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Mathematical Elegance with Biochemical Realism: The Covarion Model of
Molecular Evolution
David Penny,
1
Bennet J. McComish,
1
Michael A. Charleston,
2,
* Michael D. Hendy
2
1
Institute of Molecular BioSciences, P.O. Box 11222, Massey University, Palmerston North, New Zealand
2
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand
Received: 13 February 2001 / Accepted: 22 May 2001
Abstract. There is an apparent paradox in our under-
standing of molecular evolution. Current biochemically
based models predict that evolutionary trees should not
be recoverable for divergences beyond a few hundred
million years. In practice, however, trees often appear to
be recovered from much older times. Mathematical mod-
els, such as those assuming that sites evolve at different
rates [including a ⌫distribution of rates across sites
(RAS)] may in theory allow the recovery of some ancient
divergences. However, such models require that each site
maintain its characteristic rate over the whole evolution-
ary period. This assumption, however, contradicts the
knowledge that tertiary structures diverge with time, in-
validating the rate-constancy assumption of purely math-
ematical models. We report here that a hidden Markov
version of the covarion model can meet both biochemical
and statistical requirements for the analysis of sequence
data. The model was proposed on biochemical grounds
and can be implemented with only two additional param-
eters. The two hidden parts of this model are the propor-
tion of sites free to vary (covarions) and the rate of
interchange between fixed sites and these variable sites.
Simulation results are consistent with this approach, pro-
viding a better framework for understanding anciently
diverged sequences than the standard RAS models. How-
ever, a ⌫distribution of rates may approximate a co-
varion model and may possibly be justified on these
grounds. The accurate reconstruction of older diver-
gences from sequence data is still a major problem, and
molecular evolution still requires mathematical models
that also have a sound biochemical basis.
Key words: Covarion model — Hidden Markov
model — Rates across sites — Role of models in science
Introduction
The earliest models for the evolution of sequences as-
sumed that all variable sites evolved at the same rate.
Two approaches were introduced early in the study of
molecular evolution to relax this assumption and to
allow for rate heterogeneity between sites. One was the
well-known approach of fitting a probability distribu-
tion of rates (Uzzell and Corbin 1971); the other was
the covarion model (Fitch and Markovitz 1970; Fitch
1971). Under the Uzzell and Corbin [rates-across-sites
(RAS)] model, each site has a characteristic (or intrinsic)
rate which is maintained over the whole time period
being studied. Sites differ in these intrinsic rates and this
can be modeled by a probability distribution; for a de-
tailed analysis see Chang (1996). This general approach,
especially the ⌫distribution (see Yang 1996), has been
well developed mathematically, but several other distri-
butions (e.g., Waddell et al. 1997) and empirically mea-
sured distributions (Van de Peer et al. 1996) have been
used.
These distributions have desirable mathematical prop-
erties in that they require only one or two additional
Correspondence to: David Penny; E-mail: d.penny@massey.ac.nz
*Current address: Zoology Department, University of Oxford, Oxford,
England
J Mol Evol (2001) 53:711–723
DOI: 10.1007/s002390010258
©Springer-Verlag New York Inc. 2001
parameters, irrespective of the length of the sequences.
However, a problem with these RAS models is that there
is no biochemical explanation as to how each site main-
tains its own intrinsic rate over long evolutionary peri-
ods. This is a problem because the mathematics used for
the RAS models does require that each site maintains its
intrinsic rate over the entire time period being studied
(see Chang 1996; Tuffley and Steel 1997). Thus RAS
models require that each site maintains its own intrinsic
rate throughout evolution; our knowledge of biochemis-
try contradicts this (see later). Some features of the struc-
tural evolution of proteins are considered next, to illus-
trate the potential biochemical basis for evolutionary
models.
Structural Evolution of Proteins
It is difficult to find a biochemical mechanism that would
maintain the same intrinsic rate of evolution at each site,
irrespective of whether the gene was in eukaryotes, ar-
chaea, or eubacteria—or, indeed, within thermophiles or
mesophiles. In fact, biochemical information predicts the
opposite; homologous sites in widely different lineages
should vary in their rates. The following examples show
that one of the strongest conclusions from structural bi-
ology is that the three-dimensional (3-D) structure of
proteins varies during evolution. This has been demon-
strated, for example, by changes in the root-mean-square
(rms) difference in the position of the ␣carbon atoms
(C
␣
) along the backbone of the 3-D structure of a protein
(measured in angstroms). In an important study, Chothia
and Lesk (1986, 1987) reported on a variety of proteins
and showed that the average rms difference in 3-D struc-
ture increased with the sequence divergence—even if
considering only the core of the proteins. The effect was
nonlinear, with increasing difference in 3-D structure at
higher sequence divergence. Similar effects were found
using the structural alignment score (Levitt and Gerstein
1998). In another wide-ranging study, Pascarella and Ar-
gos (1992) also showed changes in structure resulting
from 714 insertions/deletions (see, for example, their
Fig. 6). Similarly, Carrugo and Argos (1997) have dis-
cussed, for a wide range of enzymes, the evolution of the
3-D structure of nucleotide-binding domains. They find
examples of both divergence and convergence of the
domains.
There have also been many studies on specific pro-
teins. For example, when comparing the X-ray crystal-
lographic structures of a fish and human hemoglobin the
rms difference is 1.4 Å, though the closeness of the
match varies throughout the protein (Camardella et al.
1992; e.g., their Fig. 8). Another example is studies
showing changes in 3-D structure between repeated units
of a protein. The subunits had diverged in structure over
time even if the units were identical initially in their 3-D
structure. An example is the “regulator of chromosome
condensation,” namely, RCC1. It has a seven-blade pro-
peller structure, but the seven repeating units deviate
slightly in 3-D structure (Renault et al. 1998, Fig. 3).
