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Persistent Cell Motion in the Absence of External Signals:
A Search Strategy for Eukaryotic Cells
Liang Li1, Simon F. Nørrelykke2, Edward C. Cox2*
1Department of Physics, Princeton University, Princeton, New Jersey, United States of America, 2Department of Molecular Biology, Princeton University, Princeton, New
Jersey, United States of America
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 ,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 (,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.
Citation: Li L, Nørrelykke SF, Cox EC (2008) Persistent Cell Motion in the Absence of External Signals: A Search Strategy for Eukaryotic Cells. PLoS ONE 3(5): e2093.
doi:10.1371/journal.pone.0002093
Editor: Mariko Hatakeyama, RIKEN Genomic Sciences Center, Japan
Received December 19, 2007; Accepted March 6, 2008; Published May 7, 2008
Copyright: ? 2008 Li et al. 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 credited.
Funding: This work was supported by the NIH, a Burroughs Welcome Fellowship and a stipend from the NIH/NIGMS (P50 071508) to L.L., and support from the
Carlsberg Foundation and the Lundbeck Foundation to S.F.N. There is no participation by funding agencies.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: ecox@princeton.edu
Introduction
It is generally believed that eukaryotic cells are large enough to
detect and then move toward a signal by counting receptor
occupancy. This can work because the relatively large eukaryotic
cell is not subject to Brownian motion and can therefore use spatial
differentiation to detect the direction of the signal over the relevant
time scale. How this spatial differentiating is accomplished is an
active area of research in fibroblasts, neutrophils and Dictyostelium,
where the major components of the chemotactic response system
are well known.
Of comparable interest is the behavior of these cells in the
absence of a chemotactic (or other) signal. We might imagine, for
example, that cells move about randomly in such a situation
(Fig. 1A), or that they have evolved a strategy that somehow
optimizes their chances of finding the source of the signal, even
when they cannot sense it (Fig. 1B–D).
Is there indeed such a thing as an optimum search strategy?
Recent theoretical work has suggested that a Le ´vy walk is the
optimum for revisitable targets, that is, targets that repopulate at
the same location after a period of time [1–4]. A Le ´vy walk is a
special class of random walks whose step lengths (l) are best
described by a power-law: N(l),l2awhere 2,a,3. Thus there is
no intrinsic scale to the step lengths, and very long steps can occur
(Fig. 1B). Although there was thought to be experimental evidence
for Le ´vy walk behavior in animal populations, a recent reanalysis
of the data makes this unlikely [5], but see also [6]. In a search for
non-replenishable targets, where, like hide and seek, each target
can be found only once, it has been suggested that a two-state
model optimizes the search [7–9] (Fig. 1C). A searcher alternates
between a local random search and a fast linear relocation. Target
detection does not occur during the linear phase, both phases stop
at random times, and each new phase is initiated in a random
direction. It has been suggested that such intermittent behavior
may be used by foraging animals [10,11].
Do any of these processes describe the behavior of single cells
searching for a hidden target? A great deal is known about how
neurons [12], amoebae [13], and fibroblasts [14] find their targets
once the signal has been sensed. In all three cases, more or less
linear trajectories with variable low amplitude random behavior is
the likely rule once the target is in range. But before pioneer
neurons sense and begin to move up (or away from) a graded
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signal, do they send out filopodia at random, or do they bias their
search to enhance the chances of finding the as yet undetected
target? Do neutrophils wander at random before they detect
bacterial peptides, or do they bias their motion in some fashion
that provides a more efficient search algorithm?
We studied these questions by placing well-separated Dictyoste-
lium amoebae on an agar surface free of food at a density of
,1 cell/cm2, ,1000 cell diameters between cells, a distance
chosen so that the amoebae could not sense and signal to each
other. Our results are of two kinds: First, these cells show long
directional persistence. They bias their motion by making turns
every 1–2 min, remembering their last turn and turning away
from it in a zigzag fashion. Similar results were obtained with the
distantly related slime mold Polysphondylium. We provide a model
that satisfactorily captures the turning bias of freely moving cells,
and links short and long-term cell motion persistence times.
Second, although we cannot say that this behavior has been
optimized by selection, we do demonstrate that it is only somewhat
less efficient than straight-line behavior.
We believe this is the first experimental evidence for a biased
walk in a foraging eukaryotic cell in the absence of spatial and
temporal cues. Because the machinery underlying eukaryotic cell
motion has been so highly conserved during evolution, we think it
is likely that similar behavior is characteristic of other target-
seeking eukaryotic cells.
Results
Nomenclature
Depending on which branch of science the reader hails from,
the meaning of terms such as ‘‘random walk’’ and ‘‘Brownian
motion’’ may differ. To eliminate at least this source of possible
confusion we offer our own definitions here.
