# Dynamics, stability and inheritance of somatic DNA methylation imprints.

**ABSTRACT** Recent research highlights the role of CpG methylation in genomic imprinting, histone and chromatin modification, transcriptional regulation, and 'gene silencing' in cancer development. An unresolved issue, however, is the role of stable inheritance of factors that manage epigenetic imprints in renewing or expanding cell populations in soma. Here we propose a mathematical model of CpG methylation that is consistent with the cooperative roles of de novo and maintenance methylation. This model describes (1) the evolution of methylation imprints toward stable, yet noisy equilibria, (2) bifurcations in methylation levels, thus the dual stability of both hypo- and hypermethylated genomic regions, and (3) sporadic transitions from hypo- to hypermethylated equilibria as a result of methylation noise in a finite system of CpG sites. Our model not only affords an explanation of the persistent coexistence of these two equilibria, but also of sporadic changes of site-specific methylation levels that may alter preset epigenetic imprints in a renewing cell population.

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**ABSTRACT:**We develop a new methodology for the efficient computation of epidemic final size distributions. We exploit a particular representation of the stochastic epidemic process to derive a method which is both computationally efficient and numerically stable. The algorithms we present are also physically transparent and so allow us to extend this method from the basic SIR model to a model with a phase-type infectious period and another with waning immunity. The underlying theory is applicable to many Markovian models where we wish to efficiently calculate hitting probabilities.Journal of Theoretical Biology 07/2014; 367. · 2.30 Impact Factor - SourceAvailable from: Hans Binder[Show abstract] [Hide abstract]

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**ABSTRACT:**During aging, a decline in stem cell function is observed in many tissues. This decline is accompanied by complex changes of the chromatin structure among them changes in histone modifications and DNA methylation which both affect transcription of a tissue-specific subset of genes. A mechanistic understanding of these age-associated processes, their interrelations and environmental dependence is currently lacking. Here, we discuss related questions on the molecular, cellular, and population level. We combine an individual cell-based model of stem cell populations with a model of epigenetic regulation of transcription. The novel model enables to simulate age-related changes of trimethylation of lysine 4 at histone H3 and of DNA methylation. These changes entail expression changes of genes that induce age-related phenotypes (ARPs) of cells. We compare age-related changes of regulatory states in quiescent stem cells occupying a niche with those observed in proliferating cells. Moreover, we analyze the impact of the activity of the involved epigenetic modifiers on these changes. We find that epigenetic aging strongly affects stem cell heterogeneity and that homing at stem cell niches retards epigenetic aging. Our model provides a mechanistic explanation how increased stem cell proliferation can lead to progeroid phenotypes. Adapting our model to properties observed for aged hematopoietic stem cell (HSC) clones, we predict that the hematopoietic ARP activates young HSCs and thereby retards aging of the entire HSC population. In addition, our model suggests that the experimentally observed high interindividual variance in HSC numbers originates in a variance of histone methyltransferase activity.Aging cell 01/2014; 13(2):320-328. · 7.55 Impact Factor

Page 1

Journal of Theoretical Biology 242 (2006) 890–899

Dynamics, stability and inheritance of somatic DNA

methylation imprints

Laura B. Sontaga, Matthew C. Lorinczb, E. Georg Luebeckc,?

aComputational and Systems Biology Program, Massachusetts Institute of Technology, Room 68-371, 77 Massachusetts Avenue,

Cambridge, MA 02139, USA

bDepartment of Medical Genetics, Life Sciences Institute, Room 5-507, The University of British Columbia, 2350 Health Sciences Mall,

Vancouver, BC, Canada V6T 1Z3

cFred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, P.O. Box 19024, Seattle, WA 98109-1024, USA

Received 2 December 2005; received in revised form 7 April 2006; accepted 5 May 2006

Available online 20 May 2006

Abstract

Recent research highlights the role of CpG methylation in genomic imprinting, histone and chromatin modification, transcriptional

regulation, and ‘gene silencing’ in cancer development. An unresolved issue, however, is the role of stable inheritance of factors that

manage epigenetic imprints in renewing or expanding cell populations in soma. Here we propose a mathematical model of CpG

methylation that is consistent with the cooperative roles of de novo and maintenance methylation. This model describes (1) the evolution

of methylation imprints toward stable, yet noisy equilibria, (2) bifurcations in methylation levels, thus the dual stability of both hypo-

and hypermethylated genomic regions, and (3) sporadic transitions from hypo- to hypermethylated equilibria as a result of methylation

noise in a finite system of CpG sites. Our model not only affords an explanation of the persistent coexistence of these two equilibria, but

also of sporadic changes of site-specific methylation levels that may alter preset epigenetic imprints in a renewing cell population.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: DNA methyltransferase (Dnmt); CpG dinucleotide; Hypomethylation; Bistability; Markov chain model

1. Introduction

The existence of CpG islands (i.e. cytosine–guanine

dinucleotide rich regions) in mammalian genomes that

otherwise have a roughly 5-fold lower density of CpG sites

isprescientoftheirdevelopmental

significance (Bird, 1980; Jones et al., 1992). Indeed, over

70% of human genes now appear to be flanked 50by CpG

islandsthatencompasstheir

et al., 2006). While dispersed CpGs and intragenic CpG

islands are predominantly 5-methylcytosine methylated in

soma, the majority of promoter-associated CpG islands are

hypomethylated and associated with transcriptional activ-

ity. In contrast, the formation and maintenance of

repressive heterochromatin is positively correlated with

hypermethylation. A case in point is the stable clonal

andregulatory

promoters(Saxonov

propagation of DNA methylation patterns on the inactive

X chromosome in somatic cells of female mammals

(Gartler et al., 1985; Lyon, 1988; Riggs and Pfeifer,

1992). Although the induction of a repressive chromatin

structure may not require DNA methylation per se, there is

evidence that methylation is required for the preservation

of such imprints in dividing cells (Mohandas et al., 1981).

