The simplified EC&M model has three main features. First, pseudopods appear as bursts from an excitable cortex. Second, active pseudopods globally inhibit new pseudopod formation. Third, where a pseudopod is active it makes the cortex locally more excitable. This third feature is the key insight that leads to zig-zag behavior: previous pseudopods leave a trace of their activity, which acts as a local memory and is the basis of persistence.
The model represents the cell as a one-dimensional circle, and on this circle pseudopods stochastically appear as excitable bursts. In order to keep the model as simple as possible, we do not explicitly model excitability. Rather, we assume that excitable bursts occur along the membrane as binary events, with a rate as described below. This is a reasonable approximation given both that the times over which a pseudopod begins and ends its growth periods (each about 1 second) are much shorter on average than its total growth time (about 10 seconds), and that a pseudopod's growth rate remains fairly constant while it is growing 
. We assume the bursting rate depends on three variables: a local memory M
, a local inhibitor L
, and a global inhibitor G
. The local pseudopod production rate is given by:
The parameters α
control the inhibitory strength of L
respectively, and ε
sets the overall pseudopod production rate of the system. We assume a cubic dependence of Γstart
because pseudopod activity involves much cooperative feedback 
. The choice of such a cubic dependence is common in models of biological excitable systems, e.g.
the FitzHugh-Nagumo model for spiking neurons 
, and the exact form of this equation is not critically important for model results.
Where a pseudopod is active we postulate that it creates a long-lived memory M
in the cellular cortex that temporarily makes that area of the cortex more excitable:
represents position along the circumference of the cell and
is a boxcar function that equals 1 where and when a pseudopod is active and is 0 everywhere else. The parameter k1
is the additional rate of memory production where a pseudopod is active, and to make the cortex permissive for occasional pseudopod formation everywhere, we assume that memory is formed at all locations with a low basal rate k0
. We also assume the memory decays with a lifetime τM
and diffuses with coefficient DM
. We chose a diffusion constant
/s, consistent with that measured for membrane-bound proteins 
Where a pseudopod is active it also produces a local inhibitor L
at rate kL
. This local inhibitor also decays with lifetime τL
and diffuses with diffusion constant DL
The local inhibitor serves both to create a refractory period after a pseudopod stops, and to limit pseudopod lifetimes. An active pseudopod has a stopping rate that depends on the value of L
at its center:
is a multiplicative coefficient. We chose a cubic dependence of Γstop
because this form yields a distribution of pseudopod lifetimes approximating observed lifetime distributions.
Experimentally it is found that pseudopod growth suppresses formation of new pseudopods 
, so the model further assumes that when a pseudopod is active it creates a global inhibitor G
that diffuses instantly, taking the same value at all points along the cortex. After the pseudopod stops we assume the global inhibitor instantly decays. Although we use inhibitors in this model, we note that substrate depletion could serve the same function as either the local inhibitor to limit pseudopod lifetimes and create refractory periods 
, or as the global inhibitor to suppress lateral pseudopod activity 
. We assume instantaneous diffusion of cGMP for simplicity 
, and we note that with a diffusion constant of 300 µm2
/s, cGMP could diffuse the length of a cell (~10 µm) in 0.17 seconds 
, which is far faster than the pseudopod time scale.
Together, the memory, local inhibitor, and global inhibitor all determine the rate of pseudopod formation at any point along the cortex. Where the memory is high, pseudopods are more likely to begin growing, while the local and global inhibitors both suppress pseudopod formation. M
, and G
are dimensionless quantities with arbitrary scale. Their effects on pseudopod dynamics are set by the parameters μ
, and β
in Equations 1 and 4. For further details on implementation of the model in simulations see Methods
, and for parameters see .
Parameters used for simulations of the EC&M model.
In order to define a zig-zag, one must analyze a time series of at least three successive pseudopods, which we refer to as the grandparent, parent, and child, following Bosgraaf and van Haastert 
. In our terminology, the parent of a new pseudopod is the most recently stopped pseudopod. If one or more pseudopod is still active when a new pseudopod starts, that which started most recently is the new pseudopod's parent. In the event that the above criteria do not define a single pseudopod, the parent is that which is spatially closest to the new pseudopod. A pseudopod's grandparent is the parent of its parent. Bosgraaf and van Haastert distinguished between “split” and “de novo
” pseudopods based on whether or not one pseudopod appeared to grow out of a previous one, and de novo
pseudopods were excluded from their analysis of zig-zag bias. However, both types of pseudopods are indistinguishable with respect to size and lifetime, so we include all pseudopods for analysis to avoid making a priori
assumptions about pseudopod dynamics. In our model, pseudopods appear to split as the result of a new pseudopod beginning very close to an existing one.