Brownian motion refers only to the passive random motion,
reported by Robert Brown, of particles suspended in a fluid.
Random walk is taken to mean a stochastic path consisting of a
series of steps, whose direction is chosen at random and where all
directions are equally probable. The step size can be either
random or fixed.
Random motion is the most general term and refers to any
stochastic path describing the motion of a particle. There may or
may not be a preferred spatial direction, correlations in step size,
persistence in direction of motion, oscillations in the velocity, etc.
The only demand is that there be some element of stochasticity in
the motion.
A long directional persistence
In Fig. 2A we show the behavior of 3 representative Dictyostelium
cells, each one followed for 8–10 hrs corresponding to ,300 cell
lengths, with a sampling interval of 10 sec. We found that
amoebae traveled at an average speed of 7 mm/min for up to
10 hr (Fig. 2B), demonstrating that they maintained an adequate
energy source over the course of these experiments. Thus, our
modeling is not confounded by changes in average cell speed over
time. On the time scale of minutes the speed was found to fluctuate
around this average speed (Fig. 2B, insert).
The mean-squared displacements (MSD) of the individual cells
are summarized in Fig. 3A and B as Æd(t)2æ vs t, and Æd(t)2æ/t vs t
respectively. t is the time interval between any two positions,
d t ð Þ~~ r r tzt
points for each trajectory. For a random walk, e.g. Brownian
motion, Æd(t)2æ/t is constant. The 3 cyan curves are measurements
from the trajectories shown in Fig. 2A, and the red curves are from
an additional 9 trajectories. For t,30 min, cell movement
deviates significantly from the random walk expectation and is
essentially ballistic, i.e., the cells are on average moving in a
straight line with constant velocity. For time intervals t between 10
and 100 min, the population-averaged data were well fitted by an
exponential cross-over from directed motion to a random walk,
ðÞ{~ r r t ð Þ
jj, and d(t)2was averaged over all pairs of time
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.
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Æd(t)2æ=2tpv2(t2tp(12exp(2t/tp))) [15], where v=5.460.1 mm/
min is a characteristic speed and tp=8.860.1 min is a persistence
time (Fig. 3B, yellow curve). Note that a cell displaces itself
approximately 3 full cell diameters (,50 mm) in 8.8 min. This
transition from directed to random walk is characteristic of the
entire time record of more than 10 hr, and the persistent time is
independent of where on the trajectory we begin our analysis.
We calculated cell velocities as ~ v v t ð Þ~ ~ r r t ð Þ{~ r r t{t
different values of t and plotted vxvs vy(Fig. 4). With increasing t,
bigger and bigger gaps appear in the centers of these plots.
Atverymuchlargervaluesoft,greaterthan 30 min,the distribution
of vxvs vyvalues again approaches Gaussian behavior, as expected
(Fig. 4). As we show in the discussion, these results essentially
rule out two well-understood models of random motion, worm-like-
ðÞðÞ=t for
chain (WLC) [15] and Ornstein-Uhlenbeck (OU) [16] models for
Dictyostelium cell trajectories.
Angular changes and cell motion
In order to quantify the behavior of a cell we first introduce a
measure of the cell’s instantaneous direction of motion (Fig. 5).
This measure is chosen as the cumulative angle Q, between the cell’s
velocity vector and a fixed direction in space: If, initially
Q(t=0)=Q0, and the cell at some time t later has moved through
a complete, counter-clockwise, circle, then the new direction of
motion is Q(t)=Q0+2p. That is, the angle Q tracks not only the
instantaneous direction of cell motion, but also the winding
number of the cell trajectory, and thus, to an extent, the history of
the cell’s directional changes.
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.
doi:10.1371/journal.pone.0002093.g002
Figure 3. Mean-squared displacement. (A) Log-log plot of the mean-squared displacement vs time interval t. (B) Mean-squared displacement
divided by t plotted as a function of t. 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 t
[10:100] min: Æd(t)2æ=2tpv2(t2tp(12exp(2t/tp))), where v=5.460.1 mm/min is a characteristic speed, and tp=8.860.1 min is a persistence time.
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With this definition, periods of straight-line motion correspond to
a constant angle; small zig-zags in the direction of motion show up as
oscillations in the angle (Fig. 6C); large turns as large increases and
decreases in the angle (red box, Fig.6A); and circlingbehavior adds or
subtracts 2p per circle completed in the counter-clockwise and
clockwisedirection(Fig.6B),respectively.InFig.6weseeallofthese
classes of behavior except the first one: The cells are never observed
to move in a completely straight line. However, any given cell will at
times appear to move in a certain fixed direction h, around which
the instantaneous angle Q fluctuates. In Fig. 6C, the cell maintained
an average direction h=2p/4 for about an hour and Q fluctuated
around this value with a characteristic time of 2–3 min and
amplitude near p/4.
