Eukaryotic DNA has several levels of organization.

Transcription

1 Background Eukaryotic DNA has several levels of organization. DNA is wound around nucleosomes. Nucleosomes are organized into 30-nm chromatin fibers. The fibers form loops of different sizes. These looped domains are condensed to different degrees (heterochromatin vs. euchromatin). Chromosomes may be organized into territories. Jay Taylor (ASU) APM Lecture 12 Fall / 51

2 Background Coarse-grained Models of DNA Some properties of large DNA molecules can be investigated using coarse-grained polymer models. All-atom MD simulations of chromosomes (N ) are not feasible at present Atomic-level details are ignored in favor of a bead-and-spring description. The beads can be interpreted as nucleotides, nucleosomes, or higher-order structural elements such as the 30-nm fiber. These models are best suited for studying statistical properties of extended polymeric chains that emerge when there are large numbers of weakly-interacting particles. Jay Taylor (ASU) APM Lecture 12 Fall / 51

3 Background Polymer Models: Framework Consider a linear polymer with N monomers, 1 i N located at positions X 1,, X N. The following quantities are of interest: r i = X i X i 1 is the bond vector from A i 1 to A i. The center of mass is X C = 1 N n X i. i=1 The end-to-end vector is R N = X N X 1 = N r i. i=2 The contour length is the sum N i=2 r i. Jay Taylor (ASU) APM Lecture 12 Fall / 51

4 Background The gyration tensor is the symmetric 3 3 matrix T = (T ij ) with entries T ij = 1 N (X k X C ) N i (X k X C ) j. k=1 The radius of gyration is the quantity R g defined by R 2 g = 1 N n i=1 X i X C 2 = 1 N 2 N X i X j 2. i,j=1 The eigenvalues of the gyration tensor can be used to characterize the anisotropy of the polymer. In particular, the asphericity is defined as A 3 = i<j (λ i λ j ) 2 2(λ 1 + λ 2 + λ 3 ) 2. Jay Taylor (ASU) APM Lecture 12 Fall / 51

5 Background Polymer models treat the coordinates X 1,, X N as a stochastic process. Ideal chains are described by models that assume that there are no interactions between monomers that are far apart on the chain, even when these are close together in space. Polymer models are often constructed so that the sequence X 1,, X N is a Markov process or is somehow derived from one. More realistic models can be constructed by introducing interactions between well-separated monomers, but this usually destroys the Markov structure. Jay Taylor (ASU) APM Lecture 12 Fall / 51

6 Background Freely-jointed Chains Suppose that the bond vectors are IID random variables that are uniformly distributed on the sphere of radius b 0 and centered at 0. Then from which it follows that E[r i ] = 0 and E [(r i r j )] = b 2 0δ ij E [R N ] = 0 E [ R 2 ] N = E [ R 2 ] N N = E [r i r j ] = Nb0. 2 i,j=1 Remark: In this case, the sequence X 1,, X N is just a random walk and so R N is asymptotically Gaussian. Jay Taylor (ASU) APM Lecture 12 Fall / 51

7 Background Freely-rotating Chains Suppose that the sequence of bond vectors r 1,, r n is a Markov chain such that the bond lengths are fixed: r i = b 0 ; the angle between successive bonds is fixed: r i r i+1 = b 2 0 cos(θ) all torsion angles are uniformly distributed on [0, 2π]. In this case, we say that the random polymer X = (X 1,, X N ) is a freely-rotating chain. Jay Taylor (ASU) APM Lecture 12 Fall / 51

8 Notice that Polymer Models of DNA Background r i+1 = cos(θ)r i + e i where e i = sin(θ)b 0, e i r i = 0 E [e i r 1,, r i ] = 0. It follows that E [r i+1 r 1,, r i ] = cos(θ)r i and consequently E [ r i+k r 1,, r i ] ] = E [E [r i+k r i+k 1 ] r 1,, r i = E [ ] cos(θ)r i+k 1 r 1,, r i = cos(θ) k r i. Jay Taylor (ASU) APM Lecture 12 Fall / 51