We define a child pseudopod's turning angle to be the angle from the parent's location to the child's location, i.e.
. Positive values of Δθ
define a left turn and negative values of Δθ
define a right turn. We say a zig-zag has occurred when a child turns in the opposite direction of its parent, making the grandparent-parent-child series right-left or left-right. A left-left or right-right sequence would be a non-zig-zag. The zig-zag ratios reported are the number of pseudopods that are third in a zig-zag sequence divided by the number that are third in a non-zig-zag sequence.
For an illustration of pseudopod dynamics in our model, consider the sequence of three pseudopods shown schematically in , with memory in blue and local inhibitor in red. In panel (i) a growing pseudopod (the grandparent) produces both cortical memory and local inhibitor. By panel (ii) the grandparent has ceased growing and its immediate vicinity is refractory due to local inhibition. A new pseudopod (the parent) begins growing to the right, and continues to grow in panel (iii). By the time the local inhibitor at the site of the grandparent has decayed, some memory remains. This memory locally increases cortical excitability, so in panel (iv), when the parent pseudopod stops growing and global inhibition lifts, a third pseudopod (the child) is most likely to form in the more excitable area left by the grandparent. This sequence of three pseudopods constitutes a zig-zag since the child turns left toward the previous location of the grandparent rather than turning right as the parent did.
The excitable cortex and memory model.
illustrates the evolution of simulated variables at the center of a pseudopod. The rate of pseudopod formation is in units of µm−1
, and L
are in units of
, respectively. The global inhibitor G
takes values of 0 or 1. The local inhibitor L
rises rapidly, which increases the pseudopod's stopping rate (Eq. 4). Meanwhile the cortical memory M
rises more slowly. When the pseudopod stops after 7 seconds, G
decays instantly, and L
decay according to their respective lifetimes, with τL
. The local rate of new pseudopod formation (Eq. 1) increases abruptly when G
disappears, then the pseudopod formation rate continues to rise more slowly as L
decays over the timescale τL
. The pseudopod production rate reaches a local peak and then declines toward its basal level as the long-lived memory M
decays on the longer timescale τM
The model produces a zig-zagging persistent random walk
show sample model cell trajectories, which exhibit the same qualitative features as real cell paths. The 10 minute paths in show small scale zig-zag behavior and persistence, which transitions to diffusive motion over longer times as seen in the 200-hour path in . To quantify this crossover we plot the mean squared displacement
versus time lag τ
in . When we fit this to the equation for a persistent random walk with constant speed v
and persistence time τp
, the simulated trajectories exhibit a persistence time of 4.0 minutes, which is within the range of 3.4 minutes 
and 8.8 minutes 
reported in the literature. shows the mean squared displacement divided by the time lag, a quantity which is linear with positive slope for directed motion (for which
), and which is constant for diffusive motion (for which
). In one can see clearly the transition from persistent motion on short timescales to a random walk over longer timescales as
approaches a constant. As in live cells, pseudopods simulated by our model preferentially zig-zag, with a zig-zag ratio of 2.0 (i.e.
the number of zig-zags divided by the number of non-zig-zags) for the parameters used here. Since not all pseudopods zig-zag and pseudopod angles are stochastic, over longer times, path persistence is lost.
Simulated trajectories from the excitable cortex and memory (EC&M) model.
A new hierarchical clustering algorithm to detect pseudopods
To test model results and predictions, we developed a new algorithm to track pseudopods in freely crawling cells. Dictyostelium
cells were tagged with mRFP-LimE and GFP-Myosin to aid cell outline detection and to ensure that we reliably detected both the front and the rear of each cell (see Methods
). Vegetative cells were allowed to migrate on glass coverslips under buffer with no external chemoattractant gradient, and images were captured every 2 seconds. shows successive cell outlines overlaid from one track (see also Movie S1
). The cell was moving from top to bottom and the time goes from green to red. The inset displays the fluorescent image from this sequence at 120 seconds, and the corresponding cell outline is in bold. Cell outlines were detected using an active contour method 
, and membrane extensions were found by comparing cell outlines from each time step to outlines from the previous time step (for further details, see Methods
Cell tracks and pseudopods as detected by our algorithm.
