Zeming Wu, Xin - pku.edu.cn

Journal of the Mechanics and Physics of Solids 137 (2020) 103867

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Journal of the Mechanics and Physics of Solids

journal homepage: www.elsevier.com/locate/jmps

Structures and mechanical behaviors of soft nanotubes

confining adhesive single or multiple elastic nanoparticles

Zeming Wu, Xin Yi ∗

Department of Mechanics and Engineering Science, Beijing Innovation Center for Engineering Science and Advanced Technology,

College of Engineering, Peking University, Beijing 100871, China

a r t i c l e i n f o

Article history:

Received 27 October 2019

Revised 13 December 2019

Accepted 4 January 2020

Available online 7 January 2020

Keywords:

Soft membrane nanotubes

Particle elasticity

Mechanical interaction

Tunneling nanotubes

Membrane tubulation

Tubular invagination

a b s t r a c t

The mechanical interaction between soft membrane nanotubes and confined nanoparticles

plays essential roles in numerous cellular activities such as inter- and intracellular trans-

port and signaling, membrane tubulation and pearling transformation driven by adsorbed

nanocomponents, and cooperative cell uptake. The aim of this work is to theoretically an-

alyze how structures and mechanical behaviors of particle-in-tube complexes are regu-

lated by the particle elasticity, covering the cases of a single particle and multiple particles

of spatial periodicity. Depending on the interplay of membrane tension, adhesion energy,

nanoparticle size, and the rigidity ratio between the nanoparticles and soft tubes, charac-

teristic interaction states in terms of wrapping degrees of the confined particles have been

identified, and the corresponding wrapping phase diagrams are determined. As the parti-

cles become softer, the adhesion energy required for the partial-wrapping state becomes

lower but that for full-wrapping is larger, and a rich variety of characteristic system con-

figurations are observed including the bulged or undulated cylinders, necklaces of spheres

or prolates, and sausage-string shapes. Moreover, two fundamental modes of interaction

between multiple nanoparticles and the soft nanotube are revealed, the cooperative wrap-

ping and individual wrapping. Based on perturbation analysis at small tube deformation

and configurational assumptions of catenoid, torus, and one-sheeted hyperboloid at more

general tube deformation, we have also obtained analytical solutions on the wrapping of

rigid spherical nanoparticles in soft tubes. Comparison between wrapping of elastic parti-

cles by a flat membrane and by a soft membrane tube is discussed. This study enriches

our knowledge on the diversity of membrane tubules in morphologies and structures.

© 2020 Elsevier Ltd. All rights reserved.

1. Introduction

Biomembrane-based nanotubes and similar tubular structures have been reported in many types of cells from im-

mune to neuronal cells as well as their organelles, and play important mechanical and biophysical roles in numerous

fundamental cellular activities including inter- and intracellular transport and communication, cell uptake, and cell mi-

gration. Depending on the membrane composition, the bending rigidity and tension of the nanotube membranes fall in

∗ Corresponding author.

E-mail address: [email protected] (X. Yi).

https://doi.org/10.1016/j.jmps.2020.103867

0022-5096/© 2020 Elsevier Ltd. All rights reserved.


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2 Z. Wu and X. Yi / Journal of the Mechanics and Physics of Solids 137 (2020) 103867

Fig. 1. (a,b) Axisymmetric conformations of soft membrane nanotubes containing single and multiple adhesive elastic nanoparticles in the adopted cylin-

drical coordinate ( r, φ, z ). The system is divided into three portions: inner free region of the particle (blue), contact region (red), and outer free region of

the tube (black). In the case of multiple nanoparticles (b), the system configuration is periodic along the z -axis. m 1 and m 2 , respectively, denote the semi-

major and semi-minor axes of an individual particle. The aspect ratio of the nanoparticle is defined as λ = m 1 /m 2 . (c-i) Experimental images of a single

or multiple nanoparticles confined in soft nanotubes. (c-f) Cargoes of different sizes, mechanical properties, and shapes in tunneling membrane nanotubes.

(c) A vesicle in a tunnel connecting two membrane-enclosed parts of a disintegrated red blood cell. (d) Membrane nanotubes with vesicular dilatations

(arrows) observed between human urothelial cells. (e) Multiple vesicular compartments in tunneling nanotubes connecting neuronal cells. (f) A membrane

nanotube with periodic presence of bulges in the cone-like photoreceptor cell line 661W. (g-i) Soft tubes with multiple confined particles. (g) Internaliza-

tion of virus-like nanoparticles induces deep membrane invagination and tubulation with nanoparticles (arrowheads) forming a single-row tubular chain.

(h) Discrete vesicles confined in a tubular structure formed by archaeal surface layers. (i) Encapsulated liposomes cause bulge deformation of the confining

polymer nanofibers. Figures adapted from Refs. (c) Igli c et al. (2003) , (d) Verani c et al. (2008) , (e) Sartori-Rupp et al. (2019) , (f) Scholkmann et al. (2018) ,

(g) Ewers (2010) , (h) Gill et al. (2019) , and (i) Mickova et al. (2012) (For interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article.).

a wide range from 10 k B T (1 k B T = 4 . 11 × 10 −21 J) to 150 k B T ( Boal, 2012 ) and from 0.003 mN/m to 1 mN/m ( Morris and

Homann, 2001 ), respectively. For cellular membranes, the membrane tension is low. For example, it is reported that the

tension in the endoplasmic reticulum membrane network is about 0.013 mN/m and the tension in the Golgi membrane is

around 0.005 mN/m ( Upadhyaya and Sheetz, 2004 ). A typical value of cellular membranes could be taken as 0.02 mN/m

( Morris and Homann, 2001 ). A high membrane tension in tense lipid membranes could be induced by changing pressure

difference across the vesicle membranes, which is more common in engineered rather than biological environments. As the

diameter of a lipid membrane tube in equilibrium is proportional to the square root of the ratio between the bending rigid-

ity and tension of the membrane, the membranous tubes in vitro and in vivo are of nanoscaled diameters from about tens

to hundreds of nanometers and could last for minutes to several hours in living cells. The tube length also varies in a wide

range from hundreds of nanometers up to hundreds of microns, depending on the biological circumstances. For example,

the tunneling membrane nanotubes connecting neighboring PC12 cells have a diameter in a range of 50 nm to 200 nm

and an average length of 6 μm, and for T cells a tube diameter from 180 nm to 380 nm and an average length of 22 μm

( Austefjord et al., 2014 ). In comparison, the membrane nanotubes formed in the endocytic internalization of nanoparticles

are much shorter, realized experimentally with a length in hundreds of nanometers to few micrometers depending on the

particle number ( Ewers, 2010; Gill et al., 2019 ).

Owing to their broad involvements in various cell functions, the membrane nanotubes in cells are rarely isolated and

in a freely standing state, but have regular interaction with various cellular components or engineered nanomaterials in

cells. In the cases of intra- and intercellular transport and communications, membrane nanotubes as thin and long tubular

membrane protrusions are also named as tunneling nanotubes (TNTs), and they serve as cellular connection to accomplish

transfer and exchanges of various components such as ions, proteins, pathogen (e.g., viruses and bacteria), organelles (e.g.,

lysosomes and mitochondria), cellular cargos (e.g., lipid droplets and vesicular compartments) ( Marzo et al., 2012; Auste-

fjord et al., 2014 ), and even membrane-bound quantum dots ( He et al., 2010 ). The presence of these transferred compo-

nents causes local deformation of the confining soft nanotubes. Depending collectively on the component size and elasticity

with respect to the confining nanotube, a large heterogeneity of the tube deformation has been observed experimentally

(see Fig. 1 a,b for schematic illustration and Fig. 1 c-f for selected experimental images). Among transferred components,

viruses, bacteria, mitochondria, and solid lipid nanoparticles possessing a solid lipid core stabilized by surfactants can be

regarded as stiff nanoparticles in comparison with the membrane nanotube and they basically maintain their initial shapes

while cause local nanotube bulges. For components such as lipid droplets ( Astanina et al., 2015 ), soft vesicular particles

( Igli c et al., 2003; Nasseri and Florence, 2005; Verani c et al., 2008; Sartori-Rupp et al., 2019 ), and fractions of cytoplasm

( Patheja and Sahu, 2017 ), they have comparable rigidities with respect to the confining nanotubes, and deform with the

nanotubes. As demonstrated in Fig. 1 c–e, vesicular particles of initially spherical shapes are compressed radially due to the

nanotube confinement. Moreover, recent experimental studies indicate that the deformed membrane nanotubes containing

multiple particles could adopt a configuration of periodic bulges ( Fig. 1 f, Scholkmann et al., 2018 ). In certain cases, small

internal liposomes in a large liposome are observed to be connected to each other in linear arrays via membrane tubules,

which contact the surface of the large liposome ( Osawa et al., 2009 ). In the nanoparticle-induced membrane tubulation and

deep membrane invagination in cell uptake, similar necklacelike structures but with significantly smaller bulge separation

have been observed ( Fig. 1 g,h), where the encapsulated nanoparticles form particle-chains or linear aggregates in the mem-

Z. Wu and X. Yi / Journal of the Mechanics and Physics of Solids 137 (2020) 103867 3

brane tubules ( Ewers, 2010; Gill et al., 2019 ). A structure similar to the necklaces of prolates has also been observed in

polymer nanofibers with encapsulated liposomes ( Fig. 1 i, Mickova et al., 2012 ) or bacteria ( Gensheimer et al., 2007 ).

In addition to experimental investigation, recently some theoretical studies and molecular dynamics simulations

have been conducted on the membrane nanotube interaction with encapsulated rigid nanoparticles. For example,

Chen et al. (2009) analyzed the structure for a membrane nanotube confining a single or double adhesive rigid spheri-

cal nanoparticles, which is later generalized to the case of a finite number of identical rigid particles with an assumption

of identical configurations for the membrane portions between each two neighboring particles ( Zheng and Meng, 2014 ). For

the case of multiple rigid spherical nanoparticles adsorbed on vesicles, Monte Carlo simulations demonstrate that these par-

ticles could form linear aggregation cooperatively and induce membrane tubulation ( Bahrami et al., 2012; Šari c and Cacciuto,

2012 ), consistent with experimental observation ( Fig. 1 g,h). Further theoretical analysis indicates the cooperative wrapping

of several rigid spherical and ellipsoidal particles in long tubular membrane structures with consideration of the particle-

membrane interaction range ( Raatz et al., 2014; Wang et al., 2014; Raatz and Weikl, 2017 ). In addition to tubular structures,

coarse-grained molecular dynamics simulations demonstrate that the membrane in the wrapping of multiple nanoparticles

could form a rich variety of structures including pocket-shaped and handle-shaped structures ( Yue and Zhang, 2012; Xiong

et al., 2017; Spangler et al., 2018 ).