9 Background Remark: It follows from the previous slide that the process ρ 1, ρ 2, defined by is a martingale. More importantly, ρ k = cos(θ) k r k E [r i+k r i ] = E [E [ ] ] r i+k r i ri = E [E [ ] ] r ri i+k ri = E [ cos(θ) k r i r i ] = cos(θ) k b 2, which shows that the bond vector correlations decay geometrically at rate cos(θ). Jay Taylor (ASU) APM Lecture 12 Fall / 51

10 Background This last result can be used to calculate the mean squared end-to-end distance: N ERN 2 = E [ ] r i r j i,j=1 N i 1 = Nb 2 + E [ ] r i r j + i=1 j=1 N i 1 = Nb 2 + b 2 cos(θ) i j + i=1 Nb 2 + 2b 2 N j=1 i=1 k=1 cos(θ) k = Nb 2 + 2Nb 2 cos(θ) 1 cos(θ) N j=i+1 E [ ] r i r j N cos(θ) j i j=i+1 = Nb cos(θ) 1 cos(θ). Jay Taylor (ASU) APM Lecture 12 Fall / 51

11 Background Suppose that X 1,, X N is a polymer model with constant bond length b. The Kuhn length (effective length) of the model is defined as provided that the limit exists. Flory s characteristic ratio is 1 b K = lim N N E [ RN 2 ] C = b2 K b = lim N N N cos(θ ij ) i,j=1 Example: For the freely rotating chain, the Kuhn length and the characteristic ratio are ( ) 1 + cos(θ) 1/2 b K = b 0 and C = 1 + cos(θ) 1 cos(θ) 1 cos(θ). Jay Taylor (ASU) APM Lecture 12 Fall / 51

12 Background The following quantity is also of interest: The persistence length l p determines the characteristic length scale over 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 positions 0 and L (measured in contour length). Example: The persistence length of the freely-rotating chain is b l p = ln(cos(θ)) Jay Taylor (ASU) APM Lecture 12 Fall / 51

13 Background The Worm-like Chain Model Very stiff polymers can be modeled by taking θ 1 in the freely-rotating chain model. More precisely, let θ (ɛ) = ɛ 1/2 θ and b (ɛ) = b 0 ɛ and consider the corresponding polymer model with total contour length L = N (ɛ) b (ɛ) : X (ɛ) = ( X (ɛ) 1,, X (ɛ) N (ɛ) ) X (ɛ) (l) = X (ɛ) l/ɛ ; 0 t L. Jay Taylor (ASU) APM Lecture 12 Fall / 51

14 Background If we let ɛ 0, then the interpolated processes X (ɛ) tend to a continuous-parameter stochastic process X = (X l : 0 l L). X is called the Kratky-Porod model (1949) and has the following properties: X has continuous paths; X has persistence length l p = b/θ 2 ; X is a Gaussian process with distribution [ p(x) exp 1 L ds 2 x 2l p s ] Remark: Notice that the density gives more weight to paths with smaller curvature. Jay Taylor (ASU) APM Lecture 12 Fall / 51

15 Background The mean square end-to-end distance for the Kratky-Porod model is E [ R 2 (L) ] = E [ ( L 0 [ L = E = = 2 = 2l 2 p 0 L L 0 0 L s 0 ) 2 ] Ẋ (s)ds Ẋ (s)ds L 0 ] Ẋ (r)dr ] E [Ẋ (s)ẋ (r) dsdr e (s r)/lp drds 0 ( ) L + e L/lp 1. l p Jay Taylor (ASU) APM Lecture 12 Fall / 51

16 Background There are two regimes depending on the ratio L/l p. Ballistic regime: if L l p, then E [ R 2 (L) ] ( ) L = 2lp 2 + e L/lp 1 l p ( L 2lp L + 1 L 2 ) l p l p 2 lp 2 1 = L 2. Diffusive regime: if L l p, then E [ R 2 (L) ] ( ) L = 2lp 2 + e L/lp 1 l p = 2l p L + O(L 1 ). Remark: The second identity shows that b K = 2l p. Jay Taylor (ASU) APM Lecture 12 Fall / 51

