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A New Framework for Studying the Isochore Evolution:
Estimation of the Equilibrium GC Content Based on the
Temporal Mutation Rate Model
Satoshi OOta*,1, Kazuhiro Kawamura2, Yosuke Kawai3, and Naruya Saitou4
1

Department of Biological Systems, RIKEN BioResource Center, Tsukuba, 305-0074 Japan

2

Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 Japan3

Department of Bioinformatics, College of Life Sciences, Ritsumeikan University, Kusatsu, 525-8577 Japan

4

Division of Population Genetics, National Institute of Genetics, Mishima, 411-8540 Japan*Corresponding author: E-mail: oota@riken.jp.
Accepted: 2 July 2010

Abstract
Isochore is the genome-wide mosaic structure in guanine-cytosine (GC) content. The origin of isochores is thought to haveemerged in the ancestral amniote genome, and the GC-rich isochore is eroded in the mammalian lineages. However, there
are many enigmas in the isochore evolution: 1) although all the mammalians, birds, and even reptiles, which are clearly
polyphyletic, have isochore, opossum and platypus lack GC-rich and GC-poor isochore classes; 2) although the isochore ispredicted to vanish according to a fairly robust theory, a completely opposite conclusion was led in some mammalianlineages; and 3) the major three hypotheses on the isochore evolution cannot explain observed evidences without flaws. So
far compositional evolution has been studied under the assumption that per base pair rate of GC/AT (u) and AT/GC (v)mutations are temporally constant (the constant model). With this model alone, however, it is difficult to explain the isochore
evolution. We propose a simple model for compositional evolution based on the temporal per base pair rate of mutations(the variable model). In this model, rates u and v vary depending on temporal GC contents. Mathematically, the variable
model is an expansion of the constant model. By using high-density human single nucleotide polymorphism data, we
compared the variable model with the constant model. Although the variable model gave consistent results with theconstant model, it can potentially describe the complicated isochore evolution, which the constant model cannot explain.
The versatile characteristics of the variable model may shed new light on the mysterious isochore evolution.
Key words: isochore, equilibrium, the human genome, the chimpanzee genome, SNP.

Introduction
One of the distinctive features of the amniote genome is thehuge mosaic structure in guanine-cytosine (GC) content variation known as isochore (Bernardi et al. 1985). In mammals,
GC-rich isochore has three families: that is, H1, H2, and H3(Bernardi 1993). The large-scale variability of base compositionis significantly related to various genomic features: densities ofLINE and ALU elements, level of methylation, recombination
rates, and gene densities (Mouchiroud et al. 1991; EyreWalker 1993; Duret et al. 1995; Jabbari and Bernardi 1998;
Smit 1999; Fullerton et al. 2001; Lander et al. 2001).
Biological meaning of isochore is still an open question:
why did isochore emerge and why was it maintained against

frequent mutations in putatively evolutionarily neutral regions (e.g., intergenic regions)? There are mainly three hypotheses in regarding to the isochore evolution: 1) selection(Bernardi et al. 1985), 2) the mutation bias (Sueoka 1988;
Wolfe et al. 1989), and 3) biased gene conversion(Holmquist 1992; Eyre-Walker 1993) hypotheses. So far
no evidence was found to exclusively support one of them,suggesting that the isochore evolution harbors a complicated evolutionary mechanism.
When we consider phylogenies of various organisms
whose genome data are recently available, the isochore evolution is further enigmatic. Figure 1 shows polyphyletic characteristics of isochore families. In this figure, the isochoreª The Author(s) 2010. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.5), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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New Framework for Studying the Isochore Evolution

FIG. 1.—Polyphyletic characteristics of relative amounts of isochore families. Although primates, Rodentia, and Aves have clear isochore
structures, Didelphimorphia and platypus lack GC-rich and GC-poor isochore families, respectively. Gray ellipse (indicated by a question symbol)
represents a putative origin of GC-rich isochore. Black ellipses represent alternative (multiple) origins of GC-rich isochore. Scale at bottom represents thegeological era (S, Silurian; D, Devonian; C, Carboniferous; P, Permian; Tr, Triassic; Pg, Paleogene; Ng, Neogene). Time unit is million years. Histograms
represent weighted distributions of relative amounts of isochore families of various organisms (Costantini et al. 2006). Abscissa and ordinate are GC
contents and relative amounts of sequences, respectively. Colors in the histograms represent five isochore families (blue, L1; cyan, L2; yellow, H1;
orange, H2; red, H3) (for details, see Costantini et al. 2006). Phylogenetic tree and histograms are based on Hedges et al. (2006) and Costantini et al.
(2006). Nomenclature of OTUs does not follow precise taxonomic hierarchies.

maps were used (Costantini et al. 2006, 2009) to represent
isochore structures. Instead of using mere GC content distribution in the genome, this approach employs the plateau
values reached by the standard deviation of GC levels of isochors belonging to different isochore families, that is, theresults are expected to reflect spatial isochore structuresin terms of the original definition. Duret et al. (2002) claimed
that the origin of isochore is the ancestral amniote genome(gray ellipse in fig. 1), and the GC-rich isochore is eroded in
the mammalian lineages. However, although primates (human and chimpanzee), Rodentia (mouse), and Aves
(chicken) have clear isochore structures, Didelphimorphia(opossum) and platypus lack GC-rich and GC-poor isochorefamilies, respectively. These suggest lineage-specific isochore evolution or multiple emergence of isochore (black
ellipses in fig. 1), both of which are hard to be explainedby the existing theories.
One of the difficulties underlying the isochore studies isthat there were no sufficient data to analyze the isochore
evolution with an appropriate evolutionary model. Major