Compared to extensive experimental observations of membrane nanotube bulges caused by the stiff and soft nanocompo-

nents inside and relatively few existing theoretical investigations on the quantitative examination of the nanotube structures

in the presence of confined rigid nanoparticles, so far no dedicated theoretical effort s have been paid to the effects of par-

ticle elasticity and size in the regulating the structures and mechanical behaviors of particle-in-tube complexes, only except

for few related studies on the polymer confinement in tubular channels ( Chen, 2007; Yan et al., 2018 ). In this work, we con-

duct the first theoretical analysis to probe the mechanics underlying the interplay between soft membrane nanotubes and

confined elastic nanoparticles, focusing on the cases of a single particle and multiple particles of spatial periodicity. Specific

analytical attention is paid to the cases of spherical rigid nanoparticles with configurational assumptions for the free mem-

brane portions. Collective roles of the membrane tension, adhesion energy, particle size, and the rigidity ratio between the

particles and tubes on the particle-tube structures and mechanical interplay in terms of particle wrapping are revealed. This

study enriches the understanding of tubule-related cellular activities including cargo transport inside soft tubes, morpho-

logical transformation of membrane nanotubes, and deep membrane invagination with cooperatively wrapped nanoparticles

from a mechanical viewpoint.

The rest of this theoretical study is organized as follows. In Sections 2 and 3 , we focus on the soft nanotube interac-

tion with single and multiple confined elastic nanoparticles, respectively. In each of these two sections, after setting up the

theoretical basis, we combine analytical modeling and fully nonlinear numerical methods to characterize the nanotube in-

teraction with rigid spherical nanoparticles, and then numerically investigate how the nanoparticle elasticity regulates the

nanotube-particle interplay. Further discussion of the results is presented in Section 4 followed by conclusions in Section 5 .

2. A single nanoparticle confined in a soft cylindrical tube

2.1. Theoretical modeling and numerical methods

We first build a theoretical model to study the mechanical interplay between an infinitely long cylindrical membrane

nanotube and a single confined elastic nanoparticle which can be modelled as a vesicular particle (e.g., nanovesicles and

polymersomes) of a fixed surface area A p . Both the nanoparticle and nanotube are assumed to be of zero spontaneous

curvature with vanishing osmotic pressure, and subject to axisymmetric elastic deformation. As illustrated in Fig. 1 a, the

system is divided into three portions: inner free region of the particle (blue, #1), outer free region of the tube (black, #2),

and contact region (red, #3). Hereafter we use subscripts 1, 2, and 3 to identify quantities pertaining to these three regions,

respectively. In the adopted cylindrical coordinate ( r, φ, z ) with its origin located in the center of the particle, the total

energy of the system of axial and vertical symmetries is given in terms of the Canham–Helfrich functional ( Canham, 1970;

Helfrich, 1973 ) as E tot = E el − γ A 3 , where

E el = 2 ×[ ∑

i =1 , 2 , 3

πκi

∫ l i

0

r i

(˙ ψ i +

sin ψ i

r i

)2

d s i + 2 πσ∑

i =2 , 3

∫ l i

0

r i d s i

]

(1)

is the elastic energy of the system with κ i , ψ i , r i , s i , and l i ( i = 1 , 2 , 3 ) representing the bending rigidity, tangent angle,

r -coordinate, arclength, and undetermined total arclength of each regions, respectively; the dot hereinafter denotes a deriva-

tive with respect to the arclength (e.g., ˙ ψ i ≡ d ψ i / d s i ); σ is the membrane tension of the nanotube; γ ( > 0) denotes the

adhesion energy and A 3 = 4 π∫ l 3

0 r 3 d s 3 is the contact area which defines the wrapping degree f ≡ A 3 / A p . Here the adhesion

energy γ has a unit of energy per unit area and is also often called as adhesion strength by physicists. The prefactor 2 in

Eq. (1) stems from equal energy contributions from the upper and lower portions of the system with vertical symmetry. The

arclength s i in each region is measured from the contact edge as indicated in Fig. 1 a. Due to the vertical symmetry of the

system, our analysis is focused on the upper portion ( z ≥ 0).

In Eq. (1) we assume that the confined nanoparticle and the membrane nanotube form perfect contact and they can

slide against each other freely in the contact region, so that κ = κ + κ and the energy contribution from the Gaussian
3 1 2


curvatures can be ignored upon the system deformation without boundary or topological change according to the Gauss–

Bonnet theorem ( Kreyszig, 1991 ). We also adopt the constant tension ensemble, provided that the membrane nanotube is in

contact with a large reservoir of membrane at a constant tension. In the cases of vesicle tubulation ( Gózd z, 2005; Wu et al.,

2019 ) or particle interaction with a vesicle ( Bahrami et al., 2016; Tang et al., 2016; Yi and Gao, 2016; 2017; Yu et al., 2018;

Bahrami and Weikl, 2018 ), the ensemble of a constant area of the vesicle is more appropriate owing to the high energetic

cost of the in-plane membrane stretching.

With geometric relations

˙ r i = cos ψ i and

˙ z i = sin ψ i ,

the axisymmetric configuration can be characterized by the tangent angle ψ i = ψ i (s i ) with s i ∈ [0, l i ] ( i = 1 , 2 , 3 ), and E el in

Eq. (1) can be expressed as a function of ψ i ( s i ). As the total arclength l i is unknown, we introduce a new variable t i ≡ s i / l i in region i , convert the integral interval from [0, l i ] to [0,1], and approximate ψ i (s i ) = ψ i (t i l i ) by a cubic B-spline curve

as ψ i =

∑ n j=0 c

(i ) j

N

(i ) j

( t i ) , where the control points c (i ) j

are coefficients of the basis functions N

(i ) j

defined recursively on a

non-uniform knot vector of t i ( Yi and Gao, 2015 ). The expression of the cubic B-spline basis functions can be found in

Appendix A . Here we adopt a typical choice of the knot vector as t (0) i

, t (1) i

, t (2) i

, ..., t (n +4) i

with t (k ) i

= 0 (k = 0 , 1 , 2 , 3) and

t (k ) i

= 1 (k = n + 1 , ..., n + 4) .

To analyze the mechanical interplay between the membrane nanotube and the confined nanoparticle, a reference state

is introduced and defined as follows. In the reference state, both the elastic nanoparticle and membrane nanotube stay

alone and have no interaction. The initially spherical nanoparticle of radius a =

√

A p / (4 π) and bending rigidity κ1 has a

ground state energy E p 0

= 8 πκ1 . A cylindrical membrane nanotube of radius r and length L has the elastic energy E tube (r, L ) =2 π r L [ κ2 / (2 r 2 ) + σ ] . Minimizing E tube with respect to r gives an equilibrium radius ( Derényi et al., 2002 )

r 0 =

√

κ2 / (2 σ )

and the ground state energy of the membrane nanotube E m

0 (L ) = E tube (r 0 , L ) = 2 πL √

2 σκ2 . The variation of the system en-

ergy then becomes E tot = E el − γ A 3 with E el = E el − E p 0

− E m

0 (2 L ) .

As the elastic energy change of the system is given by E el ≡ E tot + γ A 3 = E tot + γ f A p , minimizing E tot at a certain

f is equivalent to determining the minimum state of E el with an area constraint of 4 π∫ l 3

0 r 3 d s 3 = f A p . Here we employ the

interior-point approach in constrained nonlinear optimization to numerically determine the minimum energy state of the

system at a given wrapping degree f . The boundary and constraint conditions provide either input parameters or equality

constraints during energy minimization. At the remote boundary of region 2 ( t 2 = 1 or s 2 = l 2 ), we have ψ 2 = π/ 2 as the

outer free membrane becomes asymptotically vertical and require z 2 = L/ 2 with L prescribed as a certain large number to

approximate the infinitely long membrane nanotube. In most cases L = 20 a is sufficiently large to ensure the numerical

accuracy, while a significantly larger L is required at extremely small membrane tension σ . At t 1 = 1 (or s 1 = l 1 ), we have

ψ 1 = π and require r 1 = 0 . At t 3 = 1 (or s 3 = l 3 ), we have ψ 3 = −π/ 2 and require z 3 = 0 . At the contact edge ( t i = s i =0 ), the continuities of the coordinates ( r i , z i ) and tangent angles ψ i are required. Other equality constraints include the

prescribed areas 4 π∫ l 1

0 r 1 d s 1 = (1 − f ) A p and 4 π

∫ l 3 0

r 3 d s 3 = f A p . Given the elastic energy variation E el and its first and

second derivatives with respect to the control points and total arclengths of all three regions, the minimum energy state

and corresponding system configuration can be obtained. Rescaling the energy by the membrane bending rigidity κ2 and

lengths by the effective particle radius a , we have E el = E el ( f, κ1 /κ2 , σ ) and E tot = E tot ( f, κ1 /κ2 , σ , γ ) , where

σ ≡ 2 σa 2 /κ2 and γ ≡ 2 γ a 2 /κ2 .

2.2. Analytical approximation for nearly cylindrical membrane nanotube perturbed by a single rigid spherical particle

In this section, we use perturbation theory to analytically investigate how the membrane nanotube is deformed by a

spherical rigid particle. At a given wrapping degree f , the contact region is readily determined as a spherical segment.

Therefore, here we are only interested in the quantities pertaining to the outer free region and ignore the subscript 2 for

simplicity. As the whole system is vertically symmetric, our analysis is focused on the upper portion of the outer free region.

The axisymmetric shape equation for the outer free region at equilibrium can be determined through a variation of the

energy functional in Eq. (1) as ( Hu and Ou-Yang, 1993; Derényi et al., 2002 )

...

ψ

+

2 cos ψ

r ψ +

1

2

˙ ψ

3 − 3 sin ψ

2 r ˙ ψ

2 − 2 − 3 sin

2 ψ

2 r 2 ˙ ψ +

2 − sin

2 ψ

2 r 3 sin ψ − σ

κ

(˙ ψ +

sin ψ

r

)= 0 . (2)

At σ = 0 , first integral of Eq. (2) gives

r cos ψ ψ + r sin ψ

2

˙ ψ

2 + cos 2 ψ

˙ ψ − sin ψ

r +

sin

3 ψ

2 r = C,

where C is the integration constant ( Zheng and Liu, 1993 ). The membrane of a catenoid configuration, the only nonplanar

axisymmetric minimal surface (zero mean curvature everywhere), corresponds to the special case of C = 0 .


With knowledge of

˙ ψ = sin ψ ψ

′ , ψ = ψ

′′ sin

2 ψ + sin ψ cos ψ (ψ

′ ) 2 , ...

ψ

= ψ

′′′ sin

3 ψ + 4 sin

2 ψ cos ψ ψ

′ ψ

′′ − sin ψ (1 − 2 cos 2 ψ )(ψ

′ ) 3 ,

Eq. (2) becomes ( Boži c et al., 2001 )

sin

2 ψ ψ

′′′ + 2 sin ψ cos ψ

(2 ψ

′ +

1

r

)ψ

′′ +

2 − 3 sin

2 ψ

2

[(ψ

′ ) 3 − ψ

′ r 2

]+

4 − 7 sin

2 ψ

2 r (ψ

′ ) 2

+

2 − sin

2 ψ

2 r 3 − σ

κ

(ψ

′ +

1

r

)= 0 , (3)

where the primes represent differentiation with respect to z (e.g., ψ

′ ≡ d ψ /d z and ψ

′ ′ ′ ≡ d

3 ψ /d z 3 ).

