Dynamic nuclear structure emerges from chromatin crosslinks and motors
Kuang Liu, Alison E. Patteson, Edward J. Banigan, J. M. Schwarz
DDynamic nuclear structure emerges from chromatin crosslinks and motors
Kuang Liu , Alison E. Patteson , Edward J. Banigan , J. M. Schwarz , Department of Physics and BioInspired Syracuse, Syracuse University, Syracuse,NY USA, Institute for Medical Engineering and Science and Department of Physics,MIT, Cambridge, MA Indian Creek Farm, Ithaca, NY, USA (Dated: August 18, 2020)The cell nucleus houses the chromosomes, which are linked to a soft shell of lamin filaments. Ex-periments indicate that correlated chromosome dynamics and nuclear shape fluctuations arise frommotor activity. To identify the physical mechanisms, we develop a model of an active, crosslinkedRouse chain bound to a polymeric shell. System-sized correlated motions occur but require bothmotor activity and crosslinks. Contractile motors, in particular, enhance chromosome dynamicsby driving anomalous density fluctuations. Nuclear shape fluctuations depend on motor strength,crosslinking, and chromosome-lamina binding. Therefore, complex chromatin dynamics and nuclearshape emerge from a minimal, active chromosome-lamina system.
The cell nucleus houses the genome, or the materialcontaining instructions for building the proteins that acell needs to function. This material is ∼ ∼ µ mnucleus [1]. Chromatin is highly dynamic; for instance,correlated motion of micron-scale genomic regions overtimescales of tens of seconds has been observed in mam-malian cell nuclei [2–6]. This correlated motion dimin-ishes both in the absence of ATP and by inhibition of thetranscription motor RNA polymerase II, suggesting thatmotor activity plays a key role [2, 3]. These dynamics oc-cur within the confinement of the cell nucleus, which isenclosed by a double membrane and 10-30-nm thick fila-mentous layer of lamin intermediate filaments, the lamina[7–9]. Chromatin and the lamina interact through vari-ous proteins [10–12] and form structures such as lamina-associated domains (LADs) [13, 14]. Given the complexspatiotemporal properties of a cell nucleus, how do cor-related chromatin dynamics emerge and what is their in-terplay with nuclear shape?Numerical studies suggest several explanations for cor-related chromatin motions. A confined Rouse chain withlong-range hydrodynamic interactions that is driven byextensile dipolar motors can exhibit correlated motionover long length and timescales [4]. Correlations arisedue to the emergence of local nematic ordering of withinthe confined globule. However, such local nematic order-ing has yet to be observed. In the absence of activity, aconfined heteropolymer may exhibit correlated motion,with anomalous diffusion of small loci [15, 16]. How-ever, in marked contrast with experimental results [2, 3],introducing activity in such a model does not alter thecorrelation length at short timescales and decreases it atlonger timescales.Since there are linkages between chromatin and thelamina, chromatin dynamics may influence the shape ofthe nuclear lamina. Experiments have begun to inves-tigate this notion by measuring nuclear shape fluctua-tions [17]. Depletion of ATP, the fuel for many molecu-lar motors, diminishes the magnitude of the shape fluc-tuations, as does the inhibition of RNA polymerase II transcription activity by α -amanitin [17]. Other studieshave found that depleting linkages between chromatinand the nuclear lamina, or membrane, results in moredeformable nuclei [18, 19], enhanced curvature fluctua-tions [20], and/or abnormal nuclear shapes [21]. Inter-estingly, depletion of lamin A in several human cell linesleads to increased diffusion of chromatin, suggesting thatchromatin dynamics is also affected by linkages to thelamina [22]. Together, these experiments demonstratethe critical role of chromatin and its interplay with thenuclear lamina in determining nuclear structure.To understand these results mechanistically, we con-struct a chromatin-lamina system with the chromatinmodeled as an active Rouse chain and the lamina as anelastic, polymeric shell with linkages between the chainand the shell. Unlike previous chain and shell mod-els [20, 23, 24], our model has motor activity. We imple-ment the simplest type of motor, namely extensile andcontractile monopoles, representative of the scalar eventsaddressed in an earlier two-fluid model of chromatin [25].We also include chromatin crosslinks, which may be aconsequence of motors forming droplets [26] and/or com-plexes [27], as well as chromatin binding by proteins, suchas heterochromatin protein I (HP1) [28]. Recent rheolog-ical measurements of the nucleus support the notion ofchromatin crosslinks [23, 24], as does indirect evidencefrom chromosome conformation capture (Hi-C) [29]. Inaddition, we explore how the nuclear shape and chro-matin dynamics mutually affect each other by comparingresults for an elastic, polymeric shell with those of a stiff,undeformable one.
