Evolutionary Analysis Using Likelihood#

Specifying substitution models#

The available pre-defined substitution models#

In cases where code takes a substitution model as an argument, you can use the value under “Abbreviation” as a string.

from cogent3 import available_models

available_models()

Specify a model using 'Abbreviation' (case sensitive).
Model Type	Abbreviation	Description
nucleotide	BH	Barry and Hartigan Discrete Time substitution model Barry and Hartigan 1987. Biometrics 43: 261–76.
nucleotide	DT	Discrete Time substitution model (non-stationary, non-reversible). motif_length=2 makes this a dinucleotide model, motif_length=3 a trinucleotide model.
nucleotide	GN	General Markov Nucleotide (non-stationary, non-reversible). Kaehler, Yap, Zhang, Huttley, 2015, Sys Biol 64 (2): 281–93
nucleotide	ssGN	strand-symmetric general Markov nucleotide (non-stationary, non-reversible). Kaehler, 2017, Journal of Theoretical Biology 420: 144–51
nucleotide	K80	Kimura 1980
nucleotide	JC69	Jukes and Cantor's 1969 model
nucleotide	GTR	General Time Reversible nucleotide substitution model.
nucleotide	TN93	Tamura and Nei 1993 model
nucleotide	HKY85	Hasegawa, Kishino and Yano 1985 model
nucleotide	F81	Felsenstein's 1981 model
codon	CNFGTR	Conditional nucleotide frequency codon substitution model, GTR variant (with params analagous to the nucleotide GTR model). Yap, Lindsay, Easteal and Huttley, 2010, Mol Biol Evol 27: 726-734
codon	CNFHKY	Conditional nucleotide frequency codon substitution model, HKY variant (with kappa, the ratio of transitions to transversions) Yap, Lindsay, Easteal and Huttley, 2010, Mol Biol Evol 27: 726-734
codon	MG94HKY	Muse and Gaut 1994 codon substitution model, HKY variant (with kappa, the ratio of transitions to transversions) Muse and Gaut, 1994, Mol Biol Evol, 11, 715-24
codon	MG94GTR	Muse and Gaut 1994 codon substitution model, GTR variant (with params analagous to the nucleotide GTR model) Muse and Gaut, 1994, Mol Biol Evol, 11, 715-24
codon	GY94	Goldman and Yang 1994 codon substitution model. N Goldman and Z Yang, 1994, Mol Biol Evol, 11(5):725-36.
codon	Y98	Yang's 1998 substitution model, a derivative of the GY94. Z Yang, 1998, Mol Biol Evol, 15(5):568-73
codon	H04G	Huttley 2004 CpG substitution model. Includes a term for substitutions to or from CpG's. GA Huttley, 2004, Mol Biol Evol, 21(9):1760-8
codon	H04GK	Huttley 2004 CpG substitution model. Includes a term for transition substitutions to or from CpG's. GA Huttley, 2004, Mol Biol Evol, 21(9):1760-8
codon	H04GGK	Huttley 2004 CpG substitution model. Includes a general term for substitutions to or from CpG's and an adjustment for CpG transitions. GA Huttley, 2004, Mol Biol Evol, 21(9):1760-8
codon	GNC	General Nucleotide Codon, a non-reversible codon model. Kaehler, Yap, Huttley, 2017, Gen Biol Evol 9(1): 134–49
protein	DSO78	Dayhoff et al 1978 empirical protein model Dayhoff, MO, Schwartz RM, and Orcutt, BC. 1978 A model of evolutionary change in proteins. Pp. 345-352. Atlas of protein sequence and structure, Vol 5, Suppl. 3. National Biomedical Research Foundation, Washington D. C Matrix imported from PAML dayhoff.dat file
protein	JTT92	Jones, Taylor and Thornton 1992 empirical protein model Jones DT, Taylor WR, Thornton JM. The rapid generation of mutation data matrices from protein sequences. Comput Appl Biosci. 1992 Jun;8(3):275-82. Matrix imported from PAML jones.dat file
protein	AH96	Adachi and Hasegawa 1996 empirical model for mitochondrial proteins. Adachi J, Hasegawa M. Model of amino acid substitution in proteins encoded by mitochondrial DNA. J Mol Evol. 1996 Apr;42(4):459-68. Matrix imported from PAML mtREV24.dat file
protein	AH96_mtmammals	Adachi and Hasegawa 1996 empirical model for mammalian mitochondrial proteins. Adachi J, Hasegawa M. Model of amino acid substitution in proteins encoded by mitochondrial DNA. J Mol Evol. 1996 Apr;42(4):459-68. Matrix imported from PAML mtmam.dat file
protein	WG01	Whelan and Goldman 2001 empirical model for globular proteins. Whelan S, Goldman N. A general empirical model of protein evolution derived from multiple protein families using a maximum-likelihood approach. Mol Biol Evol. 2001 May;18(5):691-9. Matrix imported from PAML wag.dat file

