Budding of an Adhesive Elastic Particle out of a Lipid Vesicle - ACS

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Budding of an Adhesive Elastic Particle out of a Lipid Vesicle Xin Yi, and Huajian Gao ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.6b00815 • Publication Date (Web): 08 Feb 2017 Downloaded from http://pubs.acs.org on February 14, 2017

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ACS Biomaterials Science & Engineering

Budding of an Adhesive Elastic Particle out of a Lipid Vesicle Xin Yi1,2 and Huajian Gao2∗ 1 Beijing

Innovation Center for Engineering Science and Advanced Technology (BIC-ESAT), and

Department of Mechanics and Engineering Science, College of Engineering, Peking University, 5 Yiheyuan Road, Haidian District, Beijing 100871, China 2 School

of Engineering, Brown University, 182 Hope Street, Providence, Rhode Island 02912, United States E-mail: [email protected]

∗ To

whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract The problem of particle budding out of a lipid vesicle is of fundamental importance to our understanding of the biophysical mechanisms involved in viral budding and exocytosis. Here, we present a theoretical study on the outward budding of an adhesive elastic particle out of a lipid vesicle of different spontaneous curvatures. It is shown that a discontinuous shape transformation can occur for budding out of a vesicle with positive spontaneous curvature but not for a vesicle with zero or negative spontaneous curvature, and that softer particles require stronger adhesion energy to achieve full release from the vesicle. Calculations also indicate that the adhesion energy required for full release increases as the spontaneous curvature of the vesicle decreases. A rich variety of budding phase diagrams accounting for the stable or metastable states of no-budding, partial-budding and full-release are determined. Endocytosis, exocytosis, intracellular budding of elastic particles and related biological implications are discussed. Our results provide physical insights into the biophysical mechanisms of viral budding and exocytosis, and may also provide rational design guidelines for controlled drug delivery systems. Keywords: cell-particle interaction, vesicles, exocytosis, endocytosis, intracellular budding, budding phase diagrams, elasticity, spontaneous curvatures


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Introduction As a minimal biomimetic model of cells, lipid vesicles have been extensively employed in experiments to reveal the underlying biophysical mechanisms involved in cell interaction with nanomaterials. 1–7 For example, it has been demonstrated that adhesive nanoparticles can be transported into a vesicle via a passive internalization process regulated by particle size and surface properties, 1–3 which is also consistent with the internalization of nanoparticles in red blood cells that lack active endocytic machinery. 8 Theoretical studies and molecular dynamics simulations have also been performed to investigate the vesicle-nanoparticle interactions. 4–7,9–13 At a given adhesion energy, it is shown that a minimum nanoparticle size is required for a vesicle to wrap around a particle being internalized. 4–7,14–16 Recent theoretical studies indicate that, during incorporation into a vesicle, a rigid ellipsoidal particle undergoes a change in orientation from one with longer axis parallel to the vesicle membrane to perpendicular to the membrane. A similar conclusion has been drawn in the interaction between a nanorod and a membrane patch in molecular dynamics simulations 17,18 and theoretical studies. 19,20 This particle reorientation driven by membrane deformation sheds light on the reorientation of short rod-shaped viruses during budding. 21 Also, the spontaneous curvature of a bilayer membrane induced by the asymmetry between the two leaflets of the bilayer is recognized as a key parameter for particle engulfment, 12 especially in the case of clathrin-dependent endocytosis where the clathrin-coated membrane exhibits a finite spontaneous curvature. There is now mounting evidence that the elasticity of a particle plays an important role in regulating its interaction with a lipid vesicle. 22–25 It has been shown that softer particles require higher adhesion energy to achieve successful internalization into a vesicle. This is consistent with the existing theoretical studies and molecular dynamics simulations on wrapping of an elastic particle by a membrane patch. 20,26–31 On the other hand, a recent theoretical study on the kinetics of receptor-mediated endocytosis of elastic nanoparticles indicates that successful internalization of softer nanoparticles is kinetically faster than that of stiffer particles. 32 Compared to extensive studies on nanoparticle uptake and endocytosis, less attention has been paid to the companion process of exocytosis through which nanoparticles, molecules and waste 3 ACS Paragon Plus Environment