Similarly, genomic regions belonging to endogenous

retroviruses and transposable elements such as short/

long interspersed nuclear elements (SINEs/LINEs) are

also heavily methylated and transcriptionally silenced

(Smit, 1999). Thus, one of the critical functions of

methylation-based epigenetic imprinting is to encode the

transcriptional state of the cell and to provide a mechanism

for the controlled (de)activation of regulatory genes during

development and differentiation. Although the term

‘imprinting’ is typically used in reference to heritable

epigenetic methylation marks in the germline, here we

extend this term to include somatic epigenetic patterns

ARTICLE IN PRESS

www.elsevier.com/locate/yjtbi

0022-5193/$-see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jtbi.2006.05.012

?Corresponding author. Tel.: +12066674145; fax: +12066677004.

E-mail address: gluebeck@fhcrc.org (E. Georg Luebeck).

Page 2

whose maintenance is critical for the proper development

of the organism.

How do dividing cells manage to preserve their

epigenetic imprints? Loss of methylation by the conversion

of fully methylated CpG/GpC dyads to hemimethylated

dyads is a simple consequence of DNA replication (when

unmethylated cytosines are incorporated into the daughter

DNA strand opposite parental CpG sites, see Fig. 1). Thus

the question arises, how do dividing cells maintain

simultaneously hypo- and hypermethylated CpG regions?

Once they have been established, how is it possible that

these two states can coexist with sufficient stability in

distinct regions of the genome?

Recent experiments show that there are at least three

DNA methyltransferases (Dnmt’s) that are involved in

post-replicative methylation of CpG sites (e.g. see Okano et

al., 1999). While de novo methylation is attributed to the

action of the isoforms Dnmt3a and Dnmt3b, Dnmt1

appears mainly responsible for maintaining the parental

methylation pattern by methylating the correct daughter

CpGs, as shown in Fig. 1 (Kim et al., 2002; Chen et al.,

2003; Vilkaitis et al., 2005).

In an early paper, Otto and Walbot (1990) provided a

first description of methylation dynamics in terms of both

de novo and maintenance methylation. Although the

specific enzymes responsible for these processes had not

yet been identified, their model predicted globally stable

methylation equilibria in dividing cell populations as a

consequence of the recurrent actions of DNA replication

and joint de novo and maintenance methylation. A similar

model (in continuous time, rather than discrete time) was

put forward by Pfeifer et al. (1990). The former was

recently improved by Genereux et al. (2005) to allow for

differential de novo methylation between parental and

daughter DNA strands, including the case when de novo

methylation occurs on one strand but not the other. Their

model was also formulated to allow parameter estimation

via maximum likelihood based on counts of fully

methylated, hemimethylated, and unmethylated CpG

dyads (see also Laird et al., 2004). Their analysis of CpG

methylation within the promoter of the human gene FMR1

also confirmed previous estimates of methylation efficien-

cies that lacked multi-site information. However, no model

of methylation dynamics has this far addressed the stability

question raised above, in particular, the possibility of

metastable equilibria in dividing cell populations.

Here we recast the model by Otto and Walbot (1990) in

terms of a Markov chain process for which the steady-state

equilibrium solutions can be readily computed. In doing so,

we allow for a more general methylation dynamics,

including asymmetry of DNA strand segregation. Site-

specific transitions between the methylation states of a

CpG dyad are associated with the rates of both de novo

and maintenance methylation. The basic premise of the

model is that after DNA replication Dnmt1 methylates any

hemimethylated CpG dyad with probability r, while the

combined role of Dnmt3a/b is that of methylating both

hemi- and unmethylated CpG dyads with probability m.

Although there is evidence that members of these two

Dnmt families cooperate, the details of their interaction are

not well understood (Kim et al., 2002). However, weak de

novo CpG methylation by Dnmt1 has been demonstrated

in the absence of Dnmt3a/b in mouse embryonic stem cells

(Lorincz et al., 2002), further complicating our model.

Analysis of the steady-state solutions of the linear

methylation model, in which the actions of maintenance

and de novo methylation are assumed independent,

suggests that stable maintenance of hypomethylated states

requires an exquisite repression of de novo methylation

(i.e. mo0:01). Although it has been hypothesized that the

methyltransferases may require specific histone modifica-

tions possibly involving Lys9 methylation on H3, and

presence of HDAC and/or HP1 for their activation, it is

not clear how such complex modifications (or persistent

lack of such modifications) would be faithfully inherited

from one cell generation to the next (e.g. see the discussions

in Wolffe et al., 1999; Jones and Baylin, 2002).

Here we propose an alternative solution to this question

based on a modification of the basic Markov chain model

that assumes cooperativity of the Dnmt’s. Specifically,

we propose that the efficiency of de novo methylation, m,

is dependent upon the region-specific density of hemi-

methylated sites immediately after DNA replication.

