Open Access Article

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V. E.
Debets
^{a},
L. M. C.
Janssen
*^{a} and
A.
Šarić
*^{bc}
^{a}Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands. E-mail: l.m.c.janssen@tue.nl
^{b}Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, UK
^{c}MRC Laboratory for Molecular Cell Biology, University College London, London, UK. E-mail: a.saric@ucl.ac.uk

Received
20th April 2020
, Accepted 5th October 2020

First published on 6th October 2020

Tracing the motion of macromolecules, viruses, and nanoparticles adsorbed onto cell membranes is currently the most direct way of probing the complex dynamic interactions behind vital biological processes, including cell signalling, trafficking, and viral infection. The resulting trajectories are usually consistent with some type of anomalous diffusion, but the molecular origins behind the observed anomalous behaviour are usually not obvious. Here we use coarse-grained molecular dynamics simulations to help identify the physical mechanisms that can give rise to experimentally observed trajectories of nanoscopic objects moving on biological membranes. We find that diffusion on membranes of high fluidities typically results in normal diffusion of the adsorbed nanoparticle, irrespective of the concentration of receptors, receptor clustering, or multivalent interactions between the particle and membrane receptors. Gel-like membranes on the other hand result in anomalous diffusion of the particle, which becomes more pronounced at higher receptor concentrations. This anomalous diffusion is characterised by local particle trapping in the regions of high receptor concentrations and fast hopping between such regions. The normal diffusion is recovered in the limit where the gel membrane is saturated with receptors. We conclude that hindered receptor diffusivity can be a common reason behind the observed anomalous diffusion of viruses, vesicles, and nanoparticles adsorbed on cell and model membranes. Our results enable direct comparison with experiments and offer a new route for interpreting motility experiments on cell membranes.

Deducing the underlying molecular mechanisms from the trajectories is highly non-trivial as a large number of possible molecular mechanisms can result in similar anomalous motility.^{10,11} For instance, high-speed single-particle tracking studies have reported anomalous diffusion of functionalised nanoparticles, vesicles, and virus-like particles bound to receptors on membranes in living cells^{1} and supported bilayers.^{2–9} Various physical and chemical effects have been proposed to underlie the observed anomalous diffusion,^{12} including multivalent interactions between the nanoparticle and receptors, coupling between membrane leaflets,^{2} molecular pinning,^{2} receptor clustering,^{13} formation of transient membrane domains,^{14–16} and membrane-cytoskeleton interactions.^{17}

Here we take a reverse approach: we simulate physical interactions between a nanoobject and deformable fluctuating membranes and measure the resulting trajectories for membranes of various properties. We characterise the resulting diffusion profiles of the nanoobject and match them to the underlying molecular mechanisms that are evident in molecular simulations. It is our hope that such an approach can help to interpret experimental trajectories and identify the molecular mechanisms behind them.

We specifically focus on the role of the receptor concentration and the membrane structure/phase state in the resulting diffusion behaviour of the membrane-bound nanoparticle. In our simulations the nanoparticle can for instance represent a virus-like particle, a globular macromolecule, or an inorganic nanoparticle. The particle binds to the membrane via multivalent interactions with the membrane receptors and locally deforms the membrane underneath it. We measure the nanoparticle's diffusion profile within molecular dynamics (MD) simulations on fluid, gel-like, and fully cross-linked membranes, at varying receptor concentrations. We find a range of behaviours, from standard random walk to anomalous diffusion characterised by the particle's hopping between regions of local trapping. We find that the anomalous diffusion is not caused by mutivalent binding, as previously proposed, but by the hindered receptor diffusivity, which results in the particle trapping in regions rich in receptors. The random walk is then recovered if the membrane is fully saturated with receptors. We provide an in-depth numerical analysis of the data and a theoretical framework that characterises the observed anomalous diffusion.

The nanoparticle is represented as a spherical particle with a fixed diameter of 10σ (∼40 nm), where σ is the MD unit of length. Its size is chosen to probe the regime in which the nanoparticle does not insert in the bilayer (∼10 nm), but is still small enough to be able to diffuse (<∼10^{2} nm). This is also the size regime relevant for most viruses and drug-delivery nanoparticles.^{21,22} The nanoparticle adsorbs onto the membrane and, due to interactions with the membrane particles and thermal fluctuations, diffuses laterally over it. The adsorption is facilitated by allowing certain particles within the membrane to bind the particle. These are deemed as receptors and attract the particle via a generic truncated and shifted Morse potential. Such a potential mimics relatively weak binding of the nanoparticle to many receptors, as observed in some non-enveloped viruses,^{14} or binding of synthetic nanoparticles via screened electrostatic interactions.^{23} Both receptors and non-receptors (non-binding membrane particles) interact with the nanoparticle via volume exclusion (see Appendix B for more details).