Model:
Interphase chromatin is modeled as a Rousechain consisting of 5000 monomers with radius r c con-nected by Hookean springs with spring constant K . Weinclude excluded volume interactions with a repulsive,soft-core potential between any two monomers, ij , anda distance between their centers denoted as | (cid:126)r ij | , asgiven by U ex = K ex ( | (cid:126)r ij | − σ ij ) for | (cid:126)r ij | < σ ij , where σ ij = r c i + r c j , and zero otherwise. We include N C crosslinks between chromatin monomers by introducinga spring between different parts of the chain with thesame spring constant as along the chain. In addition to a r X i v : . [ q - b i o . S C ] A ug FIG. 1. Left: Two-dimensional schematic of the model. Cen-ter: Schematic of the two types of motors. Right: Simulationsnapshot. (passive) thermal fluctuations, we also allow for explicitmotor activity along the chain. In simulations with mo-tors, we assign some number, N m , of chain monomersto be active. An active monomer has motor strength M and exerts force F a on monomers within a fixed range.Such a force may be attractive or “contractile,” drawingin chain monomers, or alternatively, repulsive or “exten-sile,” pushing them away (Fig. 1). Since motors in vivo are dynamic, turning off after some characteristic time,we include a turnover timescale for the motor monomers τ m , after which a motor moves to another position on thechromatin.The lamina is modeled as a layer of 5000 identicalmonomers connected by springs with the same radii andspring constants as the chain monomers and an averagecoordination number z ≈ .
5, as supported by previ-ous modeling [20, 23, 24] and imaging experiments [7–9]. Shell monomers also have a repulsive soft core. Wemodel the chromatin-lamina linkages as N L permanentsprings with stiffness K between shell monomers andchain monomers (Fig. 1).The system evolves via Brownian dynamics, obeyingthe overdamped equation of motion: ξ ˙ r i = ( F br + F sp + F ex + F a ), where F br denotes the (Brownian) thermalforce, F sp denotes the harmonic forces due to chainsprings, chromatin crosslink springs, and chromatin-lamina linkage springs, and F ex denotes the force dueto excluded volume. We use Euler updating, a time stepof dτ = 10 − , and a total simulation time of τ = 500.For the passive system, F a = 0. In addition to the de-formable shell, we also simulate a hard shell by freezingout the motion of the shell monomers. To assess thestructural properties in steady state, we measure boththe radial globule, R g , of the chain and the self-contactprobability. After these measures do not appreciablychange with time, we consider the system to be in steadystate. See SM for these measurements, simulation pa-rameters, and other simulation details. Results:
We first look for correlated chromatin mo-tion in both hard shell and deformable shell sys-tems. We do so by quantifying the correlations be-tween the displacement fields at two different pointsin time. Specifically, we compute the normalized spa-tial autocorrelation function defined as C r (∆ r, ∆ τ ) = N (∆ r ) (cid:80) N (∆ r ) < d i ( r , ∆ τ ) · d j ( r + ∆r , ∆ τ ) >< d ( r , ∆ τ ) > , where ∆ τ is thetime window, ∆ r is the distance between the centers ofthe two chain monomers at the beginning of the timewindow, N (∆ r ) is the number of ij pairs of monomerswithin distance ∆ r of each other at the beginning of thetime window, and d i is the displacement of the i th chainmonomer during the time window, defined with respectto the origin of the system. Two chain monomers mov-ing in the same direction are positively correlated, whilemonomers moving in opposite directions are negativelycorrelated.Fig. 2 shows C r (∆ r, ∆ τ ) for passive and active samplesin both hard shell (Figs. 2 (a) and (b)) and soft shellcases for N C = 2500, N L = 50, and M = 5 (Figs. 2 (e)and (f)). Both the passive and active samples exhibitshort-range correlated motion when the time window issmall, i.e. , ∆ τ <
