1 Compaction and segregation of sister chromatids
2 via active loop extrusion
3
4
5 Anton Goloborodko1, Maxim V. Imakaev1, John F. Marko2, and Leonid Mirny1,3,*
6
7 Short title: Compaction and segregation of sister chromatids
8
9
10 1 Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139
11 2 Department of Molecular Biosciences and Department of Physics and Astronomy,
12 Northwestern University, Evanston IL 60208
13 3 Institute for Medical Engineering & Science, Massachusetts Institute of Technology,
14 Cambridge MA 02139
15
16 * Corresponding author: Leonid Mirny, Institute for Medical Engineering & Science,
17 Massachusetts Institute of Technology, 77 Mass Ave, Cambridge MA 02139, [email protected],
18 +16174524862
19 Keywords: prophase, polymers, condensin, structural maintenance of chromosomes
20
1 21
22 Abstract
23
24 The mechanism by which chromatids and chromosomes are segregated during mitosis
25 and meiosis is a major puzzle of biology and biophysics. Using polymer simulations of
26 chromosome dynamics, we show that a single mechanism of loop extrusion by
27 condensins can robustly compact, segregate and disentangle chromosomes, arriving at
28 individualized chromatids with morphology observed in vivo. Our model resolves the
29 paradox of topological simplification concomitant with chromosome “condensation”, and
30 explains how enzymes a few nanometers in size are able to control chromosome
31 geometry and topology at micron length scales. We suggest that loop extrusion is a
32 universal mechanism of genome folding that mediates functional interactions during
33 interphase and compacts chromosomes during mitosis.
34
2 35 Introduction
36 The mechanism whereby eukaryote chromosomes are compacted and concomitantly
37 segregated from one another remains poorly understood. A number of aspects of this
38 process are remarkable. First, the chromosomes are condensed into elongated
39 structures that maintain the linear order, i.e. the order of genomic elements in the
40 elongated chromosome resembles their order along the genome 1. Second, the
41 compaction machinery is able to distinguish different chromosomes and chromatids,
42 preferentially forming intra-chromatid cross-links: if this were not the case, segregation
43 would not occur 2. Third, the process of compaction is coincident with segregation of
44 sister chromatids, i.e. formation of two separate chromosomal bodies. Finally, originally
45 intertwined sister chromatids become topologically disentangled, which is surprising,
46 given the general tendency of polymers to become more intertwined as they are
47 concentrated 3.
48
49 None of these features cannot be produced by indiscriminate cross-linking of
50 chromosomes 4, which suggests that a novel mechanism of polymer compaction must
51 occur, namely “lengthwise compaction” 3,5,6, which permits each chromatid to be
52 compacted while avoiding sticking of separate chromatids together. Cell-biological
53 studies suggest that topoisomerase II and condensin are essential for metaphase
54 chromosome compaction 7–10, leading to the hypothesis that mitotic compaction-
55 segregation relies on the interplay between the activities of these two protein
56 complexes. Final structures of mitotic chromosomes were shown to consist arrays of
57 consecutive loops 11–13. Formation of such arrays would naturally results in lengthwise
58 chromosome compaction.
59
3 60 One hypothesis of how condensins can generate compaction without crosslinking of
61 topologically distinct chromosomes is that they bind to two nearby points and then slide
62 to generate a progressively larger loop 2. This “loop extrusion” process creates an
63 array of consecutive loops in individual chromosomes 2. When loop-extruding
64 condensins exchange with the solvent, the process eventually settles at a dynamical
65 steady state with a well-defined average loop size14. Simulations of this system at larger
66 scales 14 have established that there are two regimes of the steady-state dynamics: (i) a
67 sparse regime where little compaction is achieved and, (ii) a dense regime where a
68 dense array of stabilized loops efficiently compacts a chromosome. Importantly, loop
69 extrusion in the dense regime generates chromatin loops that are stabilized by multiple
70 “stacked” condensins 15, making loops robust against the known dynamic binding-
71 unbinding of individual condensin complexes 16., These two quantitative studies of loop
72 extrusion kinetics14,15 focused on the hierarchy of extruded loops and did not consider
73 the 3D conformation and topology of the chromatin fiber in the formed loop arrays. The
74 question of whether loop-extruding factors can act on a chromatin fiber so as to form an
75 array of loops, driving chromosome compaction and chromatid segregation, as originally
76 hypothesized for condensin in 2, is salient and unanswered. The main objective of this
77 paper is to test this hypothesis and to understand how formation of extruded loop arrays
78 ultimately leads to compaction, segregation and disentanglement (topology
79 simplification) of originally intertwined sister chromatids.