Because the activity of Dnmt1, which has a preference

ARTICLE IN PRESS

Fig. 1. Illustration of the CpG methylation model. DNA replication (left) leads to hemimethylated DNA double strands. After DNA replication,

maintenance methylation (via Dnmt1) methylates hemimethylated CpG dyads, but occasionally fails to do so (marked ‘X’). De novo methylation

(primarily via Dnmt3a/b) is assumed to act either concomitantly or in tandem, and with some probability methylates any unmethylated CpG site

(indicated by lightly shaded ovals).

L.B. Sontag et al. / Journal of Theoretical Biology 242 (2006) 890–899

891

Page 3

for hemimethylated CpG sites, may likely depend on the

degree of hemimethylation in the region of interest, the

dependence of m on this level can be viewed as a reflection

of a synergistic interaction between Dnmt1 and one or both

members of the Dnmt3 family. Note, this model does not

provide an explanation for the stable existence of CpG

islands in the mammalian genome, which clearly is a

germline issue. Rather, the model seeks to explain how

sparsely and densely methylated CpG regions can be

maintained stably in renewing or clonally expanding cell

populations in soma.

The most significant consequence of this non-linear,

non-local (i.e. region-specific) methylation model is that it

entertains two distinct methylation equilibria: stable

hypermethylation as well as stable hypomethylation.

However, for a system with a finite number of CpGs, the

two states are metastable, meaning that noise-induced

transitions may occur from one state to the other.

Metastability of the hypomethylated state may well explain

the observation of age-related gene silencing by sporadic

methylation of CpGs at the promoters of affected genes

(e.g. see Ahuja et al., 1998; Toyota and Issa, 1999).

In the following sections we introduce, explore, and

discuss both linear and non-linear versions of our methyla-

tion model. We use the term ‘non-linear’ to simply refer to

the state dependence of the transition matrix of the Markov

chain. We provide precise definitions of the transition

matrices, a ‘topological map’ of stable equilibrium states,

bifurcation analyses, and (finite-system) computer simula-

tions that validate the Markov chain model and demon-

strate the importance of methylation noise.

2. Models and methods

A convenient mathematical framework in which the

three distinct processes (DNA replication, de novo

methylation, and maintenance methylation) can be linked

together isthat ofa discrete-time

(e.g. Norris, 1997). We describe CpG dyads as being in

one of four states: both CpGs unmethylated (state 0),

hemimethylated with a methyl group on one of the

complementary DNA strands (state 1), hemimethylated

with a methyl group on the opposing strand (state 2), and

both CpGs of the dyad methylated (state 3). The two

hemimethylated states seem redundant, as they will have

the same probability under symmetric cell divisions.

However, we prefer to distinguish the two states to

accommodate the possibility of asymmetric cell divisions

that retain a preferential DNA strand (Cairns, 2002;

Tannenbaum et al., 2005).

We use the Markov chain model to keep track of

incremental changes in the probabilities of the CpG dyads

being in each of the four CpG methylation states from one

cell generation to the next. The probabilities for these four

states, pi(i ¼ 0;1;2;3), sum to 1, i.e.P3

of this state vector are independent. Transitions between

Markovchain

0pi¼ 1. Thus, for

symmetric cell divisions, only two of the four components

methylated states necessarily occur when the cell replicates

and synthesizes new (unmethylated) DNA strands. For

example,mCpG=GpCm(state 3) undergoes a transition into

mCpG=GpC (state 2) and CpG=GpCm(state 1) with equal

probability when cells divide symmetrically.

Transition matrices: The diagram in Fig. 2 shows the

possible transitions and the resulting probability flow

among the four methylation states of a CpG dyad. The

transitions follow basic probability laws and can either be

viewed to represent the (conditional) frequencies of finding

a particular CpG dyad in a specific state in a large ensemble

of identical and independently dividing cells, or alterna-

tively, as frequencies across a specific region including a

large number of independent CpG sites. For the purpose of

developing a non-linear, region-specific model (below), the

latter view is more helpful. In either view, however, the

Markov chain approach used here limits the description of

the methylation dynamics to that of sample averages based

on cell lineages obtained from independently evolving

clones and therefore ignores correlations among samples

from individual clones.

The stochastic matrix corresponding to the flow diagram

shown on the left in Fig. 2, which describes asymmetric

DNA replication that retains one of the complementary

DNA strands with probability s, is

0

B

Ds¼

1

s

1 ? s

0

s

0

0

01 ? s

0

1 ? s

s

0

0

00

B

B

@

1

C

C

C

A.

This form allows the ‘tracking’ of lineages with preferential

retention of DNA strands. The particular choice s ¼ 0:5

describes the segregation of the two parental strands with

ARTICLE IN PRESS

(0,1)

(0,1)

1-σ

µ(1-µ)

1-µ-ρ+µρ

1-µ-ρ+µρ

µ+ρ-µρ

µ+ρ-µρ

µ(1-µ)

1-σ

1-σ

(1,1)

(1,1)

(1,0)

(1,0)

(0,0)

(0,0)

σ

σ

σ

µ2

Fig. 2. Schematic diagram of the Markov chain model representing the

probability flow between the four possible methylation states of a CpG

dyad(denoted inbinaryform:

ð1;0Þ ¼ hemimethylated, ð1;1Þ ¼ fullymethylated). The diagram on the

left shows possible transitions following DNA replication. Note, the

parameter s can be used to describe departures from symmetric cell

divisions (s ¼ 0:5) toward fully asymmetric divisions with preferential

retention of one or the other complementary DNA strand. The diagram

on the right shows the combined and concomitant action of both de novo

methylation and maintenance methylation after DNA replication with

rates m and r, respectively.