part of isochore resides in noncoding genomic regions,which are in general not alignable between distantly related
species like human and mouse. Conventional evolutionarymodels are empirically based on substitution patterns incoding regions (in many cases, substitution patterns in synonymous sites and introns are employed to exclude selective
pressures). In other words, we implicitly assume that evolution of alignable genic regions fairly reflect genome-level
isochore evolution.
By using synonymous substitution patterns in coding sequences, Duret et al. (2002) and Belle et al. (2004) predictedthat mammalian isochore is vanishing. Meanwhile, AlvarezValin et al. (2004) and Gu and Li (2006) claims that the vanishing isochore theory may be wrong due to misinference ofancestral states because the ‘‘parsimony-based’’ method
may not be appropriate to analyze the long-term evolution.
Gu and Li (2006) thus applied the maximum likelihood (ML)method, which is more robust than the parsimony-based
method, using the nonhomogeneous model (Galtier and
Gouy 1998), to reconstruct ancestral states. Surprisingly,

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they led to an opposite conclusion to that of Duret et al(2002): the GC content of GC-rich genes appears to have increased in recent times at least in some lineages, for example,
the rabbit genome. Galtier and Gouy (1998) claimed that thenonhomogeneous model is able to estimate accurate ancestral compositions under a given evolutionary model. As mentioned above, however, conventional evolutionary models(including the nonhomogeneous model) were originallydevised to analyze genic regions not to analyze genomic
regions. The definition of isochore is the genome-wide
nonrandom spatial variation in GC content. Taking them into
consideration, we should be cautious to analyze isochore
evolution by using the conventional models.
Webster et al. (2003) examined the compositional evolution of noncoding DNA by using 1.8 Mb genomic alignments of human, chimpanzee, and baboon. According tomutation patterns inferred by parsimony in human single
nucleotide polymorphisms (SNPs), the equilibrium GC content of each GC content class was not significantly different.
This again suggests homogenization of the isochore structure (the vanishing isochore theory). Because it may be inappropriate to parsimoniously reconstruct ancestral GC
levels among even closely related species, they carefullycompared results by the conventional parsimony-based
method with those by the ML method and found virtuallyno difference between them. Therefore, the model-dependent mis-inference is likely avoided in the genomic substitution patterns in the human and chimpanzee lineage.
Since most of the SNP sites in intergenic regions are approximately subject to neutral evolution, human SNP data
and the human and chimpanzee genomes are ideal extant
data. Today, large amounts of reliable human SNP data are
available at the international HapMap project (International
HapMap Consortium 2004; Couzin 2006). To infer reliable
mutation patterns, we used the human genome and SNP
data obtained from the HapMap database and dbSNP as
well as the chimpanzee genome as outgroup (a reference).
In this paper, we focus on mutation patterns, which can be
estimated from the SNP data and the reference genome
data, to study the isochore evolution. Our objectives are:1) by using the high-density SNP data set to detectcharacteristic evolutionary patterns that were missedin the previous studies and 2) to develop a simple
mathematical model of the compositional evolution.

Basic Concepts for the ModelTo estimate the equilibrium GC content, in general, the perbase pair rate of GC/AT (u) and AT/GC (v) mutations
have been assumed to be temporally constant (Webster
et al. 2003): that is, a time-homogeneous Markov process
is used in the model. We call this model ‘‘the constantmodel.’’ However, as Galtier and Gouy (1998) mentioned,
this may be an unrealistic assumption. The constant model

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implicitly assumes that a genomic region has positionspecific mutation rates u and v. On the other hand, it isshown that both of rates u and v are positively correlated
to GC contents (Smith et al. 2002; Webster et al. 2003). This
raises a question: can rates u and v be temporally constant
while GC content is changing?
According to the previous studies, inferred mutation patterns (in fact, synonymous substitution patterns in manycases) have a tendency to converge toward an equilibrium
state. If it is exactly true, by using a sufficiently rich data set
and a reliable evolutionary model, we can estimate a set ofsimilar equilibrium values to each other independent frominitial GC contents.
We propose a simple model that explains compositional
evolution without a complicated parameter set. We call thismodel ‘‘the variable model’’: unlike the constant model, theper base pair rate of GC/AT (u) and AT/GC (v) mutations
vary depending on time. It is virtually impossible to estimate
functions u and v of time t by using currently available data.
Instead, we propose a more specific model under the assumption that rates u and v are constant in each GC content
class but vary in time. In the constant model, u and v are
fixed depending on ‘‘observed’’ GC content. In the variable
model, u and v are determined depending on ‘‘temporal’’GC content: that is, they are functions of temporal GC content GC(t) (GC content at time t). Therefore, this is nota time-homogeneous Markov model and is determinedby two functions u(GC(t)) and v(GC(t)). Once GC content
at time t is given, the variable model allows us to estimate
values of u and v at time t. Consequently, GC content at timet þ 1 is recursively estimated from the one at time t. Because
time t is arbitrary, we use reconstructed ancestral GC content as an initial GC content at t 5 0. The constant model
can be defined as a special case of the variable model((u(GC(t)) and v(GC(t)) are constant functions here). In thispaper, we formalize the variable model as an extension of
the constant model and evaluate the model by using an
actual rich data set: the human and chimpanzee genomes.

