Spontaneous Bending of Hydra Tissue Fragments Driven by Supracellular Actomyosin Bundles Jian Su1 2Haiqin Wang1 2Zhongyu Yan1and Xinpeng Xu1 2

2025-05-03 0 0 2.26MB 26 页 10玖币

侵权投诉

Spontaneous Bending of Hydra Tissue Fragments Driven by Supracellular Actomyosin

Bundles

Jian Su,1, 2 Haiqin Wang,1, 2 Zhongyu Yan,1and Xinpeng Xu1, 2, ∗

1Guangdong Technion - Israel Institute of Technology,

241 Daxue Road, Shantou, Guangdong, China, 515063

2Technion – Israel Institute of Technology, Haifa, Israel, 3200003

(Dated: November 22, 2022)

Hydra tissue fragments excised freshly from Hydra body bend spontaneously to some quasi-stable

shape in several minutes. We propose that the spontaneous bending is driven mechanically by

supracellular actomyosin bundles inherited from parent Hydra. An active-laminated-plate model is

constructed, from which we predict that the fragment shape characterized by spontaneous curvature

is determined by its anisotropy in contractility and elasticity. The inward bending to endoderm side

is ensured by the presence of a soft intermediate matrix (mesoglea) layer. The bending process

starts diﬀusively from the edges and relaxes exponentially to the ﬁnal quasi-stable shape. Two

characteristic time scales are identiﬁed from the dissipation due to viscous drag and interlayer

frictional sliding, respectively. The former is about 0.01 seconds, but the latter is much larger,

about several minutes, consistent with experiments.

Hydra is a multicellular fresh-water polyp, which ex-

hibits remarkable regeneration capabilities, making it an

excellent model system for studying tissue morphogen-

esis [1–3]. The Hydra body consists of a single-axis

hollow-cylindrical tube (about 5−10 mm long) [1], which

has a triple-layer structure composed of two epithelial

cell layers (endoderm and ectoderm) that are adhered

together by an intermediate soft layer of extracellular

matrix (called mesoglea) as shown in Fig. 1(b). Hydra

body shape is maintained by a contractile actomyosin

cytoskeleton [4], which shows a highly-aligned orienta-

tion over supracellular scales (see Fig. 1(b)): the ac-

tomyosin bundles are longitudinally oriented (along the

tube axis) in the ectoderm and circularly oriented in the

endoderm. Regenerating Hydras use their cytoskeleton

to regulate their cells and to guide the regeneration pro-

cess [4]. When fragments are excised from the adult Hy-

dra body, the supracellular pattern of cytoskeletal bun-

dles are inherited, survive, and ﬁnally become a part of

the new daughter Hydra. These supracellular cytoskele-

tal bundles provide structural “memory” of the align-

ment along the body axis [4]. They generate mechanical

forces and direct the alignment of cytoskeleton in regions

where the supracellular order is lacked in the early stage

of regeneration.

At the initial stage of the regeneration of Hydra tis-

sue fragments that are excised freshly from adult Hydra

body [4,5], the fragments bend spontaneously to some

quasi-stable (or mechanical-equilibrium) shape in several

minutes (see Fig. 1(a)). Subsequently, the cytoskeleton

of the fragment remodels to ﬁnd a balance between main-

taining its pre-existing organization and adapting to the

new curved conditions. After about one hour of the

bending-remodeling loop, the excised fragments fold into

small hollow spheroids [4,5].

In this Letter we focus on the initial spontaneous bend-

ing of Hydra fragments. Since the spontaneous bend-

FIG. 1. (color online). (a) Spontaneous bending of freshly

excised Hydra (plate- and rod-like) tissue fragments to quasi-

stable bent shape in several minutes. (b) Triple-layer struc-

ture of Hydra fragments: ectoderm cell layer (yellow), en-

doderm cell layer (light yellow), and intermediate soft ma-

trix (mesoglea). Two perpendicular supracellular actomyosin

bundles (red arrows) are formed on the basal sides of each

epithelial layer. Reproduced from Livshits et al. [4] with per-

mission from Elsevier.