At f = 0 and σ = 1 , containing a single rigid spherical particle of radius a , the membrane nanotube adopts an ideal

cylindrical shape of uniform radius r 0 (= a ) . As f increases or at other σ , the membrane nanotube deforms and deviates

from an ideal cylinder of uniform radius, and its perturbed contour can be written as

r(z) = r 0 + u (z) , (4)

where u ( z ) represents the radial deviation. For a weakly deformed membrane nanotube, ψ is approximately π /2, and we

have

sin ψ ≈ 1 , cos ψ ≈ u

′ , ψ ≈ π

2

− u

′ , ψ

′ ≈ −u

′′ , ψ

′′ ≈ −u

′′′ , ψ

′′′ ≈ −u

′′′′ . (5)

Substituting Eqs. (4) and (5) into Eq. (3) and retaining the leading order in u , a fourth-order linear differential equation

is obtained as

u

′′′′ +

u

r 4 0

= 0 . (6)

Here we have assumed that the derivatives of u ( z ) with respect to z are of the same order as u ( z ) itself. Similar forms to

Eq. (6) have also been reported in literature ( Boži c et al., 2001; Calladine and Greenwood, 2002; Derényi et al., 2002 ). For

readers who are familiar with the beam on elastic foundation analysis, Eq. (6) refers to the characteristic equation for a

uniform beam on a Winkler foundation ( Timoshenko, 1963 ).

Eq. (6) has four independent characteristic solutions of the form

u α = C α exp

(ξα

z − z c

r 0

)(α = 1 , . . . , 4) ,

where ξα = ±(1 ± i ) / √

2 and z c ( ≥ 0) is the z -coordinate of the contact edge. As we focus on the upper portion of the outer

free region, here z ≥ z c . To enforce the remote boundary conditions ( r = r 0 and ψ = π/ 2 as z → ∞ ), we are only interested

in damped terms which vanish as z approaches infinity ( ξα with negative real parts). The resulting expression for r ( z ) reads

r(z) = r 0 + e −(z−z c ) / ( √

2 r 0 )

[β1 cos

(z − z c √

2 r 0

)+ β2 sin

(z − z c √

2 r 0

)],

where coefficients β1 and β2 are determined by the boundary conditions at the contact edge z = z c .

For a single spherical rigid nanoparticle of radius a at a wrapping degree f , we have the boundary conditions r(z c ) =a √

1 − f 2 and r ′ (z c ) = − f/ √

1 − f 2 at z = z c = fa, which yield β1 = a √

1 − f 2 − r 0 and β2 = β1 −√

2 r 0 f / √

1 − f 2 . Selected

nanotube configurations are presented in Fig. 2 a.

Fig. 2. (a) Selected nanotube configurations at different σ and f as well as (b) elastic energy E el in the case of a single rigid spherical nanoparticle

confined in the elastic membrane nanotube. Solid and dashed lines correspond to the results based on fully nonlinear numerical solutions and perturbation

analysis, respectively.


Fig. 3. The system configurations (a) and profiles of total energy E tot (b) for a single rigid spherical nanoparticle at σ = 5 and γ = 0 , 4 , and 60.82. (c)

Wrapping phase diagram of a single rigid nanoparticle in the membrane nanotube with respect to γ and σ .

By changing parameterization ψ( s ) in Eq. (1) to r ( z ), the elastic energy of the membrane nanotube perturbed by a rigid

spherical nanoparticle can be determined as

E el = 2 ×[πκ

∫ [ r r ′′ − (r ′ ) 2 − 1] 2

r[1 + (r ′ ) 2 ] 5 / 2 d z + 2 πσ

∫ r √

1 + (r ′ ) 2 d z

]+ f × 8 πκ + f × 4 πa 2 σ (7)

over the integral interval [ z c , ∞ ), and then the elastic energy change E el and total energy change E tot = E el − γ A 3 can

be obtained ( Fig. 2 b).

In a special case of σ = 1 (or a = r 0 ), the reference nanotube has the same radius as the nanoparticle does and its energy

of deformation due to adhesive wrapping at small f could be approximated by expanding the two terms in the integrands

of Eq. (7) up to lowest orders in r ′ and r ′′ . At σ = 1 , such a small gradient expansion of Eq. (7) yields

E el = 2 ×[πκ

∫ ∞

fa

(1

r 0 − r

r 2 0

− 2 r ′′ )

d z + 2 πσ

∫ ∞

fa

(r − r 0 )d z

]+ f × 6 πκ = 2 πκ f

(

3 − 2 √

1 − f 2

)

= 2 πκ f + O ( f 3 ) ,

which means that the minimum adhesion energy γmin necessary for initial wrapping at σ = 1 is

γmin ≡d(E el /κ)

d(2 π f )

∣∣∣∣f→ 0

= 1 .

This analytical result is consistent with our numerical result in Fig. 3 c and previous numerical result by Chen (2007) . It

is interesting to note that in the case of wrapping a cylindrical rigid particle by a flat membrane, we also have γmin = 1 ,

independent of σ ( Yi et al., 2011 ).

At relatively small f , the nanotube shapes and elastic energy E el are well approximated by the perturbed solutions (see

Fig. 2 a,b). As f increases, the shape discrepancies between the linearized and fully nonlinear solutions become more signif-

icant. Having knowledge of energy profiles E el (solid lines in Fig. 2 b), the total energy E tot = E el − γ A 3 as a function

of f can then be determined, and three wrapping phases illustrated in Fig. 3 a can then be observed depending at different

values of γ from the profiles of E tot ( f ) (e.g., Fig. 3 b at σ = 5 ). In no-wrapping phase, usually correlated with small ad-

hesion energy γ , E tot increases monotonically with f and its minimum is located at f = 0 ( Fig. 3 a,b). As γ increases, a

global minimum at a state of partial-wrapping is achieved. If γ is large enough, E tot decreases monotonically with f and a

stable full-wrapping state arises. As the particle approaches the full-wrapping state, a narrow membrane neck of high cur-

vature emerges on the particle poles and the neck diameter shall be larger than the membrane thickness which is around

5 nm. Moreover, for such a narrow neck the thermal fluctuation alone can pinch the wrapped nanoparticle off the mem-

brane tube. Therefore, for simplicity we take f = 0 . 99 as the full-wrapping condition hereinafter. Summarizing γ separating

these three wrapping states at different values of σ , we then obtain the wrapping phase diagram for the single rigid par-

ticle ( Fig. 3 c). As σ increases or equivalently the nanotube radius decreases, the minimum adhesion energy γmin required

for partial wrapping decreases. The same phase boundary between the no-wrapping and partial-wrapping states has been

reported by Chen et al. (2009) . Moreover, that wrapping phase boundary given by the nonlinear numerical solutions is well

approximated by the perturbation analysis (dashed line in Fig. 3 c).

2.3. Fully nonlinear numerical solutions for the membrane nanotube containing a single elastic nanoparticle

In this section, we probe the mechanical interplay between an elastic nanoparticle and the confining soft nanotube.

Fig. 4 a shows the elastic energy change E el ( f ) as a function of the wrapping degree f at different κ1 / κ2 and σ . As κ1 / κ2

decreases or σ increases, the slope d( E el )/d f decreases in the early wrapping stage and increases in the late stage, which

means that softer particles require lower adhesion energy for partial wrapping but demand higher adhesion energy to be


Fig. 4. (a) Elastic energy E el and (b) normalized particle volume V / V 0 and aspect ratio λ = m 1 /m 2 at σ = 0 . 5 , 1 . 5 , 4 , 5 , 8 and κ1 /κ2 = 5 , 1 , 0 . 1 . Here

V 0 = 4 πa 3 / 3 is the volume of a rigid spherical particle. m 1 and m 2 denote the semi-major and semi-minor axes, respectively, as illustrated in Fig. 1 b. (c,d)

Selected morphologies of the particle-nanotube interaction configurations for different sets of ( σ , κ1 /κ2 ) at f = 0 . 1 , 0 . 5 , and 0.95. Softer particles undergo

more significant deformation.

fully wrapped. The geometric information and configurations of the deformed particles are shown in Fig. 4 b-d. As f in-

creases, the particle volume V ( f ) first varies slightly, then decreases at a much higher rate until f exceeds a certain value,

and increases till the particle becomes fully wrapped ( Fig. 4 b). Introducing the aspect ratio of the nanoparticle as λ = m 1 /m 2

with m 1 and m 2 denoting the semi-major and semi-minor axes of an individual particle, respectively ( Fig. 1 b), we can find

that the profiles of the particle aspect ratio λ( f ) also exhibit nonmonotonicity ( Fig. 4 b). The larger the σ or the smaller the

κ1 / κ2 , the smaller the minimum value of V /(4 πa 3 /3) or the larger the maximum value of λ. At σ < 1 , the particle is initially

stretched perpendicular to the tube axis ( λ < 1) and becomes axially stretched ( λ > 1) as f increases ( Fig. 4 b,c). The particle

shape transition from λ < 1 to λ > 1 is more evident for the very soft particle, and corresponds to a nonmonotonic segment

at small f in the profile of V ( f ) (e.g., κ1 /κ2 = 0 . 1 and σ = 0 . 5 in Fig. 4 b). In the cases of relatively large σ and small κ1 / κ2 ,

the energy profile exhibits a plateau characterizing the membrane wrapping during which the wrapped particle remains

the same shape with a relatively small volume and large aspect ratio ( Fig. 4 b). This is demonstrated in Fig. 4 d where the

very soft nanoparticle of κ1 /κ2 = 0 . 1 at σ = 8 exhibits the same shape at f = 0 . 1 and f = 0 . 5 . From Fig. 4 d we can also find

that, at high membrane tension, the very soft particle elongates and has a smaller enclosed volume than the stiffer particle

exhibiting an ovoid shape. A similar geometric feature has been observed in the conformational transition of polymers in a

membrane nanotube ( Chen, 2007; Yan et al., 2018 ). As the monomer number increases, the confined polymer could undergo

a conformational transition from a swollen phase of a tubular shape with a smaller volume to a globular phase of a prolate

shape with a larger volume ( Chen, 2007 ).

With the knowledge of energy profiles E el ( f ) in Fig. 4 a, the total energy profiles E tot ( f ) = E el − γ A 3 for elastic

nanoparticles can be determined. Following the same procedure for determining Fig. 3 c in the case of a rigid spherical

nanoparticle, we then obtain the wrapping phase diagram for elastic nanoparticles of different values of κ1 / κ2 ( Fig. 5 ). It

is shown that the minimum adhesion energy γmin required for partial wrapping decreases as σ increases. At large σ , the

radius of a free nanotube is significantly smaller than the particle radius, and we have γmin = 0 as the particle and the nan-

otube membrane could form tight contact spontaneously. For elastic particles, the elastic deformation is partitioned between

the particle and the nanotube. As the particle becomes softer, the nanotube deformation is weakened and γmin decreases.

