Statistical Mechanics Model for the Dynamics of Collective Epigenetic Histone Modification

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by  Hang Zhang

Statistical Mechanics Model for the Dynamics of Collective Epigenetic Histone
Modification
Hang Zhang1, Xiao-Jun Tian1, Abhishek Mukhopadhyay2, K.S. Kim3,Jianhua Xing1,2,4∗
1Department of Biological Sciences, Virginia Tech, Blacksburg, Virginia, 24061-0406, USA
2Department of Physics, Virginia Tech, Blacksburg, Virginia, 24061-0406, USA
3Lawrence Livermore National Laboratory and University of California, Livermore, California, 94550, USA
4Beijing Computational Science Research Center, Beijing 100084, China
Epigenetic histone modifications play an important role in the maintenance of different cell
phenotypes. The exact molecular mechanism for inheritance of the modification patterns over cell
generations remains elusive. We construct a Potts-type model based on experimentally observed
4
nearest-neighbor enzyme lateral interactions and nucleosome covalent modification state biased
1
enzyme recruitment. The model can lead to effective nonlocal interactions among nucleosomes
0
suggested in previous theoretical studies, and epigenetic memory is robustly inheritable against
2
stochastic cellular processes.

b
e PACSnumbers: 82.39.Rt,87.17.Aa,87.16.Yc,87.16.A
F

7
Nucleosomes are basic organizational units of chro-
2 J
matin in eukaryotic cells. A typical nucleosome has
] approximately 147 base pairs wrapped around a histone
N
octamerandareinterconnectedbylinkerDNAofvarying
G
length (see Fig. 1) [1, 2]. Covalent modifications of Nucleosome in nucleation region
. severalaminoacidresiduesonthehistonecorecanleadto
io either active or repressive gene expression activities [3]. Nucleosome eAnctziyvme em caormk mpleetxhylation Renezpyrmesesicvoemmpalrekxmethylation
b Active mark
A dynamic equilibrium in the nucleosome modification Active mark demethylation Repressive mark demethylation
- Repressive mark enzyme complex enzyme complex
q state is attained due to a ‘tug-of-war’ between the
[ associated covalent mark addition and removal enzymes
FIG. 1: (Color online) Schematic illustration of the model.
4 [4]. The system may show a bistable behavior due to ǫ denotes enzyme binding energy, J denotes enzyme lateral
v coexistence of repressive and active epigenetic states for interaction energy
2 different copies of a gene within the same cell [5].
2
Experiments suggest that at least some of the
4
nucleosome covalent patterns can be transmitted over enzyme lateral interactions lead to sustained repression
1
. a number of generations [1]. Although the actual or activation of genes, and cancer cells show mutations
1
mechanism for this epigenetic memory is unclear, a linked with such lateral interactions [14, 15].
0
4 simple rule-based model by Dodd et al. [5] shows that In this work we construct a theoretical model aiming
1 robust bistability requires cooperative effects beyond to bridge the gap between detailed molecular events
: neighboring nucleosomes, which they suggest is due to occurringatthesub-secondtimescale,andthelong-time
v
i compact chromatin structures. Subsequent theoretical scaleepigeneticchangedynamicsthatistypicallyindays
X
studies on yeast chromatin silencing [6], mouse stem cell orlonger. Tobespecificwefocusonlysine4(active)and
r differentiation [7], and plant flowering regulation [8] also lysine9(repressive)methylationonhistoneH3,although
a
concludethatthisnonlocalcooperativityisnecessaryfor we expect the mechanism discussed here can be general.
generating stable epigenetic memory. Consideragene withN nucleosomes,as showninFig.