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materials are transported out of the cells. 33–35 Similarly, compared to the widely investigated process of particle incorporation into a vesicle, less effort has been made to study the pathways of internalized particles undergoing outward budding from a vesicle. 4,7,9,23,24,36–41 It is known that larger particles in binary colloidal mixtures confined in a vesicle are more likely pinned to the vesicle surface due to the entropic depletion effect. 36 Recent experimental studies have revealed that charged colloids confined densely in a vesicle lead to significant vesicle shape transformation from spheres to tube-, discocyte-, stomatocyte- and necklace-like shapes. 38 Similar phenomena due to the depletion volume effect have also been observed in the shape transformation of giant phospholipid vesicles containing macromolecules. 42 In addition to theoretical work on the budding of a single particle, 7,9,23,24 a combination of computational simulations and experimental work have shown that particles can organize into linear aggregates and induce membrane tubulation or deep invagination, depending on the adhesion energy between the vesicle and trapped particles. 39,40 Membrane tension has been recognized as a parameter regulating a transition between tubular structures and linear aggregates of particles. 41 Two-dimensional molecular dynamics simulations have also been performed to study the budding and fission of a vesicle induced by an internal nanoparticle. 37 In spite of the above advances, there are still open questions concerning the roles of particle size and elasticity as well as membrane spontaneous curvature in particle budding out of a vesicle. To address these questions, here we perform a systematic theoretical investigation on how the particle budding depends on particle size, adhesion energy, bending rigidity ratio between the particle and vesicle membrane as well as the spontaneous curvature of the vesicle. With the help of the calculated energy profiles of the budding process, we determine a set of budding phase diagrams demarcating different stable and metastable states of no-budding, partial-budding and full-release. It will be shown that the budding of a relatively large particle from a vesicle with a finite positive spontaneous curvature leads to a discontinuous shape transition, the underlying mechanism of which will be discussed in detail.


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Theoretical Modeling

Figure 1: Axisymmetric configurations of an elastic particle budding out of an initially spherical vesicle in cylindrical coordinates (r, ϕ , z). (a) The system is divided into three regions: inner free region, contact region, and free region of the vesicle. Each region is characterized by the tangent angle ψ and normalized arclength s (scaled by the vesicle radius R); For example, the normalized arclength s in the free region of the vesicle is measured from the bottom pole of the vesicle (s = 0) to the contact edge (s = l ). (b) Schematic configurations for the states of no-budding with zero contact area and full-release in which the fully wrapped particle is completely covered by the vesicle membrane except a curved membrane neck of an infinitesimal toroid and about to pinch off from the vesicle. Partial-budding corresponds to an intermediate configuration with incomplete budding, as shown in (a).

Consider the outward budding of an adhesive spherical elastic particle of radius a from a unilamellar vesicle of radius R, as shown in Figure 1a. The elastic particle is modeled here as a vesicular particle for simplicity. Both the budding particle and the host vesicle are assumed of fixed surface areas and vanishing osmotic pressure, and subject to axisymmetric elastic deformation. Quantities pertaining to the vesicular elastic particle and the lipid vesicle are identified by subscripts 1 and 2, respectively. By adopting the Canham-Helfrich membrane theory, 43 the total system energy is



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assumed to be Etot = Eel − γ Ac , where

Eel = 2κ1

∫

M12 dA1 +

κ2 2

∫

(2M2 −C0 )2 dA2

is the elastic energy, Mi , κi , and dAi (i = 1, 2) denoting the mean curvature, bending rigidity, and surface element of the budding particle and the host vesicle, respectively; 24,44,45 C0 is the spontaneous curvature of the vesicle membrane; γ (> 0) is the adhesion energy and Ac the area of the contact region. The initial radii of the particle and vesicle are a =