ing happens very fast only in several minutes, it is rea-

sonable to assume that the bending is accompanied by

changes only in cell shape but not cellular rearrange-

ments or cell division and apoptosis [6]. Furthermore,

we propose that the spontaneous tissue bending is an

active process driven mechanically by the contractility

of the aligned, supracellular actomyosin bundles. An

active-laminated-plate model is constructed that corre-

lates the triple-layer structure and the supracellular con-

tractile bundles with the spontaneous bending of Hydra

fragments (see Fig. 1(b)). We predict that the bent shape

of plate-like Hydra fragments characterized by the spon-

taneous curvature tensor are mostly determined by their

anisotropy in both the supracellular actomyosin contrac-

tility and the elastic properties of each lamina layer, as

summarized in Fig. 2. The bending dynamics of rod-like

Hydra fragments propagates diﬀusively from the edges

arXiv:2210.11696v3 [physics.bio-ph] 19 Nov 2022

into the center (with the bent length being proportional

to the square root of time). Whereas, when the bending

is close to its ﬁnal equilibrium (quasi-stable) shape, the

end-to-end distance decays exponentially with time to-

wards its equilibrium value. Moreover, we identify that

the interlayer frictional sliding is the dominant dissipa-

tion mechanism governing the spontaneous bending of

Hydra fragments in the ﬁrst several minutes observed in

experiments [4].

Spontaneous curvature of Hydra tissue fragments. —

To construct a continuum model for Hydra tissue frag-

ments, we ﬁrst note a structural analogy between Hy-

dra fragments (with triple-layer structure) and compos-

ite laminated plates in material science [7,8]. Moreover,

we ﬁnd that in typical regeneration experiments [4], the

lateral lengths of Hydra fragments are usually ∼100 µm

much larger than their thickness h∼10 µm. We then for-

mulate a thin active laminated-plate (ALP) model [9,10]

for Hydra fragments by following the classical thin lami-

nated plate theory [8] that extends the classical thin plate

theory [7] for mono-layer isotropic materials to multi-

layer orthotropic materials. In our ALP model, the bent

shape of a Hydra fragment is characterized by sponta-

neous curvature tensor and determined by minimizing

the deformation energy of the laminate fragment upon

internal active contraction applied by the two perpendic-

ular supracellular actomyosin bundles.

Technically, we treat the Hydra fragment as a lami-

nated plate consisting of two lamina layers (see Fig. 1(b)):

the ectoderm layer-1, and the composite endoderm-

mesoglea layer-2. In this case, we ﬁrst consider Hydra

fragments with isotropic lamina, where the total bend-

ing energy density (per area) is found to be [9,10]

Fb=1

2Dp(c2

x+ 2νcxcy+c2

y)−(M(1)

pcx+M(2)

pcy),(1)

with cxand cybeing the two principal curvatures of the

neutral surface at z=z0. Here νis the Poisson ratio and

Dpis the eﬀective ﬂexural stiﬀness. M(k)

p≈τ(k)

pz0and

τ(k)

p<0 (k= 1,2) are the active torques and contractile

forces generated by supracellular actomyosin bundles in

each of the two lamina layers, respectively. Note that

contrasted with the bending energy in a quadratic form

of principal curvatures in classical thin plate theory [7],

terms coupling active actomyosin torques linearly with

curvatures [11,12] also appear in Eq. (1), which result in

non-zero spontaneous curvatures.

Minimizing Fb=RdxdyFbgives the principal sponta-

neous curvatures [10]:

cx0 =c1−νc2

1−ν2, cy0 =c2−νc1

1−ν2,(2a)

in which ck=M(k)

p/Dp≈τ(k)

pz0/Dp(k= 1,2) char-

acterizes the contractility of the supracellular bundles in

each lamina layer with the neutral surface position given

z0=E(2)

p,eﬀ h2

2−E(1)

∗h2

2hE(1)

∗h1+E(m)

∗hm+E(2)

∗(h2−hm)i.(2b)

Here E(k)

∗≡E(k)/(1−ν2); E(k)(k= 1,m,2) are Young’s

modulus of the ectoderm, mesoglea, and endoderm, re-

spectively. h1and h2are the thicknesses of the two lam-

ina layers, respectively, with hmbeing the thickness of

mesoglea. The Poisson ratio νis assumed to be con-

stant (0 < ν < 1/2) through the fragment thickness.

