Persistent cell motion in the absence of external signals: a search strategy for eukaryotic cells

Abstract

Background: Eukaryotic cells are large enough to detect signals and then orient to them by differentiating the signal strength across the length and breadth of the cell. Amoebae, fibroblasts, neutrophils and growth cones all behave in this way. Little is known however about cell motion and searching behavior in the absence of a signal. Is individual cell motion best characterized as a random walk? Do individual cells have a search strategy when they are beyond the range of the signal they would otherwise move toward? Here we ask if single, isolated, Dictyostelium and Polysphondylium amoebae bias their motion in the absence of external cues.

Methodology: We placed single well-isolated Dictyostelium and Polysphondylium cells on a nutrient-free agar surface and followed them at 10 sec intervals for approximately 10 hr, then analyzed their motion with respect to velocity, turning angle, persistence length, and persistence time, comparing the results to the expectation for a variety of different types of random motion.

Conclusions: We find that amoeboid behavior is well described by a special kind of random motion: Amoebae show a long persistence time ( approximately 10 min) beyond which they start to lose their direction; they move forward in a zig-zag manner; and they make turns every 1-2 min on average. They bias their motion by remembering the last turn and turning away from it. Interpreting the motion as consisting of runs and turns, the duration of a run and the amplitude of a turn are both found to be exponentially distributed. We show that this behavior greatly improves their chances of finding a target relative to performing a random walk. We believe that other eukaryotic cells may employ a strategy similar to Dictyostelium when seeking conditions or signal sources not yet within range of their detection system.

Figure 1. The search problem and search models.

Four characteristic types of random motion: (A) Random walk. (B) Levy walk. Step lengths were picked from a power-law distribution, and thus very long steps are possible. A Levy walk is considered to be the best strategy for searching revisitable scarce targets. (C) Two-state motion. Here a ballistic relocate phase is followed by a diffusive search phase. Switching between states occurs at random times and in random directions. This model is believed to optimize the search for low density non-revisitable targets, for example, hide and seek in the patchy environment shown here. (D) Simulation of Dictyostelium searching based on features reported in this study. The speed and number of steps is the same in A and B.

Figure 2. Cell trajectories and speeds.

(A) Three typical 10 hr cell trajectories. Boxed regime, see Fig. 5 caption. (B) Cells do not slow down over the ten-hour observation time, so we can think of them as being in a stationary (time-independent) state. However, on the time-scale of minutes the speeds do show fluctuations around their average, time-independent values (see insert). The error bars were obtained by first using a 30 min window to average each of twelve trajectories, and then, for each 30 min average, calculating the standard error of the twelve averages.

Figure 3. Mean-squared displacement.

(A) Log-log plot of the mean-squared displacement vs time interval τ. (B) Mean-squared displacement divided by τ plotted as a function of τ. Random walk would gives rise to a line with zero slope. Cyan, data from the 3 trajectories showed in Fig. 2; Red, additional 9 trajectories; Blue, average of all 12 trajectories. Yellow, fit of an exponential cross-over from directed to random walk in the interval τ [10∶100] min: 〈δ(τ)²〉 = 2t_pv ²(τ−t_p(1−exp(−τ/t_p))), where v = 5.4±0.1 µm/min is a characteristic speed, and t_p = 8.8±0.1 min is a persistence time.

Figure 4. Non-Gaussian velocity distribution.

Velocities were calculated for different τ: and v_x was plotted vs v_y with increasing τ. Larger and larger gaps at the centers of the distributions with time demonstrate that the cell velocity distributions are non-Gaussian. As expected, at very large τ, the distribution approaches a Gaussian again. Inserts, histograms of the x component of the velocities for the intervals defined by the parallel lines.

Figure 5. Cells move in a zig-zag manner.

Enlarged view of the rectangular box in Fig. 2. The magenta scale-bars are 10 µm. Turns are marked by black crosses. Motion following left turns is blue and motion following right turns is yellow.

Figure 6. Cumulative angles.

(A) Angles were corrected by +/−2π for changes larger than π. As shown by trajectories in red boxes, sharp turns correspond to large drops or rises; continuous turns in one direction appear as continuous drops or rises (box 1, enlarged view in B); and periods of directed motion are summarized as plateaus (box 2, enlarged view in C). On average, angles change but slowly with time. Box 3, see Fig. 8 caption. (B) Enlarged view of box 1 in A. The transition from 0 to −4π is continuous. (C) Enlarged view of box 2 in A. To reduce the influence of noise, angles were calculated at a larger τ (30 s) for most of the analyses. Thus from each trajectory, 3 interlaced time-series of angles were obtained. They are marked by different colors in the lower panel.

