Stochastic Dynamics of Nanoparticle and Virus Uptake
SStochastic Dynamics of Nanoparticle and Virus Uptake
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: May 6, 2019)The cellular uptake of nanoparticles or viruses requires that the gain of adhesion energy overcomesthe cost of plasma membrane bending. It is well known that this leads to a minimal particle size foruptake. Using a simple deterministic theory for this process, we first show that, for the same radiusand volume, cylindrical particles should be taken up faster than spherical particles, both for normaland parallel orientations. We then address stochastic effects, which are expected to be relevant dueto small system size, and show that, now, spherical particles can have a faster uptake, because themean first passage time profits from the multiplicative noise induced by the spherical geometry. Weconclude that stochastic effects are strongly geometry dependent and may favor spherical shapesduring adhesion-driven particle uptake.
Biological cells constantly communicate with their en-vironment by relaying signals and material across theirplasma membranes. In particular, cells routinely take upsmall particles with sizes in the order of 10 −
300 nm. Thisprocess is usually driven by receptor-ligand interactions[1], such that the adhesion energy can overcome the en-ergy required for membrane bending. It is also exploitedby viruses that have to enter host cells for replication [2].Due to the physical nature of this process, cells also takeup artificial nanoparticles of various shapes [3], whoseuptake may be either intentional or undesired, as in drugdelivery [4] or in case of microplastics [5], respectively.Viruses come in many different shapes [6], but themost paradigmatic and most frequently occuring ones arespherical and cylindrical [7, 8]. Therefore here we focusour discussion on spheres, cylinders and a combination ofboth, as shown in Fig. 1. It is generally believed that thespherical shape is optimal because it maximizes the vol-ume to surface ratio and confers high mechanical stability[9]. However, it is less clear which shapes are optimal inregard to interactions with the membrane [3]. Here weshow that stochasticity might play an important role inthis context.Due to their small size, viruses are typically coveredwith only few tens of ligands for cell surface receptors[10, 11] and thus stochastic effects are expected to berelevant. For example, the icosahedral and medium-sized(60 −
100 nm) members of the family of reoviruses haveonly 12 primary JAM-A- σ (a) (b) (c)(e)(d) FIG. 1. Uptake of a particle of (a) spherical, (b) normalcylindrical ( rocket mode ), (c) spherocylindrical and (d) par-allel cylindrical shape ( submarine mode ). In a deterministicmodel, the virus is homogeneously covered with ligands, ad-hering to the cell membrane along the adhesive area A ad . (e)Stochastic uptake of a virus, for which the surface presentsdiscrete ligands. The virus particle binds (unbinds) with for-ward rate g N (backward rate r N ) to receptors on the cellmembrane. is that in contrast to deterministic dynamics, stochasticdynamics tends to favor spherical shapes for uptake.Deterministic approaches usually start by balancingthe contributions from adhesion and bending. Becauseadhesion energy scales with the radius squared, whilebending energy is independent of radius, a minimal sizeexists for particle uptake [17]. The most investigatedaspect of receptor-mediated uptake is the role of par-ticle shape [16, 18, 19]. More detailed approaches in-clude the variational problem for finding minimal energyshapes [20, 21], the role of receptor diffusion towards thenanoparticle [22, 23], particle deformability [24, 25], thephysics of the scission step [26] and the role of the cy- a r X i v : . [ q - b i o . S C ] M a y σ (N/m) − − −
10 100 1000 − − W ( m J / m ² ) R (nm) T d e t ( s ) R ∗ crit R No uptakeNo uptake − −
10 100 R (nm)(a) T d e t ( s ) Normal cylinderParallel cylinderSphereSpherocylinder (b) (c)