5. However, for longer time windows,both the extensile and contractile active samples exhibitmore long-range correlated motion than the passive case.These correlations are visible in quasi-2d spatial mapsof instantaneous chromatin velocities, which show largeregions of coordinated motion in the active, soft shell case(Figs. 2 (c) and (g)).To extract a correlation length to study the corre-lations as a function of both N C and N L , we use aWhittel-Marten (WM) model fitting function C r ( r ) = − ν Γ( ν ) (cid:16) rr cl (cid:17) ν K ν (cid:16) rr cl (cid:17) for each time window (Fig. 2(f)) [3]. The parameter ν is approximately 0 . e.g. , by particle imagevelocimetry) and do not subtract off the global centerof mass [2, 3, 6]. However, one experiment noted thatthey observed drift of the nucleus on a frame-to-framebasis, but considered it negligible over the relevant timescales [3]. Additionally, global rotations, which we have (a) (b) (c) (d)(e) (f) (g) (h) FIG. 2. (a) The spatial autocorrelation function C r (∆ r, ∆ τ ) for passive and extensile cases at different time lags, ∆ τ , for thehard shell, while (b) shows the contractile and passive case. (c) Two-dimensional vector fields for ∆ τ = 5 (left), 50 (right) forthe passive case (top) and the contractile case (bottom). (d) The correlation length as a function of N L and N C for the twotime lags in (c). (e ∼ h): The bottom row shows the same as the top row, but with a soft shell. See SM for representative fitsto obtain the correlation length. (a) (b)(c) (d) FIG. 3. (a) MSD for the hard shell case with N C = 2500, N L = 50, and M = 5. For the inset, N C = 0. (b) Densityfluctuations for the same parameters as in (a). Figures (c)and (d) show the soft shell equivalent to (a) and (b). not considered, could yield large-scale correlations.We also study the mean-squared displacement of thechromatin chain to determine if the experimental featureof anomalous diffusion is present. Figs. 3 (a) and (c)show the mean-squared displacement of the chain with N L = 50 and N C = 2500 as measured with reference tothe center-of-mass of the shell for both the hard shelland soft shell cases, respectively. For the hard shell,the passive chain initially moves subdiffusively with anexponent of α ≈ .
5, which is consistent with an un-crosslinked Rouse chain with excluded volume interac-tions [31]. However, the passive system crosses over topotentially glassy behavior after a few tens of simulationtime units. We present N C = 0 case in the inset to Fig. 3(a) for comparison to demonstrate that crosslinks are po-tentially driving a gel-sol transition as observed in priorexperiments [32]. The active hard shell samples exhibitlarger displacements than passive samples, with α ∼ . FIG. 4. Power spectrum of the shape fluctuations with N L = 50 and N C = 2500 for the passive and both activecases. Different motor strengths are shown. The insets showsexperimental data from mouse embryonic fibroblasts with animage of a nucleus with lamin A/C stained. for a central cross-section as a function of wavenumber q for different motor strengths. We observe that the shapefluctuation spectrum is broad until saturating due to thediscretization of the system. The decrease in the shapefluctuations is less significant for both the passive andextensile systems than for the contractile system withan approximate q − scaling, characteristic of membranetension, for the former versus an approximate q − scal-ing for the latter. This difference could be due to themore anomalous density fluctuations in the contractilecase, demonstrating that chromatin spatiotemporal dy-namics directly impacts nuclear shape. We do not ob-serve a q − contribution due to emergent bending, whichwas suggested by previous experiments [17] and simula-tions [23]. However, additional experiments measuringnuclear shape fluctuations of mouse embryonic fibrob-lasts (MEFs) also do not show a bending contribution(inset to Fig. 4 and see SM for materials and methods).Additionally, the amplitude of the shape fluctuations in-creases with motor strength, N C , and N L (see SM). Discussion:
We have studied a composite chromatin-lamina system in the presence of activity, crosslinking,and number of linkages between chromatin and the lam-ina. Our model captures the correlated chromatin mo-tion on the scale of the nucleus in the presence of bothactivity and crosslinks (Fig. 2). The deformability of theshell also plays a role. We find that global translations ofthe composite soft shell system contribute to the correla-tions. We observe anomalous diffusion for the chromatin(Figs. 3 (a) and (c)), as has been observed experimen-tally [22], with a crossover to a smaller anomalous ex-ponent driven by the crosslinking [32]. Interestingly, the contractile system exhibits a larger MSD than the exten-sile one, which is potentially related to the more anoma-lous density fluctuations in the contractile case (Figs. 3(b) and (d)). Finally, nuclear shape fluctuations dependon motor strength and on amounts of crosslinking andchromatin-lamina linkages (Fig. 4). Notably, the con-tractile case exhibits more dramatic changes in