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**|**PLoS Comput Biol**|**v.6(8); 2010 August**|**PMC2920832

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PLoS Comput Biol. 2010 August; 6(8): e1000874.

Published online 2010 August 12. doi: 10.1371/journal.pcbi.1000874

PMCID: PMC2920832

Edward C. Cox, Editor^{}

Department of Cell Biochemistry, University of Groningen, Haren, the Netherlands

Princeton University, United States of America

* E-mail: ln.gur@tretsaah.nav.M.J.P

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

Received 2010 April 14; Accepted 2010 June 30.

Copyright Peter J. M. Van Haastert.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

This article has been cited by other articles in PMC.

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 *Dictyostelium* cells. We have observed that the formation of pseudopodia is highly ordered with two types of pseudopodia: First, de novo formation of pseudopodia at random positions on the cell body, and therefore in random directions. Second, pseudopod splitting near the tip of the current pseudopod in alternating right/left directions, leading to a persistent zig-zag trajectory. Here we analyzed the probability frequency distributions of the angles between pseudopodia and used this information to design a stochastic model for cell movement. Monte Carlo simulations show that the critical elements are the ratio of persistent splitting pseudopodia relative to random de novo pseudopodia, the Left/Right alternation, the angle between pseudopodia and the variance of this angle. Experiments confirm predictions of the model, showing reduced persistence in mutants that are defective in pseudopod splitting and in mutants with an irregular cell surface.

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 *Dictyostelium* cells extend protrusions called pseudopodia. These cells can extend a new pseudopod in a random direction. However, usually cells use the current pseudopod for alternating right/left splittings, by which they move in a persistent zig-zag trajectory. A stochastic model was designed for the persistent random walk, which is based on the observed angular frequencies of pseudopod extensions. Critical elements for persistent movement are the ratio of de novo and splitting pseudopodia, and, unexpectedly, the shape of the cell. A relatively round cell moves with much more persistence than a cell with an irregular shape. These predictions of the model were confirmed by experiments that record the movement of mutant cells that are specifically defective in pseudopod splitting or have a very irregular shape.

Eukaryotic cells move by extending pseudopodia, which are actin-filled protrusions of the cell surface [1]. Pseudopod formation by *Dictyostelium* cells, like many other moving cells, shows a typical pseudopod cycle: upon their initiation, pseudopodia grow at a constant rate during their first ~15 s and then stop. The next pseudopod is typically formed a few seconds later, but sometimes commences while the present pseudopod is still growing, giving rise to a cell with two pseudopodia. The fate of the pseudopod after its initial growth phase determines its role in cell movement: the pseudopod is either retracted, or is maintained by flow of the cytoplasm into the pseudopod thereby moving the cell body. The frequency, position and directions of the maintained pseudopodia form the basis of cell movement, because they determine the speed and trajectory of the cell. An important aspect of cell motility is the ability of cells to respond to directional cues with oriented movement. Gradients of chemicals give rise to chemotaxis [2]. Other directional cues that can induce oriented movement are temperature gradients (thermotaxis) or electric fields (electrotaxis) [3], [4]. These signals somehow modulate basal pseudopod extension such that, on average, cells move in the direction of the positional cues. In this respect, studies on cell movement are critical for understanding directional movement.

Cells in the absence of external cues do not move in random directions but exhibit a so-called correlated random walk [5]–[9]. This tendency to move in the same direction is called persistence, and the duration of the correlation is the persistence time. Cells with strong persistence make fewer turns, move for prolonged periods of time in the same direction, and thereby effectively penetrate into the surrounding space. Other search strategies for efficient exploration are local diffusive search and Levi walks [8], [10]. Can we understand the cell trajectory by analyzing how cells extend pseudopodia?