To continue with other types of studies, in an artificial
evolution experiment, Spiller et al. (1999) find variants
of an esterase that are optimal under different environ-
mental conditions. They show that these variants also
have differences in 3-D structure and describe their re-
sults in terms of a fitness landscape of 3-D structure
through which the enzyme evolves. Finally, a recent
study (Fisher et al. 2000) uses site-directed mutagenesis
and NMR to test (and confirm) predictions from the co-
varion model about the effect of specific mutations on
structure. A recent overview of protein structural evolu-
tion is available (Lesk 2000, Chaps. 5 and 6).
The overriding conclusion is that, although a few es-
sential sites may be invariable over long periods of evo-
lutionary time, most sites do change their functional en-
vironment during evolution. As such, the functional
constraints on sites are expected to change. This is per-
haps one of the best-substantiated facts of structural bi-
ology—individual amino acid sites are not in the same
environment over all of evolution. Thus, although the
probability distribution (RAS) approach has desirable
mathematical properties, the requirement for each site to
maintain its characteristic rate throughout evolution
(Chang 1996; Tuffley and Steel 1997) is not in agree-
ment with our current understanding of biochemical pro-
cesses. It might have been hoped that RAS models could
handle the biochemical situation if the same distribution
of rates were maintained throughout evolution—even
though individual sites varied in their rate class. For ex-
ample, it could be that the proportion of fixed sites stayed
constant, even if individual sites were variable or fixed in
different parts of the tree. Similarly, another proportion
of sites may be evolving at no more than 10% of the
maximum rate, but again the actual sites varied their
rate—though the overall distribution was still main-
tained. Unfortunately, the mathematical proofs (see ear-
lier) for the RAS model require that each site maintain its
intrinsic rate over the whole tree. Thus the biochemical
results contradict (disprove?) an essential assumption of
the RAS approach.
Covarion Model
The covarion substitution model (Fitch and Markovitz
1970; Fitch 1971) was introduced at the same time as the
RAS model of Uzzell and Corbin (1971). The covarion
model posits that, although some sites in a macromol-
ecule are critical to function and can never change, most
sites switch between being free to evolve in some taxa
and being fixed in others. This switch during evolution
would result from slight changes in secondary and ter-
712
tiary structures referred to above. For example, in an
early study of cytochrome cthe overall rate of evolution
was about 10% of the neutral rate, consistent with 10%
of sites being free to vary at any one time. However, in
mammals, 15% of sites had changed, but over a wide
range of eukaryotes, about 70% of positions had changed
(Fitch and Markowitz 1970). The conclusion drawn was
that
because of the structural restraints imposed by
functional requirements, mutations that will not be
selected against are available only for a very lim-
ited number of positions....However, as such ac-
ceptable mutations are fixed they alter the posi-
tions in which other acceptable mutations may be
fixed. Thus, only about ten codons, on the average,
in any cytochrome cmay have acceptable muta-
tions available to them but the particular codons
will vary from one species to another. We shall
term those codons at any one instant in time and in
any given gene for which an acceptable mutation is
available as the concomitantly variable codons.
(Fitch and Markowitz 1970, p. 585)
“Covarion” is a contraction of concomitably variable
codons, and of course, the principle can be applied at the
nucleotide level (Fitch 1986), as well as for proteins. We
prefer not to use the term covariotide for the nucleotide
version; the underlying concepts are identical irrespec-
tive of the number of character states, and it is highly
undesirable to increase new terms for every possible
variant (Penny 1993).
The covarion model is an extension of the earliest
explanations for the differing rates of evolution between
proteins (e.g., Dickerson 1971). (Here we are con-
sidering differences in the rates of evolution between
proteins, not differences between sites within a protein.)
We call these earliest explanations of differential rates
in protein evolution the Kimura–Dickerson model.
Under this biochemical model, changes in sequences are
both stochastic and neutral (Kimura), and variation in
the proportion of unconstrained (“free to vary”) sites
accounts for the different rates between proteins
(Dickerson). Several authors had suggested that differ-
ences in the number of sites free to vary within a protein
accounted for the differences in rates between proteins.
Dickerson (1971), an early structural biologist also in-
terested in evolution, expresses the idea clearly in rela-
tion to 3-D structure.
Figure 1 illustrates this simple biochemical model.
Each site free to change evolves at the same rate—for
nucleotides this is independent of whether it is the first,
second, or third position in the codon. Under this model
the first two positions have more sites constrained, con-
sequently their average rate is lower (though their vari-
ance would be higher). In agreement with this, cases
have been reported where there is little difference in the
rate of saturation at the three positions in the codon, for
example, in cytochrome b(Griffiths 1997). This obser-
vation is expected only when there are no changes in 3-D
structure (and, consequently, in structural constraints) of
a protein (remembering that under this model the slower
average rate at the first and second positions is because
fewer sites are variable). That equal rates of saturation at
the three codon positions is found at all is strong rein-
forcement that the basic biochemical model (the Kimu-
ra–Dickerson model) is part of the evolutionary process,
even if incomplete by itself. Under more complex (and
realistic) models, this equal rate of saturation at all three
codon positions need not occur.
The covarion model is an extension of this basic
Kimura–Dickerson model where some sites switch be-
tween “on” and “off” in different parts of the tree. Thus
it may be called a Kimura–Dickerson–Fitch model. This
name is not intended to replace the well-established co-
varion name; rather it is to show that the model is made
up of components, thus making it easier to analyze. Ka-
ron (1979), using Fitch’s cytochrome cdata, improved
the original model to account more fully for the redun-
dancy in the genetic code and used more robust statistical
methods to fit the model to the data. More recently, Fitch
and Ayala (1994) reported that the rate of evolution of
Fig. 1. The problem of explaining different evolutionary rates within
the codon. As shown in a, the average rate of change usually varies
between codon positions, with the second position the slowest and the
third the fastest. However, simple biochemical models assume that
most nucleotide changes are either neutral or lethal, and therefore sites
are either variable or constant (“on” or “off” in b). Thus, in these
simple models the first and second sites evolve more slowly on aver-
age, but each variable site saturates at the same rate (see Griffiths
1997). This predicts that the first and second positions of a codon are
basically no more reliable than the third codon positions, although the
experience of most researchers is that first and second sites are more
reliable for older divergences.