E(2)

p,eﬀ =E(2)

∗−(E(2)

∗−E(m)

∗)h2

m/h2

2, is the thickness-

averaged eﬀective modulus of the endoderm-mesoglea

layer-2. Note that the concept of spontaneous curva-

ture has been used long time ago to represent the ten-

dency of lipids to curve in lipid membranes [13] and to

describe the spontaneous bending of uniformly-heated

bimetal plates [14], in which the driving forces are chemi-

cal heterogeneity and asymmetric thermal expansion, re-

spectively. In contrast, the spontaneous bending of tis-

sue fragments proposed in this work are driven by com-

pletely diﬀerent forces due to contractile supracellular

actomyosin bundles.

FIG. 2. (a) Schematic illustration for the spontaneous bent

shape of Hydra fragments. The contour length, width, and

total thickness of the fragments are denoted by `c,w, and

h, respectively. (b) A schematic bent-shape diagram of Hy-

dra plates with diﬀerent anisotropy in actomyosin contrac-

tility (measured by c2/c1) and in elasticity (measured by

E(k)

2/E(k)

1). Here ckcharacterize the contractility of supra-

cellular bundles; E(k)

1and E(k)

2are the two principal Young’s

moduli of each (k= 1,2) orthotropic lamina layer.

From Eq. (2) we conclude that the spontaneous bend-

ing (with non-zero spontaneous curvatures) of Hydra

fragments necessitates two basic conditions: the supra-

cellular actomyosin contraction (represented by non-zero

τ(k)

p) and the internal asymmetry in the elastic properties

and/or thicknesses of the two lamina layers (character-

ized by non-zero z0). Moreover, for elastically isotropic

fragments, we ﬁnd from Eq. (2a) that the principal cur-

vatures and the equilibrium bent shape are mainly deter-

mined by the contractility anisotropy measured by c2/c1.

If ν < c2/c1< ν−1, the Hydra fragments will bend spon-

taneously to elliptical cap shape; otherwise, they will

bend to saddle shape as shown in Fig. 2.

Next, for tissue fragments consisting of strongly

anisotropic lamina layers (for example, due to the pres-

ence of aligned thick actomyosin bundles in the two ep-

ithelial layers) we ﬁnd that the spontaneous bending in

the two principal directions are almost independent [10],

and the neutral surfaces of the spontaneous bending lie

in diﬀerent layers for bending along diﬀerent directions:

in ectoderm and in endoderm for bending along the

x-direction and ˆ

y-direction, respectively. In this case,

the two principal spontaneous curvatures are given by

cx0 ≈c1>0, cy0 ≈c2<0, that is, the fragments bend

spontaneously to be saddle shape (see Fig. 2(b)).

Recent experiments [4] show that (see Fig. 1(a)) Hydra

fragments bend spontaneously to a spherical cap shape

toward the inner endoderm side, that is, cx0 ≈cy0 >0

and c1≈c2>0. From the bent-shape diagram in

Fig. 2(b), we ﬁnd such cap-like bent shape corresponds

to the case of more-or-less isotropy in both elasticity and

contractility. Moreover, in this case, we ﬁnd from Eq. (2)

that z0<0, which requires E(2)

p,eﬀ < E(1)

∗(for h1≈h2).

This is ensured by the presence of the soft mesoglea

layer with E(m)

∗E(2)

∗, which can reduce the eﬀective

endoderm-mesoglea modulus E(2)

p,eﬀ to E(2)

∗(1 −h2

m/h2

that can be smaller than both the endoderm modulus

E(2)

∗and the comparable ectoderm modulus E(1)

∗[6].

Note that such mesoglea-softening mechanism is similar

to that of connecting a stiﬀ spring to a soft spring in

series. Proper and stable inward tissue bending to endo-

derm side is essential for the formation of hollow Hydra

spheroid and the whole Hydra regeneration process.