Pseudopods detected from these extensions are shown in , which has the same scale as . The numbers shown are the pseudopod starting and stopping times, the small hash lines indicate the angle of each individual extension, and the larger arrows show the mean extension angle of each pseudopod. To define pseudopods from individual extensions we used hierarchical clustering to group the extensions based on adjacency in time, extension angles, spatial distance, and the percentage of the newer extension which grew out of the older extension (see Methods
). Consistent with the model definition, parent pseudopods were defined to be the most recently active extending pseudopod, or the most recently started if more than one pseudopod was still active, or the closest pseudopod in space if more than one was equally recent. In total we tracked 57 cells and 1764 pseudopods.
In agreement with previous results, our algorithm finds that cells extend pseudopods with a zig-zag ratio of 1.8, which is consistent with the pseudopod zig-zag ratio of 2.0 obtained from our model, and the zig-zag turning ratio of 2.1 reported by Li et al.
. Although Bosgraaf and van Haastert 
found a higher pseudopod zig-zag ratio, near 3, our results are not directly comparable because we include all pseudopods whereas they included only pseudopod series judged to be splitting, excluding pseudopods judged to be de novo
Model predicts observed dependence of zig-zag ratio on distance from parent
The first paper to analyze pseudopod zig-zagging found that a pseudopod forming very close to its parent was less likely to zig-zag than a pseudopod forming farther from its parent 
, although this behavior was not explained. Interestingly, simulations from our model reproduce this behavior as shown in , where we plot the pseudopod zig-zag ratio for pseudopods emerging at different angles from their parent. New experimental data from our pseudopod tracking algorithm () agree with both the previous observations and our model results. The zig-zag ratio in declines at 180° from the parent because of circular symmetry. Note that angular difference is directly proportional to arc-length distance for model cells, which always remain circular. Although real cells change shape as they move, their shapes remain relatively smooth and their pseudopods extend perpendicularly from the membrane 
. Therefore a larger angular difference still generally implies a larger spatial distance along the membrane.
The zig-zag ratio increases with distance from the parent pseudopod.
illustrates the origins of distance dependence using a constructed series of two pseudopods. The first pseudopod (i) formed at θ
30° at time zero and lasted for 7 seconds. After an interval of 4 seconds, the second pseudopod (ii) formed at θ
0° and lasted 7 seconds. The cortical variables are shown 0.5 seconds after the second pseudopod stopped. When the next pseudopod forms, pseudopod (i) will be its grandparent and pseudopod (ii) will be its parent.
Close to the parent (ii), the amount of local memory is high (dash-dotted blue). The memory that was generated by the grandparent (i) is lower because memory decays and diffuses over time (dashed blue). Therefore, in the area near the parent, the smaller amount of additional memory from the grandparent will make a smaller relative contribution to the total local memory (solid blue). This means that very near a parental pseudopod such as (ii) in , there will not be a large relative difference between the amount of memory on one side of the parent and the other, and there will not be a large inclination for a child pseudopod to zig-zag. Farther away from the parent, however, there is less total memory in the cortex. Additional memory from the grandparent would here make a larger relative contribution, increasing the probability that a child pseudopod would zig-zag. In addition, farther from the parent there is a larger spatial distance between a point on one side of the parent and a point at the same distance from the parent on the other side. If cortical excitability is a gradually varying function of distance as our model suggests, points that are farther from each other would be expected to have larger differences in excitability.