713
superoxide dismutase (SOD) was consistent with a mo-
lecular clock if modeled by a covarion process. Although
not commented on, their model was similar to a hidden
Markov process (Elliot et al. 1995); see below. Later
work (Miyamoto and Fitch 1995) suggested that the co-
varion model gave a better fit to the data than arbitrary
mathematical models, such as the commonly used ⌫dis-
tribution of RAS. Similarly, a new quantitative test
(Lockhart et al. 1998) shows that for their data a covarion
model fits the data better than a RAS model—or more
accurately, their test rejects any RAS model where each
site always has the same rate. Several other authors have
limited discussions of the covarion model (e.g., Koonin
and Gorbalenya 1989; Marshall et al. 1994). A review of
over a hundred papers that mention covarions revealed
only one author (Gillespie 1988) who appeared to dis-
agree with the covarion hypothesis.
Despite its sound biochemical basis and its potential
importance for evolutionary studies, the covarion model
has not been fully developed; from a statistical viewpoint
it has far too many parameters to be useful. If most
amino acid positions are constant over some portions of
the tree and variable in others, then it appears that one
could include as many parameters as desired “in order to
fit the data to the model”! In general, invoking more and
more parameters weakens the power of any model (see
Steel and Penny 2000). Indeed in the case of evolution-
ary trees, Steel et al. (1994) proved that with enough
variability of rates between sites, any data could, in prin-
ciple, be derived from any tree. Thus the covarion model
had the converse properties to the probability distribution
(RAS) approach; the covarion model had a reasonable
biochemical foundation but (as originally formulated)
lacked the required mathematical properties. However,
before proceeding with an implementation of the co-
varion model, we study the problem of saturation in the
basic Kimura–Dickerson model.
Saturation in the Basic Biochemical Model
It is expected that the basic biochemical (Kimura–
Dickerson) model, which assumes that a site is always in
the same rate class, would lead to sequences saturating
relatively quickly during evolution. This is most easily
illustrated by simulation. From previous computer simu-
lations for a variety of methods, recovering evolutionary
trees is inaccurate when there has been, on average, more
than about one change per site (Charleston et al. 1994).
However, most simulation studies use neither defined
time periods nor known rates of molecular evolution. For
the present study, we use the neutral rate of evolution for
unconstrained sites as 0.5% per site per million years (Li
1997, p. 75; Page and Holmes 1997, p. 239). There is
then an average of one change per site every 200 million
years. Using a measured rate allows the essential linking
of simulation results to elapsed time.
To illustrate this problem of saturation for ancient
divergences with simple biochemical models (that is,
neither covarion nor RAS models), we ran 300,000 simu-
lations on randomly selected nine-taxon trees. The time
periods for the simulation increased logarithmically so
that the expected numbers of changes per site (which is
linear with time) increased from 0.025 to 12.5. This is
equivalent to 5 to 2500 million years of evolution and
thus gives a relationship between simulation results and
evolutionary time. Individual lengths of edges (branches)
on the tree were selected randomly, with internal edges
being no more than one-half of the longest external edge.
Data sets were generated for two-state characters using
the Hadamard conjugation (Hendy and Charleston 1993).
Data were generated for 100–1000 variable sites (that is,
excluding any invariable sites). Trees were inferred from
each data sample by a variety of methods.
The results in Fig. 2 show the expected error in re-
covering the correct tree for nine taxa (six internal edges)
Fig. 2. Expected accuracy versus time of divergence. Data were gen-
erated under a Kimura 2-ST model for nine taxa for 5–2500 million
years (x-axis). Neighbor-joining (Phylip) was used to infer a tree from
each data set. The proportion of correct internal edges (branches) is
shown on the y-axis, ranging from 6 internal edges (branches), the fully
correct tree, to 0, where all internal edges were wrong. Each point
represents 1000 simulations with sequences 1000 nucleotides long. For
shorter times the correct tree (or a tree with only one error) was found
in virtually all trials; after the equivalent of about 400 Mya the fully
correct tree was not recovered under this model of evolution.
714
and are for 1000 variable sites using neighbor-joining
(Felsenstein 1997)—which is consistent under this
model. The increasing time of divergence (x-axis) is
equivalent to 5 to 2500 million years of evolution at this
mutation rate. The y-axis is the proportion of internal
edges (branches) inferred correctly. For shorter times it is
expected that either the entire tree (six internal edges)
will be recovered correctly or a tree with no more than a
single error. However, as the overall time increases, the
expected number of correct branches decreases (rela-
tively rapidly) until eventually the inferred tree is ex-
pected to have all six internal edges wrong! Note that
even at the extreme the results are still better than ran-
dom; the method is consistent and would eventually get
the correct tree if extremely long sequences were avail-
able.
Our conclusion is that under a simple biochemical
model (the Kimura–Dickerson model), together with re-
alistic rates of evolution, it should be difficult or impos-
sible to recover trees accurately after divergences of
more than about 300–400 million years. This conclusion
comes from the present work (where only neighbor-
joining results are shown), those of Charleston et al.
(1994), the covariance matrix calculated by Hadamard
transforms (Waddell et al. 1994), and the length of se-
quence that can be required even for maximum likeli-
hood (Steel and Penny 2000). Although in practice the
long edges on the tree may be broken up by other taxa,
the present example illustrates the problem of current
models for ancient divergences. Under these standard
models, there is no justification for expecting correct
results for ancient divergences.