In recent years molecular details on nucleosome 1. Each nucleosome can be in one of the three covalent
covalent modification dynamics have been extensively states: repressively modified (s = −1), unmodified (0),
studied. Measurements show that the typical residence and actively modified (1). Here for simplicity we only
time of a modification enzyme on chromatin is within consider one covalent modification site on each histone
sub-seconds to a few minutes [4]. Experimental octamer, and do not distinguish multiple modification
observations also suggest that a modified nucleosome (e.g., mono-,di-,andtri-methylation)states. Ourmodel
may have higher binding affinity for the corresponding is flexible enough to admit straightforward extensions
enzymes [3, 9–11]. Another interesting observation is with increasing complexity. Each nucleosome can be
that a nucleosome bound modification enzyme complex in one of the enzyme binding states with corresponding
laterally interacts with another bound to neighboring binding energies: empty (σ = 1,ǫ = ǫ = 0),
1s
nucleosomes [10, 12, 13]. Although the functional repressive modification enzyme bound (σ = 2,ǫ =
consequences of these interactions on epigenetic dynam- ǫ ), repressive modification removal enzyme bound
2s
ics are unclear, recent work suggests that increased (σ = 3,ǫ = ǫ ), active modification enzyme bound
3s
2
(σ = 4,ǫ = ǫ ), active modification removal enzyme
4s
TABLE I:Model parameters
bound (σ = 5,ǫ = ǫ ). To account for the s-
5s
dependence of binding affinity, we assume that the methylation free energy of binding within -1
binding energies for the adding/removing enzymes to a nucleation region ǫσ0,σ=2,4
nucleosome bearing the corresponding (or antagonizing) demethylation enzymefree energy of bindingwithin 1
mark are ∆ǫ lower (or higher) than those binding to an nucleation region ǫσ0,σ=3,5
methylation enzymefree energy of bindingoutside 3
unmodified nucleosome. Furthermore if two neighboring
nucleation region ǫσ0,σ=2,4
(i-th and (i+1)-th) nucleosomes are both bound, the demethylation enzymefree energy of bindingoutside 3
binding enzymes interact laterally with energy Jσiσi+1, nucleation region ǫσ0,σ=3,5
otherwise J = J = 0. The above s state related free energy of bindingbias ∆ǫ 2
σi=1,σi+1 σi,σi+1=1
background enzyme-nucleosome binding has no DNA lateral interaction between two identical enzyme
3
sequence specificity, and the corresponding binding molecules Jαα
lateral interaction between two different enzyme
energiesestimatedfromexperimentaldataareweak. Itis -2
molecules J ,α6=β
αβ
suggested that transcription factors and other molecules
enzymatic interaction rate vα→β 1.5/hour
recruitthe enzymesto bindonspecific nucleosomeswith histone exchange rate d 0.6/hour
significantlystrongerbindingaffinity[17]. DNAsequence cell cycle time 20 hours
elements,e.g. CpGislands,havealsobeenshowntohave
higher but less sequence-specific enzyme binding affinity
[18–22]. Therefore,wedenoteaspecialnucleationregion
exp(−1(ǫ +ǫ )). Fornotationalsimplicity,weomit
of nucleosomes (for H3K4me3 and H3K9me3 centered 2 αsN βs1
the s-dependence of the transfer matrices. Then the
around the transcription start site (TSS)) with lower
probability of finding site i in state σ is P ({s}) =
binding energies. We will index the middle nucleosome i σi
Tr[T ···T G T ···T ]/Z , where (G ) =
within this region as 0, those on its left negative, and 1 i−2 σi i+1 N s σi αβ
(T ) (T ) except (G ) =(T ) (T ) .
thoseonits rightpositive. Moredetailsofthe modelcan i−1 α,σi i σi,β σ1 αβ N α,σ1 1 σ1,β
befoundintheonlinesupportingtext. Specificallythere Our overall simulation procedure is as follows:
is a nucleosome-free region near the TSS [23], and some at each step, we first calculate {Pσi}({s}) of each
DNA distortion may be needed. nucleosome, then update the s state using the
The overall model has the structure of a coupled Gillespie algorithm with the possible events includ-
two-layer Potts model. Modification of the s-state ing enzymatic reaction on nucleosome i with rate
requires the corresponding enzyme bound, and the ki = δsi,0(v0→−1Pσi=2({s}) + v0→1Pσi=4({s})) +
enzyme binding energy is s-dependent. The s-state of a δsi,−1v−1→0Pσi=3({s}) + δsi,1v1→0Pσi=5({s}), where δ
nucleosome can also be changed to 0 due to stochastic is the Kronecker delta function, and histone exchange
exchange with unmarked histones in solution with a (si → 0) with rate d; at every cell cycle (20h), the s-
rate of about once per 100 minutes [24], and random state of eachnucleosomehas 50%probability to be reset
replacement with 50% probability by an unmarked to 0.