ð0;0Þ ¼ unmethylated,

ð0;1Þ

and

L.B. Sontag et al. / Journal of Theoretical Biology 242 (2006) 890–899

892

Page 4

equal probability to the daughter cells (symmetric divi-

sion). Note, all simulations and analyses reported here,

unless stated otherwise, are for scenarios with symmetric

cell divisions.

Subsequent state transitions (before the next round of

DNA replication occurs) are induced by the de novo and

maintenance methylation processes. We first consider two

simple Markov chain models that differ only in the time

ordering of maintenance and de novo methylation events.

The first model (model 1, shown in Fig. 2) assumes that

DNA replication is first followed by maintenance methyla-

tion, then by de novo methylation. This model is

mathematically equivalent to a model in which DNA

replication is followed by the concomitant, but indepen-

dent, action of both methylation processes. The second

model (model 2) assumes the reverse order of the two

methylation processes after DNA replication, i.e. de novo

methylation is followed by maintenance methylation. This

model is useful to explore the consequences of a possible

time delay in the activity of Dnmt1 as suggested by

Liang et al. (2002).

Let Nm be the transition matrix describing de novo

methylation and Mr the transition matrix representing

maintenance methylation. Models 1 and 2 can then be

described by the (composite) transition matrices T1¼

Nm? Mr? Ds and T2¼ Mr? Nm? Ds, respectively. Note,

both models begin with DNA synthesis and the segregation

of their DNA strands. Explicit expressions for Nmand Mr

can be obtained by inspection of the probability flows

depicted in Fig. 2 (right panel), representing the concomi-

tant action of de novo and maintenance methylation.

Mathematically, the joint action of these two processes can

be expressed by the stochastic matrix

0

B

Qm;r¼

ð1 ? mÞ2

mð1 ? mÞ

mð1 ? mÞ

m2

000

ð1 ? mÞð1 ? rÞ

0

1 ? ð1 ? mÞð1 ? rÞ

00

ð1 ? mÞð1 ? rÞ

1 ? ð1 ? mÞð1 ? rÞ

0

1

B

@

B

B

1

C

C

A

C

C

.

For example, the effect of joint but independent

processing of hemimethylated CpGs by de novo and

maintenance methylation can be computed from the

probability that neither of them cause a change in the

methylation state, i.e. ð1 ? mÞ ? ð1 ? rÞ. Other matrix

elements are immediately clear.

Transition matrices describing de novo (Nm) and

maintenance (Mr) methylation follow from Qm;r: Nm¼

Qm40;r¼0

stipulates precedence of maintenance methylation over de

novo methylation, it is mathematically equivalent to the

concomitant process, i.e. T1¼ Qm;r? Ds. Furthermore,

when comparing empirical data with model predictions,

we assume that DNA replication is promptly followed

by the two methylation processes and that they complete in

a time that is short compared with the duration of the cell

cycle. These assumptions are consistent with the observed

co-localization of Dnmt’s with the replication fork

and Mr¼ Qm¼0;r40. Note, while model 1

(Vilkaitis et al., 2005). Thus, we assume that measur

ements of methylation patterns are unlikely to occur in

between DNA replication and de novo and maintenance

methylation.

Steady-state solutions: Let P ¼ ðp0;p1;p2;p3Þ be the

probability state vector for a system of CpG dyads. After

n cell divisions we have PðnÞ¼ TnPð0Þ, with Pð0Þbeing the

initial CpG methylation state. For example, Pð0Þ¼

ð0;0;0;1Þ corresponds to a system that starts with all

CpGs in the region of interest fully methylated. The state

probability vector Pe, for which TPe¼ Pe, defines the

steady state (or equilibrium state) of the Markov chain.

Thus, it corresponds to an eigenvector of T with eigenvalue

1. For the two models considered here, this eigenvector can

be obtained algebraically. For the case of symmetric cell

divisions (i.e. s ¼ 0:5), and for a constant rate of de novo

methylation m, we obtain the following equilibrium states:

2

Pðe;1Þ¼

ð1 ? rÞð1 ? mÞ3

1 þ m ? r þ mr

ð1 ? mÞð2 ? mÞð1 ? rÞm

1 þ m ? r þ mr

ð1 ? mÞð2 ? mÞð1 ? rÞm

1 þ m ? r þ mr

mð?m2þ m2r ? 3mr þ 3m þ 2rÞ

1 þ m ? r þ mr

6666666666664

3

7777777777775

and

Pðe;2Þ¼

ð1 ? rÞð1 ? mÞ3

1 þ m ? r þ 3mr ? 2m2r

ð1 ? mÞð2 ? mÞð1 ? rÞm

1 þ m ? r þ 3mr ? 2m2r

ð1 ? mÞð2 ? mÞð1 ? rÞm

1 þ m ? r þ 3mr ? 2m2r

mð?m2þ m2r ? 5mr þ 3m þ 4rÞ

1 þ m ? r þ 3mr ? 2m2r

for models 1 and 2, respectively. Note, experimental

determination of any two independent components of Pe

allows the determination of the two parameters m and r.