the shapefluctuations as a function of wavenumber as compared tothe extensile case.Our short-ranged, overdamped model contrasts withan earlier confined, active Rouse chain interacting witha solvent via long-range hydrodynamics [4]. While bothmodels generate correlated chromatin dynamics, with theearlier model, such correlations are generated only withextensile motors that drive local nematic ordering of thechromatin chain [4]. Moreover, our correlation lengthsare significantly larger than those obtained in a confinedactive, heteropolymer simulation [15]. Activity in thisearlier model is modeled as extra-strong thermal noisesuch that the correlation length decreases at longer timewindows as compared to the passive case. This decreasecontrasts with our results (Figs. 2 (d) and (h)) and ex-periments [3]. In addition, our model takes into accountdeformability of the shell and the chromatin-lamina link-ages. Future experiments could potentially distinguishthese mechanisms by looking for prominent features ofour model, such as a dependence on chromatin bridgingproteins and linkages to the lamina and effects of whole-nucleus motions.Our modeling motivates further spatiotemporal stud-ies of nuclear shape. Particularly interesting would be in vivo studies with vimentin-null cells, which have min-imal mechanical coupling between the cytoskeleton andthe nucleus. Vimentin is a cytoskeletal intermediate fil-ament that forms a protective cage on the outside of thenucleus and helps regulate the nucleus-cytoplasm cou-pling and, thus, affects nuclear shape [8]. The ampli-tudes of the nuclear shape fluctuations in vimentin-nullcells may increase due to a softer perinuclear shell or maydecrease due to fewer linkages between the nucleus andthe mechanically active cytoskeleton.There are intriguing parallels between cell shape [34–36] and nuclear shape with cell shape being driven byan underlying cytoskeletal network—an active, filamen-tous system driven by polymerization/depolymerization,crosslinking, and motors, both individually and in clus-ters, that can remodel, bundle and even crosslink fila-ments. Given the emerging picture of chromatin mo-tors acting collectively [26, 27], just as myosin motorsdo [37], the parallels are strengthened. Moreover, themore anomalous density fluctuations for the contractilemotors as compared to the extensile motors could poten-tially be relevant in random actin-myosin systems typi-cally exhibiting contractile behavior, even though eitheris allowed by a statistical symmetry [38]. On the otherhand, distinct physical mechanisms may govern nuclearshape since the chromatin fiber is generally softer thancytoskeletal filaments and the lamina is stiffer than thecell membrane.We now have a minimal chromatin-lamina model thatcan be augmented with additional factors, such as dif-ferent types of motors—dipolar, quadrupolar, and evenchiral, such as torque dipoles. Chiral motors may readilycondense chromatin just as twirling a fork “condenses”spaghetti. Finally, it is now established that nuclear actinexists in the cell nucleus, yet its form is under investiga-tion [39]. We propose that short, but stiff, actin filamentsacting as stir bars can potentially increase the correlationlength of micron-scale chromatin dynamics. 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CONTENTS
A. Model 11. System and initialization 12. Parameters 1B. Simulation results 21. Radius of gyration 22. Self-contact probability 33. Mean-Squared Displacement 34. Density fluctuations 45. Correlation function and correlation length 46. Shape fluctuations 7C. Experiments 8References 9
Appendix A: Model1. System and initialization
We use a Rouse chain with soft-core repulsion between each monomer capturing excluded volume effects to representthe chromatin. Since the chromatin is contained within the lamina, modeled as a polymeric shell, we present theprotocol to obtain the initial configuration for the composite system. As shown in Fig. S1(left), we first implementa three-dimensional self-avoiding random walk in an FCC lattice for 5000 steps to generate the chain. We thensurround the chain in a large polymeric, but hard, shell. To create the shell, we generate a Fibonacci sphere with5000 nodes and identify 5000 identical monomers with these nodes. The springs between the shell monomers forma mesh and each shell monomer is connected to 4 . R s is 10. We set the monomerradius to be r c = 0 . φ is approximately 0 . φ is smaller in soft-shell cases due to expansion as the shell monomers undergo thermalfluctuations.