histone every cell division [25], which is around 20 We select the model parameters roughly representing
hours [26] for mammalian cells. The total number the gene Oct4, one of the core genes maintaining cell
of states is 15N, which is computationally prohibitive pluripotent stemness [27], and the one monitored by
for direct dynamic simulations. Given the clear time Hathaway et al. [28]. Figure 2(a) shows a typical
scale separation between the enzyme binding/unbinding simulated trajectory. Despite large fluctuations, a
events and other processes, we treat the former as an block of nucleosomes centered around the nucleation
equilibrium process, and simulate others stochastically, regionshowcollectivedynamics,withoccasionalswitches
as described below. betweenrepressive(lightgrayinprint,greenonline)and
The interactions between covalent modification en- active (dark gray in print, red online) states. Closer
zymes and a collection of nucleosomes at given s-states examination of a switching event (Fig. 2(b)) shows
can be described by the following Hamiltonian that a cluster of nucleosomes with the same type of
mark initially form around the nucleation region, then
N N−1
propagatesteadilyoutwards. Indeed,theepigeneticstate
Hs =Xǫσisi − X Jσiσi+1. (1) can be reversed by artificially changing the nucleosome
i=1 i=1
marks within the nucleation region rather than outside
Throughout this work energy is given in units of this region, consistent with the experiment done by
k T, where k is the Boltzmann constant, and T is Hathaway et al. [28].
B B
the temperature. The partition function is, Zs = The (N=40) simulation is generated by the following
P{σ}exp(−Hs) = Tr[T1···TN], where the transfer steps: the values of free energy of binding are
matrix T has elements (T ) = exp(−1(ǫ + estimated from measured enzyme bound fraction and
i i αβ 2 αsi
ǫ ) + J ) for i = 1,··· ,N − 1, and (T ) = concentrations [4], values of J are chosen to reproduce
βsi+1 αβ N αβ
3
(a) (c) fluctuations is due to randomreplacements during every
−1
Simulation time / Cell Cycle102468(55b0000045) 0 Repressive Mark Percentage00000.....12345 crbrwitsheeeeipllatfalnohrnxaercaetsoeyustxniicrtvopelaheenelc.eremtitlimolnmaAeecrecxnkyfottsctisnaelerrljcae)eebmlclieatolsaxeuucnaerdthsseeiu6cvqertichusehseismoialocalutnkerr.lnydsytt,ihfvowtioisoFrsnhiimofiagacHnuhass,rteiteniLestrtaahaeaSdleil7yacnsexoifnlsasrltcgshataoicotoan[tw3ensi1soisvts]n(.aatlteblehunosIleeesttf
56
1 0 epigenetic state against cell division perturbation. We
−7 0 8 16 24 32 −7 0 8 16 24 32
Nucleosome position Nucleosome position
also define an average dwelling time at an epigenetic
state as the average time the system stays in the
FIG. 2: (Color online) Simulation results using model
epigenetic state with one mark dominating before it
parameters corresponding to Oct4 (Table 1). (a): Heat
map representation of a typical trajectory. (b): Zoom-in switchestothestatewithanothermarkdominating;this
of the heat map in (a) showing epigenetic state transition. is calculated using the algorithm adapted from ref.[32].