Turning preference
To further investigate cell motion, we studied the occurrence of
discrete turns in the cell trajectories. Over a range of user-defined
parameters, such as the threshold for calling a turn, the results
reported below are robust (see Materials and Methods).
We observed the same frequency of left and right turns, and
thus choosing one hand over the other does not contribute to the
observed wiggling behavior. However, these data do reveal a
strong turning preference, in which cells tend to turn away from
their last turn. Fig. 7A illustrates our turn-run-turn analysis. Fig. 7B
plots data from all 12 experimental runs. The turning ratio was
biased by a factor of 2.160.1 (mean6sem, n=12 cells), obtained
by classifying 4822 consecutive turns from all trajectories. The
correlation coefficient for consecutive turning directions, for all
cells, was 20.36, with a P-value,1024, a highly significant anti-
correlation. There is a weak, but significant, positive correlation
between the last and the second-from-last turn, but cell memory
does not extend much further back (Fig. 7C). For comparison, the
insert in Fig. 7C shows the autocorrelation from a Monte Carlo
simulation of the WLC model.
Fig. 7D shows a histogram of the turn amplitudes. Its tail is
best described by an exponential distribution (characteristic
angle<0.67 rad). Small angles are rarely observed, partly
because of limitations in the turn detection algorithm, but also
because of the pseudopod-branching process we discuss later. No
significant temporal correlation was observed (upper right panel).
Fig. 7E is the histogram of time intervals between turns. It is also
an exponential distribution (characteristic time<0.67 min) and
again no significant autocorrelation was observed, consistent
with a Poisson process. Fig. 7F gives the histogram of distances
between the positions of the cells at consecutive turns. Its tail is
well fitted byan exponential
length<5 mm).
distribution (characteristic
Biased motion in Polysphondylium
These experiments were repeated with the distantly related
slime mold amoeba Polysphodylium pallidum with essentially the same
results (Figure S1, S2). Polysphodylium and Dictyostelium are in
different Genera, use different chemotactic signals, have different
fruiting body morphologies, and diverged ,500,000 years ago
[17]. This suggests that the dynamic behavior documented here is
highly conserved and suggests further that it may be a feature of all
migratory eukaryotic cells.
Discussion
Non Ornstein-Uhlenbeck (OU) and non worm-like-chain
(WLC) motion
We have learned that Dictyostelium cell motion cannot be
described by a random walk. Next, we show that two other
standard models also fail to capture our data. First, note that an
exponential crossover of the form we used to fit the MSD data
(Fig. 3B) arises in several models describing disparate physical
phenomena: Both WLC [15] models from polymer physics, and
OU [16] processes from the modeling of Brownian motion, have
this feature. Thus, an exponential crossover is not in itself enough
to pin down the dynamics of cell motion.
We observed larger and larger gaps appearing in the centers of
vxvs vyplots, with increasing t (Fig. 4). This is because at very
short time intervals the cells have moved very little, and the data
shows essentially random micron-scale jiggling about the
centroid of the cell. This Gaussian behavior at very small t also
Figure 4. Non-Gaussian velocity distribution. Velocities were
calculated for different t: ~ v v t ð Þ~ ~ r rt{~ r rt{t
with increasing t. 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 t, the distribution
approaches a Gaussian again. Inserts, histograms of the x component
of the velocities for the intervals defined by the parallel lines.
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ðÞ=t and vxwas plotted vs vy
Figure 5. Cells move in a zig-zag manner. Enlarged view of the
rectangular box in Fig. 2. The magenta scale-bars are 10 mm. Turns are
marked by black crosses. Motion following left turns is blue and motion
following right turns is yellow.
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indicates a non-WLC movement, because a WLC motion has a
constant speed, thus giving rise to a circular plot. With increasing
t, the behavior is distinctly non-Gaussian, ruling out the OU
process because it predicts a Gaussian distribution for all t. Two-
dimensional velocity histograms with a crater shape are one of
the hallmarks of self-propelled persistent motion and have been
predicted to occur when cells migrate [18]. To the best of our
knowledge, this is the first published experimental demonstration
of this effect.
Modeling angular change
To capture the behavior observed in cumulative angles (Fig. 6),
we write down dynamic equations for the angles Q and h. The
average direction of motion h is modeled as a random walk,
reflecting the experimental fact that there are no preferred
directions of motion in the absence of a chemotactic signal. The
instantaneous direction of motion Q, was observed to be somewhat
enslaved by h, exhibiting noisy oscillations around it with a
characteristic time-scale. We therefore model Q as a sum of a
random walk h, and noisy oscillations y (Eqs. 1–3).