80
81 Here we use large-scale polymer simulations to show that active loop extrusion in
82 presence of topo II is sufficient to reproduce robust lengthwise compaction of chromatin
83 into dense, elongated, prophase chromatids with morphology in quantitatively accord
84 with experimental observations. Condensin-driven lengthwise compaction, combined
85 with the strand passing activity of topo II drives disentanglement and segregation of 4 86 sister chromatids in agreement with theoretical prediction that linearly compacted
87 chromatids must spontaneously disentangle 3.
88
89
90 Model
91 We consider a chromosome as a flexible polymer, coarse-graining to monomers of 10
92 nm diameter, each corresponding to three nucleosomes (600bp). As earlier 13,17, the
93 polymer has a persistence length of ~5 monomers (3Kb), is subject to excluded volume
94 interactions and to the activity of loop-extruding condensin molecules and topo II (see
95 below, and in 17). We simulate chains of 50000 monomers, which corresponds to 30 Mb,
96 close to the size of the smallest human chromosomal arm.
97
98 Each condensin complex is modeled as a dynamic bond between a pair of monomers
99 that is changed as a function of time (Figure 1, Video 1). Upon binding, each
100 condensin forms a bond between two adjacent monomers; subsequently both bond
101 ends of a condensin move along the chromosome in opposite directions, progressively
102 bridging more distant sites and effectively extruding a loop. As in prior lattice models of
103 condensins 14,15, when two condensins collide on the chromatin, their translocation is
104 blocked; equivalently, only one condensin is permitted to bind to each monomer,
105 modeling their steric exclusion. Exchange of condensins between chromatin and
106 solution is modeled by allowing each condensin molecule to stochastically unbind from
107 the polymer. To maintain a constant number of bound condensins, when one
108 condensin molecule dissociates, another molecule associates at a randomly chosen
109 location, including chromatin within loops extruded by other condensins. This can
110 potentially lead to formation of reinforced loops (Video 2, see below) 14,15.
5 111
112 To simulate the strand-passing activity of topo II enzymes, we permit crossings of
113 chromosomal fiber by setting a finite energy cost of fiber overlaps. By adjusting the
114 overlap energy cost we can control the rate at which topology changes occur.
115
116 This model has seven parameters. Three describe the properties of the chromatin fiber
117 (linear density, fiber diameter, and persistence length); two control condensin kinetics
118 (the mean linear separation between condensins, and their processivity, i.e. the average
119 size of a loop formed by an isolated condensin before it dissociates, and defined as two
120 times the velocity of bond translocation times the mean residence time); one parameter
121 controls the length of the monomer-monomer bond mediated by a condensin (effectively
122 the size of a condensin complex), and finally we have the energy barrier to topological
123 change.
124
125 The effects of two of these parameters, linear separation of condensins and their
126 processivity, have been considered earlier in the context of a 1D model 14. These
127 parameters control the fraction of a chromosome extruded into loops and the average
128 loop length. In our simulations, we use the separation of 30kb (1000 condensins per 30
129 Mb chromosome, as measured in 18) and the processivity of 830 kb such that
130 condensins form a dense array of gapless loops with the average loop length of 100kb,
131 which agrees with available experimental observations 11–13,19. The effects of the other
132 five parameters are considered below.
133
134 Initially, chromosomes are compacted to the density of chromatin inside the human
135 nucleus and topologically equilibrated, thus assuming an equilibrium globular
136 conformation, corresponding to a chromosomal territory. In the simulations of sister 6 137 chromatid segregation, the initial conformation of one chromatid is generated as above,
138 while the second chromatid traces the same path at a distance of 50nm and winds one
139 full turn per 100 monomers (60 kb). In this state, two chromatids run parallel to each
140 other and are highly entangled, while residing in the same chromosome territory. This
141 represents to the most challenging case for segregation of sister chromatids.
142
143
144 Results
145 Loop extrusion compacts randomly coiled interphase chromosomes into
146 prophase-like chromosomes
147
148 Our main qualitative result is that loop extrusion by condensins and strand passing
149 mediated by topo II convert initially globular interphase chromosomes into elongated,
150 unknotted and relatively stiff structures that closely resemble prophase chromosomes
151 (Figure 2, Video 2, Figure 2 - figure supplement 1). The compacted chromosomes
152 have the familiar “noodle” morphology (Figure 2BC) of prophase chromosomes
153 observed in optical and electron microscopy 20,21. Consistent with the observed linear
154 organization of prophase chromosomes, our simulated chromosomes preserve the
155 underlying linear order of the genome20–24 (Figure 2C). Moreover, the geometric
156 parameters of our simulated chromosomes match the experimentally measured
157 parameters of mid-prophase chromosomes (Figure 2B): both simulated and human
158 chromosomes have a linear compaction density ~5 kb/nm 20 and the radius of 300nm
159 25–27.