To better quantify the difference between the nanoparticle motion on the fluid and gel-like membrane we have calculated several diffusive quantities, focusing primarily on ones that could be accurately obtained (well-averaged) and that could be compared to experimental results. The first of these are the (time-ensemble averaged) mean square displacement (MSD), i.e. 〈(r(t) − r(0))^{2}〉, and the corresponding instant diffusion coefficient D = MSD/4t, which have been plotted in Fig. 3. It can be noticed that after a brief increase due to ballistic motion, the instant diffusion coefficients D start to decrease over multiple orders of magnitude of time. This decrease is almost negligible for the fluid membrane which corresponds to normal diffusion, where MSD ∝ t, and thus a constant D.^{26} For the gel-like membrane the diffusion coefficient is qualitatively different since it drops approximately an order of magnitude before saturating towards a constant value. In other words, clear anomalous diffusion occurs on the observed time scales, while normal diffusion is regained in the long time limit. Interestingly, similar behavior has been observed in experiments involving the diffusive motion of a spherical gold nanoparticle binding to a model membrane via receptor–ligand interactions (see Fig. 3). It thus seems that the temporal trapping of the nanoparticle, sometimes of the order of the simulation time (explaining the relatively large variety between time averaged MSDs), results in anomalous diffusion on membranes of decreased fluidities.

Fig. 3 Diffusion on fluid and gel membranes is qualitatively different. (A) Time-ensemble averaged MSD in the lateral direction for diffusion on a fluid (in orange) and gel-like membrane (in blue). (B) The instant diffusion coefficient derived from the time-ensemble averaged MSD from (A). The shaded areas represent the variation in time averaged MSDs and corresponding instant diffusion coefficients. (C) The experimental result of a 40 nm gold particle that binds to a supported model membrane. Reprinted with permission from ref. 2. Copyright 2014 American Chemical Society. |

The analysis of the MSD is, however, limited and can only identify whether the motion is anomalous or not. It gives little information about the underlying mechanism of the observed diffusion process. To further rationalise our findings, we have therefore retrieved the (2D) cumulative density function (CDF) P(r,t), i.e. the probability that a particle starting at the origin is found within a circle of radius r at time t, for several discrete times t_{n} = nΔt. This can be achieved by counting the number of absolute displacements |r(t_{i+n}) − r(t_{i})| < r for all trajectories and normalising over the total number of considered data points.^{27} The results for t_{n} = 1000τ are shown in Fig. 4, which again demonstrate qualitatively different behavior for the fluid and the gel-like membrane settings.

Fig. 4 Diffusive modes on fluid and gel-like membranes. Plots of the cumulative density function P(r,t) for (A) fluid and (B) gel membranes. Fits to both the Gaussian (eqn (1)) and bi-Gaussian (eqn (2)) result are plotted for t_{n} = 1000τ. Insets show D_{fast}, D_{slow}, and w as a function of time resulting from a bi-Gaussian fit of the retrieved CDFs. (C) Experimental results of the cumulative density function (CDF) along with fits to the one-mobility (Gaussian), two-mobility (bi-Gaussian) and three-mobility models for the diffusion of a 20 nm gold nanoparticle attached to GM1 ganglioside in supported DOPC bilayer. Reprinted with permission from ref. 13. Copyright 2014 American Chemical Society. (D) Schematic representation an example nanoparticle trajectory indicating how it alternates between temporary confined parts and fast diffusive parts. These different segments of the trajectory can be characterised by a slow and fast diffusion coefficient respectively (D_{slow} and D_{fast}) which have also been shown in the picture. |

We assess this discrepancy more quantitatively by introducing a two-component mobility model^{27–30} in which the obtained CDFs are fitted with both a single Gaussian distribution corresponding to normal diffusion (Brownian motion)

P(r,t) = 1 −e^{−r2/4Dtt}, | (1) |

P(r,t) = 1 − we^{−r2/4Dfastt} − (1 − w)e^{−r2/4Dslowt}. | (2) |

The model thus allows us to identify if a single, ‘simple’ diffusive process with a diffusion coefficient D_{t} is sufficient to describe the (anomalous) motion or that (at least) two distinct processes, characterised by a slow D_{slow} and fast D_{fast} diffusion coefficient with a relative weight w, are required. We stress that the separation in two diffusive modes might still not be enough to fully describe the observed motion, but it does allow us to study a departure from ‘normal’ diffusive motion.