Fig. 8 illustrates the biological processes and features
corresponding to the two terms in Eqs. 1 and 2. In our model,
each cell has an ‘‘intrinsic vector’’ that establishes an angle h
relative to an arbitrary, but fixed, direction in space. We leave
the molecular components of this intrinsic vector undefined, but
remark that it could e.g. be given by a vector connecting the cell
nucleus with the centrosome [19]. This intrinsic vector is
assumed to change its orientation only slowly over time, and it
consequently does not track the instantaneous direction of
motion, which will fluctuate around h. The instantaneous
direction of motion Q, is determined by short-lived processes,
such as a stick-slip event, or the extension of pseudopods, as
illustrated in Fig. 8. Although the sum of all the fast processes is
ultimately the cause of the changes in h, we model this sum
simply as a random number, independent of Q. That is, we
model the orientation h, of the intrinsic vector as a random walk
and ignore its causal connection with Q:
dh
dt~
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2Dhgh
p
,
ð1Þ
where Dhis a diffusion coefficient and gh(t) is a normalized,
Gaussian, white noise term whose mean is zero.
Figure 6. Cumulative angles. (A) Angles were corrected by +/22p for changes larger than p. 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 24p is continuous. (C) Enlarged view of box 2 in A. To reduce the influence of noise, angles
were calculated at a larger t (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.
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Figure 7. A left turn is followed by a right turn –a Poisson process. (A) Definition of angle of turns (a), direction of turns (left or right), time
between turns and length between turns. (B) The jthturn plotted against the (j+1)thturn 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)thturn is biased by the jthturn 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 a. 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 tj+12tjis 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 mm).
Distribution of length is trivially exponential if cell averaged speed is constant and times between turns are exponentially distributed.
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The time-development of the instantaneous direction of motion
is governed by
Q t ð Þ~h t ð Þzy t ð Þ,
ð2Þ
where the second term on the right-hand-side y(t) is colored noise:
It is the solution to the second order stochastic differential
equation describing a noise-driven, harmonic oscillator with
resonance at 2pf0~
k=m
ffiffiffiffiffiffiffiffip
~{cdy t ð Þ
md2y t ð Þ
dt2
dt
{ky t ð Þzc
ffiffiffiffiffiffiffiffiffi
2Dy
p
gyt ð Þ,
ð3Þ
where m is a persistence parameter (in a mechanical model of a
block on a spring it would be the inertial mass, but mass plays
no role here), c is a dissipative parameter (friction in a mechani-
cal model), k is a restoring-force parameter (spring constant), Dy
gives the strength of the driving force (noise) gy(t), which is nor-
malized and has zero mean, but is not necessarily Gaussian, or
white.
Does this model adequately capture the essential features of
the data? We calculated the expected power spectral density
(PSD) of Q (Materials and Methods), and compared the results
to our experimental PSD (Fig. 9A). We also included the aliasing
effect of finite sampling frequency and introduced a noise term
accounting for measurement errors (dotted line; and see
Materials and Methods). Fig. 9A shows that our model fits the
experimental PSD well. Two time scales were obtained from the
fitting parameters and found to be consistent with other analyses
discussed here. The value for f0, the resonance frequency, gives
a characteristic time of 2.460.1 min, consistent with the
average oscillation period observed in the time series of Q. h is
the overall direction of motion, and its value is subject to a
random walk (Eq. 1). Given the fitted diffusion coefficient Dh,
we estimate that after 1 rad2/Dh<8 min, the MSD of h grows
large enough that we can consider cells to have lost their
original direction. This is also consistent with the persistence
time described in Fig. 3.
In order to further test our model we ran Monte-Carlo
simulations of Eqs. 1–3 with parameters obtained from a fit to
the experimental PSD. We then subjected the synthetic data to the
same analysis as the experimental data. As an example, Fig. 9B
shows the close agreement between the measured autocorrelation
function of DQ(t)=Q(t)2Q(t2t) for both experimental and
synthetic data. We also formed the histogram of DQs and found
that it follows a Laplacian (double-exponential) distribution,
Fig. 9C. The same distribution was found for the synthetic data
(data not shown). A Laplacian distribution describes the difference
between two independent, identically distributed, exponential,
random variables, implying that Q itself is exponentially distrib-
uted. Such exponential distributions for motility data are thought
to arise from the interplay of various cellular processes combined
with a finite rate of ATP production [20].
Dictyostelium cells move by continuously extending new pseudo-
pods. A restoring force exists because pseudopods cannot extend to
infinity, and thus oscillations should be restricted about h. The
driving force in our model is a sum of white noise contributed by
all lateral pseudopods, and an oscillating force applied by the
leading pseudopods. A colored noise term captures these
contributions. Normally, a pseudopod leads cell motion for 1–
2 min before a new pseudopod forms and dominates, consistent
with the observed periods of oscillations shown in Fig. 6.
