Theory of active particle penetration through a planar elastic membrane

With the rapid advent of biomedical and biotechnological innovations, a deep understanding of the nature of interaction between nanomaterials and cell membranes, tissues, and organs, has become increasingly important. Active penetration of nanoparticles through cell membranes is a fascinating phenomenon that may have important implications in various biomedical and clinical applications. Using a fully analytical theory supplemented by particle-based computer simulations, the penetration process of an active particle through a planar two-dimensional elastic membrane is studied. The membrane is modeled as a self-assembled sheet of dipolar particles, uniformly arranged on a square lattice. A coarse-grained model is introduced to describe the mutual interactions between the membrane particles. The active penetrating particle is assumed to interact sterically with the membrane particles. State diagrams are presented to fully characterize the system behavior as functions of the relevant control parameters governing the transition between different dynamical states. Three distinct scenarios are identified. These compromise trapping of the active particle, penetration through the membrane with subsequent self-healing, in addition to penetration with permanent disruption of the membrane. The latter scenario is accompanied by a partial fragmentation of the membrane and creation of a hole of a size exceeding the interaction range of the membrane components. Our analytical theory is based on a combination of a perturbative expansion technique and a discrete-to-continuum formulation. Our approach might be helpful for the prediction of the transition threshold between the trapping and penetration in real-space experiments involving motile swimming bacteria or artificial active particles.


I. INTRODUCTION
As one of the most fundamental components in biological systems, the cell membrane defines and protects the cell and is selectively permeable for ions and organic molecules, allowing to control the movement of required chemicals into the cell and of unwanted products out of the cell. It is now possible not only to reassemble cell membranes artificially 1 , but also to design synthetic membranes with properties tailored to the needs of 21st centuries societies [2][3][4] . In fact, synthetic membranes are now routinely used already for applications from water purification 5,6 to dialysis 7,8 and can be regarded as a paradigmatic success of biomimetics [9][10][11] . Future perspectives for the usage of synthetic membranes involve problems like targeted gene and drug delivery to (cancer) cells [12][13][14][15][16][17][18][19][20][21][22] or, more generally, the delivery of cargo to the interior of synthetic droplets, requiring a precise understanding of the interaction of motile particles with synthetic and biological membranes. Evidence from previous studies has shown that the physical uptake by living cells is strongly affected by the particle and membrane physicochemical and functional properties [23][24][25][26][27][28][29] .
While membranes comprising active inclusions, e.g. in a) Electronic mail: abdallah.daddi.moussa.ider@uni-duesseldorf.de b) Electronic mail: hartmut.loewen@uni-duesseldorf.de the form of embedded proteins creating a stress on the membrane, have been studied for decades [30][31][32] , the penetration of active particles through the membrane is less explored 14,33 with the few existing studies focusing on nano-and biotechnology perspectives. In particular, penetration of nanoparticles through a membrane has been studied using dissipative particle dynamics simulations, focusing on effects of particle shape 14 and surfacestructure 34 . In addition, molecular dynamics simulations have been employed to investigate the penetration of fullerens through lipid membranes 35 . Recent studies have also explored interactions of active particles with membranes, from a more physical point of view, but did not focus on particle penetration [36][37][38] . For a 1D membrane, we have recently performed a corresponding investigation 39 .
Conversely to most of the above works, here we explore the penetration of an active particle through a 2D synthetic membrane from a physics perspective, aiming at predicting overall properties such as the membrane shape or the parameter domain leading to penetration starting from coarse microscopic details. We focus on a minimal model membrane that can be realized in principle using magnetic microparticles which are popularly employed as building blocks to self-assemble chains and sheets [40][41][42][43][44][45][46][47][48][49][50][51][52] . To predict the state diagram, informing us about the parameter domains where particles can penetrate through the membrane and where they cannot, we systematically de-rive a continuum description of the membrane. We compare our results with particle-based computer simulations, finding close quantitative agreement regarding the transition between trapping and penetrating states, membrane shape and dynamics. Our analytical closed-form expressions might help to predict the properties of synthetic membranes, e.g. regarding the speed and size of particles which will be able to penetrate through them.
