Polymer Models of DNA Background Eukaryotic DNA has ... · Polymer Models of DNA Background Eukaryotic DNA has several levels of organization. DNA is wound around nucleosomes. Nucleosomes

Polymer Models of DNA Background

Eukaryotic DNA has several levels of organization.

DNA is wound aroundnucleosomes.

Nucleosomes are organizedinto 30-nm chromatin fibers.

The fibers form loops ofdifferent sizes.

These looped domains arecondensed to differentdegrees (heterochromatinvs. euchromatin).

Chromosomes may beorganized into territories.

Jay Taylor (ASU) APM 530 - Lecture 12 Fall 2010 1 / 51


Coarse-grained Models of DNA

Some properties of large DNA molecules can be investigated usingcoarse-grained polymer models.

All-atom MD simulations of chromosomes (N ≈ 108 − 109) arenot feasible at present

Atomic-level details are ignored in favor of a bead-and-springdescription.

The beads can be interpreted as nucleotides, nucleosomes, orhigher-order structural elements such as the 30-nm fiber.

These models are best suited for studying statistical properties ofextended polymeric chains that emerge when there are large numbersof weakly-interacting particles.



Polymer Models: Framework

Consider a linear polymer with N monomers, 1 ≤ i ≤ N located atpositions X1, · · · ,XN . The following quantities are of interest:

ri = Xi − Xi−1 is the bond vector from Ai−1 to Ai .

The center of mass is

XC =1

N

n∑i=1

Xi .

The end-to-end vector is

RN = XN − X1 =N∑

i=2

ri .

The contour length is the sum∑N

i=2 ||ri ||.



The gyration tensor is the symmetric 3× 3 matrix T = (Tij ) withentries

Tij =1

N

N∑k=1

(Xk − XC )i (Xk − XC )j .

The radius of gyration is the quantity Rg defined by

R2g =

1

N

n∑i=1

||Xi − XC | |2 =1

N2

N∑i ,j=1

||Xi − Xj | |2.

The eigenvalues of the gyration tensor can be used to characterize theanisotropy of the polymer. In particular, the asphericity is defined as

A3 =

∑i<j (λi − λj )

2

2(λ1 + λ2 + λ3)2.



Polymer models treat the coordinates X1, · · · ,XN as a stochastic process.

Ideal chains are described by models that assume that there are nointeractions between monomers that are far apart on the chain, evenwhen these are close together in space.

Polymer models are often constructed so that the sequenceX1, · · · ,XN is a Markov process or is somehow derived from one.

More realistic models can be constructed by introducing interactionsbetween well-separated monomers, but this usually destroys theMarkov structure.



Freely-jointed Chains

Suppose that the bond vectors are IID random variables that are uniformlydistributed on the sphere of radius b0 and centered at 0. Then

E[ri ] = 0 and E [(ri · rj )] = b20δij

from which it follows that

E [RN ] = 0

E[R2

N

]= E

[R2

N

]=

N∑i ,j=1

E [ri · rj ] = Nb20.

Remark: In this case, the sequence X1, · · · ,XN is just a random walk andso RN is asymptotically Gaussian.



Freely-rotating Chains

Suppose that the sequence of bond vectors r1, · · · , rn is a Markov chainsuch that

the bond lengths are fixed: ||ri || = b0;

the angle between successive bonds is fixed:

ri · ri+1 = b20 cos(θ)

all torsion angles are uniformly distributed on [0, 2π].

In this case, we say that the random polymer X = (X1, · · · ,XN) is afreely-rotating chain.



Notice that

ri+1 = cos(θ)ri + ei

where

||ei || = sin(θ)b0, ei · ri = 0

E [ei |r1, · · · , ri ] = 0.

It follows that

E [ri+1|r1, · · · , ri ] = cos(θ)ri

and consequently

E[ri+k |r1, · · · , ri

]= E

[E [ri+k |ri+k−1] r1, · · · , ri

]= E

[cos(θ)ri+k−1|r1, · · · , ri

]= cos(θ)kri .