Fits to eqn (1) and (2) are shown in Fig. 4 and indicate that for the fluid membrane settings both distributions provide high quality fits, confirming that particle motion on the fluid membrane is governed by normal diffusion. However, on the gel-like membrane only the bi-Gaussian is able to accurately fit the observed results, which is again in qualitative agreement with experimental results of a gold nanoparticle diffusing on a model membrane, as shown in Fig. 4. The particle motion is thus effectively split in two (or possibly more) distinct diffusive modes which, combined, are likely responsible for the anomalous diffusion manifested in the MSD.

To gain more insights from the fits and study the observed behavior over time, we have plotted the bi-Gaussian fitting parameters (D_{fast}, D_{slow}, and w) over a range of times (see insets Fig. 4). It can be seen that on the fluid membrane, as expected, normal nanoparticle diffusion occurs on all considered timescales. In particular, the fast and slow diffusion coefficient remain either equal to each other, which is the same as a single Gaussian fit, or differ only slightly. Switching to the gel-membrane settings, we find that in this case normal diffusion (approximately equal D_{slow} and D_{fast}) is only obtained for the smallest considered time (t_{n} = 10τ). Careful inspection of the trajectories (Fig. 2) suggests that this is a consequence of the nanoparticle not yet experiencing the effects of the temporal confinements. At later times, however, the values of the slow and fast diffusion coefficient become increasingly separated and a clear distinction can be made between both diffusive modes. Interestingly, both diffusion coefficients are decreasing, but the slow diffusion coefficient drops much faster in value (over two orders of magnitude). A possible explanation for this rapid drop in D_{slow} and the slower decrease in D_{fast} can be found when we link the slow diffusion coefficient to the confined parts of the trajectories (see Fig. 4 for a schematic representation). Since the MSD saturates for confined diffusion,^{31} the slow diffusion coefficient is expected to decay to zero in the long time limit, i.e. when the time is much larger than the average confinement time. Consequently, the overall diffusive motion of the nanoparticle should tend (in the same limit) towards normal diffusion with a smaller diffusion coefficient, which explains the more slowly decreasing (and possibly saturating) D_{fast}.

We finalise our analysis of the nanoparticle motion and corresponding underlying diffusion process by calculating the normalised velocity autocorrelation function (VAF). This quantity, often used as a diagnostic tool to distinguish among different mechanisms for anomalous diffusion, is defined as^{32–34}

(3) |

Here brackets denote time-ensemble averaging and the velocity of the nanoparticle at time t is calculated over a time period δ via

(4) |

The results for different times δ have been plotted in Fig. 5. On the gel-like membrane we notice the formation of clear negative peaks in the VAFs at t = δ over a long range of δ values. This indicates the existence of antipersistent behavior, which is likely a result of the temporary confinement experienced by the nanoparticle. In particular, during periods of confinement the particle seeks to escape but is often drawn back into the center of the confined area, which negatively correlates its overall velocity.

Fig. 5 Antipersistent motion on a gel-like membrane. Velocity autocorrelation function for (A) fluid and (B) gel membrane as defined by eqn (3) (using time-ensemble averaging) for different times δ that indicate the time over which the velocity is defined (eqn (4)). |

For the fluid membrane settings we cannot identify significant antipersistent behavior and the VAFs decay to zero at t = δ. This is consistent with normal diffusion for which the direction of movement is random and no correlations between the velocity at different times are expected.

In comparison to its fluid counterpart, the cross-linked membrane, due to its cross-linked nature, never encapsulates the particle, even when it consists solely of receptors. This allows us to study the diffusion of the particle across the membrane for different receptor percentages. Letting the trajectories again serve as our starting point (see Fig. 6), we find that for 30% receptors (smaller percentages lead to particle detachment) clear clustering and temporal confinement of the particle occur; these hallmarks become even more evident at a larger receptor percentage of 50% (increased membrane inhomogeneity). However, on a homogeneous membrane consisting of only receptors the trapping behavior disappears. The particle motion on the cross-linked membrane therefore seems, despite the change in membrane mechanics, to be very similar to that on a fluid (gel-like) membrane. The only difference rests in the fact that, instead of receptor diffusivity, the percentage of receptors appears to control the transition from hop-like diffusion towards a freely diffusive behavior. This clearly indicates that the membrane inhomogeneity, rather than the multivalent binding between the particle and the receptors, is responsible for the observed anomalous behaviour.

