Drawing a Waddington landscape to capture dynamic epigenetics

Biol. Cell (2013) 105, 576–584 DOI: 10.1111/boc.201300029 Scientiae forum

Drawing a Waddington landscapeto capture dynamic epigeneticsFloriane Nicol-Benoit, Pascale le Goff and Denis Michel1

Universite de Rennes 1 IRSET U1085 Transcription Environment and Cancer, Campus de Beaulieu, Rennes Cedex 35042, France

Epigenetics is most often reduced to chromatin marking in the current literature, whereas this notion was initiallydefined in a more general context. This restricted view ignores that epigenetic memories are in fact more robustlyensured in living systems by steady-state mechanisms with permanent molecule renewal. This misconception islikely to result from misleading intuitions and insufficient dialogues between traditional and quantitative biologists.To demystify dynamic epigenetics, its most famous image, a Waddington landscape and its attractors, are explicitlydrawn. The simple example provided, is sufficient to highlight the main requirements and characteristics of dynamicgene networks, underlying cellular differentiation, de-differentiation and trans-differentiation.

Alternative conceptions of epigeneticsThe dialogue between traditional and theoretical bio-logists is often limited and sometimes confused byambiguous vocabulary, such as for the term epige-netics. Specialists of chromatin (the complex thatcontains DNA in the nucleus of eukaryotic cells), as-cribed in good faith this word to their domain, basedon the fact that it contributes to specify the pheno-type, though not directly inscribed in DNA sequence.For instance, journals whose names refer to ‘epige-netics’ deal exclusively with chromatin. However, re-ducing epigenetics to the field of chromatin posesserious problems: (i) changes in chromatin are notvery strong memories; (ii) conversely, strong mem-ories can be generated without any modification ofchromatin; (iii) the concept of epigenetics has beendefined long before studies on chromatin, as a way ofstabilising a cellular state through the formation ofattractors in dynamic gene networks. In the presentcontext, attractors should be understood as compat-ible configurations of gene expression, that is to say,gene expression patterns that can remain stable inabsence of strong perturbations. A prerequisite forunderstanding dynamic epigenetics is a good per-ception of the steady state, characteristic of livingsystems and which should not be confused with adynamic equilibrium.

1 To whom correspondence should be addressed ([email protected])Key words: epigenetics, multi-stability, steady state.

Intuition is not always a good adviserDespite the existence of pedagogical reviews onattractors (Huang and Ingber, 2006; Ferrell, 2012),it remains difficult for the biologist community toadmit that non-equilibrium structures can not onlybe stable, but also ensure long-term memory. Settinga flag or a label appears at first glance as the only wayto guarantee a permanent marking, whereas dynamicfigures like vortices seem inherently fleeting. Thisimpression is reinforced by DNA, that is actually adigital memory, effectively duplicated, repaired andpropagated. Given that the different cell types in ourbody contain the same DNA, it is tempting to extendthis principle to chromatin, onto which cell type-specific memories are supposed to be printed. Butexperiences showed that changes of chromatin marksremain possible in highly compacted chromatinand are too labile and reversible over short timescales to ensure a persistent memory. In fact, sincethe discovery of enzymes of chromatin modificationand de-modification, there is no difference betweenhistone modifications and, for example, the phos-phorylation of cyclins, long established as reversible.Curiously, to regain some stability of the cell differ-entiation states, we must return to other principlesof memory engraved in moving figures, less intuitivebecause they seem more fragile. Nevertheless, wemust admit that the persistence of dynamic circuitsis the definition of life, but its understandingrequires a good vision of the concept of steadystate.

576 www.biolcell.net | Volume (105) | Pages 576–584

Draw me a landscape Scientiae forum

The steady stateThe essential property of living matter is to be main-tained in non-equilibrium states called steady states.An example of non-equilibrium structure is the vor-tex that forms above the drain of a bathtub whichis emptied. This structure can be maintained if thebathtub plug is open and the inlet water flow corres-ponds exactly to the exhaust flow. This is preciselywhat is a living cell: an open non-equilibrium systemsustained by permanent replenishment and dissipa-tion of matter and energy. In the cell, the long-termstorage of information, such as the cell differentia-tion status, is more registered in eddies than inlabels. Molecular labelling exists but paradoxicallyensures only short-term memory. This inversion oftime scales compared with our everyday experiencemay appear disconcerting, but a fundamental featureof life is that these components are replaced veryquickly. The majority of our atoms will be renewedin a few months without us having the impressionof having changed. It must be admitted that life andour identity are ‘sustained transient figures’, robustversions of the vortex of the emptying bath. If thebathtub faucet is open, the vortex may appear simi-lar on photos taken at different times, whereas thewater molecules materialising it are constantly re-newed. This continuous flow is the essence of life atall scales: at the cellular level, mRNA and proteinswhich are continuously degraded and resynthesised,and at the species level, individuals die and others areborn. The only long-term characteristic of life is basi-cally information (Michel, 2013). Matter comes alivein this information or more exactly, this informationembodies in matter.