Below, we first define our model (Sec. II), followed by a brief discussion of the relevant parameters and of the 2D membrane dynamics as induced by the active particle approaching it (Sec. III). Here, besides a trapping state in which the membrane is deformed in the final state and does not allow the particle to pass, we find two scenarios of penetration. The first of these corresponds to the particle breaking through the membrane, followed by a complete self-healing of the membrane, which might be the desired behavior when delivering cargo towards a synthetic droplet or a healthy cell. The second scenario of penetration occurs mainly for larger particles, creating a hole in the membrane with a size exceeding the interaction range of the membrane components. This situation is accompanied by a partial fragmentation of the membrane structure into isolated particles. Following this qualitative discussion, we systematically explore the corresponding state diagram using numerical simulations, showing for which parameter combinations which of these three states prevails, and we develop a detailed analytical theory (Sec. IV). The latter is able to predict essentially the entire state diagram as well as the shape and the dynamics of the membrane, in close quantitative agreement with our simulations. Interestingly, the transition between self-healing and non-healing states is sharp, suggesting that there is a critical size for particles that pass a membrane by causing significant damage. This also suggests that if one were to permanently damage the membrane in our minimal model (and perhaps similarly in practice to treat cancer cells), one needs to use particles with a certain minimal size. Finally, concluding remarks summarizing our findings are contained in Sec. V.

II. SYSTEM SETUP
We examine the penetration mechanism of a nonfluctuating membrane by an active particle moving under the action of a constant propulsion force F 0 . In this context, the penetrating particle may be viewed as a selfpropelling agent in the limit of its persistence length being large compared to the distance initially separating the particle from the membrane 53-64 . The active particle may represent a swimming microorganism [65][66][67][68][69][70] or an artificial microrobot that can be manipulated by controlled external fields 71-74 . In our model, the membrane is composed of N identical dipolar spheres of radius a and dipole moment m, uniformly arranged on a square lattice of size L × L, rotated by 45 • around the axis z, the latter directed normal to the membrane, as schematically illustrated in Fig. 1. We denote by h the lattice spacing after initialization. The dipole (a) (b) Figure 1. (Color online) Graphical illustration of the system setup. (a) An active particle of radius R moving through an effective driving force F0 toward a two-dimensional membrane composed of N dipolar particles of radius a and dipole moment m. The membrane particles are initially arranged on a square lattice of dimension L × L and spacing h, rotated by 45 • around the z axis, the latter oriented normal to the plane of the membrane. The membrane is centered about the origin and clamped at its periphery. Periodic boundary conditions are imposed in both x and y directions. We assume that the membrane particles are subject to dipolar, steric, and elastic pairwise interactions. The system is fully immersed in a Newtonian viscous fluid of shear viscosity η. (b) Schematic illustration of the lattice structure composing the model membrane. For future reference, the eight nearest neighbors of the particle at the center of the lattice are identified by numbers (1)(2)(3)(4)(5)(6)(7)(8). Here, the dipoles are initially aligned along the x direction. Elastic springs are also inserted along the lattice diagonals but are not displayed here for reasons of clarity. moments are rigidly attached to the particles and initially aligned along the x direction. The membrane is immersed in a Newtonian fluid, characterized by a constant dynamic viscosity η. We support the membrane at its periphery (the particle displacements are zero for x, y = ±L/2) and assume periodic boundary conditions in the transverse directions (x, y). Moreover, we suppose that the mutual interactions between the membrane particles are pairwise additive and described by forces that depend only on the difference of coordinates of each two neighboring particles.
