Conventional analysis of movement on non-flat surfaces like the plasma membrane makes Brownian motion appear anomalous

Bioinformatics

Diffusion depends upon the properties of both the diffusing species and the medium, so a known medium can be used to characterise an unknown molecule or vice versa. On biological surfaces, the primary interest is to characterize the interaction of different molecules with a membrane and to establish whether movement is or is not Brownian where non-Brownian movement would require the recognition of a mechanism.

When the medium is a surface like the plasma membrane and 2D imaging is used the surface must be both flat and aligned with the imaging plane to correctly characterize the diffusing species in the medium—a requirement that is rarely acknowledged and, presumably, even less frequently met. Even then 2D images only permit 2D distance measurements. Misalignment of the surface with the imaging field or non-flat surfaces compromise the measurement of distances, since the commonly used 2D, but also 3D, measurements require the physically impossible, that diffusing molecules can leave and reenter the surface, i.e., illegitimate movement.

We set out to investigate whether it is possible to obtain diffusion coefficients that are independent of surface topography, which would permit the identification of genuine non-Brownian diffusion. In our quest, we revealed that in simulations emulating simple diffusion, topographical features alone can produce super- and subdiffusion. This is a critical finding when interpreting the movement of membrane molecules.

Vector calculus and partial differential equations are traditionally formulated in a continuous setting where the solutions are analytical expressions, often in a closed form. When computers are used to derive approximate solutions to the differential equations, discretizations of well-known continuous operators are usually applied. With this approach, it has to be established that the discretized, computed approximation and the desired analytical solution do not deviate too much, typically by assuring convergence with finer discretization. However, the approach we followed, discrete calculus, is inherently discrete and treats a discrete domain (essentially a graph) as its own entity without reference to an underlying continuum31. Consequently, since the concern in traditional discretization about convergence to a continuous solution is not a consideration in discrete calculus, we do not present a convergence analysis. Using our discrete approach also means that details on the cell surface not resolved by the imaging device are not represented in the subsequent topography analysis, which means that the genuine topography effects are likely to be larger than we report.

A critical observation is that 2D measurements always underestimate the net movement on non-flat surfaces, producing diffusion coefficients that are lower than those for a flat surface. On flattish areas movement would be only slightly reduced, which could be interpreted as a small reduction in the rate of diffusion, while where the movement has a substantial Z component, the dramatic fall in the apparent local rate diffusion risks being interpreted as binding or trapping19. Still 2D tracking of movement over the plasma membrane is common and variations in topography risk being mistaken for reduced diffusion and as indications of anomalous diffusion.

It follows that at least some of the apparent reduction in the rate of movement and the trapping observed in plasma membrane diffusion studies is attributable to 2D imaging, 2D distance measurements and topography19. In addition, the use of large gold particles could lead to erroneous results12 and the reduction in the long-term diffusion, i.e. the estimated difference between diffusion in model membranes and the plasma membrane, has been shown to be halved with the small lipid probes used in STED-FCS compared with gold-particle SPT-studies11. Interestingly, the lipids studied were found to undergo confined diffusion in an Arp 2/3-dependent, i.e. actin polymerisation-dependent, manner. The confined diffusion was mostly seen at the lamellipodia- (also Arp 2/3-dependent) rich cell edge, suggesting a contribution of cell topography to the residual anomalous diffusion.

Another phenomenon that can reduce diffusion and account for differences between model and cellular membranes is molecular crowding, i.e. that the protein to lipid ratio in biological membranes is relatively high, making large membrane areas inaccessible4. That crowding plays a role was supported by a recent study comparing diffusion in giant unilamellar vesicles, giant plasma membrane vesicles (GPMVs) and the plasma membrane of intact cells, where diffusion in the former (containing mostly lipids) was about five times faster than in the GPMVs and 10–25 times faster than in cells32. The difference between the cells and GPMVs is probably caused by the cells having a more variable topography. Interestingly, the measurements were performed on the basal part of the cells, where non-flat topographical features are considered to be less prominent than at the apical cell side. However, they still exists33.

In an isotropic medium, identical diffusion coefficients could be calculated from SPT measured in one, two or three dimensions. However, membranes are continuous surfaces and not isotropic. A related question is whether on deformed surfaces switching from 2D to 3D distance measurements is sufficient to produce topography-independent diffusion coefficients. We find that while 3D measurements are clearly a substantial improvement on 2D, especially over short timescales, they ultimately fail because the 3D does not recognize that movement is confined to the surface.

Interestingly, the reduced movement recorded with 3D in deformed surfaces falls with time. This is because the 3D ignores the topography and therefore is not confined to the surface. At longer times the difference between the shortest linear distance and the actual movement increases, as more paths include non-horizontal parts of the surface. The MS(quared)D gives extra weight to longer tracks, explaining why confined diffusion becomes more apparent at longer intervals.

On a flat surface straight-line distances are always possible, in the sense that a particle could take this route, while on a non-flat surface most 3D straight lines are impossible, because they leave the surface. This created our rationale for introducing the SWSD, i.e. distances must remain within the surface. The SWSD produces diffusion coefficients that are closer to those on a flat surface and less dependent on the topography than the 3D measurements. Importantly, in our simulations a deviation of the SWSD from one suggests anomalous diffusion, but is really only reflecting apparent anomalous diffusion, since it is caused by the topography and unrelated to the interactions between the diffusing species and the surface, i.e. the primary question in most studies. The deviation reflects the prevalence of folds and deformed topographical features of which the folds are accurately measured by the SWSD, but even the SWSD understates movement on the deformed parts of the surface.