Spontaneous bending of rod-like Hydra fragments.— If,

instead of a Hydra plate, one cuts a rod-like Hydra frag-

ment, the total bending energy density (per length) in

Eq. (1) reduces to [10]: Fb=1

2Drc2−Mrcwith cbe-

ing the curvature of the neutral line at z=z0. Here Dr

is the eﬀective ﬂexural stiﬀness; τr<0 and Mr≈τrz0

are the active contractile force and active torque gener-

ated by supracellular actomyosin bundle in the ectoderm

layer-1, respectively. Minimization of Fb=RdxFbgives

the spontaneous curvature [10]: c0=Mr/Dr≈τrz0/Dr,

and the neutral line position z0is given by the same form

as Eq. (2) if replacing E(k)

∗by E(k)wwith wbeing the

width of the Hydra rod.

Using Dr∼Eh3w,z0∼h, and noting that in typical

cell experiments [6,15]: τr/hw ∼0.1 kPa, h∼10 µm,

E∼1 kPa, we estimate the spontaneous curvature c0∼

1/100 µm−1. Interestingly, for Hydra regenerating from

cell aggregates [3], the curvature cof the hollow Hydra

spheroids formed from an aggregate of a minimal num-

ber of Nc∼1000 cells [2,3] can be estimated to be the

same scale ∼1/100 µm−1by calculating the surface area

of the spheroid, 4πc−2∼NcR2

c, and using the cell size

Rc∼10 µm. Moreover, such cell-scale curvature is also

known to appear very often in many natural in vivo mi-

croenvironment [11], e.g., cylindrical shaped glands and

blood vessels [16].

Note that the supracellular contraction or the inter-

nal asymmetry in elastic and/or geometric properties are

generally non-uniform and hence the spontaneous curva-

ture can be diﬀerent at diﬀerent positions through the

fragments. In this case, some non-trivial periodic bent

shapes have been obtained as shown in Fig. S2 of SM [10].

Moreover, we would like to point out that the mechanis-

tic perspective of the spontaneous bending of tissue frag-

ments during morphogenesis conveyed in this work has

been proposed before [5,17–19] and several classical the-

ories have been developed. Among the earliest theories of

this kind, Lewis proposed (1947) [17] a mechanical model

of epithelial sheets, consisting of brass bars, tubes, and

rubber bands. We discuss this model in SM [10] and com-

pare it with our ALP model for the spontaneous bending

shape of Hydra tissue.

In addition, during the tissue bending, interlayer slide

may occur between each cell layer and the intermediate

mesoglea matrix [15], and the fragment becomes incoher-

ent [20]. In this case, we neglect the in-plane contraction

and the total energy Fttakes the following phenomeno-

logical form [12,20]

Ft=Z`c/2

−`c/2

ds Ys

22

s−χsc+Dr

2(c−˜c0)2.(3)

Here `cis the rod contour length, sdenotes the interlayer

slide or the misﬁt strain discontinuity between the two

lamina layers, and Ysis the stiﬀness for the interlayer

slide. ˜c0being the spontaneous curvature of coherent

fragments. χis the linear coupling coeﬃcient. The equi-

librium bent state of the rod is then characterized by the

spontaneous curvature c0≡˜c0/(1 −χ2/DrYs) and the

equilibrium slide s0 =χc0/Ys. Particularly, in the limit

of Ys→ ∞, we recover the case of coherent fragments

with c0= ˜c0and s0 = 0 where the strain is continuous

through the thickness.

Bending dynamics of tissue rods: eﬀects of viscous dis-

sipation and interlayer frictional sliding. — We now con-

sider the dynamic process of the spontaneous bending of

Hydra tissue rods in surrounding viscous ﬂuids starting

from initial ﬂattened state to the ﬁnal equilibrium shape

as shown in experiments (Fig. 1(a)). The bending dy-

namics of long soft rods with non-zero spontaneous cur-

vature c0and contour length `c2πR0(R0=c−1

has been explored intensively in both theory and ex-

periments [21,22]. However, in typical Hydra regener-

ation experiments [4], Hydra rods are usually short with

`c∼2πR0. In this case, the total energy Ftof the lam-

inated Hydra rod is given by Eq. (3), and the bending

dynamics is driven by the relaxation of stored elastic en-

ergy in the initial ﬂattened state. Furthermore, the dis-

sipation during the bending arises mainly from viscous

drag in the surrounding ﬂuids and the interlayer fric-

tional sliding inside the fragments, as taken into account

by the dissipation function

Φ = Z`c/2

−`c/2

ds 1

2ξv˙

r2+1

2ξs˙2

s.(4)

Here r(s) is the position vector of points (parameterized

by arc length s) on the rod, and the local curvature c

is given by its second derivatives [13]. ξvand ξsare the

viscous drag coeﬃcient per rod length and friction coef-

ﬁcient for interlayer sliding, respectively. Interestingly,

the interlayer frictional sliding has recently been found

to be able to facilitate the long-range force propagation

in tissues that usually have multi-layer structures [15].

