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Elastocytosis
Gaetano Napoli∗^ and Alain Goriely†
∗Dipartimento di Matematica e Fisica ’E. De Giorgi’, Università del Salento, Italy, †Mathematical Institute, University of Oxford, UK
August 27, 2020
Abstract
Endocytosis and exocytosis are key mechanisms in cellular systems by which a cell deforms its plasma membrane to move substances in or out of the cell. Whereas these cellular processes typically rely on active mechanisms, we study here the problem of achieving encapsulation by a purely physical process. We consider an ideal system in which a rigid particle is put in contact with a spherical deformable body. Depending on the interaction energy between both objects and the mechanical response of the body, partial or total encapsulation may reduce the total energy of the system. In this case, the system would evolve naturally so that the particle will be partially or completely engulfed by the body, a process that we refer to as partial or complete elastocytosis. We consider three systems of increasing complexity and we show that elastocytosis is a generic mechanism that can always be achieved in some parameter regimes.
1 Introduction
A key problem for biological cells is to carry material across their membranes. Indeed, while the key function of a membrane is to isolate the cell from the external world, it also needs to interact with its environment by allowing passage through its wall or membrane. A multitude of mechanisms has been developed through evolution to optimize these processes known mostly as endocytosis (inclusion) or exocytosis (exclusion), as coined by de Duve in 1963 [1]. They rely mostly on active transport and the existence of membrane reservoirs [2]. These events are typically mediated by receptor proteins sitting on the membrane and targeting specific molecules with which they interact. A particularly striking example of endocytosis is phagocytosis [3, 4]. Phagocyte cells are part of the immune systems and can undergo very large deformations to engulf harmful foreign particles or dead cells that are subsequently removed as shown experimentally [5] and theoretically [6, 7]. While the problem studied here is motivated by these biological mechanisms, the question that we address is whether encapsulation can be achieved through purely physical mechanisms related to the deformation of an elastic body and the interaction energy between particles and cells. We consider an idealized situation in which a spherical (or circular) rigid particle is presented to a deformable spherical (or circular) body. The two bodies interact through direct contact. The interaction energy is taken to be the surface tension when one of the bodies is a liquid droplet or a given adhesion energy when both objects are solids. The rigid particle is initially presented to the deformable body at a single point of contact. The first question is whether a configuration with a contact zone is energetically favorable. If it is the case, the membrane will deform and partially
encapsulate the particle, as shown in Figure 1. The second question is to find the conditions under which the particle will be completely encapsulated. Complete encapsulation is achieved when the membrane folds back and touches itself. By analogy to the biological process of engulfment, we refer to these encapsulations generated by a balance between the tension generated by the adhesion energy and the elastic forces due to the deformation as partial and complete elastocytosis.
initial contact
complete elastocytosis
partial elastocytosis
Figure 1: A rigid particle (red) is initially placed into contact with a deformable body. Depending on the system’s parameters, it will either (from left to right) remain in contact, be partially engulfed, or be (almost) completely encapsulated.
To elucidate the interplay between physical forces at the membrane level and the deformation and high curvature needed for the engulfment process to take place, we consider three systems of increasing complexity. First, we look at the problem of a small rigid sphere in contact with a spherical liquid droplet. In this case, the adhesion between the droplet and the particle is entirely determined by the relative surface tensions and the geometry of the system. Therefore, we can use methods from the theory of droplets [8] to classify different behaviors. This process of partial or complete encapsulation of a sphere inside a droplet has been proposed as a possible mechanism for the release of reactive substances in drug delivery [9]. Second, we consider the effect of curvature by looking at the ideal problem of a planar elastic circular rod in contact with a disk. Here, we rely on the theory developed for elastic rods in contact [10] to solve the problem for the shape in terms of two key parameters. Formally, this problem is
α
β
Σf
Σc
k f k c
y
R
r
x
Pressure p
Figure 2: Schematic representation of the equilibrium configuration.
where ν is the unit outward normal vector to Σf, P is the projection onto the tangent plane, kf is the outward normal to the boundary in the tangent plane and H is the mean curvature. Note that by choosing the plus in front of the second term in the right-hand side of (4), we adopt the convention that the mean curvature of a sphere of radius R is equal to 1 /R. The quantity σ (^) f = γfP can be interpreted as the tangential stress tensor. In the contact region, the only virtual displacements u˜ are tangent to rigid surface. Consequently, we have ∫
∂Σf
γf(Pkf) · u dl =
∂Σc
γc(Pkc) · u˜ dl, (5)
where kc is the outward normal to the boundary of the contact region. The virtual work done by the internal pressure is
δL =
Σf
pν · u dA. (6)
The variational principle (1) leads to the Young-Laplace law for the free part of the interface
2 γfH = p, (7)
0.5 1.0^ 1.5 2.0^ 2.5 3.