Typically, various types of interactions may occur among membrane particles including dipolar, steric, and elastic interactions. For instance, dipolar and steric interactions can be imposed by membrane phospholipids chains and other biomolecules [75][76][77] , whereas intermolecular coupling between the lipid bilayer and the cytoskeleton network gives rise to elastic interactions [78][79][80] . Accordingly, the total potential energy of the membrane here is written as a sum of three distinct contributions as wherein µ 0 is the vacuum permeability, while m represents the magnitude of the dipole moments which we assume to be constant and equal for the membrane particles. Moreover,m i = m i /m represents the unit orientation vector of the dipole moment m i of the ith particle, rigidly anchored to the particle frame. r ij = |r ij | is the distance between particles i and j, r ij = r i − r j , andr ij = r ij /r ij is the corresponding unit distance vector. In addition, is an energy scale associated with the Weeks-Chandler-Anderson (WCA) pair-potential 81 , σ = 2a is the diameter of the dipolar particles, N ij = H (r C − r ij ), with H(·) denoting the Heaviside step function, and r C = 2 1/6 σ is a finite cutoff distance beyond which the steric interactions energy vanishes. Furthermore, k is the elastic constant of the harmonic springs coupling each particle to its four nearest and four next-nearest neighbors, r 0ij is the rest length of the springs, and ξ ∈ (0, 1] is a prestress parameter. Here, we use the notation N (i) to denote the set of nearest and nextnearest neighbors of the ith membrane particle. For the sake of simplicity, we neglect throughout this work all possible hydrodynamic interactions between particles. Moreover, we assume that the particles are small enough or sufficiently matched in density to the surrounding fluid for the influence of gravity to be neglected, and large enough for the effect of thermal fluctuations to be neglected.
The corresponding interaction force acting on the ith membrane particle is obtained by differentiating the potential energy described by Eq. (1) with respect to the particle position as F i = −∂U/∂r i . Accordingly, In addition, the resulting torque acting on the ith particle is of dipolar origin and can be calculated from the potential energy of the membrane as 82 T i = −m i × (∂U/∂m i ). Defining the dimensionless vector c ij =m j − 3 (m j ·r ij )r ij , we obtain At small length scales, aqueous systems are characterized by small Reynolds numbers, so that viscous forces dominate over inertial forces. The resulting overdamped dynamics can therefore be adequately described within the framework of linear hydrodynamics 83,84 . Accordingly, the translational and rotational velocities of the membrane particles, respectively denoted as V i and Ω i , are linearly coupled to the forces and torques acting on their surfaces via the hydrodynamic mobility functions [85][86][87][88] . The latter are second-order tensors, which simply reduce to scalar quantities when considering motion in an unbounded medium and neglecting the fluid-mediated hydrodynamic interactions between the particles. Specifically, where µ and γ denote, respectively, the translational and rotational self-mobility functions of the membrane particles. These are given by the usual Stokes formulas for an isolated sphere in an infinite fluid domain as µ = 1/(6πηa) and γ = 1/(8πηa 3 ). In addition, F ext i represent the external force exerted by the active particle due to the steric interactions with the membrane particles. These pair interactions are modeled via a soft repulsive WCA potential as in Eq. (1) for which σ = R + a, with R denoting the radius of the active particle.
The equations governing the temporal evolution of the translational and rotational degrees of freedom of the ith membrane particle read The latter equation can further be reformulated after making use of Eqs. (3) and (4b) as We introduce at this point an additional cutoff length beyond which the dipolar and elastic interactions are set to zero. Accordingly, the dipolar and elastic potentials are Reduced activity Admittance Table I. Expressions of essential dimensionless numbers that characterize the system in the trapping and penetrating states with the corresponding denominations.
also shifted to this cutoff length, so as to ensure that the resulting forces are continuous. Physically, may represent, for instance, an average distance between cytoskeletonbilayer connection sites. Throughout this work, we set = 3h/2 > √ 2h, such that the pair-interactions between the membrane particles are restricted to the four nearest and four next-nearest neighbors only.

III. TRAPPING, PENETRATION, AND SELF-HEALING
Having introduced a model for our membrane and derived the corresponding equations governing the translational and rotational dynamics of the particles composing the membrane, we next study in detail the dynamical states emerging from the interaction between an active particle propelling toward the membrane. For that purpose, we solve numerically the set of ordinary differential equations in time given by Eqs. (2) -(6) using a standard 4th-order Runge-Kutta scheme with adaptive time stepping 89 . Before the active particle starts to interact with the membrane particles, we assume that the lattice spacing h is identical to the cutoff length scale r C associated with the WCA pair potential. In addition, we assume that the rest length of the elastic springs is equal to the initial interparticle separation, i.e., r 0ij = h for the pairs of particles located along the lattice axes, and r 0ij = √ 2h for the pairs along the diagonal. Under these conditions, the membrane is initially at equilibrium, on account of the periodic boundary conditions imposed along the transverse directions (x, y). We further mention that requiring h = r C is equivalent to considering a constant ratio h/a = 2 7/6 . Unless stated otherwise, we consider throughout the present article a membrane composed of N = 450 dipolar particles and set the prestress parameter as ξ = 0.9.