√

A1 /(4π ) and R =

√

A2 /(4π ),

respectively. At a certain Ac or budding degree f = Ac /A1 , the bending energy of the system is

Eel ≡ Etot + γ Ac = Etot + f γ A1 . Introducing a normalized spontaneous curvature c0 = C0 R, we can obtain Eel /κ2 as a function of a/R, κ1 /κ2 and c0 . We further assume that there is no separation between the particle and vesicle in the contact region where they can slide freely against each other, so that the bending rigidity of the contact region can be approximated as κ1 + κ2 . The variation of Eel leads to a set of governing equations for the system morphology, 24,44,45 which can be numerically solved together with boundary conditions specifying values of the tangent angles and r-coordinates at the contact edge and pole of each region, as well as the continuity of

ψ and r across the contact edge. Further details of the derivations and numerical procedure can be found in refs 24,44,45. The elastic energy Eel and the system morphology can be obtained with the help of geometrical relations r˙ = cos ψ and z˙ = sin ψ . The total system energy is determined as Etot = Eel − f γ A1 , or in reduced form Etot /κ2 = Eel /κ2 − 2π f γ¯, where γ¯ ≡ 2γ a2 /κ2 . Depending on the budding degree f , the system can exhibit three characteristic budding states: no-budding ( f = 0), partial-budding (0 < f < 1), and full-release ( f → 1).

Results Our present study on particle budding out of a lipid vesicle will consider the effects of particle stiffness and size as well as the spontaneous curvature of the vesicle. Figure 2 plots the elastic energy 0 as a function of the budding degree f , where E ≡ E + γ A denotes the change ∆Eel = Eel − Eel tot c el


1

0.6

budding degree

(b) 8

c

0 0.0

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4

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6

)

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/(

a/R

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1.0

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/(

0

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/(

el

0.5

decreasing

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decreasing

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0.8

/

2

1.0

0 0.0

/(

el

/(

4

E

el

E

el

E

el

2

2 decreasing

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/

2

0 0.0

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decreasing

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1.0

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10

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)

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1.0

)

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2

a/R

E

2

el

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E

E

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2.0

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)

a/R

a/R

el

2,

, 5, 1, 0.1

)

2

2

2

0

/

el

/(

c

1

/(

)

6

2

4

(a)

E

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/(

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decreasing

0.2

0.4

f

0.6

1

0.8

/

8 6 4 2

2

1.0

0 0.0

decreasing

0.2

0.4

f

0.6

1

0.8

/

2

1.0

Figure 2: Elastic energy ∆Eel as an adhesive soft particle undergoes outward budding from a vesicle at different size ratios a/R, bending rigidity ratios κ1 /κ2 and spontaneous curvatures c0 = 2 (a), 0 (b), and −2 (c). 0 = 8πκ + 2πκ (2 − c )2 is the reference energy before the elastic energy of the system and Eel 1 2 0

particle contacts the inner surface of the vesicle. We focus on the cases in which the magnitude of the reduced spontaneous curvature c0 is small enough that the vesicle remains spherical when free. For the incorporation of rigid particles into a vesicle of c0 = 0, the energy profiles ∆Eel for

a/R < 0.35 shown in Figure S1b are similar to those reported in refs 7,23. As the particle stiffness decreases, the slope d(∆Eel )/d f decreases in the early stage and increases in the late stage of budding (Figures 2 and S2), indicating that, compared to rigid particles, softer particles require smaller adhesion energy in the early stage of budding but larger adhesion energy toward the late stage of budding. Similar stiffness-dependence were previously reported for membrane wrapping of elastic vesicular particles or solid capsules, 26,30 as well as vesicle incorporation of elastic particles, 23,24 where the underlying mechanism has been attributed to a partition of elastic deformation energy between the particle and the receiving membrane patch or vesicle. As κ1 /κ2 decreases, the particle deforms more while the vesicle deforms less in the early stage of budding, and the reverse is true at the late stage of budding (Figures 3 and S3). In the case of c0 = 2, the energy curve ∆Eel ( f ) is smooth at relatively small a/R (Figure 2a and Figure S1a in Supporting Information), indicating that the budding configuration evolves con-