Α
1.
0.
Γc
Γf
Figure 3: Isobaric wetting. γf/γc versus the wetting angle α, for μ = 1. 5 , 2 , 3 , 5 , 10.
and the boundary condition
γfkf · kc = −γc. (8)
2.2 Analysis
2.2.1 Isobaric wetting
From (7), it follows that the free interface is a portion of a sphere of radius
R =
2 γf p
Thus, in a process where p and γf are fixed, while the adhesion strength γc changes, the radius of the liquid droplet remains constant. Using the angles α and β defined in Figure 2, (8) becomes
cos(α − β) = γc γf
with the constraint
r sin α = R sin β. (11)
By solving the last two equations, we obtain
γc γf = μ−^1 (− sin^2 α + cos α
μ^2 − sin^2 α), (12)
where μ = R/r. Therefore, we conclude that (see Figure 3) for any μ: (i) adhesion (α > 0 ) is possible if γc < γf; (ii) γc/γf is a strictly decreasing function of α; (iii) complete encapsulation (α = π) only occurs when γc = −γf. In this setting, the interplay between the internal pressure and the surface tension γf fixes the radius of the droplet independently of the adhesion strength γc. Thus, by decreasing γf the area of the contact region increases as the wetting angle increases, without a change in the droplet radius.
3.1 Energetic considerations
The elastic energy (density) of a rod shaped as an elastic circular ring of radius R is
Wcirc = kπR
R
− ζ 0
where k is the bending stiffness, while ζ 0 denotes the (signed) intrinsic curvature of the rod (if ζ 0 < 0 , the unloaded ring has been inverted and cutting a small part of it would result in an inversion of curvature). In the ideal case of complete elastocytosis, a new circular elastic ring of radius R − r is formed, while the disk is completely coated by the elastic loop to form an elastosome. The energy corresponding to these final configurations are
W± = kπ(R − r)
R − r
− ζ 0
r
± ζ 0
− 2 πr∆γ, (17)
where (+) refers to endocytosis and (−) to exocytosis, and ∆γ > 0 is the ring-disk adhesion energy density (energy per unit length). If we assume that these configurations are selected through energy minimization, we see that the sign of the spontaneous curvature selects which of the two phases is energetically favored. In fact, ∆W = W+ − W− = 4πkζ 0 and therefore exocytosis is a minimum energy state when ζ 0 > 0 , while endocytosis is favored for ζ 0 < 0. Intuitively, if ζ 0 > 0 , the ring is initially in a low energy state and the creation of two rings with the same curvature (exocytosis) is preferable to the creation of a large ring of the same curvature and a small one of inverted curvature (endocytosis). Furthermore, if we compare Equation (17) with (16), we obtain the two critical values
(∆γ)±^ = k R^2 − r(R − r)(1 ± 2 ζ 0 R) 2 r^2 R(R − r)
For adhesion values larger than (∆γ)±, the splitting of the initial rings into a smaller ring and an elastosome becomes energetically favorable: However, as the parameters in the system are varied through this bifurcation, we have no information about how this transition takes place. This information is contained in the intermediary phase where the disk partially adheres to the flexible loop, which we consider next.