160
161 The internal structure of compacted chromosomes also agrees with structural data for
7 162 mitotic chromosomes. First, loop extrusion compacts a chromosome into a chain of
163 consecutive loops 14, which agrees with a wealth of microscopy observations 11,12,28 and
164 the recent Hi-C study 13. Second, our polymer simulations show that a chain of loops
165 naturally assumes a cylindrical shape, with loops forming the periphery of the cylinder
166 and loop-extruding condensins at loop bases forming the core (Figure 2C, Video 2), as
167 was predicted in 2. Condensin-staining experiments reveal similar cores in the middle of
168 in vivo and reconstituted mitotic chromosomes 28,29. In agreement with experiments
169 where condensin cores appear as early as late prophase 27, we observe rapid formation
170 of condensin core in simulations (Figure 2, Video 2). Thus, the structure of
171 chromosomes compacted by loop extrusion is consistent with the loop-compaction
172 picture of mitotic 11,29,30 and meiotic chromosomes 31,32. However, it should be noted
173 that our model does not rely on a connected “scaffold”, and that cleavage of the DNA
174 will result in disintegration of the entire structure, as is observed experimentally 33.
175
176 Loop-extrusion generates chromosome morphology kinetics similar to those
177 observed experimentally
178 Simulations also reproduce several aspects of compaction kinetics observed
179 experimentally. First, the model shows that formed loops are stably maintained by
180 condensins despite their constant exchange with the nucleoplasm 8,16,34. This stability is
181 achieved by accumulation of several condensins at a loop base 14,15. Supported by
182 multiple condensins, each loop persists for times much longer than the residency time
183 for individual condensins 14.
184
185 Second, simulations show that chromosome compaction proceeds by two stages
186 (Figure 2AB): fast formation of a loose, elongated morphology, followed by a slow
187 maturation stage. During the initial “loop formation stage”, condensins associate with 8 188 the chromosome and extrude loops until colliding with each other, rapidly compacting
189 the chromosome (Figure 2, Video 2). At the end of this stage, that takes a fraction of
190 the condensin exchange time, a gapless array of loops is formed, the loops are relative
191 small, supported by single condensins, and the chromosome is long and thin (Figure
192 2B).
193
194 During the following “maturation stage”, due to condensin exchange some loops
195 dissolve and divide while others grow and get reinforced by multiple condensins14,15. As
196 we recently showed14, a loop can dissolve when all of its condensins dissociate; while
197 new loops are born when two condensins land inside an existing loop at about the same
198 time and split it into two loops. As the mean loop size grows during the maturation stage
199 (Figure 2A), the chromosome becomes thicker and shorter. By the end this stage,
200 which takes several rounds of condensin exchange, the rates of loop death becomes
201 equal the rate of loop division and the chromosome achieves a steady state. At steady
202 state loops get reinforced by multiple condensins and the average loop size and the
203 lengthwise compaction reach their maximal values (Figure 2). These dynamics are in
204 accord with the observation that chromosomes rapidly attain a condensed “noodle”-like
205 shape in early prophase, then spend the rest of prophase growing thicker and shorter
206 20,21,27,35 (see Discussion).
207
208 Loop extrusion separates and disentangles sister chromatids
209 In a second set of simulations, we studied whether loop extrusion and strand passing
210 simultaneously lead to spatial segregation 2 and disentanglement of sister chromatids 5.
211 We simulated two long polymers connected at their midpoints that represented sister
212 chromatids with stable centromeric cohesion. To model cohesion of sister chromatids in
213 late G2 phase we further twisted them around each other (see Methods) while 9 214 maintaining them with a chromosomal territory.
215
216 We find that the activity of loop-extruding condensins in the presence of topo II leads to
217 compactions of each of the sister chromatids into prophase-like “noodle” structures
218 (Video 3). Moreover, we observe that formed compact chromosomes (a) become
219 segregated from each other, as evident by the drastic growth of the mean distance
220 between their backbones (Figure 3A), and (b) become disentangled, as shown by the
221 reduction of the winding number of their backbones (Figure 3B). Thus, loop extrusion
222 and strand passing folds highly catenated sister chromatids into separate disentangled
223 structures. (Figure 3C, Video 3).
224
225 Despite stochastic dynamics of condensins, compaction, segregation and
226 disentanglement of sister chromatids are highly reproducible and robust to changes in
227 simulation parameters. Disentanglement and segregation were observed in each of 16
228 simulation replicas (Video 4). We also performed simulations with altered chromatin
229 polymer parameters (Video 5). Again, we observed chromatids segregation and
230 disentanglement irrespective of fiber parameters and size of a single condensin
231 molecule.
232
233 Our model makes several predictions that can be tested against available experimental
234 data. First, in agreement with the published literature36–39 the abundance of condensins
235 on the chromosome significantly affects geometry of simulated chromatids (Video 6).