To attain full wrapping, softer particles require larger γ as implied by the elastic energy profiles in Figs. 2 b and 4 a, where


Fig. 5. Wrapping phase diagram of a single nanoparticle confined in the membrane nanotube with respect to γ and σ at κ1 /κ2 = 0 . 1 , 1 , 5 , and ∞ .

softer particles exhibit larger maximum slope in the energy profiles. A similar feature in the elastic energy profiles has also

been observed in the wrapping of elastic nanoparticles by a membrane patch ( Gao, 2014; Tang et al., 2015; Yi et al., 2011,

2018; Yi and Gao, 2015 ) and the elastic nanoparticle interaction with a vesicle ( Tang et al., 2016; Yi and Gao, 2016, 2017 ).

We now take a look at the wrapping of an elastic nanoparticle by a flat membrane and by a membrane nanotube.

Comparing Fig. 3 in the work of Yi et al. (2011) and Fig. 5 here, except very soft ones (e.g., κ1 /κ2 = 0 . 1 ), the wrapping

of a single particle in the soft nanotube requires much smaller γ to achieve no-wrapping state and significantly larger γfor a full-wrapping state, both due to the confining feature of the nanotube. For a very soft particle, its wrapping in the

membrane tube requires slightly larger γ at very small σ but smaller γ at relatively large σ to achieve partial-wrapping in

comparison with the wrapping of a single particle by a flat membrane.

3. Multiple nanoparticles confined in a soft cylindrical tube

3.1. Theoretical modeling

In this section, we consider the cooperative packing of elastic nanoparticles in a soft cylindrical tube with an infinite

periodic pattern. In comparison with the modeling for a single particle in the tube, the distance L p between the centers

of two neighboring nanoparticles or the wavelength of the periodically deformed tube emerges here as a new degree of

freedom. To numerically determine L p at different κ1 / κ2 , σ , and f , different values of L p as input are chosen a priori and

the wavelength selection is eventually given from the energy minimization. Here one periodicity structure unit of vertical

symmetry is considered either that with the particle in the just middle ( Fig. 1 b) or that between the centers of two neigh-

boring nanoparticles ( Fig. 6 ). For the first periodic unit, ψ 2 = π/ 2 is required at z 2 = L p / 2 . Apart from this, the boundary

and constraint conditions and the energy minimization procedure remain the same as these in Section 2 , and the energy

profiles and wrapping phase diagrams could be obtained.

Fig. 6. Geometric approximation of the axisymmetric shapes of the outer free membrane between two neighboring nanoparticles. (a) Catenoid, (b) torus,

and (c) one-sheeted hyperboloid. (d) Three-dimensional views of the axisymmetric shapes in (a-c). Solid black and dashed gray lines in (a-c) correspond

to the membrane profiles and rest portions of the approximated surfaces not applicable for the theoretical analysis, respectively.


3.2. Analytical approximation for multiple rigid spherical nanoparticles in the membrane nanotube

To gain more insight into the particle-tube mechanical interplay, here we analytically explore the periodic packing of rigid

spherical nanoparticles in a soft tube at a finite membrane tension. Attentive comparison between the system configurations

in Figs. 1 b and 6 suggests that the outer free membrane in one periodic unit forms a rotational surface with strong visual

similarity to a catenoid at intermediate and relatively large wrapping degree f and to a one-sheeted hyperboloid or the

inner portion of a toroidal configuration at relatively small and intermediate f . In the following analysis, this observation

is confirmed analytically and numerically. As only quantities pertaining to the outer free region in one periodic unit are

focused, in this section the subscript 2 is ignored for simplicity and the origin of the rz -coordinate relocates at the middle

point of the segment connecting the centers of two neighboring particles.

3.2.1. The catenoid portion at intermediate and relatively large wrapping degree f

In the cylindrical rz -coordinate ( Fig. 6 a), the catenoid as a surface of revolution generated by a catenary is parameterized

with radius ( Kreyszig, 1991; McDargh and Deserno, 2018 )

r = b cosh

z

b ≥ b or | z | = b ln

(

r

b ±

√

r 2

b 2 − 1

)

, (8)

where b as the minimum value of r can be expressed as

b = r/ √

1 + (d r/ d z) 2 . (9)

As the principal curvatures of the catenoid are ±b −1 sech

2 (z/b) , its mean curvature vanishes ( H = 0 ) everywhere, and the

membrane bending energy of the catenoid membrane is E cat b

= 2 κ∫

H

2 d A = 0 , where d A is the surface area element.

At the lower contact edge, the radius of circular contact edge is a √

1 − f 2 and the tangent angle is ψ c = arccos f, then

from Eq. (9) we have

b =

a √

1 − f 2 √

1 + cot 2 ψ c

= a (1 − f 2 ) and r = a (1 − f 2 ) cosh

z/a

1 − f 2 , (10)

and from Eq. (8)

L p = 2 a [ sin

2 ψ c ln ( csc ψ c + cot ψ c ) + cos ψ c ] or L p = 2 a [ f + (1 − f 2 ) arsech

√

1 − f 2 ] . (11)

Here arsech represents the inverse of the hyperbolic secant function and we have used the relation ln ( csc ψ + cot ψ) −arsech ( sin ψ) = 0 for ψ ∈ [0, π /2].

As L p ≥ 2 a , we have ψ c ∈ [0, ψ 0 ] with

arsech ( sin ψ 0 ) − 1

1 + cos ψ 0

= 0 or ψ 0 ≈ 0 . 9711 .

By recognizing the above equation as a Taylor series about ψ 0 = 1 and taking the linear term, we can also have

ψ 0 ≈ 1 − sin 1

2

[1 − (1 + cos 1) arcosh ( csc 1)] ,

where arcosh represents the inverse of the hyperbolic cosine function.

In terms of the wrapping degree, we have

f = cos ψ c ≥ cos ψ 0 ≈ 0 . 5644 or f ≥ cos ψ 0 ≈ 3 +

√

3

9

(

√

3 − arcosh

√

3

2

)

. (12)

Maximum of L p is determined by d L p / d ψ c = 0 which gives a critical value ψ c1 of ψ c satisfying

arsech ( sin ψ cl ) cos ψ cl = 1 .

Numerically, ψ c1 ≈ 0.5853 or using the Taylor series as ψ c1 ≈ π/ 6 +

√

3 (−2 +

√

3 arcosh 2) / 8 . Moreover, we have

f cl = cos ψ cl ≈ 0 . 8336 or f cl ≈2

9

(2

√

6 − arcosh

√

3 ) and max L p = 2 a/ f cl ≈ 2 . 399 a. (13)

Here the potential of adhesion between the nanoparticle and membrane tube has a vanishing interaction range.

With consideration of different finite adhesive interaction ranges, slightly larger values of the maximum of L p ( > 2.4 a )

have been given in a recent theoretical study on the cooperative wrapping of spherical nanoparticles by a tensionless

membrane nanotube ( Raatz et al., 2014 ).

From Eq. (8) , the surface area of the catenoid portion in the range of z ∈ [ −z c , z c ] is given by

A cat = 2 × 2 π

∫ z c

r √

1 + (d r/ d z) 2 d z = πb 2 (

sinh

2 z c

b +

2 z c

b

).

0


With the geometric relation z c = L p / 2 − fa and Eqs. (10) and (11) , we have

A cat = πa 2 (1 − f 2 ) 2 [ sinh (2 arsech

√

1 − f 2 ) + 2 arsech

√

1 − f 2 ] = 2 πa 2 (1 − f 2 ) 2 (

f

1 − f 2 + arsech

√

1 − f 2

). (14)

In the case of two neighboring particles in touch ( L p = 2 a or f = f 0 = cos ψ 0 with (1 + f 0 ) arsech

√

1 − f 2 0

= 1 ), the sur-

face area of the catenoid portion becomes A = 2 πa 2 (1 − f 2 0 ) , equal to the total area of two circular disks formed by the

upper and lower contact circles. Therefore, the solution f 0 = cos ψ 0 corresponds to the Goldschmidt ring solution, which

also implies f ≥ f 0 = cos ψ 0 .

In the limiting case of zero membrane tension ( σ = 0 ), the membrane nanotube deformation is only governed by the

membrane bending energy. As the bending energy is positive definite, at σ = 0 the catenoid outer free region of zero bend-

ing energy ( E cat b

= 0 ) is the most energetically favorable solution and the total elastic energy arising from the adhesion

region is E el = 8 fπκ as expected.

At finite σ , the total elastic energy of the deformed membrane nanotube in one periodic unit can be given using

Eqs. (11) and (14) as

E el = E cat b + 8 fπκ + σ (A cat + 4 πa 2 f ) − 2 πL p

√

2 σκ

= 2 πκ f (4 + σ − 2

√

σ ) + πκ(1 − f 2 ) f σ + πκ(1 − f 2 ) √

σ [(1 − f 2 ) √

σ − 4] arsech

√

1 − f 2 .

At f = 1 , we have E el = 2 πκ(4 + σ − 2 √

σ ) .

The total system energy then becomes E tot /κ = E el /κ − 2 π f γ , and the minimum adhesion energy γmin necessary for

an energetically favorable wrapping at f can be determined as

γmin ( f ) =

d(E el /κ)

d(2 π f ) = 4 + 2(1 − f 2 ) σ − 4

√

σ + 2 f √

σ [2 − (1 − f 2 ) √

σ ] arsech

√

1 − f 2 .

3.2.2. The toroidal portion at relatively small and intermediate wrapping degree f

The toroidal configuration of the outer free membrane between two neighboring nanoparticles is a portion of a ring

torus, which is a surface of revolution generated by rotating a circle of radius ρ about a coplanar axis a distance R ( > ρ)

away from the circle center ( Fig. 6 b). In a Cartesian coordinate with its origin located in the middle of a line segment

connecting the centers of two neighboring nanoparticles, the position vector r of the torus portion could be parameterized

as ( Kreyszig, 1991; Deserno and Gelbart, 2002 )

r(φ, θ ) = ((R − ρ cos θ ) cos φ, (R − ρ cos θ ) sin φ, ρ sin θ )

with φ ∈ [0, 2 π ] and θ ∈ [ −θ0 , θ0 ] . Here θ0 represents the angle at which the wrapped spherical nanoparticle and the torus

portion are mutually tangent.

Then the geometric information of the torus portion such as the area and mean curvature can be determined, respec-

tively, as (see Appendix 2.1 for details)

A torus = 4 πρ[ −ρ f + ( ρ + a )

√

1 − f 2 arcsin f

] and H torus =

1

2

(1

ρ− cos θ

R − ρ cos θ

).

Therefore, the membrane bending energy of the torus portion is

E torus b = 2 κ

∫ H

2 torus d A =

4 πR

2 κ

ρ√

R

2 − ρ2 arctan

( √

R + ρ

R − ρtan

θ0

2

)

− 8 πκ sin θ0

with R = (ρ + a ) cos θ0 and f = sin θ0 in the case of rigid spherical nanoparticles.