3.2 Partial wrapping
Let s denote the arclength measured from the bottom of the circle (Figure 5). A point p(s) on the ring can be parametrized by its Cartesian coordinates r(s) = [x(s), y(s)] that obey
x′^ = cos ψ, y′^ = sin ψ, (19)
where ψ(s) is the angle between the tangent at p and the x-axis. We consider an inextensible and unshearable closed elastic rod of length L = 2πR and restrict our attention to symmetric equilibrium shapes with respect to the y axis. The bending energy functional is then
Wb[ψ] = k
∫ (^) ¯s
0
(ψ′^ − ζ 0 )^2 ds + k
∫ L/ 2
s¯
r ± ζ 0
ds, (20)
θ¯
r
x
y
s = 0
s = ¯s s = s
θ^ ¯
r
x
y
s = 0
s = ¯s
endocytosis exocytosis
Figure 5: Schematic representation of the equilibrium configuration: endocytosis (left) and exocy- tosis (right).
where s¯ denotes the arclength of the first contact point, and, as before, the plus sign applies in the case of endocytosis (minus for exocytosis). In addition, at the points where the rod is in contact with the rigid disk, there is an adhesion potential of the form
Wa[ψ] = −2∆γ
∫ L/ 2
s¯
ds. (21)
In order to enforce the inextensibility constraint, the following energy terms are added to the energy functional
Wc[ψ] = 2nx
∫ (^) ¯s
0
(cos ψ − x′)ds + 2ny
∫ (^) ¯s
0
(sin ψ − y′)ds, (22)
where nx and ny are the unknown Lagrange multipliers associated with the constraint. Physically, they are interpreted as the horizontal and vertical components of the internal force. The equilibrium shapes are stationary points of the total functional
W [ψ] = Wa[ψ] + Wb[ψ] + Wc[ψ]. (23)
The corresponding Euler-Lagrange equations associated with W yields
kψ′′^ + nx sin ψ − ny cos ψ = 0, (24)
and we recover the classical balance of torque for planar rods (see e.g. [31, p. 122]). Without loss of generality, we assume that the rigid disk is presented at the top of the membrane and we fix the
whence
¯s = F(q∗)
(c + h)(c + h cos ψ¯)
where F(·) denotes the incomplete elliptic integral of the first kind [36] with
q∗ :=
[ ¯
ψ 2
2 h c + h
]
Furthermore, Equation (19) 1 equipped with the boundary conditions (25) 1 and (26) leads to
x ¯ =
∫ (^) ψ¯
0
cos ψdψ √ 2(c + h cos ψ)
that yields
x ¯ = − c h
s¯ + E(q∗)
h
c + h, (35)
where E(·) denotes the incomplete elliptic integral of the second kind [36]. Substitution of (26) and (28) into (31), (33) and (35) leads to
c =
r
`ec
¯s = F(q∗)
(c + h)(c − h cos θ¯)
, (36b)
r sin θ¯ = − c h ¯s + E(q∗)
h
c + h. (36c)
These equations, together with Equation (27), can be solved numerically to obtain s¯, h, c and θ¯ as functions of all the parameters: `ec, r, and R.
3.3.2 Exocytosis
In the exocytosis case, (see Figure 5), the equilibrium solution ψ(s) increases for s ∈ (0, s 0 ), while it decreases for s ∈ (s 0 , ¯s). We denote ψ 0 = ψ(s 0 ) ∈ [0, π] to be the maximum value of ψ(s) in the interval (0, ¯s). Since ψ′(s 0 ) = 0, (30) gives c = −h cos ψ 0. Thus, by integrating (30) by separation of variables over the two intervals where ψ(s) is monotonic, we obtain
¯s =
2 h
∫ (^) ψ 0
0
dψ √ cos ψ − cos ψ 0
2 h
∫ (^) ψ¯
ψ 0
dψ √ cos ψ − cos ψ 0
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
a. b. 3 c.
d. e. f.
Figure 6: Planar endocytosis equilibrium shapes for μ = 5 (top) (η = 0. 94 , θ¯ = 41◦^ (left), η = 0. 97 , θ^ ¯ = 83◦^ (center), η = 0. 93 , θ¯ = 125◦^ (right)) and μ = 10 (bottom) (η = 0. 82 , θ¯ = 39◦^ (left), η = 0. 83 , θ¯ = 72◦^ (center), η = 0. 81 , θ¯ = 119◦^ (right)).