2. Parameters
In our simulations, we use the set of parameters shown in Table 1.
FIG. S1. Left: The chain is initially generated via a self-avoiding random walk on an FCC lattice. Center: The chain is thenenclosed in a Fibonacci sphere. Right: Composite system at time τ = 0. Diffusion constant D k B T dτ − Number of chain monomers N r c . N s r s . R s φ . K K ex N m M / τ m . N C / / / / / N L / / / / ξ µ m, one simulation time unit corresponds to 0.5 seconds, and one simulation energy scale corresponds to approx-imately 10 − J = k B T , T = 300 K. With this mapping, the spring constant corresponds to approximately 1 . × − µ m with a Young’s modulus for the chain of 0 .
28 Pa.
Appendix B: Simulation results1. Radius of gyration
For a polymer, the radius of gyration is defined as R g = (cid:80) ( r i − r cm ) /N , where N = 5000 is total chain monomernumber. In the hard shell case, we fix the radius of the shell to R s = 10. In the soft-shell case, the shell expandsdue to the thermal fluctuations and due to the activity of the chain inside. Fig S2 shows the radius of gyration ofthe chain (solid lines) and the average radius of shell (dashed lines) in the soft shell case as function of time. Aftera short-time initial expansion, both the chain’s and the shell’s respective radii reach a plateau by 100 τ for mostparameters, indicating that the system is reaching steady state. Only for the zero crosslinks with contractile activity,does the radius of gyration continue to increase slightly over the duration of the simulation of 500 τ . τ . . . r a d i u s [ R s ] passiveextensile contractile τ . . . r a d i u s [ R s ] passiveextensile contractile τ . . . r a d i u s [ R s ]
50 linkages, 2500 crosslinks passiveextensile contractile τ . . . r a d i u s [ R s ]
600 linkages, 0 crosslinks passiveextensile contractile τ . . . r a d i u s [ R s ]
600 linkages, 2500 crosslinks passiveextensile contractile
FIG. S2. Radius of gyration of the chain (solid lines) and average radius of the shell(dashed lines) as function of simulationtime for N C = 2500 and N L = 50 (middle figure). For contrast, R g for N L = 0 ,
600 and N c = 0 ,
2. Self-contact probability
Since the globule radius is an averaged quantity, we also look for steady state signatures in the self-contact proba-bility, which yields information about the chromatin spatial structure. More specifically, Hi-C allows one to quantifythe local chromatin interaction domains at the megabase scale [4]. Such domains are stable across different eukaryoticcell types and species [5]. To quantify such interactions in the simulations, one determines the number of monomersin the vicinity of the ith chain monomer. In other words, one creates an adjacency matrix. This adjacency matrixis shown Fig. S3 for two examples. To compute the self-contact probability, one sets a threshold distance that apair of monomers within that range is considered to be in contact. Then the fraction of contacted pairs for eachpolymeric distance 1 , , , , ... is calculated. This fraction as a function of polymeric distance is called the self-contactprobability. See Fig. S4 for the self-contact probability for N L = N C = 0 at the beginning and at the end of thesimulation for the soft shell case. While there is some change between the two, in Fig. S5, we show the self-contactprobability for different times τ to demonstrate that after τ = 50, the probability does not change with time, implyinga steady state. FIG. S3. Contact map for a soft shell, contractile system with no linkages or crosslinks at the beginning and at the end of thesimulation, i.e. τ = 0 and τ = 500.
