Models of DNA evolution

A number of different Markov models of DNA sequence evolution have been proposed. This is because evolutionary processes vary between genomes and between different regions of a genome, for example different evolutionary processes apply to coding and noncoding regions. These models mostly differ in the parametrization of the rate matrix and in the modeling of rate variation.

Continuous Time Markov Chains
Continuous-time Markov chains have the usual transition matrices which are, in addition, parameterized by time, $$ t\ $$. Specifically, if $$ E_1,\ldots,E_4\ $$ are the states, then the transition matrix


 * $$ P(t) = \big(P_{ij}(t)\big)$$ where each individual entry, $$ P_{ij}(t)\ $$ refers to the probability that state $$ E_i\ $$ will change to state $$ E_j\ $$ in time $$ t\ $$.

Example: We would like to model the substitution process in DNA sequences (i.e. Jukes-Cantor, Kimura, etc.) in a continuous time fashion. The corresponding transition matrices will look like:

$$ P(t) = \begin{pmatrix}         p_{AA}(t) & p_{AG}(t) & p_{AC}(t) & p_{AT}(t) \\ p_{GA}(t) & p_{GG}(t) & p_{GC}(t) & p_{GT}(t) \\ p_{CA}(t) & p_{CG}(t) & p_{CC}(t) & p_{CT}(t) \\ p_{TA}(t) & p_{TG}(t) & p_{TC}(t) & p_{TT}(t) \end{pmatrix} $$

where the top-left and bottom-right $$ 2\times 2\ $$ blocks correspond to transition probabilities and the top-right and bottom-left $$ 2\times 2\ $$ blocks corresponds to transversion probabilities.

Assumption: If at some time $$ t_0\ $$, the Markov chain is in state $$ E_i\ $$, then the probability that at time $$ t_0+t\ $$, it will be in state $$E_j\ $$ depends only upon $$i\ $$, $$j\ $$ and $$t\ $$. This then allows us to write that probability as $$p_{ij}(t)\ $$.

Theorem: Continuous-time transition matrices satisfy:


 * $$P(t+\tau) = P(t)P(\tau)\ $$

Deriving the Dynamics of Substitution
Consider a DNA sequence of fixed length m evolving in time by base replacement. Assume that the processes followed by the m sites are Markovian independent, identically distributed and constant in time. For a fixed site, let


 * $$ \mathbf{P}(t) = (p_A(t),\ p_G(t),\  p_C(t),\  p_T(t))^T$$

be the column vector of probabilities of states $$A, \ $$ $$\ G, \ $$  $$\ C, \ $$  and $$\ T \ $$ at time $$ t \ $$. Let


 * $$ \mathcal{E} = \{A,\ G, \ C, \ T\}$$

be the state-space. For two distinct


 * $$ x, y \in \mathcal{E}$$, let $$ \mu_{xy}\ $$

be the transition rate from state $$x\ $$ to state $$y\ $$. Similarly, for any $$ x\ $$, let:
 * $$\mu_x = \sum_{y\neq x}\mu_{xy}$$

The changes in the probability distribution $$ p_A(t)\ $$ for small increments of time $$\Delta t\ $$ are given by:
 * $$p_A(t+\Delta t) = p_A(t) - p_A(t)\mu_A\Delta t + \sum_{x\neq A}p_x(t)\mu_{xA}\Delta t $$

In other words (in frequentist language), the frequency of $$ A\ $$'s at time $$ t + \Delta t\ $$ is equal to the frequency at time $$ t\ $$ minus the frequency of the lost $$ A\ $$'s plus the frequency of the newly created $$ A\ $$'s.

Similarly for the probabilities $$p_G(t), \ p_C(t), \ \mathrm{and} \ p_T(t)$$. We can write these compactly as:
 * $$\mathbf{P}(t+\Delta t) = \mathbf{P}(t) + Q\mathbf{P}(t)\Delta t $$

where,


 * $$ Q = \begin{pmatrix} -\mu_A & \mu_{GA} & \mu_{CA} & \mu_{TA} \\

\mu_{AG} & -\mu_G & \mu_{CG} & \mu_{TG} \\ \mu_{AC} & \mu_{GC} & -\mu_C & \mu_{TC} \\ \mu_{AT} & \mu_{GT} & \mu_{CT} & -\mu_T \end{pmatrix}$$ or, alternately:
 * $$ \mathbf{P}'(t) = Q\mathbf{P}(t) $$

where, $$Q\ $$ is the rate matrix. Note that by definition, the rows of $$Q\ $$ sum to zero.