Common misconceptionsSeveral misconceptions must be ruled out: (i) Thesteady state resembles, but is not an equilibrium, be-cause it vanishes as soon as the system is disconnectedfrom its environment, as for the vortex previouslydescribed. But everyone knows that an active cellis never disconnected, except accidentally and defi-nitely. (ii) At the microscopic level, the steady stateis not more dynamic than a dynamic equilibrium. Abucket of water is a system in equilibrium where wa-ter molecules constantly move but can never sponta-neously form a macroscopic structure like a whirlpool.(iii) Finally, a major feature of the steady state com-

pared with equilibrium, is the possibility of gettingdifferent figures from the same starting ingredients,provided that some conditions are satisfied, such asthe presence of feedback circuitry. Contrary to thesteady state, a single final state is possible in equi-librium, irrespective of the number and nature of theingredients involved. This phenomenon, essential forunderstanding systems biology and cellular differen-tiation, is called multi-stability. Waddington’s epi-genetics is the most striking image of multi-stability(Slack, 2002), where different cell types can be ob-tained from the same batch of genes and withoutneed for labelling them. Non-equilibrium systemshave amazing and interesting properties. They canbe very sensitive to disturbances that can cause sud-den phenomena, asymmetries, special structures, inshort, organisation. Time and life appear out of equi-librium and as stated by the Nobel prize recipientIlya Prigogine: ‘Out of equilibrium, matter begins tosee’.

Building an elementary Waddington‘epigenetic landscape’The best way to demystify Waddington landscapesis to build one explicitly, using simple mathematicaltools. For this, let us choose an elementary networkof genes, in fact the simplest possible one, made ofa single gene encoding a transcription factor F anda single circuit, precisely a positive feedback on itsown gene (Figure 1). The formal description of thisprototype of positive loop, first proposed in Keller(1995), is modified below to be expressed as a functionof the total concentration of F. It is assumed that thisfactor binds to DNA only as a dimer, which is oftenthe case for transcription factors.

Quasi-equilibrium of DNA bindingIf the interactions between the transcription factorand the DNA are fast enough compared with geneexpression dynamics, then the approximation of‘time scale separation’ is allowed (Michel, 2009) andthe equilibrium constant can be assimilated to aconcentration ratio. The dimensionless dissociationtendency depends on the absolute equilibrium disso-ciation constant K and on the concentration of

[F2

]and corresponds to the ratio of free over bound DNA

C© 2013 Societe Francaise des Microscopies and Societe de Biologie Cellulaire de France. Published by John Wiley & Sons Ltd 577

F. Nicol-Benoit, P. le Goff and D. Michel

Figure 1 Schematic circuit used for the present model

(A) Mini network reduced to a single gene and a single circuit.

(B) Detailed circuit including the parameters necessary for

its modelling. A transcription factor is capable, as a dimer,

of stimulating the transcription of its own gene. KDNA is the

equilibrium constant of dissociation from DNA and Kdim is the

dimerisation constant. In the equations, these constants are

written K and D respectively.

as follows

K[F2

] =[DNA0

][DNA − F2

] (1)

where[DNA0

]and

[DNA − F2

]should be under-

stood as the times during which the promoter is freeor occupied respectively. DNA can be either emptyor occupied by a dimer but not a monomer of F.Hence,

[DNAtot

] = [DNA0

] + [DNA − F2

]and

its fraction of occupancy Y is the fraction of timeduring which DNA is occupied by F2

Y =[DNA − F2

][DNA0

] + [DNA − F2

] (2a)

that can be transformed using the value of[DNA − F2

]derived from Eq. (1),

Y =[DNA0

] [F2

]K

[DNA0

] + [DNA0

] [F2

] (2b)

which simplifies to

Y =[F2

]K + [

F2] (2c)

