^{*}

Conceived and designed the experiments: PJMVH. Performed the experiments: PJMVH. Analyzed the data: PJMVH. Contributed reagents/materials/analysis tools: PJMVH. Wrote the paper: PJMVH.

The author has declared that no competing interests exist.

Cell migration in the absence of external cues is well described by a correlated random walk. Most single cells move by extending protrusions called pseudopodia. To deduce how cells walk, we have analyzed the formation of pseudopodia by

Even in the absence of external information, many organisms do not move in purely random directions. Usually, the current direction is correlated with the direction of prior movement. This persistent random walk is the typical way that simple cells or complex organisms move. Cells with poor persistence exhibit Brownian motion with little displacement. In contrast, cells with strong persistence explore much larger areas. We have explored the principle of the persistent random walk by analyzing how

Eukaryotic cells move by extending pseudopodia, which are actin-filled protrusions of the cell surface

Cells in the absence of external cues do not move in random directions but exhibit a so-called correlated random walk

To obtain large data sets of extending pseudopodia we developed a computer algorithm that identifies the cell contour and its protrusions. The extending pseudopod is characterized by a vector that connects the x,y,t coordinates of the pseudopod at the beginning and end of the growth phase, respectively

In this report we investigated the theory of correlated random walks in the context of the observed ordered extension of pseudopodia. The aim is to define the descriptive persistence time or average turn angle with primary experimentally-derived pseudopod properties. First we obtained detailed quantitative data on the probability frequency distributions of the size and direction of pseudopod activity. We then formulated a model that consists of five components: pseudopod size, fraction of splitting pseudopodia, alternating right/left bias, angle between pseudopodia and variance of this angle due to irregularity of cell shape. We measured the parameter values of these components for several

The strains used are wild type AX3, _{2}HPO_{4}⋅12H_{2}O, 0.49 g KH_{2}PO_{4}, 10.0 g glucose), harvested in PB (10 mM KH2PO4/Na2HPO4, pH 6.5), and allowed to develop in 1 ml PB in a well of a 6-wells plate (Nunc). Movies were recorded at a rate of 1 frame per second for at least 15 minutes with an inverted light microscope (Olympus Type CK40 with 20× objective) and images were captured with a JVC CCD camera. Cell trajectories were recorded as the movement of the centroid of the cell as described

Images were analyzed with the fully automatic pseudopod-tracking algorithm Quimp3, which is described in detail

Movies at a rate of 1 frame per second were recorded for

Property | Symbol | Units | WT 1h | WT 3h | WT 5h | WT 7h | ||||

7/215 | 8/256 | 28/835 | 7/294 | 7/312 | 8/208 | 8/219 | 8/164 | |||

Pseudopod size | _{p} |
µm | 5.0±0.2 | 5.3±0.2 | 5.2±0.2 | 4.7±0.2 | 4.6±0.4 | 7.7±0.5 | 5.3±0.7 | 5.6±0.4 |

Splitting angle | degrees | 62 | 58 | 55 | 55 | 54 | 50 | 54 | 54 | |

SD splitting angle | _{φ} |
degrees | 26.1 | 29.7 | 27.8 | 27.5 | 26.9 | 28.5 | 27.5 | 46.5 |

Alternating | - | 0.74±0.02 | 0.74±0.06 | 0.77±0.04 | 0.82±0.06 | 0.67±0.05 | 0.68±0.05 | 0.75±0.08 | 0.75±0.03 | |

Fraction splitting | - | 0.55±0.07 | 0.60±0.05 | 0.86±0.06 | 0.89±0.05 | 0.71±0.06 | 0.67±0.10 | 0.41±0.07 | 0.82±0.09 | |

Correlation factor | _{obs} |
- | 0.46±0.11 | 0.52±0.07 | 0.74±0.09 | 0.81±0.10 | 0.58±0.11 | 0.55±0.11 | 0.35±0.08 | 0.53±0.07 |

Turn angle | degrees | 63 | 59 | 42 | 36 | 55 | 57 | 70 | 58 | |

Monte Carlo correlation factor | _{MC} |
- | 0.40 | 0.46 | 0.65 | 0.70 | 0.53 | 0.51 | 0.35 | 0.46 |

Equation (9) correlation factor | _{step} |
- | 0.40 | 0.42 | 0.64 | 0.69 | 0.50 | 0.48 | 0.31 | 0.43 |