Remark: It follows from the previous slide that the process ρ1, ρ2, · · ·defined by

ρk = cos(θ)−krk

is a martingale. More importantly,

E [ri+k · ri ] = E[E[ri+k · ri

]∣∣∣ri

]= E

[E[ri+k

∣∣ri

]· ri

]= E

[cos(θ)kri · ri

]= cos(θ)k b2,

which shows that the bond vector correlations decay geometrically at ratecos(θ).



This last result can be used to calculate the mean squared end-to-enddistance:

ER2N =

N∑i ,j=1

E[ri · rj

]

= Nb2 +N∑

i=1

i−1∑j=1

E[ri · rj

]+

N∑j=i+1

E[ri · rj

]= Nb2 + b2

N∑i=1

i−1∑j=1

cos(θ)i−j +N∑

j=i+1

cos(θ)j−i

≈ Nb2 + 2b2

N∑i=1

∞∑k=1

cos(θ)k

= Nb2 + 2Nb2 cos(θ)

1− cos(θ)= Nb2 1 + cos(θ)

1− cos(θ).



Suppose that X1, · · · ,XN is a polymer model with constant bond length b.

The Kuhn length (effective length) of the model is defined as

bK = limN→∞

1

NE[R2

N

]provided that the limit exists.

Flory’s characteristic ratio is

C∞ =b2

K

b20

= limN→∞

1

N

N∑i ,j=1

cos(θij )

Example: For the freely rotating chain, the Kuhn length and thecharacteristic ratio are

bK = b0

(1 + cos(θ)

1− cos(θ)

)1/2

and C∞ =1 + cos(θ)

1− cos(θ).



The following quantity is also of interest:

The persistence length lp determines the characteristic length scaleover which the polymer retains its directionality:

E[

cos(θ(L)

)]∼ e−L/lp

where θ(L) is the angle between tangents to the polymer at positions0 and L (measured in contour length).

Example: The persistence length of the freely-rotating chain is

lp = − b

ln(cos(θ))



The Worm-like Chain Model

Very stiff polymers can be modeled by taking θ 1 in the freely-rotatingchain model. More precisely, let

θ(ε) = ε1/2θ and b(ε) = b0ε

and consider the corresponding polymer model with total contour lengthL = N(ε)b(ε):

X(ε) =(

X(ε)1 , · · · ,X (ε)

N(ε)

)X

(ε)(l) = X

(ε)bl/εc; 0 ≤ t ≤ L.



If we let ε→ 0, then the interpolated processes X(ε)

tend to acontinuous-parameter stochastic process X = (Xl : 0 ≤ l ≤ L).X is called the Kratky-Porod model (1949) and has the followingproperties:

X has continuous paths;

X has persistence length lp = b/θ2;

X is a Gaussian process with distribution

p(x) ∼ exp

[− 1

2lp

∫ L

0ds

∣∣∣∣∂2x

∂s2

∣∣∣∣2]

Remark: Notice that the density gives more weight to paths with smallercurvature.



The mean square end-to-end distance for the Kratky-Porod model is

E[R2(L)

]= E

[(∫ L

0X (s)ds

)2]

= E[∫ L

0X (s)ds

∫ L

0X (r)dr

]=

∫ L

0

∫ L

0E[X (s)X (r)

]dsdr

= 2

∫ L

0

∫ s

0e−(s−r)/lp drds

= 2l2p

(L

lp+ e−L/lp − 1

).



There are two regimes depending on the ratio L/lp.

Ballistic regime: if L lp, then

E[R2(L)

]= 2l2p

(L

lp+ e−L/lp − 1

)≈ 2l2p

(L

lp+ 1− L

lp+

1

2

L2

l2p− 1

)= L2.

Diffusive regime: if L lp, then

E[R2(L)

]= 2l2p

(L

lp+ e−L/lp − 1

)= 2lpL + O(L−1).

Remark: The second identity shows that bK = 2lp.



Self-Avoiding Walks

In contrast to ideal chain models, which neglect interactions betweenmonomers separated by large contour distances, the properties of aself-avoiding walk are largely determined by such interactions.