To study the effect of the receptor percentage in more detail and test whether the observed particle motion is truly similar for both membranes, we have calculated the same quantities that have been used to characterise the diffusion process on fluid and gel-like membranes. The (time-ensemble averaged) MSD and related instant diffusion coefficient D = MSD/4t on the cross-linked membrane are shown in Fig. 7. They demonstrate experimentally consistent subdiffusive behavior (decreasing D) on intermediate timescales for 30% and 50% percent receptors, as opposed to normal diffusive behavior (constant D) for 100% receptors. This again corroborates that the temporal trapping of the particle underlies anomalous diffusion. Since on average the particle trajectories involve more clustering and longer trapping at 50% receptors compared to 30%, this also explains why the drop in value of D is larger for the former.

The fact that the subdiffusive behavior initially becomes more apparent with increasing receptor percentage, but diminishes when going to a membrane consisting of only receptors, hints at some non-trivial dependence of the motion on receptor percentage. We quantify this by retrieving the MSD at a number of different receptor percentages and fitting the results to MSD = K_{α}t^{α}. The obtained anomalous diffusion coefficients K_{α} and anomaly exponents α are plotted in Fig. 7. It can be seen that, starting from 30% receptors, anomalous behavior and weaker diffusion become more apparent at first (smaller values of α and K_{α}), most notably around 50–70%, but eventually fade away leading to normal diffusion for 100% receptors. These results suggest that there exists an optimum receptor percentage around ∼50% for which anomalous diffusion is most evident in completely cross-linked membranes.

Moving back to the nature of the anomalous particle diffusion, we have examined the CDFs for cross-linked membranes consisting of 50% and 100% receptors by means of the two-component mobility model (see Fig. 8). At an intermediate time of t_{n} = 900τ, we note that both a Gaussian (eqn (1)) and bi-Gaussian (eqn (2)) give identical high quality fits to the CDF on a uniform membrane (100% receptors). The accompanying fit parameters of the bi-Gaussian (D_{fast}, D_{slow}, and w) in turn show that this is the case across the entire analysed time range, since the fast and slow diffusion coefficient remain (almost) equal to each other. Realising that over time the fitted diffusion coefficients also keep an approximately constant value, which is in quantitative agreement with the one derived from the MSD, we conclude that the motion corresponds to normal diffusion.

Fig. 8 Diffusive modes and antipersistent motion on a cross-linked membrane. Plots of the cumulative density function P(r,t) at time t_{n} = 900τ for a cross-linked membrane that contains (A) 50% and (B) 100% receptors. Fits to both the Gaussian (eqn (1)) and bi-Gaussian (eqn (2)) result are plotted as well. Insets show D_{fast}, D_{slow}, and w as a function of time resulting from a bi-Gaussian fit of the retrieved CDFs for several times t_{n}. (C) and (D) Plots of the velocity autocorrelation function as defined by eqn (3) (using time-ensemble averaging) for different times δ that indicate the time over which the velocity is defined (eqn (4)) for a membrane that contains (C) 50% and (D) 100% receptors. |

On a non-uniform membrane with 50% receptors we see that at a time of t_{n} = 900τ (within the subdiffusive regime of the MSD) the CDF deviates significantly from a single Gaussian. A bi-Gaussian, in comparison, is able to accurately fit the CDF suggesting that the motion is split in two or possibly more diffusive modes. Moreover, the shape of the CDF curve clearly resembles the one obtained for a gel-like membrane (see Fig. 4 and 8). Combined with the already observed similarities between both membranes in terms of trajectories and MSD, we expect the underlying mechanism of the anomalous particle motion on a non-uniform cross-linked membrane to be the same as on the gel-like membrane. This claim is further substantiated by considering the bi-Gaussian fit parameters (D_{fast}, D_{slow}, and w), which change over time in an almost identical manner as those for the gel-like membrane (see insets Fig. 4 and 8). Applying earlier proposed reasoning, we can again explain the fast drop in D_{slow} and more moderate decrease of D_{fast} by linking the former to the confined parts of the particle trajectories.