Quasi-equilibrium of dimerisationThe dimerisation constant D can be expressed as

D =[F2

][F1

]2 (3a)

and the sum of monomeric and dimeric receptor isthe total receptor[

F]

tot= 2

[F2

] + [F1

](3b)

Eqs. (3a) and (3b) give[F2

] = D([

F]

tot− 2

[F2

])2(4)

Hence,[F2

]is the acceptable solution of a

quadratic equation that is

[F2

] =(

1 + 4D[F

]tot

−√

1 + 8D[F

]tot

)/8D

(5)

Circuit modellingThe evolution of

[F

]tot over time depends on the

ratio between its synthesis and its elimination.Its rate of synthesis is the maximum transcriptionfrequency ‘smax’, weighted by the fraction of timeduring which the factor is present in the promoter(Y). The loss of

[F

]tot is simply a linear function of its

amount, with a rate constant ‘r’ (removal). Finally, toallow switching to a high expression level, the factorshould be synthesised independently of its own autostimulatory action. This low-frequency synthesis rateis written s0. We thus obtain an ordinary differentialequation

d[F

]tot

d t= s 0 + s max

[F2

]K + [

F2] − r

[F

]tot

(6)

that can be expressed as function of[F

]tot only by

replacing[F2

]by its value defined in Eq. (5).

Draw me a landscapeA Waddington landscape resembles a potential land-scape where basins centred on attractors are sepa-rated by barriers of potential. Waddington drew his



landscape intuitively without clear mathematical idea(Wang et al., 2011) and several approaches can be pro-posed to formalise it. In particular, an analogy can bemade with the gravitational potential. Gravitationalattraction, that is inversely proportional to the squareof the distance in Newton’s law F (x ) = −M/x2 canbe designed according to Einstein, not as a force ap-plying directly between massive objects, but as a shiftto the lowest potential in a curved space described by−∫

F (x )d x = −M/ |x |, whose representation givesout the popular crater-shaped gravitational attrac-tor. Similarly, the landscape of Waddington can beconsidered as the negative integral of the gene(s) evo-lution rate(s) (Ferrell, 2012). In the present case fora unique gene, this integral gives the landscape withtwo attractors and a separating barrier representedin Figure 2. This minimalist example is sufficient toshow several essential features of gene networks:

i. The ‘spontaneous expression’ used for buildingthis landscape can be a transcriptional ‘noise’ re-lated to basal gene promoter (for instance a TATAbox).

ii. Non-linearity is essential to allow such bistabili-ty. It is in this case, related to the dimerisation ofF. Indeed, one can simply verify that in absenceof a square in the equations, a unique solutionwould have been obtained, which means that allcells would have a single gene expression state(mono-stability). Strongly non-linear effects ofcombinations of transcription factors can gene-rate nearly all-or-nothing gene activations andthresholds resembling those of Boolean gene net-works whose capacity to give rise to multiplesteady states has long been shown (Kauffman,1969).

iii. The positive feedback loop is also a sine quanon condition of multi-stability (Kaufman et al.,2007). For proof, one can replace the previousloop by a self-inhibition using the approxima-tion of the rapid equilibrium, with a repressoractive as a dimer.

The simple one-dimensional landscape describedhere (Figure 2) is purely theoretical because manytranscriptional influences interfere in the cell anda gene is rarely controlled by a single factor, butby a combination of factors. The so-called produc-tion functions that weight the maximum frequency

Figure 2 Construction of a minimalist unidimensionalepigenetic landscape

Evolution of gene products (top curve) and the correspond-

ing Waddington landscape (bottom curve), as function of the

initial condition [F]tot 0. These profiles have been computed

using the following set of parameter values: Basal expression

s0 = 0.4; dimerisation constant D = 0.2; DNA dissociation

constant K = 3; maximum expression rate smax = 20 and

degradation rate r = 1.

