Stochastic Dynamics of Nanoparticle and Virus Uptake

TL;DR

This paper investigates how stochastic effects and particle geometry influence cellular uptake, revealing that spherical particles may be taken up faster than cylindrical ones due to noise effects, contrary to deterministic predictions.

Contribution

It introduces a stochastic model showing geometry-dependent effects on nanoparticle and virus uptake, highlighting the role of noise in favoring spherical shapes.

Findings

Spherical particles can have faster uptake due to stochastic effects.

Deterministic theory predicts cylindrical particles are faster for the same volume.

Stochastic effects depend strongly on particle geometry.

Abstract

The cellular uptake of nanoparticles or viruses requires that the gain of adhesion energy overcomes the cost of plasma membrane bending. It is well known that this leads to a minimal particle size for uptake. Using a simple deterministic theory for this process, we first show that, for the same radius and volume, cylindrical particles should be taken up faster than spherical particles, both for normal and parallel orientations. We then address stochastic effects, which are expected to be relevant due to small system size, and show that, now, spherical particles can have a faster uptake, because the mean first passage time profits from the multiplicative noise induced by the spherical geometry. We conclude that stochastic effects are strongly geometry dependent and may favor spherical shapes during adhesion-driven particle uptake.

Figures

Equations

$$E_{\mathrm{total}}=-{\int }_{A_{\mathrm{ad}}}W\mathrm{d}A+{\int }_{A_{\mathrm{mem}}}2\kappa H^2\mathrm{d}A+\sigma \Delta A$$

$$\dot{\theta }={\nu }_{\mathrm{up}}-{\nu }_{\sigma }(1-\cos \theta )$$

$$T_{\det }^{\circ }\approx \frac{\pi }{{\nu }_{\mathrm{up}}\sqrt{1-\frac{2{\nu }_{\sigma }}{{\nu }_{\mathrm{up}}}}}.$$

$$\dot{\theta }={\nu }_{\mathrm{up}}-D\frac{\cos \theta }{{\sin }^2\theta }+\sqrt{\frac{2D}{\sin \theta }}\xi (t)$$

$$T_{\mathrm{mult}}^{\circ }=T_{\det }^{\circ }\left[1-{\mathrm{e}}^{-\frac{{\nu }_{\mathrm{up}}}{D}}{I}_0\left(\frac{{\nu }_{\mathrm{up}}}{D}\right)\right]<T_{\det }^{\circ }$$

$$T_{\mathrm{add}}^{\parallel }=T_{\det }^{\parallel }-\frac{D^{\parallel }}{{\nu }_{\mathrm{up}}^{\parallel 2}}\left[1-\exp \left(-\frac{\pi {\nu }_{\mathrm{up}}^{\parallel }}{D^{\parallel }}\right)\right]<T_{\det }^{\parallel }$$

Full text

Biological cells constantly communicate with their environment by relaying signals and material across their plasma membranes. In particular, cells routinely take up small particles with sizes in the order of $10-300\,{\rm nm}$. This process is usually driven by receptor-ligand interactions Alberts (2015), such that the adhesion energy can overcome the energy required for membrane bending. It is also exploited by viruses that have to enter host cells for replication Dimitrov (2004). Due to the physical nature of this process, cells also take up artificial nanoparticles of various shapes Zhang et al. (2015), whose uptake may be either intentional or undesired, as in drug delivery Panyam and Labhasetwar (2003) or in case of microplastics von Moos et al. (2012), respectively.

Viruses come in many different shapes Hulo et al. (2011), but the most paradigmatic and most frequently occuring ones are spherical and cylindrical Crick et al. (1956); Roos et al. (2010). Therefore here we focus our discussion on spheres, cylinders and a combination of both, as shown in Fig. 1. It is generally believed that the spherical shape is optimal because it maximizes the volume to surface ratio and confers high mechanical stability Zandi et al. (2004). However, it is less clear which shapes are optimal in regard to interactions with the membrane Zhang et al. (2015). Here we show that stochasticity might play an important role in this context.

Due to their small size, viruses are typically covered with only few tens of ligands for cell surface receptors Kumberger et al. (2016); Sun and Wirtz (2006) and thus stochastic effects are expected to be relevant. For example, the icosahedral and medium-sized ($60-100\,{\rm nm}$) members of the family of reoviruses have only $12$ primary JAM-A-$\sigma 1$ (junction adhesion molecule) ligands on their surfaces Barton et al. (2001). Although stochastic effects have been argued to be essential for the stability of small adhesion clusters Erdmann and Schwarz (2004); Robin et al. (2018), particle uptake is usually studied analytically only with deterministic approaches. Stochastic effects are automatically included in computer simulations with thermostats Vácha et al. (2011); Huang et al. (2013), but such approaches do not allow for a systematic study of the role of noise. Here we analytically calculate the first passage times for particle uptake of various shapes and show that they strongly depend on geometry. Our main result is that in contrast to deterministic dynamics, stochastic dynamics tends to favor spherical shapes for uptake.