Finally, for completeness, we have also calculated the VAFs corresponding to both a membrane composition of 50% and 100% receptors. The results are plotted in Fig. 8 and appear akin to the ones obtained for a gel-like and fluid membrane respectively. Specifically, we see significant negative correlations (antipersistent behavior) for the 50% membrane, which can be attributed to the temporary confinement of the nanoparticle. These results thus provide additional support that the underlying diffusion mechanism of the nanoparticle is independent of our choice of membrane model and rather seems governed by receptor patterning and diffusivity.

Fractional Brownian motion is often used to describe stochastic motion within a viscoelastic medium. In this case the particle locally deforms the medium, which results in a tendency for it to go back to locations it has visited in the past and in turn yields antipersistent behavior. Although this is consistent with the obtained simulation results, fBM also predicts a single Gaussian cumulative density function (CDF) and a subdiffusive (time-ensemble averaged) MSD in the long time limit,^{41} which are both incompatible with the obtained results. The second model, cBM, describes normal diffusion within a form of confinement (e.g. hard walls or a harmonic potential). This yields antipersistent behavior, but also a saturating MSD which is not observed in our simulations. The most intuitive mechanism to describe the simulation results with is the CTRW. This model describes the motion of a particle in terms of random jumps Δr, drawn from a distribution p(Δr). In between these jumps the particle is immobilised for a certain waiting time t_{w}, which is drawn from an independent distribution ψ(t_{w}).^{43,44} This notion of temporal confinement lines up with our particle trajectories. However, due to the assumed random jump direction in a CTRW, no antipersistent behavior is expected to occur^{33} and therefore it disagrees with our simulation results.

It thus seems that the proposed single models are inadequate to rationalise our simulation results. An alternative approach, inspired by the accurate bi-Gaussian fit of the cumulative density functions, is to segment trajectories in distinct diffusive modes and assign separate models to each of them,^{33} or to combine different anomalous diffusion models.^{45–47} We focus on the latter by invoking a so-called noisy continuous time random walk (nCTRW).^{40,45} In principle, a nCTRW combines the CTRW and cBM processes by superimposing Ornstein–Uhlenbeck (OU) noise, i.e. diffusion in a harmonic potential, upon the CTRW motion such that the particle jiggles around its CTRW position during the waiting time (see Appendix C for more details). Supported by our trajectories, we can relate these two separate mechanisms to the temporal binding and fast jumps of the nanoparticle (CTRW), and the thermal fluctuations it still experiences while being bound (OU noise). In fact, trajectories obtained from simulating a nCTRW appear akin to the ones obtained for our nanoparticle (see Appendix C). Besides linking to the observed trajectories, the nCTRW also supports the bi-Gaussian fits of the CDF (two distinct diffusive modes) and explains the observed antipersistent behavior of the VAF since the OU process tends to drive a particle back towards its ‘binding site’ set by the CTRW. Additionally, numerical simulations and an analytical derivation (Appendix C) indicate that, for strong enough OU noise, the time-ensemble averaged MSD of a particle subject to a nCTRW increases sublinearly (anomalous diffusion) over an intermediate time range before saturating towards a linear increase in time in the long time limit.^{40,45} This makes our results consistent with the theoretical description of a nCTRW, which thus provides a promising (combined) mechanism to describe the diffusive motion of temporally bound, thermal particles. Overall, we have demonstrated that, already for a coarse-grained simulation set-up, theoretical models require a superposition of stochastic processes to fully grasp the complexity often encountered in biological systems.

Our model considers weak transient binding of the nanoparticle to receptors, where receptors are often exchanged underneath the nanoparticle. Interestingly, even at higher receptor concentrations, when the particle becomes well-wrapped by the membrane, to the point of endocytosis, we have found that the diffusion retains its normal profile, albeit at a lower value of a diffusion coefficient (see movie 1, ESI†). Moreover, when we change the model such that the nanoparticle is permanently bound to a certain number of receptors in the fluid membrane, to mimic a situation of strong non-detachable receptor–ligand bonds observed in some systems,^{49} we recover normal diffusion of the nanoparticle along with its associated receptors (Fig. 10).

