 3 P = + ( )e- 4l t ii(t ) 4 4 PA(0) = 0, 1 1 P = - ( )e- 4 l t A(t ) 4 4

If the initial nucleotide is G instead of A, then

1 1 P = - ( )e- 4 l t GA(t ) 4 4

A general probability, Pij (t), that a nucleotide will become j at time t, given that it was i at time 0.

1 1 P = - ( )e- 4l t ij(t ) 4 4 where i ≠ j

In Yang (2006), he use qij to denote the instantaneous rate of substitution from nucleotide

i to nucleotide j, with i, j = T, C, A or G. Thus the substitution-rate matrix is

- 3l l l l й щ к ъ l - 3l l l Q = qij = к ъ к l l - 3l l ъ к ъ л l l l - 3l ы

where the nucleotides are ordered T, C, A and G. The total rate of substitution of any nucleotide i is 3, which is -qij

Transition probability, pij (t) is the probability that a given nucleotide i will become j

time t later. The matrix P(t) = {pij (t)} is known as transitional probability matrix.

p (t) p (t) p (t) p (t) й 0 1 1 1 щ м 1 3 - 4 l t кp (t) p (t) p (t) p (t)ъ p 0 (t) = + e Qt 1 0 1 1 п 4 4 P(t) = e = к ъ, with н p (t) p (t) p (t) p (t) 1 1 - 4 l t к 1 1 0 1 ъ п p (t) = - e к ъ о 1 4 4 лp 1(t) p1(t) p1(t) p0 (t)ы

1. P(t) sums to 1 2. P(0) = I, the identity matrix, reflect the case of no evolution (t = 0) 3. Rate  and time t occur in the transition probability only in the form of a product t. With no external information about either the time or the rate, we can estimate only the distance, but not time or rate individually. 4. When t ® Ҙ , pij(t ) =1/4, for all i and j.

Kimura’s two-parameter model (K80) Substitution between two pyrimidines (TC) or between two purines (AG) are called transitions, while those between a pyrimidine and a purines (TCAG) are called transversion. In real data, transitions often occur at higher rates than transversion.

In this model, the rate of transitional substitution at each nucleotide site is  per unit time, whereas the rate of each transversionalTime substitution is  per unit time.  A G

   

C T  Let us consider the probability that a site that has A at time 0 will have A time t. After one time unit, the probability of A changing to G is , and the probability of A changing to either C or T is 2. Thus the probability of A remaining unchanged after one time unit is

PA(1) = 1 –   

At time 2, the probability of having A at this site is given by the sum of the probabilities of four different scenarios: 1) A remained unchanged at t =1 and t =2; 2) A change to G

4 Workshop on Computational Molecular Evolution Chau-Ti Ting 12/22/2007 at t =1 and reverted by a transition to A at t =2; 3) A change to C at t =1 and reverted by a transversion to A at t =2; 4) A change to C at t =1 and reverted by a transversion to A at t =2. Hence,

PA(2) = (1 –    PA(1) + PT(1) + PC(1) + PG(1)

By extension,

PA(t+1) = (1 –    PA(t) + PT(t) + PC(t) + PG(t)

Similarly, we can obtain

PT(t+1) =  PA(t) +(1 –    PT(t) + PC(t) + PG(t)

PC(t+1) =  PA(t) + PT(t) +(1 –    PC(t) + PG(t)

PG(t+1) =  PA(t) + PT(t) + PC(t) +(1 –    PG(t)

From this set of four sequences, we arrive at the following solution: 1 1 1 P = + ( )e- 4 bt + ( )e- 2(a +b )t AA(t ) 4 4 2

As in JC69 model, PAA(t) = PGG(t) = PCC(t) = PTT(t)

1 1 1 X = + ( )e- 4bt + ( )e- 2(a +b )t (t ) 4 4 2

Let Y(t) = the probability that the initial nucleotide and the nucleotide at time t differ from each other by a transition.

