Dynamics scaling behavior and control of nuclear wrinkling Jonathan A. Jackson1 2Nicolas Romeo3 4Alexander Mietke3 5Keaton J. Burns3 Jan F. Totz3Adam C. Martin1J orn Dunkel3and Jasmin Imran Alsous6

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Dynamics, scaling behavior, and control of nuclear wrinkling

Jonathan A. Jackson,1, 2, ∗Nicolas Romeo,3, 4, ∗Alexander Mietke,3, 5 Keaton J. Burns,3

Jan F. Totz,3Adam C. Martin,1J¨orn Dunkel,3, †and Jasmin Imran Alsous6, ‡

1Department of Biology, Massachusetts Institute of Technology

2Graduate Program in Biophysics, Harvard University

3Department of Mathematics, Massachusetts Institute of Technology

4Department of Physics, Massachusetts Institute of Technology

5School of Mathematics, University of Bristol

6Center for Computational Biology, Flatiron Institute, Simons Foundation

(Dated: June 27, 2023)

The cell nucleus is enveloped by a complex membrane, whose wrinkling has been implicated in

disease and cellular aging. The biophysical dynamics and spectral evolution of nuclear wrinkling

during multicellular development remain poorly understood due to a lack of direct quantitative

measurements. Here, we combine live-imaging experiments, theory, and simulations to characterize

the onset and dynamics of nuclear wrinkling during egg development in the fruit ﬂy, Drosophila

melanogaster, when nurse cell nuclei increase in size and display stereotypical wrinkling behavior.

A spectral analysis of three-dimensional high-resolution data from several hundred nuclei reveals

a robust asymptotic power-law scaling of angular ﬂuctuations consistent with renormalization and

scaling predictions from a nonlinear elastic shell model. We further demonstrate that nuclear wrin-

kling can be reversed through osmotic shock and suppressed by microtubule disruption, providing

tunable physical and biological control parameters for probing mechanical properties of the nuclear

envelope. Our ﬁndings advance the biophysical understanding of nuclear membrane ﬂuctuations

during early multicellular development.

Wrinkling and ﬂickering of ﬂexible sheet-like structures

essentially determine mechanics and transport in a wide

range of physical and biological systems, from graphene

[1, 2] and DNA origami [3] to nuclear envelopes (NEs)

[4, 5, 12, 14] and cell membranes [8, 9]. Over the last

decade, much progress has been made through experi-

mental and theoretical work in understanding the eﬀects

of environmental ﬂuctuations on the bending behaviors

of carbon-based monolayers [10] and the shape deforma-

tions of lipid bilayer membranes of vesicles [11–13] and

cells [14, 61]. In contrast, the emergence and dynami-

cal evolution of surface deformations in NEs [12, 14, 15]

at diﬀerent length- and timescales, to which we re-

fer throughout this paper simply as ‘wrinkling’, still

pose fundamental open questions, as performing three-

dimensional (3D) observations at high spatio-temporal

resolution remains challenging under natural physiolog-

ical and developmental growth conditions. Speciﬁcally,

it is unclear how NE wrinkle formation proceeds during

cellular development, which biophysical processes govern

wrinkle morphology, and whether there exist character-

istic scaling laws for NE surface ﬂuctuations [5, 12, 17].

Addressing these questions through quantitative mea-

surements promises insights into the physics of complex

membranes and can clarify the biological and biomedical

implications of NE deformations that have been linked

to gene expression [12], cellular aging [18], and diseases

like progeria syndrome [4, 19].

∗These authors contributed equally

†dunkel@mit.edu

‡jalsous@ﬂatironinstitute.org

Here, we combine 3D confocal microscopy, theoretical

analysis, and simulations to characterize the wrinkling

morphology and dynamics of nuclear surfaces in fruit

ﬂy egg chambers. A spectral analysis of over 300 nu-

clei provides evidence for an asymptotic power-law scal-

ing of the surface ﬂuctuations, consistent with predic-

tions from renormalization calculations [48, 52] and scal-

ing arguments based on a nonlinear elasticity model for

thin shells. Although the scaling is found to be highly

robust against physical and biological perturbations, its

magnitude (prefactor) can be tuned via osmotic pressure

variation and microtubule disruption. These two diﬀer-

ent control mechanisms enable the tuning and probing

of the NE’s spectral and mechanical properties, and pro-

vide biophysical strategies for suppressing and reversing

nuclear wrinkling.