236 When we varied abundance of chromosome-bound condensin by 20-fold (5-fold
237 decrease and 4-fold increase) we observed two major effects (Figure 2 - figure
238 supplement 1). First, we noticed that condensin depletion leads to a systematic (3-fold)
239 increase in chromosome diameter. Second, we observed a nonmonotonic changes in 10 240 the linear density of chromatin in compacted chromosomes: both 5-fold increase and 4-
241 fold decrease of condensin abundance lead to about 2-fold drop in linear compaction
242 (i.e. 2-fold increase in chromosome length). This reduced compaction, however,
243 emerges due to different mechanisms: an increase of condensin abundance leads to
244 reduction of loop sizes while keeping all chromatin in the loops, creating thin
245 chromosomes with long scaffold formed by excessive condensins. Decrease of
246 condensins abundance, on the contrary, leads to formation of chromosomes where
247 some regions are not extruded into loops; such gaps between loops lead to substantial
248 increase in chromosome length and inefficient compaction. These nontrivial
249 dependences of chromosome geometry on condensin abundance can be further tested
250 experimentally.
251
252 Next, we examined the role of topological constraints and topo II enzyme in the process
253 of chromosome compaction and segregation. Activity of topo II is modeled by a finite
254 energy barrier for monomer overlap, allowing for occasional fiber crossings. We
255 simulated topo II depletion by elevating the energy barrier and increasing the radius of
256 repulsion between distant particles, thus reducing the rate of fiber crossings. In these
257 simulations segregation of sister chromatids is drastically slowed down (Video 7). This
258 effect of topological constraints on kinetics of compaction and segregation is expected
259 since each chromosome is a chain with open ends that thus can eventually segregate
260 from each other. Reduction of topo II activity (increase of the crossing barrier) prevents
261 disentanglement and segregation of sister chromatids, while having little effect of
262 diameter of individual chromatids (Figure 3 – figure supplement 1). We conclude that
263 topo II is essential for rapid chromosome segregation and disentanglement, in accord
264 with an earlier theoretical work 40 and with experimental observations 9,41–43.
265 11 266 Finally, we examined the role of cohesin-dependent cohesion of chromosomal arms of
267 sister chromatids. It is known that cohesins are actively removed from chromosomal
268 arms in prophase. Interference with processes of cohesins removal or cleavage leads to
269 compaction into elongated chromosomes that fail to segregate 44,45. We simulated
270 prophase compaction in the presence of cohesin rings 46 that stay on sister chromatids
271 and can be pushed around by loop-extruding condensins. If cohesins remain uncleaved
272 on sister chromatids (Video 8), we observe compaction of sister chromatids, that
273 remain unsegregated, in agreement with experiment 44,45.
274
275 Taken together, these results indicate that loop extrusion is sufficient to compact,
276 disentangle and segregated chromosomes from a more open interphase state, to a
277 compacted, elongated and linearly organized prophase state. Future Hi-C experiments
278 focused on prophase chromosomes may be able to make detailed structural
279 comparison with the prediction of our model.
280
281
282 Repulsion between loops shapes and segregates compacted chromatids.
283 Reorganization of chromosomes by loop extrusion can be rationalized by considering
284 physical interactions between extruded loops. A dense array of extruded loops makes a
285 chromosome acquire a bottle-brush organization with loops forming side chains of the
286 brush. Polymer bottle-brushes has been studied in physics both experimentally and
287 computationally 47–49, and has been proposed as models of mitotic and meiotic
288 chromosomes 4. A recent study also suggested a bottle-brush organization of
289 centromeres in yeast, where centromere chromatin loops lead to separation of sister
290 centromeres and create spring-like properties of the centromere in mitosis 50,51.
291 12 292 Central to properties of chromosomal bottle-brushes are excluded volume interactions
293 and maximization of conformational entropy that lead to repulsion of loops from each
294 other 4,52. Loops further stretch from the core and extend radially (Figure 4), similar to
295 stretching of side chains in polymer bottle-brushes 47–49. This in turn leads to formation
296 of the central core where loop-bases accumulate (Figure 4A). Strong steric repulsion
297 between the loops, caused by excluded volume interactions, stretches the core, thus
298 maintaining linear ordering of loop bases along the genome (Figure 2C). Repulsion
299 between loops also leads to stiffening of the formed loop-bottle-brush, as chromosome
300 bending would increase overlaps between the loops.
301
302 Steric repulsion between loops of sister chromatid leads to segregation of sister
303 chromatids that minimizes the overlap of their brush coronas (Figure 4B). Stiffening of
304 the bottle-brush, in turn, leads to disentanglement of sister chromatids. To test this
305 mechanism we performed simulations where excluded volume interactions have been
306 turned off (Video 5). In these simulations, chromosomes did not assume a “bottle-
307 brush” morphology and did not segregate, which confirms the fundamental role of steric
308 repulsion for chromosome segregation.