A quantitative picture of cell motion
The observation that cumulative angles oscillate, and the way
that cells move by extending pseudopods, inspired us to describe
cell motion as a discrete, but easily pictured turn-run-turn model.
The anti-correlations found in turn directions is consistent with
the discovery that new leading pseudopods arise mostly by
dividing old ones [21]. These correlations can be understood in
Figure 8. A model for Dictyostelium motion. (A) The blue line is hand drawn to guide the eye and represents h, a one-dimensional random walk.
The black line Q, is an enlarged view of the data in box 3 of Fig. 6. In our model, Q is the net effect of h and colored noise centered on h. The
stochastic differential equations used to describe the behavior of Q and h are explained in detail in the text. (B) h 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 [19]. 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.
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a quantitative manner with the smallest possible set of assump-
tions. Next to having no memory at all (as in the flipping of a coin),
the simplest kind of memory is one that only extends one step back
in the past. Processes that have this kind of memory are called
Markov processes. Assume that the turns are given by a Markov
process and ascribe a value of +1 to left-turns and 21 to right-
turns, or vice versa. The autocorrelation function for the turns is
then easily found to be cm=(p2q)m, where m is the time-lag
measured in turns, p is the probability of making a turn in the same
direction (left-left or right-right) and q is the probability of making
a turn in the opposite direction (left-right or right-left). Both p and
q are known to us, since p+q=1 and q/p=2.160.1 (see Results).
The near-exact equivalence of the data and the theory shown in
Fig. 7C supports this interpretation of the turn-memory in
Dictyostelium.
Taken together, the data presented in Figs. 7 & 9 support the
following descriptive model of migration: Each cell is polarized
(has an intrinsic vector) and moves (runs) in a random direction for
a random period of time by the extension of pseudopods. Because
the duration of a run is exponentially distributed and uncorrelated
with any previous run duration it is tempting to think of each run
as corresponding to an actin-polymerization event nucleated by a
protein complex anchored to the leading edge of a pseudopod. If
the association (dissociation) of this actin-nucleating protein
complex with the membrane is equally probable at all times, it
would be described by a Poisson process and the time between on/
off events would follow an exponential distribution. Why the
direction of migration follows a double-exponential distribution is
less obvious, but may reveal that there are just a few limiting steps
underlying the behavior documented in Fig. 7D.
We may compare our results to the pioneering work of others
with Dictyostelium and the results of Hartman et al with neutrophils
[22,23]. Potel and Mackay carried out a very thorough analysis of
aggregation-competent Dictyostelium cells plated at a cell density
orders of magnitude larger than the one we have used. In their
experiments, cell-cell interactions were common. Their results on
cell speed as a function of time, and change in direction between
successive times as an approximate demonstration of a persistence
time, are in broad agreement with ours. However, their time
resolution, and the fact that the cells were responding to each
other’s presence, confounded their results, and in the end they
concluded that a simple, persistent, random walk best fit their
data. Hartman et al found local non-Markov displacements in their
study with neutrophils. Although their time resolution was high,
their analyses were applied to cell traces that are typically only a
few cell-diameters long, and thus their experiments did not last
long enough to detect the features we report here.
An efficient search strategy?
In the Introduction we described three well-studied models have
been used to characterize searching behavior (Fig. 1). Dictyostelium
motion differs from each of them. Unlike the intermittent behavior
in the two-state model, no obvious searching and relocating phases
are observed. However, we note that Dictyostelium cells appear to
have gone one step further by not pausing on the search, and
detecting signals as they move. Instead of choosing turns randomly
in amplitude and direction (random walk) or sampling from a
power law distributed run length (a Le ´vy walk), both Dictyostelium
and Polysphondylium bias their motion by remembering their last
turn and employ a persistence time of ,9 min. This means that
cells cover more territory in a given number of steps than they
would in a random walk (Fig. 1A,D). Also, they pick turn
amplitudes randomly from an exponential distribution rather than
a Gaussian distribution, which has been suggested to help optimize
Figure 9. Statistics of the cumulative angles and fit to the
theory. (A) Experimental power spectral density (PSD) of Q and fit of
the theory to the data. Two time-scales were returned by the fit:
(f0)21=2.3560.08 min and 1 rad2/Dh=7.660.3 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 DQ. Qs were calculated for t=30 s. Blue: Experimental
values and standard errors. Black: Theoretical expectation value
calculated from a Monte Carlo simulation on Q based on Eqs. 1–3,
with parameters obtained from a fit to the PSD. (C) Experimental
histogram of DQs calculated for t=30 s. Insert: Same histogram shown
on a semi-logarithm scale demonstrating the non-Gaussian, exponential
tails. With increasing t, the distribution of DQ becomes more and more
Gaussian (data not shown).
doi:10.1371/journal.pone.0002093.g009
Cells Searching sans Signals
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Page 9
search efficiency when targets are randomly distributed in a patch
of finite size [24].
