A stochastic description of Dictyostelium chemotaxis

Abstract

Chemotaxis, the directed motion of a cell toward a chemical source, plays a key role in many essential biological processes. Here, we derive a statistical model that quantitatively describes the chemotactic motion of eukaryotic cells in a chemical gradient. Our model is based on observations of the chemotactic motion of the social ameba Dictyostelium discoideum, a model organism for eukaryotic chemotaxis. A large number of cell trajectories in stationary, linear chemoattractant gradients is measured, using microfluidic tools in combination with automated cell tracking. We describe the directional motion as the interplay between deterministic and stochastic contributions based on a Langevin equation. The functional form of this equation is directly extracted from experimental data by angle-resolved conditional averages. It contains quadratic deterministic damping and multiplicative noise. In the presence of an external gradient, the deterministic part shows a clear angular dependence that takes the form of a force pointing in gradient direction. With increasing gradient steepness, this force passes through a maximum that coincides with maxima in both speed and directionality of the cells. The stochastic part, on the other hand, does not depend on the orientation of the directional cue and remains independent of the gradient magnitude. Numerical simulations of our probabilistic model yield quantitative agreement with the experimental distribution functions. Thus our model captures well the dynamics of chemotactic cells and can serve to quantify differences and similarities of different chemotactic eukaryotes. Finally, on the basis of our model, we can characterize the heterogeneity within a population of chemotactic cells.

Figure 1. Experimental setup.

(A) Definition of the coordinate system. (B) Microfluidic gradient mixer, adapted from . The x-direction of the coordinate system corresponds to the direction of fluid flow in the main channel of the device, the y-direction to the direction of the chemoattractant gradient. (C) Trajectories of chemotactic Dictyostelium cells in a gradient of 0.16 nM/m cAMP. The starting point of all trajectories was shifted to (0,0). (D) Average chemotactic index as a function of the cAMP gradient. Note that the data point displayed at very low gradient values (nM/m) corresponds to an experiment where no gradient of cAMP was applied.

Figure 2. Comparison of experimental and simulated histograms.

Experimental histograms (gray boxes) and simulated histograms (red lines) of (A) , (B) , (C) , and (D) . (E) Experimental (gray boxes) and numerical (red line) distributions of as a function of . (F) Each dot marks a cells according to its mean speed and chemotactic index in the (,CI)-plane. Black symbols mark the experimental data, red dots the numerical results. The vertical and horizontal lines indicate the mean speed and chemotactic index of the entire population as obtained from the experiment. The numbers mark the subpopulations defined by the four quadrants. They are differentiated according to their directionality and speed, (1) slow non-chemotactic, (2) fast non-chemotactic, (3) slow chemotactic, and (4) fast chemotactic cells.

Figure 3. Deterministic components of the Langevin equation.

Deterministic components of (A) the parallel and (B) the perpendicular acceleration for (gradient direction), as a function of . Black dots show the experimental results, the red lines display fits according to and , respectively. (C) as a function of . The red line shows the fit . (D) as a function of . The red line shows the fit . Error bars indicate the 95% confidence interval on the values of and .

Figure 4. Stochastic components of the Langevin equation.

(A) Stochastic component of parallel acceleration. Black dots show the experimental data, the red line shows a linear fit . (B, C) and are independent of . The red lines show constant fits.

Figure 5. Evolution of the deterministic components with the gradient strength.

(A) Friction coefficient , and effective force terms (B) , and (C) as a function of the gradient. The error bars indicate the standard deviation in (A) and the 95% confidence intervals in (B) and (C). As in Fig. 1D, the data point displayed at very low gradient values (nM/m) corresponds to an experiment where no gradient of cAMP was applied.

Figure 6. Evolution of the deterministic components at a given gradient, for each subpopulations.

(A) Friction coefficient , and effective force terms (B) , and (C) for each subpopulation. The error bars indicate the standard deviation in (A) and the 95% confidence intervals in (B) and (C).

Figure 7. Schematic trajectories.

(A) Two examples of schematic trajectories are displayed that have the same chemotactic index and the same average speed, but a very different geometrical character. (B) Trajectories governed by the Langevin equation , . (C) Trajectories of an Ornstein-Uhlenbeck process with drift, , . Also the trajectories in (B) and (C) have the same chemotactic index and the same average speed. The numbers on the axes are arbitrary space units.