smax, include several factors. Depending on the modeof interference between the factors, several produc-tion functions can be defined (Bintu et al., 2005).The combined effects of the factors may be coopera-tive (sigmoidal) or not (hyperbolic) (Michel, 2010),with important consequences for multi-stability. Ifthe landscape of Waddington has a great pedago-gical value in facilitating the visualisation of attrac-tors, one should nevertheless beware of false repre-sentations. The vertical dimension of the landscapeis used to represent the potential, leaving only twodimensions to represent the gene expression levels.A landscape determined by a single gene gives a



curve like that of Figure 2 and a landscape deter-mined by two genes appears as a two-dimensionalsurface deformed in 3D. This image is extensivelystudied because it is very illuminating to illustratecell differentiation. In the case of two mutually in-hibiting and self-stimulating genes A and B, stemcells are present in an elevated attractor with co-expression of A+B, and two differentiated cell typesare found in two low elevation attractors with ex-clusive expression of either A or B (Huang, 2009;Wang et al., 2011). But landscapes correspondingto three or more genes (n-dimensional) are naturallynot representable. As such, the famous Waddingtondiagram representing a two-dimensional landscapebelow which a myriad of genes ‘pull the strings’(Slack, 2002) does not make sense. n-Dimensionallandscapes are purely mathematical objects, unrepre-sentable and rarely explicitly computable. But eventhis mathematical vision is illusory because the ther-modynamic and kinetic parameters required for thecalculation or simulation are almost impossible todetermine. The inherently multi-variable nature ofbiological systems makes them impossible to modelin a ‘bottom-up’ manner using elementary kineticingredients. Waddington landscapes are essential fortheir conceptual help in understanding biochemicalnetworks, but are not realistic modelling tools.

How to change attractorMulti-stable networks are not definitely trapped in asingle basin of attraction. There are several ways tomove from one steady state to another.

Passage from a high potential pool to a lowpotential poolThis is the most likely evolution, particularly invokedduring embryonic development. Stem, totipotent orpluripotent cells are located in elevated basins wherethe expression of many genes co-exists, whereas thedifferentiated cells are in low potential basins, wherethe expression of many genes is precluded by incom-patible circuits. Cellular switches between attractorsare generally multistep processes with a series of bi-nary choices where an upstream valley splits into twodownstream valleys (Foster et al., 2009), thereby of-fering the theoretical possibility to generate 2n celltypes after n bifurcations. If transitions accompaniedby a drop of potential are predominant, transitions

to new states of the same or even of higher potential,are not prohibited. Such jumps have been proposedto challenge the Waddington landscapes (Ladewiget al., 2013), but any transition is just a matter ofkinetics and of inputs received from the outside.

Passages between basins of the same or higherelevationLateral jumps between wells of equivalent poten-tial are involved in trans-differentiation. Jumps frombottom to top attractors underlie de-differentiation,reprogramming of differentiated cells into ‘inducedpluripotent stem cells’, and cancer (Huang, 2011). Inthis respect, primary cancers may occur without ge-nomic alterations (Brock et al., 2009) and might besecondarily consolidated by mutations (Huang andIngber, 2006). The idea of a reversible cancer with-out massive oncogenic mutations is not commonlyaccepted, but it is nevertheless suggested by the pos-sibility of reprogramming the nucleus of malignantcells (Hochedlinger et al., 2004).

The procedure for changing attractorGenes are the only molecular players in the cell whosenumber is invariant, all changes of wells result fromchanges in gene product concentrations. These fluc-tuations can be programmed or not.

The road is less important than the destinationTrajectories between two stationary states are notimportant by themselves and studying them in anattempt to find so-called molecular relays is not al-ways relevant. An enlightening study showed thatthe same final state can be reached through diffe-rent paths (Huang et al., 2005). In this study, thedifferentiation of HL60 cells into neutrophils can beinduced by treatment with either DMSO or retinoicacid. With both treatments, the gene network finallyreadjusts in the same state; but kinetic transcriptomicstudies revealed that intermediate transients are com-pletely different between the two treatments, somegenes being upregulated by one treatment and down-regulated by the other one. It is therefore clear thata gene induced at an early stage following a pertur-bation is not necessarily the hallmark of a signallingpathway. In gene networks, transient evolutions donot matter and only the stationary states reflect cel-lular activities. Accordingly, many experimental cellbiologists observed that a new cellular activity can



be induced in different ways, which have reciprocalactions. The circular relationships existing in genenetworks plunge into deep perplexity biologists whoare desperately seeking for originator molecules inorder to designate the elusive ‘therapeutic targets’necessary to obtain medical research funding.