The character of this search algorithm was examined by
comparing our results to straight-line and random walk searches
with respect to the efficiency of searching, defined as the number
of targets captured per unit time, the number of cell diameters
over which a cell can recognize a close-by target, and the target
density. In these simulations, which assume that targets are
arrayed at random on an infinite plane, straight-line searching is
the most efficient because for a given detection radius R all
deviations from a straight line cover some fraction of the area
already covered; and, as expected, the efficiency is lowest for
random walk searches. Dictyostelium cells are about half as efficient
as a straight-line search, and 1.6 to 2.4 fold more efficient than
random walk searching.
The improvement in search efficiency relative to a random walk
is due to the fairly straight motion of the amoebae. Although the
cells move by the extension of a series of discrete pseudopods, the
direction of protrusion of these processes is coordinated in a
manner that gives rise to an overall directed motion: Each new
pseudopod propels the cell in a direction slightly angled relative to
the general direction of motion, but the pseudopods are generated
in a left-right-left-right fashion such that the cell zigzags it’s way
forward. This propensity to zigzag is quantitatively described by
the anti-correlation between turns, see Fig. 7 A–C. Without the
anti-correlation, the persistence length would drop from the
observed value of 48 mm to just 20 mm (calculated from the
equivalent two-dimensional freely-rotating-chain model, with
segment length=5 mm and angle=0.67 radians).
In addition, the comparatively long persistence time of ,9 min we
discovered might help amoebae not only while foraging on their own,
but also during the earliest stages of multi-cellular life when starving
cells begin to signal to each other. The signaling system consists of
cyclic AMP waves that propagate as spirals or circles from a core of
signaling cells. These waves continue to organize morphogenesis as
cells begin to stream towards the signaling centers guided by the
traveling wave front. During the earliest stages of signaling, wave
periodicity is 6–10 min, depending on the strain and growth
conditions. This time corresponds closely to the 9 min persistence
time reported here, and suggests to us that once cells are given an
orientation in the wave, they propagate towards the center without
further information from the cyclic AMP gradient. This may also help
explain how cells discriminate between the back of a receding wave
and the front of an approaching one. They use the persistence time
once oriented to move in a more or less straight line toward the center
and away from the periphery. Because this bias is in our view
stochastic in nature, this might also help explain why cells occasional-
ly move in the wrong direction, i.e. away from the signaling center.
Summary
We have discovered that Dictyostelium and Polysphondylium cell
motion is not a simple random walk. Unlike a Le ´vy walk, no
intrinsic scale invariance in cell trajectory is apparent. Unlike an
Ornstein-Uhlenbeck process, cell velocity distributions deviate
from a Gaussian velocity distribution. Unlike a worm-like-chain
model, the observed oscillations in angles indicate a well-
developed and organized cellular mechanism driving the observed
behavior. With respect to searching strategy, a left turn tends to be
followed by a right turn. Cells move forward in a zig-zag manner
and maintain a long directional persistence. In this way, time
wasted on exhaustive back and forth searching is greatly
reduced, thereby enlarging the search area and improving search
efficiency.
Materials and Methods
Cell Culture
Dictyostelium discoideum AX4 and Polysphondylium pallidum PN500
were grown on lawns of Escherichia coli B/r at 22uC as described
[25]. Vegetative amoebae were harvested and bacteria removed
by centrifugation. The cells were suspended in PB (20 mM
KH2PO4, 20 mM Na2HPO4?7H2O) and plated at densities of
,1 cell/cm2on 2% agar in distilled deionized water. At this cell
density the ratio of cell area/agar surface area is ,461026.
Cell Tracking
Cell movement was followed by phase contrast microscopy
using a 106 objective. Movies were recorded at 10 sec intervals
for 8 to 10 hr. (Movie S1, S2).
Data Analysis
Cell locations were defined as the centroids of a cell’s contours
(Movie S1, S2). The trajectory of each centroid consisted of a
sequence of paired coordinates Dt=10 sec apart: (~ r rj~~ r r tj
jDt,j~1,2,3...). A displacement between any two cell positions
was defined as (~ s sj t ð Þ~~ s sj nDt
were then calculated (~ v vj t ð Þ~~ v vj nDt
1,2,3...). Instantaneous angles were calculated (bj=atan[(yj2
yj21)/(xj2xj21)]), then corrected by +/22p if the changes between
consecutive angles were larger than p, and added to the previous
angle: Qj=Qj21+bj. Also, if the change in two consecutive angles
was bigger than p, it was identified as a false jump. The jump was
then replaced by two random numbers from a Gaussian
distribution with zero mean and the same standard deviation as
obtained from experimental data. Further analysis and original
programming was carried out in MATLAB.