We then use Onsager’s variational principle [23,24]

and minimize Rayleighian R[˙

r,˙s] = ˙

Ft+ Φ yielding a

set of highly nonlinear partial diﬀerential equation (e.g.,

Kirchhoﬀ equations) [22]. However, here we will not solve

these nonlinear equations but carry out direct-variational

analysis [23,24] for the very initial bending stage near

the ﬂattened state and the ﬁnal bending stage close to

the equilibrium bent shape.

Initially near the ﬂattened state, we neglect the eﬀects

of interlayer slide and take a linear trial proﬁle of local

curvature (see Fig. S4(a) in SM [10]), where the dynamic

bending process is characterized by one time-dependent

parameter: the bent length a(t) of the rod. Substitut-

ing the trial proﬁle into Eqs. (3) and (4), we obtain the

Rayleighian R≈−1

3Drc2

0˙a+1

2ξva˙a2and its minimization

with respect to ˙agives

a≈2Drc2

3ξv1/2

t1/2,(5)

that is, the bending starts from the edges and propagates

diﬀusively into the center. Moreover, from Eq. (5) we

calculate the normalized end-to-end (ETE) distance ˜

`≡

(`−`eq)/(`c−`eq) scaling as 1 −˜

`∼a3∼t3/2, where `

is the ETE distance with `eq being its equilibrium value.

Interestingly, although the linear trial proﬁle has some

visible deviation from the simulated proﬁle (as shown in

Fig. S4(a) in SM [10]), the scaling law predicted above

for ETE distance agrees well with simulation results in

Fig. 3(a) and is not sensitive to the form of trial curvature

proﬁle; the same scaling is obtained by using a simpler

trial step-function proﬁle of curvature. However, we note

that the initial diﬀusive bending process happens only in

a very short period as shown in Fig. 3(a). The bending

dynamics slows down quickly from the diﬀusive regime

to a subdiﬀusive regime with 1 −˜

`∼t0.9.

In the ﬁnal stage of the bending dynamics close to

the quasi-stable bent state, we take the same strategy

as above and also choose a linear trial proﬁle of the cur-

vature (see Fig. S4(c) in SM [10]), where the dynamic

bending process is described by the two time-dependent

parameters: the normalized curvature ˜cm(t) = cm/c0and

interlayer slide ˜s,m(t) = s,m/s0 in the rod center at

s= 0. Substituting the trial proﬁle into Eqs. (3) and

(4), we obtain the Rayleighian Rand its minimization

yields

τ˙

˜cm= 1 −(1 + B)˜cm+B˜s,m, τs˙

˜s,m= ˜cm−˜s,m,(6)

where B=Ys2

s0/˜

Drc2

0characterizes the relative stiﬀ-

ness of interlayer slide and out-plane bending with ˜

Dr≡

Dr−χ2/Ysbeing the normalized ﬂexural stiﬀness, τ=

6ξvL/˜

Drc4

0and τs=ξs/Ysare the two characteristic time

scales associated with viscous drag and interlayer fric-

tional sliding, respectively. Note that Lis a nonlinear

function of rod contour length `c(changing from 0 to

order one; see its expression in SM [10]).

FIG. 3. Temporal evolution of the end-to-end (ETE) distance

`for Hydra rods: (a) the initial bending starting diﬀusively

from the edges, and (b) the ﬁnal bending relaxing exponen-

tially to the equilibrium shape. Universal scaling relations are

found for rods of diﬀerent contour lengths π/2≤˜

`c≤3π.