whence
¯s =
2F(q 0 ) − F(¯q) √ h sin ψ 20
where
q 0 :=
[
ψ 0 2
, csc^2 ψ 0 2
]
, q¯ :=
[ ¯
ψ 2
, csc^2 ψ 0 2
]
Similarly, we integrate Equation (19) 1 with the boundary conditions (25) 1 and (26):
x ¯ = ¯s cos ψ 0 +
h
(2E(q 0 ) − E(¯q)) sin ψ 0 2
3.4 Discussion
The model depends on three dimensionless parameters
μ :=
R
r , η := r `ec , ς 0 := ζ 0 R, (41)
which represent, respectively, the ratio between the radii, the gluing number, and the reduced intrinsic curvature. The gluing number, η, is a ratio of the two characteristic physical length scales: the disc curvature (associated with bending) and the elasto-capillary length. At ec constant, the ring is more likely to adhere on an almost flat surface (small curvature or large r) than on a curved surface (small r). Similarly, for a fixed disc radius, the ability of the ring to adhere increases with the adhesion strength (smallec) and decreases with the elastic stiffness (large `ec). The reduced intrinsic curvature, μ, is the ratio of the two geometric lengths of the elastica: the radius of the undeformed ring R and the spontaneous curvature radius 1 /ζ 0. The latter is the natural radius of a small piece of the ring in the absence of stress. Any deviation away from this radius has an energy cost given by (16). In particular, for ς 0 = 1, the energy (16) attains it absolute minimum. For elastocytosis, the sign of ς 0 is expected to play an important role since surfaces with curvatures of the same sign have a greater tendency to stick, while surfaces with opposite curvatures tend to separate. Figures 6 and 7 sketch equilibrium shapes obtained by varying the wrapping angle at fixed μ. During endocytosis small inclusions can self-encapsulate (Figure 6 right-bottom), while for larger ones the bottom self-contact occurs before encapsulation (Figure 6 right-top). During exocytosis the disk can be encapsulated for a sufficiently large gluing number (Figure 7 right). We see in Figure 8 that wrapping increases with the gluing number during exocytosis (red line) but not during endocytosis (blue line). The value of η corresponding to the circular solution with a single contact point can be computed by observing that s¯ = πR and ψ¯ = π. Thus, from (30), we deduce that h = 0 and, hence, c = 1/(2R^2 ). This observation, together with (31), leads to the activation thresholds
η act± = 1 ± μ−^1 √ 2
, ((+) endocytosis, (-) exocytosis). (42)
Beyond these thresholds, partial wrapping becomes energetically favored compared to the no-contact solution. We note that exocytosis is triggered at a lower threshold than that of endocytosis. Phys- ically, this difference is due to the fact that during exocytosis the curvatures of the surfaces in contact match, while during endocytosis they have opposite signs. Both curves of Figure 8 stop at the point of self-intersection. To understand the energy landscape, we introduce the dimensionless energy w = W R/k, with which Equations (16) and (17) now read
wcirc = π(1 − ς 0 )^2 , (43a)
wendo = π
μ^2 μ − 1
− 2 η^2 μ + ς 02
wexo = π
μ^2 μ − 1 − 2 η^2 μ + ς 02
, (43c)
while (18) yields the dimensionless thresholds
η±^ =
1 + μ^2 − μ ± 2 ς 0 (μ − 1) 2 μ(μ − 1)
, ((+) endocytosis, (-) exocytosis). (44)
0.5 0.6^ 0.7^ 0.8^ 0.9^ 1.
4
2
0
2
4
WR/k
w circ
w endo^ w exo
η-act^ η^ - + η+act^ η
Figure 9: Energy landscape for μ = 5 and ς 0 = 0. The green line represents the no-contact solution, while the solid black line corresponds to complete endo/exocytosis. The dashed curves are the energy of partial endocytosis (blue) and partial exocytosis (red).