FIG. 2. Deterministic uptake dynamics. (a) Uptake times for sphere (red), normal cylinder (blue), spherocylinder (dashed blue)and parallel cylinder (green) as function of radius R at equal particle volume. (b), (c) Uptake times for a sphere as a function ofadhesion energy W and radius R or membrane tension σ , respectively. In (b) the critical (optimal) radius for spherical uptakeis shown in blue (green). Parameter values (if not varied) are R = 90 nm, W = 4 . · − mJ / m and σ = 0 . · − N / m. toskeleton [11]. In order to develop a stochastic descrip-tion, here we start with a minimal deterministic model,which we then extended to the stochastic case.We assume that ligands to the cellular adhesive recep-tors are homogeneously distributed on the viral surface.The total free energy can be written as [27] E total = − (cid:90) A ad W d A + (cid:90) A mem κH d A + σ ∆ A (1)where W is the adhesion energy per area, κ is the bendingrigidity, H is the mean curvature, σ is the membranetension and ∆ A the excess area due to the deformation(compared to the flat case). Typical parameter valuesare W = 10 − mJ / m , κ = 25 k B T and σ = 10 − N / m[28, 29]. For example, a reovirus has 12 major and 60minor ligands [12, 30]. With a binding energy of around15 k B T each, one estimates W = 4 . · − mJ / m for ahomogeneous adhesion model.The two membrane parameters define a typical lengthscale λ = (cid:112) κ/σ ≈
32 nm. As shown schematically inFig. 1, we consider a spherical or cylindrical particle ofradius R adhering to the membrane. Then the bend-ing energy in Eq. (1) has contributions both from theadhering ( A ad ) and free part ( A mem − A ad ) of the mem-brane. It has been previously shown [18, 28, 29, 31] thatthe contributions from the free part can be neglected intwo special regimes. In the loose regime ( R (cid:28) λ ), thefree membrane can adopt the shape of a minimal surfaceand thus its bending contribution vanishes. In the tenseregime ( R (cid:29) λ ), the free part is essentially flat and thecontribution from the adhered membrane is much largerthan the one from the free membrane. To arrive at ananalytical model, here we neglect the contributions fromthe free membrane also for the intermediate case.Eq. (1) can now easily be analyzed for the geometriessketched in Fig. 1, namely (a) for a sphere ( ◦ ), (b) for acylinder oriented normally to the membrane ( ⊥ ), (c) fora spherocylinder ( ∩ ) and (d) for a cylinder oriented par- allel to the membrane ( (cid:107) ). Although it has been shownin coarse-grained molecular dynamics simulations thatspherocylinders might undergo a dynamical sequence offirst lying down and then standing up [16], the two cylin-drical modes considered here are the paradigmatic con-figurations encountered during wrapping [18]. In case ofthe normal cylinder, the top and bottom surfaces are ne-glected as they do not contribute to the uptake force. Tokeep our calculations as transparent as possible, we ne-glect them also for the parallel cylinder. In both cases,we neglect the bending energies at the kinked edges.The uptake forces F up are calculated by taking thevariation of the energy with respect to opening angle θ or height z , respectively, and are balanced by the fric-tion force they experience, F up = F friction [21]. For aspherical particle, the membrane covered area is a spher-ical cap of radius R and opening angle θ , i.e. A ◦ ad =2 πR (1 − cos θ ), cf. Fig. 1(a). The friction force is F ◦ friction = η πR sin( θ ) R ˙ θ , i.e. an effective membrane mi-croviscosity of order η = 1 Pa s times the change of themembrane-covered particle surface. The resulting equa-tion of motion reads˙ θ = ν up − ν σ (1 − cos θ ) (2)with ν w = W/ ( Rη ), ν κ = 2 κ/ ( R η ), ν σ = σ/ ( Rη ) and ν up = ν w − ν κ . The uptake time can be calculated as [32] T ◦ det ≈ πν up (cid:113) − ν σ ν up . (3)Note that it diverges for ν up = 2 ν σ , defining a crit-ical radius R crit , below which uptake is not possible.In the limit of vanishing σ , we recover the classical re-sult R crit = (cid:112) κ/W ≈
44 nm [17]. We note that ourtheory predicts normal uptake forces of around ten pN,which agrees well with results from recent atomic forcemicroscopy experiments [33, 34].Analogous calculations can be performed for the threecases with cylinders at equal volumes [32]. Fig. 2(a) com- N u p ( t ) N (t)480 80400 t (ms) (a) t (s) TrajectoryMean bound receptorsDet. uptake timeSim. mean uptake time R crit ( s ) s i m T (b) W ( m J / m ²)