In analogy to our previous work on a 1D model membrane 39 , we introduce four relevant dimensionless numbers that characterize the system behavior. We define the reduced dipole strength which quantifies the importance of the dipolar forces (∼ µ 0 m 2 /a 4 ) relative to the steric forces (∼ /a). Based on dimensional considerations, one might suppose that the membrane is endowed with an effective bending stiffness of the form κ B ∝ E 1 , where the proportionality coefficient depends upon the membrane curvature and the number of particles constituting the membrane, as rigorously shown by Goriely and coworkers 90,91 . In addition, we introduce the reduced activity which represents a balance between the magnitude of the active driving force F 0 and the steric forces. Further, we define the reduced stiffness to represent the ratio between the elastic forces (∼ ka) and dipolar forces. Finally, we introduce the size ratio to denote the radius of the active particle relative to that of the membrane particles. An additional dimensionless parameter, that we denominate as "admittance", is introduced to quantify the penetration capability of the active particle. It is defined based on the above definitions of E 1 and E 2 and expresses the ratio between active and dipolar forces. Specifically, The prefactor in front of the ratio E 2 /E 1 follows from theoretical considerations as will be shown in the sequel. Here, the admittance serves to quantify a criterion of whether or not the active passes through the membrane. For ease of reference, the explicit expressions of the key dimensionless numbers characterizing the states of the system are listed in Tab. I. To get a first intuition of the possible membrane dynamics, we display the different observed scenarios in Fig. 2. For low admittance (P 0 0.7, top row and movie S1 in the Supporting Information), the membrane starts to deform when the motile active particle comes close, but only up to some point, reaching a steady state of constant membrane shape and fixed position of the active particle [see Fig. 2, panels (c) and (d)]. When increasing the admittance to P 0 2 (second row and movie S2), the membrane of the membrane in Fig. 2 (l) remain isolated because the range of the internal membrane interactions is shorter than the separation distance of these four particles from the rest of the membrane. Such a behavior is even more pronounced for significantly larger particles (δ = 11) (bottom row and movie S4). In Fig. 2 (p), the membrane around the center is fragmented into 24 particles after the active particle has penetrated through the membrane.
In Fig. 3, we present state diagrams indicating the system behavior in the parameter space (E 1 , E 2 ) for two different values of the reduced stiffness κ, namely, (a) κ = 1 and (b) κ = 10. As already mentioned, the membrane is composed of N = 450 dipolar particles. Here, we set δ = 1. Depending on the ratio between the control parameters E 1 and E 2 , we observe that the active particle either passes   (2) - (6). In addition to trapping (blue squares) and penetration with healing (red triangles), penetration events without subsequent healing (green symbols) occur in some parameter ranges for large values of the size ratio δ. These penetration scenarios are accompanied by the creation of a permanent hole of a size exceeding the interaction range of the membrane particles in addition to the partial fragmentation of the membrane into isolated particles.
through the membrane to reach the other side (red triangles) or remains trapped (blue rectangles). The transition between the two states can be described by a linear hypothesis of the form P 0 = κ. Accordingly, penetration events occur when the membrane restoring forces consisting of dipolar and elastic contributions become weaker than the damaging force resulting from the steric interactions with the active particle. After full penetration has occurred, the membrane self-heals and relaxes back to its initial equilibrium configuration. We have systematically checked that an analogous state digram holds for strongly elastic membranes as well, namely for κ = 10 2 and κ = 10 3 .
In Fig. 4, we show dynamical state diagrams in the plane of the control parameters (δ, E 2 ) for E 1 = 10 −2 . To limit the parameter space, we set the reduced stiffness to κ = 1. We observe that the transition between the trapping and penetration states can also be enabled by varying the aspect ratio δ. Accordingly, the penetration capability through a membrane is not only determined by the system admittance, but also by the size of the active particle relative to that of the membrane particles. This is in agreement with earlier experimental investigations indicating that particle size may strongly affect the uptake efficiency and kinetics [92][93][94][95][96] . Consequently, an active particle with a size larger than that of the membrane particles is more likely to remain trapped. It is worth noting that, in the considered range of parameters, the transition has been found to only depend on the admittance P 0 for our simplistic 1D model membrane studied in a previous work 39 . For large values of the size ratio, the penetration process may also occur without subsequent self-healing of the membrane. This situation is accompanied by partial fragmentation of the membrane, during which a number of particles around the center remain isolated, creating a permanent hole in the membrane. The number of fragments largely depends on the propulsion speed and the size ratio. This effect points to an interesting size effect of the membrane behavior and shows that motile particles can be used to permanently damage the considered type of membrane.