25 rows x 3 columns

Getting a substitution model with `get_model()`#

from cogent3.evolve.models import get_model

hky = get_model("HKY85")
print(hky)

TimeReversibleNucleotide(name='HKY85'; params=['kappa']; num_motifs=4; motifs=['T', 'C', 'A', 'G']))

Rate heterogeneity models#

We illustrate this for the gamma distributed case using examples of the canned models displayed above. Creating rate heterogeneity variants of the canned models can be done by using optional arguments that get passed to the substitution model class.

For nucleotide#

We specify a general time reversible nucleotide model with gamma distributed rate heterogeneity.

from cogent3.evolve.models import get_model

sub_mod = get_model("GTR", with_rate=True, distribution="gamma")
print(sub_mod)

TimeReversibleNucleotide(name='GTR'; params=['A/C', 'A/G', 'A/T', 'C/G', 'C/T']; num_motifs=4; motifs=['T', 'C', 'A', 'G']))

For codon#

We specify a conditional nucleotide frequency codon model with nucleotide general time reversible parameters and a parameter for the ratio of nonsynonymous to synonymous substitutions (omega) with gamma distributed rate heterogeneity.

from cogent3.evolve.models import get_model

sub_mod = get_model("CNFGTR", with_rate=True, distribution="gamma")
print(sub_mod)

TimeReversibleCodon(name='CNFGTR'; params=['A/C', 'A/G', 'A/T', 'C/G', 'C/T', 'omega']; num_motifs=61; motifs=['TTT', 'TTC', 'TTA', 'TTG', 'TCT', 'TCC', 'TCA', 'TCG', 'TAT', 'TAC', 'TGT', 'TGC', 'TGG', 'CTT', 'CTC', 'CTA', 'CTG', 'CCT', 'CCC', 'CCA', 'CCG', 'CAT', 'CAC', 'CAA', 'CAG', 'CGT', 'CGC', 'CGA', 'CGG', 'ATT', 'ATC', 'ATA', 'ATG', 'ACT', 'ACC', 'ACA', 'ACG', 'AAT', 'AAC', 'AAA', 'AAG', 'AGT', 'AGC', 'AGA', 'AGG', 'GTT', 'GTC', 'GTA', 'GTG', 'GCT', 'GCC', 'GCA', 'GCG', 'GAT', 'GAC', 'GAA', 'GAG', 'GGT', 'GGC', 'GGA', 'GGG']))

For protein#

We specify a Jones, Taylor and Thornton 1992 empirical protein substitution model with gamma distributed rate heterogeneity.

from cogent3.evolve.models import get_model

sub_mod = get_model("JTT92", with_rate=True, distribution="gamma")
print(sub_mod)

Empirical(name='JTT92'; num_motifs=20; motifs=['A', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'K', 'L', 'M', 'N', 'P', 'Q', 'R', 'S', 'T', 'V', 'W', 'Y']))

