Modeling and measuring signal relay in noisy directed migration of cell groups

Abstract

We develop a coarse-grained stochastic model for the influence of signal relay on the collective behavior of migrating Dictyostelium discoideum cells. In the experiment, cells display a range of collective migration patterns, including uncorrelated motion, formation of partially localized streams, and clumping, depending on the type of cell and the strength of the external, linear concentration gradient of the signaling molecule cyclic adenosine monophosphate (cAMP). From our model, we find that the pattern of migration can be quantitatively described by the competition of two processes, the secretion rate of cAMP by the cells and the degradation rate of cAMP in the gradient chamber. Model simulations are compared to experiments for a wide range of strengths of an external linear-gradient signal. With degradation, the model secreting cells form streams and efficiently transverse the gradient, but without degradation, we find that model secreting cells form clumps without streaming. This indicates that the observed effective collective migration in streams requires not only signal relay but also degradation of the signal. In addition, our model allows us to detect and quantify precursors of correlated motion, even when cells do not exhibit obvious streaming.

Figure 1. Time lapse images during the chemotaxis of wild-type and mutant cells in linear cAMP gradient.

(A) Wild-type cells can relay the signal by secreting cAMP from their tails. They form streams which are unstable towards swirling clumps. (B) The mutant cells (aca-) lacking the ACA enzyme cannot secrete cAMP and thus undergo uniform motion in the direction of the external cAMP gradient. (C) Some representative tracks of aca- cells obtained with the tracking algorithm. Vector displacements along the tracks are color coded according to real time. (D) Distributions of the angle representing the displacement of cells exposed to different constant gradient amplitudes with respect to the vertical axis. The panel labels (5 nM to 5 µM) denote the cAMP concentration in the reservoir.

Figure 2. The time autocorrelation and variance of θ.

versus τ for three different imposed cAMP gradient strengths corresponding to cAMP concentrations of 50 nM (black bullet), 0.5 µM (black square) and 5 µM (black triangle) in the reservoir on the cell exit side of the gradient chamber. The solid lines are best fits to yielding values for of min, 0.94 min and 1 min. Autocorrelations are obtained from , , and cells, respectively. Error bars represent the standard deviation. (D) The variance , versus the distance from the cell input side of the gradient chamber for the three gradient strengths in Figs. 2A–C is plotted using the same symbols black bullet, black square and black triangle.

Figure 3. Cell tracks from simulations for the three representative modes of collective motion, uncorrelated motion, streaming, and aggregation.

(A) For a relatively slow cAMP secretion rate () the cells move independently, showing no sign of collective motion. (B) If the cAMP secretion is moderate () cells form streams. (C) For high relative cAMP secretion rate () cells exhibit aggregation and therefore form clumps. Figs. (D–F) are snapshots from the same simulations exhibiting the spatial organization of the cells.

Figure 4. Mean progression speed and the cell density are used in quantifying collective motion.

(A) is used to compare experimental data (aca- with ) with a representative single run that is obtained with model parameters that mimic the experimented aca- mutant cells. (B) and (C) show respectively, , and as a function of the distance from the cell reservoir for , and three different cAMP secretion rates. Error bars are obtained from different realizations with the same simulation parameters for each curve and represent the standard error of the mean. (D) The maximum in the region is plotted against its corresponding . Each point corresponds to a single numerical run. For (A), when the cells enter the chamber at , we initialize the cell orientation vectors for cell according to a distribution of the angle with respect to the , where this distribution is uniform in the range, . This is done so as to roughly match the experimental at .

Figure 5. Mean progression , as a function of relative signaling rate , and relative degradation rate .

(A) as a function of . Error bars are obtained from many numerical realizations (between ) and represent the standard error of the mean. In the top panel, the degradation rate is comparable to the experimentally obtained degradation of the phosphodiesterase. In the bottom panel, we used small cAMP degradation rate, which models the mutant PDE1- cells, incapable of secreting the enzyme that degrades cAMP. (B) as a function of the relative cAMP secretion and relative cAMP degradation rates. The red regions correspond to uncorrelated motion. The dynamically unstable regions, where streams are likely to form, of the (, ) phase space is labeled with yellow and white. Blue regions are associated with aggregate formation.