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 [11]. A picture of ordered cell movement has emerged from the analysis of ~6000 pseudopodia that are extended by wild type and mutant cells in buffer [12]. *Dictyostelium* cells, as many other eukaryotic cells, may extend two types of pseudopodia: *de novo* at regions devoid of recent pseudopod activity, or by splitting of an existing pseudopod [12], [13]. Pseudopod splitting occurs very frequently alternating to the right and left at a relatively small angle of ~55 degrees. Therefore, pseudopod splitting may lead to a persistent zig-zag trajectory [14]. In contrast, de novo pseudopodia are extended in all directions and do not exhibit a right/left bias, suggesting that de novo pseudopodia induce a random turn of the cells. We observed strong persistence for cells that extend many splitting pseudopodia. Conversely, mutants that extend mostly de novo pseudopodia have very short persistence time and exhibit a near Brownian random walk [12].

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 *Dictyostelium* mutants with defects in signaling pathways or cytoskeleton functions. Subsequently, we used these observed parameters in Monte Carlo simulations of the model and compared the predicted trajectories with the observed trajectories of the mutants. The results demonstrate two critical components in these correlated random walks: the ratio of pseudopod splitting relative to de novo pseudopodia, and the shape of the cell.

The strains used are wild type AX3, *pi3k*-null strain GMP1 with a deletion of *pi3k1* and *pi3k2* genes [15], *pla2*-null with a deletion of the *plaA* gene [16], *sgc/gca*-null cells (abbreviated as *gc*-null cells) with a deletion of *gca* and *sgc* genes [17], *sgc/pla2*-null cells with a deletion of *sgc* and *pla2A* genes [18], and *ddia2*-null cells lacking the *forH* gene encoding the *Dictyostelium* homologue of formin [19]. Cells were grown in HG5 medium (contains per liter: 14.3 g oxoid peptone, 7.15 g bacto yeast extract, 1.36 g Na_{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 [20].

Images were analyzed with the fully automatic pseudopod-tracking algorithm Quimp3, which is described in detail [11]. Briefly, the program uses an active contour analysis to represent the outline of the cell using ~150 nodes [21]. Extending pseudopodia that satisfied the user-defined minimum number of adjacent convex nodes and the minimum area change were identified. The direction of each extending pseudopod was identified by the x,y and time coordinates of the central convex node of the convex area at the start and end of growth, respectively. The tangent to the surface at the node where the pseudopod started was calculated using the position of the adjacent nodes. The automated algorithm annotates each pseudopod as de novo versus splitting using the criterion that the convex area of the new pseudopod exhibits overlap with the convex area of the current pseudopod or is within a user-defined distance. The output files containing the x,y-coordinates of the start and end position of the pseudopod, the tangent of the surface and the annotation of the pseudopod were imported in Excel to calculate pseudopod size, interval, direction to gradient, direction to tangent, etc for de novo and splitting pseudopodia (see Fig. 1), as well as fraction *s* of pseudopod splitting and alternating Right/Left bias *a* (RL +LR)/total splitting; Table 1).

The cell shape parameter *Ψ* was determined as follows: Using the outline of the cell with ~150 nodes, two ellipsoids were constructed, the largest ellipse inside the cell outline and the smallest ellipse outside the cell outline. Then an intermediate ellipse was constructed by interpolation of the inner and outer ellipse. This intermediate ellipse makes several intersections with the cell outline, thereby forming areas of the cell that are outside the intermediate ellipse (with total surface area *O*), and areas of the intermediate ellipse that do not belong to the cell (with total surface area *I*; see Fig. S4). The intermediate ellipse was positioned in such a way that (this also implies that the surface area of the cell (T) is identical to the surface area of the intermediate ellipse). The cell shape parameter is defined as ; it holds that . For a cell with a regular shape that approaches a smooth ellipsoid, the surface areas *O* and *I* are very small and *Ψ* approaches zero. In contrast, *O* and *I* are larger for a cell with a very irregular shape; the largest value observed among ~600 cells was *Ψ*=0.92.

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 Table 1.