An N-step self-avoiding walk on Zd is a sequence of sites(ω(0), ω(1), · · · , ω(N)

)such that

|ω(j)− ω(j − 1)| =∑d

i=1 |ωi (j)− ωi (j − 1)| = 1ω(i) 6= ω(j) for all i 6= j .

Let ΩN be the set of N-step SAWs starting at 0 and let cN = |ΩN |. Arandom N-step SAW is a random vector X = (X (0), · · · ,X (N))which is uniformly distributed on ΩN :

P(X = ω) =1

cN, ω ∈ ΩN .



Properties of Self-Avoiding Walks

There are relatively few rigorously proven results concerning the behaviorof SAW in dimensions 2 ≤ d ≤ 4. When d = 3, it is conjectured that

cN ∼ AµNNγ−1, with γ ≈ 1.162

E[X 2

N

]∼ DN2ν , with ν ≈ 0.59

These are supported by scaling arguments and MCMC.

Notice that the SAW moves away from the origin more rapidly than arandom walk.



The Rouse Model

Ideal chain models have also been used to investigate the dynamicalproperties of dilute polymer solutions. The Rouse model is a Brownianperturbation of a linear system of N coupled harmonic oscillators:

dXn = −k0

ζ(2Xn − Xn−1 − Xn+1) dt +

√2kBT

ζdW

(n)t

where

Xn(t) is the location of the n’th monomer;

W1, · · · ,WN are independent 3-dim Brownian motions;

k0 = 3kBT/b2 is the spring constant for bonded monomers;

ζ = 6πηa is the friction constant (Stoke’s law).



If XC (t) = 1N

∑Ni=1 Xi (t) denotes the center of mass of the molecule at

time t, then XC satisfies the following SDE:

dXC =

√2kBT

ζ

N∑i=1

dW(n)t

d=

√2NkBT

ζdWt ,

where Wt is a 3-dim Brownian motion. This shows that the center of massof a Rouse polymer follows Brownian motion with diffusion coefficient

DC =kBT

Nζ,

i.e., the diffusivity of the polymer is inversely proportional to the polymermass.



In fact, the Rouse model can be solved explicitly. Notice that the SDE canbe written as

dXt = −k

ζAXtdt +

√2kBT

ζdWt ,

where Xt =(X1(t), · · · ,XN(t)

), Wt is a 3N-dim Brownian motion, and

A is a 3N × 3N matrix. Since this is a linear SDE, the solution is given by

Xt = e−kζ

AtX0 +

√2kBT

ζ

∫ t

0e−

kζ

A(t−s)dWs .



When N is large, the eigenvalues of A are approximately equal to

λp ≈π2p2

N2, p ≥ 0,

with λ0 = 0 exact (corresponding to the motion of the center of mass).The time scale for relaxation of the conformation of the polymer isdictated by the smallest positive eigenvalue and is called the Rouse stressrelaxation time:

τR ≈ζ

k0

N2

π2=

ζN2b2

3π2kBT.

Empirically, one has τR ≈ N3/2 at the Θ temperature.

This discrepancy arises from the neglect of hydrodynamics in theRouse model.


Polymer Models of DNA Chromosome Territories

Rosa, A. and Everaers, R. 2008. Structure and Dynamics of InterphaseChromosomes. PLoS Comput. Biol. 4: e1000153.

Motivation:

Interphase chromosomes are organized intoterritories in some species (humans, rats,Drosophila), but not in budding yeast.

The territories of individual chromosomestend to re-appear after cell division, butchange over time and differ between cells.

Figure from Bolger et al. (2005).



Chromosomes are condensed during mitosis and decondensedduring interphase.



Chromosome structure can be studied quantitatively by imagingmarked chromosomes.

Chromosomes can be marked(painted) using FISH orimmunofluorescent DNA-bindingproteins.

Distances between markers can bemeasured using laser scanningconfocal microscopy.

Image from Bystricky et al. PNAS (2004).



Chromatin fibers can be modeled as worm-like chains over shortlength scales.