The cell shape parameter

With the exception of 5h starved cells, each database contains information from 200–300 pseudopodia, obtained from 6–10 cells, using two independent movies. For 5h starved cells, we collected a larger database containing 835 pseudopodia from 28 cells using 4 independent movies, and typical databases for each mutant. The data are presented as the means and standard deviation (SD) or standard error of the means (SEM), where n represents the number of pseudopodia or number of cells analyzed, as indicated in

The probability density functions of angles can not be analyzed as the common distribution on a line. Angular distributions belong to the family of circular distributions, which are constructed by wrapping the usual distribution on the real line around a circle. The data were analyzed with two circular distributions, the von Mises distribution (vMD), which matches reasonably well with the wrapped normal distribution, and the wrapped Cauchy distribution (WCD), which has fatter tails _{0}(κ

Pseudopod extension is an ordered stochastic event _{p}_{φ}_{i,n}^{th} pseudopod: _{1,n }_{2,n}_{3,n}_{φ}^{2}_{φ}^{2}_{p}

Please note that in the simulations the direction of the simulated de novo pseudopodia is random; consequently, a small fraction of de novo pseudopodia are in the same direction of the previous pseudopod, which would be recognized in experiments as splitting pseudopodia. Conversely, a small fraction of the simulated splitting pseudopodia have angles much larger than 55 degrees and would be recognized in experiments as de novo pseudopodia. From the geometry of the cell, we estimate that the number of simulated de novo in the current pseudopod and the number of splitting pseudopodia outside the current pseudopod are approximately the same, suggesting that the simulations represent the observed ratio of splitting and de novo pseudopodia.

The angles between pseudopodia were analyzed in detail and the results are presented in _{1,2}) has a clear bimodal distribution (_{1,2} = +/−55) that have the same variance _{φ}^{2}_{φ}_{1,2} = 28 degrees). _{1,3}), which is best described by a single vMD with a mean of _{1,3} = 2 degrees and _{φ}_{1,3} = 42 degrees.

_{1,2} between current and next splitting pseudopod, yielding a bimodal von Mises distribution with mean _{1,3} between current and next-next pseudopod exhibit a single distribution with mean

The angle between a de novo pseudopod and the previous pseudopod shows a very broad distribution (

To investigate the consequence of the observed ordered extension of pseudopodia for cell movement on a coarse time scale for many pseudopodia we recorded the movement of _{p}_{p}_{obs}) with the corresponding turn angle (

The trajectories of wild type and mutant cells (see _{obs} as indicated in

How is pseudopod extension related to the observed correlation factor of dispersion _{obs}? As previously stated (see _{obs} = 0 (turn angle _{obs} is expected to depend on the ratio _{obs} = 0.921_{obs} = −0.044) giving a turn angle _{obs} = 0.88) yielding a small turn angle (

The alternating right/left extension of splitting pseudopodia can be used to simplify a description of the movement of

_{φ}^{2} (see _{φ}^{2}). Since all splitting pseudopodia show the same variance this can be further reduced to_{φ}_{φ}

The diagram shows the probabilities, angles and sizes of pairs of pseudopod extensions.

Finally, by considering movement in pairs of steps,

We used Monte Carlo simulations to investigate how _{p}_{φ}^{2}. These simulations are also useful to inspect whether step size _{φ}^{2}. The Monte Carlo simulation starts with a random angle _{φ}^{2} to stochastically simulate the angle of the next pseudopod (see

To investigate how the correlation factor _{φ}_{MC}. The symbols in

The trajectories of 100,000 cells were obtained by Monte Carlo simulation; the displacement was analyzed with Eq. 4 to obtain the correlation _{MC} and step size _{p} = 1; _{φ}_{φ}

We first investigated the angle _{MC} decreases sharply as _{MC} remains relatively high as long as the angle between pseudopodia is below 60 degrees. The results of the MC simulation appear to be described very well by the simplified model (Eqs. 8–10). Furthermore, at the observed angle of _{MC} is 0.88 (see asterisk in

The fraction of splitting pseudopodia has a major impact on the persistence factor _{φ}^{2} of the splitting angle to the persistence factor _{MC} following an approximately linear relationship with cos(_{φ}_{φ}

We also used these Monte Carlo simulations to obtain an estimate of the step size

In summary, the obtained correlation factor from the MC simulation (_{MC}) are nearly identical to the correlation factor calculated with Eq. 9 (_{step}). This suggests that the movement of _{p}_{φ}