(c): Probability of observing repressive marks at different Figure S6 shows that it increases with the cell cycle
nucleosome sites. The nucleation region is at nucleosomes time. That is, shorter cell cycle makes the epigenetic
-1, 0, and 1.
state less stable. This is consistent with experimental
findings that increasing cell division rate accelerates the
epigenetic reprogramming from differentiated cells to
thebell-likeshapedhistonemethylationpatterncentered induced pluripotent stem cells [33].
around the nucleation region with a half-height width of To further analyze the dependence of the model
about 10 nucleosomes (Fig. 2(c)) [28], and k is chosen bistable behavior onparameters,we explore the bistable
to reproduce the observed ∼4 days of transition time region in the ∆ǫ-J plane. Figure 4(a) shows that a
from an active to a silent gene state [28]. This set of finite value of J is necessary for generating bimodal
model parameters, as summarized in Table 1, serves as distributions of the fraction of histones with repressive
the starting point for analyzing the model dependence marks. Belowacriticalvalue∼2,thesystemonlyshows
on parameters. For simplicity, in this work we assume unimodal distribution even with very large ∆ǫ values.
that the boundaries of the nucleosome region under The required value of J also increases sharply upon
study are occupied by insulating elements [29, 30] which decreasing∆ǫ. With∆ǫ=0,thesystemcannotgenerate
could prevent spreading of the epigenetic modifications a bimodal distribution with an arbitrarily large value of
beyond the region. The qualitative results of this work J. While one should be cautious of results with large
are not affected by using alternative periodic boundary (possibly unphysical) values of J and ∆ǫ since the time-
conditions. scale separation argument then becomes questionable,
To understand the molecular mechanism underlying the results in Fig. 4(a) suggest that both J and ∆ǫ are
the dynamics, we hypothesize that the enzyme lateral necessary to generate bimodal distributions.
interactions are essential for collective nucleosome The above results demonstrate that the model,
modification. Indeed, Fig. 3 shows that with Jαα = 0 which is based on only nearest-neighbor enzyme lateral
the percentage of nucleosomes with repressive marks interactions without direct correlation of s-state update
fluctuatesbutshowsaunimodaldistribution. Acellwith dynamics between two nucleosomes, can generate the
this dynamical property cannot maintain a memory of observed inheritable epigenetic bistability. Does it
its epigenetic state over generations. With Jαα = 2.5, contradict with the nonlocal interaction requirement of
however, one can identify clearly a two-state dynamics previous studies [5–8]? To gain mechanistic under-
from the trajectories, which is further evidenced by the standing, we present a statistical analysis on possible
bimodal distribution of the fraction of time the system correlations between different nucleosomes. First we
stays at a collective epigenetic state. A cell gaining define the correlation function for the σ states of two
certainepigeneticpatterncanpropagatetheinformation nucleosomes i and j with a given set of s configurations,
to its progenies for several generations before losing it. i.e., thecorrelationbetweennucleosomeiinstateσ =α
i
With an even larger Jαα = 3.5, a cell can stay in one and nucleosome j in σj =β,
epigenetic state over many cell cycles, and the states
with high and low fraction of repressive marks are well C (σ ,σ ;{s})= hδσi,αδσj,βis−hδσi,αishδσj,βis
separated. Inthesecalculationstheparametersforactive α,β i j hδ δ i −hδ i hδ i
σi,α σi,α s σi,α s σi,α s
and repressive modifications are the same, therefore the
P ({s})−P ({s})P ({s})
behavior of active marks is similar but anti-correlated = σi=α,σj=β σi=α σj=β .
P ({s})(1−P ({s}))
with that of the repressive marks. σi=α σi=α
Close examination of the trajectory in Fig. 3(c) For j−i=1,
reveals that a major contribution to nucleosome mark Pσi,σj({s}) = Tr[T1···Ti−2G3σi=α,σi+1=βTi+2···TN]/Zs,
4
1 0.01.52 (a) 4 (nb)0.6 J=3.5
(((abc)))Histone state percentage000...5550011 obability of Repressive Mark00..000000....55114600 Jαα123...555230 1 Δ2ε 3 4 σ orrelatios correlation00000N.....2462400u0 cleosome1 0position (2b0egin frJJo==m213 ..0T50S S)
Pr 0.2
0 0
0 20 40 60 80 100 0 10 20 30 40 FIG. 4: (Color online) Mechanism of bistability. (a): Phase
Modification Time/cell cycle Number of Repressive Nucleosome
diagramonthe∆ǫ-J plane. Allotherparameterstakevalues
FIG. 3: Typical trajectories of the fraction of nucleosomes in Table 1. (b): Correlation functions C¯1,1(σ0,σL) (upper)
withrepressivemarks(left)andthecorrespondingprobability and C1,1(s0,sL) (lower).