The NE is a double membrane that separates the cell’s

nuclear interior from the surrounding cytoplasm. The

two concentric ∼4 nm-thick lipid bilayers are ∼20-50 nm

apart and are supported by the nuclear lamina, a non-

contractile meshwork of intermediate ﬁlaments that lie

adjacent to the inner nuclear membrane, conferring me-

chanical stability and aﬀecting essential cellular processes

through regulation of chromatin organization and gene

expression [23]. Among other proteins, the NE contains

nuclear pore complexes, multi-protein channels that pri-

marily regulate passage of macromolecules between the

nucleus and the cytoplasm [24, 25]. Recent in vitro stud-

ies have provided key insights into the role of lamins,

cytoplasmic structures, and the physical environment in

aﬀecting NE morphology, as well as evidence for the crit-

ical importance of nuclear shape for many cellular and

nuclear functions [4, 17], including transcriptional dy-

arXiv:2210.11581v4 [physics.bio-ph] 25 Jun 2023

e f

(i)

(ii)

(iii)

(iv)

Older egg chambers

Younger egg chambers

Time proxy

100 150 200

10-3

10-2

10-1

Roughness

100101

Power spectrum

Angular number

10-6

10-4

10-2

-1

Time proxy

100 120 140 160 180 200

Nurse cell nuclei Oocyte

Follicle cell nuclei

Spectral

reconstruction

0 min 10 min

20 min 30 min

Nup107

Data

80 109 140 166 186 217

FIG. 1. Dynamic wrinkling of nurse cell nuclear envelopes during Drosophila egg development. a, Maximum-

intensity projection (MIP) of a 3D image of an egg chamber expressing GFP-labeled Nup107, a component of the nuclear pore

complex. The wrinkled nuclei of the 15 nurse cells are substantially larger than those of the surrounding follicle cells. b, MIP of

four egg chambers showing an increase in nurse cell nuclear size and nuclear surface deformation as egg chambers age. Curved

arrows indicate developmental progression from youngest (i) to oldest (iv). c, MIPs of individual nurse cell nuclei from six egg

chambers spanning all ages included in our dataset, showing an increase in nuclear radius and NE wrinkling with age. Note

that scale bar is the same size for each image; oldest nucleus shown is about 2.3 times the diameter of youngest shown. d,

Spectral reconstruction of NE surfaces shown in cfrom 3D microscopy data using spherical harmonics with an angular number

up to lmax = 25 (Eq. (1), Methods). Time proxy values for each nucleus are included above the reconstructions. e, Power

spectra normalized by average radius for N= 78 nuclei from 39 egg chambers in nurse cells directly connected to the oocyte

(results are qualitatively similar for nuclei farther away from the oocyte; Supp. Fig. S4). Hashed area indicates approximate

noise threshold for young nuclei; color indicates the time proxy (corresponding to the color bar in d) as deﬁned in the text and

detailed in SI Sec. III 1. f, NE roughness R=Pl≥3(2l+ 1)Plfor the same nuclei as in eincreases exponentially with time

proxy (see also Supp. Fig. S4). g, Snapshots of the same nucleus at four diﬀerent time points illustrate that NE wrinkling

is a dynamic process (Supplementary Video 2). Blue and orange arrowheads point to wrinkles that disappear and appear,

respectively, between subsequent frames. Scale bars: 50 µm (a, b), 10 µm (c, g).

namics [12]. Despite notable progress, a quantitative

understanding of how wrinkling phenomenology and 3D

spectral properties of nuclear surfaces evolve in time and

during cellular development has remained elusive.

To investigate the biophysical dynamics, scaling be-

haviors, and reversibility of nuclear wrinkling, we used

the egg chamber of the fruit ﬂy Drosophila melanogaster,

a powerful system amenable to 3D high-resolution live

imaging and targeted biological and physical perturba-

tions [26]. The egg chamber contains 15 nurse cells and

the oocyte (the immature egg cell), all connected via cy-

toplasmic bridges and enclosed by a thin layer of hun-

dreds of follicle cells (Fig. 1a, with schematics in Supp.