Figure 3: Selected budding configurations at different a/R and κ1 /κ2 . tinuously, as illustrated in Figure 3a,b. As the size ratio a/R reaches a critical value ρc , the energy curve ∆Eel ( f ) exhibits a kink at a critical budding degree fc , corresponding to a discontinuous shape transformation of the vesicle from an elongated pear shape into a double-gourd shape (Figure 3c). For κ1 /κ2 → ∞ and c0 = 2, no kink is observed until a/R exceeds ρc ≈ 0.49 (Figure S1a). Further numerical analysis indicates that softer particles correlate with larger ρc and fc (Figure 2a), and ∆Eel ( fc ) is insensitive to κ1 /κ2 . Compared to the case of c0 = 2, there is no kink in energy curve or discontinuous shape transformation for c0 = 0 and −2 (Figures 2b,c and S3). Besides particle budding, the incorporation of relatively large particles into a vesicle also involves discontinuous vesicle shape transformation, where the kink in energy curve is followed by a flat energy plateau characterizing the propagation of a membrane protrusion during the incorporation process. 24 However, different from the budding case, there are kinks in the energy curves of particle incorporation into a vesicle of zero spontaneous curvature, i.e. when c0 = 0. 1

/

(b) 2

/

(c) 2

V/(4 R /3)

0.90

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, 5, 1, 0.1 0.4

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decreasing

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V/(4 R /3)

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decreasing

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(a) 1.00

V/(4 R /3)

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0.88 0.0

0,

a/R

0.2

0.4

1

/

0.92

2

0.90

0.3

f

0.6

decreasing

c

2,

0

0.8

1.0

0.88 0.0

0.2

1

a/R 0.4

f

/

2

0.3 0.6

0.8

1.0

Figure 4: Reduced volume V /(4π R3 /3) of the vesicle during outward budding of an elastic particle for different spontaneous curvatures of the vesicle: c0 = 2 (a,b), 0 (c) and −2 (d).


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Besides the energy profiles ∆Eel ( f ), the particle budding process could also be characterized by the evolution of vesicle volume V , as shown in Figure 4. As f increases, the vesicle volume decreases first and then increases as f exceeds a certain value. The larger the a/R and κ1 /κ2 , the smaller the minimum value of V /(4π R3 /3). In the cases of c0 = 0 and −2, V evolves continuously and smoothly (Figures 4c,d and S4); while for c0 = 2 there is a discontinuous jump at f = fc for relatively large particles with a/R ≥ ρc (Figure 4b). As indicated in Figure 4, budding of a softer particle leads to a larger volume V in the early and middle stages of budding, but a smaller V in the late stage of budding. This stiffness-dependent evolution of vesicle volume is consistent with the configurational evolution of the vesicle. As indicated in Figure 3, a softer particle adopts a more flattened morphology parallel to the vesicle membrane in the early and middle stages of budding. Since the vesicle is under vanishing osmotic pressure and can adjust its volume freely, the flattened softer particle would cause the vesicle to undergo less deviation from the initial spherical shape, which corresponds to a larger V . In the late stage of budding, the soft particle undergoes a shape transition from a parallel flattened morphology to an elongated morphology perpendicular to the vesicle membrane. Compared to a stiff particle, both the soft particle and the free region of the bulging vesicle undergo more deviation from spherical shapes. Therefore, budding of a softer particle leads to a smaller V in the late stage of budding. As f → 1, the vesicle adopts a doublegourd shape and V /(4π R3 /3) → [1 − (a/R)2 ]3/2 + (a/R)3 , independent of the particle stiffness, as shown in Figures 4 and S4. Figure 5 shows typical profiles of the total system energy ∆Etot ( f ) = ∆Eel − γ Ac at six selected sets of system parameters (κ1 /κ2 , a/R, γ¯, c0 ). The behavior of ∆Etot as a function of the budding degree f indicates that there exist five possible budding phases (I to V), depending on the stability of no-budding, partial-budding and full-release states. In phase I (no-budding state), the adhesion energy γ¯ is usually small and ∆Etot increases monotonically with f . As γ¯ increases, phase II arises in which a stable state of no-budding and a metastable state of partial-budding coexist. Further increase in γ¯ could lead to a global minimum at a state of partial-budding (phase III). In phase IV there is a metastable state of partial-budding with an energy barrier to reach the stable full-release