In contrast with theory, many researchers (because of
agreement with other data) appear confident with many
aspects of evolutionary trees for older divergences. This
is the basic problem—simple biochemical models with
neutral changes, and sites always having the same rate of
change, predict that ancient divergences would be poorly
handled by current methods. This difference between
theory and practice must be addressed.
In defense of theory, it is well known that the order of
divergences of the main avian and mammalian groups is
still controversial. This result is consistent with theory
and is found even for eutherian mammals that diverged
within the last 130 million years (relatively recently on
the scale in Fig. 1). There has been good progress in
resolving this question (Waddell et al. 1999) but the
point at issue is, why are we more confident of much
older divergences when we know that we cannot guar-
antee more recent ones?
In defense of practice, it is repeatedly found that there
is considerable agreement between different data sets
(both molecular and morphological). Bird sequences do
not come out among mammals or invertebrates, mosses
among flowering plants or fungi, and so on. There are
many difficulties with differences between data sets (es-
pecially with the oldest divergences) but there is no sug-
gestion (as in Fig. 2) that all internal edges of a tree are
incorrect. Nevertheless, there are major difficulties be-
tween data sets for ancient divergences. It is difficult to
see why researchers are so confident in their results when
the relatively recent divergences within mammals, birds,
or flowering plants are only now being resolved. The
covarion model offers a possible resolution to this fun-
damental problem of molecular evolution. But first we
consider a range of alternatives as to why some sites
evolve more slowly.
Alternative Models
A standard answer is just that some sites “change more
slowly” or that “some sites are more constrained.” How-
ever, this is a description of what is observed, not an
explanation. What mechanisms would result in a site in
a protein “changing more slowly”? Possibilities include
the following:
(1) mutation rates are lower at first and second posi-
tions;
(2) mutations fixed at first and second positions are
slightly deleterious and therefore are less likely to be
fixed (giving a slower rate);
(3) fewer mutations are viable at first and second posi-
tions (but sites are always variable or always fixed—
the Kimura–Dickerson model);
(4) first and second positions can switch between being
“on” and being “off” with slight changes in second-
ary and tertiary structure (a covarion model); and
(5) a new model will explain the observation.
A difference in mutation rate between codon positions in
a single gene is not considered likely as a general expla-
nation (though mutation rate may vary in different parts
of the genome). DNA polymerases, proofreading, and
error correction all function independently of the reading
frame. The second explanation is the fixation of some
slightly deleterious mutations in a population. Such sites
will change more slowly than the neutral rate—because
negative selection still eliminates some mutations. By
itself, this is not a general solution because it implies a
slow continual decline in fitness over time. A different
version of this model is due to Zuckerkandl (1976),
where positive selection was invoked for fixation of vir-
tually all changes. In this model proteins were subopti-
mal (because some of the desirable properties were in-
compatible). There was a continual cycle of fixation of
new mutations that improved one aspect of the protein
but led to decreased functionality in another. Although
the model was a selectionist explanation for a molecular
clock, it never received direct biochemical support and
there was increasing support for the view that most
715
changes were neutral. Indeed, positive selection for simi-
lar changes on different lineages would only make it
harder to recover the correct tree.
We demonstrate here that the third explanation, dif-
ferences in the number of viable nucleotides at a site,
does not work either. The basic Kimura–Dickerson
model of molecular evolution allows from one to four
nucleotides at a site to be viable. On this explanation,
negative selection eliminates from zero to three muta-
tions. If most mutations at a site were lethal, would this
maintain a phylogenetic signal longer? This can be
checked by calculation.
Consider the possibility that only two nucleotides (or
amino acids) were viable at a site. The rate of loss of
phylogenetic signal was calculated on a four-taxon tree
that was rooted on the central edge and had equal rates of
evolution [so there were no problems with inconsistency
(Hendy and Penny 1989)]. Using the method of Hendy et
al. (1994), calculations were made for all nucleotide
changes equally likely (the Jukes–Cantor model) and
with 5% change on the internal edge. The external edges
(equal to time) were then made longer and longer and are
shown as the x-axis in Fig. 3. Time is on a logarithmic
scale up to 2 billion years (a time routinely used when
studying the tree of life). “Correc_2” and “correc_4”
represent the probability of a site supporting the correct
tree with two, or four, nucleotides viable at that site.
Conversely, “wrong_2” and “wrong_4” are the prob-
abilities of a site supporting one of the two incorrect trees
(values should be doubled if considering the probability
of a site supporting either incorrect tree). Note that with
only two of the four nucleotides viable, there are (for
t⳱4 taxa) only 8 possible patterns at a site; there are 64
patterns for four nucleotides. Consequently, as sites be-
come randomized each value converges to 0.125 for two
viable nucleotides [1 of 8 (2
t−1
) patterns] and to 0.0469
for four nucleotides [3 patterns of 64 (4
t−1
)].
In Fig. 3 there is little difference in the rate of ran-
domization (loss of phylogenetic signal) whether two or
four nucleotides are viable at a site. Under the parameters
used, sites with either two or four nucleotides viable are
approaching randomization by 300–400 million years.
With only two nucleotides viable there is a 1–3% slower
approach to randomization—essentially no difference.
Another conclusion from the results in Fig. 3 (not shown)
was that the proportion of sites that had not changed,
although they were free to vary, decreased to zero rela-
tively quickly. This reinforces the conclusion that sites
that are constant for anciently diverged trees are func-
tionally constrained (that is, genuinely invariable). Such
sites should not be used when estimating either the num-
ber of multiple changes or the nucleotide compositions at
other sites that are free to vary. Retaining such sites in an
analysis means that the number of changes is underesti-
mated (Palumbi 1989), and even maximum likelihood is
no longer consistent (Lockhart et al. 1996; Steel and
Penny 2000).