17 Background Self-Avoiding Walks In contrast to ideal chain models, which neglect interactions between monomers separated by large contour distances, the properties of a self-avoiding walk are largely determined by such interactions. An N-step self-avoiding walk on Z d is a sequence of sites ( ω(0), ω(1),, ω(n) ) such that ω(j) ω(j 1) = d i=1 ω i(j) ω i (j 1) = 1 ω(i) ω(j) for all i j. Let Ω N be the set of N-step SAWs starting at 0 and let c N = Ω N. A random N-step SAW is a random vector X = (X (0),, X (N)) which is uniformly distributed on Ω N : P(X = ω) = 1 c N, ω Ω N. Jay Taylor (ASU) APM Lecture 12 Fall / 51

18 Background Properties of Self-Avoiding Walks There are relatively few rigorously proven results concerning the behavior of SAW in dimensions 2 d 4. When d = 3, it is conjectured that c N Aµ N N γ 1, with γ E [ XN 2 ] DN 2ν, with ν 0.59 These are supported by scaling arguments and MCMC. Notice that the SAW moves away from the origin more rapidly than a random walk. Jay Taylor (ASU) APM Lecture 12 Fall / 51

19 Background The Rouse Model Ideal chain models have also been used to investigate the dynamical properties of dilute polymer solutions. The Rouse model is a Brownian perturbation of a linear system of N coupled harmonic oscillators: where dx n = k 0 ζ (2X n X n 1 X n+1 ) dt + X n (t) is the location of the n th monomer; 2k B T dw (n) t ζ W 1,, W N are independent 3-dim Brownian motions; k 0 = 3k B T /b 2 is the spring constant for bonded monomers; ζ = 6πηa is the friction constant (Stoke s law). Jay Taylor (ASU) APM Lecture 12 Fall / 51

20 Background If X C (t) = 1 N N i=1 X i(t) denotes the center of mass of the molecule at time t, then X C satisfies the following SDE: dx C = 2k B T ζ N i=1 dw (n) t d = 2Nk B T d W t, ζ where W t is a 3-dim Brownian motion. This shows that the center of mass of a Rouse polymer follows Brownian motion with diffusion coefficient D C = k BT Nζ, i.e., the diffusivity of the polymer is inversely proportional to the polymer mass. Jay Taylor (ASU) APM Lecture 12 Fall / 51

21 Background In fact, the Rouse model can be solved explicitly. Notice that the SDE can be written as dx t = k ζ AX tdt + 2k B T dw t, ζ where X t = ( X 1 (t),, X N (t) ), W t is a 3N-dim Brownian motion, and A is a 3N 3N matrix. Since this is a linear SDE, the solution is given by X t = e k ζ At X 0 + 2k B T t e k ζ A(t s) dw s. ζ 0 Jay Taylor (ASU) APM Lecture 12 Fall / 51

22 Background When N is large, the eigenvalues of A are approximately equal to λ p π2 p 2 N 2, p 0, with λ 0 = 0 exact (corresponding to the motion of the center of mass). The time scale for relaxation of the conformation of the polymer is dictated by the smallest positive eigenvalue and is called the Rouse stress relaxation time: τ R ζ k 0 N 2 π 2 = ζn2 b 2 3π 2 k B T. Empirically, one has τ R N 3/2 at the Θ temperature. This discrepancy arises from the neglect of hydrodynamics in the Rouse model. Jay Taylor (ASU) APM Lecture 12 Fall / 51

23 Chromosome Territories Rosa, A. and Everaers, R Structure and Dynamics of Interphase Chromosomes. PLoS Comput. Biol. 4: e Motivation: Interphase chromosomes are organized into territories in some species (humans, rats, Drosophila), but not in budding yeast. The territories of individual chromosomes tend to re-appear after cell division, but change over time and differ between cells. Figure from Bolger et al. (2005). Jay Taylor (ASU) APM Lecture 12 Fall / 51

24 Chromosome Territories Chromosomes are condensed during mitosis and decondensed during interphase. Jay Taylor (ASU) APM Lecture 12 Fall / 51