IV. ANALYTICAL THEORY
To rationalize our numerical results, we derive in the following an analytical theory based on a perturbative expansion technique that describes the system behavior in the small-deformation regime. Particularly, we are interested to determine theoretically the membrane displacement and orientation fields of the dipolar particles in the trapping state. Our analytical calculations proceed through the linearization of the governing equations of motion, followed by prescribing the relevant fields using a discreteto-continuum approach 97,98 .

A. Linearized equations of motion
In the following, we neglect for simplicity the steric interactions between the membrane particles and assume that the mutual distance between neighboring particle is within the interaction range of the dipolar and elastic forces, i.e., r ij ∈ [h, ], with j ∈ N (i), for i = 1, . . . , N .
The dynamical equations governing the evolution of the ith membrane particle displacement and dipole orientation fields can be cast in the forṁ wherein the superposed dot represents a temporal deriva-tive, and F D i and F E i are dimensionless forces stemming from the dipolar and elastic interactions, respectively. Here, we have defined for convenience the parameters which have the dimension of a diffusion coefficient and inverse time, respectively. Assuming that the active particle has a radius comparable to that of the membrane particles, i.e., for δ ∼ 1, it can readily be verified that the resistive force due to the steric interactions with the active particle vanishes only for the four particles located near the center of the membrane, the initial coordinates of which are given in the Cartesian coordinate system by (x, y) = (± √ 2h/2, 0) and (x, y) = (0, ± √ 2h/2).
Adopting a local spherical coordinate system, the origin of which coincides with the center of the ith dipolar particle, the orientation vector can be represented aŝ where φ i ∈ [0, 2π) denotes the azimuthal angle, and ψ i = π/2 − θ i stands for the complement of the polar angle, with θ i ∈ [0, π]. In the initial configuration, φ i = ψ i = 0, for i = 1, . . . , N . Accordingly, the rotation rate of the ith dipolar particle can be obtained using Euler representation as Following a linear elasticity theory approach 99,100 , we express the position vectors of each dipolar particle relative to the laboratory frame as r i = (U i + u i )ê x +(V i + v i )ê y + w iêz , for i = 1, . . . , N , where U iêx + V iêy is the position vector in the undeformed state of reference, and u iêx + v iêy + w iêz is the displacement of the membrane particles relative to the initial configuration. The linearized magnetic force acting on the ith dipolar particle reads where we have defined p j = (u j − u i ) /h, q j = (v j − v i ) /h, and r j = (w j − w i ) /h to denote the displacement gradients.
Here, the numbers j = 1, . . . , 8 appearing in subscript denote the index of a nearest or next-neighbor particle on the lattice, as schematically illustrated in Fig. 1 (b). Moreover, we have used the shorthand notations S α = α 2 + α 4 + α 6 + α 8 , Analogously, the elastic force acting on the ith particle can be presented in a linearized form as Notably, the inplane components of the dipolar and elastic forces involve gradients of the lateral displacements p j and q j in addition to the azimuthal orientation φ j . In contrast to that, the normal components are found to depend on the displacement gradient r j and the complement of the polar angle ψ. Consequently, a decoupling between the lateral and normal displacements is found for planar membranes, in a way analogous to what has previously been observed for 2D elastic membranes that are modeled as a continuum hyperelastic material featuring resistance toward shear and bending [101][102][103][104] . Particularly, for a non-prestressed membrane (ξ = 1), the elastic forces are purely tangential (oriented along the plane of the membrane) and depend solely on the inplane displacement gradients p j and q j .
