1. Lecture 11 – Increasing Model Complexity
Differences in functional and structural constraints across sites leads to different sites evolving at different rates.
3rd-codon positions evolve fastest, followed by 1st
positions, & 2nd-position sites evolve the slowest.
Among-site rate variation exacerbates the loss of
historical information caused by multiple hits.
10 substitutions, and they are distributed randomly across 50 sites, there should only
rarely be more than a single substitution per site.
However if those 10 substitutions are distributed across 50 sites in a non-random
fashion, say concentrated to 1/3 of them, many more will occur at multiply hit sites.

2. Among-site rate variation
Discrete methods – assign sites to a series of rate categories (or partitions).
Add a relative rate parameter, r, to our models.
There are c rate categories, and wr is the probability that site i belongs to a particular rate
category; these are binary (0 or 1) if we’re assigning sites to rate classes

3. Site-Specific Rates (or SSR) model
In the SSR models, the theoretical limit to the number of rate categories is the number of sites in the alignment, but usually these are determined a priori and often they follow codon structure.
So in this case, w1, w2, and w3 are fixed to 0 or 1, and we just use a different
relative rate for each class (e.g., codon position).
The relative rate parameters then can be assigned or they can be optimized
numerically, which is what is usually done.
Advantage: one also can use a different transformation matrix (Q) for each class.
Disadvantage: that all sites within a category are assumed to be evolving a uniform rate.

4. Invariable Sites Model
This is based on observations that there are sites in alignments of conserved
genes which all life seem to have the same state.
This allows two rate categories and in one of these, the relative-rate parameter is zero.
We can think about this model in two ways.
Mixture model
winvar = pinvar This is the probability that a site is in the class where r = 0.
wvar = pvar The probability that the site is in the class where r ≠ 0.
wvar = 1 - winvar
Sites that are observed to vary have winvar = 0.
Invariable sites
pinvar is the proportion of sites across an alignment that are not free to vary.
≤ the proportion of sites that are observed constant.

5. and we can use rate-mixture models to deal with this.
Continuous Methods
There’s no biological reason to expect rates to fall into discrete categories,
and we can use rate-mixture models to deal with this.
Gamma distributed rates.
shape parameter (a) & scale parameter (b); mean = ab
we set the mean of the G-distribution equal to 1 by constraining b = 1/a.

6. Discretizing the G-distribution
Gamma distribution with shape parameter (a) = 0.49
Cut-points and category rates for discrete G approximation with ncat (or c) = 4.
cut-points
cat lower upper rate (mean)
infinity
So r1 = , r2 = , r3 = , & r4 = 2.907; note these sum to 4 (=ncat).
Each site has some non-zero probability of belonging to each rate class:
wr is optimized for each of the c classes.

7. Discretizing the G-distribution
Cut-points and category rates for dG w/ncat = 8
cut-points
cat lower upper rate (mean)
infinity
The more highly we discretize the G, the shorter the runs, but: a
ncat
Poor estimates of a
with ncat < 12

8. Properties of G models
This is done across the entire data set, so essentially we take the same transformation
matrix (Q) for each site and scale it by the average rate for each category.
This has the large advantage of being able to accommodate such a high diversity of rates with just a single parameter, a. Some sites can be so slowly evolving to have a high probability of stasis, yet others (perhaps adjacent) may be free to evolve rapidly.
It has the disadvantage that we apply the same transformation matrix uniformly
across a data set – one set of base frequencies and one R matrix.
We may get a much better fit if we allow very different Q matrices.
Furthermore, SSLs for each site are calculated many times (ncat times), so the
better we approximate a continuous G, the longer our run times.
It’s important to note that this is actually a rather constrained mixture model.

9. I+G Models
This is intuitively very appealing when one considers that, at least from some genes,
there’s a set of sites that are constant across essentially the tree of life.
pinvar= 0
I+G = G
a = ∞
I+G = I
pinvar= 0 & a = ∞
I+G = ER
There are some issues with it that are sometimes not appreciated.

10. I+G Models
First, both the mixed model and the gamma alone expect there to be many constant
sites. It can be very difficult to discern the sites that are truly invariable from those
potentially variable sites that are evolving slowly enough to have a high probability of stasis

11. The GTR+I+G family of models
There are 1624 possible
special cases of GTR+I+G
There are 10 parameters in the
full model:
3 free b.f. (p)
5 relative rates (R)
2 rate variation parameters
(pinvar & a)

12. Another look at a GTR+SSR3
pA1 pA2 pA3
pC1 pC2 pC3
pG1 pG2 pG3
pT1 pT2 pT3
r(AC)1 r(AC)2 r(AC)1
r(AG)1 r(AG)2 r(AG)3
r(AT)1 r(AT)2 r(AT)3
r(CG)1 r(CG)2 r(CG)3
r(CT)1 r(CT)2 r(CT)3
r(GT)1 r(GT)2 r(GT)3
9 free base frequencies
15 relative rate parameters
pA1 = pA2 = pA3
pC1 = pC2 = pC3
pG1 = pG2 = pG3
pT1 = pT2 = pT3
r(AC)1 = r(AC)2 = r(AC)1
r(AG)1 = r(AG)2 = r(AG)3
r(AT)1 = r(AT)2 = r(AT)3
r(CG)1 = r(CG)2 = r(CG)3
r(CT)1 = r(CT)2 = r(CT)3
r(GT)1 = r(GT)2 = r(GT)3
Single GTR

13. Estimate relative rates for each site on a starting tree.
GTR+CAT in RAxML
Estimate relative rates for each site on a starting tree.
Lump sites with similar relative rates into categories.
So now, wr = 0 or 1, & r is set to value of highest lnL site in category.

14. rRNA Models Non-independence of sites.
a priori partitioning of sites into stem and loop regions, and sites
in the loops partition are treated with some variant of the
GTR+I+G family.
Sites in the stem regions are treated using a doublet model.
Doublets are treated as characters rather than
nucleotides and there are 16 states.
So there are 120 reversible substitution types.
This is very parameter rich (how many parameters?) and we’re forced to use empirical models.

15. Again, empirical matrices can be used.
Codon Models
In-frame triplets are used as characters and there are 61 possible character states.
Thus the transformation matrix has 3660 rate parameters (or 1830 in the reversible case).
Again, empirical matrices can be used.
Alternatively, cells of the transformation matrix can be restricted so that there are only, say, two substitution types.
E.g., TTT  TTC both code for Phe, so this is a silent substitution.
apC, where a is the rate of silent substitutions and pC is (as before) the frequency
of nucleotide C.
TTT TTA results in an amino acid replacement.
bpA, where b is the rate of amino acid replacement substitutions.
In this example there are 4 free parameters (3 b.f. and a rate ratio).