Making a likelihood function#

You start by specifying a substitution model and use that to construct a likelihood function for a specific tree.

from cogent3 import make_tree
from cogent3.evolve.models import get_model

sub_mod = get_model("F81")
tree = make_tree("(a,b,(c,d))")
lf = sub_mod.make_likelihood_function(tree)

Providing an alignment to a likelihood function#

You need to load an alignment and then provide it a likelihood function. I construct very simple trees and alignments for this example.

from cogent3 import make_aligned_seqs, make_tree
from cogent3.evolve.models import get_model

sub_mod = get_model("F81")
tree = make_tree("(a,b,(c,d))")
lf = sub_mod.make_likelihood_function(tree)
aln = make_aligned_seqs(
    [("a", "ACGT"), ("b", "AC-T"), ("c", "ACGT"), ("d", "AC-T")]
)
lf.set_alignment(aln)

Scoping parameters on trees – time heterogeneous models#

For many evolutionary analyses, it’s desirable to allow different branches on a tree to have different values of a parameter. We show this for a simple codon model case here where we want the great apes (the clade that includes human and orangutan) to have a different value of the ratio of nonsynonymous to synonymous substitutions. This parameter is identified in the precanned CNFGTR model as omega.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
print(tree.ascii_art())

          /-Galago
         |
-root----|--HowlerMon
         |
         |          /-Rhesus
          \edge.3--|
                   |          /-Orangutan
                    \edge.2--|
                             |          /-Gorilla
                              \edge.1--|
                                       |          /-Human
                                        \edge.0--|
                                                  \-Chimpanzee

sm = get_model("CNFGTR")
lf = sm.make_likelihood_function(tree, digits=2)
lf.set_param_rule(
    "omega",
    tip_names=["Human", "Orangutan"],
    outgroup_name="Galago",
    clade=True,
    init=0.5,
)

We’ve set an initial value for this clade so that the edges affected by this rule are evident below.

lf

CNFGTR

number of free parameters = 78

Global params
A/C	A/G	A/T	C/G	C/T
1.00	1.00	1.00	1.00	1.00

Edge params
edge	parent	length	omega
Galago	root	1.00	1.00
HowlerMon	root	1.00	1.00
Rhesus	edge.3	1.00	1.00
Orangutan	edge.2	1.00	0.50
Gorilla	edge.1	1.00	0.50
Human	edge.0	1.00	0.50
Chimpanzee	edge.0	1.00	0.50
edge.0	edge.1	1.00	0.50
edge.1	edge.2	1.00	0.50
edge.2	edge.3	1.00	1.00
edge.3	root	1.00	1.00

Motif params
AAA	AAC	AAG	AAT	ACA	ACC	ACG	ACT	AGA	AGC	AGG	AGT	ATA
0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02

continuation
ATC	ATG	ATT	CAA	CAC	CAG	CAT	CCA	CCC	CCG	CCT	CGA	CGC
0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02

continuation
CGG	CGT	CTA	CTC	CTG	CTT	GAA	GAC	GAG	GAT	GCA	GCC	GCG
0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02

continuation
GCT	GGA	GGC	GGG	GGT	GTA	GTC	GTG	GTT	TAC	TAT	TCA	TCC
0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02

continuation
TCG	TCT	TGC	TGG	TGT	TTA	TTC	TTG	TTT
0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02

A more extensive description of capabilities is in Allowing substitution model parameters to differ between branches.