10 100 R (nm) Slow uptake (cid:0) ¹10 (cid:0) ² 10 (cid:0) ³10 (cid:0) ²10 (cid:0)✁ (c) ✂✄ ✄ ✄ Slow uptake ( s ) s i m T ☎ ✆ ¹10 ✆ ² (N/m) ✝ (cid:0)✞ (cid:0)✁ (cid:0) ³ FIG. 3. Stochastic uptake dynamics for a spherical particle (reovirus with N max = 12). (a) Histogram of uptake times withobtained mean T ◦ sim ≈
16 ms (green line; standard deviation is ≈
11 ms) as compared to the deterministic value T ◦ det ≈
96 ms(red line). (Inset) Two example trajectories (black) of the number of bound receptors as a function of time and the meannumber (cid:104) N sim (cid:105) obtained from simulating the master equation (blue). (b), (c) Mean uptake time T ◦ sim as a function of W andeither R or σ . In the dark colored region, uptake may still occur beyond the used simulation time. In (b) the blue line is thecritical radius R crit of the deterministic model. Parameter values as in Fig. 2. pares the resulting uptake times. For the normal and par-allel cylinders, we take the same radius as for the sphereand adjust the aspect ratio. For the spherocylinder, wetake the same aspect ratio as for the cylinders and adjustthe radius. All four geometries show similar behaviors:a critical radius R crit exists, below which uptake is notpossible. The parallel cylinder has half the critical radiusof the sphere because its mean curvature is half as largeat equal radius. Moreover an optimal radius R ∗ exists, atwhich the uptake time is minimal [32]. Interestingly, thecritical and optimal values are very close to each other,and the cylindrical particles are taken up faster than thespherical ones. The spherocylinder is the slowest case,because at equal volumes, the aspect ratio is modest andthe spherical part dominates. Fig. 2(b) and (c) displaythe uptake time for a spherical particle as a function of W and R or σ , respectively, showing that a smaller adhe-sion energy has to be compensated by either larger radiusor smaller membrane tension. Importantly, in the deter-ministic case certain parameter values do not lead to anyuptake.We now turn to the stochastic variant of our uptakemodel (cf. Fig. 1(e)). For the sphere, we map the mem-brane covered area onto the number of bound receptors N [32], assuming axial symmetry. Using Eq. (2), we find adynamical equation for N through ˙ N = (d N/ d θ ) ˙ θ . Fromthis discrete equation a one-step master equation [35] canthen be deduced, with the forward rate g N = ν w N E andthe backward rate r N = ν κ N E + 2 ν σ N E ( N − / ( N max − N E ( N ) = (cid:112) ( N − N max − − ( N − tra-jectories. Fig. 3(a) shows the resulting distribution ofuptake times and the results for the number of boundreceptors as a function of time (inset). Clearly, the mean uptake time is substantially smaller than the uptake timefrom the deterministic approach. Fig. 3(b) and (c) dis-play the simulated mean uptake times as a function of W and R or σ , respectively. Comparing to the determin-istic approach, cf. Fig. 2(b) and (c), we see that now up-take is possible for any parameter value, although it canbe rather long in regions where the deterministic modeldoes not allow for uptake. However, the parameter regionwith uptake in experimentally accessible uptake times isstrongly increased and now also extends below the blueline indicating the critical radius R crit of the determinis-tic model. This expansion of the parameter regime thatallows the process to complete is also known from thestochastic dynamics of small adhesion clusters [13, 14].We next discuss the interplay between shape andstochastic dynamics in analytical detail. For simplic-ity we set the membrane tension to zero in the follow-ing ( σ = 0). We approximate the master equation by aFokker-Planck equation (FPE) using a Kramers-Moyalexpansion [35]. The equivalent stochastic differentialequation can be transformed to angle space using Itˆo’sLemma [37] ˙ θ = ν up − D cos θ sin θ + (cid:114) D sin θ ξ ( t ) (4)where D = ( ν w + ν κ ) / ( N max − ξ ( t ) is as-sumed to be Gaussian with (cid:104) ξ ( t ) (cid:105) = 0 and (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ). From Eq. (4) one can directly read the driftterm (or the corresponding potential) of the correspond-ing FPE in angle space and its diffusion coefficient D [32]. Because for the spherical case this diffusion con-stant depends on the state variable θ , here we deal withmultiplicative noise.In marked contrast, for the two cylindrical casesone obtains additive noise. For example, for the par-allel cylinder we find ˙ θ (cid:107) = ν (cid:107) up + √ D (cid:107) ξ ( t ) where ν (cid:107) up = W/ ( ηR ) − κ/ (2 R η ) and D (cid:107) = ( ν (cid:107) up + (a) T (ms) N max (b) T =12=120 N max N max T ◦ det T ◦ det T ◦ sim T ◦ mul T det T ana T add α
