Dynamics of particle uptake at cell membranes
DDynamics of particle uptake at cell membranes
Felix Frey, Falko Ziebert, and Ulrich S. Schwarz
Institute for Theoretical Physics, Heidelberg University, Philosophenweg 19,69120 Heidelberg, Germany and BioQuant, Heidelberg University,Im Neuenheimer Feld 267, 69120 Heidelberg, Germany (Dated: October 29, 2019)Receptor-mediated endocytosis requires that the energy of adhesion overcomes the deformationenergy of the plasma membrane. The resulting driving force is balanced by dissipative forces,leading to deterministic dynamical equations. While the shape of the free membrane does notplay an important role for tensed and loose membranes, in the intermediate regime it leads to animportant energy barrier. Here we show that this barrier is similar to but different from an effectiveline tension and suggest a simple analytical approximation for it. We then explore the rich dynamicsof uptake for particles of different shapes and present the corresponding dynamical state diagrams.We also extend our model to include stochastic fluctuations, which facilitate uptake and lead tocorresponding changes in the phase diagrams.
I. INTRODUCTION
The plasma membrane presents a physical barrier thatseparates the interior of the cell from its environment.Therefore, the ability of cells to exchange informationand material across their plasma membrane is of centralimportance for their function [1, 2]. On the one hand,these uptake processes are vital for nutrient influx andsignal transduction [3]. On the other hand, pathogenslike viruses hijack cellular uptake processes to enter hostcells during infections [4]. In addition, uptake of artificialparticles at cell membranes can be desired, as e.g. in thecontext of drug delivery [5], or undesired, as e.g. in thecontext of microplastics [6].In receptor-mediated endocytosis particles with sizesbetween 10 −
300 nm are taken up because the energy gainupon particle binding to cell surface receptors overcomesthe deformation energy of the membrane [7]. Membraneshape is of central importance for this process. It is fixedby particle shape at the adhered part, but follows fromminimization of the membrane energy for the free part,compare Fig. 1. Very importantly, cargo particles cancome with a huge diversity in shape, including the caseof viruses. The most frequent virus shape is the spher-ical or icosahedral shape, followed by filamentous andthen by more complex shapes. To name a few examples,reovirus, causing respiratory or gastrointestinal illnesses,has icosahedral shape [8], Marburg or Ebola viruses havefilamentous shape [9], and rabies virus has bullet-likeshape [10]. Apart from shape, stochastic fluctuations inreceptor binding might also play a role, as the cargo par-ticles are small and typically covered by only few tens ofligands.The uptake of small particles has been previously stud-ied both analytically and by computer simulations. De-terministic approaches usually focus on calculating min-imal energy shapes for the plasma membrane and theattached particle to deduce dynamical state diagrams[11–13], investigate uptake dynamics and the role of re-ceptor diffusion within the plasma membrane [7, 14],study the consequences of particle elasticity during up-
Unwrapped Fully wrappedPartially wrappedFree membrane Adheredmembrane Stochastic bondingCell Particle
FIG. 1. During receptor-mediated endocytosis, the parti-cle goes from unwrapped through partially wrapped to fullywrapped. While membrane shape is fixed by particle shapein the region of the adhered membrane (red part of the con-tour), the shape of the free membrane follows from an energyminimization (grey part of the contour). take [15, 16] or interactions of the particle and the cy-toskeleton [17]. Stochastic approaches usually focus onthe effect of ligand-receptor binding [18]. Computer sim-ulations complement these studies by considering espe-cially the role of non-spherical particle geometries [19–23] or the role of scission, when the wrapped particle isseparated from the membrane [24].Despite this plethora of different approaches, ana-lytical approaches are rare that allow us to study theinterplay of particle shape, free membrane shape andstochasticity in one transparent framework. Recently weshowed that in a deterministic model, spherical parti-cles are taken up slower compared to cylindrical parti-cles, whereas the situation can reverse in a stochasticdescription, because spherical particles profit more fromthe presence of noise [25]. However, in this earlier workwe have neglected the exact role of the free part of themembrane and did not investigate the possibility that thedynamics stops with a partially wrapped particle (Fig. 1).Earlier it has been suggested that the free membranemight act as an effective line tension [11] and exact for-mulae for the elastic energy barrier provided by the freemembrane have been derived for the limits of tense andloose membranes [26]. Here we present a comprehen-sive study of these important effects and show that in a r X i v : . [ q - b i o . S C ] O c t the general case, a simple term that is similar to butdifferent from a line tension term describes this energybarrier well. With this simplification, we are able to per-form a comprehensive analysis of the uptake dynamicsat membranes, including the effect of particle shape andstochastic fluctuations. By combining models for mem-brane mechanics, overdamped membrane dynamics andstochastic dynamics of receptor-ligand binding, we find avery rich scenario of possible uptake dynamics.Our work is organized as follows. In section II wefirst calculate the shape and energy of the free membraneby numerically solving the shape equations that give theminimal energy shapes of the free membrane. By com-paring this energy to the energy of the adhered membranewe can identify the parameter regime in which the freemembrane cannot be neglected, as it contributes up to20% of the total energy. We also show that this energycontribution is similar to but different from the effect ofa line tension; in particular, it leads to faster initial up-take and the associated energy barrier is located at highercoverage. These effects can be described well by a simpleanalytical ansatz introduced here. In section III we de-rive and analyze the deterministic dynamical equationsfor cylindrical, spherical and spherocylindrical particles.Here the free membrane is not yet taken into accountexplicitly, but its potential effect can already be appreci-ated on the level of a line tension. We find that sphericalparticles are taken up slower than cylindrical particles.In addition, we find that short cylinders are taken upfaster in normal orientation, whereas long cylinders aretaken up faster in parallel orientation. For spherocylin-drical particles we find that they are taken up fastest inparallel orientation. In section IV we include the ana-lytically exact energy of the free membrane for parallelcylindrical particles into our dynamical model and cal-culate dynamical state diagrams. In addition, we alsoinclude free membrane effects by the phenomenologicalapproach. As the phenomenological approach gives sim-ilar results, we then apply our approach also to spheri-cal particles. In section V we calculate dynamical statediagrams for spherical particles using the phenomeno-logical ansatz. We identify three regimes: full uptake,partial uptake and no uptake determined by membraneelasticity, adhesion energy and the free membrane. Insection VI we then extend our model to a stochastic de-scription, which in contrast to earlier work [25] now in-cludes the effect of the free membrane. We find thatfluctuations decrease uptake times and expand uptake toparameters regions where uptake is not possible in thedeterministic model. In addition, we find that sphericalparticles can be taken up faster with fluctuations com-pared to parallel cylindrical particles. II. MEMBRANE ENERGIES
To arrive at a dynamical model for particle uptake atcell membranes, we follow the standard approach and ( )( ) ( )
FIG. 2. Parametrization of membrane shape for a sphericalparticle. The adhered part of the membrane is shown in red,the free part of the membrane is shown in grey and the particleis shown in blue. R is the particle radius, θ is the uptake angleand s the contour length of the free membrane. Tangent angle ψ , radial distance r and height z are functions of s . first compare the energy gain due to adhesion and theenergy cost due to bending and tension [1, 11–13]. Laterwe will introduce dynamics by also considering dissipa-tive forces. We are interested in ligand-receptor interac-tions and for simplicity assume that they are distributedhomogeneously over the particle surface. The total en-ergy of the membrane is then described by the followinggeneralization of the Helfrich bending Hamiltonian [27] E = − (cid:90) A ad W d A + (cid:90) A mem κH d A + σ ∆ A + ζ E . (1)The first term is the gain in adhesion energy, where theadhesion energy per area W is defined to be positive.The second term is the bending energy with κ beingthe bending rigidity and H the mean curvature of themembrane. The third term is the tension energy, with σ being the membrane tension and ∆ A the excess areacompared to the flat membrane. It is important to notethat only the part A ad of the membrane that adheresto the particle contributes to the adhesion energy, whileboth the adhered and the free parts of the membrane, A mem = A ad + A free , contribute to bending and tension(compare Fig. 1). The last term in Eq. (1) results froma possible line tension ζ , with E being the length of theedge between the membrane adhering to the particle andthe free cell membrane. We note that in general Eq. (1)also includes a term that depends on Gaussian curvature.However, since we do not consider topological changes,it can be neglected in our context [12].The two membrane parameters κ and σ define a typ-ical length scale λ = (cid:112) κ/σ . Using this scale, the mem-brane can be classified as tense ( λ/R (cid:28)