Specifying a parameter as constant#

This means the parameter will not be modified during likelihood maximisation. We show this here by making the omega parameter constant at the value 1 – essentially the condition of selective neutrality.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
sm = get_model("CNFGTR")
lf = sm.make_likelihood_function(tree, digits=2)
lf.set_param_rule("omega", is_constant=True)

Providing a starting value for a parameter#

This can be useful to improve performance, the closer you are to the maximum likelihood estimator the quicker optimisation will be.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
sm = get_model("CNFGTR")
lf = sm.make_likelihood_function(tree, digits=2)
lf.set_param_rule("omega", init=0.1)

Setting parameter bounds for optimisation#

This can be useful for stopping optimisers from getting stuck in a bad part of parameter space. The following is for omega in a codon model. I’m also providing an initial guess for the parameter (init=0.1) as well as a lower bound. An initial guess that is close to the maximum likelihood estimate will speed up optimisation.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
sm = get_model("CNFGTR")
lf = sm.make_likelihood_function(tree, digits=2)
lf.set_param_rule("omega", init=0.1, lower=1e-9, upper=20.0)

Setting an upper bound for branch length#

If the branch length estimates seem too large, setting just an upper bound can be sensible. This will apply to all edges on the tree.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
sm = get_model("F81")
lf = sm.make_likelihood_function(tree)
lf.set_param_rule("length", upper=1.0)

Note

If, after optimising, the branch lengths equal to the upper value you set then the function has not been fully maximised and you should consider adjusting the boundary again.

Specifying rate heterogeneity functions#

We extend the simple gamma distributed rate heterogeneity case for nucleotides from above to construction of the actual likelihood function. We do this for 4 bins and constraint the bin probabilities to be equal.

from cogent3 import load_tree
from cogent3.evolve.models import get_model

sm = get_model("GTR", with_rate=True, distribution="gamma")
tree = load_tree("data/primate_brca1.tree")
lf = sm.make_likelihood_function(tree, bins=4, digits=2)
lf.set_param_rule("bprobs", is_constant=True)

For more detailed discussion of defining and using these models see Analysis of rate heterogeneity.

Specifying Phylo-HMMs#

from cogent3 import load_tree
from cogent3.evolve.models import get_model

sm = get_model("GTR", with_rate=True, distribution="gamma")
tree = load_tree("data/primate_brca1.tree")
lf = sm.make_likelihood_function(tree, bins=4, sites_independent=False, digits=2)
lf.set_param_rule("bprobs", is_constant=True)

For more detailed discussion of defining and using these models see Evaluate process heterogeneity using a Hidden Markov Model.

Fitting likelihood functions - Choice of optimisers#

There are 2 types of optimiser: simulated annealing, a global optimiser; and Powell, a local optimiser. The simulated annealing method is slow compared to Powell and in general Powell is an adequate choice. I setup a simple nucleotide model to illustrate these.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")
sm = get_model("F81")
lf = sm.make_likelihood_function(tree, digits=3, space=2)
lf.set_alignment(aln)
lf.optimise(show_progress=False)

The default is to use Powell. For Powell, it’s recommended to set the max_restarts argument since this provides a mechanism for Powell to attempt restarting the optimisation from a slightly different spot which can help in overcoming local maxima.

lf.optimise(local=True, max_restarts=5, show_progress=False)

We might want to do crude simulated annealing following by more rigorous Powell. To do this we first need to use the global optimiser, setting local=False setting a large value for global_tolerance.

lf.optimise(local=False, global_tolerance=1.0, show_progress=False)

Followed by a standard call to optimise().

lf.optimise(show_progress=False, max_restarts=5, tolerance=1e-8)

How to check your optimisation was successful#

There is no guarantee that an optimised function has achieved a global maximum. We can, however, be sure that a maximum was achieved by validating that the optimiser stopped because the specified tolerance condition was met, rather than exceeding the maximum number of evaluations. The latter number is set to ensure optimisation doesn’t proceed endlessly. If the optimiser exited because this limit was exceeded you can be sure that the function has not been successfully optimised.

We can monitor this situation using the limit_action argument to optimise. Providing the value raise causes an exception to be raised if this condition occurs, as shown below. Providing warn (default) instead will cause a warning message to be printed to screen but execution will continue. The value ignore hides any such message.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")
sm = get_model("F81")
lf = sm.make_likelihood_function(tree, digits=3, space=2)
lf.set_alignment(aln)
try:
    lf.optimise(
        show_progress=False,
        limit_action="raise",
        max_evaluations=10,
        return_calculator=True,
    )
except ArithmeticError as err:
    print(err)

FORCED EXIT from optimiser after 10 evaluations

Note

We recommend using limit_action='raise' and catching the ArithmeticError error explicitly (as demonstrated above). You really shouldn’t be using results from such an optimisation run.