80 0.5
FIG. 4. (a) Geometry-dependent mean uptake times for spheres (red) and parallel cylinders (green) as a function of themaximum number of receptors. Shown are the analytical results for the deterministic case (solid) and for multiplicative(additive) noise corresponding to the spherical (cylindrical) geometry (dashed). The result from the simulations of the masterequation is shown for sphere (cylinder) as symbols. (b) The case with membrane tension can be treated with computersimulations. Shown are the mean uptake times for a sphere as a function of the dimensionless parameter α = ν σ /ν up (byvarying σ ) for two different numbers of receptors and the deterministic case (diverging at α = 1 / κ/ ( R η )) π/ (2( N max − θ because the length of the the moving contactline is constant [32]. The different quality of the noisesuggests that the uptake dynamics change in a funda-mental manner in the different geometries.The mean uptake times can be obtained analyticallystudying the mean first passage time problem usingthe backwards FPE [37, 38] with reflecting (adsorbing)boundary condition at θ = 0 ( θ = π ). Neglecting the an-gle dependent drift term (as it is large only for the firstand last step) but keeping the multiplicative noise, themean uptake time evaluates to [32] T ◦ mult = T ◦ det (cid:104) − e − ν up D I (cid:16) ν up D (cid:17)(cid:105) < T ◦ det (5)where I is the modified Bessel function of the first kind.In the limit of small fluctuations compared to the up-take frequency, we recover the deterministic limit. Inthe opposite limit of large fluctuations, the uptake timeapproaches T ◦ mult ≈ π/D .For the parallel cylinder the noise is additive and themean uptake time is readily obtained as [32] T (cid:107) add = T (cid:107) det − D (cid:107) ν (cid:107) (cid:34) − exp (cid:32) − πν (cid:107) up D (cid:107) (cid:33)(cid:35) < T (cid:107) det (6)where T (cid:107) det = π/ν (cid:107) up . In the limit of small fluctuationsone again recovers the deterministic uptake time, whilefor large fluctuations one finds T (cid:107) add ≈ π / (2 D (cid:107) ). Hence,for both geometries the mean uptake time is alwayssmaller than the deterministic one. In general, fluctua-tions in small systems combined with a reflecting bound-ary should always decrease the mean first passage time,since the stochastic process profits from the presence ofthe boundary, while the deterministic process does not. We now consider a particle with R = 180 nm, i.e. inthe region of Fig. 2(a) where the deterministic uptaketimes of sphere and parallel cylinder are similar. Fig. 4(a)shows the mean uptake times for different geometries atequal volume as a function of the number of receptors.We note that for the parallel cylinder it is also possi-ble to compute the uptake time directly from the masterequation T (cid:107) ana [37]. The agreement between simulations(symbols) and analytical results (lines) is very good forcylinders and rather good for spheres. For small num-ber of receptors, i.e. strong fluctuations, the uptake of asphere is faster than the one of a cylinder. We concludethat uptake of spherical particles dynamically benefitsfrom the noise. In fact, using in Eq. (6) ν up and D in-stead of ν (cid:107) up and D (cid:107) , we find that T ◦ mult < T ◦ add < T ◦ det always holds.While membrane tension could not be treated analyti-cally, it can be included in the simulations, and we get thesame results, i.e. the uptake times are reduced by increas-ing stochasticity. Fig. 4(b) shows the mean uptake timesas a function of the dimensionless parameter α = ν σ /ν up for different N max and R = 90 nm. Although stochas-ticity is most important for small numbers of receptors,nevertheless, even for substantial numbers on the orderof one hundred receptors, the stochastic effects survive.In conclusion, we found that the uptake of sphericalparticles profits from the presence of noise. Our resultssuggest yet