Fig. S1a, [27]). For most of the ∼3 days of oogenesis, the

nurse cells supply proteins, mRNAs, and organelles to

the oocyte through diﬀusion and microtubule-mediated

directed transport [31, 33, 34, 60]. To provide the prodi-

gious amount of material and nutrients that the oocyte

needs, each nurse cell replicates its DNA ∼10 times with-

out undergoing cell division, thereby notably increasing

its nuclear and cell sizes [35]. In the ∼30-hour window

studied here, the diameter of nurse cell nuclei in the cells

directly connected to the oocyte increases from approxi-

mately 16 to about 40 micrometers [33, 35], accompanied

by the progressive appearance of fold-like deformations in

the NE, providing an ideal test bed for studying the onset

and evolution of NE wrinkling (Fig. 1b,c).

To compare nurse cell nuclei within the same egg cham-

ber and across diﬀerent egg chambers, we deﬁned a proxy

measurement for developmental time (referred to here as

the ‘time proxy’) based on the geometric average of the

egg chamber’s length and width (Methods, SI Sec. III 1,

Supp. Fig. S1b,c). Since egg chamber geometry cor-

relates closely with developmental progression, adopt-

ing this continuous geometric characterization oﬀers ﬁner

temporal resolution than the traditional approach of dis-

tinguishing 14 discrete morphological stages [33, 34] (for

a comparison between the time proxy and developmen-

tal stage, see Supp. Fig. S1c). By time-ordering nu-

clei according to this metric, we could more accurately

determine the time of emergence of nuclear wrinkling

and reconstruct its evolution (Fig. 1b,c). To track the

NEs of the nurse cells in space and time, we used a

ﬂuorescently-tagged version of the nuclear pore complex

protein Nup107 that delineates the nucleus (Supplemen-

tary Video 1; qualitatively similar wrinkling patterns

were observed using a diﬀerent labeled protein in the

NE and via label-free imaging, see Supp. Fig. S2); note

that this label allows observation only of deformations

that include both membranes of the nuclear envelope,

but is unlikely to label deformations that include only

the inner membrane, such as Type I nucleoplasmic retic-

ula [34, 35]. Having acquired highly resolved 3D imag-

ing data (Fig. 1c, Supp. Fig. S3), we reconstructed the

nuclear surface radius R(θ, ϕ) relative to the geometric

center of the nucleus, where θand ϕare the spherical

polar angles.

To obtain a compact 3D spectral representation of

the nuclear surface deformations, we computed the real

spherical harmonic coeﬃcients flm, deﬁned by

R(θ, ϕ) =

lmax

l=0

m=−l

flmYlm(θ, ϕ),(1)

where Ylm is the spherical harmonic with angular num-

ber land order m(Methods). Equation (1) allows for

a continuous reconstruction of the NEs (Fig 1d, Supp.

Fig. S3), with the mode-cutoﬀ lmax setting the angular

resolution of the spectral representation (Methods). The

coeﬃcient values {flm}depend on the choice of coordi-

nate system, that is, the orientation of the nuclei. To

obtain a rotation-invariant characterization of the sur-

face wrinkles, we consider the power spectrum of radial

out-of-plane deformations

Pl=4π

(2l+ 1)f2

m=−l

lm,(2)

normalized by the average radius of the shell ⟨R⟩=

f00/√4π. The non-negative numbers Plmeasure the

average power in a mode of angular wavenumber l. A

single-valued summary statistic of surface wrinkling can

be given in terms of the ‘roughness’ parameter R=

Pl≥3(2l+ 1)Pl, the total power contained in angular

numbers l≥3. By ignoring the long-wavelength modes

l < 3, Rmeasures the contribution of ﬁner-scale wrinkles

to NE deformations. Our analysis of over 300 nurse cell

nuclei shows that the power spectrum of NEs maintains

an approximately constant shape as development pro-

gresses, but with a steadily-increasing amplitude (Fig. 1e;

Supp. Fig. S4), reﬂecting the fact that wrinkling becomes

more pronounced as nuclei increase in size. Rincreases

exponentially with the time proxy (Fig. 1f), suggesting

that nurse cell nuclei transition smoothly from an un-

wrinkled to a wrinkled state.