Deterministic approaches usually start by balancing the contributions from adhesion and bending. Because adhesion energy scales with the radius squared, while bending energy is independent of radius, a minimal size exists for particle uptake Lipowsky and Döbereiner (1998). The most investigated aspect of receptor-mediated uptake is the role of particle shape Huang et al. (2013); Dasgupta et al. (2014); Bahrami et al. (2014). More detailed approaches include the variational problem for finding minimal energy shapes Deserno (2004); Agudo-Canalejo and Lipowsky (2015), the role of receptor diffusion towards the nanoparticle Gao et al. (2005); Decuzzi and Ferrari (2007), particle deformability Yi et al. (2011); Zeng et al. (2017), the physics of the scission step McDargh et al. (2016) and the role of the cytoskeleton Sun and Wirtz (2006). In order to develop a stochastic description, here we start with a minimal deterministic model, which we then extended to the stochastic case.

We assume that ligands to the cellular adhesive receptors are homogeneously distributed on the viral surface. The total free energy can be written as Helfrich (1973)

[TABLE]

where $W$ is the adhesion energy per area, $\kappa$ is the bending rigidity, $H$ is the mean curvature, $\sigma$ is the membrane tension and $\Delta A$ the excess area due to the deformation (compared to the flat case). Typical parameter values are $W=10^{-1}\,{\rm mJ/m}^{2}$, $\kappa=25\,{\rm k_{\mathrm{B}}T}$ and $\sigma=10^{-4}\,{\rm N/m}$ Foret (2014); Kumar and Sain (2016). For example, a reovirus has $12$ major and $60$ minor ligands Barton et al. (2001); Veesler et al. (2014). With a binding energy of around $15\,{\rm k_{\mathrm{B}}T}$ each, one estimates $W=4.4\cdot 10^{-2}\,{\rm mJ/m}^{2}$ for a homogeneous adhesion model.

The two membrane parameters define a typical length scale $\lambda=\sqrt{\kappa/\sigma}\approx 32\,{\rm nm}$. As shown schematically in Fig. 1, we consider a spherical or cylindrical particle of radius $R$ adhering to the membrane. Then the bending energy in Eq. (1) has contributions both from the adhering ($A_{\mathrm{ad}}$) and free part ($A_{\mathrm{mem}}-A_{\mathrm{ad}}$) of the membrane. It has been previously shown Foret (2014); Kumar and Sain (2016); Dasgupta et al. (2014); Sadeghi et al. (2018) that the contributions from the free part can be neglected in two special regimes. In the loose regime ($R\ll\lambda$), the free membrane can adopt the shape of a minimal surface and thus its bending contribution vanishes. In the tense regime ($R\gg\lambda$), the free part is essentially flat and the contribution from the adhered membrane is much larger than the one from the free membrane. To arrive at an analytical model, here we neglect the contributions from the free membrane also for the intermediate case.

Eq. (1) can now easily be analyzed for the geometries sketched in Fig. 1, namely (a) for a sphere ($\circ$), (b) for a cylinder oriented normally to the membrane ($\bot$), (c) for a spherocylinder ($\cap$) and (d) for a cylinder oriented parallel to the membrane ($\parallel$). Although it has been shown in coarse-grained molecular dynamics simulations that spherocylinders might undergo a dynamical sequence of first lying down and then standing up Huang et al. (2013), the two cylindrical modes considered here are the paradigmatic configurations encountered during wrapping Dasgupta et al. (2014). In case of the normal cylinder, the top and bottom surfaces are neglected as they do not contribute to the uptake force. To keep our calculations as transparent as possible, we neglect them also for the parallel cylinder. In both cases, we neglect the bending energies at the kinked edges.