? ?,tj~
ðÞ~~ r rj{~ r rj{n,n~1,2,3...). Velocities
ðÞ~~ s sj nDt
ðÞ=nDt~~ s sj t ð Þ?t,n~
Recognizing turns
Cell trajectories were first smoothed by a moving average over 5
consecutive positions(~ r rj~ ~ r rj{2z~ r rj{1z~ r rjz~ r rjz1z~ r rjz2
Then the time series of angles (Qj) and changes in consecutive
angles (DQj=Qj2Qj21) were calculated with t=10 s using
smoothed positions. DQjrepresent cell turning rates as a function
of time. When its amplitude goes above a threshold value, a cell is
considered to be making a turn and the corresponding time points
were marked. Next, the marked time points were clustered: First,
all consecutive points were clustered and the largest cluster-
member was picked to represent this turn; Second, if the time
interval between any two consecutive turns was less than 30 sec, it
was considered to be part of the same turn and again the largest
value was picked. Each cell had an individual threshold value,
chosen as the average amplitude of that cell’s DQ series. If the turn
had a positive value for DQ it was recorded as a left turn
(counterclockwise), otherwise, as a right turn (clockwise).
???5).
Cumulative angle analysis: PSD, autocorrelation and
histogram
To reduce noise in the analysis of cumulative angles (Fig. 9), Q
was calculated with t=30 s using unprocessed original positions.
Thus, from each cell trajectory, 3 interlaced time series of Q were
obtained and analyzed.
Simulating Search Efficiency
A Monte-Carlo simulation was employed to compare
Dictyostelium trajectories to random walks and straight-line
searches. Targets were distributed randomly on an infinite
plane with a characteristic average density. Searchers using
Cells Searching sans Signals
PLoS ONE | www.plosone.org9May 2008 | Volume 3 | Issue 5 | e2093
Page 10
different strategies were compared: Amoeboid motions were
parameterized using our experimental data, random walks were
simulated using the average speed for Dictyostelium, and straight-
line movement at a constant speed used the mean speed of
these amoebae. If a target site lay within a ‘detection radius’ (R)
from the searcher, then this target was scored as found and
removed. Efficiency was defined as the ratio of the mean
number of targets found to the total traveling time. In each
simulation, cells were allowed to search for 10 hr. The
simulation was repeated with a range of target densities
(1022mm22–1025mm22), and the detection radius was varied
from 5 mm to 75 mm, 1/3 to 5 cell diameters, respectively.
Simulation of cumulative angles
We simulated the harmonic noise by first solving Eq. 3 on the
discrete time-interval Dt (=30 sec) to obtain the recursive rela-
ftion:
yjz1
ujz1
??
~e{MDt
yj
uj
??
z
Dyj
Duj
??
,
M~t
0{1
c=m
k=m
s,
where u=dy/dt, and we demand that Dyjand Dujbe exponentially
distributed random numbers of zero mean and flat power
spectrum. Note that the distributions for Dyjand Dujdepend on
Dt because exponential distributions, unlike Gaussian ones, are not
stable under integration. That is, as Dt is increased, these
distributions will tend to Gaussians.
Power spectral analysis
The frequency content of Q was examined by Fourier-
transforming the two independent equations of motion, Eqs. 1 & 3,
{2pifk~h hk~
ffiffiffiffiffiffiffiffiffi
2Dh
p
~ g gh,k
and
{2pifk
ðÞ2m~ y yk~2pifkc~ y yk{k~ y ykz
ffiffiffiffiffiffiffiffiffi
2Dy
p
c~ g gy,k
where
~ x xk~
ðtmsr=2
{tmsr=2
x t ð Þei2pfktdt,
x=h, y, gh, or gQ, and tmsris the measurement time.
From which we form the power spectral density:
Pk:S ~ Q Qk
jj2T
tmsr
~Dw
?
2p2
f2
k
??
z
Dy
?2
?
f2
2p2
??
2pm
c
?
0{f2
k
??2zf2
k
:
Where we have used the following characteristics of the noise
terms
S~ g gh,kT~S~ g gy,kT~0;
S~ g g?
h,k~ g gh,lT~S~ g g?
y,k~ g gy,lT~tmsrdk,l;
S~ g gh,k~ g gy,lT~0;
We see that the PSD consist of two terms, one that decays as f22
(first term on the right-hand-side corresponding to free diffusion)
and one that, potentially, has a resonance peak at fk=f0(second
term on the right-hand-side).