Y(t) = PAG(t) = PGA(t) = PTC(t) = PCT(t)

1 1 1 Y = + ( )e- 4bt - ( )e- 2(a +b )t (t ) 4 4 2 The probability, Z(t), that the initial nucleotide and the nucleotide at time t differ by a specific type of transversion is given by

1 1 Z = - ( )e- 4bt (t ) 4 4

Note that each nucleotide subject to two types of transversion, but only one type of transition. Also

X(t) + Y(t) + 2 Z(t) = 1

The rate matrix is as follows

й- (a + 2b) a b b щ к ъ a - (a + 2b) b b Q = qij = к ъ к b b - (a + 2b) a ъ к ъ л b b a - (a + 2b)ы

where the nucleotides are ordered T, C, A and G. The total rate of substitution of any nucleotide is a + 2b , and the distance between two sequences separated by time t is d=

(a + 2b)t. Note that at is the expected number of transitions per site and 2bt is the expected number of transversions per site. It is convenient to use distance d and the transition/transversion rate ratio k = a /b .

Transition probability, pij (t) is the probability that a given nucleotide i will become j

time t later. The matrix P(t) = {pij (t)} is known as transitional probability matrix.

йp 0(t) p1(t) p2(t) p2 (t)щ

кp (t) p (t) p (t) p (t)ъ P(t) = eQt = к 1 0 2 2 ъ кp 2(t) p2 (t) p0(t) p1(t)ъ к ъ лp 2(t) p2 (t) p1(t) p0 (t)ы

where the three distinct elements of the matrix are

1 1 1 1 1 1 p (t) = + e- 4bt + e- 2(a +b )t = + e- 4d /(k +2) + e- 2d(k +1)/(k +2) 0 4 4 2 4 4 2 1 1 1 1 1 1 p (t) = + e- 4bt - e- 2(a +b )t = + e- 4d /(k +2) - e- 2d(k +1)/(k +2) 1 4 4 2 4 4 2 1 1 1 1 p (t) = - e- 4bt = - e- 4 d /(k +2) 2 4 4 4 4

Note that p0(t) + p1(t) + 2p2 (t) =1

Number of nucleotide substitutions between two DNA sequences If two sequences of length N differ from each other at n site, then the proportion of differences, n/N, is referred to as the degree of divergence or Hamming distance.

If the degree of divergence is substantial, then the observed number of differences is likely to be smaller than the actual number of substitutions due to multiple substitution or multiple hit at the same site.

ACTGAACGTAACGC Â Ê A A C C ®A single substitution T T G G A ®C ®T ®G A sequential substitution A A C ®G C ®A Coincidental substitution G G T ®A T ®A Parallel substitution A A A ®C ®T A ®T Convergent substitution C C G G C C ®T ®C Back substitution

Number of nucleotide substitutions between two noncoding sequences Let us start with JC69 model. In this model, it is sufficient to consider only I(t), which is the probability that the nucleotide at a given site at the time t is the same in both sequences. Suppose that the nucleotide at a given site was A at time 0. At time t, the probability that a descendant sequence will have A at this site is PAA(t), and consequently 2 the probability that two descendant sequences have A at this site is P AA(t). Similarly, the 2 2 2 probabilities that both sequence have T, C, G at this site are P AT(t) P AC(t) P AG(t), respectively. Therefore,

2 2 2 2 I(t ) = PAA(t ) + PAT (t ) + PAC(t ) + PAG(t ) 1 3 I = + e- 8l t (t ) 4 4

2 Note that the probability that t PAA(t)he two sequences are different at a site at time t is p = 1 I(t). Thus, 3 p = (1- e- 8l t ) 4 or 4 8l t = - l n(1- p) 3

The time of divergence between two sequences is usually given not known, and thus we can not estimate . Instead, we compute d, which is the number of substitutions per site since the time of divergence between two sequences. In the case of the one parameter model, d= 2(3t), where 3t is the number of substitutions per site in a single lineage. We can calculate d as

3 4 d = - l n(1- p) 4 3 Where p is observed proportion of different nucleotides between two sequences. For sequence length n, the sampling variance is approximately given by

p(1- p) var(d) = 4 n(1- p)2 3

In the case of two-parameter model, the differences between two sequences are classified into transitions and transversions. Let S and V be the proportion of transitional and transversional differences between two sequences, respectively. Then the number of nucleotide substitutions per site between two sequences, d, is estimated by

1 1 d = - l n(1- 2S - V ) - l n(1- 2V ) 2 4 2 l n(1- 2S - V ) k = ҙ - 1 l n(1- 2V )

Equivalently the transition distance and the transversion distance are estimated as

1 1 at = - l n(1- 2S - V ) + l n(1- 2V ) 2 4 1 2bt = - l n(1- 2V ) 2 Aslo, the variance of d is

var(d) = [a2S + b2V - (aS + bV )2 ]/n

a = (1- 2S - V )- 1 1 b = [(1- 2S - V )- 1 + (1- 2V )- 1] 2

Violation of assumptions Several assumptions have been made that are not necessary met by the sequences under study.