Overview of the fitted likelihood function#

In Jupyter, the likelihood function object presents a representation of the main object features.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

sm = get_model("GTR")
tree = load_tree("data/primate_brca1.tree")
lf = sm.make_likelihood_function(tree)
aln = load_aligned_seqs("data/primate_brca1.fasta")
lf.set_alignment(aln)
lf.optimise(local=True, show_progress=False)
lf

GTR

log-likelihood = -6992.7690

number of free parameters = 16

Global params
A/C	A/G	A/T	C/G	C/T
1.2316	5.2534	0.9585	2.3159	5.9700

Edge params
edge	parent	length
Galago	root	0.1731
HowlerMon	root	0.0449
Rhesus	edge.3	0.0216
Orangutan	edge.2	0.0077
Gorilla	edge.1	0.0025
Human	edge.0	0.0061
Chimpanzee	edge.0	0.0028
edge.0	edge.1	0.0000
edge.1	edge.2	0.0034
edge.2	edge.3	0.0120
edge.3	root	0.0076

Motif params
A	C	G	T
0.3757	0.1742	0.2095	0.2406

Log likelihood and number of free parameters#

Reusing the optimised lf object from above, we can get the log-likelihood and the number of free parameters.

lnL = lf.lnL
lnL

-6992.768994254335

nfp = lf.nfp
nfp

Warning

The number of free parameters (nfp) refers only to the number of parameters that were modifiable by the optimiser. Typically, the degrees-of-freedom of a likelihood ratio test statistic is computed as the difference in nfp between models. This will not be correct for models in which a boundary conditions exist (rate heterogeneity models where a parameter value boundary is set between bins).

Aikake Information Criterion#

Reusing the optimised lf object from above.

lf.get_aic()

14017.53798850867

We can also get the second-order AIC.

lf.get_aic(second_order=True)

14017.732482609492

Bayesian Information Criterion#

Reusing the optimised lf object from above.

lf.get_bic()

np.float64(14112.61578431146)

Getting maximum likelihood estimates#

Reusing the optimised lf object from above.

One at a time#

We get the statistics out individually. We get the length for the Human edge and the exchangeability parameter A/G.

a_g = lf.get_param_value("A/G")
a_g

np.float64(5.253426081232493)

human = lf.get_param_value("length", "Human")
human

np.float64(0.006060124932976551)

Just the motif probabilities#

mprobs = lf.get_motif_probs()
mprobs

T	C	A	G
0.2406	0.1742	0.3757	0.2095

As tables#

tables = lf.get_statistics(with_motif_probs=True, with_titles=True)
tables[0]  # just displaying the first

global params
A/C	A/G	A/T	C/G	C/T
1.2316	5.2534	0.9585	2.3159	5.9700

1 rows x 5 columns

Testing Hypotheses - Using Likelihood Ratio Tests#

We test the molecular clock hypothesis for human and chimpanzee lineages. The null has these two branches constrained to be equal.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")
sm = get_model("F81")
lf = sm.make_likelihood_function(tree, digits=3, space=2)
lf.set_alignment(aln)
lf.set_param_rule(
    "length",
    tip_names=["Human", "Chimpanzee"],
    outgroup_name="Galago",
    clade=True,
    is_independent=False,
)
lf.set_name("Null Hypothesis")
lf.optimise(local=True, show_progress=False)
null_lnL = lf.lnL
null_nfp = lf.nfp
lf

