Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to $0$, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least $2$. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1

Deformability and collision-induced reorientation enhance cell topotaxis in dense microenvironments

In vivo, cells navigate through complex environments filled with obstacles. Recently, the term 'topotaxis' has been introduced for navigation along topographic cues such as obstacle density gradients. Experimental and mathematical efforts have analyzed topotaxis of single cells in pillared grids with pillar density gradients. A previous model based on active Brownian particles has shown that ABPs perform topotaxis, i.e., drift towards lower pillar densities, due to decreased effective persistence lengths at high pillars densities. The ABP model predicted topotactic drifts of up to 1% of the instantaneous speed, whereas drifts of up to 5% have been observed experimentally. We hypothesized that the discrepancy between the ABP and the experimental observations could be in 1) cell deformability, and 2) more complex cell-pillar interactions. Here, we introduce a more detailed model of topotaxis, based on the Cellular Potts model. To model persistent cells we use the Act model, which mimicks actin-polymerization driven motility, and a hybrid CPM-ABP model. Model parameters were fitted to simulate the experimentally found motion of D. discoideum on a flat surface. For starved D. discoideum, both CPM variants predict topotactic drifts closer to the experimental results than the previous ABP model, due to a larger decrease in persistence length. Furthermore, the Act model outperformed the hybrid model in terms of topotactic efficiency, as it shows a larger reduction in effective persistence time in dense pillar grids. Also pillar adhesion can slow down cells and decrease topotaxis. For slow and less persistent vegetative D. discoideum cells, both CPMs predicted a similar small topotactic drift. We conclude that deformable cell volume results in higher topotactic drift compared to ABPs, and that feedback of cell-pillar collisions on cell persistence increases drift only in highly persistent cells

Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph G1Rd is the infinite graph whose nodes are points in Rd, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G1R2 of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of G1Rd to closed convex subsets X of Rd. We show that the graph G1Rd[X] is connected precisely when the radius of r(X) of X is equal to 0, or when r(X) ≥ 1 and the affine dimension of X is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.Optimizatio

Deformability and collision-induced reorientation enhance cell topotaxis in dense microenvironments

In vivo, cells navigate through complex environments filled with obstacles such as other cells and the extracellular matrix. Recently, the term “topotaxis” has been introduced for navigation along topographic cues such as obstacle density gradients. Experimental and mathematical efforts have analyzed topotaxis of single cells in pillared grids with pillar density gradients. A previous model based on active Brownian particles (ABPs) has shown that ABPs perform topotaxis, i.e., drift toward lower pillar densities, due to decreased effective persistence lengths at high pillar densities. The ABP model predicted topotactic drifts of up to 1% of the instantaneous speed, whereas drifts of up to 5% have been observed experimentally. We hypothesized that the discrepancy between the ABP and the experimental observations could be in 1) cell deformability and 2) more complex cell-pillar interactions. Here, we introduce a more detailed model of topotaxis based on the cellular Potts model (CPM). To model persistent cells we use the Act model, which mimics actin-polymerization-driven motility, and a hybrid CPM-ABP model. Model parameters were fitted to simulate the experimentally found motion of Dictyostelium discoideum on a flat surface. For starved D. discoideum, the topotactic drifts predicted by both CPM variants are closer to the experimental results than the previous ABP model due to a larger decrease in persistence length. Furthermore, the Act model outperformed the hybrid model in terms of topotactic efficiency, as it shows a larger reduction in effective persistence time in dense pillar grids. Also pillar adhesion can slow down cells and decrease topotaxis. For slow and less-persistent vegetative D. discoideum cells, both CPMs predicted a similar small topotactic drift. We conclude that deformable cell volume results in higher topotactic drift compared with ABPs, and that feedback of cell-pillar collisions on cell persistence increases drift only in highly persistent cells

Computational modelling of cell motility modes emerging from cell-matrix adhesion dynamics

International audienceLymphocytes have been described to perform different motility patterns such as Brownian random walks, persistent random walks, and Le ´vy walks. Depending on the conditions, such as confinement or the distribution of target cells, either Brownian or Le ´vy walks lead to more efficient interaction with the targets. The diversity of these motility patterns may be explained by an adaptive response to the surrounding extracellular matrix (ECM). Indeed, depending on the ECM composition, lymphocytes either display a floating motility without attaching to the ECM, or sliding and stepping motility with respectively continuous or discontinuous attachment to the ECM, or pivoting behaviour with sustained attachment to the ECM. Moreover, on the long term, lymphocytes either perform a persistent random walk or a Brownian-like movement depending on the ECM composition. How the ECM affects cell motility is still incompletely understood. Here, we integrate essential mechanistic details of the lymphocyte-matrix adhesions and lymphocyte intrinsic cytoskeletal induced cell propulsion into a Cellular Potts model (CPM). We show that the combination of de novo cell-matrix adhesion formation, adhesion growth and shrinkage, adhesion rupture, and feedback of adhesions onto cell propulsion recapitulates multiple lymphocyte behaviours, for different lymphocyte subsets and various substrates. With an increasing attachment area and increased adhesion strength, the cells' speed and persistence decreases. Additionally, the model predicts random walks with short-term persistent but long-term subdiffusive properties resulting in a pivoting type of motility. For small adhesion areas, the spatial distribution of adhesions emerges as a key factor influencing cell motility. Small adhesions at the front allow for more persistent motility than larger clusters at the back, despite a similar total adhesion area. In conclusion, we present an integrated framework to simulate the effects of EC