We consider the following two limits of the dynamic

equation (6) that are particularly interesting and may

be relevant to the bending dynamics during tissue mor-

phogenesis and regeneration. Firstly, in the limits of ei-

ther coherent Hydra rods or incoherent Hydra rods with

small friction, we have τs/τ 1 and the viscous drag

from the surrounding ﬂuids is the dominant dissipation

mechanism. In this limit, we ﬁnd from Eq. (6) the nor-

malized ETE distance ˜

`(t) follows ˜

`∼exp (−t/τ), which

have been justiﬁed for rods with various contour lengths

by numerical simulations using bead-spring model and

shown in Fig. 3(b). In addition, we have used bead-spring

model to numerically examine the bending dynamics of

Hydra rods under some diﬀerent boundary conditions

(see Figs. S5-S7 in [10]). We ﬁnd that the bending dy-

namics predicted above is very robust: the same scaling

laws (including the nonlinear contour-length dependence

of relaxational time τdue to L) are found for most dif-

ferent boundary conditions in both the initial diﬀusive

bending and the ﬁnal relaxational bending processes.

Secondly, in the limit of incoherent Hydra rods with

large friction, we have τs/τ 1, we obtain from Eq. (6)

the ETE distance ˜

`(t) following the same exponential

form:

`∼exp [−t/τs(1 + B)] ,(7)

in which B ∼ 1 so that all energy terms in Eq. (3) are

comparable. That is, in either limits, the characteristic

time for the spontaneous bending of tissue rod is con-

trolled by the slower dissipative dynamics and the longer

time scale. Particularly for Hydra tissue rods, we can

estimate the magnitude of the two time scales as fol-

lows. Using ˜

Dr∼Eh3w,ξv∼η,w∼h,Ys∼Ehw,

and ξs=ξh2wwith ξbeing the interlayer friction coeﬃ-

cient [15], we obtain

τ∼ξv

Drc4

∼η

Eh4c4

∼0.01 s, τs∼ξh

E∼1 min,(8)

where we have taken typical parameter values measured

in cell experiments [6,15]: h∼10 µm, c0∼1/100 µm−1,

η∼10−3Pa ·s, E∼1 kPa, and ξ∼1010 Pa ·s/m.

Therefore, the spontaneous bending of Hydra rods lies in

the limit of τs/τ 1 and is characterized by τs∼1 min in

consistent with the time duration of initial spontaneous

bending observed in Hydra regeneration experiments (see

Fig. 1(a)) [4].

The proposal here on the spontaneous bending of Hy-

dra tissue fragments driven by contractile supracellu-

lar actomyosin bundles should be generically present in

a broad range of cell aggregates and tissue fragments

during tissue regeneration or morphogenesis during em-

bryo development. Our mechanical model of Hydra frag-

ments can be easily extended to study the morphogen-

esis of other tissues and to include the couplings be-

tween the organization of cytoskeleton and the deforma-

tion/curvature of the tissue [25]. The eﬀects of the per-

meation of water or other small molecules on the dynam-

ics of tissue bending can be explored similarly by model-

ing the tissue fragments as active-laminated gels [26,27].

Additional aspects of the cell biology, such as cell divi-

sion/apoptosis and cell morphological changes [18,19], or

active behaviors such as migration and oscillations, could

be incorporated as well [17,28].

The authors thank Samuel Safran, Kinneret Keren,

Zhihao Li, Yonit Maroudas-Sacks, Lital Shani-Zerbib,

and Anton Livshits for fruitful discussions. X.X. is

supported in part by National Natural Science Foun-

dation of China (NSFC, No. 12004082), by Guang-

dong Province Universities and Colleges Pearl River

Scholar Funded Scheme (2019), and by 2020 Li Ka

Shing Foundation Cross-Disciplinary Research Grant

(No. 2020LKSFG08A). J. Su and H. Wang contributed

equally to this work.

∗E-mail address: xu.xinpeng@gtiit.edu.cn

[1] T. Fujisawa, Dev. Dyn. 226, 182 (2003).

[2] Y. Takaku, T. Hariyama, and T. Fujisawa, Mech. Dev.

122, 109 (2005).