In contrast, if the membrane fluidity is decreased such that the receptors cannot freely diffuse anymore, the anomalous diffusion naturally arises. The particle diffuses normally until it hits a region with a higher local receptor concentration where it gets trapped, resulting in a lower local diffusion coefficient. Occasionally, the particle hops between such regions, which gives rise to a second, higher diffusion coefficient, and overall a trajectory that is inconsistent with a random walk. This behaviour is more pronounced at higher receptor concentrations, in good agreement with the experimental study of Hsieh et al.^{13} for a system of a gold nanoparticle bound to GM1 ganglioside or DOPE lipids in supported DOPC bilayers. Specifically, they also reported trajectories of strong transient confinements and overall anomalous diffusion, which could be decomposed into two effective diffusion coefficients. Furthermore, their study showed that the anomalous diffusion is more pronounced at higher receptor concentrations, consistent with our simulations. Very similar results to ours were also reported in high-speed single-particle tracking of a gold nanoparticle bound to GM1 receptors in model membranes.^{2} Interestingly, a comparison of the observed behaviour with previous numerical work also yielding subdiffusion at intermediate timescales,^{12} hints at the possibility of mapping at least, due to its solid nature, our cross-linked membrane system onto a particle diffusing through a fixed energy valley landscape.

Our study suggests that the change in the membrane structure, and/or the inability of the receptor to freely diffuse, can possibly explain a plethora of previously reported experimental results and drive new studies. The conclusion can be easily tested in membranes of controlled fluidities.^{50} Our approach is different to previous numerical approaches e.g.ref. 10, 12, 17 and 51 as it is particle-based: it incorporates explicit molecular ingredients, their interactions and mechanics. Our simulation set-up can easily incorporate additional effects regularly found in more biologically-realistic settings, such as heterogeneity in the lipid composition,^{52} direct interactions between membrane proteins,^{53} asymmetry of the nanoparticle ligand arrangement,^{54} or the presence of membrane-deforming machinery.^{55} We hope that our study will inspire future feedback between experimental studies of membrane-adhering components and coarse-grained simulations.

Fluid model.
In the fluid model the membrane particles interact with one another through a pairwise interparticle potential given by^{18,19}

where the repulsive branch (r < r_{m}) is given by a ‘soft’ 4-2 Lennard-Jones (LJ) potential. The attractive branch (r_{m} < r < r_{c}) is a cosine function that decays to zero at the cutoff radius r_{c}. The exponent ζ determines how rapidly the function tends to zero and serves as a measure of the diffusivity of the membrane particles. Its value is set at either ζ = 4.0 to obtain a fluid state where particles can freely diffuse through the membrane, or ζ = 2.5 resulting in a more gel-like membrane state where diffusion of particles is severely limited.^{18,19} More precisely, this results in particle diffusion coefficients of the order of ∼0.1σ^{2}/τ and ∼10^{−4}σ^{2}/τ respectively with τ = (mσ^{2}/k_{B}T)^{1/2} the time unit of the system.^{18} The orientation-dependent function yields

and drives the membrane particles towards a flat configuration with its direct surroundings. Note that this form neglects spontaneous curvature and we thus assume the membrane to be locally flat on the investigated length scales, i.e. only a small part of the cell membrane is modelled. The parameter μ = 3.0 can be interpreted as a weight of the energy penalty when the particles are deviating from a flat configuration, and therefore relates to the bending rigidity of the model membrane.

(A1) |

Here r_{ij} denotes the distance vector between particles i and j with r_{ij} = |r_{ij}| and _{ij} = r_{ij}/r_{ij}, while the unit vectors n_{i} and n_{i} represent the axes of symmetry of particle i and j respectively. We set ε = 4.34k_{B}T, r_{m} = 2^{1/6}σ, and r_{c} = 2.6σ. The potential U is separated in a distance-dependent part u(r) that forces particles to stick together and an orientation-dependent part ϕ(_{ij},n_{i},n_{j}) which substitutes the hydrophobic effects of the lipids. The former of these is described by

(A2) |

ϕ(_{ij},n_{i},n_{j}) = 1 + μ[(n_{i} × _{ij})·(n_{j} × _{ij}) − 1], | (A3) |

Cross-linked model.
The cross-linked model is created by placing all particles on the nodes of a standard triangulated mesh with hexagonal symmetry.^{56} To avoid overlap, each pair of particles repels one another through a Weeks–Chandler–Andersen (WCA) potential, i.e.

where ε_{0} = 5.0k_{B}T and r denotes the distance between the centers of two membrane particles. We then enforce surface fixed connectivity to the membrane by connecting each particle with its six nearest neighbours via a harmonic spring potential (see Fig. 1)

which is described in terms of the spring constant K_{s} = 18k_{B}T/σ^{2} and an equilibrium bond length r_{B} = 1.23σ. These model the stretching rigidity and the equilibrium configuration of the membrane respectively. To ensure an energetically favorable flat membrane, the model includes a bending rigidity in terms of a dihedral potential between adjacent triangles on the mesh

(A4) |

U_{stretching}(r) = K_{s}(r − r_{B})^{2}, | (A5) |

U_{bending}(ϕ) = K_{b}[1 + cos(ϕ)]. | (A6) |

Here K_{b} = 20k_{B}T is the bending constant and ϕ the dihedral angle between opposite vertices of any two triangles sharing an edge.