Nuclear surface wrinkling is a highly dynamic pro-

cess [12]. By imaging individual nurse cells at ∼40 s inter-

vals, we too observed that NE surface shapes ﬂuctuate

substantially, with smaller features appearing and dis-

appearing faster than larger ones (Fig. 1g, Supplemen-

tary Video 2). Speciﬁcally, power spectra Plof repeat-

edly imaged nuclei changed on timescales of minutes or

faster (Supp. Fig. S4 and Supp. Fig. S5). The rotational

invariance of spectra implies that these ﬂuctuations are

not the result of whole body rotations, but instead reﬂect

a rapid shape dynamics of NE surfaces. Experimental

limitations prevented quantiﬁcation of timescales for the

entire 3D surface, but our observations are qualitatively

consistent with ﬁndings that smaller wrinkles typically

decay faster [36, 61]. Furthermore, the fact that the de-

formation spectrum is monotonically decreasing (Fig. 1e)

implies that there is no preferred wavelength, suggesting

that the observed NE shapes do not correspond to ﬂuc-

tuations about the steady-states of buckled shells, but

instead reﬂect dynamic wrinkling across all experimen-

tally resolved angular scales.

Angular number

Power spectrum

100101

Normalized Lamin C intensity

“Young”

“Old”

LamC

Nup107

80 100 120 140 160

Developmental coordinate

100101

Angular number

Power spectrum

60 100 140 180

0.1

0.2

0.3

0.4

0.5

0.6

Time proxy

60 100 140 180

0-1 1

100102.5 25

10-6

10-5

10-4

10-3

10-2 0.1 0.40.05 0.2

0.05 0.1 0.2 0.4

3.104

10-6

10-5

10-4

10-3

10-2

FIG. 2. Fluctuating elastic shell theory predicts a scaling law with exponent ≈3for the wrinkle power

spectrum, in agreement with experiments.a, Equilibrium simulation snapshots of nuclei at temperature Teﬀ = 10Teq ,

undeformed radius R= 25 µm and Rc/R = 20, at ﬁxed FvK number γ= 3 ×104for varying elastic moduli controlled by

kTeﬀ/κ. Color indicates the normalized deviation of the surface from the mean shell radius. b, Time-averaged spectra of

simulated NEs of undeformed radius R= 25 µm, Rc/R = 20, Teﬀ = 10Teq for diﬀerent moduli κ, Y at ﬁxed γ= 3 ×104,

showing the transition from weak nonlinearity to strong nonlinearity as bending rigidity decreases. Color bar matches the dots

from a.c, Binned averages of spectra from nuclei in nurse cells directly connected to the oocyte reveal that shape ﬂuctuations

follow a scaling law with an exponent between −3.2 and −8/3 that is obeyed throughout development. ‘Young’ nuclei have a

time proxy between 80 −140, N= 29 nuclei, from 12 egg chambers; ‘Old’ nuclei have a time proxy between 160 −220, N= 40

nuclei, from 22 egg chambers. Bars show extremal values. Hashed area indicates approximate noise threshold for young nuclei.

(See Supp. Fig. S4 for comparison between nuclei at diﬀerent positions in the egg chamber) d, Fixed egg chambers expressing

Nup107::RFP and stained for Lamin C, showing a decrease in Lamin C intensity in nurse cell nuclei as egg chambers increase

in age. In contrast, Nup107::RFP intensity stays relatively constant. The same trend is observed in live imaging of ex vivo egg

chambers expressing LamC::GFP and Nup107::RFP (Supp. Fig. S6a). Wrinkling of nuclei in younger egg chambers (all but

the rightmost) is a result of ﬁxation and is not observed in live imaging until later stages. Arrows indicate increasing age; egg

chamber boundaries are shown in dashed outlines. Scale bar: 50 µm. e, Normalized Lamin C ﬂuorescence intensity decreases

by approximately 5-fold over time. Normalization details are speciﬁed in the Methods. N= 337 nuclei from 23 egg chambers;

colored dots show means for each egg chamber.