3. Mean-Squared Displacement
To quantify the dynamics of the chain, we compute its mean-squared displacement (MSD) measured with respectto the center of mass of the shell. Fig. S6 plots the MSD of the chain during the duration of the simulation. At shorttime scales, the chain undergoes sub-diffusive motion and the MSD follows an exponent around α ≈ . N C = 2500and N L = 50. At longer time scales, the MSD crosses over to a smaller exponent. The value of the exponent dependson N C and N L . In all cases, the active systems diffuse faster than the passive system, and contractile motors enhancediffusion more than extensile motors. The insets in Fig. S6 show the MSD for the center of mass of the chromatinchain for the soft shell. For the crosslinked, active chain, this MSD is slightly faster than diffusive. polymeric distance10 − − − C o n t a c t p r o b a b ili t y passiveextensilecontractile polymeric distance10 − − − C o n t a c t p r o b a b ili t y passiveextensilecontractile FIG. S4. Self-contact probability for the hard shell case (left) and the soft shell case (right) with the latter corresponding toright figure in previous Fig. S3. polymeric distance10 − − − C o n t a c t p r o b a b ili t y passivecontractile polymeric distance10 − − − C o n t a c t p r o b a b ili t y passivecontractile FIG. S5. Self-contact probability at τ = 0 , , , , , τ (left) and for τ = 50 and τ = 300 (right) for soft shell passive andcontractile systems with N C = 2500 and N L = 50.
4. Density fluctuations
The density fluctuations are computed in the following way: • Select a spherical region in the system with radius r d and count the number of monomers in that region. • Randomly select spherical regions at other places with the same radius and count the monomers included. • Compute the variance of counted monomer amount σ for this radius r d . • Vary r d and repeat the above three steps and obtain the variance for each r d .We plot σ as a function of r d . Typically, for a group of randomly distributed monomers in three dimensions,the density fluctuations scale as σ ∼ r − d . From Fig. S7 we see that the overall density fluctuations are broader inthe active cases, as compared to the passive cases. Contractile motors induce more anomalous density fluctuations,particularly in the soft shell case.
5. Correlation function and correlation length
To evaluate the spatial and temporal correlation motion along the chain, we compute the spatial autocorrelationfunction. Suppose (cid:126)d ( (cid:126)r, ∆ τ ) is the displacement of monomer at (cid:126)r over time, (cid:126)d ( (cid:126)r + ∆ r, ∆ τ ) is the displacement ofanother monomer, which is located a distance ∆ r away and over the same time window. We then use the functionbelow to compute the correlation function: C (∆ r, ∆ τ ) = (cid:104) (cid:126)d ( (cid:126)r, ∆ τ ) · (cid:126)d ( (cid:126)r + ∆ r, ∆ τ ) (cid:105)(cid:104) (cid:126)d ( (cid:126)r, ∆ τ ) (cid:105) . . FIG. S6. MSD as a function of time for N C = 2500, N L = 50, and M = 5 (middle column) and for four extreme cases (0 or600 linkages, 0 or 2500 crosslinks) in the hard shell (top row) and the soft shell (bottom row). Insets are MSD plots of thecenter of mass of the chain. From Ref. [6] we assume the correlation function follows C r ( r ) = − ν Γ( ν ) (cid:16) rr cl (cid:17) ν K ν (cid:16) rr cl (cid:17) , where r cl is the extractedcorrelation length, K ν is the Bessel of the second type of order ν , and ν is a smoothness parameter. Larger ν denotes that the underlying spatial process is smooth, not rough, in space. In Fig. S8 we show the correlated functioncomputed from numerical simulations (dots) and the fitted correlation function from the above formula (lines) fordifferent parameters. Lines from light to dark represent time windows from short to long (1 τ , 2 τ , 5 τ , 10 τ , 20 τ , 50 τ ,100 τ , 200 τ ). We see that the numerical results with shorter time windows fit the formula better.In Fig. S9, we plot the