25 Chromosome Territories Chromosome structure can be studied quantitatively by imaging marked chromosomes. Chromosomes can be marked (painted) using FISH or immunofluorescent DNA-binding proteins. Distances between markers can be measured using laser scanning confocal microscopy. Image from Bystricky et al. PNAS (2004). Jay Taylor (ASU) APM Lecture 12 Fall / 51

26 Chromosome Territories Chromatin fibers can be modeled as worm-like chains over short length scales. The mean square end-to-end distance under the WLC model is ( ) L R 2 (L) = 2lp 2 + e L/lp 1. l p This study estimated the persistence length of chromatin to be l p 197 ± 62 nm. Markers separated by N bp are separated by approximately L = N 10µm Mbp 1. Figure from Bystricky et al. PNAS (2004). Jay Taylor (ASU) APM Lecture 12 Fall / 51

27 Chromosome Territories The fit to the WLC model breaks down at longer length scales. Black line is the WLC model prediction of mean square end to end distance. Figure shows FISH data for yeast (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). Jay Taylor (ASU) APM Lecture 12 Fall / 51

28 Chromosome Territories Reptation and Entanglement of Polymer Chains In semidilute solutions, polymers can become mutually entangled. Entanglement only occurs if the chain contour length exceeds a characteristic value called the entanglement length L e. L e depends on both the stiffness of the polymer and its contour length density. For chromatin at typical nuclear densities during interphase, L e 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. Jay Taylor (ASU) APM Lecture 12 Fall / 51

29 Chromosome Territories Entangled linear polymers can move by reptation: each polymer is confined to move within a tubular neighborhood which it traverses by diffusion of stored lengths along its contour. The characteristic timescale for disentanglement via reptation is τ d 32(L/L e ) 3. yeast: τ d s; fly: τ d s 5 yr; human: τ d s 500 yr. Thus, reptation is unlikely to lead to equilibration of human or Drosophila chromosomes during interphase. Jay Taylor (ASU) APM Lecture 12 Fall / 51

30 Chromosome Territories Alternatively, topoisomerases may accelerate equilibration of decondensing chromosomes. Type II toposiomerases catalyze strand passage of one DNA duplex through another. The maximal effect of topo II molecules on chromosome decondensation can be estimated by considering a solution of phantom (non-interacting) chains. In this case the equilibration time is approximately τ R 32(L/L e ) 2. yeast: τ d s; fly: τ d s 10 days; human: τ d s 250 days. Fig. from Fass et al., Nat. Struct. Biol. (1999). Jay Taylor (ASU) APM Lecture 12 Fall / 51

31 Chromosome Territories Motivated by the preceding calculations, Rosa & Everaers propose that: Interphase chromosomes of sufficient length are not equilibrated in dividing cells. In particular, they remain largely unentangled. Chromosome territories are a consequence of the segregation of the unentangled chromosomes due to topological barriers. Territories can form in the absence of protein crosslinking within chromosomes or between chromosomes and other structural elements within the nucleus (i.e., the nuclear cytoskeleton). They investigate these proposals by carrying out MD simulations of groups of decondensing chromosomes. Jay Taylor (ASU) APM Lecture 12 Fall / 51

32 Chromosome Territories 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 comprises three effects: A shifted Leonard-Jones potential: U LJ = 4ɛ [ (σ/r ij ) 12 (σ/r ij ) 6 + 1/4 ] if r ij < σ2 1/6. A connectivity potential between nearest neighbors: U FENE = 0.5kR 2 0 ln [ 1 (r ij /R 0 ) 2] if r ij < R 0. A bending stiffness potential between consecutive bonds: U stiff = βk 0 (1 cos(θ)). Jay Taylor (ASU) APM Lecture 12 Fall / 51

33 Chromosome Territories Simulations were performed in a periodic rectangular box, with either a constant isotropic pressure (human, Drosophila) or a constant volume, leading to a chromatin volume fraction of 10 %. The initial conformations of the chromosomes were either linear or ring-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 days of real time. Jay Taylor (ASU) APM Lecture 12 Fall / 51