Figure 9 gives an example of the energy landscape. The partial contact branches, represented by the dashed lines, have been obtained numerically. The green line represents wcirc, while the solid black line represents wendo and wexo for ς 0 = 0. For sufficiently low gluing numbers wcirc is the energy minimum. As η exceeds the critical value η+act (η− act, respectively) given by (42) partial endocytosis (exocytosis, respectively) becomes energetically favored. At the critical value given by (44)
η±^ =
1 + μ^2 − μ 2 μ(μ − 1)
the minimum is attained by complete exocytosis or endocytosis (the solid black line) with equal energy at ς 0 = 0. Thus, for exocytosis, the transition from no contact to partial contact is contin- uous, while the transition to the complete expulsion is discontinuous. For endocytosis there is a discontinuous transition between the no-contact branch and the complete engulfment of the disk. In the terminology of statistical physics, these transitions are, respectively, first- and second-order phase transitions. The role of the spontaneous curvature is evident from Equation (44): negative values of ς 0 promote endocytosis, since from (44), we have η+^ < η−^ ; while positive values of ς 0 favor exocytosis, (η+^ < η−). Whenever ς 0 6 = 0 the black solid line (44) splits into two branches, one relative to exocytosis and the other to endocytosis, where the one with the lowest energy branch is determined by the sign of ς 0. Using Equations (43b) and (43c), the energy gap between these two branches is ∆w = wendo − wexo = 4πς 0. From Equation (42), we also observe that the activation threshold does not depend on the spontaneous curvature. Therefore, by tuning ς 0 one can go from a situation with no contact to complete engulfment or budding.
ρ(s) and z(s), such that p − 0 = ρer + zez (Figure 10). Consequently, we have
Hf =
sin ψ ρ
, Kf =
ψ′^ sin ψ ρ
where ψ(s) denotes the angle between the tangent to the generating curve and the plane orthogonal to the symmetry axis (see Sec. 4.6 of [38]). Once the value of ψ(s) is known, the shape is given by integrating
ρ′^ = cos ψ, z′^ = sin ψ. (49)
For definition, we set the origin so that the free region is in the interval s ∈ [0, ¯s], while the contact region corresponds to value of s ∈ [¯s, ]. Unlike the planar case, where the total length is known, in the three-dimensional case both ¯s and are unknown a priori. The generating curve for the contact region is an arc of circumference of radius r, hence we have
ρ(s) = r sin θ, z(s) = z 0 ± (−r cos θ), ψ′^ = ±
r
, ((+) endocytosis, (-) exocytosis) (50)
where z 0 is the z-coordinate of the rigid sphere center and θ = (` − s)/r ∈ [0, θ¯], where θ¯ represents, as before, the wrapping angle
θ^ ¯ := `^ −^ s¯ r
By replacing (47), (48) and (50) into (46), we obtain
Wb = 4πk
∫ (^) ¯s
0
[
sin ψ ρ
− c 0
] 2
ρds + 4πr^2 k
r
− c 0
(1 − cos θ¯), (52)
while the adhesion energy is assumed to be proportional to the contact area
Wc = −Area(Σc)∆γ = − 2 πr^2 (1 − cos θ¯)∆γ. (53)
The total area of the membrane A 0 and (possibly) the enclosed volume V 0 are prescribed, thus the following constraints must hold
A 0 = Area(Σf) + Area(Σc) = 2π
∫ (^) ¯s
0
ρ(s)ds + 2πr^2 (1 − cos θ¯), (54a)
V 0 = Vol(Σf) ± Vol(Σc) = π
∫ (^) s¯
0
ρ^2 sin ψ ds ±
πr^3 (2 + cos θ¯) sin^4 θ¯ 2
. (54b)
In Equation (54b), V 0 denotes the volume of the membrane enclosed liquid added to the volume of rigid particle in endocytosis, while it denotes the volume of the liquid only in endocytosis. The plus/minus refers to endocytosis/exocytosis, respectively. The effective energy must include further terms with their Lagrange multipliers, besides Wb and Wc, whenever the constraints (49) and (54) are imposed.
4.1 Equilibrium equations
For the derivation of equilibrium equations, it is useful to split the total energy W into two terms: W = Wf + Wc. The first term Wf denotes the energy of the free (non-contact) membrane
Wf =
∫ (^) ¯s
0
wf ds, (55)
where
wf 2 π
= 2kρ
[
sin ψ ρ
− c 0
] 2
ρ^2 sin ψ + λ 1 (ρ′^ − cos ψ) + λ 2 (z′^ − sin ψ). (56)
Here λ and p are constant Lagrange multipliers related to the global constraints (54), while λ 1 (s) and λ 2 (s) are related to the local constraints (49). The term Wc is the energy of the contact region
Wc 2 π
[
2 k
r
]
r^2 (1 − cos θ¯) +
pr^3 (2 + cos θ¯) sin^4 θ¯ 2
The Euler-Lagrange equations derived from (55), valid for s ∈ [0, s¯], are
ψ′′^ = cos ψ ρ
sin ψ ρ
− ψ′
λ 1 kρ
sin ψ + p 2 k
ρ cos ψ, (58a)
λ′ 1 = k 2
[
(ψ′^2 − 2 c 0 )^2 − sin^2 ψ ρ^2
]
λ′ 2 = 0. (58c)
These equations are coupled to Equations (49) and (54).