1) or loose( λ/R (cid:29) κ = 25 k B T and σ = 10 − − − N / m [26, 28],one has λ = 10 −
100 nm. As typical sizes for virus ornanoparticles range from R = 10 −
100 nm, it holds that λ/R ∼ . −
10 and the biologically most relevant regimeis hence intermediate between tense and loose.The line tension term in Eq. (1) could result e.g. fromthe localization of certain lipids or proteins to the curvedmembrane at the border between the adhered and freemembrane [29]. More importantly in our context, how-ever, such a term could potentially also be used to de-scribe the effective behaviour of the free membrane, evenin the absence of a microscopic line tension [11]. In thiscase, one could restrict the integration of the bendingenergy over the adhered part of the membrane. For di-mensional reasons, one then would expect the effectiveline tension to scale as ζ = √ κσ and the typical rangewould be ζ = 1 −
10 pN [12, 26].In order to discuss the role of the free membrane shapein more detail, let us consider the uptake of a sphericalparticle. Fig. 2 shows the used parametrization [26, 30],where θ is the uptake angle, measured with respect to thesymmetry axis (along the z -direction). The membranecontour is parameterized by its arc length s relative tothe point where the adhering membrane is connected tothe free part. Furthermore, ψ ( s ) is the angle between theradial axis normal to the z -axis and the contour tangent, r ( s ) is the radial distance to the z -axis, and z ( s ) theheight. Note that r and z can be obtained by integrationover ψ .For the spherical particle, the area adhering to themembrane as a function of particle radius R and uptakeangle θ is given by A ad = 2 πR (1 − cos θ ) and thus theadhesion energy would be E W = − πW R (1 − cos θ ).Similarly, also the contributions from bending and ten-sion of the adhered part can be given explicitly. In thefollowing, we non-dimensionalize energies by the bendingrigidity. Then the total mechanical energy of the adheredpart, E totad = E κ ad + E σ ad , reads E totad κ = 4 π (1 − cos θ ) + π R λ (1 − cos θ ) . (2)For the energy of the free parts, E totfree = E κ free + E σ free ,one has [26, 30] E totfree κ = π (cid:90) ∞ (cid:18) ˙ ψ + sin ψr (cid:19) r d s + 2 πλ (cid:90) ∞ (1 − cos θ ) r d s . (3)This energy has to be minimized with respect to the freemembrane shape at given uptake angle θ . Together withthe geometrical relations between ψ , r and z , one getsthe following shape equations [11, 12, 26, 31, 32]:¨ ψr cos ψ + ˙ ψr cos ψ + 12 ˙ ψ r sin ψ −
12 (cos ψ + 1) sin ψ − r λ sin ψ = 0 , ˙ r − cos ψ = 0 , ˙ z + sin ψ = 0 . (4)This set of ordinary differential equations has to be solvedwith the boundary conditions r (0) = R sin θ, ψ (0) = θ, ψ ( ∞ ) = 0 , ˙ ψ ( ∞ ) = 0 (5)and an additional one, z ( ∞ ) = 0, where other choices arealso possible. / R = 0.1/ R = 0.3/ R = 1.0/ R = 3.0/ R = 10 π π π π θ E / λ / R = 0.1 E ad E free (a) π π π π θ (b) E t o t f r ee / ( E t o t f r ee + E t o t a d ) E t o t f r ee / ( E t o t f r ee + E t o t a d ) FIG. 3. (a) The bending and tension energy of the freemembrane relative to the total bending and tension energy,i.e. of the adhered and free parts of the membrane for differentvalues of λ/R . To calculate the energy of the free parts weuse Eq. (A3) in the case of λ/R < λ/R > λ = R . Theinset shows the bending and tension energy of the adheredand free parts of the membrane for λ/R = 0 .
1. (b) The sameas in (a) but for the phenomenological description of the freemembrane given in Eq. (9).
We solved the boundary-value problem for the givensystem of ODEs from Eqs. (4) using a 4-th order collo-cation algorithm with matched asymptotics [33]. Detailscan be found in appendix A. We then evaluated the en-ergy contributions from the free membrane. In addition,we compared to asymptotic expressions that have beengiven previously [26] for the limit of a tense ( λ/R (cid:28) λ/R (cid:29) λ/R . Here the energies of the free parts werecalculated using Eq. (A3) for the tense regime ( λ/R < λ/R >
1) and numeri-cally for the intermediate regime ( λ = R ). The analy-sis demonstrates that in the limit of a loose membrane( λ/R (cid:29)
1) the relative energies of its free parts are verysmall. The underlying reason is that in this case, the (a) (b) (c) E f r ee / / R = 0.1 / R = 0.3 / R = 1.0 E totfree EE phenofree E tenseForet E looseForet FIG. 4. Energies of the free membrane for (a) λ/R = 0 .
1, (b) λ/R = 0 . λ/R = 1 .
0. The numerically calculatedtotal energy of the free membrane is shown in green, the line tension approximation E ζ is shown in blue, the phenomenologicalapproximation E phenofree from Eq. (9) is shown in orange and the analytical limits for tense (from Eq. A3 [26]) and loose (fromEq. A4 [26]) membranes are shown as dashed curves in red and cyan, respectively. (Top) Numerically calculated shapes of themembrane. membrane assumes the shape of a minimal surface [26]and both bending and tension contributions become verysmall. In case of a tense membrane ( λ/R (cid:28) λ/R ≈ λ/R ∼ . − , (6)which is the relevant regime for biological systems [12].The procedure of solving the shape equations is numer-ically involved and makes it difficult to proceed with acomprehensive study of uptake dynamics. A simple phe-nomenological expression that represented the energy ofthe free membrane well would allow for analytical insightand for a transparent discussion of the effect of the freemembrane on the uptake process. It has been suggestedearlier [11] that the effects of the free membrane may beseen as an effective line tension. Considering a sphericalparticle, this corresponds to an additional energy contri-bution E ζ = ζ E = ζ πR sin θ , (7)with ζ the effective line tension and E the length of theedge. However, this simple form has been shown to bestrictly true only in the double limit of high tension andlarge uptake angle, where E totfree scales like a line tensionwith ζ = √ κσ , (8) as proposed in the works of Deserno [12] and Foret [26].In order to arrive at such a phenomenological ap-proach, we first note that the analytical expressionsfor the tense and loose regimes in leading order are E totfree = πζRθ sin θ and E totfree = πζR ( R/λ ) sin θ , respec-tively. Compared to the expression for a line tensionfrom Eq. (7), which has a symmetric barrier at θ = π/ θ .We therefore explored the effects of analytically simpleterms scaling as ∼ θ sin θ and ∼ θ sin θ and found thatthe second choice works very well. Fig. 3(b) shows theresults for the relative contribution of the free membranefor different values of λ/R with the phenomenologicalansatz E phenofree = ζπRθ sin θ . (9)They are surprisingly similar to the exact results shownin Fig. 3(a).To further explore the validity of this ansatz, in Fig. 4we compare the numerically computed total energy of thefree membrane E totfree (green) to a line tension E ζ (blue)and the phenomenological approximation E phenofree (or-ange). Both for the line tension, defined in Eq. (7), andthe phenomenological approximation, defined in Eq. (9),we use the value of ζ defined in Eq. (8). In the tensedcase (Fig. 4(a)), one sees the excellent agreement be-tween the numerical calculations (green curve) and theanalytical result given as Eq. (A3) (dashed red). Nextwe note that Eq. (A3) (dashed red) is a rather poor de-scription for λ/R = 1 (Fig. 4(c)) as it gets negative forlarge uptake angles, whereas Eq. (A4) (dashed cyan) iscompletely off for λ/R < λ/R ∼ . −
1. As it is also hard tointerpolate between the two analytical limits, the phe-nomenological approximation is much more convenient.Very importantly, it performs better than the line tension(cf. blue curve), which is not only off quantitatively, butalso places the barrier at too small values of θ . We con-clude that the phenomenological approximation E phenofree represents a good description of the qualitative behaviorin the biologically relevant regime and that it works wellover the whole range of angles, different from the ansatzof an effective line tension. III. DETERMINISTIC DYNAMICSA. Dynamical equations
We now discuss the deterministic dynamics ofadhesion-mediated particle uptake, with a focus on therole of particle shape, which next to size is the parti-cle feature of largest interest [19–23]. The focus of thissection is an analytical treatment of the dynamics andtherefore here we include the effect of the free membraneonly on the level of a line tension. Our reference shape isalways the spherical one due to the dominance of icosa-hedral viruses. Cylindrical particles may make contactto the cell membrane at any orientation. However, ashort (long) cylinder would position itself normal (par-allel) to the membrane to maximize the initial adheredcontact area. It is therefore reasonable to compare thesetwo configurations (orientation normal and parallel) tothe spherical (icosahedral) shape. We note that for sim-plicity here we neglect the top and bottom faces of thecylindrical shape and the bending energy of the kinkededges. In order to study the effect of the edges, we alsoconsider spherocylindrical particles in normal and par-allel orientation (Fig. 5). While our approach is wellsuited for all these shapes which obey axial symmetry,more complex shapes as for example cubes have to be (a) (b) (c)(d) (e)