309
310 Discussion
311 Our results show that chromosomal compaction by loop extrusion generates the major
312 features of chromosome reorganization observed during prophase 2,15. Loop extrusion
313 leads to lengthwise compaction and formation of cylindrical, brush-like chromatids,
314 reducing chromosome length by roughly 25-fold, while at the same time removing
315 entanglements between separate chromatids. Unlike models of prophase condensation
316 based on crosslinking of randomly colliding fibers 52,53, chromatid compaction by loop
13 317 extrusion exclusively forms bridges between loci of the same chromosome (Figure 1A).
318 Similarly, loop-extrusion in bacteria, mediated by SMC proteins, was suggested to form
319 intra-arm chromosomal bridges leading to lengthwise chromosome compaction 54,55.
320
321 Our computations indicate that it takes an extended period of time for loop-extruding
322 enzymes to maximally compact chromosomes. Maximal compaction is achieved by
323 slow maturation of the loop array over multiple rounds of condensin exchange, rather
324 than a single round of loop expansion. Loop length and the number of condensins per
325 loop gradually increase during the maturation stage. Assuming that condensin II in
326 prophase exchanges with the fast rate of condensin I in metaphase (~200 sec) 16 and
327 using an average loop size of 100 kb, and an average spacing of condensins of 30 kb, it
328 should take ~5 condensin exchange times, or about 17 minutes to achieve maximal
329 compaction, which roughly agrees with the duration of prophase in human cells 56,57.
330 This gradual compaction also may explain why chromosomes of cells arrested in
331 metaphase continue shrinking for many hours after the arrest (condensin II metaphase
332 exchange time ~hours 16).
333
334 Our model is idealized and does not aim to describe all of aspects of chromosome
335 folding during mitosis. First, our model does not fully capture the specifics of later
336 stages of mitosis. We model only the action of condensin II in prophase 58. Simple
337 geometric considerations, Hi-C studies and mechanical perturbation of mitotic
338 chromosomes show that, in metaphase, cells use additional mechanisms of lengthwise
339 (axial) compaction.
340
341 Loop extrusion does not require specific interaction sites (e.g. loop anchors or
342 condensin binding sites) and is therefore robust to mutations and chromosome 14 343 rearrangements. This robustness also explains how a large segment of chromatin
344 inserted into a chromosome of a different species can be reliably compacted by mitotic
345 machinery of the host 59,60. At the same time, this model allows the morphology of
346 mitotic chromosomes to be tuned. Most importantly, the diameter and length of the
347 simulated chromosomes depend on chromatin fiber properties as well as numbers of
348 condensins (Figure 2 - figure supplement 1); those dependences invite experimental
349 tests.
350
351 In summary, we have shown that loop extrusion is a highly robust mechanism to
352 compact, segregate and disentangle chromosomes. In a recent study 17 we also
353 demonstrated that loop extrusion by another SMC complex, cohesin, could lead to
354 formation of interphase topological association domains (TADs). We suggest that during
355 interphase, cohesins are sparsely bound, extruding 50-70% of DNA into highly dynamic
356 loops, while metaphase condensins are bound more densely and more processive
357 forming a dense array of stable loops. Taken together these studies suggest that loop
358 extrusion can be a universal mechanism of genome folding, to mediate functional
359 interactions during interphase and to compact during mitosis.
360
15 361 Materials and methods
362 Methods
363 Our model of loop extrusion acting on a polymer fiber consists of (a) a 1D model that
364 governs the dynamics of intra-chromosomal bonds formed by condensins over time and
365 (b) a 3D polymer model of chromosome dynamics subject to the condensins bonds
366 The 1D model of loop extrusion
367 The dynamics of loop extruding condensins on a chromatin fiber is simulated using a 1D
368 lattice simulations as it was described previously for generic loop-extruding factors
369 14,15 In the lattice, each position corresponds to one monomer in the 3D polymer
370 simulation. We model a condensin molecule very generally as having two chromatin
371 binding sites or, “heads”, connected by a linker. Each head of a condensin occupies one
372 lattice position at a time, and no two heads can occupy the same lattice position. To
373 simulate the process of loop extrusion, the positions of the two condensin heads
374 stochastically move away from each other over time (simulated with the Gillespie
375 algorithm 61).
376
377 We initialize the simulations by placing condensin molecules at random positions along
378 the polymer chain, with both heads in adjacent positions. To simulate the exchange of
379 condensins between chromatin and solution, condensins stochastically dissociate from
380 the chromatin fiber. Every dissociation event is immediately followed by association of
381 another condensin molecule with the chromatin fiber so that the total number of
382 condensins bound to the chromatin stays constant.