Specifically, for model 1, we obtain the point estimates

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

and

2

6666666666664

3

7777777777775

^ m ¼ 1 ?

¯ p0

¯ p0þ ¯ ph

r

^ r ¼2ð¯ p0þ ¯ phÞ2? ð¯ p0þ ¯ phþ 1Þ

ð¯ p0þ ¯ ph? 1Þ

where ¯ p0 is the observed fraction of unmethylated CpG

dyads and ¯ ph¼ ¯ p1¼ ¯ p2 the observed fraction of hemi-

methylated CpG dyads (assuming symmetric cell divisions

and steady-state conditions).

For example, using data from mouse embryonic stem

(ES) cells published by Liang et al. (2002) (their Fig. 5, CI-f

M1/3A/3B) we have ¯ p0¼ 0:12 and ¯ ph¼ 0:025. This yields

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¯ p0ð¯ p0þ ¯ phÞ

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¯ p0ð¯ p0þ ¯ phÞ

p

,

ARTICLE IN PRESS

L.B. Sontag et al. / Journal of Theoretical Biology 242 (2006) 890–899

893

Page 5

^ m ¼ 0:09 and

promoter region in human lymphocytes, Laird et al.

(2004) find ¯ p0¼ 0:155 and ¯ ph¼ 0:032 which yields the

estimates ^ m ¼ 0:09 and ^ r ¼ 0:95, surprisingly similar to the

estimates from mouse ES cells. Model 2 yields identical

estimates for ^ m and only slightly lower estimates for ^ r

compared to model 1, suggesting that the time-ordering of

de novo and maintenance methylation has little effect on

inference of m and r.

Modeling the stimulation of de novo methylation by

maintenance methylation: Direct support for a synergistic

interaction between the Dnmt’s comes from in vitro

experiments using mouse ES cells (e.g. Chen et al., 2003;

Liang et al., 2002) and experiments with human Dnmt’s in

bacteria (Kim et al., 2002). These studies show that the rate

of de novo methylation is substantially increased (up to 15-

fold) when the de novo methyltransferases Dnmt3a/b act in

concert with Dnmt1. This cooperativity between the

Dnmt’s suggests a model extension in which the efficiency

of de novo methylation depends on the density of

hemimethylated sites after DNA replication but before

maintenance and/or de novo methylation occurs. In terms

of the parameters of the Markov chain process, this

assumption is akin to assuming that the de novo efficiency

m is a function of the current methylation state P

immediately after DNA replication. For simplicity, and

consistent with our current assumption of symmetric

segregation of DNA strands, we assume the symmetric form

^ r ¼ 0:97. Interestingly, for the FMR1

m ¼ m0þ ðmmax? m0Þðp1þ p2Þr

with rX0 and 0pm05mmax.

This choice constrains m to values between m0and mmax. The

dependency of m on the mean level of hemimethylation

immediately after DNA replication, essentially p1or p2as

one DNA strand will be completely unmethylated at that

time, can be interpreted as reflecting an interaction between

Dnmt1 and one or the other member of the Dnmt3 de novo

methylation family. In this picture, the site-specific activity

of de novo methylation via Dnmt3a/b is coupled to the

activity of Dnmt1, which we assume is proportional to the

frequency (or fraction) of hemimethylated sites immediately

after DNA replication.

As we will show here, this modification of the linear

DNA methylation model may lead to bifurcations in the

methylation equilibria. Under the assumed methylation

dependence of m, one of the two equilibria may be

considered hypomethylated. This is obvious for m0¼ 0:

the unmethylated state (i.e. p0¼ 1) leads to m ¼ 0 and

therefore is a fix-point. For the general case, when m040,

closed form expressions for the steady state(s) are difficult

to compute. However, a simple iterative method provides a

tool to efficiently search for bifurcations and to predict

numerically the associated methylation states: (1) compute

the linear steady-state (equilibrium) solution Pe using

either m0or mmaxas starting values for the parameter m, (2)

update m by setting m equal to m0þ ðmmax? m0Þðpe;2þ pe;3Þr,

the equivalent rate of de novo methylation under the linear

model prior to DNA replication, (3) re-compute Pe,

and iterate steps 2 and 3 until a specified level of precision

is achieved.

We have tested this method numerically for a number of

scenarios (results not shown). The method works well and

significantly reduces computing time compared to explicitly

iterating the non-linear Markov chain over a number of

cell generations. For example, for the parameter set used in

Fig. 5 (bottom panel) the fix-point method converged in

20 iterations (with six digits of precision) versus 300 explicit

iterations using the Markov chain to evolve the hyper-

methylated state, and 5 versus about 500 iterations for

the hypomethylated state. In case a bifurcation does not

occur, as may be the case for smaller values of r (e.g. see

Fig. 5, top panel), an exploration based on both fix-point

method and explicit computation of the Markov chain

may be helpful.

Parameter identifiability: Assuming symmetric segrega-

tion of the parental DNA strands (i.e. s ¼ 0:5), the two

parameters of the linear methylation model, r and m, can

be determined directly using the above expressions for ^ r

and

^ m with measurements of any two independent

components of the equilibrium state (say, the respective

frequencies of unmethylated and hemimethylated CpG

dyads). For the ‘non-linear’ models, however, the two

equilibrium states (assuming a bifurcation does occur) are

not independent, in the sense that one state cannot be

controlled independently of the other. This is immediately

clear considering that, under equilibrium conditions, the

maintenance methylation rate r can be expressed in terms

of components of either one of the two equilibrium states

(see the equation for ^ r). Thus, the two states are related

and only two of the three m-related parameters can be

determined.