The results in Figs. 2 and 3 illustrate a fundamental
problem with inferring ancient evolutionary trees from
sequence data; current biochemically based models are
not encouraging regarding our ability to recover deep-
branching phylogenetic signals. However, our working
hypothesis was that, with real sequence data, processes
such as the continuous operation of a simple covarion
model could make the inference of older divergences
more accurate. We now report results that support this
expectation. Data were generated under a simple co-
varion model and then analyzed by standard (nonco-
varion) models.
Covarion Model with Two Additional Parameters
Given the failure of the first three suggestions for slower
average evolution at the first two positions in a codon,
we now consider covarion models. These allow sites to
vary in their rate of evolution as the 3-D structure of the
macromolecule evolves. Such models may be useful if
we can combine biochemical realism with mathematical
rigor. We have been involved in a long-term study of
covarion models and the present work is the background
to this wider study. The areas of interest include math-
ematical analysis (Tuffley and Steel 1997) and a dem-
onstration that sites evolve at different rates over the tree
(Lockhart et al. 1996, 1998). The work has led to tests
(Lockhart et al. 1998, 2000) that distinguishes a covarion
model from one that predicts sites are always in the same
rate class. This present paper gives the basic reasoning
Fig. 3. Rate of saturation when two or four nucleotides are viable at
a site, for increasing times. The probabilities of a site supporting the
correct tree are correc_2 (for two viable nucleotides) and correc_4 (for
four viable nucleotides). Conversely, probabilities for sites supporting
the wrong tree are wrong_2 and wrong_4. The results show that sites
saturate at about the same rate, irrespective of whether two or four
nucleotides are viable. For two viable nucleotides, there are only 8
patterns at a site, compared with 64 when all nucleotides are viable
(three of which directly support the correct tree).
716
behind our studies and reports results from the applica-
tion of a partially hidden Markov process to model the
covarion process. The model requires only two param-
eters additional to the commonly used Markov models
(Tuffley and Steel 1997). It thus solves the main problem
in the past, that the original covarion model appeared to
require several parameters per site. We give the formal
model here for nucleotide evolution but it is readily ex-
tended to amino acids.
The hidden Markov model has two main processes.
The first is a standard Markov model for molecular evo-
lution and is implemented here with the Kimura (1981)
3ST model—which contains the 2ST and Jukes–Cantor
models as special cases. The second (hidden) process has
the two additional parameters: , the proportion of sites
that are free to vary (the covarions); and ␦, the rate of
interchange between the covarions and sites that are in-
variable (cannot change because of biochemical con-
straints). In this simple version, all sites have the same
probability of being in either rate class, and thus the
model is still stationary and i.i.d (independent and iden-
tically distributed). The model is “stationary” in the
sense that the basic process and frequencies of rate
classes are unchanged over the whole tree. At a variable
site, a mutation may be fixed in the population either by
random genetic drift (neutral) or by positive selection
[although the model has no enhanced rate of fixation for
positive selection (see also Ohta and Kimura 1971)].
Whether a site in a sequence is fixed or variable at a
particular point in time is unknown—hence the name
“hidden” Markov model. The variability status is repre-
sented by a superscript: + when the states are free to
change (A
+
,G
+
,C
+
, and T
+
) and − when they are fixed
(A
−
,G
−
,C
−
, and T
−
). For proteins, the 40 character states
are A
+
,C
+
,D
+
,E
+
,F
+
,...Y
+
for the potentially variable
sites and A
−
,C
−
,D
−
,E
−
,F
−
,...Y
−
for sites that are
invariant at a particular point in time. The rate of inter-
change between the fixed and the variable states is set to
maintain the proportion of variable (covarion) and
fixed sites (see below).
Figure 4 illustrates the difference between the co-
varion model (Fig. 4A) and the distribution of RAS
model. In the covarion model, a site can change in the
tree between at least two rate classes. In the distribution
of RAS model, each site (or category) has its own rate,
which is maintained over the entire tree, for example,
either fast as in Fig. 4B or slow as in Fig. 4C.
Kimura’s 3ST model is used by both RAS and co-
varion models and is described by an instantaneous rate
matrix, K, which has three substitution types, ␣,, and
␥; hence the name 3ST. The Hadamard matrix Hdiago-
nalizes K:
K=
A+
G+
C+
T+
冤
∗␣␥
␣∗␥
␥∗␣
␥␣∗
冥
H=
冤
1111
1−11−1
11−1−1
1−1−11
冥
⌳=H
−1
KH =
冤
1
000
0
2
00
00
3
0
000
4
冥
where *⳱−(␣++␥) (so that the rows sum to zero);
H
−1
⳱
1
⁄
4
H; and
1
⳱0,
2
⳱−2(␣+␥),
3
⳱−2(+␥),
and
4
⳱−2(␣+) are the eigenvalues for K(for ␣>
ⱖ␥). This diagonalization enables the ready calculation
of the exponent of K, exp(K)⳱I+K+(K
2
/2!) +
(K
3
/3!) + (K
4
/4!) +...⳱H䡠exp(⌳)H
−1
, and exp(⌳)is
the diagonal matrix whose entries are exp(
i
). The tran-
sition matrix M⳱exp(Kt) expresses the probabilities of
the substitutions of each type during an interval of time
tand the values from Mare used to predict the amount
and types of nucleotide changes during evolution.