Predicted pseudopod dynamics are statistically significant in the experimental data
In agreement with model predictions, all five variables discussed above appear correlated with the probability that a pseudopod will zig-zag, including the angle of a pseudopod from its parent, the angle of the parent from the grandparent, whether or not the parent zig-zagged, the size of the grandparent, and delay time since the grandparent ceased growing (). To test whether the observed trends are statistically significant, we performed multiple logistic regression. The particular equation we fit was:
is the probability of zig-zagging (note that
is the zig-zag ratio), Δθ
is the angle of a pseudopod from its parent, Δθparent
is the angle of the parent from the grandparent, Zparent
takes the value 1 if the parent zig-zagged and 0 otherwise, Agp
is the area of the grandparent, Δt
is the delay time since the grandparent stopped growing, and the corresponding β'
s are the fit coefficients. We used the deviation of pseudopod angles from 90° because zig-zag ratios are highest at approximately 90° for both Δθ
() and Δθparent
The resulting best fit to experimental data shows that four of the five variables are significantly correlated with the probability of zig-zagging (). Pseudopods are less likely to zig-zag as Δθ or Δθparent deviates from 90° (P<10−6 and P<10−10 respectively), pseudopods are less likely to zig-zag if the parent had zig-zagged (P<0.005), and pseudopods are less likely to zig-zag as the delay time since the grandparent increases (P<0.01). Pseudopods are more likely to zig-zag as the area of their grandparent increases, but this is not a significant trend in the multiple regression analysis (P<0.14). However, when the grandparental area is used as the sole predictor in a separate logistic regression, the grandparental area does significantly predict zig-zagging (P<0.05).
Multiple logistic regression results evaluating the effects of observed trends on pseudopod zig-zag probabilities.
The fact that grandparental area significantly predicts zig-zagging when used as the sole predictor, but not when used in combination with the other variables in Eq. 5, suggests that the grandparental area may be correlated with some of these other variables. Indeed, the grandparental area is negatively correlated both with Zparent (P<5×10−4 using a single logistic regression), and with Δt (P<10−7 using either Kendall's or Spearman's rank correlation). When Zparent and Δt are excluded from Eq. 5, the area of the grandparental pseudopod is significant as a predictor of zig-zagging (P<0.05).
The EC&M model is consistent with these observed negative correlations between Agp and both Zparent and Δt. The first correlation is equivalent to saying that a larger parent makes the child less likely to zig-zag. Larger parents are generally longer-lived, and longer-lived parents generally increase the time interval between grandparent and child. Since such a longer grandparent-child interval decreases the likelihood that the child will zig-zag, it follows that larger parents should have children that are less likely to zig-zag.
The second correlation – that there is a smaller grandparent-child interval when the grandparent is larger after – supports our hypothesis that pseudopod activity increases the excitability of the cellular cortex. In the framework of the EC&M model, larger grandparents create more cortical memory, which makes the cortex more excitable. This more excitable cortex then leads to shorter intervals between pseudopod bursting events. Simulation results recapitulate this finding: longer-lived grandparents are followed by children after shorter intervals.
A potential concern is that the observed trends may be influenced by difficulty in determining small turning angles. To confirm that our conclusions were not influenced by noise in extracting pseudopods with small turning angles, we excluded from analysis any pseudopods for which Δθ
was less than 30°. When we performed logistic regression on this smaller dataset, we still found all variables to be significant predictors of zig-zagging (Fig. S2
, again separating the grandparental area because of its correlations with the other variables).
Thus all five trends predicted by the EC&M model are statistically significant in experimental data after accounting for correlations between the trends. In addition, the observation that larger pseudopods are followed by their children and grandchildren with shorter delay times supports our hypothesis that higher pseudopod activity increases cortical excitability.
Model accounts for chemotactic behavior
For many crawling cells external chemical gradients can modulate motility to produce a directed path. A chemotactic gradient creates a gradient of bound cell-surface receptors, which in turn stimulates a gradient of downstream signalling activity, which feeds into the basal motility circuit 
. We can extend our model to include chemotaxis along external gradients by spatially varying k0
, the basal production rate of M
. displays typical paths and chemotactic indices from simulations in which k0
is varied, resulting in chemotactic behavior (see Methods
). We find statistically significant chemotaxis in response to variation of k0
by as little as 3% across the model cell, with the chemotactic index increasing as the total variation of k0
(the “gradient”) becomes larger. Experimentally, the threshold for significant chemotaxis in Dictyostelium
cells occurs at a receptor occupancy difference of 1% to 16% across the cell, depending on conditions 
. Our model is consistent with this range of thresholds, even without assuming internal sharpening of gradients downstream of receptors. Importantly, model cells are still able to perform chemotaxis even when the assumed gradient in k0
is much smaller than the order of magnitude difference between k0