The mean square end-to-enddistance under the WLC model is

R2(L) = 2l2p

(L

lp+ e−L/lp − 1

).

This study estimated thepersistence length of chromatinto be lp ≈ 197± 62 nm.

Markers separated by N bp areseparated by approximatelyL = N × 10µm Mbp−1.

Figure from Bystricky et al. PNAS (2004).



The fit to the WLC model breaks down at longer length scales.

Black line is the WLC modelprediction of mean square endto end distance.

Figure shows FISH data foryeast (brown), humans (blue),and fly (orange, green).

Discrepancy may be due to:

confinement;slow reptation dynamics.

Axis goes from 10 kb to 100 Mb.

Fig. 1a from Rosa & Everaers (2008).



Reptation and Entanglement of Polymer Chains

In semidilute solutions, polymers can become mutually entangled.

Entanglement only occurs if the chain contour length exceeds acharacteristic value called the entanglement length Le .

Le depends on both the stiffness of the polymer and its contourlength density. For chromatin at typical nuclear densities duringinterphase, Le ≈ 1.2µm.

The contour lengths of the chromosomes considered in this study are

yeast chr. 6: 3 µm; chr. 14: 8 µm;fly chr. 2: 440 µm;human chr. 4: 1.8 mm.



Entangled linear polymers can move by reptation: each polymer isconfined to move within a tubular neighborhood which it traverses bydiffusion of stored lengths along its contour.

The characteristic timescale for disentanglement via reptation isτd ∼ 32(L/Le)3.

yeast: τd ∼ 2× 104 s;

fly: τd ∼ 2× 104 s ∼ 5 yr;

human: τd ∼ 2× 1010 s ∼ 500 yr.

Thus, reptation is unlikely to lead to equilibration of human or Drosophilachromosomes during interphase.



Alternatively, topoisomerases may accelerate equilibration of decondensingchromosomes.

Type II toposiomerases catalyze strand passage of one DNA duplexthrough another.

The maximal effect of topo II molecules on chromosomedecondensation can be estimated by considering a solution ofphantom (non-interacting) chains. In this case the equilibrationtime is approximately τR ∼ 32(L/Le)2.

yeast: τd ∼ 2× 103 s;

fly: τd ∼ 2× 106 s ∼ 10 days;

human: τd ∼ 2× 1010 s ∼ 250 days.

Fig. from Fass et al., Nat. Struct. Biol. (1999).



Motivated by the preceding calculations, Rosa & Everaers propose that:

Interphase chromosomes of sufficient length are not equilibrated individing cells. In particular, they remain largely unentangled.

Chromosome territories are a consequence of the segregation of theunentangled chromosomes due to topological barriers.

Territories can form in the absence of protein crosslinking withinchromosomes or between chromosomes and other structural elementswithin the nucleus (i.e., the nuclear cytoskeleton).

They investigate these proposals by carrying out MD simulations of groupsof decondensing chromosomes.



Kremer-Grest Model

Their model represents each chromosome as a chain of interacting beads,each of diameter σ = 30 nm (3000 bp). The interaction energy comprisesthree effects:

A shifted Leonard-Jones potential:

ULJ = 4ε[(σ/rij )

12 − (σ/rij )6 + 1/4

]if rij < σ21/6.

A connectivity potential between nearest neighbors:

UFENE = −0.5kR20 ln

[1− (rij/R0)2

]if rij < R0.

A bending stiffness potential between consecutive bonds:

Ustiff = βk0 (1− cos(θ)) .



Simulations were performed in a periodic rectangular box, with eithera constant isotropic pressure (human, Drosophila) or a constantvolume, leading to a chromatin volume fraction of 10 %.

The initial conformations of the chromosomes were either linear orring-shaped helices.

Human and yeast chromosomes were initially randomly oriented,while the fly chromosomes were aligned along a common axis.

Simulations were carried out for periods roughly comparable to 3 daysof real time.



Fig. 3 in Rosa & Everaers.

The simulation time is sufficient to mix the yeast chromosomes, butthe fly and human chromosomes are confined to territories.

