7.4: The Extended Mk Model
The Mk model assumes that transitions among all possible character states occur at the same rate. However, that may not be a valid assumption. For example, it is often supposed that it is easier to lose a complex character than to gain one. We might want to fit models that allow for such asymmetries in rates.
For models of DNA sequence evolution there are a wide range of models allowing different rates between distinct types of nucleotides (Yang 2006) . Unequal rates are usually incorporated into the Mk model in two ways. First, one can consider the symmetric model (SYM; Paradis et al. 2004) . In the symmetric model, the rate of change between any two character states is the same forwards as it is backwards (that is, rates of change are symmetric; q i j = q j i ). The rate for a particular pair of states might differ from other pairs of character states. Note that when k = 2 the symmetric model is identical to the basic Mk model. The rate matrix for this model has as many free rate parameters as there are pairs of character states:
(eq. 7.11)
\[ p = \frac{k(k-1)}{2} \]
However, in general symmetric models will not have stationary distributions where all character states occur at equal frequencies, as noted above for the Mk model. We can account for these uneven frequencies by adding additional parameters to our model:
(eq. 7.12)
\[ \pi_{SYM} = \begin{bmatrix} \pi_1 & \pi_2 & \dots & 1 - \sum_{i=1}^{n-1} \pi_i \end{bmatrix} \]
Note that we only have to specify n − 1 equilibrium frequencies, since we know that they all sum to one. We have added n − 1 new parameters, for a total number of parameters:
(eq. 7.13)
\[ p = \frac{k(k-1)}{2} + n-1 \]
To obtain a Q -matrix for this model, we combine the information from both the relative transition rates and equilibrium frequencies:
(eq. 7.14)
\[ \mathbf{Q} = \begin{bmatrix} \cdot & r_1 & \dots & r_{n-1} \\ r_1 & \cdot & \dots & \vdots \\ \vdots & \vdots & \cdot & r_{k(k-1)/2} \\ r_{n-1} & \dots & r_{k(k-1)/2} & \cdot \\ \end{bmatrix} \begin{bmatrix} \pi_1 & 0 & 0 & 0 \\ 0 & \pi_2 & 0 & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & \pi_n \\ \end{bmatrix} \]
In this equation I have left the diagonal of the first matrix as dots. The final Q -matrix must have all rows sum to one, so one can adjust the values of that matrix after the multiplication step.
In the case of a two-state model, for example, we can create a model where the forward rate is double the backward rate, and the equilibrium frequency of character one is 0.75. Then:
(eq. 7.15)
\[ \mathbf{Q} = \begin{bmatrix} \cdot & 1 \\ 2 & \cdot \\ \end{bmatrix} \begin{bmatrix} 0.75 & 0 \\ 0 & 0.25 \\ \end{bmatrix} = \begin{bmatrix} \cdot & 0.25 \\ 1.5 & \cdot \\ \end{bmatrix} = \begin{bmatrix} -0.25 & 0.25 \\ 1.5 & -1.5 \\ \end{bmatrix} \]
It is worth noting that this approach of setting parameters that define equilibrium state frequences, although borrowed from molecular evolution, is not completely standard in the comparative methods literature. One also sees equilibrium frequencies treated as a fixed property of the model, and assumed to be either equal across states or tied directly to the parameters in the Q -matrix.
The second common extension of the Mk model is called the all-rates-different model (ARD; Paradis et al. 2004) . In this model every possible type of transition can have a different rate. There are thus k ( k − 1) free rate parameters for this model, and again n − 1 parameters to specify the equilibrium frequencies of the character states.
The same algorithm can be used to calculate the likelihood for both of these extended Mk models (SYM and ARD). These models have more parameters than the standard Mk. To find maximum likelihood solutions, we must optimize the likelihood across the entire set of unknown parameters (see Chapter 7).