383
384 This 1D model has four parameters: the size of the lattice, number of condensins bound
385 to chromatin, speed of extrusion, average residency time of a condensin on the
16 386 chromatin fiber. We model a 30 Mb chromosome with a lattice of 50000 sites, 600 bp
387 each. The chromosome is bound by 1000 condensins. Without loss of generality, we set
388 the speed of extrusion to be 1 step per unit time and the condensin residency time at
389 692 units of time, such that the resulting average loop length is equal 167 monomers,
390 or, 100kb, close to previous observations in vivo.
391
392 3D simulations of chromosomes
393 To perform Langevin dynamics polymer simulations we use OpenMM, a high-
394 performance GPU-assisted molecular dynamics API 62,63. To represent chromatin fibers
395 as polymers, we use a sequence of spherical monomers of 1 unit of length in diameter.
396 Here and below all distances are measured in monomer sizes (~3 nucleosomes,
397 ~10nm), density is measured in particles per cubic unit, and energies are measured in
398 kT. We use the following parameters of the Langevin integrator: particle mass = 1 amu,
399 friction coefficient = 0.01 ps^-1, time step = 1ps, temperature = 300K.
400
401 Neighboring monomers are connected by harmonic bonds, with a potential =
402 100( − 1) (here and below in units of kT). We model polymer stiffness with a three
403 point interaction term, with the potential =5 (1−cos( )), where alpha is the angle
404 between neighboring bonds.
405
406 To allow chain passing, which represents activity of topoisomerase II, we use a soft-
407 core potential for interactions between monomers, similar to 13,64. All monomers interact
408 via a repulsive potential
= 2.0(−1 + ( /√(6 ⁄ 7))^12 ⋅ (( /√(6 ⁄ 7))^2 − 1) ⋅ (823543 )/46656)
409
17 410 This is a fast and efficient potential designed to be constant at 2.0 kT up to r=0.7-0.8
411 and then quickly go to zero at r=1.00.
412
413 To connect 1D LEF simulations with 3D polymer simulations, we first run 1D LEF
414 dynamics for a total period of 10 condensin residence times, recording the state of the
415 systems each unit of time. We then assign bonds to the monomers in polymer
416 simulations according to the current position of condensins’ heads. The two monomers
417 held by the two heads of each condensin are connected by a harmonic bond with the
418 potential = 100( − 1) . For each position of condensins’ heads from the 1D model,
419 we perform 40000 steps of Langevin dynamics. 3D conformations are recorded every
420 20000 steps.
421
422 We allow an overlap of the heads of collided condensins at the loop bases (i.e. two
423 heads of collided condensins could occupy the same monomer instead of two adjacent
424 monomers). We implement this by shifting the positions of the downstream heads of all
425 condensins by 1 monomer downstream. This allows us to achieve the maximally
426 possible compaction of the chromosomal core (one loop per one monomer of the axis).
427
428 We generate the initial conformation of a single chromosome as following: a polymer
429 chain was spherically compacted to a density of 0.01 particle per unit length^3, then
430 allowed to equilibrate over 4,000,000 steps of Langevin dynamics, with a gradual
431 increase of repulsion energy to equilibrate both the topology and the distribution of
432 density inside the confining sphere. We generate the initial conformations of two sister
433 chromatids by winding of two polymer chains along the conformation of a single
434 chromosome, with one full turn each 100 monomers.
435 18 436 We simulate topo II depletion by adjusting two factors. First, we increase the energy of
437 monomer overlap to 20 kT. Since this measure alone proved to be inefficient to prevent
438 chain passing, we additionally increase the radius of repulsion up to 3 length units. In
439 order to maintain the contour length of the polymer, we keep the length of a monomer
440 bond at 1 and ignore repulsion between pairs of neighboring monomers, up to 3
441 monomers distance along the chain.
442
443 To study how the parameters of simulations affect the geometry of compacted
444 chromosomes, we alter the following parameters: (a) increase 2x and (b) decrease 0.5x
445 the number of condensins, (c) increase 2x and (d) decrease 0.5x the bending energy,
446 (e) disallow overlaps of condensins at loop bases, thus simulating wide condensins; (f)
447 reduce linear DNA density of the chromatin fiber to 400 bp/10nm to model compaction
448 of a 10nm fiber of stacked nucleosomes and (e) increase linear DNA density of the
449 chromatin fiber to 2400 bp/10nm and increased the fiber thickness to 30nm to model
450 compaction of a 30nm fiber.
451
452 Acknowledgements
453 Work at NU was supported by the NSF through Grants DMR-1206868 and MCB-
454 1022117, and by the NIH through Grants GM105847 and CA193419. Work at MIT was
455 supported by the NIH through Grants GM114190, R01HG003143, and DK107980. We
456 thank Elnaz Alipour, Job Dekker, Nancy Kleckner, the MIT Biophysics students and the
457 members of Mirny Lab: Geoff Fudenberg, Nezar Abdennur, Christopher McFarland and,
458 especially, E. M. Breville for stimulating and productive discussions.