The uptake forces $F_{\mathrm{up}}$ are calculated by taking the variation of the energy with respect to opening angle $\theta$ or height $z$, respectively, and are balanced by the friction force they experience, $F_{\mathrm{up}}=F_{\mathrm{friction}}$ Agudo-Canalejo and Lipowsky (2015). For a spherical particle, the membrane covered area is a spherical cap of radius $R$ and opening angle $\theta$, i.e. $A_{\mathrm{ad}}^{\circ}=2\pi R^{2}(1-\cos\theta)$, cf. Fig. 1(a). The friction force is $F_{\mathrm{friction}}^{\circ}=\eta 2\pi R\sin(\theta)R\dot{\theta}$, i.e. an effective membrane microviscosity of order $\eta=1\ {\rm Pa\;s}$ times the change of the membrane-covered particle surface. The resulting equation of motion reads

[TABLE]

with $\nu_{\mathrm{w}}=W/(R\eta)$, $\nu_{\mathrm{\kappa}}=2\kappa/(R^{3}\eta)$, $\nu_{\sigma}=\sigma/(R\eta)$ and $\nu_{\mathrm{up}}=\nu_{\mathrm{w}}-\nu_{\mathrm{\kappa}}$. The uptake time can be calculated as sup

[TABLE]

Note that it diverges for $\nu_{\mathrm{up}}=2\nu_{\mathrm{\sigma}}$, defining a critical radius $R_{\mathrm{crit}}$, below which uptake is not possible. In the limit of vanishing $\sigma$, we recover the classical result $R_{\mathrm{crit}}=\sqrt{2\kappa/W}\approx 44\,{\rm nm}$ Lipowsky and Döbereiner (1998). We note that our theory predicts normal uptake forces of around ten pN, which agrees well with results from recent atomic force microscopy experiments Alsteens et al. (2017); Pan et al. (2017).

Analogous calculations can be performed for the three cases with cylinders at equal volumes sup . Fig. 2(a) compares the resulting uptake times. For the normal and parallel cylinders, we take the same radius as for the sphere and adjust the aspect ratio. For the spherocylinder, we take the same aspect ratio as for the cylinders and adjust the radius. All four geometries show similar behaviors: a critical radius $R_{\mathrm{crit}}$ exists, below which uptake is not possible. The parallel cylinder has half the critical radius of the sphere because its mean curvature is half as large at equal radius. Moreover an optimal radius $R_{*}$ exists, at which the uptake time is minimal sup . Interestingly, the critical and optimal values are very close to each other, and the cylindrical particles are taken up faster than the spherical ones. The spherocylinder is the slowest case, because at equal volumes, the aspect ratio is modest and the spherical part dominates. Fig. 2(b) and (c) display the uptake time for a spherical particle as a function of $W$ and $R$ or $\sigma$, respectively, showing that a smaller adhesion energy has to be compensated by either larger radius or smaller membrane tension. Importantly, in the deterministic case certain parameter values do not lead to any uptake.

We now turn to the stochastic variant of our uptake model (cf. Fig. 1(e)). For the sphere, we map the membrane covered area onto the number of bound receptors $N$ sup , assuming axial symmetry. Using Eq. (2), we find a dynamical equation for $N$ through $\dot{N}=(\mathrm{d}N/\mathrm{d}\theta)\dot{\theta}$. From this discrete equation a one-step master equation van Kampen (1984) can then be deduced, with the forward rate $g_{N}=\nu_{\mathrm{w}}N_{\mathcal{E}}$ and the backward rate $r_{N}=\nu_{\mathrm{\kappa}}N_{\mathcal{E}}+2\nu_{\mathrm{\sigma}}N_{\mathcal{E}}(N-1)/(N_{\mathrm{max}}-1)$, where $N_{\mathcal{E}}(N)=\sqrt{(N-1)[(N_{\mathrm{max}}-1)-(N-1)]}$ corresponds to the number of receptors at the advancing edge.

We first solved the master equation numerically using the Gillespie algorithm Gillespie (1977) and averaging over $10^{4}$ trajectories. Fig. 3(a) shows the resulting distribution of uptake times and the results for the number of bound receptors as a function of time (inset). Clearly, the mean uptake time is substantially smaller than the uptake time from the deterministic approach. Fig. 3(b) and (c) display the simulated mean uptake times as a function of $W$ and $R$ or $\sigma$, respectively. Comparing to the deterministic approach, cf. Fig. 2(b) and (c), we see that now uptake is possible for any parameter value, although it can be rather long in regions where the deterministic model does not allow for uptake. However, the parameter region with uptake in experimentally accessible uptake times is strongly increased and now also extends below the blue line indicating the critical radius $R_{\mathrm{crit}}$ of the deterministic model. This expansion of the parameter regime that allows the process to complete is also known from the stochastic dynamics of small adhesion clusters Erdmann and Schwarz (2004); Robin et al. (2018).