distribution of observing given number of nucleosomes with
repressive marks (right). All simulations are performed with
∆ǫ = 2, but different Jαα values, (a): Jαα = 0, (b): Jαα = alsoshowsimilarcorrelations. Thisnonlocalnucleosome-
2.5, C: Jαα = 3.5. Other parameters values are from Table nucleosome s state correlations are mediated through
1. The dwelling time distribution is obtained by averaging enzyme binding.
over 100 trajectories, each started with a randomly selected
In summary, our model analysis shows that the
initial histone modification configuration, simulated for 103
experimentally observed nearest-neighbor interaction
Gillespie steps, then followed by another 2× 103 Gillespie
and modification state biased enzyme recruitment of in-
steps for sampling.
dividualnucleosomesworksynergeticallyandsufficiently
to result in collective active and repressive epigenetic
and for j−i>1, states. Unlike a simple 1-Dmodel with nearestneighbor
Pσi,σj({s})=Tr[T1···Ti−2GσiTi+1···GσjTj+1···TN]/Zs, interactions that shows no phase transition, the coupled
where h·i means ensemble average with a given set two-layer model here gives rise to bistability due to
{s}
of s configurations, δ is the Kronecker delta function, positive feedback of nucleosome mark state to enzyme
G3 is obtained by replacing all the elements recruitment. The model supports the proposal of Dodd
inσTi=α,σTi+1T=β containingno termrelatedto σ ,σ by et al. [5] that nonlocal ’effective interactions’ among
i−1 i i+1 i i+1
zero, i.e., only keeping the (σ )-th column of T , the nucleosomes affect the covalent modification rates (as
i i−1
(σi)-th row and (σj)-th column of Ti and (σj)-th row of evidenced by the dependence of Pσi on s states of all
T nonzero. nucleosomes) and are necessary for generating robust
i+1
Averaging over N consecutive samples of Gillespie bistable epigenetic states. In the supporting text we
s
simulations with the waiting time at each step (the time compare the two models. Our analysis demonstrates
it takes for the next Gillespie move) τ , and the total a possible molecular mechanism of generating these
l
simulation time t = Ns τ , we obtain the correlation effective interactions, and epigenetic memory, mediated
Pl=1 l
functions averagedover the s states, through nearest-neighbor enzyme lateral interactions.
Let’s focus on a specific unmarkednucleosome. Without
Ns interactions from other nucleosomes, with a set of
C¯α,β(σi,σj) = XCα,β(σi,σj;{s})τl/t. (2) symmetrically chosen parameters the nucleosome has
l=1 equal probability of being actively or repressively
Similarly we define the s state correlationfunctions as modified. The term ∆ǫ determines what types of
enzymes are likely to bind on other nucleosomes within
C (s ,s ) = hδsi,aδsj,bi−hδsi,aihδsj,bi, (3) the correlation region. The enzyme lateral interactions
a,b i j hδ δ i−hδ ihδ i (J) result in the stabilization of enzyme binding on this
si,a si,a si,a si,a
tagged site by the binding events at other nucleosomes
where with N samples, hδ i = Ns δ τ /t, with withinthecorrelationregion. Thisallowsthenucleosome
s si,a Pl=1 si(l),a l
corresponding definitions for other terms. to “read” the majority epigenetic mark type of these
The nucleosome enzyme binding states show correla- nucleosomes, bias its recruitment of the corresponding
tionsfromthesmallestlengthscale,nearestneighborsfor enzyme and “write” on itself accordingly. As shown
small J values, to the larger length scales spanning the in the online supporting text, this mechanism is robust
whole region for sufficiently large J values (Fig. 4(b)). with different choices of model parameter values, with
It is not surprising for a Potts-type model with nearest- the essential requirement that the time scale for mark
neighbor interactions to give rise to beyond-nearest- restoration must be faster than that of perturbations,
neighbor correlations of σ states. Because the σ and s mainly from mark removal reactions, cell division and
state dynamics are coupled, the s states of nucleosomes histone exchange, all of which may vary significantly
5
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