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 [22]. The vMD is given by

(1)

where *I _{0}(κ*) is the modified Bessel function of the first kind of order zero

(2)

The WCD is given by

(3)

Pseudopod extension is an ordered stochastic event [12]. The position of the tip of the formed pseudopodia depends on pseudopod size *λ _{p}*, splitting fraction

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 Fig. 2. For splitting pseudopodia, the angle between the current and next pseudopod (*ϕ*
_{1,2}) has a clear bimodal distribution (Fig. 2A). A probability density function (PDF) of angles belongs to the family of circular or wrapped distributions. The data reported in this study were all fitted well by a von Mises distribution (vMD), which is the circular analog of the normal distribution. The wrapped Cauchy distribution has fatter tails and provided a poorer fit of the data (data not shown). The bimodal vMD presented in Fig. 2A is symmetric, yielding two means (*ϕ*
_{1,2}=+/−55) that have the same variance *κ*=1/*σ _{ϕ}^{2}*;

The angle between a de novo pseudopod and the previous pseudopod shows a very broad distribution (Fig. 2E). Nearly all angles between −180 and +180 are well represented with a somewhat lower abundance of angles around 0 degrees. This suggests that a de novo pseudopod can be extended in any direction, but with slightly lower probability of the direction of the current pseudopod.

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 *Dictyostelium* cells during 15 minutes; in this period about 30 pseudopodia are extended. Previously we have presented the cell trajectories for several strains and developmental stages [12] (see also Fig. S1). The mean square displacement as a function time, , exhibits a slow approach to a linear function (Fig. 3A), which is typical for a transition of a correlated random walk at short times to a Brownian random walk after longer times [6], [23]. Previously, the often used equation for a correlated random walk were fit to the data points to estimate persistence time and speed of the cells [12]. The aim of the present study is to analyze the mechanism of cell movement from the perspective of the extending pseudopodia, which have a specific length and direction. A correlated random walk in two dimensions can also be described with steps and turns [24], [25]. With the replacement of the number of steps (n) in Eq. 7 in reference [25] for n=Ft we obtain

(4)

where *λ* is the step size in µm, F is the step frequency, and *γ* is the correlation factor of dispersion (0<*γ*<1), defined as the arithmetic mean of the cosine of the turn angle *θ* between steps

(5)

With three variables (F, *λ*, *γ*) the estimates of the parameters become uncertain. Fortunately, the step size can be deduced accurately from experimental data. As will be shown below in Eq. 10, the step size is given by *λ*=*λ _{p}*cos(

How is pseudopod extension related to the observed correlation factor of dispersion *γ*
_{obs}? As previously stated (see Fig. 2), *Dictyostelium* cells may extend either *de novo* pseudopodia in nearly random directions, or splitting pseudopodia in a direction similar to the previous direction. Therefore, cells that extend exclusively de novo pseudopodia are expected to exhibit a random walk with *γ*
_{obs}=0 (turn angle *θ*=90 degrees), whereas cells extending exclusively splitting pseudopodia will exhibit strong persistence with large *γ* and small turn angle *θ*. As a consequence, *γ*
_{obs} is expected to depend on the ratio *s* of splitting/de novo pseudopodia. Fig. 3B demonstrates that within experimental error this relationship is approximately linear; this holds true for the mutants as well as for wild type cells at different stages of development. The linear regression of all data yields *γ*
_{obs}=0.921*s*−0.044. Thus, when all pseudopodia are de novo (*s*=0) the correlation factor is small (*γ*
_{obs}=−0.044) giving a turn angle =93 degrees, close to the expected value of 90 degrees for random turns. In contrast, when all pseudopodia are the result of splitting (*s*=1) the correlation factor is large (*γ*
_{obs}=0.88) yielding a small turn angle (=29 degrees). The implication of small turn angles for splitting pseudopodia will be discussed later.

The alternating right/left extension of splitting pseudopodia can be used to simplify a description of the movement of *Dictyostelium* cells over longer distances. In this approach, the simplification may be valid for movement on a longer time scale only, as we study here, but may not be appropriate over shorter time scales of a few pseudopodia. Because pseudopodia are frequently extended alternating right/left, we consider movement by pairs of two pseudopodia.

