A model for spread of epigenetic marks

Amogh Sood∗

Massachusetts Institute of Technology,Department of Chemistry

(Dated: May 18, 2019)

Herein we develop a stochastic model that describes the dynamics of epigenetic marks along agiven DNA region and attempt to mathematically rationalise emergence of bistable and persistentepigenetic states from cooperative recruitment. We attempt to relate the rate of spreading of epige-netic marks from one site to another to power law contact probabilities between sites typically seenin polymers. The case of asymmetric interactions is also examined. Through stochastic simulation,we see how the bistability is affected by different power-law scalings. Some parallels are drawnbetween the dynamical system proposed herein and a 1D Ising model with long-range interactions.

I. INTRODUCTION

Cells are known to carry information handed downfrom their ancestors, and to pass it on to their descen-dants, primarily in two ways-either this information isencoded in the base pair sequence in the genome (“ge-netic”), or is mediated via a series of chemical mod-ifications to nucleosomes, a basic unit of DNA pack-aging in eukaryotes, consisting of a segment of DNAwound in sequence around eight histone protein cores(“epigenetic”)[1]. The former is the primary mecha-nism for information transfer over longer, evolutionarytimescales, while the latter mediates memory over shortertime scales. Nucleosomes package eukaryotic DNA, witha density of about one nucleosome per 200 bp of DNA [2][3], and the core nucleosome is composed of two moleculeseach of four core histones (H2A, H2B, H3, and H4)around which c.a.150 bp of DNA is wrapped [3].

Chemical modifications of nucleosomes come in manyflavours (e.g., acetylation, methylation, or phosphoryla-tion), and occur at different amino acid positions on thedifferent histones. This confers a large information ca-pacity on each nucleosome.

Additions and removals of these modifications are me-diated out by classes of enzymes, like histone acetyltrans-ferases (HATs), histone methyltransferases (HMTs), his-tone deacetylases (HDACs), histone demethylases [4] [5]

These modifications influence the activity of nearbygenes in many ways. These modifications can affect thebinding of regulatory proteins to nucleosomes, or theycan influence the three-dimensional structure of the chro-matin, encouraging it to exist in either a more compact(heterochromatin) or more open (euchromatin) confor-mation [3]. These effects can set up a positive feedbackloop allowing nucleosomes that carry a particular mod-ification to recruit either directly enzymes that catalysesimilar modifications, or indirectly, by being simply moreaccessible to modifying enzymes by existing in a moreopen conformation.

∗ amogh@mit.edu

The physical mechanisms that govern these biologicalprocesses behind the initiation, spreading, and inheri-tance of epigenetic states are extremely variegated andrich; they can involve changes in the molecular propertiesassociated with the chemical modifications of DNA andhistone proteins, such as methylation and acetylation,but also the physics that governs the three-dimensionalorganisation of the genome in cell nuclei. In order toachieve stability and heritability of epigenetic states, cellsexploit several different physical principles, ranging fromthe universal behaviour of polymers, general features ofdynamical systems, to the electrostatic and mechanicalproperties related to chemical modifications of DNA andhistones. For an in-depth review of the subject one isencouraged to refer to [6]

II. THE MODEL

An attempt at modelling needs to capture three salientfeatures: a) multistability (since epigenetic marks act asswitches between different functional states). b) spatialpatterns, and coupling between spatial conformation andreorganisation and spread of epigenetic marks. c) Andfinally, heritability [6]. In particular, we look at bistabil-ity, wherein different patterns of epigenetic marks allowone to switch between two states that have a well-definedfunctional characterisation. We consider a system of Nnucleosomes that can be in ns different states. The en-tire system is then consider as a polymeric chain of Nnucleosomes, whose state is defined by the set of vari-ables {s1, s2 . . . sN}, with si denoting the state of i-thnuclesome in the chain, and we impose an overall con-

straint on the number of nuclesomes,ns∑j=1

nj = N , with

ni denoting the number of nucleosomes in state j. Themodel proposed herein, is a modification of work done byDodd et al.[4] For instance, ns = 3 corresponds to a nu-cleosome that can be either, “modified”(M, or si = +1),“unmodified”(U, or si = 0), or “anti-modified” (A, orsi = −1). The analysis is indifferent to the actual iden-tity of modifications involved. Nucleosomes can be ac-tively inter-converted via four symmetrical positive feed-

2

back loops (see FIG. 1. (a)), with one rate parameterα. The term recruitment is used in a general sense toimply that the presence of a modified nucleosome makesit more likely that another nucleosome in the region willcarry the same modification. Nucleosomes can also beinter-converted in an independent recruitment manner(“noisy”). Again, we propose four symmetrical randomnoisy inter-conversions in the model (see FIG. 1.) withthe parameter, 1 − α. We define, F = α

1−α as a ratiowhich is the critical parameter in the system.