459
460
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27 657 FIGURES
658 Figure 1. Model of loop extrusion by condensins. Top row, the update rules used in
659 simulations: (A) a condensin extrudes a loop by moving the two ends along the chromosome in
660 the opposite directions, (B) collision of condensins bound to chromosomes blocks loop
661 extrusion on the collided sides, (C) a condensin spontaneously dissociates and the loop
662 disassembles; (D) a condensin associates at randomly chosen site and starts extruding a loop.
663 Bottom row, (E) we study how combined action of many loop extruding condensins changes the
664 conformation of a long chromosome using polymer simulations.
665
666 Figure 2. Loop-extruding condensins convert globular interphase chromosomes into
667 elongated structures. (A) Dynamics of loop array formation. Left, the fraction of the
668 chromosome extruded into loops (loop coverage, gray) and the mean loop size (black). The
669 time is measured in the units of condensin residence time. Data for each curve was obtained
670 using 10 simulation replicas; the thickness of each line shows the difference between the 10th
671 and 90th percentile at each time point. Right, a phase diagram showing loop coverage vs mean
672 loop length. The curve is averaged over the simulation replicas and colored by log-time. Two
673 stages of dynamics are separated by the two vertical dashed lines corresponding to 95%
674 coverage and 95% of the steady state mean loop length. The distinction between the two stages
675 of compaction is especially easy to see on the phase curve. (B) Dynamics of chromosome
676 morphology. Left: chromosome length (blue) and the mean chromosome diameter (red) shown
677 as a function of time; the thickness of each line shows the difference between the 10th and 90th
678 percentile at each time point. Right, a phase diagram showing chromosome length vs diameter;
679 the curve is averaged over the simulation replicas and colored by log-time. Note that most of
680 chromosome linear compaction is achieved during the first phase, while the second is
681 characterized by widening of the chromosome and further three-fold shortening. (C) Snapshots
682 of polymer simulations. Top row: the conformations of chromosomes at the beginning of
683 simulations (left), slightly before the transition between the loop formation and maturation stages
684 (middle), and in the steady state (right). Color (rainbow: from red to blue) shows positions of 28 685 individual monomers along the chain and demonstrates that loci in compacted chromosomes
686 are arranged linearly according to their genomic location. Bottom row: same as above, with
687 chromosomes shown in semitransparent gray and condensins as red spheres.
688
689 Figure 2 - figure supplement 1. The geometrical parameters of compacted chromosomes
690 in simulations with altered parameters. (A) Simulations with altered condensin abundance:
691 “0.2x condensin”, “0.5x condensin”, “2x condensins” and “4x condensins” - simulations with 200,
692 500, 2000 and 4000 condensins, correspondingly. (B) Simulations with altered parameters of
693 the chromatin fiber and condensins: “Stiffer fiber” – simulations with 2x bending energy of the
694 fiber, “softer fiber” – simulations with 0.5x bending energy; “wide condensins” – simulations with
695 increased spacing between condensins at loop bases; “10nm fiber” - simulations with reduced
696 linear DNA of the fiber; “30nm fiber” – simulations with increased linear DNA density of the fiber
697 and increased fiber thickness. The vertical gray lines show the experimental observations on the
698 linear density of DNA along the human prophase chromosome I (6.25 kb/nm) and chromosome
699 XXII (5.0 kb/nm) 20. The horizontal gray lines show the experimental data on the diameter of
700 prophase chromosomes: 400 nm 27, 600 nm 26 and 679 nm 25.
701
702 Figure 3. Segregation and disentanglement of sister chromatids. (A) The median
703 separation between condensin-rich axes of two sister chromosomes (blue) and the mean
704 chromosome diameter (red) as a function of time, measured in condensin residence times, as
705 above. The blue band shows the range between 25th and 75th percentile across 10 simulation
706 replicas at each time point; the light blue shows 10%--90% range. (B) Entanglement of sister
707 chromatids measured by the linking number of the condensin cores of sister chromatids; the
708 blue bands show variability between simulation replicas, same as above. (C) Conformations of
709 two sister chromatids in the polymer simulations (shown in red and blue) illustrate the observed
710 segregation and disentanglement.
711
29 712 Figure 3 - figure supplement 1. Segregation and disentanglement of sister chromatids in
713 simulations with reduced Topo II concentration. (A) The median separation between
714 condensin-rich axes of two sister chromosomes (solid lines) and the mean chromosome
715 diameter (red) as a function of time, measured in condensin residence times, as above.
716 Depletion of topo II was simulated by a simultaneous increase of the energy barrier of fiber
717 overlap (shown by the line color) and a 3x increase in of the radius of repulsion. (B)
718 Entanglement of sister chromatids in simulations of topo II depletion, measured by the linking
719 number of the condensin cores of sister chromatids.