Ergodicity
If all the transition probabilities, $$ \mu_{xy}\ $$ are positive, i.e. if all states $$ x, y \in \mathcal{E}\ $$ communicate, then the Markov chain has a stationary distribution $$ \mathbf{\Pi} = \{\pi_x, \ x \in \mathcal{E} \} $$ where each $$ \pi_x \ $$ is the proportion of time spent in state $$ x\ $$ after the Markov chain has run for infinite time, and this probability does not depend upon the initial state of the process. Such a Markov chain is called, ergodic. In DNA evolution, under the assumption of a common process for each site, the stationary frequencies, $$ \pi_A, \pi_G, \pi_C, \pi_T \ $$ correspond to equilibrium base compositions. Definition A Markov process is stationary if its current distribution is the stationary distribution, i.e. $$ \mathbf{P}(t) = \Pi\ $$ Thus, by using the differential equation above,
 * $$ \frac{d\Pi}{dt} = Q\Pi = 0 $$

Time Reversibility
Definition: A stationary Markov process is time reversible if (in the steady state) the amount of change from state $$ x\ $$ to $$ y\ $$ is equal to the amount of change from $$ y\ $$ to $$ x\ $$, (although the two states may occur with different frequencies). This means that:
 * $$ \pi_x\mu_{xy} = \pi_y\mu_{yx} \ $$

Not all stationary processes are reversible, however, almost all DNA evolution models assume time reversibility, which is considered to be a reasonable assumption.

Under the time reversibility assumption, let $$ s_{xy} = \mu_{xy}/\pi_y\ $$, then it is easy to see that:
 * $$ s_{xy} = s_{yx} \ $$

Definition The symmetric term $$ s_{xy}\ $$ is called the exchangeability between states $$x\ $$ and $$ y\ $$. In other words, $$ s_{xy}\ $$ is the fraction of the frequency of state $$ x\ $$ that results as a result of transitions from state $$ y\ $$ to state $$ x\ $$.

Corollary The 12 off-diagonal enteries of the rate matrix, $$ Q\ $$ (note the off-diagonal enteries determine the diagonal enteries, since the rows of $$ Q\ $$ sum to zero) can be completely determined by 9 numbers; these are: 6 exchangeability terms and 3 stationary frequencies $$\pi_x\ $$, (since the stationary frequencies sum to 1).

JC69 model (Jukes and Cantor, 1969)
JC69 is the simplest substitution model. There are several assumptions. It assumes equal base frequencies ($$\pi_1 = \pi_2 = \pi_3 = \pi_4 = {1\over4}$$) and equal mutation rates. The only parameter of this model is therefore $$\mu$$, the overall substitution rate.


 * $$Q = \begin{pmatrix} {*} & {\mu\over 4} & {\mu\over 4} & {\mu\over 4} \\ {\mu\over 4} & {*} & {\mu\over 4}& {\mu\over 4}\\ {\mu\over 4}& {\mu\over 4}& {*} & {\mu\over 4}\\ {\mu\over 4}& {\mu\over 4}& {\mu\over 4}& {*} \end{pmatrix}$$


 * $$ P= \begin{pmatrix} {{1\over4} + {3\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} \\\\ {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} + {3\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} \\\\ {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} + {3\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} \\\\ {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} - {1\over4}e^{-t\mu}} & {{1\over4} + {3\over4}e^{-t\mu}} \end{pmatrix}$$

Distance between two sequences is given by
 * $$d=-{3\over4} \ln({1-{4\over3}p})$$

where $$p$$ is the proportion of sites that differ between the two sequences.

K80 model (Kimura, 1980)
Distinguish between Transition(A <-> G within purines or T <-> C within pyrimidines) and Transversion(between purines and pyrimidines) (&alpha;/&beta;)

Equal base frequencies ($$\pi_T = \pi_C = \pi_A = \pi_G = {1\over4}$$)

Rate matrix $$Q= \begin{pmatrix} {*} & {\kappa} & {1} & {1} \\ {\kappa} & {*} & {1} & {1} \\ {1} & {1} & {*} & {\kappa} \\ {1} & {1} & {\kappa} & {*}  \end{pmatrix}$$

The Kimura two-parameter distance is given by:
 * $$d = {1\over2}\ln(1-2P-Q) - {1\over4} \ln (1-2Q)$$

where $$P$$ is the proportion of sites that show transitional differences and $$Q$$ is the proportion of sites that show transversional differences.