We then introduce distance dependent cooperativityby making the rates of recruited inter-conversion reac-tions at each of nucleosome decay as a power-law de-pendence typical of the three-dimensional probability ofcontact, r ∝ Pij(γ) ≡ 1

|r(i)−r(j)|γ , where r(k) defines the

location k-th nucleosome. We then attempt to system-atically look at the effect of γ on the bistability in oursystem.

A. Stochastic Simulation: Implementation

The stochastic simulation of the proposed model is car-ried out by iterating the following process of attemptedmodification of a nucleosome, as proposed by[4]

Step 1A: A random nucleosome n1 is selected for mod-ification from N = 60 total nucleosomes. With proba-bility α, a recruited conversion of n1 is attempted (Step2A), OR with probability 1−α, a noisy conversion of n1is attempted (Step 2B).

Step 2A: Recruited conversion: A second random nu-cleosome n2 is from within the N = 60, d positions awayfrom n1 with probability decays as 1/dγ , and if n2 is ineither the M or the A state, n1 is changed one step to-wards n2. The reaction schemes can be represented asfollows:

n1 n2 Conversion (n1 → n′1)+1 +1 ∅-1 -1 ∅0 0 ∅

+1 -1 1k1−→ 0

0 -1 0k2−→ −1

-1 +1 -1k3−→ 0

0 +1 0k4−→ +1

Step 2B: Nucleosome n1 is changed one step towardeither of the other types, (barring direct si = −1↔ si =+1) interconversions with a probability 1/3. (see FIG. 1.(b)) Namely, when noisy conversion is being attemptedthe transition probabilities are given by pM,U = pU,M =pA,U = pU,A = pU,U = 1/3 and pM,M = pA,A = 2/3.

1/3 probabilities in the previous step ensure that forα = 1−α = 0.5, the numbers of each type of nucleosomeA = U = M , where we use A,U,M to denote the numberof elements equal to s = +1, s = 0, s = −1 .

FIG. 1. (a) Schematic cartoon showing recruited conversion[4](b) Schematic showing transition probabilities during noisyconversion (c) G plotted against model parameter F for γ =0, 1, 1.5, 2, 3

A feedback-to-noise ratio, F = α1−α is defined, which

contains information about the relative activities of thepositive-feedback and noise-conversion processes. Al-though, one notes that for a given F , the actual ra-tios of recruited and noise conversions vary depending onthe numbers of the three nucleosomes types at the time.Since F describes the tendency to order versus disorder,it can be thought of as a temperature.

We also need a metric to quantify the bistability by thebroadness of the probability distributions of the numberof M nucleosomes. We thus introduce a gap measure

G ≡⟨|M−A|M+A

⟩which is the absolute difference between

the number of M nucleosomes and the number of A nu-cleosomes (normalised to the maximum possible differ-ence). G values close to one signify a close-to-maximalspread of the distribution, indicating strong bistability,and in this sense, it behaves like a magnetisation (abso-lute value). Again we have used A,U,M to denote thenumber of elements equal to s = +1, s = 0, s = −1.

B. Results

As can be seen from FIG. 1. (c), the model is stronglybistable for γ = 0, 1, 1.5. Coincidentally, the curves forγ = 0, 1 coincide perfectly; γ = 1.5 coincides with theformer only in limits of high noise (small F ) or high re-cruitment (large F ), with intermediate values showingsome deviations. One notes the critical Fc is c.a. 1, withthe Fc showing a tendency to increase with increasing γ.γ = 2, 3 is indicative of a system that is not bistable atall.

This behaviour is suggestive and leads one to drawparallels with an Ising model governed by the followingHamiltonian,

3

−βH = K∑|i− j|−γsisj (1)

As first conjectured by Kac and Thomson [7], andshown by Thouless[8], Ruelle[9], and Dyson[10], such aHamiltonian admits a phase transition at a finite temper-ature if and only if 1 < γ ≤ 2 (the case γ = 2 is specialand requires some care). Perhaps there exists some cor-respondence, or mapping between the dynamical systemprescribed herein and an Ising-like model with power-lawlong-range interactions and spin variables si = {±1, 0}.

Our model with γ = 0 is essentially a high dimen-sional model since each site is coupled to every other site(during recruited conversion the second site is chosen uni-formly at random from any of the other remaining sites).With increasing γ, the coupling between sites becomesmore and more short-ranged, and eventually, it decaystoo fast to support bistability.