Requiring measurements over a surface to remain in the surface is an improvement on 2D and 3D and the SWSD provides the best measure of movement in every simulation. On folded surfaces the SWSD is accurate, but on deformed surfaces, it still underestimates the actual movement. This arises because the MSD is based on the net movement and ignores the path, which on a deformed surface does not correctly characterize the actual movement. The SWSD finds the shortest route, analogously to taking a pass through a mountain range, which is representative of the efficient route chosen by hikers, but underreports the relatively directionless and longer routes taken by mountain goats. This makes the SWSD an imperfect proxy for the actual movement. When a shortcut is present, initially the calculated diffusion increases since the shortcut provides access to larger areas and increased spreading dominates. Later, the relatively small actual flux through the shortcut is overwhelmed by the majority of the diffusing species whose routes did not utilise the shortcut. Net movement is then underreported by the shortened SWSD, producing a fall in the measured diffusion. This caveat means that, even if the precise topography of a biological membrane is known and the SWSD used, molecular movement and simulations however analysed cannot correctly report the rate of diffusion or differentiate between topography-induced and other causes of anomalous diffusion.

Our results suggest that the decrease in diffusion with time that is generally interpreted as confined diffusion, e.g., hop diffusion, could be caused by topography as demonstrated with the 2D measurements on cells. It should be noted that it has been argued that the MSD versus Δt is not ideal for spotting anomalous diffusion and is also incapable of assessing fractions of multicomponent populations of tracks and therefore the cumulative probability distribution of the square displacements rather than individual tracks should be analysed34,35. However, regardless of the method used to assess anomalous diffusion, the effect of topography must not be ignored, if the aim is to assess the interaction of molecules with the surface rather than characterising their spread over an unknown topography.

The SWSD provides the most accurate diffusion coefficients but still underestimates and distorts the pattern of movement. The ultimate solution lies in simulating simple diffusion over the surface to show the pattern that Brownian motion alone would produce and then comparing this with the experimentally observed pattern from SPTs or FRAP. It should then be possible to establish whether the experimentally observed non-Brownian movement exceeds that caused by topography. The major practical problem is how to obtain sufficiently detailed maps of the surface of living cells. In their absence, SPT and other techniques where diffusion is assessed, FCS and FRAP, should be interpreted with caution and interpretations involving topography considered before more elaborate models are invoked.

In our simulations, the topography of the surface per se does not alter the interaction between the particle and surface and therefore the mechanism underlying movement is identical on flat and non-flat surfaces. There are, however, scenarios where topography could directly affect diffusion for instance by blocking the entry of large and/or inflexible particles from areas of high curvature, as reported for proteins and thin membrane tubes36,37. Particles excluded from topographical features could diffuse more rapidly, which may explain the differential diffusion behaviour of synthetic lipids observed in cells at super lateral resolution38 and the remarkable fast diffusion of a cholesterol analogue39. It has also been reported that diffusion along the longitudinal direction in membrane nanotubes slows as the radius of the tube is reduced40,41, but note that this involves curvatures tighter than those found in cell membranes and the effect of the diffusion may be caused by crowding or changes in membrane tension and lipid packing that also restrict diffusion42,43. However, in cases where topography does effect the diffusion our protocol would identify the diffusion as anomalous. The exception being when the surface covers a thin tube, that although it is a foldable surface the SWSD would, like in the case with the shortcut, underestimate the actual distance moved around the axis of the tube with a full circle being measured as no net movement. A remedy could be to consider only the longitudinal movement which, at least when using the MSD versus Δt approach, would show Brownian movement37.

To determine the contribution of topography to any deviation of the diffusion from Brownian, the topography must be established, which is non-trivial given the dynamics, folding potential and thickness (<5 nm) of biological membranes. Our findings suggest a scheme for disentangling the topographical component from other sources of anomalous diffusion (Fig. 8). Firstly, create a 3D model of the surface. Secondly, run simulations with multiple start points covering the area of the SPT tracks producing probability distributions. Finally, assess whether or not the recorded tracks are consistent with the distribution generated in the simulations44. Note that, as visualised in Suppl. Figure 1, the variation between tracks is substantial, meaning that many tracks over the same surface are required to determine whether anomalous diffusion is indeed taking place29. That the start position is important for the simulations further emphasises the need for a large number of starting points for proper surface characterisation.

When topography has not been considered as a cause of anomalous diffusion, the norm, the resulting plasma membrane models are questionable since the primary objective is to understand the interaction of molecules with a biological surface, not to characterise spreading patterns over an unknown and varying topography. Establishing and factoring out topography needs to be done before invoking more complex explanations for reduced and anomalous diffusion – Occam rules.

In conclusion membrane topography has been identified as a cause of both the consistent underestimation of diffusion and the overestimation of apparent anomalous diffusion. Measurements are more accurate when the topography is known and distance measurements are kept within the surface, i.e., using the SWSD on a high-resolution 3D surface. Membrane topography itself can also cause apparent anomalous diffusion by allowing increased or decreased spread of the diffusion species. To factor out the topographically induced anomalous diffusion simulations of simple diffusion over the surface is required.

Fig. 8
Fig. 8

Scheme for disentangling the contribution of topography from other sources of anomalous diffusion

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