Materials and Methods
Analysis of Mutation Patterns on the Human
Lineage
A total of 3,100,436 SNPs of the human genome wereobtained from the HapMap database (The International
HapMap Consortium 2003) and dbSNP (Sherry et al.
1999). By using Blast program (Altschul et al. 1990), we
mapped the SNPs on the chimpanzee genome (National
Center for Biotechnology Information [NCBI] PanTro1.1)
by using 5# and 3# flanking regions of the SNPs as queries.
To distinguish coding and transcribed regions from the othergenomic regions, we used the Ensembl annotation (release41) of the human genome from the UCSC genome annotation database for the March 2006 GenBank freeze

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New Framework for Studying the Isochore Evolution

we categorized the known mutations in human into 50
GC classes by 49 GC content values (0.02, 0.04, . . . , and
0.998). We performed bootstrap resampling from theeffective bins, assuming that mutations independently
occurred in different bins.

Estimation of the Equilibrium GC Contents by theConstant and Variable Models
We consider expected GC content transitions with discrete
time t 5 0, 1, 2, . . . in the following description.

FIG. 2.—Schematic representation of inference of a mutation
pattern by using human SNP and the chimpanzee genome asa reference. In this figure, the human genome has a polymorphism
adenine (A) and guanine (G) in a site. A homolog site in the chimpanzee
genome has G. Therefore, an ancestral state of the human genome isinferred to be G, according to the parsimonious method. Note that the
coalescence time of the human SNP is regarded as a unit time in thefollowing formalization. Expected length of the unit time is approximately 2NeX20 years because the average time to coalescence for two
randomly chosen SNPs is 2Ne generations, where Ne is an expected
effective population size of the human lineage.

assembled by NCBI (hg18, Build 36.1). We also excluded putative CpG islands and repetitive regions from the humangenome, which were predetected by the UCSC genome database. The human genome was divided into 2 kb windows(nonoverlapped bins). Only biallelic SNPs, in which the chimpanzee root was the same as one of the two human alleles(fig. 2), and windows greater than or equal to 100 bp wereused in this analysis.
The genome data are mixture of derived and ancestral
states in polymorphic loci. To estimate accurate per base pairrate of GC/AT (u) and AT/GC (v) mutations (Sueoka
1988, 1993), we need to reconstruct nucleotide compositions of the ancestral genome of the human population. Ancestral and derived nucleotide compositions are generally
nearly identical to each other in the human genome. However, especially when the number of SNP data is huge (i.e.,
SNP density is high) and/or an effective window (bin) size
can happen to be small, an effect of the heterogeneity is
not negligible. We reconstructed ‘‘ancestral’’ chromosomesof the human population by the parsimony-based method.
We basically followed Webster et al. (2003) to estimate
the per base pair rate of mutations for various GC content
classes. They estimated equilibrium GC contents by using
five GC content classes (GC1 , 0.3, 0.3  GC2 , 0.4,
0.4  GC3 , 0.5, 0.5  GC4 , 0.6, and GC5  0.6, whereGCi is the i th GC content class) with parsimony-inferredmutation patterns. Owning to the high-density SNP data,

The Constant Model. The constant model is based on thefollowing two assumptions:
Assumption 1. The mutation rates u and v are constant ineach GC class and further.
Assumption 2. The mutation rates u and v are constant intime as well.
 GC ðtÞ be the expected number of G þ C sites at time
Let N
 GC ðtÞ satisfy
 GC ðt þ 1Þ and N
t. Under these assumptions, N
the following formula GC ðtÞ þ N AT/GC ðtÞ  N
 GC/AT ðtÞ;
 GC ðt þ 1Þ 5 N
Nð1Þ GC/AT ðtÞ are the expected numbers
 AT/GC ðtÞ and N
where Nof transitions AT-.GC and GC-.AT at time t, respectively.
 GC/AT ðtÞ are AT/GC ðtÞ and N
In the constant model, Nwritten as NAT/GC ðtÞ5vNAT ðtÞ and NGC/AT ðtÞ5uNGC ðtÞ,respectively.
 GC ðtÞ[L, the total number of sites.
 AT ðtÞ þ N
Note that N
From these we obtain GC ðtÞ þ v N
 AT ðtÞ  uN
 GC ðtÞ
 GC ðt þ 1Þ 5 N
N
 GC ðtÞÞ  uN
 GC ðtÞ
 GC ðtÞ þ vðL  N
5N
 GC ðtÞ þ v L:
5 ð1  u  vÞN
Dividing both sides by
 GC ðtÞ=L, we obtainGCðtÞ5NL

andnoticingGCðt þ 1Þ 5 ð1  u  vÞGCðtÞ þ v;

that
ð2Þ

where GC(t) is GC content at time t. Suppose that GC(t) has
reached an equilibrium f* at time te. Then f* 5 GC(te) 5
GC(te þ 1) 5 . By equation (2), we obtain the equilibriumv.
formula (Sueoka 1988, 1993) f  5 uþv

The Variable Model.
The Derivation of the Variable
Model by Generalizing the Constant Model. Now we retain
Assumption 1 and replace Assumption 2 by:Assumption 3. The mutation rates u and v vary in time.
Furthermore, we make a stronger assumption:
Assumption 3a.The mutation rates u and v are functions u 5u(GC(t)) and v 5 vGC(t) of the GC content at time t, GC(t).
We do not claim that Assumption 3a is theoretically justified. First of all, there is no a priori reason to assume that
mutation rates u and v depend only on GC(t). Secondly, theGenome Biol. Evol. 2:558–571. doi:10.1093/gbe/evq041 Advance Access publication July 8, 2010