34 Chromosome Territories Fig. 3 in Rosa & Everaers. The simulation time is sufficient to mix the yeast chromosomes, but the fly and human chromosomes are confined to territories. The chromosome territories are Rabl-like in flies, but ellipsoidal in humans, reflecting the initial configurations. Jay Taylor (ASU) APM Lecture 12 Fall / 51

35 Chromosome Territories Human chromosomes form stable territories in the Kremer-Grest model. Figure shows trajectories of the centers of mass of the four simulated human chromosome 4 s. Motion is confined to regions of radius 0.1 µm. In contrast, the dimensions of the decondensed chromosomes are µm. Fig. 6 Jay Taylor (ASU) APM Lecture 12 Fall / 51

36 Chromosome Territories Simulated chromosome territories are stabilized by a chain crossing barrier. Fig. 4c Jay Taylor (ASU) APM Lecture 12 Fall / 51

37 Chromosome Territories Individual sites are dynamic within chromosome territories. Figure shows the growth of the MSD of six internal beads over time. Individual sites explore regions 1 µm over 5 hour periods. Purple dots show experimental data for a marked gene in yeast. Reptation dynamics appear to be observed at long time scales for all three organisms. Fig. 2 Jay Taylor (ASU) APM Lecture 12 Fall / 51

38 Chromosome Territories Figures show R 2 (L) averaged over three time windows of growing size (red, magenta, cyan). Gray curves show the initial end-to-end distances. Notice that the initial state rapidly unfolds (within 40 min). Black curve is the WLC prediction. Drosophila chromosomes deviate from the WLC curve even after 3 days at contour lengths > 0.2Mb. Simulated yeast chromosomes equilibrate within 7 hours. Fig. 1b, c Jay Taylor (ASU) APM Lecture 12 Fall / 51

39 Chromosome Territories Simulated human chromosomes also differ from the WLC predictions for long contour lengths even after 3 days. However, the simulated end-to-end distances also differ from data even over modest contour lengths. At intermediate contour lengths, the data is closer to the WLC predictions than to the simulations. At longer contour lengths, the data also falls below the simulated end-to-end lengths. The authors suggest that the discrepancies have two different causes. Fig. 1d Axis goes from 10 kb to 100 Mb. Jay Taylor (ASU) APM Lecture 12 Fall / 51

40 Chromosome Territories The blue circles come from data from the end of the p-arm. Comparison with distances measured from the ends of the simulated chromosome is much better over intermediate contour lengths (Fig. 1e). Equilibration is more rapid near chromosome ends. Comparison with simulations started from an initial ring configuration show better agreement at longer contour lengths (Fig. 1f). Such a configuration may result from bending at the centromere. Fig. 1e, f Jay Taylor (ASU) APM Lecture 12 Fall / 51

41 Loop Polymers Bohn, M. et al Random loop model for long polymers. Phys. Rev. E 76: Motivation: Chromosomes are known to form loops stabilized by protein-protein contacts. Bohn et al. propose that loop formation could account for the leveling off of end-to-end distance over long contour distances. Figure from Ong and Corces, J. Biol. 8: 73 (2009). Jay Taylor (ASU) APM Lecture 12 Fall / 51

42 Loop Polymers Gaussian Chains with Non-Local Harmonic Interactions Consider a polymer with N + 1 monomers at positions X 0 = 0 and X 1,, X N. The authors first assume that there are non-random harmonic interactions between all monomer pairs: U = κ 2 N X j X j j=1 N κ ij X i X j 2 i<j 1 where κ is the backbone spring constant; κ ij = κ ji are spring constants for loop attachment points. Jay Taylor (ASU) APM Lecture 12 Fall / 51

43 Loop Polymers Since the potential is additive over spatial dimensions, the Boltzmann distribution factors into the product of three independent distributions over each dimension. The distribution over the x-coordinates is ( P x (X 1,, X N ) = C x exp 1 ) 2 U(X 1,, X N ) ( = C x exp 1 ) 2 XT KX where X = (X 1,, X N ) T and K = N j=0 κ 1j κ 12 κ 1N κ N 21 j=0 κ 2j κ 2N.. κ N1 κ N2. N j=0 κ Nj. Jay Taylor (ASU) APM Lecture 12 Fall / 51

