# Kosiol’s AIS, almost invariant sets¶

This is for grouping amino acids into sets where there is a high probability of exchange within the set but exchange between sets has a lower probability. The method is described in Kosiol et al (2004) J. Theoret. Biol. 228: 97–106. The program is available here.

The program wants the equilibrium frequencies for the model, the q-matrix (bigQ), and the eigenvectors. Here is an example using the wag model, with wag frequencies:

# The easiest way to get the bigQ and such from p4 is to just set up
# the appropriate protein model as usual, and then calculate a
# likelihood.

# Read in some data.  You could use your own data if you wanted to use
# empirical comps.
seqs = """ 2 20
one       arndcqeghilkmfpstwyv
two       rndcqeghilkmfpstwyva
"""

# a tree, with the corresponding taxa.
ts = "(one, two);"

d = Data()
t.data = d
t.newComp(partNum=0, free=0, spec='wag')    # or maybe you want empirical for your own data?
t.newRMatrix(partNum=0, free=0, spec='wag')
t.setNGammaCat(partNum=0, nGammaCat=1)
t.setPInvar(partNum=0, free=0, val=0.0)
t.calcLogLike()
#t.model.dump()

# Write the aa freqs
f = file('equi', 'w')
f.write('20\n')
for i in range(20):
f.write("%f\n" % t.model.parts[0].comps[0].val[i])
f.close()

bigQ = t.model.getBigQ()

# Write the bigQ
f = file('q', 'w')
f.write('20\n')
for i in range(20):
for j in range(20):
f.write("%5g  " % bigQ[i][j])
f.write('\n')
f.close()

# Get the eigensystem
import numpy
import numpy.linalg
evals,evecs = numpy.linalg.eig(bigQ)

# look, if you want
if 0:
numpy.set_printoptions(precision=4, linewidth=300)
print evals
print evecs

# According to the web page, "The right eigenvectors should be ordered
# according to the absolute value of their eigenvalues."  Well, the
# output from numpy, which uses lapack, is not so ordered.  So do it.
sorter = numpy.argsort(evals)
sorter = sorter[::-1]   # reverse
#print sorter

f = file('evec', 'w')
f.write('20\n')
f.write('20\n')
for colNum in sorter:
for rowNum in range(20):
f.write("%5g\n" % evecs[rowNum][colNum])
f.close()


Kosiol’s program ais asks for these 3 files made above, and the number of groups that you want, and it suggests a grouping.

When I did that with Dayhoff 78, with D78 composition, I got these groups —

• Set 0 = { R N D Q E H K T }
• Set 1 = { C V }
• Set 2 = { A G P S }
• Set 3 = { I L M }
• Set 4 = { W }
• Set 5 = { F Y }

While the groups from the log odds table are

1. c
2. stpag
3. ndeq
4. hrk
5. milv
6. fyw

So both similar and different. Maybe it differs because the groups from the log odds are from a PAM250 matrix, which would correspond to highly diverged sequences?