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mutation rates u and v, if they vary in time, should be
regarded as stochastic processes and the distributions ofu(GC(t)) and v(GC(t)) should play an important role.
Assumption 3a states that u and v follow deterministic rules
defined by the functions u(GC(t)) and v (GC(t)).
On the basis of Assumption 3a, the formula (2) ischanged as followsGCðt þ 1Þ 5 GCðtÞ  GCðtÞfuðGCðtÞÞþ vðGCðtÞg þ vðGCðtÞÞ:ð3Þ

When the GC content (at the site we are examining) has
reached the equilibrium fN at time te, then we obtain fN 5
GC(te) 5 GC(te þ 1) 5 GC(te þ 2) 5 . . .. From this withequation (3), we obtain fN 5 fN  fNfu(GC(te)) þ v(GC(te)g
þ v(GC(te)) and Sueoka’s formula in this context takes ananalogs formfN 5vðGCðte ÞÞ:uðGCðte ÞÞ þ vðGCðte ÞÞð4Þ

We still do not know explicit formulas of u(GC(t)) andv(GC(t)). These functions are to be estimated from the actual
mutation patterns (AT-.GC and GC-.AT, respectively) byusing the 50 GC classes described above. Because formula
3 is concerned with the GC contents at time t and time t þ 1,
we choose the initial time t 5 0 as an initial state or t 5 1 asan observed state.
By the generalization, the model no longer follows thestationery Markov process because the transition matrices
changes depending on time. Before proceeding further,
we make a digression to an auxiliary consideration.
An Auxiliary Consideration on Potential Equilibria. We
introduce an auxiliary function f(x) defined byf ðxÞ 5 ð1  uðxÞ  vðxÞÞx þ vðxÞ;0  x  1;

ð5Þ

where u(x) and v(x) are smooth functions, such that 0 , u(x),v(x) ,, 1. The change of variable x 5 GC(t) gives equation(3) and the function f(x) stands for a transition rule of the GC
content. Because f(x) is given by the two mutation rates u(x)and v(x), a temporal GC content GC(t) is also given by thetwo mutation rates in a recursive way. We have f(0) 5 v(0) .
0 and f(1) 5 1 – u(1) , 1. These two inequalities imply that
there is at least a value xe such that f(xe) 5 xe. Such value isalso called a ‘‘fixed point.’’ There may be several fixed points,
each of which corresponds to a potential equilibrium.
The behavior of iterations of the function f near a potential equilibrium xe is described by the absolute value of thederivative, jf’(xe)j (most likely ,1 or .1 and not exactly
equal to 1 in our context). Let f(2)(x) 5 f(f(x)), f (3)(x) 5f(f(f(x))), and so on.
Class I: 0 , f (xe) , 1: When we start with a value x0 , xe,
which is ‘‘sufficiently close’’ to xe, then we have x0 , f(x0) ,562

FIG. 3.—Schematic representations of a typical transition of GC
content under the f(GC(t)) framework (red lines). The diagonal broken
line represents f(GC(t)) 5 GC(t þ 1) 5 GC(t), that is, a set of fixed points
and, in this case, the equilibrium states. The intersection of GC(t) andGC(t þ 1) represents an equilibrium. GC content transition (red lines)
asymptotically approaches the equilibrium (GCe). The variable model
predicts asymptotical convergence of transitions on potential equilibrium GC(te), where te is equilibrium time. The total derivative f#(GC(t))
satisfies condition 0 , f‘ (GC (te)) , 1 (Class I).

f (2)(x0) , . . . , f (n)(x0) , / xe as n / N. Similarly when we
start with a value x0 . xe, which is sufficiently close to xe,
then we have x0 . f(x0) . f (2)(x0) . . . . . f (n)(x0) . / xe as
n / N (see fig. 3).
Class II:1 , f’(xe) , 0: When we start with a value x0 , xe,
which is sufficiently close to xe, then we have an‘‘oscillation’’ x0 ,xe. f(x0) áxeñ . f(2)(xe) ,xe. f(3)(xe),. . . and f(n)(x0) / xe as n / N (see fig. 4a).
Class III: jf’(xe)j . 1: Here, the behavior of iterations of f exhibits a sharp contrast with Classes I and II. Even if we startwith a value x arbitrarily close to xe, the iterated values f (n)(x)are ‘‘repelled’’ from xe, and it is not possible to predict their
asymptotic behavior in general.
Although sometimes there may be periodic values xp inthe sense that after some iteration of the function f, xp
comes back to itself. In formula, this is expressed as: xp 5
f (n)(xp): 5 f (f (f . . . (n times) (xp))).
If the derivative (f (n))‘(xp) 5 f’ (f (n  1)(xp)) f‘(f(n  1)(xp)) . . .
f‘(xp) falls into Class I or Class II, we may derive similar
conclusions on the periodic value xp (see fig. 4b).
In our application to the actual data set, f(x) is considerably close to x. It is convenient to introduce a function g(x) 5
x – f(x). Noticing that g‘(x) 5 1 – f‘(x), we haveClass I : 0,f # ðxe Þ,1

if and only if

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0,g# ðxe Þ,1;
ð6:1Þ

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FIG. 4.—Schematic representations of ‘‘atypical’’ transitions of GC content under the f(GC(t)) framework (red lines). (a) Class II: oscillatory
convergence of transitions on a potential equilibrium GC(te). The total derivative f#(GC(t)) must satisfy condition  1 , f#(GC(te)) , 0. (b) Class III:
divergence of transitions from fixed points.