Finally, the linearized dipolar torque has components only along the y-and z-axes and can be expressed in terms of the displacement gradients and dipole orientation angles in a scaled form as Having derived linearized expressions for the forces and torques governing the evolution of the membrane particles, we next consider the dynamics of the active particle. The latter is subject to the active driving force F 0 = F 0êz in addition to the repulsive steric forces resulting from the interaction with the nearby membrane particles. In the overdamped regime, the translational motion of the active particle along the z direction is governed by wherein z P denotes the z-position of the active particle, and F stands for the magnitude of the steric force exerted by one of the four particles located around the membrane center, and α denotes the angle this force makes with the vertical.
Equations. (12) form 2N ordinary differential equations in the time variable for the unknown membrane displacement and dipole orientation fields. These equations are subject to the initial conditions of vanishing membrane displacement and reorientation angles of the dipoles, in addition to vanishing displacement at the membrane periphery and periodic boundary conditions along the x and y directions. In the steady state, the problem is equivalent to searching for the solution of linear recurrence relations coupling the positions and orientations of all the membrane particles initially located on a lattice. Due to the somewhat complicated nature of the resulting equations, an analytical solution is far from being trivial. To handle this difficulty and to obtain a quantitative insight into the system behavior in the small-deformation regime, we will approach the problem differently. Our solution methodology will be based on a continuum description of the linearized equa-tions of motion as detailed below.

B. Continuum theory
The core idea of discrete-to-continuum analysis, is to express the membrane displacements and dipole orientations following the standard approach as where D α = ∂/∂α, α ∈ {x, y} represents the differential operator and (s, r) ∈ {0, ±1/2, ±1}. Here, the fraction at subscripts i ± 1/2 refer to the nearest-neighboring particle on the lattice axes, namely, the ones identified by even numbers in Fig. 1 (b). The integer subscripts i ± 1 refer to the next-nearest-neighboring particles located on the lattice diagonals.
The exponential argument in Eq. (20) can be expanded up to the second order in power series using a twodimensional Taylor expansion as 105 Applying this transformation rule to Eq. (12a), the partial differential equation governing the translational degrees of freedom of the membrane particles can be rewritten in vector form as where we have approximated the steric force exerted on the particles near the center of the membrane by a two-dimensional Dirac delta function δ(x, y) = δ(x)δ(y). We have checked that taking alternative forms for the steric force, such as a 2D rectangle function centered around the origin, does not alter our results significantly. Therefore, a Dirac delta function has been adopted here for simplicity. As for the rotational degree of freedom given by Eq. (12b), a discrete-to-continuum transformation yields In the steady-state limit, it follows readily from Eq. (23) that ψ − w ,x = 0 and u ,y + v ,x = 0. Accordingly, the steady polar orientation of the dipolar particles only dependents on the displacement gradient. Interestingly, this situation is in analogy with the Kirchhoff-Love theory for elastic beams or sheets 99 wherein P 0 = µhF 0 /A, is the system admittance defined above by Eq. (11). We recall that P 0 ∼ E 2 /E 1 . In Eq. (24), we explicitly observe that P 0 controls how far the active particle can penetrate through the initial planar membrane and leads to a deflection of the membrane. We note that the 2D Dirac delta function has the dimension of inverse length squared. In the following, we attempt to obtain closed analytical expressions for the displacement and orientation fields not only for the steady state but also for transient dynamics situations.

C. Steady solution
Because of the already-mentioned decoupling between the lateral and normal displacements, the solution for the in-and out-of-plane deformations can be obtained independently. Since the external force is exerted normal to the plane of the membrane, deformation will predominantly occur along the z direction. In the following, we assume that |w| L, for our approximate equations of motion derived above to be valid.
By projecting Eq. (24) onto the z direction, the normal displacement is governed by a second-order partial differential equation of the form where k = c − 5 8 + 6(1 − ξ)κ , (26a) are dimensionless numbers that comprise both dipolar and elastic contributions. For future reference, we have defined the constant By rescaling the spatial variables, x = X k and y = Y √ k ⊥ , Eq. (25) can be presented in the form of a steady diffusion equation with a source term as As k < k ⊥ , the deformation profile is expected to be broader (high variance distribution) along the y direction than along the x direction, when considering weakly elastic membranes for which κ 1. For strongly elastic membranes (with ξ = 1), it follows that k k ⊥ , and thus the system behavior is primarily determined by membrane elasticity and approximately isotropic.