A basic rate matrix (M) for our hidden Markov model
is
K⬘=
冋
K0
00
册
+␦
冋
−II
kI −I
册
where K⬘isan8×8matrix and Kand Iare4×4
matrices. To make the model more general (allowing
different proportions of fixed and variable sites), we
choose a value for kso that the proportions of variable
and constants sites is constant during evolution. The in-
stantaneous rate matrix (K⬘) for the hidden Markov
model we actually use, based on a Kimura three-
parameter model (K), is shown in Fig. 5. From this point
our use of the rate matrix K⬘is standard; for specific
values of ␣,,␥,␦, and k, we take the exponential to get
Fig. 4. A difference between the covarion model
(A) and rates-across-sites models (Band C). Under
the covarion model, a site can change from the
standard rate (solid line) to zero (dashed line), and
vice versa. Under the rates-across-sites model (B
and C), each site maintains its own intrinsic rate over
the whole tree, though there is a distribution of sites
evolving at different rates (faster in B, slower in C).
717
a Markov transition matrix—which is used for simulat-
ing sequence data (see below).
Our working hypothesis was that the covarion model
allows older divergences to be recovered more accu-
rately. The trees chosen to test the covarion model (Fig.
6) are inconsistent under uncorrected parsimony (Hendy
and Penny 1989) and are especially difficult to recover
without accurate corrections for multiple changes. The
four-taxon tree (t
1
,(t
2
,(t
3
,t
4
))) (Fig. 6A) has one slowly
evolving lineage (t
2
), and the other lineages fit a molecu-
lar clock. The correct tree becomes the longest (not the
shortest) tree for uncorrected parsimony. In the five-
taxon tree ((t
1
,t
2
),(t
3
,t
4
),t
5
) all lineages fit the molecular
clock (Fig. 6B). However, with uncorrected parsimony
the correct tree (even with infinitely long sequences) is
the 12th longest (of 15). For each tree, the internal part of
the tree was fixed, while the external edges were allowed
to increase in length, representing longer and longer
times.
Results from Modeling the Covarion Process
Our test was the frequency of recovery of the correct tree
as the total period of evolution increased. Sequences
were generated under a covarion model and then a
Kimura model in Phylip (Felsenstein 1997) used to infer
trees. The covarion process was based on the Kimura
2ST model with ␣⳱0.005 (the transversion rate) and 
⳱␥⳱0.0025 (the transition rate) and with equal
nucleotide frequencies at the root. For the two additional
parameters for the covarion model, (the proportion of
variable sites) was set at 0.5 and ␦(the rate of inter-
change between fixed and variable sites) was varied. For
the results reported here, sequences were of length 1000.
DNAML was used for maximum likelihood and NEIGH-
BOR for neighbor-joining on distances.
The computational time for maximum-likelihood cal-
culations with a thousand simulations for even a single
data point on the five-taxon tree is large. Consequently
the simulation and tree-building steps were performed on
50 Pentium PC computers running in parallel and con-
trolled remotely over the network. This process was
largely automated by using batch and input files to con-
trol each set of simulations and to transfer output to the
tree-building programs.
Results for the five-taxon tree (Fig. 6B) are shown in
Fig. 7 (the four-taxon tree in Fig. 6A has similar results;
data not shown). The x-axis is time (on a logarithmic
scale) and the y-axis the probability of correctly recov-
ering the generating tree. Figure 7A is for neighbor-
joining and Fig. 7B for maximum likelihood, both using
Phylip. The successive curves are for increasing values
of ␦(the rate of interconversion between fixed and vari-
able sites). The general conclusion from Fig. 7 is that
increasing ␦increases the ability to recover the tree cor-
rectly. ␦⳱0 is equivalent to 50% of sites being fixed
and 50% being variable (is 0.5). As ␦increases to ⬁,
the model becomes equivalent to all sites being variable,
but at half the rate of change as with ␦⳱0. If the
covarion model is a realistic biochemical description of
molecular evolution, then current methods for inferring
trees (that is, methods not using a covarion model) are
expected to perform better than the results in Fig. 2.
The extent of the improved performance is shown by
the observation that the model tree can still be identified
after longer periods of evolution—if a covarion process
is operating. Increasing the rate of interchange (␦) be-
tween fixed and variable sites increases the chance of
selecting the correct tree. DNAML was more successful
than neighbor-joining with the four-taxon tree, but with
the five-taxon tree, the results were more ambiguous.
Both, however, did better as ␦increased in value. A
covarion model with these parameters increases by 50–
100% the time over which current methods of tree build-
ing are reliable. This need not be the limit for increased
performance. Many other combinations of parameters
could be tested, though it is preferable to explore theo-
retical properties first to test predictions more construc-
tively. Note, again, that the covarion process was used
only to generate the data, not when inferring the tree
from the data.
Consequences for Ancient Divergences
It is clear that simulation studies need to use both real-
istic times and real rates of evolution (not rates averaged
across variable and invariable sites). Most simulation
studies use only relatively short evolutionary periods and
Fig. 5. Parameters for a hidden Markov model for nucleotide evolu-
tion. AThe instantaneous rate matrix Kⴕ.BA graphical representation.
The diagonals (labeled *) are given values so that each row of the rate
matrix sums to 0. The arrows in the graphical representation (B) cor-
respond to the positive entries in the rate matrix. For example, there are
four rates to or from A
+
(␣,,␥, and ␦) and only one (␦) from A
−
.
718
do not even use using measured rates of evolution. Con-
sequently, it is impossible to apply any conclusions to
real data and real times of divergence. In many ways, the
conclusion in Fig. 2 is self-evident; current methods
should fail for ancient divergences under the standard
assumptions of the models. It is not scientific just to
assume that models will work for the oldest divergences;
they must be tested explicitly.
Before examining ways in which covarion-type mod-
els may help ancient divergences, it is necessary to be
cautious of existing knowledge about deep divergences
in the tree of life. There is no good biochemical reason
for current models to be reliable for inferring the early
branches of the tree of life. It was for this reason that we
used relics of the RNA world as a possible alternative for
rooting the tree of life (Poole et al. 1998). Given the
difficulties in inferring the order of divergence of mam-
mals correctly, it is certainly premature to be confident in
ancient divergences. However, the caution works both
ways; just because two genes give different trees does
not mean that one (or both!) have been subject to lateral
transfer. It is expected that genes can differ in the trees
they predict; they do even for mammals (Penny et al.