Class II :  1,f # ðxe Þ,0

if and only if

1,g# ðxe Þ,2;
ð6:2ÞClass III : jf # ðxe Þj.1 if and only if g# ðxe Þ,0 or 2,g# ðxe Þ:
ð6:3Þ
A Preliminary Assessment. To briefly assess characteristics ofthe variable model, we first performed three kinds of computations by using human chromosome 21 (HSA21): 1) quadratic fitting and numeric: u and v were given by nonlinear
regression to fit the five sets of replicated data to quadraticmodels, and the equilibrium GC contents were estimated bygiving a large value of time t (say, 2,000 unit times); 2) quadratic fitting and analytic: the equilibrium GC contents wereanalytically estimated under assumption that GC(te 1)’GC(te) 5 GC(te þ 1) holds, where te is time when the
GC content reaches the equilibrium; 3) GC class: u and v
are estimated according to discrete GC classes (as a step
function), say, 0  GC , 0.3, 0.3  GC , 0.4, 0.4 GC , 0.5, 0.5  GC , 0.6, and 0.6  GC , 1.0. Six initial
GC contents were used (0.1, 0.3, 0.4, 0.5, 0.6, and 0.8).
Note that the 2,000 unit times are 2,000 times as long as
the average time to coalescence for two randomly chosen
SNPs: approximately 800 My, assuming that the effective
population size for the human lineage is 10,000 and thegeneration time for human is 20 years.

Results
Figure 5 shows transitions of GC content under the constantmodel with various initial GC content classes (0.2, 0.3, 0.4,

0.5, 0.6, 0.7, and 0.8). To examine overall characteristics oftransitions of GC content, u(GC(t)) and v(GC(t)), are simplified by nonlinear regression: û50:002014 0:000735 ĜC 2
and v̂50:003486  0:000812 ĜC2 , respectively. The fitting
was better in the quadratic models than the linear models:R-squared values of rates u and v are 0.623 and 0.547 in
the linear models, respectively, whereas those in thequadratic models are 0.700 and 0.551, respectively. In
the constant model, each ĜC was given as an initial GC content for each GC content class. Although it is known that GC
contents under the constant model take extremely long time
to reach the equilibria (Sueoka 1962), an estimated convergence time is around 350 My under the given conditions: 1)
the intergenic regions we used are subject to neutral evolution, 2) the effective population size of the human lineages
is 10,000, and 3) the generation time for human is 20 years.
It is approximately equal to or rather shorter than 400 My,
which corresponds to the divergence time of amniotes: a putative age of the origin of isochore (Duret et al. 2002) (fig. 1).
Table 1 shows comparison of GC content equilibria of
HSA21 between the constant and variable models. The
equilibrium GC contents under the constant model werefairly similar. Meanwhile, owing to the high-density SNP
data, the small differences appeared to be significant (fig.
6a). Although Webster et al. (2003) did not report any significant differences of the equilibrium among GC classes,
our results were consistent with theirs in terms of those similar equilibria, suggesting homogenization or ‘‘vanishing’’
isochore under the constant model. Meanwhile, underthe variable model, all the GC content categories convergedon the exactly same equilibrium (table 1 and fig. 6b), which

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FIG. 5.—Transitions of CG content of chromosome 21 (HSA21) under the constant model. Italic bold figures represent initial GC content; Italic
figures represent GC content values at t 5 2,000 (putative unit times to reach the equilibria). A unit time represents approximately 400,000 years.

is still close to the equilibrium values under the constantmodel.
Next, we examined transition behaviors of GC content
based on the f(GC(t)) framework, the one which describes
GC(t þ 1) by a function f(GC(t)) of GC(t). We used thefunctions u(GC(t)) and v (GC(t)) instead of the simplified
quadratic models and the stepwise functions of GC content
classes. Figure 7 shows the per base pair rate of GC/AT (u)and AT/GC (v) mutations corresponding to GC content in
human chromosome 3 (HSA3). Note that figure 7a andb shows the relationships between GC content and rates
u and v, respectively, instead of the net mutation rates.
By using synonymous substitution patterns, Smith et al.
(2002) showed that both u and v are positively correlated
to GC content. However, detailed relationships estimated
Table 1Comparison of GC Content Equilibria of HSA21 between theConstant and Variable Models
Equilibrium
Variable
Quadratic Fitting
Initial GC

Constant

Numeric

0.1
0.3
0.4
0.5
0.6
0.8

0.36590.36340.36110.35800.35390.34230.36200.36200.36200.36200.36200.3620Analytic}

0.3620GC Class
0.36830.36830.36830.36830.36830.3683

NOTE.—Initial GC: GC content at time t 5 0, which corresponds to GC content
classes; Constant, the constant model; Variable, the variable models; Quadratic fitting,u and v are given as quadratic models of local GC content; Numeric, numericestimation of the equilibrium GC content by giving a large value to time t; Analytic,
analytical estimation of the equilibrium GC content; GC class, u and v are estimatedaccording to discrete GC classes (as a step function). For detail, see text.

