Abstract
Cell membrane-bound receptors control signal initiation in many important cellular signaling pathways. In many such systems, receptor dimerization or cross-linking is a necessary step for activation, making signaling pathways sensitive to the distribution of receptors in the membrane. Microscopic imaging and modern labeling techniques reveal that certain receptor types tend to co-localize in clusters, ranging from a few to tens, and sometimes hundreds of members. The origin of these clusters is not well understood but they are likely not the result of chemical binding. Our goal is to build a simple, descriptive framework which provides quantitative measures that can be compared across samples and systems, as groundwork for more ambitious modeling aimed at uncovering specific biochemical mechanisms. Here we discuss a method of defining clusters based on mutual distance, applying it to a set of transmission microscopy images of VEGF receptors. Preliminary analysis using standard measures such as the Hopkins’ statistic reveals a compelling difference between the observed distributions and random placement. A key element to cluster identification is identifying an optimal length parameter \(L^*\). Distance based clustering hinges on the separation between two length scales: the typical distance between neighboring points within a cluster vs. the typical distance between clusters. This provides a guiding principle to identify \(L^*\) from experimentally derived cluster scaling functions. In addition, we assign a geometric shape to each cluster, using a previously developed procedure that relates closely to distance based clustering. We applied the cluster [support] identification procedure to the entire data set. The observed particle distribution results are consistent with the random placement of receptors within the clusters and, to a lesser extent, the random placement of the clusters on the cell membrane. Deviations from uniformity are typically due to large scale gradients in receptor density and/or the emergence of “mega-clusters” that are very likely the expression of a different biological function than the one behind the emergence of the quasi-ubiquitous small scale clusters.
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Notes
- 1.
For the nearest neighbor distance, the mean is \(\langle r \rangle _\mathsf {_{NN}}= b{/}2\) and the mode (maximum probability) \(r^*_\mathsf {_{NN}} = b /\sqrt{2\pi } \) corresponds to the radius of a circle of area A / N, \(\pi {r^*_\mathsf {_{NN}}}^2 = b^2 = 1 /\lambda \).
- 2.
We use “optimal length parameter” (\(L^*\)) to distinguish from the specific choice of [5].
- 3.
From Eqs. (1) and (S4):\(\langle r \rangle _\mathsf {_{NN}} = \frac{1}{2}b\), \(\langle r \rangle _\mathsf {_{NN2}} = \frac{3}{4}b\) where \(b^2=\frac{A}{N}\); all images have the same area.
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Experimental details, mathematical definitions and methodologies, and their justification are provided as supplementary materials available at https://www.dropbox.com/s/8bggjrzhzr1vsne/Supplementary-Materials-Clustering-Paper.pdf?dl=0.
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Güven, E., Wester, M.J., Wilson, B.S., Edwards, J.S., Halász, Á.M. (2018). Characterization of the Experimentally Observed Clustering of VEGF Receptors. In: Češka, M., Šafránek, D. (eds) Computational Methods in Systems Biology. CMSB 2018. Lecture Notes in Computer Science(), vol 11095. Springer, Cham. https://doi.org/10.1007/978-3-319-99429-1_5
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