Figure 4 shows four possibilities of pairs of splitting pseudopodia, which are the RL, LR, RR and LL, each with corresponding probabilities and angles as indicated. In addition to these splitting pairs, three combinations with de novo pseudopodia are possible: split-de novo, de novo-split, and de novo-de novo. The correlation factor of dispersion yields for the seven pairs:

(6)

De novo pseudopodia are extended in a random direction, i.e. , and equal zero. The turn angles of the four splitting pairs are 0, *ϕ* and 2*ϕ*, as indicated in Fig. 4A, and the variance is approximately 2*σ _{ϕ}*

(7)

where denotes the expected value of the cosines of the angles on a circle with weights given by the vMD with mean *ϕ* and variance given by *κ*=1/(2*σ _{ϕ}*

(8)

In this equation is obtained by calculating the probabilities of all turn angles on a circle with the vMD using Eqs. 1 and 2 and then taking the weighted average of the cosines of these angles. Although this procedure is straightforward, Eq. 8 can be further simplified, because for *σ _{ϕ}* smaller than ~50 degrees a good approximation is (see Fig. S2). Finally, on a longer time scale and averaged over many steps, the correlation factor of pairs is related to the correlation factor of its underlying two steps by . With these replacements we obtain the analytical expression for the correlation factor

(9)

Thus, the correlation factor *γ* is the product of three terms: the splitting ratio *s*, a noise term with the variance *σ _{ϕ}*, and a term with right/left bias

Finally, by considering movement in pairs of steps, Fig. 4 reveals that the step size of the displacement is given by

(10)

We used Monte Carlo simulations to investigate how *λ* and *γ* depend on the pseudopod parameters size *λ _{p}*, splitting fraction

To investigate how the correlation factor *γ* depends on pseudopod parameters, the displacement was calculated from 100,000 trajectories obtained by MC simulation using a unit pseudopod size and different values of *s*, *a*, *ϕ* and *σ _{ϕ}*. The obtained displacement was then fitted to Eq. 4 to obtain estimates for the step size

We first investigated the angle *ϕ* between splitting pseudopodia and the alternating right/left bias *a*. When all splitting pseudopodia are alternating (*a*=1), the cells make a nearly perfect zig-zag trajectory, and therefore the angle *ϕ* has very little effect on the persistence factor *γ* (Fig. 5A). When splitting pseudopodia are extended in a random fashion to the right or left (*a*=0.5), the persistence factor *γ*
_{MC} decreases sharply as *ϕ* becomes larger than ~30 degrees. At an intermediate right/left bias (*a*=0.75) the persistence factor *γ*
_{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 *ϕ*=55 degrees and alternating factor of *a*=0.77, the deduced persistence factor *γ*
_{MC} is 0.88 (see asterisk in Fig. 5A).

The fraction of splitting pseudopodia has a major impact on the persistence factor *γ*. In the MC simulations, the value of *γ* declines approximately linearly with the value of *s* (Fig. 5B), as was also observed experimentally (Fig. 3B), and obtained in Eqs. 8 and 9. Finally, we investigated the contribution of the variance *σ _{ϕ}*

We also used these Monte Carlo simulations to obtain an estimate of the step size *λ*. It appears that *λ* does not to depend on *s* and *a*, but depends on *ϕ* according to (Inset Fig 5A), as was obtained in Eq. 10.

In summary, the obtained correlation factor from the MC simulation (*γ*
_{MC}) are nearly identical to the correlation factor calculated with Eq. 9 (*γ*
_{step}). This suggests that the movement of *Dictyostelium* cells is qualitatively and quantitatively described very well by the model of persistent steps and random turns with the observed pseudopod parameters *λ _{p}*,

How does the movement of pseudopodia relate to the movement of the centroid of the cell? The data presented in Table 1 reveal that the observed correlation factor *γ*
_{obs} of the centroid for different cell types correspond well with the deduced correlation factors of the pseudopods (*γ*
_{MC} and *γ*
_{step}), but is always larger by ~15% (Table 1). Apparently, the observed turn angle of the cell's centroid is smaller than the turn angle of the extending pseudopod. Inspection of movies of 5h starved AX3 cells confirm this notion: the average angle between splitting pseudopodia is 55±28 degrees (Fig. 2A), while the centroid moves during period at an angle of only 31±23 degrees (mean and SD). Equation 9 reveals that the correlation factor *γ*
_{step} increases by 15% when *ϕ*=55±28 degrees for the pseudopod is replaced by *ϕ*=31±23 degrees for the cells centroid.