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from the rich SNP data were more complicated in theintergenic regions. Owing to the high-density SNP data,
we could detect multimodal structures of relationships
between GC and per base rates u and v. These multimodal
structures of estimated u(GC(t)) and v(GC(t)) were observedin all chromosomes (data not shown). According to the resampled data, the structural relationships between GC
content and the per base pair rates are moderately stable
across the chromosome, suggesting that the relationships
are universal at intrachromosomal level. Figure 8 shows
relationships between GC(t) and g(GC(t)) 5 GC(t) f(GC(t)) 5 GC(t)  GC(t  1) under the variable model(HSA21). A root of g(GC(t)) 5 0 is a fixed point (see section,
An auxiliary consideration on potential equilibria) of f(GC(t))
and, in this case, a potential equilibrium value since the derivative at the fixed point satisfied the condition given byformulas (6.1) and (6.2) (note that g(GC(t)) is the functionof GC(t), not t, in our context). This value belongs to Class I
in the transition behavior of GC content (fig. 3). As shown infigure 8, while g(GC(t)) varies depending on resampled SNP
data from HSA21, the variation is moderate (for details, see
figure legends).
We furthermore elaborated on characteristics of g(GC(t))
for each replicated data set. Figure 9 shows distribution of
fixed points of f(GC(t)) and the derivatives at fixed points ofg(GC(t)) in chromosome 1 (HSA1). There are mainly threeclusters: low (;0.2), medium (;0.37), and high (;0.7).
The derivatives at the fixed points have a tendency to be
negative, positive (or nearly zero), and negative for thelow, medium, and high clusters, respectively. Because manyof the fixed points of the medium cluster satisfy the conditions given by formulas (6.1) and (6.2), they are suggested to
be potential equilibria. However, there are a considerable
number of fixed points that do not follow the tendency: for

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FIG. 6.—Close-ups of transitions of CG content of HSA21 under (a) the constant model and (b) the variable model. Although all the GC categories
showed similar but different equilibria under the constant model, all the GC content categories converged on the same equilibrium under the variable
model. For legend, see figure 5.

example, some of the fixed points at low and high GC content clusters satisfy the condition (6.3), many of which arethought to be ‘‘repulsive’’ fixed points leading to the adjacent potential equilibria. Figure 10 shows distribution of
fixed points of g(GC(t)) in all chromosomes. CpG andnon-CpG SNPs were distinguished. Overall, there are 2–4
clusters of fixed points for each of the data sets, suggestingthat plural GC content equilibria and/or divergent behaviors
under the variable model. GC content equilibria under theconstant model (indicated by dashed and dotted lines forthe all SNP and non-CpG SNP data, respectively) are wellmatched in mean GC values of one of the clusters, manyof which are potential equilibria under the variable model.
Surprisingly, except for these clusters, there were virtually no
fixed points between them. This means that randomly sampled data from each chromosome share the characteristics,
suggesting that the f(GC(t)) framework is considerably stable in the given data set.

Figure 11 shows the genome-level derivatives of g(GC(t))
at the fixed points appeared in figure 10. It is obvious that all
chromosomes share similar characteristics in terms of distribution of the clusters. Note that we did not elaborate upon
Class III except for repulsive fixed points because it was impossible to predict their asymptotic behavior in general.

Discussion
There are evolutionary enigmas in compositional evolution:1) the opossum (Mikkelsen et al. 2007) and platypus(Warren et al. 2008) genomes lack GC-rich and GC-poor
isochore families, respectively, whereas all the other mammalians, birds, and even reptiles, which are polyphyletic,
have clear isochore structures (Duret et al. 2002); 2) rat
and mouse show reduction of GC content variation (murine
shift); 3) the ML-based inference can sometimes lead to acompletely opposite conclusion against that by the

FIG. 7.—The per base pair rate of GC-.AT (u) (a) and AT-.GC (v) (b) mutations corresponding to GC content in chromosome 3 (HSA3). Thick(gray) and thin (black) regions indicate 95% confidence intervals based on a normal distribution fitted to 1,000 bootstrap replications of HSA3 data andfor their means, respectively.

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FIG. 8.—An example of relationships between GC(t) and g(GC(t))
5 GC(t)  f(GC(t)) 5 GC(t)  GC(t þ 1) in the variable model (HSA21).
The upper and lower arrows indicate an equilibrium GC value and theslope of g(GC(t)) at the fixed point, respectively. For legend, see figure 7.

maximum parsimony-based inference (Gu and Li 2006); 4)
the major three hypotheses, which are all sophisticated andplausible, cannot explain the isochore evolution without
flaw (Eyre-Walker and Hurst 2001). These suggest thatthe existing compositional evolution frameworks need tobe reconsidered.
The constant model (the time-homogeneous/stationary
Markov model) is simple and intuitive. Many compositional
evolution studies, explicitly or implicitly, have used thismodel: that is, the per base pair rate of mutations is assumed
to be temporally constant. As a consequence, transitions of
GC content are necessarily gradual and asymptotic to equilibrium states. However, with this model alone, it is difficult
to elucidate the above four enigmatic phenomena.

FIG. 9.—Distribution of fixed points of f(GC(t)) in chromosome 1(HSA1). Left axis: frequencies of fixed points of f(GC(t)) in HSA1
(histogram); Right axis: the derivative at of fixed points of g(GC(t))(yellow dots). Classes I and II: ‘‘attractive’’ fixed points or potential
equilibria; Class III: ‘‘divergent’’ fixed points (green). Note that, in actual
computation, the fixed points were given by roots of g(GC(t)) 5 0.
Therefore, the three classes in terms of behaviors of GC content
transition were determined as follows: (a) 0  g’(GC(te)) , 2:convergence, suggesting an equilibrium (purple); (b) g’(GC(te)) . 2 org’(GC(te)) , 0: divergence (green). The constant model shows equilibria
around GC content ;0.4, which are consistent with those of theconstant model with substitution patterns of GC3.