The chromosome territories are Rabl-like in flies, but ellipsoidal inhumans, reflecting the initial configurations.



Human chromosomes form stable territories in the Kremer-Grestmodel.

Figure shows trajectories of thecenters of mass of the foursimulated human chromosome 4’s.

Motion is confined to regions ofradius ∼ 0.1 µm.

In contrast, the dimensions of thedecondensed chromosomes are≈ 5× 2× 1 µm.

Fig. 6



Simulated chromosome territories are stabilized by a chain crossingbarrier.

Fig. 4c



Individual sites are dynamic within chromosome territories.

Figure shows the growth of theMSD of six internal beads overtime.

Individual sites explore regions≈ 1 µm over 5 hour periods.

Purple dots show experimentaldata for a marked gene in yeast.

Reptation dynamics appear tobe observed at long time scalesfor all three organisms. Fig. 2



Figures show R2(L) averaged overthree time windows of growing size(red, magenta, cyan).

Gray curves show the initialend-to-end distances. Notice thatthe initial state rapidly unfolds(within 40 min).

Black curve is the WLC prediction.

Drosophila chromosomes deviatefrom the WLC curve even after 3days at contour lengths > 0.2Mb.

Simulated yeast chromosomesequilibrate within 7 hours.

Fig. 1b, c



Simulated human chromosomes alsodiffer from the WLC predictions forlong contour lengths even after 3days.

However, the simulated end-to-enddistances also differ from data evenover modest contour lengths.

At intermediate contour lengths,the data is closer to the WLCpredictions than to the simulations.

At longer contour lengths, the dataalso falls below the simulatedend-to-end lengths.

The authors suggest that thediscrepancies have two differentcauses.

Fig. 1d

Axis goes from 10 kb to 100 Mb.



The blue circles come from datafrom the end of the p-arm.

Comparison with distances measuredfrom the ends of the simulatedchromosome is much better overintermediate contour lengths(Fig. 1e).

Equilibration is more rapid nearchromosome ends.

Comparison with simulations startedfrom an initial ring configurationshow better agreement at longercontour lengths (Fig. 1f).

Such a configuration may resultfrom bending at the centromere.

Fig. 1e, f


Polymer Models of DNA Loop Polymers

Bohn, M. et al. 2007. Random loop model for long polymers. Phys. Rev.E 76: 051805.

Motivation:

Chromosomes are known to formloops stabilized by protein-proteincontacts.

Bohn et al. propose that loopformation could account for theleveling off of end-to-end distanceover long contour distances.

Figure from Ong and Corces, J. Biol. 8: 73 (2009).



Gaussian Chains with Non-Local Harmonic Interactions

Consider a polymer with N + 1 monomers at positions X0 = 0 andX1, · · · ,XN . The authors first assume that there are non-randomharmonic interactions between all monomer pairs:

U =κ

2

N∑j=1

||Xj − Xj−1| |2 +1

2

N∑i<j−1

κij ||Xi − Xj | |2

where

κ is the backbone spring constant;

κij = κji are spring constants for loop attachment points.



Since the potential is additive over spatial dimensions, the Boltzmanndistribution factors into the product of three independent distributionsover each dimension. The distribution over the x-coordinates is

Px (X1, · · · ,XN) = Cx exp

(−1

2U(X1, · · · ,XN)

)= Cx exp

(−1

2XT KX

)where X = (X1, · · · ,XN)T and

K =

∑N

j=0 κ1j −κ12 · · · −κ1N

−κ21∑N

j=0 κ2j · · · −κ2N...

......

−κN1 −κN2 · · ·∑N

j=0 κNj

.



Furthermore, the marginal distribution of the distance between beads iand j is just the Maxwell distribution with density

P (rij ) = C · r2ij exp

(−1

2

1

σii − 2σij + σjjr2ij

)=

4√π

Γ3/2r2ij exp

(−Γr2

ij

)where Σ = (σij ) = K−1 and Γ = 1

21

σii−2σij+σjj. It then follows that the

mean squared end-to-end distance is:

R2ij =

∫R3

r2ij P(rij )dr = 3 (σii − 2σij + σjj ) .