FIG. 5. Shapes considered here: cylinders in (a) normal orien-tation (rocket mode) and (b) parallel orientation (submarinemode), (c) spherical particles, spherocylinders in (d) normalorientation and (e) parallel orientation. In a deterministicmodel, the particle/virus is homogeneously covered with lig-ands (blue) that can adhere to the cell membrane. The ad-hered area A ad is marked in red. investigated numerically [21].During uptake the particle adheres to the membranealong the adhesive area A ad . We describe the progress ofuptake by the uptake height z for the normal cylinder,see Fig. 5(a), or by the uptake angle θ for the parallelcylinder or sphere, see Fig. 5(b), (c). One can calculatethe thermodynamic uptake force by taking the variationof the energy E with respect to the uptake variable F up = − ∂E/∂x, (10)where x = z or x = θR . The uptake force is balancedby a friction force, F up = F friction . Here we assume thatthe dynamics of the uptake process is dominated by onelocal timescale. As a lower limit to all possible choices,we identify the microviscosity η of the membrane, whichis known to dominate uptake times for vesicles and whichhas a typical value η = 1 Pa s [13]. Hence F friction = η E ( x ) ˙ x. (11)Particle shape will enter our results through the variableedge length E ( x ). If one was interested in another rate-limiting dissipative process for uptake that was also dom-inated by one single local time scale acting at the inter-face between the adhered and free membrane parts, onecould simply rescale all our results to the desired scale.This however would not change the relative sequence ofuptake and the phase diagrams presented below. Solvingthe force balance equation for ˙ x , we obtain the dynamicequation for particle uptake. In the next paragraphs webriefly summarise the results for the two cylinder orienta-tions, the sphere and the two spherocylinder orientations. B. Cylinder with normal orientation ( ⊥ ) For a cylinder (radius R , length L ) oriented normallyto the membrane one has adhesive area A ⊥ ad = E ⊥ z , edgelength E ⊥ = 2 πR and mean curvature H ⊥ = 1 / (2 R ),hence E ⊥ = − W πRz + κ πzR + σ πRz + ζ πR. (12)We again note that the top and bottom surfaces of thecylinder would not contribute to the uptake force and areneglected here for simplicity. The differential equationfor the uptake then simply reads˙ z = − ∂E ⊥ /∂zη E ⊥ . (13)One sees that the line tension does not affect the uptakedynamics as the length of the edge does not change with z . Thus the uptake rate is a constant, ˙ z = ν ⊥ up , with ν ⊥ up = ν ⊥ w − ν ⊥ κ − ν ⊥ σ = W/η − κ/ (2 R η ) − σ/η. (14)In case ν ⊥ w overcomes the counteracting terms ν ⊥ κ + ν ⊥ σ from bending and tension, uptake progresses at constant V () (a) = 0.3= 0.8 0 0.2 0.4 0.6 0.8 1.005101520 N D u p t a k e t i m e L/R=1.0L/R=2.0L/R=5.0L/R=10.0 (b)
Cylinder Cylinder L / R (c) Cylinder Uptake No uptake0 0.5 1.0 1.50510 L / R T > TT < T (d) Cylinder Partial uptake
FIG. 6. Cylindrical uptake. (a) Shape of the uptake potential for parallel orientation of the cylinder. Full uptake is possiblefor reduced membrane tension α (cid:107) < / θ = π is a boundary minimum, green solid curve). For α (cid:107) > /
2, onlypartial uptake can take place (minimum for finite uptake angle θ < π , green dashed). (b) Non-dimensionalised uptake times ofparallel (green) and normal cylinder (blue) as a function of α (cid:107) at equal radius but different aspect ratios (i.e. different volume).For the normal cylinder the uptake time increases with aspect ratio whereas it stays constant for the parallel cylinder. (c)Dynamical state diagram of normal cylindrical uptake. Orange region: full uptake, green region: no uptake. (d) Dynamicalstate diagram of parallel cylindrical uptake. Orange region: full uptake, blue region: partial uptake. Above the dashed red linethe parallel cylinder and below the dashed red line the normal cylinder is taken up fastest. speed and the uptake time is given by T ⊥ det = L/ν ⊥ up .Otherwise, the particle does not get taken up at all( ν ⊥ up ≤ R ⊥ crit = (cid:112) κ/ (2( W − σ )) . The optimal radius (for whichuptake is fastest) is given by R ⊥∗ = √ R ⊥ crit . Below we will compare all particle shapes at equal vol-ume and equal radius because it is a natural questionto ask which shape performs best at a given volume oftransported cargo. Taking the sphere as the referenceshape, the normally oriented cylinder then has a length L = 4 R/ C. Cylinder with parallel orientation ( (cid:107) ) Completely analogously, one obtains the energy for theparallel cylinder, now as a function of the uptake angle(see Fig. 5(b)) E (cid:107) = − W θ LR + κ θLR + σ ( θ − sin θ )2 LR + ζ L . (15)Again the bottom and top faces of the cylinder are ne-glected for simplicity. Then, the dynamic equation foruptake is given by˙ θ = ν (cid:107) up − ν (cid:107) σ (1 − cos θ ) , (16)where ν (cid:107) up = ν (cid:107) w − ν (cid:107) κ = W/ ( Rη ) − κ/ (2 R η ) and ν (cid:107) σ = σ/ ( Rη ). Again, the line tension does not affect the up-take dynamics.It is insightful to non-dimensionalize the dynamicequation by introducing the characteristic time 1 /ν (cid:107) up to get d θ d τ = 1 − α (cid:107) (1 − cos θ ) . (17)We will assume ν (cid:107) w > ν (cid:107) κ (since otherwise there is no up-take anyways) and hence the reduced membrane tension α (cid:107) = ν (cid:107) σ /ν (cid:107) up = 2 σR W R − κ (18)is positive and uptake is expected to be the faster thesmaller α (cid:107) is. Eq. (17) can be integrated analyticallywith initial condition θ ( t = 0) = 0 to obtain θ ( t ). It is,however, simpler, to write it in potential form,d θ d τ = − dV ( θ ) dθ , V ( θ ) = − θ + α (cid:107) ( θ − sin θ ) , (19)highlighting the dynamic behaviour: first, the slope forsmall angles is always negative. For 0 < α (cid:107) ≤ / θ = π , hence complete uptake(although the uptake time diverges at α (cid:107) = 1 / α (cid:107) > / θ < π and hence one hasonly partial uptake, cf. Fig. 6(a). Integrating Eq. (17)leads to the uptake time T (cid:107) det ≈ πν (cid:107) up √ − α (cid:107) , (20)for α (cid:107) < / θ ( t → ∞ ) = 2 arctan (cid:18) √ α (cid:107) − (cid:19) , (21)for α (cid:107) > /
2. The critical radius is determined by α (cid:107) =1 /
2, yielding R (cid:107) crit = (cid:112) κ/ (2( W − σ )) . D. Comparison of cylinder with normal andparallel orientation
To compare the uptake times of the normally and par-allelly oriented cylinders, we can express both in the re-duced membrane tension α (cid:107) to get T ⊥ det Rη/σ = LR α (cid:107) − α (cid:107) , T (cid:107) det Rη/σ = π α (cid:107) √ − α (cid:107) , (22)where Rη/σ is again a characteristic time scale.Fig. 6(b) shows the rescaled uptake times for the nor-mal (blue) and parallel cylinder (green) as a functionof α (cid:107) at equal radius but different aspect ratios L/R (i.e. different volume). While for the parallel cylinderthe uptake time is a constant, for the normally orientedcylinder it naturally increases with length and hence theaspect ratio. Consequently, the normal cylinder is fasteronly as long as it is rather short, explicitly as long as LR ≤ π − α (cid:107) √ − α (cid:107) . (23)The optimal uptake orientation of a cylinder hence de-pends on the aspect ratio.Fig. 6(c) and (d) show the dynamical state diagramsfor normal and parallel cylinder uptake as a function ofaspect ratio and α (cid:107) . While normal cylinders are eithertaken up completely or not at all, parallel cylinders canalso be taken up partially. We note that the state dia-gram for the the parallel cylinder is in agreement withthe one which was previously computed by Mkrtchyanand coworkers [34]. In Fig. 6(d), we also investigate thespeed of uptake. Below the dashed red line the normalcylinder is taken up fastest, whereas above the dashedred line the parallel cylinder is taken up fastest. E. Sphere ( ◦ ) For a sphere the total energy reads E ◦ = (cid:0) − W πR + κ π + σπR (1 − cos θ ) (cid:1) × (1 − cos θ ) + ζ πR sin θ , (24)and hence the differential equation for uptake reads˙ θ ◦ = ν ◦ up − ν ◦ σ (1 − cos θ ) − ν ◦ ζ cot θ , (25)where we have introduced ν ◦ up = ν ◦ w − ν ◦ κ = W/ ( Rη ) − κ/ ( R η ), ν ◦ σ = σ/ ( Rη ) and ν ◦ ζ = ζ/ ( R η ).Neglecting line tension, the dynamic equation has thesame form as for the parallel cylinder, albeit with dif-ferent expressions for the rates. Introducing the reducedmembrane tension α ◦ = ν ◦ σ /ν ◦ up = σR W R − κ , (26) α N D u p t a k e t i m e λ/R = 0.3 λ/R = 1.0 Cylinder ⊥ Cylinder Sphere
FIG. 7. Non-dimensional uptake times for normal cylinder(blue), parallel cylinder (green) and sphere (red) for λ/R =0 . λ/R = 1 . α (cid:107) and for equal volumeat equal radius. one hence again has for the uptake time T ◦ det ≈ π/ ( ν ◦ up √ − α ◦ ), implying a critical radius R ◦ crit = (cid:112) κ/ ( W − σ ).To compare to the uptake times for the normal andparallel cylinder at equal volume and radius, we can againexpress the uptake time as a function of α (cid:107) by means of α ◦ = 1 α (cid:107) − λ R . (27)A comparison of the uptake times of the two cylinder ori-entations and the sphere is shown in Fig. 7. In case of theparticle being large compared to the characteristic lengthscale of the membrane ( λ/R ≤
1, tense membrane case),the parallel cylinder and the sphere have very similar up-take times, as also evident from Eq. (27). For instance,for λ/R ≤ .