We next discuss the interplay between shape and stochastic dynamics in analytical detail. For simplicity we set the membrane tension to zero in the following ($\sigma=0$). We approximate the master equation by a Fokker-Planck equation (FPE) using a Kramers-Moyal expansion van Kampen (1984). The equivalent stochastic differential equation can be transformed to angle space using Itô’s Lemma Gardiner (1985)

[TABLE]

where $D=(\nu_{\mathrm{w}}+\nu_{\mathrm{\kappa}})/(N_{\mathrm{max}}-1)$. The noise $\xi(t)$ is assumed to be Gaussian with $\left<\xi(t)\right>=0$ and $\left<\xi(t)\xi(t^{\prime})\right>=\delta(t-t^{\prime})$. From Eq. (4) one can directly read the drift term (or the corresponding potential) of the corresponding FPE in angle space and its diffusion coefficient $D$ sup . Because for the spherical case this diffusion constant depends on the state variable $\theta$, here we deal with multiplicative noise.

In marked contrast, for the two cylindrical cases one obtains additive noise. For example, for the parallel cylinder we find $\dot{\theta}^{\parallel}=\nu_{\mathrm{up}}^{\parallel}+\sqrt{2D^{\parallel}}\,\xi(t)$ where $\nu_{\mathrm{up}}^{\parallel}=W/(\eta R)-\kappa/(2R^{3}\eta)$ and $D^{\parallel}=(\nu_{\mathrm{up}}^{\parallel}+\kappa/(R^{3}\eta))\pi/(2(N_{\mathrm{max}}-1))$, where the latter does not depend on $\theta$ because the length of the the moving contact line is constant sup . The different quality of the noise suggests that the uptake dynamics change in a fundamental manner in the different geometries.

The mean uptake times can be obtained analytically studying the mean first passage time problem using the backwards FPE Gardiner (1985); Redner (2001) with reflecting (adsorbing) boundary condition at $\theta=0$ ($\theta=\pi$). Neglecting the angle dependent drift term (as it is large only for the first and last step) but keeping the multiplicative noise, the mean uptake time evaluates to sup

[TABLE]

where $I_{0}$ is the modified Bessel function of the first kind. In the limit of small fluctuations compared to the uptake frequency, we recover the deterministic limit. In the opposite limit of large fluctuations, the uptake time approaches $T_{\mathrm{mult}}^{\circ}\approx\pi/D$.

For the parallel cylinder the noise is additive and the mean uptake time is readily obtained as sup

[TABLE]

where $T_{\mathrm{det}}^{\parallel}=\pi/\nu_{\mathrm{up}}^{\parallel}$. In the limit of small fluctuations one again recovers the deterministic uptake time, while for large fluctuations one finds $T_{\mathrm{add}}^{\parallel}\approx\pi^{2}/(2D^{\parallel})$. Hence, for both geometries the mean uptake time is always smaller than the deterministic one. In general, fluctuations in small systems combined with a reflecting boundary should always decrease the mean first passage time, since the stochastic process profits from the presence of the boundary, while the deterministic process does not.

We now consider a particle with $R=180\,{\rm nm}$, i.e. in the region of Fig. 2(a) where the deterministic uptake times of sphere and parallel cylinder are similar. Fig. 4(a) shows the mean uptake times for different geometries at equal volume as a function of the number of receptors. We note that for the parallel cylinder it is also possible to compute the uptake time directly from the master equation $T_{\mathrm{ana}}^{\parallel}$ Gardiner (1985). The agreement between simulations (symbols) and analytical results (lines) is very good for cylinders and rather good for spheres. For small number of receptors, i.e. strong fluctuations, the uptake of a sphere is faster than the one of a cylinder. We conclude that uptake of spherical particles dynamically benefits from the noise. In fact, using in Eq. (6) $\nu_{\mathrm{up}}$ and $D$ instead of $\nu_{\mathrm{up}}^{\parallel}$ and $D^{\parallel}$, we find that $T_{\mathrm{mult}}^{\circ}<T_{\mathrm{add}}^{\circ}<T_{\mathrm{det}}^{\circ}$ always holds.

While membrane tension could not be treated analytically, it can be included in the simulations, and we get the same results, i.e. the uptake times are reduced by increasing stochasticity. Fig. 4(b) shows the mean uptake times as a function of the dimensionless parameter $\alpha=\nu_{\mathrm{\sigma}}/\nu_{\mathrm{up}}$ for different $N_{\mathrm{max}}$ and $R=90\,{\rm nm}$. Although stochasticity is most important for small numbers of receptors, nevertheless, even for substantial numbers on the order of one hundred receptors, the stochastic effects survive.

In conclusion, we found that the uptake of spherical particles profits from the presence of noise. Our results suggest yet another advantange for viruses to be spherical. Similar effects arising from the interplay of stochastic dynamics and geometry might also exist for other biologically relevant first passage problems, e.g. phagocytosis Tollis et al. (2010), the closure of transient pores on spherical vesicles Sandre et al. (1999) or the fusion of tissue over circular holes Nier et al. (2015).