1. The rate of substitution was assumed to be the same at all sites. This assumption might not hold, as the rate may vary greatly from site to site. 2. The substitution occur in an independent manner. 3. The substitution matrix was assumed not to change in time, so that the nucleotide frequencies are maintained at a constant equilibrium value throughout their evolution.

The transition/transversion rate ratio Three definitions of the ‘transition/transversion rate ratio’ are in use 1. The ratio of numbers of transitional and transversional differences between the two sequences, without correcting multiple hits. (E(S)/E(V)) 2. k = a /b , with k =1 meaning no rate difference between transitions and transversions 3. Average transition/transversion ratio (R): same as the first one but with correction

Overall, R is convenient to use for comparing estimates under different models, while k is more suitable for formulating the null hypothesis of no transition/transversion rate difference.

Distance estimation under different substitution model At small distance, the different assumptions about the structure if the Q matrix do not make much difference, and simple models such as JC69 and K80 produce very similar estimates to those under more complex models

At intermediate distance (20%~30%), different model assumptions become more important. It may be favorable to use realistic models for distance estimation if the sequences are not too short.

At large distance (>40%), the different methods often produce very different estimates, and the estimates involve large sampling errors.

Models of amino acid and codon substitution Introduction With protein coding genes, we have the advantage of being able to distinguish synonymous or silent substitutions from the nonsynonymous or replacement substitutions.

Synonymous substitutions: nucleotide substitutions that do not change the encoded amino acid Nonsynonymous substitutions: nucleotide substitutions that do change the encoded amino acid

Synonymous and nonsynonymous mutations are under very different selection pressures and are fixed at very different rates. Thus, comparison between synonymous and nonsynonymous substitution rates provides a means to understand the effect of natural selection on the protein. This comparison does not require estimation of absolute substitution rates or knowledge of the divergence time.

Models of amino acid replacement Empirical models Empirical models attempts to describe the relative rates of substitution between two amino acids without considering explicitly factors that influence the evolutionary process. They are often constructed by analyzing large quantities of sequence data, as compiled from database.

Mechanistic models consider the biological process involved in amino acid substitution, such as mutation biases in the DNA, translation of the codons into amino acid after filtering by natural selection. Mechanistic models have more interpretative power and are particular useful for study the forces and mechanisms of gene sequence evolution.

Empirical models of amino acid substitution are all constructed by estimating relative substitution rates between two amino acids under general time-reversible model.

p iqij = p jq ji , for any i №j

The first empirical amino acid substitution matrix was constructed by Dayhoff and colleagues. They compiled and analyzed protein sequences available at the time, using a parsimony argument to reconstruct ancestral protein sequences and tabulating amino acid changes along branches on the phylogeny. Dayhoff et al. approximated the transition- probability matrix for an expected distance of 0.01 changes per site, call 1 PAM (for point-accepted mutations).

Features of these matrices: 1. amino acids with similar physico-chemical properties tend to interchange with each other at high rates than dissimilar amino acids. (DE or IV) 2. The “mutational distance” between amino acids determined by the structure of the genetic code. Amino acids separated by differences of two or three codon positions have lower rates than amino acids separated by a difference of one codon position. (RK for nuclear proteins or for mitochondrial proteins) Both factors may be operating at the same time.

Estimate synonymous and nonsynonymous substitutions rates Two distances are usually calculated between protein-coding DNA sequences, for synonymous and nonsynonymous substitutions, respectively. dS or KS: the number of synonymous changes per synonymous site dN or KN: the number of nonsynonymous changes per nonsynonymous site

Two classes of methods: heuristic counting methods and the ML method

Counting Methods Three steps:

1. Count synonymous and nonsynonymous sites 2. Count synonymous and nonsynonymous differences 3. Calculate the proportion of differences and correct for multiple hits