III. ATTEMPTS AT ANALYTICAL SOLUTION

We consider the system is divided into small cells de-noted by the index i. Every cell can be occupied onlyby a restricted number of particles, which are subjectedto special dynamical rules depending on the situation inmind. The starting point is the master equation, writtenas

∂tP (n, t) = L′P (n, t) (2)

where P is the probability that a certain configura-tion characterized by the vector n at time t is realized.The evolution operator L′ has to be specified by the dy-namics of the model. Following the work of Doi[11] andothers[12] the probability distribution P (n, t) can be re-lated to a state vector ||F (t)〉 in a Fock space accord-ing to P (n, t) = 〈n|F (t)|n|F (t)〉 with the basis vectors|n〉 ≡ |n1n2 . . . nI . . .〉. As a consequence, the masterequation is is re-written as an equation in Fock space

∂t |F (t)〉 = L |F (t)〉 (3)

.Some other useful relations [11],

|F (t)〉 =∑ni

P (n, t) |n〉 (4)

where in order to derive these relations one notes thefollowing 〈s| =

∑ni

〈n|, 〈s|F |s|F 〉 = 1 and 〈s| L = 0 Typi-

cally, L is expressed using creating and annihilation op-erators which obey Bose commutation rules. However,to avoid double occupancy, we use the extension to thecase of restricted occupation numbers per lattice site byintroducing Pauli operators which commute at differentpoints and anticommute at the same lattice point. Fol-lowing methods prescribed in the literature [11–16] weattempt to cast our system in this formalism.

The dynamical system prescribed herein can be sum-marised by the following processes, at a lattice site in-dexed by i,

Ai(si = +1)k1−→ Ui(si = 0)

Mi(si = −1)k2−→ Ui(si = 0)

Ui(si = 0)k3−→ Ai(si = +1)

Ui(si = 0)k4−→Mi(si = −1)

Here once again we note that spin-states correspondingto si = +1, si = 0, si = −1 are labbeled as states Ai, Ui,Mi for notational convenience. Now we introduce cre-ation and annihilation operators a†i , u

†im†i and ai, ui,mi.

These commute at different points and anticommute atthe same lattice point.

[bαi , b†αi ]+ = 1 (5)

[bαi , b†βj ]− = 1 α 6= β or i 6= j (6)

where b1i , b2i , b

3i corresponds to ai, ui,mi respectively.

Similar correspondence holds for the creation operators.The lower indices reference lattice points. One also de-fines number operators or indicators as,

Ai = a†iai

Mi = m†imi

Ui = u†iui

(7)

The rates of these process for the model proposed hereinαkq are the rates of recruited conversion given as,

k1 =∑i 6=j

Pij(γ)Mj

k2 =∑i 6=j

Pij(γ)Aj

k3 =∑i 6=j

Pij(γ)Aj

k4 =∑i 6=j

Pij(γ)Mj

4

Aj is an indicator (or number operators) such that theytake the value 1 if spin at sj = +1 (i.e. is in the state Ai)

and zero otherwise. Mj , Uj are defined similarly. Whatthese rates imply is a particle at site i flips only if itinteracts with a particle at site j according to the rulesprescribed and appropriate weight. One then writes thetime evolution equations as follows,

∂t〈Ai〉 = α

∑i 6=j

Pij(γ)〈UiAj〉 −∑i 6=j

Pij(γ)〈AiMj〉

+

1− α3

(〈Ui〉 − 〈Ai〉

)(8)

∂t〈Mi〉 = α

∑i6=j

Pij(γ)〈UiMj〉 −∑i6=j

Pij(γ)〈MiAj〉

+

1− α3

(〈Ui〉 − 〈Mi〉

)(9)

∂t〈Ui〉 = −〈Ui〉

2(1− α)

3+∑i 6=j

Pij(γ)〈Aj〉+∑i6=j

Pij(γ)〈Mj〉

+〈Ai〉

α∑i 6=j

Pij(γ)〈Mj〉+1− α

3

+〈Mi〉

α∑i6=j

Pij(γ)〈Aj〉+1− α

3

(10)

A. Mean Field Results

Under the mean field approximation terms like〈OkPl〉 ≈ 〈Ok〉〈Pl〉 [15].

1. Case γ = 0, Mean field and uniform densityapproximation

For a second if one forgets about the discrete lattice oursystem lives on, which seems reasonable for the case γ =0 since we are interacting uniformly at random with anyof our neighbours. In this case one can simply write downchemical rate equations using the principle of mass action(neglecting all spatial information). Here, x, y denotesthe density of A,M nucleosomes respectively. Notingthe density of U , A and M must add up to 1, we haveonly two independent chemical rate equations to worry

about.