720
721 Figure 4. Steric repulsion between chromatin loops shapes the compacted chromosomes
722 and leads to segregation of sister chromatids. (A) Left, conformation of a single
723 chromosomes with chromatin fiber shown in semitransparent gray and condensins shown in
724 red; two loops are highlighted with blue. Right: the diagram showing that loop repulsion
725 straightens chromosomal cores and extends individual loops radially. (B) Left, cross section of
726 two parallel loop brushes placed at a short distance from each other. Right, the diagram shows
727 that steric interactions between loops of sister chromatids lead to chromatid repulsion and
728 segregation.
729
730 VIDEO CAPTIONS
731
732 Video 1: An illustration of the model of loop extrusion by condensins. When a condensin
733 (red sphere) binds to chromatin (grey fiber), it binds to two adjacent loci and then slides the two
734 contact points in the opposite directions, effectively extruding a loop (highlighted in orange). The
735 diagram on the bottom shows the position of the two loci brought together by the condensin
736 over time (red arc). Condensins occasionally unbind and then rebind to a randomly chosen site,
737 starting a new loop. When two condensins (loops shown in orange and purple) meet on the
738 chromosome, they stop extrusion on the collided sides. Condensins can also bind within already
739 extruded loops. If the growth of the outer loop is blocked by other loops or barriers (not shown), 30 740 the inner condensin can re-extrude it, forming a single loop reinforced by multiple condensins
741 (the number above the arc). Such a loop is stabilized against unbinding of individual condensins
742 and disassembles only upon unbinding of the last condensin. Available at
743 https://www.youtube.com/watch?v=xikxZ0yF1yU&list=PLnV-
744 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit&index=1
745
746 Video 2: A representative polymer simulation of compaction of 30 Mb chromosome by
747 loop extruding condensins in presence of strand-passing topo II. Top left, fiber color
748 corresponds to the position of each monomer along the genome. Top right, fiber shown in
749 semitransparent gray, condensins are shown as red spheres. Bottom, the diagram of the
750 crosslinks (red arcs) formed by condensins. The numbers above the arcs show the number of
751 condensins stacked at the bases of reinforced loops. We sped up the second half of the video
752 20 times to illustrate the slow phase of loop maturation and the long-term stability of the
753 compacted state. Available at https://www.youtube.com/watch?v=_Vc7__xfnfc&list=PLnV-
754 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit&index=2
755
756 Video 3: A polymer simulation of segregation and disentanglement of two sister
757 chromatids by loop extruding condensins in presence of strand-passing topo II. The two
758 sister chromatids (30Mb each) are shown in red and blue. Initially, the two chromatids were
759 compacted and twisted around each other. Second half of the video is sped up 20x. Available at
760 https://www.youtube.com/watch?v=stZR5s9n32s&index=3&list=PLnV-
761 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit
762
763
764 Video 4: Sixteen simulation replicas of chromatid segregation and disentanglement.
765 Available at https://www.youtube.com/watch?v=UXXLnGGit-8&index=4&list=PLnV-
766 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit
767
31 768 Video 5: Polymer simulation of segregation and disentanglement of two sister
769 chromatids with altered model parameters. Top row: left, simulations with a more flexible
770 fiber (0.5x bending energy); right – a stiffer fiber (2x bending energy); bottom row, left -
771 simulations with wider condensins at loop bases; right – simulations with disabled excluded
772 volume interactions. Available at
773 https://www.youtube.com/watch?v=YhgSHppJXVI&index=5&list=PLnV-
774 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit
775
776
777 Video 6: Polymer simulation of segregation and disentanglement of two sister
778 chromatids with modified condensin abundance. Top row, left - 2000 condensins per
779 chromatid, middle - 500 condensins per chromatid; right - 200 condensins; bottom row, left -
780 4000 condensins, left - 100 condensins; right, 50 condensins. Available at
781 https://www.youtube.com/watch?v=oq9TS5OXO1Q&index=6&list=PLnV-
782 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit
783
784
785 Video 7: A polymer simulation of segregation and disentanglement of two sister
786 chromatids upon topo II depletion. Depletion of topo II was simulated by a simultaneous
787 increase of the energy barrier of fiber overlap and the increased radius of repulsion, which
788 decreased, but not fully prevented chain crossings. Available at
789 https://www.youtube.com/watch?v=YO3Qx8liK3s&list=PLnV-
790 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit&index=7
791
792 Video 8: A polymer simulation of segregation and disentanglement of sister chromatids
793 in the presence of cohesins connecting the chromatids. We simulated cohesins as
794 crosslinks between the chromosomes (shown with the blue lines on the bottom arc diagram)
795 that can be pushed along each of the chromatids by loop-extruding condensins. Available at
32 796 https://www.youtube.com/watch?v=vmwNbz41pqM&list=PLnV-
797 JMzdy0ZMOvyK6Z1bzU7C2Jt8Brbit&index=8
798
33