At finite σ , the total elastic energy of the deformed membrane nanotube in one periodic unit is

E el = E torus b + 8 πκ f + σ (A torus + 4 πa 2 f ) − 2 πL p

√

2 σκ.

Energy minimization of E el = E el ( f, ρ/a ) upon a geometric constraint of L p = 2(ρ + a ) f ≥ 2 a or equivalently ρ/a ≥1 / f − 1 leads to the determination of ρ/ a , E torus

b , and E el at a given f . Numerical analysis indicates that there exists a

critical value of f = f c at which the single minimum of E el determined by d(E el ) / d(ρ/a ) = 0 locates at ρ/a = 1 / f c − 1 .

Substituting ρ/a = 1 / f c − 1 into d(E el ) / d(ρ/a ) = 0 , we have

f c +

√

2 f c

1 − f c (3 f c − 1) arctan

√

1 − f c −√

1 + f c √

2 f c +

2(1 − f c )

1 + f c

[(2 − f c ) arcsin f c

f c

√

1 − f 2 c σ − 2 f c √

σ − 2(1 − f c ) σ

]= 0 .

(15)

At zero membrane tension ( σ = 0 ), we have f c ≈ 0.5638, slightly smaller than the lower bound of the wrapping degree

( f ≈ 0.5644) based on the catenoid configuration.


Fig. 7. Energy analysis and the determination of the morphological phase diagram for the periodic wrapping of rigid spherical nanoparticles based on the

toroidal and catenoid configurations. (a) Morphological phase diagram with respect to the normalized membrane tension σ and wrapping degree f . Above

the dashed (red) line, catenoid configurations; below the dashed line, toroidal configurations. (b) Typical profiles of E el / κ as functions of ρ/ a at different

σ and f (symbols in (a)). Solid circles correspond to the local minima with d(E el /κ) / d(ρ/a ) = 0 . Vertical dash-dotted lines correspond to the locations of

ρ/a = 1 / f − 1 separate the unphysical (dashed curves) and physical (solid curves) regimes of ρ/ a .

At f ≤ f c , the minimum of E el locates at ρ/a = 1 / f − 1 as

E el

2 πκ=

(1 − f ) √

1 − f 2 arcsin f

f 2 σ +

(2 − 1

f

)σ − 2

√

σ +

√

2 (1 + f ) √

f (1 − f ) arctan

√

1 + f −√

1 − f √

2 f (16)

and L p = 2(ρ + a ) f = 2 a .

At f ≥ f c , the minimum of E el locating at ρ/ a satisfying d(E el ) / d(ρ/a ) = 0 .

The minimum adhesion energy γmin necessary for an energetically favorable wrapping at f is γmin = d(E el /κ) / d(2 π f ) .

At f ≤ f c , we have

γmin ( f ) =

1

2 f (1 − f ) +

√

2 (1 − 3 f )

2[(1 − f ) f ] 3 / 2 arctan

√

1 − f −√

1 + f √

2 f +

(

2 − f

f 2 −

√

1 − f

1 + f

2 + f

f 3 arcsin f

)

σ . (17)

In the limiting case of f = 0 , we obtain the minimum adhesion energy required for partial wrapping γmin = (4 − σ ) / 3 at

σ < 4 and γmin = 0 at σ ≥ 4 .

Based on the above analysis and with a combination of Eqs. (16) and (17) , at σ = 0 we have

E tot

2 πκ=

⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

1 , at γ ∈ [ 0 , 4 / 3 ] with f = 0 ,

( 1 + f ) − f γ(3 − 3 f − 2 f 2

)1 − 3 f

, at γ ∈ ( 4 / 3 , 4 ) with f = f ( γ ) ∈ ( 0 , 0 . 564 ) ,

4 − γ , at γ ≥ 4 with f ∈ [ 0 . 564 , 1 ] ,

(18)

where f = f ( γ ) at γ ∈ (4 / 3 , 4) is obtained from Eq. (17) .

Similar analysis on the energy profiles and instability can be made based on the hyperboloidal configuration assumption

(see Appendix 2.2 and supplementary plots in Figs. S1 and S2 for more details).

3.2.3. Numerical results for periodic wrapping of multiple rigid spherical nanoparticles in a soft membrane nanotube

Based on the previous theoretical analysis, we determine a morphological phase diagram in terms of f and σ ( Fig. 7 a).

The comparison of the elastic energy change E el between the catenoid and toroidal configurations leads to the dashed (red)

curve in Fig. 7 a, above which the catenoid configurations are energetically favored and below which toroidal configurations

prevail. Depending on the energy stability ( Fig. 7 b), the torus regime can be further divided into two parts ( Fig. 7 a): f ≤ f cwith the minimum of E el locating at ρ/a = 1 / f − 1 and f > f c with the global minimum of E el locating at ρ/a > 1 / f − 1 .

At f ≤ f c , we have L p = 2 a .

In addition to analytical prediction, we also perform fully nonlinear numerical studies on the periodic wrapping of multi-

ple rigid spherical nanoparticles in a soft nanotube. As shown in Fig. 8 at zero membrane tension, there are no distinguish-

able differences between the theoretical and numerical results on the system configurations, elastic energy change E el ,

and other relevant wrapping information. At σ = 0 , two neighboring rigid spherical particles maintain contact ( L p = 2 a ) as

f ≤ cos ψ 0 ≈ 0.564 ( Eq. (12) ) and the outer free membrane can be extremely well approximated by a torus portion in terms

of the profile and deformation energy. At a larger f ( > 0.564), two neighboring particles become separated with L p given in


Fig. 8. (a) System configurations, (b) elastic energy change E el , and (c) other wrapping information for the periodic wrapping of rigid spherical nanopar-

ticles at σ = 0 . Thin and thick lines in (b,c) correspond to the results based on fully nonlinear numerical solutions and analytical prediction with torus and

catenoid assumptions, respectively.

Fig. 9. Evolution profiles of the elastic energy E el (a) and particle distance L p (b) during the periodic wrapping of multiple rigid spherical nanoparticles

at different σ = 0 . 5 , 1 . 5 , 4 , 5 , and 8. (c) L p as a function of γ at different σ . Open symbols from left to right in each curve in (a) correspond to the solid

(blue) and dashed (red) curves in Fig. 7 a, respectively. Dashed (gray) lines in (a,b) represent results based on the catenoid and torus assumptions. Inset in

(b) compares solid lines in Figs. 2 b and 9 a (comparing E el in the cases of single and multiple rigid particles) with the (gray) dash-dotted curve indicating

the relationship between the kinks in E el ( f ) for wrapping multiple nanoparticles and the corresponding wrapping degree f (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.).

Eq. (11) that reaches a maximum value of max L p ≈ 2.399 a at f ≈ 0.8336 ( Eq. (13) ). The outer free membrane connecting

the separated particles adopts a catenoid configuration with its shape given in Eq. (10) and zero elastic energy; the elastic

energy change for a whole periodicity unit is E el ( f ) = 8 fπκ arising from the spherical segment of the adhesion region

( Fig. 8 b,c). Due to the linear feature of E el as a function of f , once γ > 4 = d(E el /κ) / d(2 π f ) the full-wrapping state is

energetically favorable and two neighboring particles become fully wrapped and connected by a narrow membrane neck of

high curvature. In other words, there is a discontinuous jump in the wrapping degree f from 0.564 to 1 at γ = 4 ( Fig. 8 c).

At a finite membrane tension ( σ > 0 ), the outer free membrane at the intermediate and relatively large f no longer

adopts exactly the catenoid configuration due to the existence of the membrane tension energy. This expectation is con-

firmed comparing the numerical results and analytical approximation shown in Fig. 9 a,b. As σ increases, the energy profiles

obtained from numerical and theoretical studies exhibit slight discrepancies ( Fig. 9 a), while more apparent discrepancies are

observed in the particle distance profiles ( Fig. 9 b). At σ ∈ (0 , 1) , profiles of E el ( f ) and L p ( f ) are continuous as numerical

results indicate, which means that the wrapping process is continuous; at σ > 1 , the energy curve E el ( f ) exhibits a kink

( Fig. 9 a), which corresponds to a discontinuous shape transformation of the particle-tube system from particles in touch to

separated particles (discontinuous jump of L p in Fig. 9 b,c). As σ increases, the onset of the discontinuous shape transition

is delayed to a higher wrapping degree f .

Comparing the elastic energy change E el ( f ) of wrapping single and multiple rigid nanoparticles (inset in Fig. 9 b), we

can immediately see that E el of one periodic unit in the multiple-particle case is always smaller than E el in the single-

particle case. Moreover, these two cases at certain σ (> 1) always exhibits an indistinguishable energy difference at the

critical wrapping degree f c associated with the kink in E el ( f ) of wrapping multiple nanoparticles, and the energy differ-

ence is only significant at small and large wrapping degrees. Such an observation indicates that there are two fundamental

modes of interaction between the multiple rigid spherical nanoparticles and the soft nanotube, the cooperative wrapping

and individual wrapping. In the case of σ > 1 , particles at small ( f < f c ) and large f are cooperatively wrapped by the mem-

brane nanotube with a relatively short particle distance ( L p equal to or slightly larger than 2 a ), and as f exceeds f c the

particles become to be wrapped individually with a relatively large particle distance and maintains that conformation at

intermediate f . At σ < 1 , E el in the multiple-particle case is significantly smaller than that in the single-particle case, and

the cooperative wrapping with a short L p is preferred, consistent with recent theoretical studies on the cooperative wrap-


Fig. 10. The system configurations (a), profile of total energy E tot (b) and wrapping phase diagram (c) in the case of the periodic wrapping of rigid

spherical nanoparticles in the soft membrane nanotube. Configurations in (a) correspond to the cases of γ = 0 , 4 , and 19.22 in (b).

ping of spherical and prolate nanoparticles by a membrane nanotube of zero tension ( Raatz et al., 2014; Raatz and Weikl,

2017 ).

Combing Fig. 9 a,b we can determine the profiles of L p ( γ ) as well as the relation between max L p and γ ( Fig. 9 c). It is

shown that, as γ increases, initially the particles form tight contact and then become separated gradually at σ ∈ (0 , 1) or

sharply at σ > 1 . In most cases of σ , a wavelength of the order of the particle diameter is observed. Similar featured profiles

of L p ( γ ) have been reported in the study of cooperative wrapping of nanoparticles by a tensionless membrane nanotube

at various contact potentials ( Raatz et al., 2014; Raatz and Weikl, 2017 ). A maximum particle distance or the maximum

wavelength of the periodic structure exists and its relation with σ is shown in the inset in Fig. 9 c.

In Section 2.2 we perform perturbation analysis on the mechanical interaction between a single rigid particle and the

soft confining nanotube. A similar approach based on perturbation analysis is also employed to study the periodic wrapping

of multiple rigid spherical particles in a soft tube (Fig. S3). In comparison with the geometric approaches in Section 3.2 ,

perturbation analysis gives rise to the results of accuracy to a similar extent but underestimates the particle distance L p(Fig. S3).