4.1.1 Boundary conditions
At s = 0, the following Dirichlet boundary conditions hold:
ρ(0) = 0, z(0) = 0, ψ(0) = 0. (59)
At the contact point s = ¯s, we have
ρ(¯s) = r sin θ,¯ ψ(¯s) = π ± θ¯ ((+) endocytosis, (-) exocytosis). (60)
The arbitrariness of z(s) variation at s = ¯s gives
∂wf ∂z′
s=¯s
= 0 ⇔ λ 2 (¯s) = 0, (61)
which, together with (58c), implies λ 2 (s) = 0.
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
-1 0 1
0
1
2
3
Figure 11: Equilibrium shapes for partial endocytosis (top) and exocytosis (bottom) for completely permeable vesicles (μ = 5, σ 0 = 0). From left to right θ¯ = 60◦, 120 ◦, 164 ◦. For endocytosis η = 1. 62 , 1. 15 , 1. 71 , while for exocytosis η = 1. 29 , 1. 52 , 2. 00.
the parallels, leading to very small mean curvatures. Therefore, highly localized deformations with low energetic costs are allowed and preferred. For a vesicle area A 0 and in the absence of external forces, a sphere of radius R is an equilibrium shape, with reduced energy
wsph = 4π[2 (1 − σ 0 )^2 + κ]. (67)
The energy corresponding to complete endocytosis is given by the energy of a spherical vesicle of area Af = 4π(R^2 − r^2 ), added to the energy of the endosome of area Ae = 4πr^2 :
wendo = 8πμ−^1
[
μ^2 − 1 − 1)σ 0 + μ(2 + σ 0 )^2
]
Observe that the dependance of wendo on the saddle-splay modulus is 8 πκ whereas it is 4 πκ in wsph. This difference is easily explained by applying the Gauss-Bonnet theorem and by the obvious fact that two separate spheres have a Euler-Poincaré characteristic twice that of a single sphere. Hence, during a change of topology corresponding to complete endocytosis, this part of the energy
1.0 (^0 50 100 )
η^ 3.
θ
1.0 1.2 1.4 1.6 1. 5
10
15
20
WR/k^25
w sph
w endo^ w exo
η
Figure 12: Permeable vesicles (μ = 5, σ 0 = 0, κ = − 1 / 2 ). (Top) Gluing number as a function of the wrapping angle. For exocytosis (red) the angle is strictly increasing with the gluing number, while for exocytosis (blue) it is not monotonic. (Bottom) Energy landscape: reduced energy versus gluing number. The green line is wsph. The solid black line represents the reduced energy for complete endocytosis and exocytosis. The red and blue dashed curves refer to partial exocytosis and endocytosis, respectively.
is reduced by this topological change whenever κ is negative. A similar formula holds for the case of complete exocytosis:
wexo = 8πμ−^1
[
μ^2 − 1 + 1)σ 0 + μ(2 + σ 0 )^2
]
In the following, the reduced energy relative to partial endocytosis and exocytosis is computed by numerically solving the equilibrium equations. Direct inspection of Equations (67), (68), and (69) reveals that the spontaneous curvature σ 0 plays a role in the energy landscape, and that if κ is negative, the contribution of the saddle-splay energy promotes the fission of the membrane. In Figure 12 we show the wrapping angle θ¯ as a function of the gluing number η and the energy landscape for μ = 5 and σ 0 = 0. The graphs η(θ¯) show that as a consequence of a continuous binding a continuum (discontinuous) engulfment transition in exocytosis (endocytosis) occurs. Furthermore, numerical simulations show that the gluing number diverges as the wrapping angle approaches total