U_{Morse}(R) = B(e^{−2a(R−R0)} − 2e^{−a(R−R0)} + C), | (B1) |

r(t) = r_{α}(t) + ηr_{OU}(t), | (C1) |

〈r^{2}(t)〉 = 〈r_{α}^{2}(t)〉 + η^{2}〈r_{OU}^{2}(t)〉. | (C2) |

Fig. 9 Plots of (A) a sample trajectory 1D x(t) and (B) the time-ensemble averaged MSD of a nCTRW with Ornstein–Uhlenbeck noise. MSDs are retrieved for increasing noise strengths η, while the trajectory corresponds to the largest noise strength η = 0.1. Figures are adapted from ref. 45, with the permission of AIP Publishing. |

Here brackets denote either an ensemble or a time-ensemble average (which we label with subscripts ‘e’ and ‘te’ respectively), and the averaging for both processes is done with respect to their individual probability density functions (PDFs), i.e. P_{α}(r_{α},t) and P_{OU}(r_{OU},t).

The OU process is retrieved by integrating the overdamped Langevin equation^{26,45}

(C3) |

(C4) |

The CTRW process is described in terms of random jumps Δr and waiting times t_{w} in between these jumps, which are drawn from independent distributions p(Δr) and ψ(t_{w}) respectively. In the Fourier–Laplace domain (depicted with the transformation variables k, s) the PDF of the CTRW relates to the jump and waiting time distributions via the Montroll–Weiss equation^{43,57}

(C5) |

Based on this relation and following the work of, ref. 57–59 one can show that for a long-tailed waiting time distribution (asymptotic behavior ψ(t) ∼ (t/τ)^{−(1−α)} and ψ(s) ∼ 1 − (τs)^{α}, with τ a decay time scale), the ensemble averaged MSD grows subdiffusively in time according to

(C6) |

(C7) |

Here K_{α} is the anomalous diffusion coefficient, Γ the Gamma function, and T ≫ t the total time used for time-averaging (the length of individual trajectories).

Combining these results yields an expression for the time-ensemble averaged MSD of the nCTRW:

(C8) |

Inspection of this formula reveals three different regimes. Using 〈r^{2}(t)〉_{te} ∼ 4D_{app}t we note that for small times kt ≪ 1 the MSD grows linearly in time with an apparent diffusion coefficient

(C9) |

(C10) |

Thus, at intermediate times kt ≈ 1 and for strong enough noise η there exists a third regime where D_{app} decreases in value and as a result the MSD grows subdiffusively in time. Moreover, the expression for the MSD (eqn (C8)) has been confirmed with numerical simulations, indicating the emergence of separate diffusive regimes upon increasing the noise strength (see Fig. 9). Finally, we mention that the time dependence of the time-ensemble averaged MSD is independent of the asymptotic behavior of ψ(t), i.e. α only enters through the total length of the simulation T and thus in our case it cannot be used to determine whether the CTRW contribution to the motion in itself is also anomalous (α < 1) or describes normal diffusion (α = 1).

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## Footnote |

† Electronic supplementary information (ESI) available: Supplementary movie 1. Caption: Visualisation of the motion and trajectory of a nanoparticle (colored in magenta) on a fluid membrane consisting of 60% receptors (colored in green). The total time span is on the order of ∼12000τ. Supplementary movie 2. Caption: Visualisation of the motion and trajectory of a nanoparticle (colored in magenta) on a fluid membrane consisting of 20% receptors (colored in green). The total time span is on the order of ∼12000τ. Supplementary movie 3. Caption: Visualisation of the motion and trajectory of a nanoparticle (colored in magenta) on a gel-like membrane consisting of 20% receptors (colored in green). The total time span is on the order of ∼12000τ. See DOI: 10.1039/d0sm00712a |

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