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Alvarez-Valin et al. (2004) and Gu and Li (2006) used the
ML method with the nonhomogeneous model (i.e., the‘‘time-nonhomogeneous’’ Markov model) (Galtier and Gouy1998) to reconstruct ancestral states and obtained the opposite conclusion from the ‘‘vanishing isochore’’ hypothesis
at least in certain lineages. We should note that their nonhomogeneous model is composed of different kinds of thetime-homogeneous Markov models, which are branch specific in a phylogenetic tree of the compared organisms.
Therefore, if we consider only one branch between ancestraland descend nodes, this model is identical to the time-homogeneous Markov model for estimating the ancestral basecompositions.
The variable model is essentially different from the‘‘nonhomogeneous’’ model. Our model is literally ‘‘timeheterogeneous’’ even on a branch. An a priori assumptionthat the variable model requires is that each GC content
class has a moderately fixed per base pair rate as shown insection The Variable Model (Materials and Methods). This
assumption may look like too strong. However, as long as
GC content homogenization is predicted under the constant model, this assumption is implicitly required also
in the constant model: if this assumption is violated, thesame GC content class will have different equilibria,
namely, the homogenization will not occur. Therefore,both the constant and variable models share the same
assumption at this point.
One might be confused by the complicated characteristics of the variable model. Overall, the variable model has 2fold characteristics: 1) if mutation patterns belong to Classes
I and II, the variable model has a unique equilibrium regardless of initial GC content values and 2) if mutation patterns
belong to Class III, the variable model can have plural equilibria. In the constant model, it is required that mutation patterns are temporally constant in a genomic region: that is,the constant model assumes that initial GC content-specific(or rather position-specific) mutation patterns last forever.
This is a special case of the variable model when u(GC(t))
and v(GC(t)) are constant functions. Therefore, the variable
and constant models are not antagonistic to each other, but
the former is an extension of the latter.
In the net compositional evolution, the variable model
provided consistent results with the constant model (fig.
10). Meanwhile, there exist plural clusters of fixed points,suggesting plural equilibria. For example, data belonging
to Classes I and II in figure 9 indicate potential GC content
equilibria, according to formulas (6.1) and (6.2). A question
arising here is whether the isochore structure could reflect
this multiplicity of equilibrium GC contents. The observed
distribution of GC content across genomic regions in humanis apparently unimodal. However, this does not contradict
the multiplicity of equilibrium GC contents. Note that 1)
the definition of isochore is the heterogeneity of spatial distribution of GC content not multimodality of the gross

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distribution (see the isochore maps in fig. 1) and that 2) weobserve a snapshot of a GC content transition. As the previous studies (Webster et al. 2006; Duret and Arndt 2008)
suggested, GC content is probably not at equilibrium inmany mammalian lineages. Furthermore, unlike the constant model, the variable model allows a change of a mutation pattern at a genomic region. This makes it difficult to
intuitively follow a behavior of GC content distribution. For
example, if the transition is oscillatory (Class II), it is quitedifficult to predict how the gross GC content distribution
behaves. Besides that, it is not obvious that each chromosome shows a unimodal GC content distribution if we extract genomic regions putatively subject to the neutral
evolution: for example, chromosomes 17 and 22 are apparently exceptional.
We assume that each chromosome has a pair of specificfunctions u(GC(t)) and v(GC(t)). The bootstrapped data
share the similar characteristics in terms of distribution of
fixed points overall (fig. 10; for the whole genome perspectives, see also supplementary figs. S1 and S2, Supplementary Material online). Although this assumption ismoderate for the net compositional evolution, the functionsu(GC(t)) and v(GC(t)) vary depending on positions in a chromosome. The nonlinearity of the variable model opens another possibility on the transitions of GC contents. As wasshown in Li and Yorke (1975), a function f(x) (0  x 1) satisfying a certain ‘‘global’’ condition (roughly illustrated in fig.
12; for detail, see Li and Yorke [1975]) exhibits a rather complicated behavior on its iterations. For example:i) Arbitrarily small difference of initial conditions may
cause a large deviation of long-term iterations;ii) There may be many periodic points with almost
arbitrary period.
This type of iteration behavior is generally called ‘‘chaotic,’’ and many mathematical studies have been madeon chaotic dynamical systems. Although no such transition
function is clearly observed in our data, there could be a possibility that future studies find transition functions that exhibit chaotic behaviors. This might shed another light on theisochore evolution.
The variable model can explain both the gradual (and/or
asymptotic) and the drastic (and/or chaotic) compositionaltransitions. By considering the present genomic compositions as a snapshot of dynamic transitions, the paradoxesof isochore are explained under the variable model: that
is, a local genomic composition can occasionally converge
on a certain GC content value and occasionally can diverge
with tiny changes of mutation patterns. Under those conditions, GC content transitions are more rapid than those predicted by the constant model because tiny fluctuations ofu(GC(t)) and v(GC(t)) lead to drastic difference in GC content transitions.
How did isochore evolve? The previous study (Duret et al.
2002) suggests the origin of isochore is the ancestral amniote genome (fig. 1). Although their theory is plausible formany extant genome data, there are counterexamples:for example, the opossum (Mikkelsen et al. 2007) and platypus (Warren et al. 2008) genomes. As mentioned above, homogenization of isochore (vanishing isochore) is supportedby the constant model. Because the homogenization takes
extremely long time to reach the equilibrium (Sueoka 1962),
we are tempted to consider that the observed isochore isjust remnants of the original isochore of the ancestral amniote genome. However, under the constant model, theconvergence time of vanishing isochore is around 350 My(see figs. 5 and 6). Because the estimated divergence time
between the amniote and its counterparts is around 400 My(Hedges et al. 2006), it is hard to think that the extant isochore is mere remnants of the ancient isochore. Furthermore, we should note that the estimated convergencetime (350 My) may be an overestimation according to thefollowing reason. We set the unit time to 2 Ne  20 5
800,000 years, assuming that the effective population sizeof the human lineage is 10,000. Because an effective population size is the harmonic mean of the actual numbers, thelong-term effective population size tends to be dominatedby the smallest temporal population size in a lineage: that is,
the human effective population size (10,000) is putatively anoverestimation as the long-term effective population size.
Therefore, it is reasonable to think that the 800,000 years
is an upper limit of the unit time and that the ancestral amniote genome would have reached the equilibrium by today
under the constant model.
To solve this contradiction, we need to assume lineagespecific isochore evolution and/or multiple emergence ofisochore. Those phenomena are hard to be explained under
the constant model.
According to the variable model, tiny fluctuation of parameters (u(GC(t)) and v(GC(t))), which are easily observedin real data, can lead to drastic changes in the genome compositions under the variable model. On the other hand, if
mutation patterns in a genome do not satisfy any conditions
resulting oscillatory, plural, or chaotic transition behaviors(Classes II and III), the genome does not have clear isochorein our context. Therefore, the genome that does not haveclear GC-rich isochore (e.g., the opossum genome) is predicted to harbor fixed points belonging to only Class I(we need to wait for high-density SNP data of the opossum
genome to confirm this prediction as well as closely related
species as a reference).