Null Hypothesis

log-likelihood = -7177.4403

number of free parameters = 10

Edge params
edge	parent	length
Galago	root	0.167
HowlerMon	root	0.044
Rhesus	edge.3	0.021
Orangutan	edge.2	0.008
Gorilla	edge.1	0.002
Human	edge.0	0.004
Chimpanzee	edge.0	0.004
edge.0	edge.1	0.000
edge.1	edge.2	0.003
edge.2	edge.3	0.012
edge.3	root	0.009

Motif params
A	C	G	T
0.376	0.174	0.209	0.241

The alternate allows the human and chimpanzee branches to differ by just setting all lengths to be independent.

lf.set_param_rule("length", is_independent=True)
lf.set_name("Alt Hypothesis")
lf.optimise(local=True, show_progress=False)
alt_lnL = lf.lnL
alt_nfp = lf.nfp
lf

Alt Hypothesis

log-likelihood = -7175.7756

number of free parameters = 11

Edge params
edge	parent	length
Galago	root	0.167
HowlerMon	root	0.044
Rhesus	edge.3	0.021
Orangutan	edge.2	0.008
Gorilla	edge.1	0.002
Human	edge.0	0.006
Chimpanzee	edge.0	0.003
edge.0	edge.1	0.000
edge.1	edge.2	0.003
edge.2	edge.3	0.012
edge.3	root	0.009

Motif params
A	C	G	T
0.376	0.174	0.209	0.241

We import the function for computing the probability of a chi-square test statistic, compute the likelihood ratio test statistic, degrees of freedom and the corresponding probability.

from scipy.stats.distributions import chi2

LR = 2 * (alt_lnL - null_lnL)  # the likelihood ratio statistic
df = alt_nfp - null_nfp  # the test degrees of freedom
p = chi2.sf(LR, df)
print(f"LR={LR:.4f} ; df={df}; p={df:.4f}")

LR=3.3294 ; df=1; p=1.0000

Testing Hypotheses - By parametric bootstrapping#

If we can’t rely on the asymptotic behaviour of the LRT, e.g. due to small alignment length, we can use a parametric bootstrap. Convenience functions for that are described in more detail here Performing a parametric bootstrap.

In general, however, this capability derives from the ability of any defined evolve likelihood function to simulate an alignment. This property is provided as simulate_alignment method on likelihood function objects.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")

sm = get_model("F81")
lf = sm.make_likelihood_function(tree, digits=3, space=2)
lf.set_alignment(aln)
lf.set_param_rule(
    "length",
    tip_names=["Human", "Chimpanzee"],
    outgroup_name="Galago",
    clade=True,
    is_independent=False,
)
lf.set_name("Null Hypothesis")
lf.optimise(local=True, show_progress=False)
sim_aln = lf.simulate_alignment()
sim_aln[:60]

	0
Chimpanzee	AATAGAACACTAGTATTGCAAACGAACCTACTTAAATTGAGTTAAATGGAACATTAAAAA
Galago	....A.......A....A.....A............GA.......T........A.....
HowlerMon	............A.......................G........T..............
Rhesus	......C.....A.......................G........T..............
Orangutan	............................................................
Gorilla	............................................................
Human	............................................................

7 x 60 dna alignment

Determining confidence intervals on MLEs#

The profile method is used to calculate a confidence interval for a named parameter. We show it here for a global substitution model exchangeability parameter (kappa, the ratio of transition to transversion rates) and for an edge specific parameter (just the human branch length).

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")
sm = get_model("HKY85")
lf = sm.make_likelihood_function(tree)
lf.set_alignment(aln)
lf.optimise(local=True, show_progress=False)
kappa_lo, kappa_mle, kappa_hi = lf.get_param_interval("kappa")
print(f"lo={kappa_lo:.2f} ; mle={kappa_mle:.2f} ; hi={kappa_hi:.2f}")
human_lo, human_mle, human_hi = lf.get_param_interval("length", "Human")
print(f"lo={human_lo:.2f} ; mle={human_mle:.2f} ; hi={human_hi:.2f}")

lo=3.78 ; mle=4.44 ; hi=5.22
lo=0.00 ; mle=0.01 ; hi=0.01

Saving results#

The best approach is to use the json string from the to_json() method. The saved data can be later reloaded using cogent3.util.deserialise.deserialise_object(). The json data contains the alignment, tree topology, substitution model, parameter values, etc..

