Bimodal Analysis of Mammary Epithelial Cell Migration in Two Dimensions

Abstract

Cell migration paths of mammary epithelial cells (expressing different versions of the promigratory tyrosine kinase receptor Her2/Neu) were analyzed within a bimodal framework that is a generalization of the run-and-tumble description applicable to bacterial migration. The mammalian cell trajectories were segregated into two types of alternating modes, namely, the “directional mode” (mode I, the more persistent mode, analogous to the bacterial run phase) and the “re-orientation mode” (mode II, the less persistent mode, analogous to the bacterial tumble phase). Higher resolution (more pixel information, relative to cell size) and smaller sampling intervals (time between images) were found to give a better estimate of the deduced single cell dynamics (such as directional-mode time and turn angle distribution) of the various cell types from the bimodal analysis. The bimodal analysis tool permits the deduction of short-time dynamics of cell motion such as the turn angle distributions and turn frequencies during the course of cell migration compared to standard methods of cell migration analysis. We find that the 2-h mammalian cell tracking data do not fall into the diffusive regime implying that the often-used random motility expressions for mammalian cell motion (based on assuming diffusive motion) are invalid over the time steps (fraction of minute) typically used in modeling mammalian cell migration.

Appendix

A key finding in bacterial chemotaxis is that the turn angle distribution is unaffected by the presence of a chemoattractant, while the run time distribution is modulated (bacteria extend their run times when moving in directions of increasing chemoattractant concentration),7 thus resulting in biased movement toward increasing chemoattractant concentration. Moreover, by demonstrating that the same run time increases were induced by a spatially homogeneous but time-varying chemoattractant concentration, Berg10 showed that E. coli were responding to the substantial derivative of the chemoattractant concentration (Dc/Dt, where c is the attractant concentration), so that the chemosensing mechanism in bacteria is related to the time rate of change in bound receptors on the cell surface. This rules out the possibility that in E. coli the chemosensory mechanism is based on the differences in the number of bound receptors over the cell surface (i.e., a direct sensing of the chemoattractant gradient by spatial comparison). A mathematical analysis of chemosensing by Berg and Purcell9 shows that despite the small cell size (∼1 μm) spatial sensing of a chemoattractant gradient is possible for E. coli; specifically, taking into account fluctuations in chemoattractant concentration on the spatial scale of a cell, Berg and Purcell derived expressions for the minimum time required for temporal sensing, T ^temporal_sensing , and for spatial sensing, T ^spatial_sensing , given by

$$ T_{{\text{sensing}}}^{{\text{temporal}}} > \frac{1} {2}\left[ {\pi aD\left( {\frac{{Ns}} {{Ns + \pi a}}} \right)\left( {\frac{{\bar cc_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} {{\bar c + c_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right)\left( {\frac{1} {{\bar c}}\frac{{\partial \bar c}} {{\partial t}}} \right)^2 } \right]^{ - 1/3} $$

(A1)

$$ T_{{\text{sensing}}}^{{\text{spatial}}} > \left[ {\frac{1} {2}\pi a^3 D\left( {\frac{{Ns}} {{Ns + \pi a}}} \right)\left( {\frac{{\bar cc_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} {{\bar c + c_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right)\left( {\frac{1} {{\bar c}}\frac{{\partial \bar c}} {{\partial x}}} \right)^2 } \right]^{ - 1} $$

(A2)

where a is the radius of the cell, D is the self-diffusion coefficient of the chemoattractant, N is the number of receptors on the cell surface, s is the cell-receptor radius, $\bar c$ is the equilibrium concentration of the chemoattractant, c _1/2 is the dissociation constant for the receptor-chemoattractant binding, and x is the direction in which the chemoattractant gradient exists. For temporal gradients created by the movement of the cell, $(1/\bar c)(\partial \bar c/\partial t) = (v/\bar c)(\partial \bar c/\partial x),$ where v is the cell speed. For typical values of these parameters for E. coli responding to an aspartate gradient, Berg and Purcell found $T_{{\text{sensing}}}^{{\text{temporal}}} > 0.4 - 1.4\;{\text{s}}$ (depending on magnitude of chemoattractant gradient) and for spatial sensing, $T_{{\text{sensing}}}^{{\text{spatial}}} > 1.7\;{\text{s}}{\text{.}}$ Since the run lengths of flagellated bacteria are typically of the order of 1 s and longer, this analysis suggests that a bacterium could use either temporal or spatial sensing; however, because the swimming motion of a bacterium causes the cell body to rotate, the resulting disturbance to the surrounding liquid medium would create fluctuations in the chemoattractant gradient much larger than the gradient itself, thus ruling out the spatial sensing mechanism. To perform a similar analysis for eukaryotic cells, we use the experimental conditions of Sai et al.47 for the study of chemotaxis of HL60 cells stably expressing CXCR2 receptor in a microfluidic device-generated gradients of CXCL8 chemokine. For these cells in this chemotaxis assay, $a = 7.5\,\mu {\text{m}},\;D = 10^{ - 6} \,{\text{cm}}^{\text{2}} {\text{/s}},\;\bar c = 1.25\,{\text{nM}},\;c_{1/2} = 1.5\,{\text{nM}},\;(1/\bar c)(\partial \bar c/\partial t) = (v/\bar c)(\partial \bar c/\partial x) = 2 \times 10^{ - 4} \,{\text{s}}^{ - 1} .$Taking Ns/(Ns + πa) = 0.5 (a typical value), and using these values in Eqs. (1) and (2), we find that $T_{{\text{sensing}}}^{{\text{temporal}}} > 1\,{\text{min}}$ and $T_{{\text{sensing}}}^{{\text{spatial}}} > 18\,{\text{s}} .$ We note that the larger size of these cells (compared to bacteria) results in the time threshold for spatial sensing being less than that of temporal sensing; this is the reverse of the situation for bacteria, in which the time threshold for spatial sensing is greater than that of temporal sensing. From the bimodal analysis of the MCF-10A cells reported here, we find that the mean directional-mode time duration of these cancer cells ranges in several minutes compared to bacterial run times of seconds. Hence both temporal and spatial sensing mechanisms remain feasible for these eukaryotic cells.