FIG. 10.—Distribution of fixed points of g(GC(t)) in human chromosomes (open bars: all SNP data; shaded bars: non-CpG SNP data). Subsets ofthe fixed points are potential equilibria. Abscissa and ordinate are GC content and frequency of fixed points, respectively. Dashed and dotted lines
indicate estimated GC content equilibria in each chromosome under the constant model by using all SNP data and non-CpG SNP data, respectively.

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FIG. 11.—The genome-level derivatives of g(GC(t)) at the fixed points (open bars: all SNP data; shaded bars: non-CpG SNP data). The three classes
are indicated by roman numerals: convergence or potential equilibrium (Classes I and II); divergence (Class III). Part of histograms are truncated forcomparison. Note that each replicated data set has plural fixed points in general (the sum of fixed points are greater than 1,000).

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class. If plural fixed points are involved (and this is a commoncase), a global behavior of GC content transitions will be
more complicated. Further theoretical studies of f(GC(t)) will
provide insight to the compositional evolution under the variable model. We used high-density SNP data of the humanpopulation and closely related species, chimpanzee, as a reference. Recently, high-speed sequencing technologies are
emerging. More abundant and reliable SNP data of various
species will be available soon. At a point of view on polymorphism, the isochore evolution is still open to further discussion due to lack of the data. New rich data will make itpossible to shed light on dynamics of the isochore evolution.
With accumulation of reliable SNP data not only of human
but also of the other animals (e.g., laboratory mouse, zebra
fish, opossum, and platypus), studies on compositional evolution will reach a new era.

Supplementary Material
FIG. 12.—A schematic representation of a chaotic behavior of GC
content transitions. Red lines represent the transitions of GC content.
Diagonal broken lines represent GC(t þ 1) 5 GC(t): that is, the conditionof fixed points (potential equilibria).

Supplementary figures S1–S2 are available at Genome
Biology and Evolution online (http://www.oxfordjournals.
org/our_journals/gbe/).

Note that the variable model requires a pair of functionsu(GC(t)) and v(GC(t)), which are derived only from mutation
patterns of unfunctional intergenic regions. They are (putatively) selectively neutral. Although, the neutrality is notessential in our model, it is unlikely that vast amounts ofthe genomic regions that we used in the analyses are understrong selection. The variable model may be able to explain
emergence and maintenance of isochore under the neutral
theory.
The previous studies (Eyre-Walker 1999; Duret et al.
2002; Smith et al. 2002; Webster et al. 2003; Duret andGaltier 2009) suggested that the fixation process in mammals is biased toward G and C. The fixation biases are, however, primarily discussed in the context of evolutionarychanges in synonymous sites and introns as well as intergenic regions. The data we used (SNP data only from unfunctional intergenic regions) showed a different view:for example, Webster et al. (2003) estimated the overall
equilibrium GC content 0.41 based on the parsimonyinferred substitution data since human–chimpanzee andthe common ancestor (95% confidence interval is 0.40–0.42). Those values are consistent with our equilibrium estimations both by the variable and constant models when
the non-CpG SNP data were used (see fig. 10).
We categorized fixed points of f(GC(t)) derived from thereplicated data into three classes: Classes I, II, and III. Eachclass theoretically shows a characteristic behavior in transitions of GC content: asymptotic convergence (Class I), oscillatory convergence (Class II), and divergence (Class III).
Note that those transitions described here are local
behaviors: that is, a single fixed point is considered in eachAcknowledgments

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The authors wish to thank Keiichi Kodaira of Japan Society
for the Promotion of Science Bonn Office, who gave us an
opportunity to present a preliminary version of this work in
his workshop, Japan-German Colloquium on Evolutionary
Genetics (8-11 March, 2009, Bad Honnef, Germany).
William Martin kindly advised on our project. We are also
grateful to Kosuke Hanada for fruitful discussion. Kuniya
Abe supported us by providing computing resources. We
would like to express our gratitude to anonymous reviewers.
We could significantly improve our manuscript owing totheir constructive comments. This work was supported by
KAKENHI from Japan Society for the Promotion of Science:
Grant-in-Aid for Scientific Research (B) (20310118).

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Associate editor: Yoshihito Niimura

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