correlation length a function of linkage number N L and crosslink number N C over the shorttime window 5 τ and the long time window 50 τ . We observe that active motors clearly enhance the correlation length.It is also clear that presence of crosslinks also enhance correlation length. The correlation length is larger for the softshell case. In the soft shell case, without subtracting the diffusion of the center of mass, the correlation length for thelong time window spans almost the radius of the system. We note that the correlation length is reduced if we subtractthe center of mass shell motion, however, it still remains larger than the hard shell case. A quasi-two-dimensionalcorrelation length is computed from a slab-like region and is also shown for potential comparison to experimental − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile − length scale [ R S ]10 − − − − − − d e n s i t y fl u c t u a t i o n − − passiveextensilecontractile FIG. S7. Density fluctuations for N C = 2500 and N L = 50 (middle column) and four extreme cases (0 or 600 linkages, 0 or2500 crosslinks) in the hard shell (top row) and in the soft shell (bottom row). The arrangement of parameters is the same asin the previous figure. . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r ) . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
50 linkages, 2500 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 0 crosslinks . . . . r [ R S ]0 . . . . . . c o rr e l a t i o n C r ( ∆ r )
600 linkages, 2500 crosslinks
FIG. S8. Correlation functions for N C = 2500 and N L = 50 (middle column) and four extreme cases (left column: 0 linkages and0 crosslinks; second from left column: 0 linkages and 2500 crosslinks; second from right column: 600 linkages and 0 crosslinks;right column: 600 linkages and 2500 crosslinks). Top two rows: The three-dimensional correlation function for the hard shell;Middle two rows: The three-dimensional correlation functions for the soft shell; Bottom two rows: Two-dimensional correlationfunctions for the soft shell. Color varies from light to dark as time lag equals 1 τ , 2 τ , 5 τ , 10 τ , 20 τ , 50 τ , 100 τ , 200 , τ ,respectively. Symbols denote the numerical results, while the dashed line represent the fitted correlation functios. Greyscale:passive. Bluescale: active with extensile motors. Redscale: active with contractile motors. results since, in the experiments, the correlated length is extracted using this method. There is not much differencebetween the three-dimensional correlation length and the two-dimensional correlation length with the center of massof the shell subtracted. We also show the corrrelation length as a function of shell stiffness (with the COM of shellsubtracted) to demonstrate the direct effect of shell stiffness on the correlated chromatin motion (see Fig. S10). N L . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N L . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N L . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N L . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N C . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N C . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N C . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile N C . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile FIG. S9. Plot of correlation length as function of linkage number N L (top row) or crosslink number N C (bottom row) for timewindows 5 τ (light) and 50 τ (dark). From left to right columns: The three-dimensional correlation length for the hard shell;the three-dimensional correlation length for the soft shell; three-dimensional correlation length for the soft shell with the COMmotion subtracted; two-dimensional correlation length for the soft shell with the COM motion subtracted. . . . . . . . . . c o rr e l a t i o n l e n g t h [ R s ] passiveextensile contractile FIG. S10. Plot of the correlation length as a function of shell stiffness for time windows ∆ τ = 5 ,