Similar to what we have in the case of a single particle in a soft nanotube ( Fig. 5 ), depending on the value of γ ,

three phases of wrapping states based on the stability of E tot ( f ) are observed for the periodic wrapping of rigid spherical

nanoparticles in the soft nanotube ( Fig. 10 a,b). As γ increases, the system evolves from the state of no-wrapping to partial-

wrapping and eventually to full-wrapping. A corresponding wrapping phase diagram can then be determined ( Fig. 10 c). As

σ increases, γmin required for partial-wrapping decreases. Note that the boundary between the no- and partial-wrapping

phases determined from the analytical prediction in Eq. (17) is slightly shifted leftward in comparison with phase boundary

based on numerical results, though the same value of γmin = 4 / 3 for partial-wrapping at σ = 0 is obtained numerically and

theoretically.

3.3. Numerical results for periodic wrapping of multiple elastic nanoparticles in a soft membrane nanotube

For elastic nanoparticles, there are no simple analytical solutions for the elastic energy change E el ( f ) as well as the

system configurations, and numerical studies are performed ( Figs. 11 , 12 , and S4). Similar to the case of wrapping an elastic

particle in a soft tube in Section 2.3 , the slope d( E el )/d f decreases in the early wrapping stage and increases in the late

stage as κ1 / κ2 decreases or σ increases, and at relatively large σ and small κ1 / κ2 the profile of E el ( f ) exhibits a plateau

( Fig. 11 a). Except very soft particles, the particle distance L p maintains at a value no less than 2 a initially and then increases

to a peak followed by a relatively smooth fall off; the softer the particle is, the larger L p is ( Figs. 11 b and 12 ). For very

soft particles (e.g., κ1 /κ2 = 0 . 1 ) which can be easily deformed, they adopt their initial configurations adaptively with large

deformation but very low energy cost ( Figs. 11 and 12 ). At σ < 1 , very soft particles initially are stretched perpendicular to

the tube axis and adopt oblate-like shapes ( λ < 1). Therefore, the corresponding L p could be smaller than 2 a initially. As γincreases (or equivalently f increases), the particles are stretched significantly along the tube axis ( λ > 1) and L p increases

and gradually saturates to a plateau value. At relatively large σ , the distance between the centers of very soft particles

does not vary significantly with changes in γ and f ( Figs. 11 b and 12 d). By comparing E el ( f ) for the cases of single and

multiple elastic nanoparticles (comparing Figs. 4 a and 11 a), one can find that at σ < 1 the cooperative wrapping mode

prevails, while at σ > 1 both cooperative and individual wrapping modes are observed, except for very soft particles (e.g.,

κ1 /κ2 = 0 . 1 ) which always have strong mutual interaction due to their large deformation and undergo cooperative wrapping.

The volume V and aspect ratio λ of individual particle as functions of f are nonmonotonic as shown in Fig. 11 c, similar to

the single-particle case ( Fig. 4 b).

Following the similar procedure in Section 2.3 , the wrapping phase diagram for multiple adhesive nanoparticles in the

soft nanotube is determined ( Fig. 13 ) which has a similar structure to Fig. 5 on the wrapping of a single particle. The softer

the confined nanoparticle is, the smaller (larger) γ required for the partial- (full-)wrapping state is. For the tube wrapping


Fig. 11. Profiles of the elastic energy E el (a), particle distance L p (b), and individual particle volume V / V 0 and aspect ratio λ = m 1 /m 2 (c) during the

periodic wrapping of multiple elastic nanoparticles of κ1 /κ2 = 5 , 1 , and 0.1 at σ = 0 . 5 , 1 . 5 , 4 , 5 , and 8. Here V 0 = 4 πa 3 / 3 .

Fig. 12. Selected particle-nanotube configurations ( f = 0 . 1 , 0 . 5 , 0 . 95 ) at κ1 /κ2 = ∞ , 5 , 1 , 0 . 1 and σ = 0 . 5 , 2 , 8 . For stiff particles, the confining nanotube

changes from a slightly deformed cylinder to unduloidlike cylinder and eventually to a necklace of spheres or prolates; as particles become softer, the

nanotube is less deformed; for very soft particles at large membrane tension, the nanotube exhibits a sausage-string structure.

of single or multiple nanoparticles ( Figs. 5 and 13 ), the periodic wrapping requires significantly smaller γ to achieve the

full-wrapping state regardless of the particle rigidity, while for the partial-wrapping state the periodic wrapping requires

smaller γ for very soft particles. For stiff particles, their periodic wrapping requires smaller γ at small σ but larger γ at

large σ to achieve partial-wrapping in comparison with the wrapping of a single particle by a soft membrane nanotube.


Fig. 13. Phase diagram for periodic wrapping of multiple nanoparticles by a soft nanotube with respect to γ and σ at κ1 /κ2 = 0 . 1 , 1 , 5 , and ∞ .

4. Further discussion

For the tube interaction with confined elastic nanoparticles, there exists a partition of elastic deformation energy between

the particles and soft tubes. As the particles become stiffer, the confined particles deform less and the local membrane

bulges are more evident. One of the analogous examples on the basis of similar geometries is the formation of periodic

tubular structures of inhomogeneous membranes undergoing phase separation. For instance, multicomponent membranes

of ternary lipid mixtures could undergo phase separation into two coexisting fluid phases, and exhibit tubular structures of

periodic bulges associated with the cholesterol-enriched phases of a higher bending rigidity than the other phase ( Baumgart

et al., 2003; Campelo et al., 2007 ). The formation of cholesterol-enriched local bulges results from a reduction in the mem-

brane bending energy by relieving the mean curvature of the membrane through relative radial expansion. Other related

cases include the formation of pearled structures in vesicles which could be caused by spontaneous curvatures or area

difference of two lipid monolayers ( Bar-Ziv and Moses, 1994; Yue et al., 2014; Jeler ci c and Gov, 2015; Bahrami and Hum-

mer, 2017 ), owing to the assembly of membrane proteins or aggregates of adsorbed nanocomponents on membranes

( Lipowsky, 2013 ).

Comparing the elastic energy changes of wrapping single and multiple nanoparticles by a soft membrane tube at σ > 1 ,

the inset in Fig. 9 b indicates that multiple rigid spherical nanoparticles at small and large wrapping degree f are coopera-

tively wrapped with a relatively short particle distance L p , and at intermediate f the particles are wrapped individually with

a relatively large L p . In addition to wrapping multiple particles by a membrane tube here, a recent theoretical study indicates

that a vesicle wrapping Janus nanoparticles strongly adsorbed at its outside surface undergoes a similar energetically favor-

able discontinuous shape transformation in terms of the particle distance, cooperative wrapping at a relatively short L p and

individual wrapping at a relatively large L p ( Bahrami and Weikl, 2018 ). For Janus particles strongly adsorbed to the vesicle

outside, the curvature-mediated interaction can be attractive; in contrast, the curvature-mediated interaction between Janus

particles inside a vesicle is repulsive ( Bahrami and Weikl, 2018 ). Recalling that the vesicle-particle interaction depends on

the nanoparticle position with respect to the vesicle ( Bahrami et al., 2016; Yi and Gao, 2016; 2017; Yu et al., 2018; Bahrami

and Weikl, 2018 ), the interaction between a soft membrane nanotube and adhesive nanoparticles at its outside surface calls

for further detailed investigations in the future.

For periodic wrapping of multiple nanoparticles, the particle distance, deformation and wrapping degree are identical for

particles at different positions. Their dependence on the normalized membrane tension σ , normalized adhesion energy γand rigidity ratio κ1 / κ2 between the nanoparticles and soft tube are shown in Figs. 9b,c, 11b,c, and S4. In a more general

case of a finite number of elastic particles confined in a soft membrane nanotube, there is no spatial periodicity along

the tube axis, and the particle-nanotube interplay depends on the particle position. Open questions on how the system

configuration and particle-nanotube interplay depend on the finite particle number require further detailed analysis.

Note that the thermal fluctuation of the membrane is not considered here. Previous theoretical studies indicate that

the thermal undulation of a free membrane nanotube lowers the membrane tension ( Barbetta and Fournier, 2009 ). In the

presence of an enclosed cylindrical rod, the undulations of the nanotube membrane is hindered by the rod, which decreases

the entropy, gives rise to steric repulsion, and leads to a larger average nanotube radius ( Daniels and Turner, 2005 ). In our

cases of nanotube-particle interaction, the entropic repulsion owing to the membrane undulations is also expected, and

tends to prevent close contact between the nanoparticles and membrane nanotubes. Therefore, ( σ , γ ) in the absence of


thermal fluctuation as our focus here effectively becomes smaller in the presence of thermal fluctuation, which would shift

the phase boundary between the no- and partial-wrapping states rightward.

Besides the nanoscaled particle-tube interaction considered here, the bulged soft tubes have also been observed at larger

scales. For example, cells and vesicle-like structures of characteristic sizes on the order of tens of microns are reported

migrating within tubular connections formed by bronchial epithelial cells ( Zani et al., 2010 ). As the diameters of these

epithelial bridges are around ten to twenty micrometers, the (hydrostatic) pressures across the bridge surfaces or cell layers

cannot be ignored. Moreover, determining effects of tension and contraction of epithelial cells in dynamics on the bridge

structures and mechanical behaviors requires more sophisticated theoretical modeling approaches. Similar issues exist in the

pearling or beading of myelinated nerve fibers ( Markin et al., 1999 ).

5. Conclusions

Adopting the Helfrich membrane model, we have performed theoretical studies on the mechanical interplay between

soft membrane nanotubes and confined adhesive elastic nanoparticles modeled as vesicular particles. In terms of the parti-

cle number, two limiting cases are considered, wrapping of a single particle and wrapping of multiple particles in the soft

tube with a periodic structure. For both cases, depending on the collective influence of membrane tension, adhesion energy,

nanoparticle size, and the bending rigidity ratio between the nanoparticles and soft tubes, wrapping phase diagrams con-

sisting of the no-, partial-, and full-wrapping states and characteristically different system configurations such as the bulged

or undulated cylinders, necklaces of spheres or prolates, or tubes of sausage-string shape have been determined. It is found

that, as the particle becomes softer, more elastic deformation associated with the particles is partitioned from that of the

particle-tube complexes, which leads to lower adhesion energy for the partial-wrapping state but higher adhesion energy

for full-wrapping. In comparison with the single-particle wrapping by a flat membrane, the wrapping of a single particle of

moderate and large stiffness in the soft nanotube requires much smaller adhesion energy to achieve the no-wrapping state

and significantly larger adhesion energy for the full-wrapping state. By comparing the elastic energy changes for wrapping

single and multiple nanoparticles by the soft nanotube, the cooperative wrapping and individual wrapping are revealed as

two fundamental modes of interaction between multiple nanoparticles and the soft nanotube. At small membrane tension,

the cooperative wrapping mode prevails; while both wrapping modes are observed at large membrane tension, except for

very soft nanoparticles which prefer cooperative wrapping regardless of the membrane tension.