Unplanned transitions between basinsThe frequency of the jumps out of an energy welldepends on the height to climb from the bottomof the well and the top of the surrounding bar-riers. By analogy with statistical mechanics, if we callthe total potential difference Ea, statistical physicspredicts that the output frequency has the formk = A exp(−E a/kBT ) where kB is the Boltzmannconstant, T is the temperature and A is said to bethe pre-exponential factor giving the unit of a rate(t−1). This frequency depends on random molecularfluctuations in the cell, like a wave higher than theothers at the surface of the ocean. These events canbe forbidden when Ea is high enough, or have an ex-tremely low frequency but may however accidentallyoccur in a large population of cells.

Scheduled runs between basinsTo ensure a stereotyped embryonic development withrobust cellular states and hierarchical transitions be-tween states, deep wells and acute valleys in thelandscape of Waddington were selected during evo-lution. In contrast, the shallower ponds separated bylow barriers offer degrees of freedom to the cells,with positive effects on phenotype plasticity but alsoadverse consequences by allowing the cells to pro-gressively reach an unwanted fatal attractor (Huang,2011). Exits from energy wells depend on mechanicalor chemical signals that affect the transcriptional andpost-translational modifications. For example, in thesimple case of two attractors in Figure 2, temporarydegradation of F suffices to trigger the switch fromthe right well to the left one. Once the transitionis completed, the degradation can be stopped andthe system self-stabilises in its new attractor. Proteindegradation seems to actually play a major role in thereprogramming of eukaryotic cells, as suggested bythe large number of genes encoding ubiquitin ligasesrevealed by genome sequencing. For example, pro-tein phosphorylation triggered by exogenous signalscan make the protein a substrate for pre-existing ubi-quitin ligases. This general principle can take many

other forms. Degradation may be replaced by proteinactivation or inactivation, for example, once againregulated by phosphorylation. The resulting distor-tion of biochemical network forces the gene networkto readjust over a new basin. Through this principle,we see that the cell is in tune with its environmentby receiving a wide range of external signals that acton key target proteins capable of destabilising attrac-tors. These signals include the embryonic inducers ofWaddington and Spemann (Slack, 2002) and moregenerally all chemical or mechanical signals that re-program all or part of cellular activities. The mainlesson of the vision of Waddington is that DNA andgenes only set the collection of all possible cellularfates, but that the precise state of a cell is ultimatelydictated by the conjunction of chance and of exter-nally imposed conditions. These states can then bemaintained by dynamic circuitry only without needfor structural marks as long as antagonistic stimuliare not provided.

How to insert chromatin epigenetics inthis pictureChromatin modifications are dispensable fortranscriptional memory imprintingCells committed to a particular lineage duringembryonic development, maintain their identityover cell divisions. In addition, some cells also havethe capacity to memorise exposure to chemicals suchas hormones. A famous example is the memory of theestrogen-dependent vitellogenesis by liver cells fromegg-laying vertebrates, where the response to estra-diol is much faster in animals previously treated withestroadiol in the past. Interestingly, if vitellogeninexpression is associated with chromatin remodellingof its gene, this remodelling is not persistent enoughto explain the memory effect (Burch and Evans,1986). This example shows that if chromatin remo-delling is indeed correlated with the gene expressionstatus, it does not necessarily have a causal influenceon its transcriptional memory. Instead, the vitel-logenesis memory effect could be purely dynamic(Nicol-Benoit et al., 2011). The examples of bacterialmemories, such as that of the lactose operon induc-tion status perpetuated over cellular generations, alsoproved that life did not await eukaryotic chromatinto make epigenetic memories. Moreover, the originof this bacterial phenotype maintenance has long



been identified as a positive feedback loop (Cohn andHoribata, 1959). ‘Chromatin epigenetics’ are neithermore nor less epigenetic than any other enzymaticmodification. They contribute to biochemicalsteady states and thus in sculpting the landscape ofWaddington.