Figure 7. A left turn is followed by a right turn –a Poisson process.

(A) Definition of angle of turns (α), direction of turns (left or right), time between turns and length between turns. (B) The j^th turn plotted against the (j+1)^th turn for the data from all 12 trajectories. There are 3263 data points in the second and fourth quadrant, 1559 in the first and third, and thus the (j+1)^th turn is biased by the j^th turn by a factor of 2.1. (C) Autocorrelation function for the turn directions (see text for details). Blue: Experimental values and standard errors. Black: Theoretical expectation value for a Markov process with probabilities taken from panel B (see text for details). Insert: Verification that turn-correlations are real and not an artifact of the turn-detection algorithm. Blue: Autocorrelation function for synthetic data. The angle-dynamics was simulated by a worm-like-chain model (WLC) with parameters taken from the MSD of the real data. A small, negative, artifactual correlation is detected which extends for around 3 turns. Black: Same as the main-panel, shown for comparison. (D) Histogram of turn amplitudes. Its tail is well fitted by an exponential distribution (characteristic angle = 0.67 rad). The rounding off at small values is caused by thresholding in the turn-detection algorithm and this sharp cut-off is smoothed by the coarse-graining applied when calculating the angles. Lower left panel: Histogram of α. Upper right panel: Autocorrelation function for turn amplitudes, no correlation was observed. The positive value at time-lag one is a verified artifact of the turn-detection algorithm. (E) Histogram of time intervals between detected turns. These data are well fitted by an exponential distribution (characteristic time = 0.67 min). Data is from all 12 trajectories. Lower left panel: Same histogram but on linear scale. The smallest detected value for t_j+1−t_j is 40 sec, the cut-off shown by the grey bar. Upper right panel: Normalized autocorrelation function for time between turns. No significant correlations were observed, consistent with a Poisson process. (F) Histogram of length between turns. Its tail is well fitted by an exponential distribution (characteristic length = 5 µm). Distribution of length is trivially exponential if cell averaged speed is constant and times between turns are exponentially distributed.

Figure 8. A model for Dictyostelium motion.

(A) The blue line is hand drawn to guide the eye and represents θ, a one-dimensional random walk. The black line ϕ, is an enlarged view of the data in box 3 of Fig. 6. In our model, ϕ is the net effect of θ and colored noise centered on θ. The stochastic differential equations used to describe the behavior of ϕ and θ are explained in detail in the text. (B) θ is the angle assumed to be fixed by a cell's intrinsic polarity, possibly a vector directed from the center of nucleus (light blue) to the position of the centrosome . Black arrow, pseudopod extensions and retractions lead to stochastic oscillations. New pseudopods bifurcate from old, and they swing back and forth about the internal vector. (C) A cartoon describing directional control.

Figure 9. Statistics of the cumulative angles and fit to the theory.

(A) Experimental power spectral density (PSD) of ϕ and fit of the theory to the data. Two time-scales were returned by the fit: (f ₀)⁻¹ = 2.35±0.08 min and 1 rad²/D _θ = 7.6±0.3 min, approximately the duration of a pseudopod and the time it takes for a cell to lose its sense of direction, respectively. Twelve individual PSDs, one for each cell, are shown in yellow. The average over the cells is shown in blue. The solid black line is a fit of the theory to the averaged signal. Dashed, dash-dotted, and dotted lines indicate the contribution to the PSD for the colored noise, the random walk, and the tracking-error terms, respectively (see Materials and Methods for details). (B) Autocorrelation function for Δϕ. ϕs were calculated for τ = 30 s. Blue: Experimental values and standard errors. Black: Theoretical expectation value calculated from a Monte Carlo simulation on ϕ based on Eqs. 1–3, with parameters obtained from a fit to the PSD. (C) Experimental histogram of Δϕs calculated for τ = 30 s. Insert: Same histogram shown on a semi-logarithm scale demonstrating the non-Gaussian, exponential tails. With increasing τ, the distribution of Δϕ becomes more and more Gaussian (data not shown).