To illustrate this, I create a very simple likelihood function. The json variable below is just a string that can be saved to disk.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

aln = make_aligned_seqs(data=dict(a="ACGG", b="ATAG", c="ATGG"))
tree = make_tree(tip_names=aln.names)
sm = get_model("F81")
lf = sm.make_likelihood_function(tree)
lf.set_alignment(aln)
json = lf.to_json()
json[:60]  # just truncating the displayed string

'{"model": {"alphabet": {"motifset": ["T", "C", "A", "G"], "g'

We deserialise the object from the string.

from cogent3.util.deserialise import deserialise_object

newlf = deserialise_object(json)
newlf

F81

log-likelihood = -14.2727

number of free parameters = 3

Edge params
edge	parent	length
a	root	1.0000
b	root	1.0000
c	root	1.0000

Motif params
A	C	G	T
0.3333	0.0833	0.4167	0.1667

Reconstructing ancestral sequences#

We first fit a likelihood function.

from cogent3 import load_aligned_seqs, load_tree
from cogent3.evolve.models import get_model

tree = load_tree("data/primate_brca1.tree")
aln = load_aligned_seqs("data/primate_brca1.fasta")
sm = get_model("F81")
lf = sm.make_likelihood_function(tree, digits=3, space=2)
lf.set_alignment(aln)
lf.optimise(show_progress=False)

We then get the most likely ancestral sequences.

ancestors = lf.likely_ancestral_seqs()
ancestors[:60]

	0
edge.0	TGTGGCACAAATACTCATGCCAGCTCATTACAGCATGAGAACAGTTTATTACTCACTAAA
edge.1	............................................................
edge.2	............................................................
edge.3	...............................................G............
root	...............................................G............

5 x 60 dna alignment

Or we can get the posterior probabilities (returned as a DictArray) of sequence states at each node.

ancestral_probs = lf.reconstruct_ancestral_seqs()
ancestral_probs["root"][:5]

	T	G
0	0.1816	0.0000
1	0.0000	0.1561
2	0.1816	0.0000
3	0.0000	0.1561
4	0.0000	0.1561

There’s nothing that improves performance quite like being close to the maximum likelihood values. So using the set_param_rule method to provide good starting values can be very useful. As this can be difficult to do one easy way is to build simpler models that are nested within the one you’re interested in. Fitting those models and then relaxing constraints until you’re at the parameterisation of interest can markedly improve optimisation speed.

Evolutionary Analysis Using Likelihood#

Specifying substitution models#

The available pre-defined substitution models#

Getting a substitution model with get_model()#

Rate heterogeneity models#

For nucleotide#

For codon#

For protein#

Making a likelihood function#

Providing an alignment to a likelihood function#

Scoping parameters on trees – time heterogeneous models#

CNFGTR

Specifying a parameter as constant#

Providing a starting value for a parameter#

Setting parameter bounds for optimisation#

Setting an upper bound for branch length#

Specifying rate heterogeneity functions#

Specifying Phylo-HMMs#

Fitting likelihood functions - Choice of optimisers#

How to check your optimisation was successful#

Overview of the fitted likelihood function#

GTR

Log likelihood and number of free parameters#

Aikake Information Criterion#

Bayesian Information Criterion#

Getting maximum likelihood estimates#

One at a time#

Just the motif probabilities#

As tables#

Testing Hypotheses - Using Likelihood Ratio Tests#

Null Hypothesis

Alt Hypothesis

Testing Hypotheses - By parametric bootstrapping#

Determining confidence intervals on MLEs#

Saving results#

F81

Reconstructing ancestral sequences#

Getting a substitution model with `get_model()`#