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

Abstract

Cells are neither flat nor smooth, which has serious implications for prevailing plasma membrane models and cellular processes like cell signalling, adhesion and molecular clustering. Using probability distributions from diffusion simulations, we demonstrate that 2D and 3D Euclidean distance measurements substantially underestimate diffusion on non-flat surfaces. Intuitively, the shortest within surface distance (SWSD), the geodesic distance, should reduce this problem. The SWSD is accurate for foldable surfaces but, although it outperforms 2D and 3D Euclidean measurements, it still underestimates movement on deformed surfaces. We demonstrate that the reason behind the underestimation is that topographical features themselves can produce both super- and subdiffusion, i.e. the appearance of anomalous diffusion. Differentiating between topography-induced and genuine anomalous diffusion requires characterising the surface by simulating Brownian motion on high-resolution cell surface images and a comparison with the experimental data.

Fig. 1

Different measures of a distance. Arrows mark the 2D, the 3D and the shortest within the surface distance between two points in a folded membrane. Note that both the 2D and the 3D distances require that the molecules leave the membrane

Fig. 2

Surfaces comprise nodes with differing numbers of neighbours. Single nodes with three neighbours (blue/grey) are found at external corners. Nodes with four neighbours make up the bulk of the surface, a single example is shown in blue. Single nodes with five neighbours are found at internal corners (orange). The neighbouring nodes are all shown in yellow

Fig. 3

The topography of the surface affects the spread of particles. a Four different surfaces (flat, ridges, pillars and invaginations) and b the probability of finding a particle at any node after 1600 iterations. The starting position was in the centre where the four different surfaces meet. The probability distribution is displayed as a summed Z-projection contour plot, using a log-scale to display the wide range of probabilities

Fig. 4

The shortest within surface distance accurately measures diffusion in folded, and outperforms current methods on deformed surfaces. a A flat and horizontal surface. b A surface with uniformly sized and spaced parallel ridges of four nodes height and four nodes width with two nodes spacing. c A surface with regularly spaced pillars (5 × 5 node base with the corners indented that rise 15 nodes above a horizontal surface on a hexagonal grid with 12 node spacing). Probability distribution simulations were launched at the positions indicated. Diffusion was measured using 2D Euclidean, 3D Euclidean and the shortest within surface distance, in a shown as the MSD/t and in b and c D_rel expressed relative to the measurements on a flat surface

Fig. 5

The SWSD greatly improves diffusion measurements on cell surfaces. A high-resolution topographical map of A6 epithelial cells from hopping ion conductance microscopy. A probability distribution simulation was launched close to the crown of a cell and iterated 2000 times. a The probability of finding a particle at any node after the indicated number of steps displayed as a summed Z-projection using a log-scale with low intensity made transparent to visualise the underlying cells is shown as a contour plot. b The diffusion expressed relative to a flat and horizontal surface (D_rel). Scale bar 25 μm

Fig. 6

Gaps in a folded surface creates short-lived superdiffusion. a A surface with uniformly sized parallel ridges seven nodes high and three nodes wide at 12 node intervals in a 1023,1023,9 volume. Simulations run with a single defect in otherwise perfectly folded surfaces, either a one node wide slot or notch. The start point for the diffusion simulation was midway between the defect and the next ridge or for the control midway between two ridges, illustrated with asterixes. The diffusion coefficients for the ridges with a slot (b) and notch (c) measured using 2D, 3D and the SWSD expressed relative to a flat and horizontal surface (D_rel)

Fig. 7

Topographical features can cause sub- and superdiffusion. Detailed examination of the SWSD diffusion measurement with the surfaces shown in Fig. 6. a Distances measured with the SWSD from the start point for the three surfaces. The stepped false colour look up table, which can also be described as a contour plot, covers the first 0–39 nodes and the ridges are indicated by vertical lines. The start point was midway between ridges −1 and +1 and at the top of the image, being the lowest intensity node in a and at the maximum probability in b, the centre of the white area. The defect is in the +1 ridge to the right of the start point. b Probability distributions at increasing numbers of iterations corresponding to the peak superdiffusion (104), the return to normal diffusion (216) and the maximum subdiffusion (1024) in Fig. 6. To include a wide range of probabilities the look up table uses the square root of the raw data. The ridges appear as vertical lines. c The number of nodes included in the spreading probability distribution increase with iterations. d Comparison of the probability for different distances from the start position at 104, 216 and 1024 iterations

Fig. 8

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