1982). Lateral transfer is undoubtedly an important fea-
ture in evolution, but the results presented here caution
against invoking it every time two genes differ in the tree
they predict.
The present results (Fig. 7) are consistent with our
working hypothesis—the covarion model predicts that
trees for ancient divergences will be better than expected
from simple biochemical models. In a sense, the co-
varion model increases the “effective number” of vari-
able sites. The covarion model could also explain why a
particular molecule might have a range of times for
which it is most suitable (e.g., Graybeal 1994; Whitfield
and Cameron 1998). This is because the length of time it
takes a particular protein to saturate depends on the rate
of evolution of its tertiary structure. If the tertiary struc-
ture does not change, the protein is expected to saturate
sooner (see Griffiths 1997). Other authors (e.g., Simon et
al. 1994) suggest that, in practice, some macromolecules
lose resolution (as expected) at intermediate dates of di-
vergence but improved again for divergences that were
even older. Such a result could occur if some slight
changes to secondary and tertiary structure occurred only
very occasionally (that is, low values of ␦, or no longer
a stationary model). In such circumstances, new invari-
ant positions that helped recovery of the tree would arise
occasionally. An alternative may be an occasional, but
larger, change in covarion structure (see Lockhart et al.
1996; Wolfe and dePamphilis 1998). For ancient diver-
gences, individual proteins might not allow resolution
because their 2-D and 3-D structure is too highly con-
served. It is possible too that, on average, ribosomal
RNA does better than expected because its secondary
structure does vary considerably. We have recently
shown that RNA secondary structure itself can be used to
reconstruct trees, even when the sequences cannot be
aligned with any confidence (Collins et al. 2000).
Some of the most interesting recent applications of a
covarion model are studies by Brinkmann and Philippe
(1999) and Lopez et al. (1999). They use concepts from
the covarion model to identify the slowest-evolving sites.
This differs from the RAS approach, which does not
identify which sites are faster or slower, just the distri-
bution of rates. Lopez et al. report that the slower- and
faster-evolving sites support different trees! Thus the
faster sites appear to be more strongly affected by the
long-edges-attract problem (Hendy and Penny 1989).
Although the covarion model in general could be con-
sidered “good news,” there are times when a nonstation-
ary version of the covarion model could be “positively
misleading.” In these cases, a covarion process could
reinforce support for an incorrect tree, and a possible
example has been discussed (Lockhart et al. 1996, 1998;
Steel et al. 2000). One case has a tree with five major
Fig. 6. The trees used in simulations of the covarion model: (A)
four-taxon; (B) five-taxon. The lengths of internal edges of the trees
were held constant, and the external edges (except for the edge to taxon
2 in A) ranged over different lengths (times) for successive simulations.
Data were simulated on these trees using the parameters in Fig. 5, and
then trees inferred from the data using Phylip with a Kimura 2ST
model. Results for the five-taxon tree are shown in Fig. 7.
719
branches, each with many sequences (Lockhart et al.
1996). On two branches, there have been major (but
different) changes in the function of the gene, and the
covarion set has changed independently in these two
groups. The consequence is that branches where genes
retain their initial function tend to group together, even
though they may not have been adjacent on the original
tree. That example of a covarion model differs from the
stationary version presented here in that it has occa-
sional, but large, changes in the covarion set; it is a
nonstationary model. In our implementation, the contin-
ued small covarion changes are independent of position
on the tree; the process does not change across the tree.
The Lockhart et al. results emphasize again the need to
Fig. 7. Results for neighbor-joining (A) and maximum likelihood (B)
for data simulated on the tree in Fig. 6B and under a covarion model.
Each point represents 1000 simulations for sequences of 1000 nucleo-
tides. The x-axis shows the relative times on a logarithmic scale, and
the y-axis the probability of recovering the generating tree correctly. In
each figure, the seven curves are ␦(d) values—the rate of intercon-
version between fixed and variable sites. Increasing ␦enhances the
chance of recovering the correct tree, allowing the correct tree to be
inferred for 50–100% longer divergence times.
720
understand the changes that are occurring in a gene dur-
ing evolution.
The results in Figs. 2 and 7 are for ideal cases: no
changes in nucleotide composition, no positive selection
for similar changes on different lineages, no correlation
between sites, no lateral transfer, etc. These additional
factors are expected to make it even more difficult to
recover trees accurately from real data. For example, it
has recently been shown that some data sets (Cao et al.
1998) show contradictions between different proteins in
the mitochondrial genome. This shows that there are dif-
ferent signals in the data, and even longer sequences may
be required than lengths estimated from simulations.
Nevertheless, the results (Fig. 7) are consistent with the
original hypothesis that the covarion model gives a bio-
chemical basis for why tree reconstruction may, in prin-
ciple, do better for ancient divergences than simple bio-
chemical models predict.
Relationships Between Models
One aspect of our model makes it more general than that
of Fitch and Ayala (1994)—the interconversion between
fixed and variable states is continuous in our formula-
tion. That is, they are not limited to a change in the
sequence of the macromolecule. Our formulation is sim-
pler to analyze because the two processes (changes be-
tween nucleotides and interconversion between fixed and
variable states) are independent. This additional flexibil-
ity is realistic. The set of variable sites may be altered by
both intramolecular and intermolecular interactions
(Lockless and Ranganathan 1999), as well as by envi-
ronmental changes such as in temperature. Although ei-
ther formulation (Fitch and Ayala’s, or ours) can be jus-
tified biologically, the calculation and analysis are more
straightforward for our hidden Markov process.