1, the uptake time for the sphere almostcoincides with the green curve in Fig. 7. For smaller par-ticles or a looser membrane, the uptake times increas-ingly separate, and the sphere is increasingly disfavored,as seen by the red curves in Fig. 7. In general we thusfind that spheres are taken up slower than cylinders inthe deterministic description [25].
F. Spherocylinder
We now consider a spherocylindrical particle either innormal or parallel orientation with respect to the mem-brane. For the normal orientation the uptake time isgiven by the sum of the uptake times for a sphere andfor a normal cylindrical particle T ∩ det = T ◦ det + T ⊥ det . (28)We note that at equal volume the radius of a sphe-rocylinder ( R ∩ ) is always smaller compared to the ra-dius of a sphere ( R ◦ ). Let us first assume that we -2 -1 Spherical volume ( µ m )020406080100 U p t a k e t i m e ( m s ) R ∩ R ∩ > R ∩ R ∩ > R ∩ SphereNormalSpherocylinderParallelSpherocylinder
FIG. 8. Uptake time as a function of spherical volume forthe sphere (solid red) and spherocylinders with different radiiin normal (solid blue) and parallel (solid green) orientation.The dashed blue line marks the uptake times of normal sphe-rocylindrical particles at equal volume. Starting from the redcurve, the length of the cylindrical part increases while reduc-ing the particle radius. compare a sphere and spherocylinder, which both havesmaller radii compared to the optimal radius of thesphere at equal volume. In this case, it holds that T ∩ det ( R ∩ ) = T ◦ det ( R ∩ ) + T ⊥ det ( R ∩ ) > T ◦ det ( R ◦ ), since T ⊥ det ( R ∩ ) > T ◦ det ( R ◦ ) − T ◦ det ( R ∩ ) < L (cf. Fig. 5(d)) the uptake times of thespherocylinders and sphere are of course identical. Forincreasing volume, i.e. increasing cylindrical length wefind that the sphere is always taken up faster comparedto the normal spherocylinders. In the case when the ra-dius and length are changed at equal volume, one moveson vertical lines through the diagram (dashed blue line).By reducing the cylindrical length, the particle radiushas to increase, while keeping the volume fixed. Uponreducing the length one finally gets back onto the uptakecurve of a sphere (which is of course identical to a sphe-rocylinder with vanishing cylindrical length). Since thisargument can be repeated for all radii, the spherocylin-der is always taken up slower compared to the sphere atequal volume.For the spherocylindrical particle in parallel orienta-tion the uptake time is given by the slowest mode as thespherical part and the cylindrical part are taken up simul-taneously. As the uptake time for the parallel cylindricalparticle is always faster than the uptake time of a spher-ical particle at equal volume and radius the uptake timeof the spherocylindrical particle in parallel orientation is given by the uptake time of the spherical particle alone T ⊃ det = T ◦ det . (29)Hence, the uptake time of the spherocylindrical parti-cle in parallel orientation is constant while adapting thevolume by only increasing the length of the cylindricalpart. In Fig. 8 the uptake times for a spherocylindri-cal particles in parallel orientation with fixed but dif-ferent spherical radii are shown in solid green. Whilefor a sphere the relation between volume and radius isunambiguous, for a spherocylinder there exist differentcombinations of radii and lengths with the same volume.While the lowest green curve corresponds to a parallelspherocylinder with optimal spherical radius, the curvestarting at smaller (larger) volume corresponds to a par-allel spherocylinder with smaller (larger) radius. We seethat spherical particles present clear optima in terms ofuptake times at intermediate volumes, while spherocylin-ders are faster for large volumes, because there additionalvolume does not increase their uptake times when imple-mented by increased length.To conclude, at equal volume a normal spherocylinderis taken up slower compared to a sphere, independent of E t o t f r ee / ( E t o t f r ee + E t o t a d ) (a) λ/R = 0.1 λ/R = 0.3 λ/R = 1.0 λ/R = 3.0 λ/R = 10 π π π π θ E t o t f r ee / ( E t o t f r ee + E t o t a d ) (b) FIG. 9. (a) The bending and tension energy of the free mem-brane of a cylindrical particle in parallel orientation relative tothe total bending and tension energy, i.e. of the adhered andfree parts of the membrane for different values of λ/R . (b)Similar to (a) but now for the phenomenological descriptionof the free membrane parts. π π π π θ V ( θ ) (a) π π π π θ ˙ θ (b) α = 0 . β = 0 . α = 0 . β = 0 . α β (c) UptakePartial uptake
FIG. 10. Deterministic uptake dynamics of a cylinder in parallel orientation including the exact solution for the free membrane.(a) Uptake potential, leading to uptake (orange) and partial uptake (blue). Parameters are given in (b). (b) Phase portrait ( ˙ θ vs. θ ) for different parameter values corresponding to uptake (orange) and partial uptake (blue). (c) Dynamical state diagramof the final steady states as a function of β (cid:107) vs. α (cid:107) . Below the red curve the cylinder is taken up completely (orange region)and above we find only partial uptake (blue region). length. For a parallel spherocylinder the order of uptakedepends on the aspect ratio. For the same volume andaspect ratio parallel spherocylinders are taken up fastercompared to normal spherocylinders. Comparing the up-take times of a (normal or parallel) cylinder to a (normalor parallel) spherocylinder at equal volume and radius,we find that cylinders are always taken up faster. Thereason is that it is more time consuming to wrap thespherical caps compared to the cylindrical parts with thesame volume. IV. STATE DIAGRAMS FOR PARALLELCYLINDERA. Free membrane energies
We now explicitly consider the effect of the free mem-brane. We first note that for a cylindrical particle innormal orientation, the shape equations are similar tothe case of a spherical particle with θ = π/
2. However,the energy of the free membrane is a constant since itdoes not depend on the invagination depth z . Therefore,the free membrane will not contribute to the dynamicsof particle uptake. The first non-trivial case therefore isthe parallel cylinder. In the next section, we will thendiscuss the spherical particle.For the parallel cylinder the energy of the free mem-brane compared to the flat case can be written as [34] E tot (cid:107) free κ = 8 Lλ (cid:18) − cos θ (cid:19) . (30)Details can be found in appendix B. Considering the freemembrane on both cylinder sides, the scaling of the freeenergy is then given by E tot (cid:107) free ∼ ζLθ = E pheno (cid:107) free , which is the analogous of the phenomenological approach ofEq. (9) for spheres. The energy of the free membranerelative to the energy of both the free and adhered mem-brane parts are shown in Fig. 9(a) for different values of λ/R . In Fig. 9(b) the same is shown for the phenomeno-logical description. The agreement is very good and weconclude that the phenomenological approach works verywell in this case. B. Dynamics with free membrane
We now study the uptake dynamics for the parallelcylinder including the free membrane. The dynamicalequation in non-dimensional form readsd θ d τ = 1 − α (cid:107) (1 − cos θ ) − β (cid:107) sin θ/ , (31)with τ = tν (cid:107) up and the reduced line tension β (cid:107) = ν (cid:107) ζ ν (cid:107) up = 4 √ κσR W R − κ and ν (cid:107) ζ = 2 √ κσηR , (32)which incorporates the effect of the free membrane. Im-portantly, the uptake dynamics does not depend on thelength of the parallel cylinder.In potential form, the dynamics d θ d τ = − dV ( θ ) dθ followsfrom V ( θ ) = − θ + α (cid:107) ( θ − sin θ ) − β (cid:107) cos θ . (33)This potential is shown in Fig. 10(a). One either hasa boundary minimum corresponding to full uptake (or-ange) or a minimum in between corresponding to par-tial uptake (blue), depending on the parameter choice.0 π π π π θ V ( θ ) (a) α = 0 . β = 0 . α = 0 . β = 0 . π π π π θ ˙ θ (b) α β (c) Uptake Partial uptake
FIG. 11. Deterministic uptake dynamics of a cylinder in parallel orientation including the free membrane by the phenomeno-logical description. (a) Uptake potential for different parameter values corresponding to uptake (orange) and partial uptake(blue). (b) Phase portrait ( ˙ θ vs. θ ). Parameters are given in (a). (c) Dynamical state diagram of the final steady states as afunction of β (cid:107) vs. α (cid:107) . Below the red curve the cylinder is taken up completely (orange region) and above we find only partialuptake (blue region). From the shape of the potential and the phase portraitin Fig. 10(b) we can deduce the dynamical state diagramof uptake in the β (cid:107) - α (cid:107) -plane in Fig. 10(c) for a parallelcylinder. We find that the parallel cylinder either getstaken up completely (orange) or only partially (blue).The boundary between these two states is given whenEq. (31) becomes zero. As the second and third term inEq. (31) become minimal for θ = π , we find the boundarybetween the partial and full uptake state whend θ d τ (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:107) = π = 0 → β (cid:107) = 1 − α (cid:107) . (34) C. Dynamics with phenomenological approach
We now consider the phenomenological ansatz adaptedto the case of a parallel cylinder. In general, the en-ergy of the phenomenological approach can be writtenas E phenofree = 1 / ζθ E , where E is the length of the linethat connects the adhered membrane and the free mem-brane. For the parallel cylinder we have E (cid:107) = 2 L andthus E pheno (cid:107) free = ζLθ , where ζ = √ κσ . The dynamicsfor the phenomenological approach is then given byd θ pheno d τ = 1 − α (cid:107) (1 − cos θ ) − β (cid:107) θ . (35)In potential form, the dynamics now reads V pheno ( θ ) = − θ + α (cid:107) ( θ − sin θ ) + β (cid:107) θ . (36)The boundary between these two states is given whenEq. (35) becomes zero. As the second term in Eq. (35)becomes minimal for θ = π we findd θ pheno d τ (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:107) = π = 0 → β (cid:107) = 2 − α (cid:107) π . (37) The potential in Fig. 11(a), the uptake dynamics inFig. 11(b) and the state diagram of uptake in Fig. 11(c)for the dynamics in the phenomenological description canbe easily compared to the exact case of the free membranein Fig. 10. We see that the dynamics are very similar. V. STATE DIAGRAMS FOR SPHEREA. Line tension