Given an observed bifurcation and the description of the

two equilibrium states in terms of their respective estimates

¯ p0and ¯ ph, it is straightforward to obtain estimates of r and

at least two m-related parameters given either m0, or mmax,

or r. In short, the above expression for ^ m allows us to derive

the condition

?

where ^ m1ð2Þrefers to the estimate of m under equilibrium

1(2) and ¯ m1ð2Þrefers to ¯ p2þ ¯ p3, the relevant frequency of

full- and hemimethylation prior to DNA replication for

equilibrium 1(2). Thus, if m0is known, we can determine r

and mmax. Experimentally, m0may be identified as the de

novo methylation rate in the absence of Dnmt3a/b,

possibly due to infrequent methylation errors of Dnmt1

that result in de novo methylations.

Note, our model assumes that the maintenance methyla-

tion rate r, unlike the de novo methylation rate m, is state-

independent. Therefore, assuming a bifurcation occurs, the

model stipulates a relationship between the two resulting

equilibrium states such that ^ r1¼ ^ r2, i.e. the estimate of r

under equilibrium 1 equals the estimate under equilibrium 2.

ln

^ m1? m0

^ m2? m0

?

¼ rln

¯ m1

¯ m2

??

,

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Thus, this requirement provides a test for the validity of the

present model, in particular the validity for the assumed

state-independence of r.

Simulations: Stochastic fluctuations in the methylation

state from one generation to the next are best visualized by

explicit simulations of a finite system of CpG dyads. The

simulations are based on random generation of state

transitions at individual CpG sites in accordance with the

assumptions of the Markov models described above.

However, rather than using the expected methylation

frequencies of the Markov chain model, the simulation

makes use of the finite ensemble frequencies for each

methylation state in randomly generated lineages of a

dividing clone. Thus, for example, in the simulations the

hemimethylation probabilities in the expression of m are

replaced by corresponding ‘polled’ frequencies, which

equal the numerical fractions of hemimethylated sites

among an assumed number of CpG dyads. Note, for

simplicity, we only show simulations based on the non-

linear extension of model 1. Respective simulations based

on model 2 are qualitatively similar. Opensource R-code

for these simulations can be obtained upon request from

the corresponding author.

3. Results

The Markov chain formalism adopted here allows us to

explore the consequences of DNA replication and Dnmt

processing on the ‘population dynamics’ of CpG dyads in

terms of their methylation states. A characteristic property

of the linear models is that they approach unique

equilibria, consistent with the early findings by Otto and

Walbot (1990) and Pfeifer et al. (1990). Fig. 3 shows the

expected level of CpG methylation projected onto a single

DNA strand as a function of the number of cell generations

for various values of m (the de novo methylation efficiency)

and incremental initial levels of methylation on that strand.

Stable maintenance of hypomethylation requires suppres-

sion of de novo methylation: For the linear Markov chain

models (i.e. when r ¼ 0) the equilibrium states are

completely determined by the two parameters m ¼ mmax

and r. Fig. 4 shows topographies for various percentages

of fully methylated CpGs (component p3) and their

dependence on the two model parameters, both for models

1 and 2. For this case (r ¼ 0), we see that stable

maintenance of a hypomethylated state (with less than

10% of CpGs fully methylated) can only be achieved if

mo0:2. However, lower levels (o1%) of methylation

require more stringent suppression of de novo methylation

activity, especially in the presence of active maintenance

methylation (r40:9), as suggested by experimental data

(see Models and methods). Although increases in r result

in higher methylation levels, a substantial degree of full

methylation is not achieved unless the value of r is well

above 0.9.

Simultaneous stability of hypo- and hypermethylated

patterns can be achieved by extending the linear model to

include a de novo methylation efficiency that is dependent

on the degree of CpG methylation immediately after DNA

synthesis (see Models and methods). This extension of the

model can lead to bifurcations in the methylation

equilibria. Fig. 5 shows the flow of methylation levels as

a function of cell generation starting with stepwise

increasing amounts of initial methylation for two scenarios

that differ in the dependence of m on the current level of

methylation via different choices for the parameter r. Fig. 5

also shows that a quadratic dependency of m on the current

level of hemimethylation leads to a stronger attraction

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0 2040 6080100

100

generation

methylation level (%)

µ=0.2

µ=0.1

µ=0.05

µ=0.025

80

60

40

20

0

Fig. 3. Mean levels of CpG methylation for the linear Markov chain

model as a function of time (cell generations) for various values of de novo

methylation, m, and for stepwise increasing levels of initial methylation.

Computed values are summarized to reflect single strand observations (i.e.

states 2 and 3 combined) and r ¼ 0:95.

de novo methylation efficiency µ

maintenance methylation efficiency ρ

0.00.2 0.40.60.8 1.0

1.0

0.8

0.6

0.4

0.2

0

Fig. 4. Topographies for full methylation (state 3) at various percentage

levels (1–99%) and their dependence on the two methylation parameters m

and r for models 1 (dashed lines) and 2 (solid lines) for the case that m is

state-independent, i.e. r ¼ 0.