/

2

,

a / R, , c

0

)

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Etot

1

Etot

(

(

, 0.3, 3.5, 0)

III : partial

(

, 0.3,1.7, 2)

1

f

0

1

Etot

0

, 0.3,1.9, 2)

II : no & partial

0 0

(

, 0.3, 2, 2)

III : partial

1

1

0

1

f

f

f

Etot

(

I : no-budding

Etot

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Etot


0

1

f

f (

, 0.3, 2.1, 2)

IV : partial & full

(

, 0.3, 2.2, 2)

V : full-release

Figure 5: Typical profiles of the total energy change ∆Etot as a function of f at six selected sets of system parameters (κ1 /κ2 , a/R, γ¯, c0 ). On the basis of the stability of no-budding, partial-budding and full-release states, the behavior of ∆Etot ( f ) can be categorized into five budding phases, I to V. Phase I, a stable no-budding state with a single energy minimum at f = 0; phase II, coexistence of a stable no-budding state and a metastable partial-budding state; phase III, a stable state or coexistence of two stable states of partial-budding; phase IV, coexistence of a metastable state of partial-budding and a stable full-release state; phase V, a stable full-release state with a single energy minimum at f = 1. In each profile, the underlined budding state indicates the phase of lower system energy. Magnitudes of ∆Etot are rescaled to a unified value. Depending on the values of the set parameters, energy minima in the phase III of two stable states of partial-budding do not have to be located very close to f = 0 or f = 1. As indicated in Figure 6, some budding phases might not be observable for some values of κ1 /κ2 and c0 . state. If γ¯ is large enough, a stable state of full-release arises with a single energy minimum at

f = 1 (phase V). With the help of energy profiles ∆Etot ( f ), the budding phase diagrams at different κ1 /κ2 and

c0 are determined in the parameter space of a/R and γ¯(≡ 2γ a2 /κ2 ) (Figures 6 and 7). For very small rigid particles, the minimum adhesion energy necessary for partial budding is γ¯min = 4. The same value has been obtained in the cases of particle interaction with a membrane patch and particle incorporation into a vesicle, 4,5,24,26 which is expected as γ¯min only depends on the particle size and local deformation of the (vesicle) membrane in the vicinity of the contact region. As κ1 /κ2 decreases, γ¯min decreases because the deformation of the vesicle is reduced by the flattening of the budding particle. Note that stiffer particles require less adhesion energy γ¯ to attain full release 10 ACS Paragon Plus Environment

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(a) 4

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Figure 6: Budding phase diagrams with respect to the normalized adhesion energy γ¯ and particlevesicle size ratio a/R at different bending rigidity ratios κ1 /κ2 = ∞, 5, 1 and 0.1 in the case of c0 = 2. Typical system energy profiles of phases I to V can be found in Figure 5. 60 55

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, 5, 1, 0.1

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10 5 no-budding 0 0.01 0.1 0.2