We now study the uptake dynamics for the sphere byincluding the free membrane by a line tension or by aphenomenological ansatz. We first consider a line ten-sion. The dynamical equation in non-dimensional formreads d θ d τ = 1 − α (1 − cos θ ) − β cot θ , (38)with τ = tν ◦ up and the reduced line tension β = ν ◦ ζ /ν ◦ up = ζRW R − κ , (39)and the reduced surface tension α = α ◦ as before. Inpotential form, the dynamics reads d θ d τ = − dV ( θ ) dθ , nowwith V ( θ ) = − θ + α ( θ − sin θ ) + β ln(sin θ ) . (40)From the shape of the potential, visualized in Fig. 12(a),one can clearly see that the line tension creates diver-gences towards −∞ for both θ = 0 and θ = π . Thedivergence at small θ is well known from classical nucle-ation theory and implies that a fluctuation is required tostart the process. The divergence at large θ reflects the1 π π π π θ V ( θ ) (a) π π π π θ ˙ θ (b) α = 0 . β = 0 . α = 1 . β = 0 . α = 1 . β = 0 . α β (c) Uptake Partial uptakeNo uptake
FIG. 12. Deterministic uptake dynamics of spherical particles with line tension. (a) Uptake potential, leading to uptake(orange), partial uptake (blue) and no uptake (green). Parameters are given in (b). (b) Phase portrait ( ˙ θ vs. θ ) for differentparameter values corresponding to uptake (orange), partial uptake (blue) and no uptake (green). (c) Dynamical state diagramof the final steady states as a function of β vs. α . Below the red curve the sphere is taken up only partially (blue region) andabove we find either uptake (orange region) or no uptake (green region). fact that a line tension accelerates the process once theequator is passed.Since the potential varies between −∞ and −∞ for0 ≤ θ ≤ π , it must have at least one maximum, corre-sponding to an unstable steady state. In case this is theonly steady state, we now assume that the initial fluctu-ations bring the system over the initial barrier and theresult will be full uptake. However, as a function of re-duced membrane and line tension α and β , additional ex-trema in the potential and hence additional steady statescan emerge. We identify two scenarios. If the potentialdisplays two maxima separated by a minimum, then wehave another steady state that we interpret as partialuptake. If the potential displays a saddle for values of θ smaller then those for the maximum, then we concludethat no uptake is possible since ˙ θ < θ . Fig. 12 visualizes our three scenarios for the dy-namics both via the potential (a) and using the phaseportrait ˙ θ vs. θ (b).The steady states of Eq. (38) as a function of α and β can be studied analytically. In fact, their numbercan change only if the curvature of a given stationarypoint changes sign. One can reformulate the problemto find the particular value of β where the two func-tions f ( θ ss ) = 1 − α (1 − cos θ ss ) and g ( θ ss ) = β cot θ ss are equal, f ( θ ss ) = g ( θ ss ) (defining a steady state), andwhere their derivatives are equal, f (cid:48) ( θ ss ) = g (cid:48) ( θ ss ) (defin-ing the change in the sign of the curvature). The secondcondition implies that sin θ ss = β/α and insertion intothe first equation results in β ( α ) = α (cid:34) − (cid:18) α − (cid:19) (cid:35) . (41)Fig. 12(c) shows the dynamical state diagram for the up-take of a sphere in the β - α -plane. Eq. (41) is shown as the red curves. Below these curves one has partial up-take (blue region) and above we find either uptake (or-ange region) or no uptake (green region), according tothe definition discussed above. Note that the red curvetends to α = 1 / β →
0, as expected.Interestingly, a moderate reduced line tension β is pro-ductive as it increases the range of reduced membranetension α for which full uptake occurs. This effect how-ever saturates at β = 1. Stronger values of β are counter-productive as they transform partial uptake into no up-take. We note that for typical parameter values (using ζ = √ κσ , i.e. assuming that the line tension is a resultof the free membrane effects) one estimates α = 0 . β = 0 .
5, for which one would expect uptake.In general, since the uptake dynamics depends on fourparameters ( R , W , κ and σ ) and α and β are combi-nations of those, one would not move along horizontalor vertical lines in an experiment, where typically onlyone parameter is varied. However, since β/α = λ/R one could move on straight lines through the origin inthe phase diagram when changing W while keeping λ/R constant. B. Phenomenological description
We now discuss the case when the simple line tension,Eq. (7), is replaced by the phenomenological description,Eq. (9), distilled out from the shape equations. The ad-ditional factor of θ / θ d τ = 1 − α (1 − cos θ ) − βθ (cid:18) θ cot θ (cid:19) , (42)with the same reduced membrane tension α and reducedline tension β as before. Now the last term cannot be2 π π π π θ V ( θ ) (a) α = 0 . β = 0 . α = 0 . β = 0 . π π π π θ ˙ θ (b) α β (c) UptakePartial uptake
FIG. 13. Deterministic uptake dynamics of spherical particles with the phenomenological contribution of the free membrane.(a) Uptake potential, leading to uptake (orange) and partial uptake (blue). (b) Phase portrait ( ˙ θ vs. θ ) for different parametervalues corresponding to uptake (orange) and partial uptake (blue). Parameters as in (a). (c) Dynamical state diagram of thefinal uptake states as a function of β vs. α . Full uptake is achieved in the orange region and partial uptake in the blue region. integrated in closed analytical form as before. The po-tential can be, however, easily obtained numerically asdisplayed in Fig. 13(a). One clearly sees that the diver-gence at θ = 0, as occurring for the line tension, is nowabsent while the speed-up of uptake for large angle per-sists. Now only full uptake (orange) and partial uptake(blue) can be observed. However, we note that a trivialno uptake state occurs for insufficient adhesion energy.Fig. 13(b) shows the phase portrait and (c) the dynami-cal state diagram for the uptake in the β - α -plane. Inter-estingly, the latter displays a re-entrance phenomenon at α -values slightly above 1 /
2, meaning that upon increas-ing β the system can display partial uptake, full uptakeand again partial uptake. We note that the re-entrancephenomenon could be potentially observed when chang-ing R while keeping W , σ and κ constant.Let us now compare and discuss the cases of no linetension vs. line tension vs. the phenomenological treat-ment of the free membrane. We will take a dynamicalsystems point of view, which turns out to be especiallyinstructive. Fig. 14 shows the steady states as a functionof the reduced membrane tension α , for the three cases.As shown in Fig. 14(a), without line tension ( β = 0) thedynamics is quite simple: for α < / θ = π isthe only attractor. As soon as α > / θ ss < π , corresponding to partial uptake,decreasing further with increasing α . The arrows markthe flow of the system, towards the stable attractors.Starting from the first adhesion formed, correspondingto θ = 0, the system hence always evolves towards full orpartial uptake, depending on reduced membrane tension α .The reduced line tension, as visible in Fig. 14(b), hastwo main effects: first, θ = 0 becomes a steady state,due to the divergence of the potential. Second, another steady state emerges at intermediate angle, which has un-stable (dashed) parts, but also a stable (solid) region cor-responding to partial uptake. The bifurcation structureis the one of a pair of saddle-nodes. While one assumesthat the unstable branch for low reduced membrane ten-sion α can be overcome by the inherent fluctuations inreceptor-ligand bond formation, hence still leading to fulluptake, the unstable branch for large reduced membranetension α is at such a large angle that it should be inter-preted as no possible uptake. Finally, the stable branchin between the two saddle-nodes corresponds to the at-tractor of partial uptake.Using the phenomenological description of the freemembrane effects leads to the scenario shown inFig. 14(c). Here, the ambiguity of overcoming a bar-rier at small angle is absent, as θ = 0 is not a steadystate anymore and no other branch prevents the flow fromreaching θ = π . Only a single saddle-node emerges forlarger reduced membrane tension α , separating uptakefrom partial uptake. In a certain sense, the phenomeno-logical description is an intermediate case between havingno line tension and a simple line tension.Fig. 15 shows the dynamics and the dependence of thesteady states on the reduced line tension β , describingthe strength of the line tension (a) or free membraneeffects (b), (c), respectively. One can clearly see thatthe emergence of saddle-nodes (one of whose brancheshas to be stable) directly corresponds to the regions ofpartial uptake. Fig. 15(b) shows an example of the re-entrance phenomenon, i.e. where partial uptake emergestwice when varying the parameter β , cf. also Fig. 13(c).To conclude, both a line tension and the approximatedfree membrane effects considerably enrich the uptake dy-namics. They share similarities, for instance they accel-erate uptake as soon as the circumference of the mem-brane edge decreases, i.e. in the second half of uptake( θ > π/ α π π ππ θ ss (a) β = 0 α π π ππ θ ss (b) β = 0 . α π π ππ θ ss (c) β = 0 . FIG. 14. Steady states as a function of the reduced membrane tension α , in case of (a) no line tension ( β = 0), (b) for a linetension with β = 0 . β = 0 .