Analytical studies on the wrapping of single or multiple rigid spherical nanoparticles in soft nanotubes have also been

provided. It is shown that at small nanotube deformation the perturbation analysis works well in characterizing the elastic

behaviors of the deformed nanotubes. In the case of wrapping multiple rigid spherical nanoparticle in the soft nanotube

at general deformation, analytical solutions based on the nanotube configurational assumptions of catenoid, torus, and one-

sheeted hyperboloid have been obtained and well predict the system configurations as well as the tube deformation. More-

over, morphological phase diagrams of the free membrane portions between neighboring particles are determined with

respect to the wrapping degree and membrane tension.

Our results clearly indicate that particle elasticity plays an essential role in governing the system structures and me-

chanical behaviors, and reveal the physical mechanisms underlying the particle-tube interaction which is of fundamental

importance to tubule-related cellular activities including inter- and intracellular transport and communications, membrane

tubulation, morphological transformation in membrane nanotubes, and cooperative wrapping and membrane invagination

in cell uptake.

Analytical and numerical approaches used here can be readily extended to the cases of wrapping rigid non-spherical

nanoparticles such as prolate particles or particles of different sizes in membrane nanotubes ( Raatz and Weikl, 2017; Xiong

et al., 2017; Tang et al., 2018 ), polymer confinement in tubular channels where polymers could exhibit elongated, compact,

and blobbed states ( Chen, 2007; Yan et al., 2018 ), and pearling in membrane tubes driven by adsorbed curved membrane

proteins or due to phase separation ( Campelo et al., 2007; Yue et al., 2014; Jeler ci c and Gov, 2015 ).

Declaration of Competing Interest

The authors have no conflicts of interest to disclose.

CRediT authorship contribution statement

Zeming Wu: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft.

Xin Yi: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - review & editing, Supervision,

Project administration, Funding acquisition.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11872005 ). Computation re-

sources supported by the High-performance Computing Platform of Peking University are acknowledged.

https://doi.org/10.13039/501100001809


Appendix A. Cubic B-spline basis functions

A cubic B-spline curve y = y (x ) is defined as y =

∑ n j=0 c j N j (x ) , where the cubic B-spline basis functions N j ( x ) defined on

a knot vector x 0 , x 1 , . . . , x k , . . . , x n +4 ( x k ≤ x k +1 ) can be expressed as

N j ( x ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(x j+4 − x j

) j+4 ∑

m = j+1

[

( x m

− x ) 3 ∏ j+4

q = j,q = m

( x m

− x q )

]

, x ∈

[x j , x j+1

),

(x j+4 − x j

) j+4 ∑

m = j+2

[

( x m

− x ) 3 ∏ j+4

q = j,q = m

( x m

− x q )

]

, x ∈

[x j+1 , x j+2

),

(x j+4 − x j

) j+4 ∑

m = j+3

[

( x m

− x ) 3 ∏ j+4

q = j,q = m

( x m

− x q )

]

, x ∈

[x j+2 , x j+3

),

(x j+4 − x

)3 (x j+4 − x j+1

)(x j+4 − x j+2

)(x j+4 − x j+3

) , x ∈

[x j+3 , x j+4

),

0 , otherwise .

In the case of a uniform knot vector x = x k +1 − x k > 0 , we have

N j ( x ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(x − x j

)3

6 ( x ) 3

, x ∈

[x j , x j+1

),

(x j+2 − x

)3

( x ) 3

−2

(x j+3 − x

)3

3 ( x ) 3

+

(x j+4 − x

)3

6 ( x ) 3

, x ∈

[x j+1 , x j+2

),

−2

(x j+3 − x

)3

3 ( x ) 3

+

(x j+4 − x

)3

6 ( x ) 3

, x ∈

[x j+2 , x j+3

),

(x j+4 − x

)3

6 ( x ) 3

, x ∈

[x j+3 , x j+4

),

0 , otherwise .

Appendix B. Approximation of the outer free membrane between two neighboring nanoparticles

Appendix B.1. Toroidal configuration

In the case of multiple nanoparticles at relatively small and intermediate wrapping degree f , the configuration of the

outer free membrane between two neighboring nanoparticles is approximated as a portion of a ring torus ( Fig. 6 b), gener-

ated by revolving a full circle of radius ρ about a coplanar axis a distance R ( > ρ) away from the circle center. In a Cartesian

coordinate with its origin located in the middle of a line segment connecting the centers of two neighboring nanoparticles,

the position vector r of the torus portion is parameterized as ( Kreyszig, 1991; Deserno and Gelbart, 2002 )

r (φ, θ ) = ((R − ρ cos θ ) cos φ, (R − ρ cos θ ) sin φ, ρ sin θ )

with φ ∈ [0, 2 π ] and θ ∈ [ −θ0 , θ0 ] .

The coefficients g torus of the first fundamental form of the torus portion in the notation of the metric tensor is

(g torus ) =

(r φ · r φ r φ · r θr θ · r φ r θ · r θ

)=

((R − ρ cos θ ) 2 0

0 ρ2

),

where r φ ≡ ∂ r / ∂ φ and r θ ≡ ∂ r / ∂ θ .

The surface area element is

d A =

√

det (g torus ) d φd θ = ρ(R − ρ cos θ )d φd θ,

and the surface area of the torus portion is

A torus =

∫ d A =

∫ θ0

−θ0

∫ 2 π

0

ρ(R − ρ cos θ )d φd θ = 4 πρ(Rθ0 − ρ sin θ0 ) .

With the unit normal vector n to the torus pointing toward the z -axis

n = − r φ × r θ∣∣r φ × r θ∣∣ = (− cos θ cos φ, − cos θ sin φ, sin θ ) ,


the coefficients b torus of the second fundamental form of the torus portion is

(b torus ) =

(n φ · r φ n φ · r θn θ · r φ n θ · r θ

)=

(−(R − ρ cos θ ) cos θ 0

0 ρ

),

where n φ ≡ ∂ n / ∂ φ and n θ ≡ ∂ n / ∂ θ .

Since g torus 12

= b torus 12

= 0 , the coordinate curves on the torus portion are lines of curvature. Consequently, the principal

curvatures are b torus 11

/g torus 11

and b torus 22

/g torus 22

, respectively, and the mean curvature is

H torus =

1

2

(b torus

11

g torus 11

+

b torus 22

g torus 22

)=

1

2

(1

ρ− cos θ

R − ρ cos θ

).

Therefore, the membrane bending energy of the torus portion is

E torus b = 2 κ

∫ H

2 torus d A =

4 πR

2 κ

ρ√

R

2 − ρ2 arctan

( √

R + ρ

R − ρtan

θ0

2

)

− 8 πκ sin θ0 ,

where κ is the membrane bending rigidity.

In the case of rigid spherical nanoparticles, A torus and E torus b

become A torus = 4 πρ[ −ρ f + (ρ + a ) √

1 − f 2 arcsin f ] and

E torus b

= E torus b

( f, ρ/a ) , respectively, with the knowledge of f = sin θ0 and R = (ρ + a ) cos θ0 . Then the total elastic energy of

one periodic unit is

E el = E torus b + 8 fπκ + σ (A torus + 4 πa 2 f ) − 2 πL p

√

2 σκ.

Energy minimization of E el at a given wrapping degree f upon a geometric constraint of L p = 2(ρ + a ) f ≥ 2 a or equiv-

alently ρ/a ≥ 1 / f − 1 leads to the determination of ρ/ a and E torus b

.

Appendix B.2. One-sheeted hyperboloid configuration

Besides the toroidal geometry belonging to quadric surfaces, an alternative quadric surface approximation of the outer

free membrane configuration in the case of multiple rigid spherical nanoparticles is the hyperboloid of one sheet ( Fig. 6 c)

which in the same coordinate in Appendix 2.1 could be parameterized as ( Kreyszig, 1991 )

x 2 + y 2

b 2 − z 2

d 2 = 1 or r (φ, t) =

(b √

1 + t 2 cos φ, b √

1 + t 2 sin φ, dt

),

with φ ∈ [0, 2 π ] and t ∈ [ −t 0 , t 0 ] with dt 0 indicating the half-height of the hyperboloid portion.

The coefficients of the first fundamental form of the hyperboloid portion is

(g hyp ) =

(r φ · r φ r φ · r t r t · r φ r t · r t

)=

(b 2 (1 + t 2 ) 0

0 d 2 + b 2 t 2 / (1 + t 2 )

).

With the unit normal vector n to the hyperboloid portion pointing toward the z -axis

n = − r φ × r t ∣∣r φ × r t ∣∣ =

1 √

d 2 + (b 2 + d 2 ) t 2 (−d

√

1 + t 2 cos φ, −d √

1 + t 2 sin φ, bt) ,

the coefficients of the second fundamental form is

(b hyp ) =

(n φ · r φ n φ · r t n t · r φ n t · r t

)=

bd √

d 2 + (b 2 + d 2 ) t 2

(−(1 + t 2 ) 0

0 1 / (1 + t 2 )

).

The surface area element is d A =

√

det (g hyp ) d φd θ = b √

d 2 + (b 2 + d 2 ) t 2 d φd t, and the surface area of the hyperboloid

portion is

A hyp =

∫ d A =

∫ t 0

−t 0

∫ 2 π

0

b √

d 2 + (b 2 + d 2 ) t 2 d φd t = 2 πb

[h

4 d 2

√

4 d 4 + (b 2 + d 2 ) h

2 +

d 2 √

b 2 + d 2 arsinh

h

√

b 2 + d 2

2 d 2

].

(A1)

where h = 2 dt 0 is the height of the hyperboloid portion of the outer free membrane between two neighboring nanoparticles,

and arsinh represents the inverse of the hyperbolic sine function.

The mean curvature is

H hyp =

1

2

(b hyp

11

g hyp +

b hyp 22

g hyp

)= −d[ b 2 (t 2 − 1) + d 2 (1 + t 2 )]

2 b[ d 2 + (b 2 + d 2 ) t 2 ] 3 / 2 .

11 22


Therefore, the membrane bending energy of the hyperboloid portion in one periodic unit is

E hyp

b = 2 κ

∫ H

2 hyp d A =

2 πd 2 κarsinh

(t 0

√

b 2 + d 2 /d )

b √

b 2 + d 2 +

2 πbt 0 κ[2 b 4 t 2 0 + b 2 d 2

(3 − 4 t 2 0

)− 6 d 4

(1 + t 2 0

)]3 d 2

[(b 2 + d 2

)t 2

0 + d 2

]3 / 2

=

κA hyp

b 2 − πκh

2 bd 2 + 4 πbh κ ×

2 b 2 d 4 +

(b 2 − 3 d 2

)2

3 d 2 3 , (A2)

where =

√

4 d 4 + (b 2 + d 2 ) h 2 .