A Random Loop Model

Since the locations of the loops in real chromosomes are largely unknown,the authors consider the following randomized version of the above model.

The backbone spring constants are held fixed at κ;

The non-local interaction constants are independent scaled BernoulliRVs:

κij =

κ with probability P

if l1 ≤ |i − j | ≤ l20 with probability 1− P.

The constants l1 < l2 determine the range of possible loop sizes.



Quenched vs. Annealed Averages

Two kinds of averaging can be done within the random loop model,depending on the time scales of thermal fluctuations and loop formation.

The quenched average is appropriate if thermal equilibrium of thechromosome is reached before the number and locations of the loopshave changed significantly.

In this case, we average first over the thermal noise conditional on aset of loops and then average over the distribution of loops:

〈R2ij 〉quenched ≡ 〈〈R2

ij 〉thermal〉loops

= 3〈σii − 2σij + σjj〉loops.



Alternatively, the annealed average is more appropriate if the numberand locations of the chromosome loops change repeatedly beforethermal equilibrium can be reached.

In this case, the partition function depends on both the conformationof the molecule and the locations of the loops:

〈Z〉ann ≡∑κij

Z (Xk, κij) P (κij)

=

∫dX1 · · · dXNe−Ubb

∏i<j−1

[P(

e−1/2κ||Xi−Xj ||2 − 1)

+ 1].

In this paper, the authors focus on the quenched average, which theyevaluate through Monte Carlo simulations. Whether the quenched orthe annealed average is more biologically appropriate is unclear.



Figure 2 shows quenched mean squarespatial distance plotted against contourlength for chains of N monomers.

In each case, P was chosen so that theexpected number of loops is 100.

When only short loops are allowed, thenR2(N) ∼ N (Fig. 2a).

In contrast, if only long loops areallowed, then R2(N) is a unimodalfunction of N (Fig. 2b).

When loops of all sizes are allowed,R2(N) is constant over most contourlengths (Fig. 2c).



Figure 3 shows how the predictedmean squared end-to-end distancecompares with that measured forhuman chromosomes 1 and 11.

Here, each bead corresponds to∼ 150 kb.

Reasonable match between theoryand data is obtained whenP ∼ 5× 10−5.

The kurtosis of the end-to-enddistance is compared betweenmodel and data in Figure 4:

c4 =〈R4〉〈R2〉2

.



Summary

Random loop models can explain the leveling off of end-to-enddistances with contour length if loops can exist on all length scales.

This remains true if the RLM is modified to account for excludedvolume interactions and regional homogeneity in loop formation.(See Mateos-Langerak et al. 2009, PNAS 106: 3812 for results.)

However, the RLM does assume that chromosomes are at thermalequilibrium during interphase, a claim that Rosa & Everaers’ 2008study contradicts.

The two papers lead to the following question: Does looping promotethe formation of chromosome territories, or do chromosome territoriespromote the formation of loops?


Polymer Models of DNA References

References

Bolzer, A. et al. (2005) Three-Dimensional Maps of All Chromosomesin Human Male Fibroblast Nuclei and Prometaphase Rosettes. PLoSBiol. 3: e157.

Bystricky, K. et al. (2004) Long-range compaction and flexibility ofinterphase chromatin in budding yeast analysed by high-resolutionimaging techniques. PNAS 101: 16495-16500.

Doi, M. and Edwards, S.F. (1986). The Theory of Polymer Dynamics.Oxford University Press.

Mateos-Langerak et al. (2009) Spatially confined folding of chromatinin the interphase nucleus. PNAS 106: 3812-3817.

Ong, C.-T. and Corces, V. G. (2009) Insulators as mediators of intra-and inter-chromosomal interactions: a common evolutionary theme.J. Biol. 8: 73.


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Polymer Models of DNA Background Eukaryotic DNA has ... · Polymer Models of DNA Background Eukaryotic DNA has several levels of organization. DNA is wound around nucleosomes. Nucleosomes