How does the movement of pseudopodia relate to the movement of the centroid of the cell? The data presented in _{obs} of the centroid for different cell types correspond well with the deduced correlation factors of the pseudopods (_{MC} and _{step}), but is always larger by ∼15% (_{step} increases by 15% when

Probably two phenomena are responsible for the difference between pseudopod and centroid: extension of multiple pseudopodia and geometry of cells. When cells extend multiple pseudopodia it is likely that at any given instant of time, the front of the cell moves with a fixed fraction of the vector sum of velocities possessed by the pseudopodia active at that instant in time. The temporal overlap of two pseudopodia was deduced from the measured probability distributions of pseudopod extensions (_{MC} and _{step}) and observed centroid correlation factor (_{obs}).

The ^{th} step _{p},

Pseudopod formation and trajectories were recorded for 5h starved

The variation in pseudopod direction _{φ}^{2} plays an important role in Eqs. 8–11 describing cell dispersal. Previously _{φ}^{2} indeed depends on the variance of the tangent and the normal to the tangent. Second, we show that wild type or mutant cells with irregular shape exhibit increased variance _{φ}^{2}. Finally we show that, due to the increased variance, the mutant exhibits poor dispersal.

Quimp3 was used to construct the tangent to the surface curvature at the position where the pseudopod emerges. We first determined for wild-type cells the angle _{t}_{t}_{φ}^{2} is derived from the variance of the tangent _{t}^{2}, which is related to the local shape of the cell.

In the collection of _{φ}_{φ}_{φ}^{2}. Importantly, the distance

Pseudopod formation and trajectories were recorded for 5h starved wild type and mutant _{t} of the tangent relative to the current pseudopod and the angle _{φ}_{t} are significantly larger for _{d}_{β}_{φ}_{φ}^{2} = 0.965.

Using the observed values for _{φ}_{step} = 0.43, significantly lower compared to _{step} = 0.69. for wild type cells. _{obs} = 0.53 (

In summary, these and previous results

The movement of many organisms in the absence of external cues is not purely random, but shows properties of a correlated random walk. The direction of future movement is correlated with the direction of prior movement. For organisms moving in two dimensions, such as most land-living organisms, this implies that movement to the right is balanced on a short term by movement to the left to assure a long-term persistence of the direction. In bipedal locomotion, the alternating steps with the left and right foot will yield a persistent trajectory. Amoeboid cells in the absence of external cues show ordered extension of pseudopodia: a new pseudopod emerges preferentially just after the previous pseudopod has stopped growth

The model for pseudopod-based cell dispersion depends on five parameters, the pseudopod size (_{φ}^{2}). With these parameters the experimental data on mean square displacement and directional displacement are well-explained using Eqs. 9 and 11, respectively. Pseudopodia are the fundamental instruments for amoeboid movement. The notion that the trajectories are described well by the five pseudopod parameters probably implies that we have identified the basic concept of the amoeboid correlated random walk: persistent alternating pseudopod splitting and formation of de novo pseudopodia in random directions.

The cells may modify one or more of these five pseudopod parameters in order to modulate the trajectories (see _{p}

The variance of the angle of pseudopod extension (_{φ}^{2}) plays an important role in movement. In wild type cells, as well as in many mutant strains, _{φ}_{φ}_{φ}_{φ}^{2} can be regarded as the noise of the system. It indicates how fast a cell that extends only alternating splitting pseudopodia (_{φ}_{φ}

The correlation factor _{φ}

In summary, the correlated random walk of amoeboid cells is well described by the balanced bipedal movement, mediated by the alternating right/left extension of splitting pseudopodia. Cells deviate from movement in a straight line due to noise and because cells occasionally hop or make random turns. The turns in particular are used by the cells to modulate the persistence time, thereby shifting between nearly Brownian motion during growth and strong persistent movement during starvation.

Trajectories. Movies were recorded during 15 minutes and the trajectories of the centroid of ten cells were determined.

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Analysis of the noise equation _{φ}^{2}_{φ}^{2}. The figure reveals that _{φ}

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Movement of pseudopod and centroid of a cell. A. The cell is drawn as an ellipse with short and long axes of 3 and 6 µm, respectively. A pseudopod of 5 µm is extended perpendicular to the ellipse at 55 degrees relative to the long axes of the ellipse, which define the starting point and direction of the pseudopod. The position of the centroid is indicated by an asterisk.

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Determination of the shape parameter

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