Chromatin modifications are fleeting executantsThe stationary level of epigenetic ‘marks’ car-ried by histones results from an incessant balletof modifications–de-modifications, as for any post-translational modification. Accordingly, antagonis-tic modifying and de-modifying enzymes are oftenpresent simultaneously in the cells. The fact thatchromatin modifications are not set in stone couldhave been anticipated from old observations such asthose of (Thomas et al., 1975), showing that the levelsof histone methylation rapidly stabilise at interme-diate levels, reflecting specific ratios between methy-lases and demethylases in the cell. The exchanges ofmethyl-groups in DNA (CpG methylation) are alsovery dynamic (Yamagata et al., 2012). The dynamiccircuits described previously are recovered for histonemodifications for which many positive feedbacks havebeen established. The most famous feedback mecha-nism described in this context is the recruitment ofhistone-modifying enzymes by neighbouring nucle-osomes already modified. This allows to strengthenand spread these changes in space along a chromoso-mal region, and over time through mitosis after dis-persion of parental nucleosomes on daughter DNAmolecules (Zhu and Reinberg, 2011). This hypothe-sis has been proposed for the H4K16 deacetylaseSir2 (Imai et al., 2000), the H3K27 methyltrans-ferase of PRC2 (Hansen et al., 2008; Margueronet al., 2009), the SUV39H1/2 H3K9 methyltrans-ferase (Nakayama et al., 2001; Dodd et al., 2007).All these examples of positive loops reconnect chro-matin modifications to the more general field of bio-chemical network circuitry. In addition to transcrip-tional memory, some very important mechanismsmay result from cycles of chromatin modifications–de-modifications, such as the ultra-sensitivity of pro-gram changes (Nicol-Benoit et al., 2012). Permanentmodifications–de-modification cycles seem unneces-sary and wasteful for the cell, and they have thereforebeen called ‘futile cycles’. But the pioneers of mod-elling realised that these cycles are far from futileand that this apparent waste has an essential role

in cellular decision making, ultra-sensitivity mecha-nisms such as the ‘zeroth order’ mechanism (Xu andGunawardena, 2012), operating for one modification(Goldbeter and Koshland, 1981) or cascades of modi-fications (Ferrell, 1996). The mechanisms establishedfor the cell cycle kinases are naturally valid in thecase of histone methylation. These incessant cycles,costly for the cell (Goldbeter and Koshland, 1987),are the price to be paid for a discerning cellular be-haviour (Michel, 2011). It is obvious that the samemechanisms are at work for chromatin modifications(Nicol-Benoit et al., 2012), but this field of investi-gation has not yet opened and may suffer from a toostatic view of the famous ‘epigenetic marks’.

Technical difficultiesThe most common techniques for studying chro-matin modifications are not suitable for measuringtheir rate of recycling. The results of chromatin im-munoprecipitation assays (ChIP) reflect only station-ary states with no information on the dynamics ofmolecule exchanges at the microscopic level. Fromthis point of view, a common mistake is to assumethat changes in chromatin marks observed by ChIPmay correspond to microscopic turnovers of thesemarks, which is false. For example, a delayed neg-ative feedback effect usually causes large oscillationsof the considered biological activity (in the mannerof a slow thermostat), but the periods of these cyclesare completely unrelated to the underlying molecularturnovers. To get an idea of the actual frequency ofmicroscopic turnovers, we must turn to more com-plicated techniques such as pulse-chase experiments.Such approaches have been developed to study the re-placement of nucleosomes (Deal and Henikoff, 2010),DNA methylation (Yamagata et al., 2012) and his-tone modifications (Thomas et al., 1975; Ng et al.,2009; Zee et al., 2010). The latter, which requirecomplex techniques, give renewal times in the rangeof day(s), unable to ensure static epigenetic memoryfor years.

Chromatin modifications: Receivers or decisionmakers?A specificity of the network concept is that the func-tions of decision and execution are not separable. Inessence, a network operates democratically and eachnode in the network is just a cog. This is obviouslythe case for chromatin modifications that are boththe results and regulators of gene expression, taking



place in circular interrelations. As opposed to a deci-sion making role of the cellular phenotype ‘above thegenome’ at the top of the pyramid, chromatin couldsometimes play a simple role of receptor of pheno-type changes imposed exogenously. A recent studyshowed that a mechanically induced cell stretch-ing could both initiate actin polymerisation andreprogram gene expression by decompaction of chro-matin (Iyer et al., 2012). We knew that the expres-sion of specific genes can cause a reorganisation ofthe cytoskeleton, for example through induction ofserum responsive factor-responsive genes. Now wesee that the reverse is also true. Another role of chro-matin modifications in signal integration is emerg-ing in the field of cellular metabolism. Given thatmost substrate precursors of enzymatic chromatinmodifications are by-products of metabolism (includ-ing NADH, acetyl-CoA or SAM), their level candictate the extent of chromatin modifications (Kimet al., 2005, Wallace and Fan, 2010; Sassone-Corsi,2013).