In the case of spherical rigid nanoparticles, the height h at the contact edge can be given as h = L p − 2 fa, and the

radius of the cross section formed by the contact edge is R = b √

1 + h 2 / (4 d 2 ) . The smooth contact between the particle and

hyperboloid portion at the contact edge further requires the continuities of the radial coordinate and tangent angle as

R = a √

1 − f 2 and

b 2 h

2 d 2 R

=

f √

1 − f 2 .

From the above four equations, we have at a given f and L p

b =

√

a (2 a − f L p )

2

, d =

1

2

√

f

√

(L p − 2 a f )(2 a − f L p ) , and b 2 + d 2 =

L p (2 a − f L p )

4 f ,

or L p =

2

f

(a − b 2

a

), d =

b

fa

√

a 2 (1 − f 2 ) − b 2 , and h =

2

fa [ a 2 (1 − f 2 ) − b 2 ] .

Eqs. (A1) and (A2) can further reduce to A hyp = A hyp ( f, b/a ) and E hyp

b = E

hyp

b ( f, b/a ) , respectively, as

A hyp ( f, b/a ) =

2 πa 2

f

(1 − f 2 − b 2

a 2

)[

1 +

( b/a ) 2 √

1 − ( b/a ) 2

arsinh

√

a 2

b 2 − 1

]

,

E hyp

b ( f, b/a ) = 2 πκ

[

f 3

3

2 + ( b/a ) 2

1 − f 2 − ( b/a ) 2

− 2 f +

1 − f 2 − ( b/a ) 2

f √

1 − ( b/a ) 2

arsinh

√

a 2

b 2 − 1

]

.

Then the total elastic energy of one periodic unit is

E el = E hyp

b + 8 fπκ + σ (A hyp + 4 πa 2 f ) − 2 πL p

√

2 σκ.

Energy minimization of E el = E el ( f, b/a ) at a given wrapping degree f upon a geometric constraint of

L p = 2(a 2 − b 2 ) / ( fa ) ≥ 2 a or equivalently b/a ≤√

1 − f leads to the determination of b / a , E hyp

b , and E el .

Numerical analysis indicates that there exists a critical value of f = f c at which the single minimum of E el determined

by d(E el ) / d(b/a ) = 0 locates at b/a =

√

1 − f c . Substituting b/a =

√

1 − f c into d(E el ) / d(b/a ) = 0 , we have

√

f c [3 − f c (6 + 3 f c − 2 f 3 c )] + 3(1 − f c ) 2 (1 + f c ) arsinh

√

f c

1 − f c

+

3(1 − f c ) 2

2

[(3 − f c )

√

f c + (1 − f c ) 2 arsinh

√

f c

1 − f c

]σ − 12(1 − f c ) 2

√

f c √

σ = 0 .

(A3)

At zero membrane tension, we have f c ≈ 0.5659, slightly larger than the lower bound of the wrapping degree ( f ≈ 0.5644)

based on the catenoid configuration.

At f ≤ f c , the minimum of E el locates at b/a =

√

1 − f as

E el = 2 πκ

[

f (6 − 3 f − f 2

)3 ( 1 − f )

+

1 − f √

f arsinh

√

f

1 − f − 2

√

σ

]

+ π( 1 + f ) κσ +

π( 1 − f ) 2 κσ√

f arsinh

√

f

1 − f (A4)

and L p = 2 a (1 − b 2 /a 2 ) / f = 2 a .

At f ≥ f c , the minimum of E el locating at b / a satisfying d(E el ) / d(b/a ) = 0 .

The total system energy then becomes E tot /κ = E el /κ − 2 π f γ , and the minimum adhesion energy γmin necessary for

an energetically favorable wrapping at f can be determined via γmin = d(E el /κ) / d(2 π f ) . At f ≤ f c , we have

γmin ( f ) =

1

6

[

8 + 4 f +

3

f +

4

(1 − f ) 2 − 3(1 + f )

f 3 / 2 arsinh

√

f

1 − f

]

+

1

4 f 3 / 2

[ √

f (1 + f ) − (1 − f )(1 + 3 f ) arsinh

√

f

1 − f

]

σ .

In the limiting case of f = 0 , we have γmin = (4 − σ ) / 3 , consistent with the analytical results based on the toroidal

configuration and the fully nonlinear numerical results in the phase diagram.


Fig. 14. Energy analysis and the determination of the morphological phase diagram for the periodic wrapping of rigid spherical nanoparticles based on

the one-sheeted hyperboloid and catenoid configurations. (a) Morphological phase diagram with respect to σ and f . Above the dashed (red) line, catenoid

configurations; below the dashed line, hyperboloid configurations. (b) Selected profiles of E el / κ as functions of b / a at different σ and f (symbols in (a)).

Solid circles correspond to the local minima with d(E el /κ) / d(b/a ) = 0 . Vertical dash-dotted lines correspond to the locations of b/a =

√

1 − f separate

the unphysical (dashed curves) and physical (solid curves) regimes of b / a .

Comparing E el between the catenoid and one-sheeted hyperboloid configurations and performing stability analysis of

E el for hyperboloid configurations, we determine a morphological phase diagram ( Fig. 14 a). Above the red curve therein

the catenoid configurations prevail and below which hyperboloid configurations are energetically favored. Depending on the

energy stability in Fig. 14 b, the hyperboloid regime can be further divided into f ≤ f c and f > f c parts ( Fig. 14 a). At f ≤ f c ,

the minimum of E el locates at b/a =

√

1 − f and L p = 2 a ; at f > f c , the global minimum of E el locates at b/a <

√

1 − f .

Numerical comparison between the fully nonlinear numerical results and the analytical prediction based on the hyper-

boloid configuration can be found in Figs. S1 and S2 in Supplementary material.

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.jmps.2020.

103867 .

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Supplementary Materials for “Structures and mechanical behaviors of soft nanotubes confining adhesive single or multiple elastic nanoparticles”

Zeming Wu and Xin Yi

Department of Mechanics and Engineering Science, BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China

E-mail address: [email protected] (X. Yi) Note 1. Comparison between the fully nonlinear numerical results and the analytical prediction based on the hyperboloid and catenoid configurations for the periodic wrapping of multiple rigid spherical nanoparticles

Fig. S1. Elastic energy change ∆Eel for the periodic wrapping of rigid spherical nanoparticles at 0=σ . Thin and thick lines correspond to the results based on fully nonlinear numerical solutions and analytical prediction with hyperboloid and catenoid assumptions, respectively.

Fig. S2. Profiles of elastic energy ∆Eel (a) and particle distance Lp (b) during the wrapping of multiple spherical rigid nanoparticles at σ = 0.5, 1.5, 4, 5, and 8. Solid lines indicate the fully nonlinear numerical solutions. Dashed (gray) lines represent results based on the catenoid and hyperboloid assumptions.

Note 2. Perturbation analysis in the case of periodic wrapping of multiple rigid spherical nanoparticles

Fig. S3. Comparison between the fully nonlinear solutions and perturbation analysis for the evolution profiles of the elastic energy ∆Eel (a) and particle distance Lp (b) during the wrapping of multiple rigidi nanoparticles at different normalized membrane tension σ = 0.5, 1.5, 4, 5 , and 8. (c) Lp as a function of γ at different σ. Dashed lines in (b,c) represent results based on perturbation analysis. In the case of multiple nanoparticles confined in the membrane nanotube of a periodic configuration perturbed from an ideal cylinder of uniform radius, the nanotube contour can be expressed as

r(z) = ρ0 + w(z), where ρ0 is the radius of a reference cylinder and w(z) denotes the radial deviation. Note that ρ0 in general does not have to be the characteristic size 0 2 / (2 )r κ σ= , and is taken as the r-coordinate of the contact edge hereinafter. Following the similar procedure in subsection 2.2, the governing shape equation of the outer free region of the membrane nanotube expanded in terms of the leading order in w(z) is be given as

2 2 2 2 2 2 20 0 0 0 0 0 0 0

1 1 1 1 3 1 1 1 1 02 2 2

w w wr r rρ ρ ρ ρ ρ

′′′′ ′′+ − + − − − =

. (S1)

Eq. (S1) is a fourth-order nonhomogeneous linear differential equation with four independent characteristic solutions of the form

2 2 24 2 2 40 0 0 0 c

0 0 0 02 2 2 20 0 0 0 0

( ) 1 1exp 1 23 63 2

r z zw D i r rr r rα α

ρ ρ ρ ρ ρρ ρ

− −= + ± − ± − −

− ( 1, , 4α = ).

For multiple spherical rigid nanoparticles of radius a at a wrapping degree f, we have four boundary conditions r(zc) = r(Lp–zc) = ρ0 and r′(zc) = –r′(Lp–zc) = –zc/ρ0 with zc = fa and

20 1a fρ = − , from which the coefficients Dα can be determined.

Substituting the nanotube contour r(z) = ρ0 + w(z) into Eq. (7) over the integral interval [zc,Lp/2–zc], we can obtain the elastic energy change ∆Eel(f,Lp). Then a state of minimum energy with respect to Lp is further determined at given f.

At the limiting case of zero membrane tension σ = 0 (r0→∞), Eq. (S1) reduces to

2 4 30 0 0

1 3 1 02 2 2

w w wρ ρ ρ

′′′′ ′′+ + − = ,

which has four independent characteristic solutions of the form

0 c

0

exp3

z zw Dα α αρ ζ

ρ −

= +

( 1, , 4α = ),

where 1 2( )b ibαζ = ± ± with 1 ( 1 2 6) / 8b = − + and 2 (1 2 6) / 8b = + . Then the

resulting expression for r(z) (z∈[zc,Lp/2–zc]) reads

c1

0

c1

0

0 c c1 2 2 2

0 0

c c3 2 4 2

0 0

4( ) cos sin3

cos sin

z zb

z zb

z z z zr z e b b

z z z ze b b

ρ

ρ

ρ η ηρ ρ

η ηρ ρ

−

−−

− −= + +

− −

+ +

,

where

1

1

1

1 c

2 c 2 c

3 1

4 2 c c

1 c 2 c 2

2 0 2 2 0 1

0 1 0 1 0 2 2

0

0 1 0 1 0 2 2

1 p 0 2 p 0 1 2

cos ) sin (3 )] / ,

(3 ) cos (3 ) sin ] / ,/ 3

3 (3 ) sin( 2 ) / , ( 2 ) / , 6( sin

[(

[,

[cos ( ) ]si

/n

,h

c

c

c

c b c z b

z b z b b c

z b z b b cb L z b L z b

e

e c

c ec bc c

η

ηη η

η

ρ ρ

ρ ρ ρρ

ρ ρ ρρ ρ

−

−

− + Ω

− + +

= −

== − −

− Ω

− + Ω− −

=Ω = +

− + −= = 1).c

Note 3. Numerical studies on periodic wrapping of multiple elastic nanoparticles

Fig. S4. The elastic energy change ∆Eel and wrapping degree f as functions of the normalized adhesion energy γ at κ1/κ2=∞, 5, 1, 0.1 and σ = 0.5, 1.5, 4, 5, 8.

Zeming Wu, Xin - pku.edu.cn

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