L.B. Sontag et al. / Journal of Theoretical Biology 242 (2006) 890–899

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toward the hypomethylated state than a sub-quadratic

dependence (ro2). This model, therefore, allows stable

maintenance of both hypo- and hypermethylated states.

The extent of stochastic fluctuations of methylation in

a finite system of 50CpG dyads can be seen in Fig. 6.

The scenarios shown in Fig. 6 also demonstrate the

role of stochastic fluctuations of methylation levels in

inducing random transitions between the two equilibria, in

particular spontaneous transitions from hypo- to hyper-

methylation.

4. Discussion

How do dividing cells maintain stable levels of CpG

methylation in regions of their genome that are critical to

transcriptional regulation, and what does ‘stable’ exactly

mean? Here we present a composite Markov chain model

for CpG methylation that accounts for three distinct

processes: DNA replication and cell division, de novo

methylation (primarily via Dnmt3a/b), and maintenance

methylation (via Dnmt1). Our approach, although similar

in concept to the approach taken by Otto and Walbot

(1990), differs in crucial aspects. They did not acknowledge

the specific roles of Dnmt’s, nor did they allow for the

possibility of sequential de novo and maintenance methy-

lation processing. However, they pointed out the existence

of a methylation equilibrium in which ‘‘the proportion of

sites which become newly methylated equals the proportion

of sites that become demethylated in a cell generation’’.

Their analysis also included quantitative estimates of the

ratio of m versus r, which suggested that the efficiency of

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100

80

60

40

20

0

100

80

60

40

20

0

methylation level (%)

0 50100 150 200

generation

methylation level (%)

µ=0.0005+0.12π2 ρ=0.95

µ=0.0005+0.12π1.4 ρ=0.95

Fig. 5. Mean methylation levels projected onto a single DNA strand

(calculated as in Fig. 3) as a function of time (in cell generations) for

stepwise increasing levels of initial methylation for two scenarios that

differ in the state-dependence of the de novo methylation efficiency m (see

text for details). Upper panel: r ¼ 1:4, lower panel: r ¼ 2; both panels with

r ¼ 0:95.

0 200400 600800 1000

100

80

60

40

20

0

100

80

60

40

20

0

100

80

60

40

20

0

methylation level (%)

r=1.0

0 200400600 8001000

methylation level (%)

r=1.4

0 200 400 6008001000

generation

methylation level (%)

r=1.8

Fig. 6. Simulations based on the non-linear extension of model 1 with de

novo methylation efficiency m ¼ 0:0005 þ 0:12ðp1þ p2Þrfor r ¼ 1.0 (top

graph), 1.4 (middle graph), and 1.8 (bottom graph). Each color-coded

trace represents an independent simulation and shows the percentage of

fully methylated CpGs in a hypothetical system of 50 CpGs over the time-

span of 1000 generations. All simulations start with 2% of CpGs fully

methylated.

L.B. Sontag et al. / Journal of Theoretical Biology 242 (2006) 890–899

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maintenance methylation had to be much larger than the

efficiency of de novo methylation for the model predictions

to be consistent with observations of the relative propor-

tions of homo- versus hemimethylated CpG dyads.

The existence of noisy methylation equilibria in con-

stantly renewing cells, such as stem cells in the crypt of the

colon, has interesting consequences. Even when a region of

CpGs is in equilibrium, individual CpGs may continue to

undergo frequent state transitions. Consequently, region-

specific methylations levels (averaged over a finite number

of CpGs) are predicted to undergo stochastic fluctuations

(e.g., as seen in Fig. 6). Therefore, differences in region-

specific methylation patterns retrieved from a pool of

renewing cells may be less a reflection of epigenetic

divergence (for example due to an accumulation of

sporadic methylation errors), but may rather reflect

naturally occurring fluctuations around the equilibrium

state. Methylation tags may therefore not be useful as

molecular clocks unless the state associated with the

corresponding methylation pattern is sufficiently separated

from the equilibrium state. Another consequence of a

methylation equilibrium is the requirement of substantial

Dnmt suppression to maintain a stable hypomethylated

state (further discussed below), although such suppression

may be facilitated by mechanisms other than DNA

methylation. For example, specific histone tail modifica-

tions such as acetylation or methylation of the H3 tail

(Jones and Baylin, 2002; Bird, 2002) may prevent the

activation of de novo Dnmt’s.

Redundancy: The topographical ‘methylation map’ in

Fig. 4 shows that, for high levels of full methylation

(p3490%), a symmetry emerges between de novo and

maintenance methylation that suggests a functional re-

dundancy. If de novo or maintenance methylation fails, the

other methylation process can compensate for the defect

and is capable of maintaining the same level of methyla-

tion. Fig. 4 also shows that a high level of methylation does

not require perfect maintenance.

In any cell population that undergoes steady cell

turnover, the absence of de novo methylation combined

with imperfect fidelity of maintenance methylation (i.e.

ro1) would ultimately lead to complete loss of methyla-

tion. Our simulations show that methylation impurities

caused by imperfect maintenance methylation can be

‘stochastically’ cured by de novo methylation. The

combined action of these two methylation processes

therefore leads to a dynamic equilibrium state character-

ized by random fluctuations in the state of individual CpG

dyads, both in time and across lineages of a clone.