To solve Eq. (25), we exploit periodicity of the system along the transverse directions by expressing the membrane normal displacement w in terms of a Fourier series 106 . Then, with p, q = 1, 2, . . . denoting the positive integers that set the coordinates in Fourier space. Here, we have defined the basis function c p (x) = cos (H p x), where H p = (2p − 1)π/L, and analogously for c q (y). In addition,ŵ(p, q) denotes the Fourier coefficients of w, defined aŝ It is worth mentioning that the solution form given by Eq. (29) follows from the prescribed boundary conditions, so as to ensure that w(x = ±L/2, y) = w(x, y = ±L/2) = 0. Moreover, the basis functions c p (x) satisfies the orthogonality relation By substituting Eq. (29) into Eq. (25) and making use of the orthogonality property given by Eq. (31), we readily obtainŵ For the torque balance equation, it follows from Eq. (23) that the steady orientation of the membrane dipoles is obtained as where we have defined the basis functions s p (x) = sin (H p x). Particularly, ψ(x = 0, y) = 0, as required by the symmetry of the system. Finally, by writing the solution for the transverse displacements u(x, y) and v(x, y) in terms of Fourier series in a way analogous to Eq. (29) and the azimuthal orientation of the dipoles as it follows that u, v, and φ must vanish to satisfy the boundary conditions imposed at the membrane extremities, considering the present approximate equations. Figure 5 shows the steady-state variations of (a) the nor- and κ = 1. Symbols indicate the numerical solution of the full nonlinear problem given by Eqs.
(2) -(6) and solid lines are the analytical predictions obtained from the solution of the continuum equations using finite Fourier transforms. Good agreement is found between the theory and simulations. The small discrepancy observed for ψ around the origin and near the membrane periphery is a drawback of the present analytical approach based on a perturbation technique. All in all, our predictive model requires no fitting parameters and thus can conveniently be applied to describe the steady-state membrane displacement and dipole orientation in the small-deformation regime considered here.

D. Transient dynamics
Having investigated the system behavior in the steady trapping limit, we next turn our attention to the transient dynamics under the action of the force exerted by an active particle pushing against the membrane. To be able to make an analytical progress, we assume that δż P µ 0 F , such that F 0 is balanced by the steric interaction with the membrane, not by friction with the fluid. Accordingly, we setż P = 0 in Eq. (22) for t > 0.
By projecting Eq. (23) onto the x direction, it follows that φ necessarily vanishes. In this way, the projected equations of motion governing the temporal evolution of the normal displacement field w and orientation ψ read Using a similar solution procedure as for the steady dynamics that is based on Fourier transforms, we obtain whereψ denotes the Fourier coefficients of ψ, defined aŝ ψ(x, y)s p (x)c q (y) dx dy .
In Fig. 6, we present the transient evolution of (a) the membrane normal displacement and (b) the dipole orientation before reaching the steady-state at three scaled times, where t S = ηL 3 / denotes the simulation time. Here, curves are shown in the plane y = 0 using the membrane parameters (E 1 , E 2 ) = (10 −1 , 10 −1.5 ) and κ = 1. Although the analytical theory involves no fitting parameters, very good agreement is obtained between full numerical simulations (symbols) and analytical predictions (solid lines).

V. CONCLUSION
In the present work, we have discussed the interaction of an active particle with a minimal 2D membrane which could be realized, e.g., using synthetic particles of controlled interactions. We have identified three different scenarios, one corresponding to a permanent trapping of the particle by the membrane and the remaining two implying penetration of the particle through the membrane. The first type of penetration is characterized by a complete subsequent healing of the membrane which relaxes towards its equilibrium configuration once the particle has passed. In stark contrast, we have shown that much larger particles can create a hole in the membrane that is large enough to prevent such a self-healing dynamics, resulting in a permanently damaged membrane. This behavior is accompanied by the expulsion of membrane particles into isolated fragments. Our result suggests that if one were to effectively damage a synthetic vesicle, or perhaps a cancer cell membrane, one would need to use particles of a certain minimal size. Complementary to simulations, we here provide a detailed analytical theory allowing to predict the entire state diagram, the shape and the dynamics of the membrane. Our approach might be useful to predict transitions between trapping, penetration with and without self-healing in experiments.