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 (Fig 2F in [26]), which reveal that ~25% of the pseudopodia overlap with another pseudopod during on average ~40% of their extension time. This suggest that the tip of the cell moves at an angle that is ~6 degrees smaller than 55 degrees. Secondly, geometry predicts that the rear of the cell makes smaller changes of direction than the tip of the cell, comparable to the differences in curvature made by the front and rear wheels of a car. Figure S3 indicates that for a stereotypic pseudopod at 55 degrees the directional change of the centroid is ~40 degrees (see Fig. S3). Together, multiple pseudopodia and cell geometry can explain observed difference between pseudopod and centroid changes of direction, leading to the small 15% difference between deduced pseudopod correlation factor (*γ*
_{MC} and *γ*
_{step}) and observed centroid correlation factor (*γ*
_{obs}).

The *directional* displacement is the displacement after n steps in the direction of the first step. An expression for the directional displacement is especially relevant when the organism is exposed to positional cues leading to a drift in one direction, such as during chemotaxis. The directional displacement of a cell after extending one pseudopod at an angle *θ* is , and for a population of cells . By Eqs. 3 and 10, the displacement at the first step may be written as , and at the i^{th} step , see Eq. 6 of reference [25]. The cumulative displacement after n steps is

(11a)

which at is given by

(11b)

In essence, this equation describes the displacement of a cell population in which all cells extend the first pseudopod in the same direction. Subsequent pseudopodia are extended with a bias, which reduces geometrically with each step; the correlation factor *γ* indicates how many pseudopodia have correlated direction and therefore how far the population will disperse in the direction of the first pseudopod. Figure 6 presents the directional displacement as observed experimentally in wild type cells. The displacement in the direction of the first pseudopod slowly decreases at each subsequent pseudopod, approaching random movement after ~10 pseudopodia. On average a cell moves ~15 µm in the direction of the first pseudopod, which is the equivalent of about 3 pseudopodia (given a pseudopod size of ~5 µm). This figure also presents the directional displacement as modeled by Eq. 11a with observed data for *λ*
_{p}, *ϕ* and *γ*, which is in very close agreement with experimental data, again suggesting that the movement of a cell is satisfactory described by the model with five pseudopod parameters.

The variation in pseudopod direction *σ _{ϕ}*

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 *α _{t}* of this tangent relative to the previous pseudopod (

In the collection of *Dictyostelium* mutants, we selected a strain with an irregular shape. Mutant *ddia2*-null with a deletion of the *forH* gene encoding the formin dDia2 has a star-like shape (Fig. 7C). In this mutant, new pseudopodia are extended at about the same frequency and distance from the present pseudopodia as in wild type cells, pseudopodia also grow perpendicular to the surface, and are extended roughly in the same direction of *ϕ*=55 degrees as wild type cells (Fig. 7A). However pseudopodia exhibit much more variation in direction (*σ _{ϕ}*=47 degrees compared to

Using the observed values for *s*, *a*, *ϕ*, and *σ _{ϕ}* for

In summary, these and previous results [12] suggest that a splitting pseudopod is induced at some distance *d* from the tip of the current pseudopod, and then grows perpendicular to the surface. In a cell with a regular shape, the tangent and therefore pseudopod direction can be approximated using the distance *d*; alternating R/L extensions lead to a relative straight zig-zag trajectory, providing strong persistence of movement. In a cell with a very irregular shape, the local curvature of the membrane at distance *d* is unpredictable. Consequently, alternating R/L splitting occur with large variation of directions, leading to frequent turns and poor persistence.