3. Stablesteady states are marked in solid, unstable ones as dashed. The arrows display the flow of the system. As usual, regions of fulluptake are marked in orange, partial uptake in blue and no uptake in green. Note that for (a) there is only one stable steadystate and the dynamics is simple. Line tension (b) and the free membrane effects (c) introduce new steady states undergoingsaddle-node bifurcations (see text). β π π ππ θ ss (a) α = 0 . β π π ππ θ ss (b) α = 0 . β π π ππ θ ss (c) α = 0 . FIG. 15. Steady states for line tension/free membrane effects as a function of reduced line tension β . Stable steady statesare marked in solid, unstable ones as dashed. The arrows display the flow of the system. As usual, regions of full uptake aremarked in orange and partial uptake in blue. (a) Case of line tension and α = 0 .
8. The partial uptake for small β is due tothe saddle-node, whose stable branch prevents the system from reaching full uptake. Increasing β (and assuming the unstablebranch can be overcome by fluctuations) induces full uptake. (b) Phenomenological treatment for the free membrane with α = 0 .
6. Here, the system displays partial uptake, full uptake and again partial uptake, i.e. a “re-entrance” of partial uptakeoccurs. (c) Same as (b) but for α = 0 . β . alone due to the interplay of adhesion energy and mem-brane tension that can generate a boundary minimum ofthe potential for spherical particles, here another partialuptake state can emerge. This partial uptake state iscaused by a non-boundary minimum of the potential dueto the saddle-node structures of the new steady statesthat is generated either by line tension or approximatedfree membrane effects. While the line tension can intro-duce no uptake states that are caused if a repellor atlarge angle values emerges, no uptake states only emergein the phenomenological description in the trivial caseof insufficient adhesion energy. The main difference be- tween them, however, is that the line tension introducesan energy barrier at small angles, that has to be over-come by fluctuations. Therefore a line tension is not anadequate description for the free membrane, which doesnot show this effect. We stress that this conclusion re-mains true beyond the approximation used in Eq. (9):Eq. (A4) leads to E totfree ∝ sin θ for small angles, display-ing the same behavior, i.e. no barrier.4 VI. STOCHASTIC DYNAMICSA. General approach
For the uptake of both nanoparticles or viruses, fluc-tuations are expected to be important since the parti-cles are small and typically only covered by few tensof ligands, rendering ligand-receptor binding a discretestochastic process. Based on the continuous determinis-tic modeling approach, we now explicitly model the dis-crete stochastic dynamics of receptor-ligand binding assketched schematically in Fig. 16. Our stochastic modelis defined from the deterministic one via two steps. First,the system is discretized by mapping the continuous de-terministic equation onto a discretized version. Secondly,this equation is interpreted as a stochastic rate equation.The considered stochastic process is designed to obtainthe deterministic result in the limit of small noise, i.e. inthe continuous deterministic limit. However, we do notintroduce temperature and our stochastic model is notdesigned to obtain the deterministic result in the limitof vanishing temperature. The underlying reason is thatwe do not want to make assumptions about the processesthat drive the membrane forward and backward. Mod-elling the growth of adhered membrane patches based onreceptor-ligand binding is a mature and challenging fieldby itself [35, 36] and might even include active processes[37]. Therefore we do not anchor our model in an equi-librium model, but derive our stochastic rates from thedeterministic theory without enforcing detailed balance,similar to stochastic processes used in population or evo-lutionary dynamics [38, 39].In detail, we first map the adhered membrane area ontothe number of bound ligands N , in order to deduce a dis-crete differential equation d N/ d t = d N/ d x · d x/ d t , where x = { z, θ } corresponds to the uptake variable (height z for the cylinder in normal orientation and invaginationangle θ for the parallel cylinder and the sphere) [25].From the deterministic framework, d x/ d t is known fromEq. (13), Eq. (16) and Eq. (25) for normal cylinders, par-allel cylinders and spheres, respectively. The next stepis to deduce the corresponding one-step Master equation(ME) [40] for the probability p N to have N ligands boundto receptors,d p N d τ = g N − p N − + r N +1 p N +1 − ( g N + r N ) p N . (43)Here g N is the forwards and r N the backwards rate bywhich ligands bind (unbind) from state N . While com-plete probability distributions can be calculated analyt-ically only for some special cases, the stochastic uptaketimes T sto for a reflecting boundary at the unwrappedstate and an absorbing boundary at the fully wrappedstate can be calculated analytically as [40] T sto = N max − (cid:88) ν =1 ν (cid:88) µ =1 r ν r ν − · · · r µ +1 g ν g ν − · · · g µ . (44) g N r N R FIG. 16. Modeling particle uptake as a discrete stochasticprocess. The particle (blue) is covered with ligands (smalldiscs) that stochastically bind to cell surface receptors (half-circles) with rate g N , leading to an advancement of the ad-hered membrane area, or unbind with rate r N . Note thataxial symmetry is assumed. Alternatively, one can use the Gillespie algorithm [41]to simulate both the probability distributions and theuptake times.
B. Cylinder with normal orientation ( ⊥ ) For a cylinder oriented perpendicularly to the mem-brane, the membrane covered area A ⊥ ad = A ( z ) is mappedonto the number of bound receptors N ⊥ ( z ) by using A ( z ) /A ⊥ max = ( N ⊥ ( z ) − / ( N max − . (45)Here we assumed that initially, the particle is alreadybound to the membrane by one ligand, yielding N ⊥ ( z ) =( N max − z/L + 1. The corresponding discrete equationthen readsd N ⊥ d t = N max − L (cid:0) ν ⊥ w − ν ⊥ κ − ν ⊥ σ (cid:1) , (46)and the corresponding rates of the ME are hence easilydeduced by g N = ( N max − ν ⊥ w /L and r N = ( N max − ν ⊥ κ + ν ⊥ σ ) /L . Finally, to implement a reflecting bound-ary condition at N = 1, we put r =0. C. Cylinder with parallel orientation ( (cid:107) ) In this case we proceed similarly and map the mem-brane covered area A (cid:107) ad = A ( θ ) onto the number of boundreceptors N (cid:107) ( θ ) via A ( θ ) /A (cid:107) max = ( N (cid:107) ( θ ) − / ( N max − N (cid:107) ( θ ) = ( N max − θ/π + 1. The corresponding discrete equation readsd N (cid:107) d t = N max − π (cid:16) ν (cid:107) w − ν (cid:107) κ − ν (cid:107) σ (1 − cos θ ) (cid:17) , (47)5and hence the corresponding rates of the ME amount to g N = ( N max − ν (cid:107) w /π and r N = ( N max − ν (cid:107) κ + ν (cid:107) σ (1 − cos θ )) /π . Finally, to implement a reflecting boundarycondition at N = 1, we again put r =0. D. Sphere ( ◦ ) Finally, for a spherical particle we map A ◦ ad = A ( z )onto N ◦ ( θ ) by A ( θ ) /A ◦ max = ( N ◦ ( θ ) − / ( N max − , (48)and one initially bound ligand implies N ◦ = ( N max − − cos( θ )) / N ◦ d t = (cid:0) ν ◦ w − ν ◦ κ − ν ◦ σ (1 − cos θ ) − ν ◦ ζ cot θ (cid:1) N E ( N ) . (49)Here N E ( N ) = (cid:112) ( N − N max − − ( N − g N = ν ◦ w N E ( N )and r N = ( ν ◦ κ + ν ◦ σ (1 − cos θ ) + ν ◦ ζ cot θ ) N E ( N ) for θ < π/ g N = ( ν ◦ w − ν ◦ ζ cot θ ) N E ( N ) and r N =( ν ◦ κ + ν ◦ σ (1 − cos θ )) N E ( N ) for θ ≥ π/ r N and g N is replacedby ν ◦ ζ θ (1 + θ/ θ ) in the intervals θ < .