Global stability of hypo- and hypermethylated CpG

regions: The existence of a unique methylation equilibrium

for non-cooperative models (in this context models that

assume state-independent action of de novo and main-

tenance methylation) raises the question of how the Dnmt’s

might cooperate to maintain both hypo- and hypermethy-

lated CpG regions with sufficient stability. From Fig. 4 we

see that, in the absence of any cooperativity of the Dnmt’s,

the stable maintenance of a hypomethylated state would

require a strong repression of de novo methylation (i.e.

m ? 0). It is unclear how the methyltransferases can

selectively and persistently be blocked from operating on

unmethylated CpGs in promoter regions of transcription-

ally active genes while subjected to hundreds of DNA

replications and cell divisions without some preservation of

epigenetic information.

Our results show that global stability of hypo- and

hypermethylated CpG regions can be achieved by a

functional dependency of the site-specific de novo methyla-

tion efficiency on the degree of methylation immediately

after DNA replication (see Fig. 5). Thus, according to our

model, the site-specific efficiency m increases with the

overall methylation ‘workload’ of Dnmt1, possibly by

strengthening the synergistic association between the

Dnmt’s as methylation levels increase and weakening this

association as methylation levels decrease. Although the

interaction among the Dnmt’s is still only poorly under-

stood, our model posits a mechanism that is consistent with

the experimentally observed synergistic effect of Dnmt1

with Dnmt3a and/or Dnmt3b (Kim et al., 2002).

Hypomethylated state is predicted to be metastable:

Explicit computer simulations (see Fig. 6) demonstrate

that, in the presence of even very small background noise in

de novo methylation (i.e. m051), the hypomethylated state

is only metastable. In this context it is noteworthy that, in

experiments using Dnmt3a=b?=?mouse embryonic stem

cells with DNA probes that were initially unmethylated,

very low levels of de novo methylation could indeed be

detected after several rounds of cell division (Lorincz et al.,

2002). A weak de novo methylation capacity was therefore

attributed to Dnmt1. Recent experiments using human

cancer cells with genetically engineered deficiencies in

different Dnmt’s arrive at similar conclusions concerning

a role of Dnmt1 in de novo methylation (Jair et al., 2006).

In terms of our model, this property of Dnmt1 would

contribute to the predicted metastability of the hypo-

methylated state as shown in Fig. 6. The role of stochastic

fluctuations in the propagation of methylation imprints

and the development of bifurcations into unmethylated

(transcriptionally active) and methylated (transcriptionally

silent) CpG constructs has also been demonstrated

experimentally in mouse erythroleukemia (MEL) cells

(Lorincz et al., 2002). Further evidence for a bimodal

distribution of methylation profiles comes from a study of

DNA methylation levels within the human major histo-

compatibility complex where the vast majority of analysed

regions were either hypo- or hypermethylated (Rakyan

et al., 2004). However, additional experiments are needed

to demonstrate a shift in the fraction of unmethylated to

methylated constructs with increasing rounds of cell

division to provide direct evidence for the metastability

of the hypomethylated equilibrium.

According to our model, random fluctuations in the

overall level of methylation, due to the assumed finite

number of CpG sites in a specific region, translate into

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fluctuations in the de novo methylation rate m. This, in

turn, gives rise to sporadic transitions from an attractor of

low entropy—the almost ordered hypomethylated state—

to an attractor of higher entropy, such as a partially

methylated equilibrium state. The simulations shown in

Fig. 6 also illustrate the dependence of the relative stability

of hypo- and hypermethylated equilibria on the exponent r

in the expression of m. As r decreases, the hypomethylated

state becomes less stable and fluctuation-induced hyper-

methylation more frequent. This type of metastability may

therefore explain the observed age-related increase in CpG

methylation in initially unmethylated promoter regions of

various regulatory genes implicated in carcinogenesis

(Kondo and Issa, 2004).

Spreading of CpG methylation across methylation bar-

riers: Progressive region-specific de novo methylation has

been observed in a number of experiments (e.g. Wong

et al., 1999; Yates et al., 2003). The mechanism proposed

here predicts such spreading. Assume two separate CpG

regions, each at equilibrium, one hypermethylated and the

other hypomethylated (as in Fig. 5, bottom panel). If these

two regions were brought in contact, our model would

predict gradual hypermethylation of the hypomethylated

region. In our state-vector notation, the initial methylation

state would roughly equal ðð0;0;0;1Þ þ ð1;0;0;0ÞÞ=2 ¼

ð0:5;0;0;0:5Þ. Thus, for the two models shown in Fig. 5,

a system starting off with 50% of their CpG dyads being

fully methylated tends to evolve toward hypermethylation.

However, the relative range of attraction of each equili-

brium state depends upon the particular values of m0, mmax,

r and the power r.

In summary, the most important aspect of the proposed

model is the quasi-stable inheritance (in soma) of both

hypo- and hypermethylated CpG imprints in CpG rich

domains/islands of the genome. This property results from

the assumption of a methylation-dependent yet CpG site-

specific de novo methylation efficiency. It is intriguing to

interpret this dependency as a consequence of the

synergistic interaction of Dnmt1 and members of the

Dnmt3 family.

Acknowledgments

We would like to thank Diane Genereux and Charles

Laird (University of Washington, Department of Biology)

for valuable comments and the Leukemia and Lymphoma

Society for supporting this research through a fellowship to

one of the authors (M.C.L.).

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