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 [12]. Importantly, the position at the cell surface where this new pseudopod emerges is highly biased. When the current pseudopod has been extended to the left (relative to the previous pseudopod), the next pseudopod emerges preferentially nearby the tip at the right side of the current pseudopod. Since pseudopodia are extended perpendicular to the cell surface, this next pseudopod is extended at a small angle relative to the current pseudopod [12]. Therefore, this (imperfect) alternating right/left pseudopod splitting resembles bipedal locomotion. Cells may also extend a de novo pseudopod somewhere at the cell body, which is extended in a random direction. In starved *Dictyostelium* cells, the probability of extending a de novo pseudopod is ~10-fold lower than of pseudopod splitting (probability calculated per µm circumference of the cell [12]).

The model for pseudopod-based cell dispersion depends on five parameters, the pseudopod size (*λ*), the fraction of split pseudopodia (*s*), the alternating left/right bias (*a*), the angle between pseudopodia (*ϕ*) and the variance of this angle (*σ _{ϕ}*

The cells may modify one or more of these five pseudopod parameters in order to modulate the trajectories (see Table 1). Nearly all mutants, as well as wild type cells at different stages of starvation and development, have approximately the same average pseudopod size *λ _{p}*. In addition, the alternating right/left bias (

The variance of the angle of pseudopod extension (*σ _{ϕ}*

The correlation factor *γ* is the product of three terms (see Fig. 5 and Eq. 9), namely: splitting fraction (*s*), alternating pseudopod angles (*a* and *ϕ*), and the SD of the pseudopod angle (*σ _{ϕ}*). Strong persistence of cell movement is attained when all three terms are large and about equal in magnitude. Starved wild type cells follow this strategy: each term is ~0.9, resulting in the observed correlation factor of 0.74. Mutants in which one of these terms is compromised, such as reduced splitting in

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. **A**. Wild type Dictyostelium cells at different times after removing of food. The frequency and size of pseudopod extension is not very different, but starved cells extend predominantly splitting pseudopodia (see (16)). **B**. Trajectories of 5 hour starved mutant cells with deletions of genes encoding guanylyl cyclases (sGC and GCA) and PLA2. **C**. Monte Carlo simulations of the trajectories calculated with the pseudopod parameters that were obtained experimentally for the mutants as indicated in Table 1.

(0.07 MB PDF)

Click here for additional data file.^{(65K, pdf)}

Analysis of the noise equation . Pseudopodia are extended with a variance σ* _{ϕ}^{2}*. In this equation, the notation is the average of the cosines of the angles on a circle with weights given by the von Mises Probability Distribution (vMD) with mean of 0 degrees and variance given by

(0.03 MB PDF)

Click here for additional data file.^{(32K, pdf)}

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. **B**. In *Dictyostelium* cells a pseudopod usually extends during ~12 seconds, and then the cytoplasm moves into the pseudopod and the rear is retracted. The open headed arrows indicate that the front of the cell moves to the tip of the pseudopod, and the rear of the cell moves in the direction of the old axis of the cell. **C**. Schematic after a few Right/Left pseudopod extensions. The centroid makes smaller turns than the pseudopod, ~40 degrees.

(0.07 MB PDF)

Click here for additional data file.^{(67K, pdf)}

Determination of the shape parameter *Ψ*. The cell outline is used to draw two ellipses, the largest possible ellipse inside the cell and the smallest possible ellipse outside the cell. Then an intermediate ellipse is constructed by interpolation of the inner and outer ellipse; the figure shows the intermediate ellipse. This ellipse intersects the outline, thereby forming the blue areas *O* of the cell that are outside the intermediate ellipse, and yellow areas *I* of the intermediate ellipsoid that do not belong to the cell. The intermediate ellipse was interpolated in such a way that *O*=*I*. The parameter of cell shape is defined as *Ψ*=(*O*+*I*)/*T*, where *T* is the surface area of the cell (grey + blue). The minimal value is *Ψ*=0 when the cell is an ellipse, and the maximal value is *Ψ*=2 for a cell with extreme long extensions; the maximal value observed among ~600 *Dictyostelium* cells is *Ψ*=0.92. The neuron cell is shown for comparison.

(0.13 MB PDF)

Click here for additional data file.^{(131K, pdf)}

The author has declared that no competing interests exist.

The author received no specific funding for this work.

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