In conclusion, to date, only the sequence of nu-cleotides in DNA can be reasonably regarded as astructural memory. All others, including chromatinmarks, belong to the dynamic epigenetic landscapeof Waddington. Waddington’s vision is not obsolete,as sometimes heard in the introduction of lectures onchromatin, but it is instead the general framework inwhich chromatin should be inserted.

Conflict of interest statementThe authors have declared no conflict of interest.

ReferencesBintu, L., Buchler, N.E., Garcia, H.G., Gerland, U., Hwa, T., Kondev,

J. and Phillips, R. (2005) Transcriptional regulation by the numbers:models. Curr. Opin. Genet. Dev. 15, 116–124

Brock, A., Chang, H. and Huang, S. (2009) Non-geneticheterogeneity—a mutation-independent driving force for thesomatic evolution of tumours. Nat. Rev. Genet. 10, 336–342

Burch, J.B. and Evans, M.I. (1986) Chromatin structural transitionsand the phenomenon of vitellogenin gene memory in chickens.Mol. Cell. Biol. 6, 1886–1893

Cohn, M. and Horibata, K. (1959) Inhibition by glucose of the inducedsynthesis of the beta-galactoside-enzyme system of Escherichiacoli. Analysis of maintenance. J. Bacteriol. 78, 601–612

Deal, R.B. and Henikoff, S. (2010) Catching a glimpse of nucleosomedynamics. Cell Cycle 9, 3389–3390

Dodd, I.B., Micheelsen, M.A., Sneppen, K. and Thon, G. (2007)Theoretical analysis of epigenetic cell memory by nucleosomemodification. Cell 129, 813–822

Ferrell, J.E. Jr. (1996) Tripping the switch fantastic: how a proteinkinase cascade can convert graded inputs into switch-like outputs.Trends Biochem. Sci. 21, 460–466

Ferrell, J.E. Jr. (2012) Bistability, bifurcations, and Waddingtonsepigenetic landscape. Curr. Biol. 22, R458–R466

Foster, D.V., Foster, J.G., Huang, S. and Kauffman, S.A. (2009) Amodel of sequential branching in hierarchical cell fatedetermination. J. Theor. Biol. 260, 589–597

Goldbeter, A. and Koshland, D.E. Jr. (1981) An amplified sensitivityarising from covalent modification in biological systems. Proc.Natl. Acad. Sci. U.S.A. 78, 6840–6844

Goldbeter, A. and Koshland, D.E. Jr. (1987) Energy expenditure in thecontrol of biochemical systems by covalent modification. J. Biol.Chem. 262, 4460–4471

Hansen, K.H., Bracken, A.P., Pasini, D., Dietrich, N., Gehani, S.S.,Monrad, A., Rappsilber, J., Lerdrup, M. and Helin, K. (2008). Amodel for transmission of the H3K27me3 epigenetic mark. Nat.Cell Biol. 10, 1291–1300

Hochedlinger, K., Blelloch, R., Brennan, C., Yamada, Y., Kim, M.,Chin, L. and Jaenisch, R. (2004) Reprogramming of a melanomagenome by nuclear transplantation. Genes Dev. 18, 1875–1885

Huang, S., Eichler, G., Bar-Yam, Y. and Ingber, D.E. (2005) Cell fatesas high-dimensional attractor states of a complex gene regulatorynetwork. Phys. Rev. Lett. 94, 128701

Huang, S. and Ingber, D.E. (2006) A non-genetic basis for cancerprogression and metastasis: self-organizing attractors in cellregulatory networks. Breast Dis. 26, 27–54

Huang, S. (2009) Reprogramming cell fates: reconciling rarity withrobustness. Bioessays 31, 546–560

Huang, S. (2011) On the intrinsic inevitability of cancer: from foetal tofatal attraction. Semin. Cancer Biol. 21, 183–199

Imai, S., Armstrong, C.M., Kaeberlein, M. and Guarente, L. (2000)Transcriptional silencing and longevity protein Sir2 is anNAD-dependent histone deacetylase. Nature 403, 795–800

Iyer, K.V., Pulford, S., Mogilner, A. and Shivashankar, G.V. (2012)Mechanical activation of cells induces chromatin remodelingpreceding MKL nuclear transport. Biophys. J. 103, 1416–1428

Kauffman, S.A. (1969) Metabolic stability and epigenesis in randomlyconstructed genetic nets. J. Theor. Biol. 22, 437–467

Kaufman, M., Soule, C. and Thomas, R. (2007) A new necessarycondition on interaction graphs for multistationarity. J. Theor. Biol.248, 675–685