∂tx = (1− x− y)

(xα+

1− α3

)−x(yα+

1− α3

)

∂ty = (1− x− y)

(yα+

1− α3

)−y(xα+

1− α3

)Which can be re-written in terms of our model param-

eter F = α1−α

∂tx = −3Fx(x+ 2y − 1)) + 2x+ y − 1

3(F + 1)≡ p(x, y, F )

∂ty = −3Fy(2x+ y − 1)) + x+ 2y − 1

3(F + 1)≡ q(x, y, F )

(11)

To obtain the stationary state solutions we set p = q =0. This yields the fixed points. The system always ad-mits a fixed point, at x∗ = y∗ = 1/3 regardless of thevalue of F . The fixed point at x∗ = y∗ = 1/3 changesfrom a stable fixed point to an unstable one as we changeF . Above a certain Fc This is accompanied by the ap-pearance of two new fixed points which are now stable.So we perform a linear stability analysis, by diagnolisingthe matrix of derivatives and computing its eigenvaluesabout the fixed point of interest.

M =

(∂xp ∂yp∂xq ∂yq

)x=x∗,y=y∗

One of its eigenvalues is always negative, the secondλ = F−1

3(F+1) changes sign when Fc = 1, which very

roughly agrees with γ = 0 case observed in the stochasticsimulation.

2. Parallels with Ising Model and phase transitions

We have alluded to this previously. From a theoreti-cal perspective, our model seems very similar to an Isingmodel where nucleosomal states represent spins (for ex-ample, M = 1, U = 0, A = +1), recruitment correspondsto coupling between spins (J), and ”noisy” transitionscorrespond to thermal fluctuations (kBT ). We define anobservable which behaves like magnetisation, G = (x−y),and plot this versus F , which captures the ratio betweenthe tendency to organise and tendency to thermalise. Forthe mean field uniform density case, we see a supercriti-cal pitchfork bifurcation at Fc = 1 one would expect for

5

FIG. 2. Steady state solutions to (11) (Nullclines given byp(x, y, F ) = 0 and q(x, y, F, ) = 0 )(a) F = 0.3 (b) F = 1 (c)F = 2 (d) G against F

a 1D Ising chain with long-range contacts. Our modelwith γ = 0 is essentially a high dimensional model sinceeach site is coupled to every other site (during recruitedconversion the second site is chosen uniformly at randomfrom any of the other remaining sites). With increasingγ, the coupling between sites becomes more and moreshort-range, and eventually, it decays too fast to supportbistability. All of this, however, is suggestive and is nota concrete mathematical mapping.

3. Non-uniform case, γ 6= 0

Introducing variables x(i) = 〈Ai〉, y(i) = 〈Mi〉 and

noting 1 − x(i) − y(i) = 〈Ui〉. We only have 2 indepen-dent equations to worry about. The density x(i) maybethought of as the probability of i-th site to be in stateAi for example (since it is an expectation of an indicatorfunction). Moreover, since Pij(γ) = |i−j|−γ decays as wemove further away from i, we can expand x(j) and y(j)about the point i using the discrete analog of a Taylorseries (since our system lives on a discrete lattice).

∑i¬j

Pij(γ)x(j) ≈ x(i)∑i 6=j

|i− j|−γ + ∆1[x](i)∑i 6=j

(i− j)|i− j|−γ

+∆2[x](i)1

2

∑i6=j

(i− j)(i− j − 1)|i− j|−γ + . . .

≈ x(i)2L∑d≥1

d−γ + ∆2[x](i)L∑d≥1

d2−γ + . . .

(12)

Here, ∆n[f ] =n∑k=0

(nk

)(−1)n−kf(x + k) is the finite for-

ward difference. A similar expansion can be carried outfor

∑i 6=j

Pij(γ)y(j) term. The sums over d range up to

some cutoff L where L is the maximum possible separa-tion between two lattice sites. On a periodic lattice of Nsites L ∼ N/2.

For now, let us focus on the first order correction. Thesum adds up to 1 since we are dealing with contact prob-abilities. Focussing on the first order term alone is tanta-mount to a uniform density approximation we have previ-ously discussed for γ = 0. This is tantamount to washingaway all spatial information.

The second order term, +∆2[x](i)∑Ld≥1 d

2−γ+ ≡ ηevaluates to some number ηi, which we can treat as aperturbation to our particle density. This allows us torewrite one pair of density terms appearing in the equa-tion as, x′(j) → x(i) + η(i), y′(j) → y(i) + η(i). In gen-eral, this now depends on local variables x(i) and y(i)but for the sake of argument we assume this correctionis sufficiently small that we are still close to the uniformdensity case. This should not be hard to realise nearsteady state.