Both maximum-intensity projections and spectral re-

constructions show that NE wrinkles and creases are

sharp, with narrow bent regions separated by ﬂatter areas

(Fig. 1). This morphology is reminiscent of the nonlinear

stress-focusing characteristic of crumpled elastic sheets

and shells such as ordinary paper sheets, which are much

more easily bent than stretched [1, 37, 38]. In partic-

ular, these geometric nonlinearities lead to anisotropic

responses when point forces are applied to the shell [38].

To rationalize the experimentally observed wrinkle mor-

phology at spatial scales larger than the NE thickness, we

constructed a minimal eﬀective elastic model, describing

the NE as a deformed spherical shell (equilibrium radius

R). In spherical coordinates r= (θ, ϕ), the shell has an

isotropic elastic free energy [42, 48]

Fshell =Zd2rκ

2(∇2f)2+λ

2ϵ2

ii +µϵ2

ij ,(3)

where i, j ∈ {θ, ϕ}and using the Einstein summation

convention. The energy functional (3), accounts for

bending stiﬀness through a Helfrich-like bending term

that penalizes out-of-plane deformation f(positive when

pointing inwards), and the stretching of the membrane

through the nonlinear strain tensor ϵij . The 2D Lam´e

parameters λ, µ are proportional to the 2D Young’s mod-

ulus Y. The strain combines contributions from fand

from the in-plane deformation u(r) (SI Sec. IV). We also

allow for a preferred radius of curvature Rcof the shell

mismatched with the radius Rof the shell Rc≥R, which

in the large-F¨oppl-von K´arm´an (FvK) regime leads to a

strain tensor ϵij =1

2(∂iuj+∂jui+∂if∂jf)−δij f/Rc

(SI Sec. IV). Previous work [44, 45, 67] has shown the

NE to be stiﬀer than most biological membranes and to

be well described as a thin membrane of a 3D isotropic

elastic material with an eﬀective 3D Young’s modulus

E≈1 kPa and thickness of h∼10 −100 nm (for a

more detailed discussion of limitations of ﬂuid membrane

models, see SI Sec. IV 5), leading to a bending rigidity of

κ= 100−300 kTeq ≈10−18 J, where Teq is the room tem-

perature, and a stretching rigidity, captured by the 2D

Young’s modulus, of Y≈10−4N/m [43]. By construc-

tion, these moduli are approximately related through the

eﬀective thickness h∼pκ/Y [48]. Note that Yis a fac-

tor of 103smaller than the stretching rigidity of a lipid

bilayer, potentially explained by the presence of ‘area

reservoirs’ in NEs and by transmembrane protein confor-

mational changes [44]. For a shell of radius R, one can

deﬁne the FvK number γ=Y R2/κ which describes the

relative propensity of the material to bend rather than to

stretch. Using the above values, we ﬁnd that the NE has

a large FvK number γ∼104−106, comparable to that

of a sheet of paper or graphene [1]. Accordingly, the NE

is more amenable to bending than to stretching, and de-

formations are expected to appear as sharp wrinkles and

creases, in agreement with our observations (Fig. 1).

To compare the surface shapes and ﬂuctuation pre-

dicted by Eq. (3) with our experimental data, we sim-

ulated the equilibrium Langevin PDE derived from this

free energy (see Methods and SI Sec. IV 4 for simulation

details). The simulations account for hydrodynamic cou-

pling and both passive and active ﬂuctuations, which are

modeled by an eﬀective temperature kTeﬀ . Despite the

model’s minimal character and theoretical limitations of

Eq. (3) at long wavelengths where l→0 (SI Sec. IV), the

numerically obtained shapes (Fig. 2a) are qualitatively

similar to those in the experiments (Fig. 1d). In the

experimentally accessible range of low-to-intermediate

angular wave numbers 3 ≲l≲11, the angular spec-

tra extracted from the simulations at diﬀerent ratios of

kTeﬀ /κ ∈[0.05,0.5] (Fig. 2b) and experimental data

(Fig. 1e) also show an approximately similar decay, sug-

gesting that the minimal elastic shell model in Eq. (3)

captures relevant features of the NE, providing a basis

for further analysis and predictions.