44 Loop Polymers Furthermore, the marginal distribution of the distance between beads i and j is just the Maxwell distribution with density ( P (r ij ) = C rij 2 exp 1 ) 1 rij 2 2 σ ii 2σ ij + σ jj = 4 Γ 3/2 r 2 π ij exp ( Γrij 2 ) where Σ = (σ ij ) = K 1 and Γ = 1 2 σ ii 2σ ij +σ jj. It then follows that the mean squared end-to-end distance is: Rij 2 = rij 2 P(r ij )dr = 3 (σ ii 2σ ij + σ jj ). R 3 1 Jay Taylor (ASU) APM Lecture 12 Fall / 51

45 Loop Polymers 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 Bernoulli RVs: κ with probability P κ ij = if l 1 i j l 2 0 with probability 1 P. The constants l 1 < l 2 determine the range of possible loop sizes. Jay Taylor (ASU) APM Lecture 12 Fall / 51

46 Loop Polymers 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 the chromosome is reached before the number and locations of the loops have changed significantly. In this case, we average first over the thermal noise conditional on a set of loops and then average over the distribution of loops: Rij 2 quenched Rij 2 thermal loops = 3 σ ii 2σ ij + σ jj loops. Jay Taylor (ASU) APM Lecture 12 Fall / 51

47 Loop Polymers Alternatively, the annealed average is more appropriate if the number and locations of the chromosome loops change repeatedly before thermal equilibrium can be reached. In this case, the partition function depends on both the conformation of the molecule and the locations of the loops: Z ann {κ ij } Z ({X k }, {κ ij }) P ({κ ij }) = dx 1 dx N e U bb i<j 1 [ ( ) ] P e 1/2κ X i X j In this paper, the authors focus on the quenched average, which they evaluate through Monte Carlo simulations. Whether the quenched or the annealed average is more biologically appropriate is unclear. Jay Taylor (ASU) APM Lecture 12 Fall / 51

48 Loop Polymers Figure 2 shows quenched mean square spatial distance plotted against contour length for chains of N monomers. In each case, P was chosen so that the expected number of loops is 100. When only short loops are allowed, then R 2 (N) N (Fig. 2a). In contrast, if only long loops are allowed, then R 2 (N) is a unimodal function of N (Fig. 2b). When loops of all sizes are allowed, R 2 (N) is constant over most contour lengths (Fig. 2c). Jay Taylor (ASU) APM Lecture 12 Fall / 51

49 Loop Polymers Figure 3 shows how the predicted mean squared end-to-end distance compares with that measured for human chromosomes 1 and 11. Here, each bead corresponds to 150 kb. Reasonable match between theory and data is obtained when P The kurtosis of the end-to-end distance is compared between model and data in Figure 4: c 4 = R4 R 2 2. Jay Taylor (ASU) APM Lecture 12 Fall / 51

50 Loop Polymers Summary Random loop models can explain the leveling off of end-to-end distances with contour length if loops can exist on all length scales. This remains true if the RLM is modified to account for excluded volume 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 thermal equilibrium during interphase, a claim that Rosa & Everaers 2008 study contradicts. The two papers lead to the following question: Does looping promote the formation of chromosome territories, or do chromosome territories promote the formation of loops? Jay Taylor (ASU) APM Lecture 12 Fall / 51

51 References References Bolzer, A. et al. (2005) Three-Dimensional Maps of All Chromosomes in Human Male Fibroblast Nuclei and Prometaphase Rosettes. PLoS Biol. 3: e157. Bystricky, K. et al. (2004) Long-range compaction and flexibility of interphase chromatin in budding yeast analysed by high-resolution imaging techniques. PNAS 101: Doi, M. and Edwards, S.F. (1986). The Theory of Polymer Dynamics. Oxford University Press. Mateos-Langerak et al. (2009) Spatially confined folding of chromatin in the interphase nucleus. PNAS 106: Ong, C.-T. and Corces, V. G. (2009) Insulators as mediators of intraand inter-chromosomal interactions: a common evolutionary theme. J. Biol. 8: 73. Jay Taylor (ASU) APM Lecture 12 Fall / 51