Keller, A.D. (1995) Model genetic circuits encoding autoregulatorytranscription factors. J. Theor. Biol. 172, 169–185

Kim, J.H., Cho, E.J., Kim, S.T. and Youn, H.D. (2005) CtBP repressesp300-mediated transcriptional activation by direct association withits bromodomain. Nat. Struct. Mol. Biol. 12, 423–428

Ladewig, J., Koch, P. and Brustle, O. (2013) Leveling Waddington: theemergence of direct programming and the loss of cell fatehierarchies. Nat. Rev. Mol. Cell Biol. 14, 225–236

Slack, J.M.W. (2002) Conrad Hal Waddington: the last Renaissancebiologist? Nat. Rev. Genet. 3, 889–895

Thomas, G., Lange, H.W. and Hempel, K. (1975) Kinetics of histonemethylation in vivo and its relation to the cell cycle in Ehrlichascites tumor cells. Eur. J. Biochem. 51, 609–615

Margueron, R., Justin, N., Ohno, K., Sharpe, M.L., Son, J., Drury,W.J. 3rd, Voigt, P., Martin, S.R., Taylor, W.R., De Marco, V., Pirrotta,V., Reinberg, D. and Gamblin, S.J. (2009) Role of the polycombprotein EED in the propagation of repressive histone marks. Nature461, 762–767

Michel, D. (2009) Fine tuning gene expression through shortDNA–protein binding cycles. Biochimie 91, 933–941

Michel, D. (2010) How transcription factors can adjust the geneexpression floodgates. Prog. Biophys. Mol. Biol. 102, 16–37

Michel D. (2011) Basic statistical recipes for the emergence ofbiochemical discernment. Prog. Biophys. Mol. Biol. 106,498–516

Michel, D. (2013) Life is a self-organizing machine driven by theinformational cycle of Brillouin. Orig. Life Evol. Biosph. 43,137–150



Nakayama, J., Rice, J.C., Strahl, B.D., Allis, C.D. and Grewal, S.I.(2001) Role of histone H3 lysine 9 methylation in epigenetic controlof heterochromatin assembly. Science 292, 110–113

Ng, S.S., Yue, W.W., Oppermann, U. and Klose, R.J. (2009) Dynamicprotein methylation in chromatin biology. Cell. Mol. Life Sci. 66,407–422

Nicol-Benoit, F., Amon, A., Vaillant, C., Le Goff, P., Le Dran, Y.,Pakdel, F., Flouriot, G., Valotaire, Y. and Michel, D. (2011) Adynamic model of transcriptional imprinting derived from thevitellogenesis memory effect. Biophys. J. 101,1557–1568

Nicol-Benoit, F., Le-Goff, P., Le-Dran, Y., Demay, F., Pakdel, F.,Flouriot, G. and Michel, D. (2012) Epigenetic memories: structuralmarks or active circuits? Cell. Mol. Life Sci. 69,2189–2203

Sassone-Corsi, P. (2013) When metabolism and epigeneticsconverge. Science 339, 148–150

Wallace, D.C. and Fan, W. (2010) Energetics, epigenetics,mitochondrial genetics. Mitochondrion 10, 12–31

Wang, J., Zhang, K., Xu, L. and Wang, E. (2011) Quantifying theWaddington landscape and biological paths fordevelopment and differentiation. Proc. Natl. Acad. Sci. U.S.A.108, 8257–8262

Xu, Y. and Gunawardena, J. (2012) Realistic enzymology forpost-translational modification: zero-order ultrasensitivity revisited.J. Theor. Biol. 311, 139–152

Yamagata, Y., Szabo, P., Szuts, D., Bacquet, C., Aranyi, T. and Paldi,A. (2012) Rapid turnover of DNA methylation in human cells.Epigenetics 7, 141–145

Zee, B.M., Levin, R.S., Xu, B., LeRoy, G., Wingreen, N.S. and Garcia,B.A. (2010) In vivo residue-specific histone methylation dynamics.J. Biol. Chem. 285, 3341–3350

Zhu, B. and Reinberg, D. (2011) Epigenetic inheritance: uncontested?Cell Res. 21, 435–441

Received: 14 May 2013; Accepted: 10 September 2013; Published online: 22 October 2013; Accepted article online: 24 Septem-

ber 2013


Documents

Drawing a Waddington landscape to capture dynamic epigenetics