A main feature of the experimentally measured spec-

tra is that both younger and older nuclei exhibit a sim-

ilar asymptotic power law decay in the limit of small

angular numbers l≤10 (Fig. 2c). To rationalize this

observation, we ﬁrst note that the scaling behavior in

our experiments deviates from the basic linear response

theory predictions, which is expected because, even for

younger nuclei, the radial ﬂuctuations ftypically exceed

the NE thickness h∼10−3R(Fig. 1c-f). More precisely,

for small ﬂuctuations (f≪h≪R) and small thermo-

dynamic pressure (p≪pc= 4√κY /R2

c, where pcis the

critical buckling pressure of the sphere), linear response

theory predicts that the power spectrum Plexhibits a

plateau for l≤lcand falls of as l−4for l≫lcwith

a crossover value lc≈γ1/4pR/Rc(SI Sec. IV)[45, 48],

which is not seen in our experiments (Figs. 1e and 2c).

Indeed, classical shell theory [2] states that nonlinear ef-

fects become important when the out-of-plane deforma-

tions fbecome comparable to or exceed the shell thick-

ness h, which is generally the case in our data where

h≪f≪R(Fig. 1c,d,g). Nonlinear analysis of elastic

plates and shells has a long history [46, 52] and has seen

major advances in the last decade [42, 48], motivated

in part by the discovery of graphene [47]. As demon-

strated above, the FvK number of the NE is comparable

to that of graphene, so we can borrow and apply recent

theoretical results to understand the ﬂuctuation spectra

of the NE. Speciﬁcally, a detailed renormalization group

(RG) analysis [48] of Eq. (3) showed that, for suﬃciently

small plate ﬂuctuations, elastic nonlinearities lead to a

modiﬁed asymptotic decay of Pl∝l−3.2, consistent with

our experimental and simulated data (Figs. 1e and 2b,c)

and with previous experiments in red blood cell spec-

trin networks [49]. Notably, earlier studies [42, 48, 52]

also predicted that the interplay of elastic nonlinearities

and ﬂuctuations can cause the spontaneous collapse of

suﬃciently large shells, suggesting a physical mechanism

that could contribute to the eventual breakdown of the

nurse cell NE when these cells donate their contents to

the oocyte [21, 60].

The previously mentioned RG methods can give rise

to divergences in large deformation regimes, where non-

linearities dominate the shell’s response (SI Sec. IV,

Fig. S9). To obtain an analytical prediction for the scal-

ing in the larger-deformation regime h≪f≪R < Rc,

relevant to older nuclei, we performed an asymptotic di-

mensional analysis that provides additional insight into

how NE wrinkling can be controlled. To that end, we

added to the elastic free energy Fshell an eﬀective pres-

sure term Fp=−Rd2rpeﬀf, where peﬀ accounts for a

normal load, which may arise from osmotic pressure dif-

ferences or microtubule-induced local stresses. Denoting

by Lthe characteristic surface variation length scale and

omitting numerical prefactors that depend on details of

the adopted thin-shell modeling approach (SI Sec. IV),

one ﬁnds for shells of thickness h∼pκ/Y that the vari-

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Dynamics,scalingbehavior,andcontrolofnuclearwrinklingJonathanA.Jackson,1,2,∗NicolasRomeo,3,4,∗AlexanderMietke,3,5KeatonJ.Burns,3JanF.Totz,3AdamC.Martin,1J¨ornDunkel,3,†andJasminImranAlsous6,‡1DepartmentofBiology,MassachusettsInstituteofTechnology2GraduatePrograminBiophysics,HarvardUniversity3Departm...

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Dynamics scaling behavior and control of nuclear wrinkling Jonathan A. Jackson1 2Nicolas Romeo3 4Alexander Mietke3 5Keaton J. Burns3 Jan F. Totz3Adam C. Martin1J orn Dunkel3and Jasmin Imran Alsous6.pdf

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Dynamics scaling behavior and control of nuclear wrinkling Jonathan A. Jackson1 2Nicolas Romeo3 4Alexander Mietke3 5Keaton J. Burns3 Jan F. Totz3Adam C. Martin1J orn Dunkel3and Jasmin Imran Alsous6

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