50. Here N C = 2500, N L = 50,and M = 5.
6. Shape fluctuations
To evaluate shape fluctuations of the shell, we compute it in two ways. First, in order to compare with experimentalmeasurements, we select a random slab through the center and project the coordinates of the shell monomers in theslab to the plane where slab lies. Then, we compute the fast-Fourier-transform (FFT) for spatial deviations of thesemonomers from the average radius with the deviations with h q denoting the Fourier transform of the deviation with q < h q > − q < h q > − q < h q > − q < h q > − q < h q > − q < h q > − FIG. S11. Power spectrum of the shape fluctuations of a random slab for different boundary linkages (top row) and crosslinks(bottom row). Left column: Extensile motor case. Middle column: Contractile motor case. Right column: Passive case. respect to wavenumber q . In Fig. S11, the power spectrum of the shape fluctuations for the passive and extensilecases follow a decay exponent of −
2, as expected for a stretchable shell [7]. The spectrum of the shape fluctuationsincreases monotonically with the number of crosslinks. The specturm varies more dramatically with contractile motorsas compared to extensile motors. Moreover, the shape fluctuation spectrum also eventually saturates as a function ofchromatin-lamina linkage number. In S12 we compute the spectrum of the shape fluctuations as characterized by thespherical harmonic functions (the Y lm s with l as the dimensionless spherical wavenumber). We obtain similar trendsas in Fig. S11. Finally, in Fig. S13, we plot the spectrum for different motor strengths and different shell stiffnesses. Appendix C: Experiments
To measure nuclear shape fluctuations in live cells, the wild-type mouse embryonic fibroblasts (mEFs) were kindlyprovided by J. Eriksson, Abo Akademi University, Turku, Finland. Cells were cultured in DMEM with 25 mM Hepesand sodium pyruvate supplemented with 10% FBS, 1% penicillin/streptomycin, and nonessential amino acids. Thecell cultures were maintained at 37 degrees C and 5% CO .Cell nuclei were fluorescently labeled by transient transfection with pEGFP-C1-NLS, 48 h before imaging. Cellnuclei were imaged at 2-min increments for 2 h by using wide-field fluorescence with a 40 × objective. To quantifythe structural features of nuclei, we traced the contour, r ( θ ), of the NLS-GFP labeled nuclei at each time point. Theshape of the nucleus was identified using a custom-written Python script, and its contour was interpolated from 0 to2 π by 150 points. Next, the shape fluctuations were calculated as h ( θ ) = r ( θ ) − r , where r is the average radiusfor each cell at each time point. The wave number-dependent Fourier modes of the fluctuations, h q , were obtained asFourier transformation coefficients, as described in Ref [8].The shape fluctuations were quantified for each cell by computing the Fourier mode magnitude square h ( q ) andaveraging over each time point. The average shape fluctuations as shown in Fig. 4 in the main text was taken as the l − − − − < h l > − l − − − − < h l > − l − − − − < h l > − l − − − − < h l > − l − − − − < h l > − l − − − − < h l > − FIG. S12. Power spectrum of the shape fluctuations in spherical harmonics, where l is the dimensionless spherical wavenumberfor different chromatin-lamina linkages (top row) or crosslinks (bottom row). Left column: Extensile motor case. Middlecolumn: Contractile motor case. Right column: Passive case. q < h q > − M = 5, extensile M = 5, contractile M = 0, passive M = 25, extensile M = 25, contractile q < h q > − K shell = K , extensile K shell = K , contractile K shell = 20 K , extensile K shell = 20 K , contractile K shell = 0 . K , extensile K shell = 0 . K , contractile l − − − − < h l > − M = 5, extensile M = 5, contractile M = 0, passive M = 25, extensile M = 25, contractile l − − − − < h l > − K shell = K , extensile K shell = K , contractile K shell = 20 K , extensile K shell = 20 K , contractile K shell = 0 . K , extensile K shell = 0 . K , contractile FIG. S13. Power spectrum of the shape fluctuations for different motor strengths or shell stiffnesses. Left two: q plot. Righttwo: Y lm plot. average over 15 cells per condition from two independent experiments. [S1] H. D. Ou, S. Phan, T. J. Deerinck, A. Thor, M. H. Ellisman, and C. C. O’Shea, Science , (6349):eaag0025 (2017).[S2] M. Falk, Y. Feodorova, N. Naumova, M. Imakaev, B. R. Lajoie, H. Leonhardt, B. Joffe, J. Dekker, G. Fudenberg, I. Solovei,and L. A. Mirny, Nature , 395 (2019).[S3] H. Kang, Y.-G. Yoon, D. Thirumalai, and C. Hyeon, Phys. Rev. Lett. , 198102 (2015).[S4] E. Lieberman-Aiden, N. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. R. Lajoie, P. J. Sabo,M. O. Dorschner, R. Sandstrom, B. Bernstein, M. A. Bender, M. Groudine, A. Gnirke, J. Stamatoyannopoulos, L. A.Mirny, E. S. Lander, and J. Dekker, Science , 289 (2009).[S5] J. Dekker and L. Mirny, Cell , 1110 (2016).[S6] H. A. Shaban, R. Barth, and K. Bystricky, Nucleic Acids Res. , 11202 (2018). [S7] L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity , 3rd ed. (Butterworth-Heinemann,2012).[S8] A. E. Patteson and et al., J. Cell Biol.218