∂tx(i) ≈ (1− x(i)− y(i))

((x(i) + η)α+

1− α3

)−x(i)

((y(i) + η)α+

1− α3

)

∂ty(i) ≈ (1− x(i)− y(i))

((y(i) + η)α+

1− α3

)−y(i)

((x(i) + η)α+

1− α3

)Introducing our model paramter F = α

1−α , these dif-ferential equations can be recast into,

∂tx = −3Fηx(2x+ y − 1)) + x(3F (x+ 2y − 1) + 2) + y − 1

3(F + 1)≡ f(x, y, F, η)

∂ty = −3Fx(η + 2y) + 3Fη(2y − 1) + y(3F (y − 1) + 2) + y − 1

3(F + 1)≡ g(x, y, F, η)

(13)

One then finds steady-state solutions, by settingg(x, y, F, η) = f(x, y, F, η) = 0. The system admits afixed points, at x∗ = y∗ = 1/3 regardless of the value ofF, η. The fixed point at x∗ = y∗ = 1/3 changes from astable fixed point to an unstable one as we change F, η.This is accompanied by the appearance of two new fixedpoints which are now stable. So we perform linear sta-bility analysis, by diagnolising the matrix of derivativesand computing its eigenvalues about the fixed point ofinterest.

M =

(∂xf ∂yf∂xg ∂yg

)x=x∗,y=y∗

The relevant eigenvalues changes sign when Fc = 11−3η .

This corresponds to the fixed point at x∗ = y∗ = 1/3

6

changes from a stable to unstable fixed point, and twonew fixed points appear. η = 0 recovers the mean fielduniform density solution we derived for γ = 0. Also, sinceF cannot be negative based on its definition, our pertur-bation takes values between 0 < η < 1

3 . For some finiteacceptable value of η, Fc increases as expected. Since apower law decay is equivalent to reducing the couplingbetween sites, we need stronger feedback-to-noise ratiosto achieve bistability. Qualitatively we have observed thisin our stochastic simulation.

Unfortunately, these calculations failed to shed lighton why γ ≥ 2 fails to show bistability.

4. Asymmetric interactions

Suppose we push this argument it a little bit, namelythat due to power law contact probabilities is similar tointroducing an additive perturbation to our particle den-sity. Also, assume for the sake of argument that marksAi, Mi do not interact symmetrically. Namely, the ratesare modified as follows.

k1 =∑i 6=j

Pij(γ1)Mj

k2 =∑i 6=j

Pij(γ2)Aj

k3 =∑i 6=j

Pij(γ2)Aj

k4 =∑i 6=j

Pij(γ1)Mj

This would correspond to regions having predomi-nantly A or M marks existing in more open or collapsedpolymeric conformations, and thus have different powerlaw contact probabilities between sites. This modifies ourrate equations,

∂tx =

(1− α

3+ α(ηx + x)

)(1− x− y)−

(1− α

3+ α(ηy + y)

)x

∂ty =

(1− α

3+ α(ηy + y)

)(1− x− y)−

(1− α

3+ α(ηxx+)

)y

(14)

One now has three parameters to tune F, ηx, ηy. Therelative ratios of these parameters allows the system to

switch between monostable and bistable regimes. Atthe steady state, the system has, at most, three fixedpoints, and the dynamical system is either monostableor bistable (with an unstable fixed point). For a given Fwhich gives the balance between recruited and noisy con-versions, depending on the asymmetry between ηx andηy the chromatin persists in a particular epigenetic state

FIG. 3. Steady state solutions with asymmetric interactionsgiven F = 2 (a) ηx = 0.3, ηy = 0.16 (b) ηx = 0.16, ηy = 0.16(c) ηx = 0.16, ηy = 0.3

over the other. A similar model with asymmetric re-action rates, which exhibits similar behaviour has beenconsidered previously[17]

IV. DISCUSSION

In this work, we proposed and analysed a simple, mi-croscopic model for the spread of epigenetic marks. Werelated the dynamics to power law contact probabilitiesthat are typical for a polymer chain, and through math-ematical analysis were able to show that system couldexist in either monostable or bistable states by tuningthe model parameters. Through stochastic simulation,we also observed that the system is only bistable whenthe scaling exponent is γ < 2. Unfortunately, we were un-able to rationalise this in our analytical solutions. Someparallels with a 1D Ising Model were observed, and itwould be interesting to make a concrete mathematicalmapping between the two, provided such a mapping isindeed possible.

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