another advantange for viruses to be spheri-cal. Similar effects arising from the interplay of stochasticdynamics and geometry might also exist for other biolog-ically relevant first passage problems, e.g. phagocytosis[39], the closure of transient pores on spherical vesicles[40] or the fusion of tissue over circular holes [41].F.F. acknowledges support by the Heidelberg Gradu-ate School for Fundamental Physics (HGSFP). The au-thors thank Steeve Boulant, Ada Cavalcanti-Adam andTina Wiegand for helpful discussions on reovirus, andwe acknowledge the Collaborative Research Centre 1129for support. U.S.S. acknowledges support as a memberof the Interdisciplinary Center for Scientific Computingand the cluster of excellence CellNetworks. [1] B. Alberts, Molecular Biology of the Cell , 6th ed. (Gar-land Science, New York, 2015).[2] D. S. Dimitrov, Nat. Rev. Microbiol. , 109 (2004).[3] S. Zhang, H. Gao, and G. Bao, ACS Nano , 8655 (2015).[4] J. Panyam and V. Labhasetwar, Adv. Drug Deliv. Rev. , 329 (2003).[5] N. von Moos, P. Burkhardt-Holm, and A. K¨ohler, Env-iron. Sci. Technol. , 11327 (2012).[6] C. Hulo, E. De Castro, P. Masson, L. Bougueleret,A. Bairoch, I. Xenarios, and P. Le Mercier, Nucleic AcidsRes. , D576 (2011).[7] F. H. Crick, J. D. Watson, et al. , Nature (London) ,473 (1956).[8] W. Roos, R. Bruinsma, and G. Wuite, Nat. Phys. , 733(2010).[9] R. Zandi, D. Reguera, R. F. Bruinsma, W. M. Gelbart,and J. Rudnick, Proc. Natl. Acad. Sci. , 15556 (2004).[10] P. Kumberger, F. Frey, U. S. Schwarz, and F. Graw,FEBS Lett. , 1972 (2016).[11] S. X. Sun and D. Wirtz, Biophys. J. , L10 (2006).[12] E. S. Barton, J. C. Forrest, J. L. Connolly, J. D. Chap-pell, Y. Liu, F. J. Schnell, A. Nusrat, C. A. Parkos, andT. S. Dermody, Cell , 441 (2001).[13] T. Erdmann and U. Schwarz, Phys. Rev. Lett. , 108102(2004).[14] T. Robin, I. M. Sokolov, and M. Urbakh, Phys. A ,398 (2018).[15] R. V´acha, F. J. Martinez-Veracoechea, and D. Frenkel,Nano Lett. , 5391 (2011).[16] C. Huang, Y. Zhang, H. Yuan, H. Gao, and S. Zhang,Nano Lett. , 4546 (2013).[17] R. Lipowsky and H.-G. D¨obereiner, Europhys. Lett. ,219 (1998).[18] S. Dasgupta, T. Auth, and G. Gompper, Nano Lett. ,687 (2014). [19] A. H. Bahrami, M. Raatz, J. Agudo-Canalejo, R. Michel,E. M. Curtis, C. K. Hall, M. Gradzielski, R. Lipowsky,and T. R. Weikl, Adv. Coll. Interf. Sci. , 214 (2014).[20] M. Deserno, Phys. Rev. E , 031903 (2004).[21] J. Agudo-Canalejo and R. Lipowsky, ACS Nano , 3704(2015).[22] H. Gao, W. Shi, and L. B. Freund, Proc. Natl. Acad.Sci. USA , 9469 (2005).[23] P. Decuzzi and M. Ferrari, Biomaterials , 2915 (2007).[24] X. Yi, X. Shi, and H. Gao, Phys. Rev. Lett. , 098101(2011).[25] C. Zeng, M. Hernando-P´erez, B. Dragnea, X. Ma, P. VanDer Schoot, and R. Zandi, Phys. Rev. Lett. , 038102(2017).[26] Z. A. McDargh, P. V´azquez-Montejo, J. Guven, andM. Deserno, Biophys. J. , 2470 (2016).[27] W. Helfrich, Z. Naturforsch. C , 693 (1973).[28] L. Foret, Eur. Phys. J. E , 42 (2014).[29] G. Kumar and A. Sain, Phys. Rev. E , 062404 (2016).[30] D. Veesler, K. Cupelli, M. Burger, P. Gr¨aber, T. Stehle,and J. E. Johnson, Proc. Natl. Acad. Sci. USA , 8815(2014).[31] M. Sadeghi, T. R. Weikl, and F. No´e, J. Chem. Phys. , 044901 (2018).[32] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.122.088102 ,which includes details on the deterministic and stochasticcalculations, as well as three additional figures.[33] D. Alsteens, R. Newton, R. Schubert, D. Martinez-Martin, M. Delguste, B. Roska, and D. J. M¨uller, Nat.Nanotechnol. , 177 (2017).[34] Y. Pan, F. Zhang, L. Zhang, S. Liu, M. Cai, Y. Shan,X. Wang, H. Wang, and H. Wang, Adv. Sci. , 1600489(2017).[35] N. G. van Kampen, Stochastic Processes in Physics andChemistry (North-Holland, Amsterdam, 1984).[36] D. T. Gillespie, J. Phys. Chem. , 2340 (1977).[37] C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin Heidelberg New York, 1985).[38] S. Redner,
A Guide to First-Passage Processes (Cam-bridge University Press, Cambridge, England, 2001).[39] S. Tollis, A. E. Dart, G. Tzircotis, and R. G. Endres,BMC Sys. Biol. , 149 (2010).[40] O. Sandre, L. Moreaux, and F. Brochard-Wyart, Proc.Natl. Acad. Sci. USA , 10591 (1999).[41] V. Nier, M. Deforet, G. Duclos, H. G. Yevick, O. Cochet-Escartin, P. Marcq, and P. Silberzan, Proc. Natl. Acad.Sci. USA112