289 and θ ≥ . g , which otherwisewould be zero. We here choose g = ν ◦ w √ N max , since (i)it should be proportional to ν w and (ii) the transitiontime from state N = 1 to state N = 2 should vanishfor large N max . Since r = 0, N = 1 is a reflectingpure boundary, i.e. a particle always stays attached tothe membrane. E. Stochastic uptake times
We numerically solve the MEs corresponding to thedifferent particle shapes and orientations by means ofthe Gillespie algorithm [41]. The used parameter valuesare summarized in Table I. For the number of ligands we
TABLE I. Simulation parameters.Parameter Used value Ref.Bending rigidity κ = 25 k B T [28]Membrane tension σ = 1 · − N / m [26]Energy density W = 0 .
04 mJ / m [13]Membrane microviscosity η = 1 Pa s [13]Line tension ζ = √ κσ Receptor-ligand pairs N max = 20 U p t a k e t i m e ( m s ) Normal cylinderParallel cylinderSphereSphere phenoSphere line Radius (nm)020406080100 U p t a k e t i m e ( m s ) (a)(b) FIG. 17. (a) Simulated mean uptake times (symbols) fora normally oriented cylinder (blue triangles), a parallelly ori-ented cylinder (green lozenges), a sphere (red circles), a sphereincluding the phenomenological approximation for the freemembrane (orange pentagons) and a sphere with line tension(cyan stars) as function of radius at equal volume and radiusfor the parameter values in Table I. In addition, the numer-ically computed mean uptake times calculated by Eq. (44)are shown as solid lines for the different shapes in the cor-responding colors. (b) For the normal cylinder, the parallelcylinder and the sphere the deterministic uptake times areshown as dashed lines in corresponding colors (for vanishingline tension/free membrane) and compared to the numericallycalculated mean uptake times (solid lines). chose a typical value of N max = 20. Stochastic effectsfurther increase upon decreasing N max , and they prevailfor N max of the order of one hundred [25], depending onparameters.In order to obtain the stochastic uptake times,we introduce an absorbing boundary at full coverage.Fig. 17(a) shows the uptake time as a function of theradius of the particle. The results of our simulations(symbols), averaged over 10 trajectories each, perfectlyagree with the results of the numerical calculations bymeans of Eq. (44) (solid lines), which verifies our simula-tions. To compare cylinders to spheres, their radius werefixed to the one of the respective sphere and the length L trajectories each) for the nor-mally oriented cylinder (blue triangles), the parallelly ori-ented cylinder (green lozenges), the sphere (red circles),the sphere including the phenomenological treatment ofthe free membrane, according to Eq. (8) and Eq. (9) (or-ange pentagons) and the sphere including a line tension(cyan stars), according to Eq. (7) and Eq. (8).In Fig. 17(b) the mean uptake times obtained by nu-merical calculations for the different particle shapes arecompared to the deterministic results (without line ten-sion or free membrane effects), which are shown as thedashed curves in the corresponding colors. We see thatfor all considered particle shapes both the determinis-tic and the stochastic dynamics show similar behavior.First, a critical radius exists below which uptake is notpossible anymore (cf. the analytical results in section IIIB,C,D) [11]. Second, for larger radii the uptake timeincreases with increasing particle size. And third, in be-tween an optimal radius exists, having minimal uptaketime [7, 20, 42, 43]. The underlying reason for this be-havior is that the bending energy is independent of parti-cle size, while the energy contributions for adhesion andtension increase with size [11].From Fig. 17 we further see that stochastic uptakeis always faster than deterministic uptake. In addition,stochasticity extends uptake towards regions beyond thecritical radius of deterministic uptake. This is due tofluctuations being able to drive particles above energybarriers during uptake. Another interesting observa-tion is that in the deterministic description, cylindersin the parallel orientation are taken up faster than (orat least equally fast as) spherical particles for all radii.In contrast, the stochastic description states that paral-lelly oriented cylinders are only faster than spheres be-low a certain radius: for the given parameters, at around R = 150 nm, the situation reverses and spheres are takenup faster [25]. In addition, we note that the typical timescale for uptake is between a few ms to a few tens of ms,which is in agreement with previous simulation results[20].We now discuss the effects of the line tension and thefree membrane. As can be judged from Fig. 17(a) whencomparing red circles, orange pentagons and cyan stars,when a line tension is used to model the free membraneparts [11], suggesting the scaling ζ = √ κσ [12], or whenincorporating the phenomenological approximation givenby Eq. (8) and Eq. (9), the effects are relatively modestin regard to the uptake times. One nevertheless can seethat the line tension slightly hinders uptake, while thephenomenological treatment slows down the uptake ofsmall particles even more. For increasing particle sizesthe effect vanishes as ν ζ ∼ /R . Furthermore, as theeffect of the free membrane only slightly affects uptaketimes and contributes up to 20% to the total deformationenergy of the membrane, neglecting the free membrane asdone in Ref. [25] is justified in regard to uptake times and for typical parameters for nanoparticle and virus uptake.However, a line tension can also originate from a local-ization of lipids or curvature generating proteins at theborder between adhered and free membrane [29]. Then ζ can possibly be larger and hence the effect of the slowingdown of the uptake would be more pronounced. F. Stochastic state diagrams
We finally make contact between our stochastic modeland the state diagrams for the deterministic dynamicspresented above. Fig. 18 shows the stochastic uptakedynamics of spherical particles with line tension with N max = 20 ligand-receptor pairs. As we here simulatedthe non-dimensionalized Eq. (38) we use the followingrates g N = N E and r N = ( α (1 − cos θ )+ β cot θ ) N E for θ <π/ g N = (1 − β cot θ ) N E and r N = α (1 − cos θ ) N E for θ ≥ π/
2. In addition, we now implement reflect-ing boundary conditions both for θ = 0 by g = √ N max , r = 0, and θ = π by g max = 0, r max = α (1 − cos θ ) √ N max to study the occupation probabilities of different statesfor long times.Fig. 18(a) shows an example of an uptake trajectoryfor N = 10 time steps. The parameters are chosensuch that uptake is expected. From uptake trajectoriesof N = 10 time steps the occupation probabilities ofthe different states are computed (b)-(d). To classifythe uptake state we calculate the state of the largest oc-cupation probability which is shown by the red verticalline. We have uptake when the N = 20 state has thelargest probability (b). Similarly, we find no uptake ifthe N = 1 state has the largest occupation probability(c). Highest probability for any other state correspondsto partial uptake (d). Using this classification we calcu-lated the dynamical state diagram of stochastic uptakeof trajectories of N = 10 time steps in the β - α -plane(e). The white lines correspond to the state boundariesof Fig. 12(c) and we see surprisingly similar behaviour.Compared to Fig. 12(c), we find that the parameter re-gion where uptake is possible is slightly extended to thepartial uptake region. In addition, we confirmed that,as assumed above, fluctuations allow the system to crossthe initial barrier caused by the line tension as discussedin section V. In order to determine the maximum anglewhich can be overcome by fluctuations we used the factthat uptake is possible in Fig. 18(a) as long as α ≤ α = 1 the root of Eq. (38) is given by π/ β .Finally, we study stochastic uptake where we includethe free membrane effects by our phenomenological de-scription. We now analyze the non-dimensionalizedEq. (42). In this case the forward and backward ratechange to g N = N E , r N = ( α (1 − cos θ ) + βθ (1 + θ/ θ )) N E for θ < .