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Figure 7: Budding phase diagrams at different κ1 /κ2 in the cases of c0 = 0 and −2. (phase V). This phenomenon is reflected in the elastic energy profiles in Figure 2, where softer particles exhibit larger maximum slope in the energy profiles. Similar behavior has been previously reported in membrane wrapping of vesicular elastic nanoparticles 26 or solid thin-walled nanocapsules, 30 and particle incorporation into a vesicle. 23,24 If γ itself is employed, instead of γ¯, we would find that larger particles requires less adhesion energy γ to achieve full release. Besides the above properties shared by vesicles of different c0 (c0 = 0 and ±2), there are also some interesting c0 -dependent budding features. As c0 decreases, the curvature of the middle downward concave region of energy curves becomes weaker, and that of the initial upward concave region at small f becomes stronger (comparing Figures 2a and 2b,c). Consequently, the structures of the budding phase diagram in Figure 7 are quite different from those in Figure 6. For example, in the cases of c0 = 0 and −2, only phases I, III and IV are observed and the adhesion energy required for full release increases as a/R increases (Figure 7). In the case of c0 = 2, all five phases (I to IV) are observed for stiff particles, and γ¯ required for full release decreases first and then 11 ACS Paragon Plus Environment


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remains around a specific value as a/R increases. As κ1 /κ2 decreases, phase III expands at the expenses of phases I, II and IV. For the softest particle (κ1 /κ2 = 0.1) under consideration, phases II and IV vanish completely (Figure 6). Moreover, a comparison among the budding phase diagrams in Figures 6 and 7 indicates that the adhesion energy γ¯ required for full release increases as c0 decreases from 2 to −2. In contrast to the case of outward budding, the adhesion energy required for full wrapping of a particle into a vesicle decreases as c0 decreases. 24 In other words, a positive vesicle spontaneous curvature facilitates outward budding but suppresses inward engulfment of an elastic particle. A similar conclusion has been drawn in a recent theoretical study on the endocytic and exocytic engulfment of rigid spherical particles. 12

Discussion 20 60 50

full release

full release

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0.5

Figure 8: Adhesion energy required for full release or full wrapping of an elastic particle at c0 = 0. The adhesion energy required for full wrapping can be found in our previous work, 24 and is presented here for comparison with the case of full release shown in Figure 7a.

Figure 8 compares the adhesion energy required for full release of an internal elastic particle and that required for full wrapping of an external elastic particle at c0 = 0. It is seen that almost the same adhesion energy is required for a rigid particle to achieve full release or full wrapping, and as κ1 /κ2 decreases larger adhesion energy is required for the particle to reach the state of full release than full wrapping. This dependence of adhesion energy on the interaction modes is more


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striking for softer particles, and can be useful in designing controlled drug delivery systems. If the affinity between cell membrane and nanoparticles serving as drug agents is tuned to a specific value between the adhesion energy required for full wrapping and that for full release, the drug agents internalized through endocytosis should then remain in the cell, instead of being released out of the cell through exocytosis. During intracellular trafficking processes, nanoparticles internalized into cells are often delivered to intracellular organelles such as endosomes or lysosomes of approximately zero spontaneous curvature, 46,47 and might be trapped there followed by possible degradation if the adhesion energy between the particles and the membrane of these organelles is not large enough for full release. Figure 8 indicates that the adhesion energy required for full release decreases as R increases at

c0 = 0. This result implies that nanoparticles delivered to endosomes or lysosomes might bud out of these intracellular compartments as the size of endosomes or lysosomes increases via vesicle fusion. From a biophysical point of view, this explains that the size of endosomes or lysosomes should be limited to prevent internalized particles from being released to the cytoplasm. Typical elastic spherical nanoparticles encountering biological interactions include biological particles such as spherical viruses and engineered particles such as synthetic colloidosomes and drug carriers including polymeric capsules, polymersomes, conventional liposomes, niosomes, and ethosomes. Experimental studies have shown that their bending rigidity can vary considerably as summarized in our previous work. 30 For example, the bending stiffness of viral capsids could be around 70 kB T (hepatitis B virus and cowpea chlorotic mottle virus), 140 kB T (bacteriophage λ ),