Our formulation of the covarion model can be con-
sidered within an expanded class of i.i.d models (see
Penny et al. 1992). Changes are independent, both be-
tween sites and on different lineages of the tree, and in
addition, each site is drawn from the same (identically
distributed) distribution. Models more general than the
Kimura 3ST can be implemented under a covarion
model, but then the Hadamard matrix cannot be used
for diagonalizing matrices—though other methods are
available. We consider our present model a member
of the Kimura–Dickerson–Fitch class of models. It in-
cludes the following elements: it is a stochastic model;
most changes are neutral; a restricted number of sites is
free to vary at any one time; but that set changes
with time.
It is interesting to note that the covarion model gives
some biochemical justification for the use of, for ex-
ample, a ⌫distribution of rates. Operationally, the ⌫
distribution compensates, in part, for some sites being
invariant (Waddell and Steel 1997). Theoretically, Tuf-
fley and Steel (1997) report that for pairs of sequences,
a covarion model can always be matched by a ⌫(or more
general) distribution. This does not generalize to higher
numbers of sequences, and tests are available that can
distinguish RAS and covarion models (Tuffley and Steel
1997; Lockhart et al. 1998, 2000). The tests compare the
numbers of constant (or varied) sites in different parts of
the tree, but little is known of the power of the tests. Thus
the equivalence of covarion and RAS models is only for
pairs of taxa, and further work is required to determine
when the ⌫distribution is, in practice, a useful approxi-
mation to a covarion model. At present, we are more
interested in exploring the usefulness of identifying
faster and slower sites (see Brinkmann and Philippe
1999; Lopez et al. 1999), rather than assuming that a site
is sampled from a distribution of RAS. Finally, the co-
varion model is also an explanation for the common
practice of discarding sites that are difficult to align.
Such difficult-to-align sites are expected to occur where
there have been changes in the 3-D structure of the mac-
romolecule.
Role of Mathematical Models
This relationship between the covarion model and an
approximation to it by a ⌫distribution (Tuffley and Steel
1997) raises another interesting question: the role in sci-
ence of formal mathematical descriptions and underlying
physical models. The most mathematically developed as-
pects of biology include population genetics, ecology,
physiology, and biochemical kinetics. In each case, the
mathematical model is a formalization of the underlying
biological mechanisms. However, with RAS models
there has been little attempt to consider the biochemistry
that might “justify” a ⌫(or other) distribution. In general,
scientists prefer the mathematics to be based on a physi-
cal (biological) model, not just be an arbitrary math-
ematical description. Should this also be the case in evo-
lutionary analysis?
There is one established viewpoint, instrumentalism,
that accepts that mathematical models may be useful
“instruments” for calculation.
There is no need for these hypotheses to be true, or
even to be at all like the truth; rather, one thing is
sufficient for them—that they should yield calcu-
lations which agree with the observations. [See dis-
cussion by Popper (1963, pp. 97ff)].
The best-known case in science is the opportunity given
to Galileo to use this reasoning as an alternative to the
heliocentric hypothesis of the planets orbiting the sun.
The church hierarchy accepted that Galileo’s calcula-
tions were more accurate—it would be acceptable just to
deny that the equations described the solar system. Sober
721
(1998) discusses some aspects of modern science where
instrumentalism is still influential, and a case can be
made for some statistical models that successfully iden-
tify a pattern, without identifying the underlying pro-
cesses.
In the case of the distribution across sites model, some
authors may be satisfied if the distribution aids in getting
the correct tree, regardless of any biochemical process
that may, or may not, underlie the calculation. We prefer,
however, that more consideration be given to the rela-
tionship between mathematical models and the underly-
ing biochemical mechanisms. It is little more than a tau-
tology just to say, “Sites evolve at different rates,”
without understanding the mechanisms involved. It is
more satisfying to have a biochemical mechanism de-
scribed mathematically, rather than a convenient math-
ematical description not based on any biological mecha-
nism. It is interesting that more biochemical realism is
being introduced into models of molecular evolution;
examples include Goldman and Yang (1994), Lio` and
Goldman (1999), Thorne et al. (1992), and Scho¨niger
and von Haeseler (1994).
Extensions to the present hidden Markov implemen-
tation of the covarion model are straightforward. The
present version already allows a proportion of sites to be
permanently fixed or permanently variable. A RAS
model could be included so that the variable sites evolve
at different rates (though a biochemical explanation is
unclear). It is straightforward to add another layer of
invariable sites with only one class of invariant sites able
to become covarions directly (that is, invariant 2 ↔in-
variant 1 ↔covarions). Other sites may always be fixed;
others may always be variable and will saturate relatively
quickly. In such cases, it would be necessary to detect the
slower-evolving sites to study ancient divergences
(Brinkman and Philippe 1999; Lopez et al. 1999). An-
other extension is a maximum-likelihood implementa-
tion of the covarion model, including estimating the op-
timal values for and ␦(A. Rambaut, personal
communication). Hidden Markov models have been used
in other aspects of recovering evolutionary information
(Baldi et al. 1994; Felsenstein and Churchill 1996; Krogh
et al. 1994). Such models are still relatively underex-
plored in molecular evolution and will probably turn out
to be as useful here as in many other areas of science.
It is premature to decide how useful our covarion-like
model will be in practice. The present paper is just one
contribution focusing on the underlying molecular biol-
ogy of molecular evolution models. There are serious
problems in studying ancient divergences, both theoret-
ical and practical. We think that a reasonable case has
been made to take the covarion model seriously, and
there may be other ways of including basic molecular
biology knowledge into evolutionary models. Future
work requires an improved synthesis of mathematics and
biochemical realism.
Acknowledgments. We thank Ted Drawneek for assistance with the
parallel use of multiple computers, P.J. Lockhart and two anonymous
referees for many useful comments, and the New Zealand Marsden
Fund for financial support.
Note Added at Proof
Galtier (2001) has recently described a maximum-
likelihood implementation of the covarion-like model
described here.
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