Nei and Gojobori (1986) 1. Count synonymous and nonsynonymous sites: S and N 2. Count synonymous and nonsynonymous differences: Sd and Nd 3. Calculate the proportion of differences (pS and pN) as

pS = Sd /S

pN = Nd /N

apply the JC69 correction for multiple hits

3 4 d = - l n(1- p ) S 4 3 S 3 4 d = - l n(1- p ) N 4 3 N

Transition/transversion rate difference and codon usage According to Li et al. (1985), we first classify the nucleotide sites into nondegenerate, twofold degenerate and fourfold degenerate site.

nondegenerate (L0): all the possible changes at this site are nonsynonymous twofold degenerate (L2): one of the three possible changes is synonymous fourfold degenerate(L4): all possible changes at the site are synonymous

The nucleotide differences in each class are further classified into transitional (Si) and transversional (Vi) differences, where i = 0, 2, and 4 denoted nondegerneracy, twofold degeneracy and fourfold degeneracy, respectively. All the substitutions at nondegenerate sites are nonsynonymous. All the substitutions at fourfold degenerate sites are synonymous. At twofold degenerate site, transitional changes are synonymous, whereas transversitional changes are nonsynonymous.

The proportion of transitional differences at i-fold degenerate sites between two sequences is calculated as

Si Pi = Li

Similarly, the proportion of transversional differences at i-fold degenerate sites between two sequences is calculated as

Vi Qi = Li

Kimura’s two-parameter method is used to estimate the number of transitional (Ai) and transversional (Bi) substitutions per ith type site. 1 1 A = l n(a ) - l n(b ) i 2 i 4 i 1 B = l n(b ) i 2 i

Where ai =1/(1– 2 Pi –Qi), bi = 1/(1– 2Qi) The total number of substitutions per ith type of degenerate site, Ki, is given by

Ki = Ai +Bi

A2 and B2 denote the numbers of synonymous and nonsynonymous substitutions per twofold degenerate site, respectively.

K4 = A4 +B4 denote the numbers of synonymous substitutions per fourfold degenerate site.

K0 = A0 +B0 denote the numbers of nonsynonymous substitutions per nondegenerate site. then, the number of synonymous substitutions per synonymous site (dS) and the number of nonsynonymous substitutions per nonsynonymous site (dN) can be obtained by

L2 A2 + L4 A4 dS = (L2 /3) + L4

L2B2 + L0 A0 dN = (2L2 /3) + L0

Li (1993) and Pamilo and Bianchi (1993) proposed to calculated the number of symnonymous substitution by taking (L2 A2 + L4 K4 )/ (L2 + L4) as an estimate of the transition component of nucleotide substitution at twofold and fourfold degenerate site

L2A2 + L4 A4 dS = + B4 L2 + L4

L2B2 + L0 A0 dN = A0 + L2 + L0

Maximum likelihood method Define number of sites and substitutions on a per codon basis.

1. The expected number of substitution per codon from codons i to j, i  j, over any time t is p ijqij t . Thus the numbers of synonymous and nonsynonymous substitutions per codon between two sequences separated by time t are

S = tr = p q t d S е i ij i№j,aai = aa j N = tr = p q t d N е i ij i№j,aai №aa j

Where rS and rN are the proportions of synonymous and nonsynonymous substitutions

2. Count sites: For w =1, the number of synonymous and nonsynonymous sites per codon are 1 S = 3r S N = 3r 1 N

3. The distances are given by

dS = Sd /S

dN = Nd /N

1 1 w = dN /dS = (r N /r S )/(r N /r S ) measures the perturbation in the proportion of synonymous and nonsynonymous substitutions caused by natural selection on the protein.

Ex: tobacco rbcL genes 1. Assume equal transition and transversion rates (k =1) and equal codon frequency ( p j =1/61) and estimate t and  by ML method

t = 0.363

w = 0.096

2. Calculate S, N and Sd, Nd 3. Calculate distances

Comparison of methods 1. Ignoring the transition and transversion rate difference leads to underestimation of S,

overestimate dS, and underestimation of the  ratio

2. Ignoring the codon usage bias leads to overestimation of S, underestimate dS, and overestimation of the  ratio

3. Different method or model assumptions can produce very different estimates even when two sequences are highly similar.

4. Assumptions appear to matter more than methods.

Advantages of the likelihood method 1. Conceptual simplicity 2. It is much simpler to accommodate more realistic models of codon substitution