240 kB T (minute virus of mice) and up to the order of 103 kB T (mature HIV, herpes simplex virus type 1, and mature murine leukemia virus) (1 kB T = 4.1×10−21 J is the unit of thermal energy). The bending stiffness of engineered particles such as liposomes, polymersomes, and niosomes could be as low as tens of kB T . Depending on the membrane composition, the bending rigidity of cell membrane can vary from 20 kB T to 150 kB T as discussed in our previous work. 30 For example, the bending rigidity κ2 of pure dimyristoyl phosphatidylcholine (DMPC) lipid bilayers is about 30 kB T , and κ2 of DMPC membranes containing 50 mol% cholesterol could be 150 kB T . In comparison



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with the cell membrane, the biological particles and engineered particles of bending stiffness much larger than 150 kB T could be regarded as rigid particles when budding out of vesicles. A budding soft particle (κ1 /κ2 = 0.1) is seen to adopt a flattened morphology parallel to the vesicle membrane in the early budding stage, then undergoes a shape transition to an elongated morphology perpendicular to the vesicle in the late budding stage, and eventually resumes to the spherical shape when it pinches off from the vesicle (Figures 3 and S3). The morphological change of the soft particle from a flattened shape parallel to the vesicle membrane to an axially elongated shape perpendicular to the membrane is similar to the reorientation of short rod-shaped viruses in their budding. 21 For example, short rod-shaped fowlpox and pigeonpox viruses align their long axes parallel to the membrane of infected chick embryo fibroblasts in the early stage but reorient their long axes perpendicular to the membrane in the late stage of budding. 21 The adhesion energy due to the specific receptor-ligand binding can be expressed as γ ≡

eRL kB T ξRL , where eRL kB T represents the binding energy per receptor-ligand bond of 10 kB T to 25 kB T , 48,49 and ξRL is the surface density of receptor-ligand bonds in the contact region which is usually proportional to the receptor density ξR and ligand density ξL . Depending on the types of the cells and nanoparticles, both ξR and ξL can vary considerably from hundreds to thousands of numbers per square µ m as discussed in our previous work. 32 Taking κ2 = 20 kB T and particle radius a to vary from tens of nanometers to 1 µ m, the normalized adhesion energy γ¯(≡ 2γ a2 /κ2 ) could vary up to a few hundreds which fully cover the range of γ¯ reported in the budding phase diagrams in Figures 6, 7 and 8. The spontaneous curvature introduced by Helfrich 43 describes the preferred curvature of a nondeformed/unstressed piece of membrane and can be used to characterize asymmetries between the two leaflets of the bilayer membrane. Molecular mechanisms underlying such asymmetries include compositional lipid asymmetry between the inner and outer membrane leaflets, asymmetric membrane adsorption or binding of ions, small molecules or macromolecules such as polymers and proteins or protein networks, insertion or penetration of proteins or molecules perturbing lipid packing and generating local membrane deformation. 50,51 Depending on the specific mechanisms,