[3] M. K¨ucken, J. Soriano, P. A. Pullarkat, A. Ott, and

E. M. Nicola, Biophys. J. 95, 978 (2008).

[4] A. Livshits, L. Shani-Zerbib, Y. Maroudas-Sacks,

E. Braun, and K. Keren, Cell Rep. 18, 1410 (2017).

[5] M. Krahe, I. Wenzel, K.-N. Lin, J. Fischer, J. Goldmann,

M. K¨astner, and C. F¨utterer, New J. Phys. 15, 035004

(2013).

[6] J. A. Carter, C. Hyland, R. E. Steele, and E.-M. S.

Collins, Biophys. J. 110, 1191 (2016).

[7] L. D. Landau and E. M. Lifshitz, Theory of Elasticity,

3rd ed., Course of Theoretical Physics, Vol. 7 (Pergamon

Press, Oxford, UK, 1986).

[8] J. N. Reddy, Mechanics of laminated composite plates and

shells: theory and analysis (CRC press, 2003).

[9] H. Wang, B. Zou, J. Su, D. Wang, and X. Xu, Soft

Matter 18, 6015 (2022).

[10] See the Supplementary Material [url].

[11] Y. Biton and S. Safran, Physical Biology 6, 046010

(2009).

[12] B. M. Friedrich and S. A. Safran, Soft Matter 8, 3223

(2012).

[13] S. A. Safran, Statistical thermodynamics of surfaces, in-

terfaces, and membranes (CRC Press, 2018).

[14] S. Timoshenko, J. Opt. Soc. Am. 11, 233 (1925).

[15] Y. Lou, T. Kawaue, I. Yow, Y. Toyama, J. Prost, and

T. Hiraiwa, arXiv:2107.03074 (2021).

[16] S. J. Callens, R. J. Uyttendaele, L. E. Fratila-Apachitei,

and A. A. Zadpoor, Biomaterials 232, 119739 (2020).

[17] A. ˇ

Siber and P. Ziherl, Cellular Patterns (CRC Press,

2017).

[18] E. Hannezo, J. Prost, and J.-F. Joanny, Proceedings of

the National Academy of Sciences 111, 27 (2014).

[19] J. Ackermann, P.-Q. Qu, L. LeGoﬀ, and M. Ben Amar,

Eur. Phys. J. Plus 137, 1 (2022).

[20] D. Srolovitz, S. Safran, and R. Tenne, Phys. Rev. E 49,

5260 (1994).

[21] L. Tadrist, F. Brochard-Wyart, and D. Cuvelier, Soft

Matter 8, 8517 (2012).

[22] A. Callan-Jones, P.-T. Brun, and B. Audoly, Phys. Rev.

Lett. 108, 174302 (2012).

[23] M. Doi, Prog. Polym. Sci. 112, 101339 (2021).

[24] H. Wang, T. Qian, and X. Xu, Soft Matter 17, 3634

(2021).

[25] S. He, Y. Green, N. Saeidi, X. Li, J. J. Fredberg, B. Ji,

and L. M. Pismen, J. Mech. Phys. Solids 137, 103860

(2020).

[26] A. C. Callan-Jones and F. J¨ulicher, New J. Phys. 13,

093027 (2011).

[27] Z. Ding, P. Lyu, A. Shi, X. Man, and M. Doi, Macro-

molecules 55, 7092 (2022).

[28] M. Popovi´c, A. Nandi, M. Merkel, R. Etournay, S. Eaton,

F. J¨ulicher, and G. Salbreux, New J. Phys. 19, 033006

(2017).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

SpontaneousBendingofHydraTissueFragmentsDrivenbySupracellularActomyosinBundlesJianSu,1,2HaiqinWang,1,2ZhongyuYan,1andXinpengXu1,2,1GuangdongTechnion-IsraelInstituteofTechnology,241DaxueRoad,Shantou,Guangdong,China,5150632Technion{IsraelInstituteofTechnology,Haifa,Israel,3200003(Dated:November22,202...

展开>> 收起<<

Spontaneous Bending of Hydra Tissue Fragments Driven by Supracellular Actomyosin Bundles Jian Su1 2Haiqin Wang1 2Zhongyu Yan1and Xinpeng Xu1 2.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Spontaneous Bending of Hydra Tissue Fragments Driven by Supracellular Actomyosin Bundles Jian Su1 2Haiqin Wang1 2Zhongyu Yan1and Xinpeng Xu1 2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: