Quantitative relationships between single‐cell and cell‐population model parameters for chemosensory migration responses of alveolar macrophages to C5a

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Abstract
Phenomenological parameters from a mathematical model of cell motility [1] are used to quantitatively characterize chemosensory migration responses of rat alveolar macrophages migrating to C5a in the linear under‐agarose assay, simultaneously at the levels of both single cells and cell populations. This model provides theoretical relationships between single‐cell and cell‐population motility parameters. Our experiments offer a critical test of these theoretical linking relationships, by comparison of results obtained at the cell population level to results obtained at the single‐cell level.Random motility of a cell population is characterized by the random motility coefficient, μ (analogous to a particle diffusion coefficient), whereas single‐cell random motility is described by cell speed, s, and persistence time, P (related to the period of time that a cell moves in one direction before changing direction). Population chemotaxis is quantified by the chemotactic sensitivity, χo, which provides a measure of the minimum attractant gradient necessary to elicit a specified chemotactic response. Single‐cell chemotaxis is characterized by the chemotactic index, CI, which ranges from 0 for purely random motility to 1 for perfectly directed motility. Measurements of cell number versus migration distance were analyzed in conjunction with the phenomenological model to determine the population parameters while paths of individual cells in the same experiment were analyzed in order to determine the single‐cell parameters.The parameter μ shows a biphasic dependence on C5a concentration with a maximum of 1.9 × 10−8 cm2/sec at 10−11 M C5a and relative minima of 0.86 × 10−8 cm2/sec at 10−7 M C5a and 1.1 × 10−8 cm2/sec in the absence of C5a; s and P remain fairly constant with C5a concentration, with s ranging from 2.1 to 2.5 μm/min and P varying from 22 to 32 min. χo is equal to 1.0 × 10−6 cm/receptor for all C5a concentrations tested, corresponding to 60% correct orientation for a difference of 500 bound C5a receptors across a 20 μm cell length. The maximum CI measured was 0.2.Values for the population parameters μ and χo were calculated from single‐cell parameter values using the aforementioned theoretical linking relationships. The values of μ and χo calculated from single‐cell parameters agreed with values of μ and χo determined independently from population migrations, over the full range of C5a concentrations, confirming the validity of the linking equations. Experimental confirmation of such relationships between single‐cell and cell‐population parameters has not previously been reported.