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the spontaneous curvature C0 of the membrane could vary considerably. For example, C0 of the clathrin-coated membranes is estimated to be around 1/50 nm−1 . 52 The membrane remodeling superfamily of BAR domain proteins is also well known for their ability to form and stabilize membrane tubules. BAR/N-BAR proteins preferentially bind to highly curved membrane structures and the radius of the induced membrane tubules varies from 10 nm to 30 nm; while the radius of membrane tubular membrane invaginations induced by F-BAR proteins falls in a range from 30 nm to 50 nm. In contrast, I-BAR (inverse-BAR) proteins induce membrane protrusions. 50 For DOPC membranes with anchored DNA molecules of average end-to-end distance 0.65 µ m at high DNA surface concentrations, C0 of the membrane is estimated to be around 1 µ m−1 from the analysis of vesicle budding. 53 For vesicles of DPPC/DOPC/cholesterol ternary mixtures enclosing aqueous polymer mixtures of dextran and polyethylene glycol (PEG), spontaneous curvatures of liquid-ordered and liquid-disordered membranes induced by weak PEG adsorption onto the vesicle membranes are experimentally determined as −1/125 nm−1 and −1/600 nm−1 , respectively. 54 As indicated in the above literature, the radius of the membrane spontaneous curvature could vary considerably from tens of nanometers to the order of a few micrometers, which is also the typical size range of engineered vesicles as well as biological vesicles such as endosomes and lysosomes. 55 Accordingly, representative values of 0 and ±2 are selected for the normalized spontaneous curvature c0 = C0 R. As demonstrated in Figures 2a and 3, a vesicle of finite positive spontaneous curvature c0 with an internal particle of a relatively large a/R undergoing budding exhibits a discontinuous shape transformation from an elongated pear shape into a double-gourd shape, and eventually evolves to a vesicle composed of two spheres connected by a narrow membrane neck at full release. As c0 decreases, the discontinuous shape transformation of the vesicle vanishes for all a/R under consideration (see Figures 2b, S2 and S3). This c0 - and a/R-dependent configurational transformation of the vesicle can be understood as follows. Theoretical analysis shows that a free-standing vesicle of c0 around 2 tends to adopt a double-gourd shape (or pear shape as used in ref 44) at reduced volumes larger than V /(4π R3 /3) ≈ 0.7 (Figure 10 in ref 44). As indicated in Figure 2, V /(4π R3 /3) in the budding of a particle of relatively large a/R are indeed around that range; meanwhile the



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sizes of the particle and the membrane portion bulges out of the vesicle match each other. Therefore, we can conclude that the double-gourd shaped vesicle predicted from our study is indeed in an energetically favorable state. As c0 decreases, theoretical analysis suggests that a bulging vesicle would continuously adopt a prolate-shape, 44 and discontinuous shape transformation is not observed. Detailed analysis on the budding behavior at c0 in a wider range requires further investigation. In contrast to the budding case, discontinuous shape transformation of the vesicle during particle incorporation could occur at zero and relatively small negative spontaneous curvatures. 24

Conclusions We have performed a theoretical study on outward budding of an adhesive elastic particle from a lipid vesicle of different spontaneous curvatures under fixed surface area and vanishing osmotic pressure. It is shown that the vesicle volume decreases first and then increases during the budding process. At zero or negative spontaneous curvature, the volume of the vesicle evolves continuously as it undergoes smooth shape transformation. In contrast, a sufficiently large particle budding out of a vesicle of positive spontaneous curvature can induce a discontinuous shape transformation of the vesicle from an elongated-pear shape into a double-gourd shape. For a softer particle, such discontinuous shape transformation occurs at a larger budding degree and a larger particlevesicle size ratio. It is shown that softer particles are more prone to partial budding but require stronger adhesion energy to achieve full release. Further analysis indicates that the adhesion energy required for full release is smaller at the positive spontaneous curvature under consideration, and increases as the spontaneous curvature decreases. Depending on the adhesion energy, size and rigidity ratios between the particle and vesicle, as well as the spontaneous curvature of the vesicle, a rich variety of budding phase diagrams consisting of the stable or metastable states of no-budding, partial-budding and full-release have been determined. These results can be used to explain that the size of endosomes or lysosomes should be limited to prevent internalized particles from being released back to the cytoplasm. Our results shed light on the effects of particle size and


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elasticity and the spontaneous curvature of a vesicle on budding of internalized particles and may provide rational design guidelines for controlled viral budding, intracellular budding and exocytosis.

Acknowledgements This work was supported by the National Science Foundation under Grant CMMI-1562904.

Supporting Information Available Supplemental figures are presented on additional morphologies of particle-vesicle system, energy profiles, evolution of the vesicle volume in the cases of zero and negative spontaneous curvatures.

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For Table of Contents Use Only Title: Budding of an Adhesive Elastic Particle out of a Lipid Vesicle Authors: Xin Yi and Huajian Gao

budding degree

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rigid particle

soft particle

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Budding of an Adhesive Elastic Particle out of a Lipid Vesicle - ACS

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