F81 model (Felsenstein 1981)
Unequal base frequencies ($$\pi_T \ne \pi_C \ne \pi_A \ne \pi_G $$)

Rate matrix $$Q= \begin{pmatrix} {*} & {\pi_C} & {\pi_A} & {\pi_G} \\ {\pi_T} & {*} & {\pi_A} & {\pi_G} \\ {\pi_T} & {\pi_C} & {*} & {\pi_G} \\ {\pi_T} & {\pi_C} & {\pi_A} & {*}  \end{pmatrix}$$

HKY85 model (Hasegawa, Kishino and Yano 1985)
Distinguish between Transition and Transversion (&alpha;/&beta;)

Unequal base frequencies ($$\pi_T \ne \pi_C \ne \pi_A \ne \pi_G $$)

Rate matrix $$Q= \begin{pmatrix} {*} & {\kappa\pi_C} & {\pi_A} & {\pi_G} \\ {\kappa\pi_T} & {*} & {\pi_A} & {\pi_G} \\ {\pi_T} & {\pi_C} & {*} & {\kappa\pi_G} \\ {\pi_T} & {\pi_C} & {\kappa\pi_A} & {*}  \end{pmatrix}$$

T92 model (Tamura 1992)
One frequency only $$\pi_{GC} $$

$$\pi_G = \pi_C = {\pi_{GC}\over 2} $$

$$\pi_A = \pi_T = {(1-\pi_{GC})\over 2}$$

Rate matrix $$Q= \begin{pmatrix} {*} & {\kappa\pi_{GC}/2} & {(1-\pi_{GC})/2} & {\pi_{GC}/2} \\ {\kappa(1-\pi_{GC})/2} & {*} & {(1-\pi_{GC})/2} & {\pi_{GC}/2} \\ {(1-\pi_{GC})/2} & {\pi_{GC}/2} & {*} & {\kappa\pi_{GC}/2} \\ {(1-\pi_{GC})/2} & {\pi_{GC}/2} & {\kappa(1-\pi_{GC})/2} & {*}  \end{pmatrix}$$

The evolutionary distance between two noncoding sequences according to this model is given by
 * $$d = -h \ln(1-{p\over h}-Q)-{1\over2}(1-h)\ln(1-2Q)$$

where $$h = 2\theta(1-\theta)$$ where $$\theta \in (0,1)$$ is the GC content.

TN93 model (Tamura and Nei 1993)
Distinguish between two different types of Transition (A <-> G) is different to (C<->T)

Unequal base frequencies ($$\pi_T \ne \pi_C \ne \pi_A \ne \pi_G $$)

Rate matrix $$Q= \begin{pmatrix} {*} & {\kappa_1\pi_C} & {\pi_A} & {\pi_G} \\ {\kappa_1\pi_T} & {*} & {\pi_A} & {\pi_G} \\ {\pi_T} & {\pi_C} & {*} & {\kappa_2\pi_G} \\ {\pi_T} & {\pi_C} & {\kappa_2\pi_A} & {*}  \end{pmatrix}$$

GTR: Generalised time reversible
GTR is the most general neutral, independent, finite-sites, time-reversible model possible. It was first described in a general form by Simon Tavaré in 1986.

The GTR parameters consist of an equilibrium base frequency vector, $$\Pi = (\pi_1 \pi_2 \pi_3 \pi_4)$$, giving the frequency at which each base occurs at each site, and the rate matrix


 * $$Q = \begin{pmatrix} {-(x_1+x_2+x_3)} & x_1 & x_2 & x_3 \\ {\pi_1 x_1 \over \pi_2} & {-({\pi_1 x_1 \over \pi_2} + x_4 + x_5)} & x_4 & x_5 \\ {\pi_1 x_2 \over \pi_3} & {\pi_2 x_4 \over \pi_3} & {-({\pi_1 x_2 \over \pi_3} + {\pi_2 x_4 \over \pi_3} + x_6)} & x_6 \\ {\pi_1 x_3 \over \pi_4} & {\pi_2 x_5 \over \pi_4} & {\pi_3 x_6 \over \pi_4} & {-({\pi_1 x_3 \over \pi_4} + {\pi_2 x_5 \over \pi_4} + {\pi_3 x_6 \over \pi_4})} \end{pmatrix} $$

Therefore, GTR (for four characters, as is often the case in phylogenetics) requires 6 substitution rate parameters, as well as 4 equilibrium base frequency parameters. However, this is usually eliminated down to 9 parameters plus $$\mu$$, the overall number of substitutions per unit time. When measuring time in substitutions ($$\mu$$=1) only 9 free parameters remain.

In general, to compute the number of parameters, you count the number of entries above the diagonal in the matrix, i.e. for n trait values per site $${{n^2-n} \over 2} $$, and then add n for the equilibrium base frequencies, and subtract 1 because $$\mu$$ is fixed. You get


 * $${{n^2-n} \over 2} + n - 1 = {1 \over 2}n^2 + {1 \over 2}n - 1.$$

For example, for an amino acid sequence (there are 20 "standard" amino acids that make up proteins), you would find there are 209 parameters. However, when studying coding regions of the genome, it is more common to work with a codon substitution model (a codon is three bases and codes for one amino acid in a protein). There are $$4^3 = 64$$ codons, but the rates for transitions between codons which differ by more than one base is assumed to be zero. Hence, there are $${{20 \times 19 \times 3} \over 2} + 64 - 1 = 633$$ parameters.