289 and g N = (1 − βθ (1 + θ/ θ )) N E , r N = α (1 − cos θ ) N E for θ ≥ . (a) (e) α =0.3 β =0.3 α β N Normalized time N (b) O cc u p a t i o n p r o b a b ili t y N (d) α =1.7 β =0.3 O cc u p a t i o n p r o b a b ili t y N (a),(b) (d)(c) N α =1.7 β =1.7 O cc u p a t i o n p r o b a b ili t y (c) α =0.3 β =0.3 FIG. 18. Stochastic uptake dynamics of spherical particles with line tension. (a) Uptake trajectory of a spherical particlewith α = 0 . β = 0 . N = 10 time steps. (b)-(d) Occupation probability during uptake with N max = 20 receptorsfor different parameter combinations for an uptake trajectory of N = 10 time steps. The receptor with largest occupationprobability is shown in red. (b) Uptake. (c) No uptake. (d) Partial uptake. (e) Dynamical state diagram of stochastic uptakeshowing the maximum occupation probability of a trajectory of N = 10 time steps. The white lines correspond to the stateboundaries of the deterministic model, cf. Fig. 12(c). trajectories of N = 10 time steps in the β - α -plane usingthe same classification procedure as before. Comparedto Fig. 13(c) we find that now uptake also extends be-yond the state boundaries of the deterministic calculation(white line), because fluctuations drive the system abovethe intermediate barrier, corresponding to a partial up-take state. Without fluctuations the partial uptake statecould not be overcome in this parameter region. In ad-dition, we find that similar to Fig. 13(c) either uptake orpartial uptake occurs. VII. CONCLUSION
Here we have studied the uptake dynamics ofnanometer-sized particles such as viruses at cell mem-branes mediated by ligand-receptor binding. Thefocus was on a simple but complete deterministicmodel, amenable to analytical insight, complemented bystochastic simulations.By considering the adhesion, the bending and the ten-sion energies of the cell membrane, including the con-tributions of the free, i.e. non-adhered membrane partsand calculating minimal energy membrane shapes numer-ically, we found that in the parameter regime which istypically relevant for biological systems, the free mem-brane contributes up to 20% to the total energy duringuptake. It thus cannot be neglected a priori . We henceincorporated the free membrane effects within a simpledynamical model for uptake by either a line tension or byan effective phenomenological description. This allowedus to study the deterministic uptake dynamics for spher-ical, cylindrical and spherocylindrical particles (both ori- α β N FIG. 19. Dynamical state diagram of stochastic uptakeof spherical particles using the phenomenological descriptionof free membrane effects showing the maximum occupationprobability of a trajectory of N = 10 time steps. The whiteline corresponds to the state boundary of the deterministicmodel, cf. Fig. 13(c). ented either perpendicular or parallel to the membrane).Similar to [20, 21], where the uptake of spherocylindri-cal particles was studied, we find that the aspect ratio ofcylindrical particles dictates the uptake pathway. Whileshort cylinders are taken up fastest in normal mode, longcylinders are taken up fastest in parallel mode. For longcylinders at large reduced membrane tension one couldspeculate that they are taken up initially in parallel modebut might reorient driven by fluctuations to the perpen-8dicular position in order to complete the uptake process[20]. Calculating the uptake times, spherical particleswere found to be always taken up slower than cylinders.For spherocylindrical particles we found that they are al-ways taken up fastest in parallel mode at equal volumeand aspect ratio. When comparing them to spheres wefound that while normal spherocylinders are taken upslower, for parallel spherocylinder the result depends onaspect ratio at equal volume. Our results regarding cylin-ders might change if one would include detailed modelsfor adhesion and membrane shapes at the bottom andtop faces. Here these effects are neglected to keep themodel transparent.For spherical particles, the free membrane effects in-fluence the dynamics. In accordance with earlier work[12, 15, 19, 21, 44], we could identify in the analyticalmodel three scenarios for both the line tension and thephenomenological description of the free membrane: fulluptake, partial uptake and no uptake, dictated by mem-brane elasticity, adhesion energy and the free membrane.There are, however, differences: the line tension inducesan energy barrier (when considering the total energy) forsmall uptake angles for all parameters, hence uptake isonly possible if assisted by fluctuations. In contrast, forthe phenomenological description, the existence of thisbarrier depends on the parameters α and β . This is simi-lar to the earlier work by Deserno and Gelbart [44], wherethe shape of the free membrane shape is represented bya torus, that prevents uptake depending on particle sizeand the elasticity of the membrane. Considering the up-take speed, both the line tension and the phenomenolog-ical description speed up the process when passing overthe equator. Overall we conclude that a line tension is notthe best approach to describe the effects of the free mem-brane in the intermediate regime between tense and loosemembranes. The phenomenological approach suggestedhere is clearly a better approximation. We presented acomplete analysis of this case, focusing on steady statesand dynamical state diagrams as a function of a reducedmembrane tension ( α ) and a reduced line tension ( β ).The occurrence of parameter regions of partial/no up-take could be traced back to new steady states emergingvia saddle-node bifurcations.Finally, we included stochasticity into the model asreceptor-ligand binding is a discrete process in a smallsystem such that fluctuations are expected to be impor-tant. This was achieved by mapping the deterministicmodels onto one-step master equations. In a first step,we used these to simulate and calculate uptake times. Wecould show that the effect of spheres profiting from noiseand getting taken up faster than parallel cylinders, as de-scribed recently [25], survives when free membrane effectsare included. In a second step, we calculated stochasticstate diagrams and again found surprisingly good agree-ment with the deterministic results. In both cases, itbecame clear that stochasticity enlarges the parameterregion in which uptake is possible. Thus, fluctuations areexpected to help the system to transverse barriers, cor- responding to no uptake or partial uptake states, whichcould not be overcome in a deterministic model. In thefuture, our model could be extended by more detailedassumptions regarding the way the membrane moves for-ward and backward at the contact line. Such a descrip-tion then could also describe the effect of temperature,which is not treated here, and of active membrane fluc-tuations, which would break detailed balance again.Our uptake times are a lower bound to experimentallymeasured uptake times because we only consider the dis-sipative forces resulting from membrane microviscosity.Other mechanisms that might be rate-limiting includereceptor diffusion within the plasma membrane, yieldinguptake times in the few seconds range [7], the assem-bly of clathrin lattices, with typical uptake times of theorder of 60 seconds [45], dissection of the cytoskeletonbeneath the plasma membrane [46], and scission, with atimescale of minutes [47]. However, there is an increas-ing amount of experimental results showing that uptaketimes can indeed occur on the timescale of tens of mil-liseconds, including 250 ms for human enterovirus 71 [48],around 80 ms for gold nanoparticles (20 nm) [49], below40 ms for the uptake of micrometer-sized latex beads byGUVs [50] and even below 20 ms for silicon nanopar-ticles of (4 nm) diameter [51]. More importantly, how-ever, is that our theory predicts very interesting effectsregarding relative uptake times for different shapes. Ourpredictions in regard to the relative sequence of uptakeand the phase diagrams do not depend on the absolutescale of the uptake time. They would only change if thedissipative uptake dynamics were not dominated by onesingle local time scale, for example if the advancement ofthe adhesion front was strongly limited by receptor diffu-sion [7, 52]. Finally our work suggests that both duringbiological evolution and for particle design in materialsscience, stochasticity might play an important role foroptimal performance, which here is identified with fastuptake. ACKNOWLEDGMENTS
F.F. acknowledges support by the Heidelberg Gradu-ate School for Fundamental Physics (HGSFP). We ac-knowledge the members of the Collaborative ResearchCentre 1129 for stimulating discussions on viruses. U.S.S.acknowledges support as a member of the Interdis-ciplinary Center for Scientific Computing (IWR) andthe clusters of excellence CellNetworks, Structures and3DMM2O.
Appendix A: Details on solving the shape equations
To numerically solve the boundary value problem givenby Eq. (4), we rewrite it as a system of four first orderordinary differential equations. Three boundary condi-tions in Eq. (5) are given for s → ∞ . We hence use9the asymptotic solution to shift the boundary conditionsfor the numerical problem to a finite arc length s max .For weak membrane deformations ( ψ (cid:28)
1) the linearizedshape equations are solved by [26] r ( s ) = s , ψ ( s ) = βK ( s/λ ) , z ( s ) = βλK ( s/λ ) , (A1)where β = θ/K ( R/λ sin θ ) is a parameter and K n are the modified Bessel functions of the second kind.Then the numerical boundary conditions using matchedasymptotics read r (0) = R sin( θ ) ,ψ (0) = θ,ψ ( s max ) = βK ( s max /λ ) , ˙ ψ ( s max ) = − β λ ( K ( s max /λ ) + K ( s max /λ )) ,z ( s max ) = βλK ( s max /λ ) . (A2)The matching point s max is then varied such that thecomputed solution fulfills ψ ( s max ) (cid:28) R (cid:29) λ ) one has E tenseForet κ = 4 π (cid:26) Rλ (cid:112) x (1 − x )(1 − √ − x ) − x − (cid:18) √ − x (cid:19) − − √ − x ) (cid:27) , (A3)and for a loose membrane ( R (cid:28) λ ) E looseForet κ = 4 π (cid:18) Rλ x (1 − x ) (cid:19) ×× (cid:26) − γ + x − x ) − ln (cid:18) Rλ (cid:112) x (1 − x )(1 − x ) (cid:19)(cid:27) , (A4)where x = (1 − cos θ ) / γ ≈ . Appendix B: Shape equations and energy of the freemembrane for cylindrical particles with parallelorientation
For the parallel spherical particle the energy of the freemembrane compared to the flat case can be written as E totfree κ = (cid:90) ∞ (cid:32) ˙ ψ − cos ψλ (cid:33) L d s . (B1)where we use a similar parametrisation and the samegeometric relations as for the sphere˙ r − cos ψ = 0 , ˙ z + sin ψ = 0 . (B2)We construct a Lagrangian L = ˙ ψ L Lλ (1 − cos ψ ) , (B3)and find the Euler-Lagrange equations to equal L ¨ ψ − Lλ sin ψ = 0 (B4)From Eq. (B3) we construct a Hamiltonian by H = ˙ r∂ ˙ r L + ˙ ψ∂ ˙ ψ L − L = ˙ ψ L − Lλ (1 − cos ψ ) . (B5)Since L does not depend on s , H is conserved. Sincethe membrane is asymptotically flat H ( s → ∞ ) = 0 andthus H = 0 [30]. Then,˙ ψ = ± (cid:114) λ (1 − cos ψ ) . (B6)Using Eq. (B6) in Eq. 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