Aliasing/finite sampling frequency
For reasons of mathematical ease, the above treatment
implicitly assumed continuous sampling. In an experiment,
however, data are taken at discrete time-intervals, leading to
aliasing. This means that, for frequencies near fsample, the
measured power can be more than 100% larger than predicted
from the above theory [26]. Taking aliasing into account is
straightforward and introduces no extra fit-parameters, as shown
below.
The aliased version of the first term, corresponding to free
diffusion, is:
Palias
k
ðÞ
~
Dh
.
f2
sample
?fsample
1{cos 2pfk
??
The correct aliased version of the second term is considerably
more involved and we refer the reader to (Nørrelykke & Flyvbjerg,
unpublished). For completeness, we also include a term for the
measurement error in the fit. The measurement error is assumed
to be white noise and thus has a flat power spectrum, so its shape is
unaltered by aliasing.
We zero-padded time-traces of Q to the nearest power of two
greater than the longest trace, before fast Fourier transforming.
Zero-padding artificially increases the frequency resolution of the
PSD. This results in the PSD values no longer being independent
of each other, so the error-bars in the residual plot underestimate
the true standard error.
Least squares fitting
We fitted the above expression to the experimental PSD in the
least-squares sense (via the built-in lsqnlin procedure in MatLab),
using as weights Pk(alias). That is, we minimized:
x2~
X
Palias
k
Palias
k
ðÞ
{Pex
.
ð
k
Þ
ðÞ
ffiffiffin
p
0
@
1
A
2
Least squares fitting presupposes each data point to be drawn from
a Gaussian distribution – which is not the case here! Rather, the
power at a given frequency, averaged over n individual PSDs, is
Gamma-distributed and tends to a Gaussian distribution as nR‘
(Nørrelykke & Flyvbjerg, unpublished).
A Gamma-distribution is skewed, and this skewness leads to an
overestimate of some of the fit-parameters by a factor of 1/n when
doing least-squares fitting (Nørrelykke & Flyvbjerg, unpublished).
But this effect is well understood, and was taken into account.
The fit parameters obtained were: f0=0.0072 rad21s216
3.5%; c/m=0.067 s2169.4% ; Dc=0.012 rad2s2163.2%; and
Dh=0.023 rad2s2164.3%. A harmonic oscillator has three
qualitatively different solutions, depending on whether the
parameter f~c? ffiffiffiffiffiffiffiffiffi
oscillations occur; in the third (f,1) the system is under-damped
and displays exponentially decaying oscillations; when f=1 the
system is poised at a critical point separating the two classes of
behavior. In our case f=0.74612.9%, that is, we are in the
under-damped, oscillating regime, but still rather close to the
critical point, which is why we observe no clear resonance peak in
the power spectrum.
The error-bars cited on the fit parameters were taken from the
formal covariance matrix, calculated as the inverse of the Jacobian
matrix of x2multiplied by its own complex conjugate. Since the
4km
p
is greater than, equal to, or smaller than
unity. In the first case (f.1) the system in over-damped and no
Cells Searching sans Signals
PLoS ONE | www.plosone.org10May 2008 | Volume 3 | Issue 5 | e2093
Page 11
residuals are neither independent nor Gaussian distributed these
variances are only estimates of the true ones.
Supporting Information
Movie S1
telium cells moving on a 2% agar surface. The cells are ,15 mm in
diameter. The movie frames were captured at 10 sec intervals and
these clips are 20 min long. Right panels: finding and tracking the
centroids for the cells shown in the left panels.
Found at: doi:10.1371/journal.pone.0002093.s001 (0.64 MB
MOV)
Left panels: phase contrast movies of single Dictyos-
Movie S2
telium cells moving on a 2% agar surface. The cells are ,15 mm in
diameter. The movie frames were captured at 10 sec intervals and
these clips are 20 min long. Right panels: finding and tracking the
centroids for the cells shown in the left panels.
Found at: doi:10.1371/journal.pone.0002093.s002 (0.69 MB
MOV)
Left panels: phase contrast movies of single Dictyos-
Figure S1
squared displacement plotted for 17 cells. Yellow, fit of an
exponential cross-over from directed to random motion in the
interval t [15:150] min: ,D(t)2.=2tpv2(t2tp(1-exp(-t/tp))), where
v=3.160.1 mm/min is a characteristic speed, and tp=11.76
0.2 min is a persistence time. (B) Cell velocity distributions are
non-Gaussian.
Found at: doi:10.1371/journal.pone.0002093.s003 (3.47 MB TIF)
Polysphondylium motion on an agar surface. (A) Mean-
Figure S2
turn plotted against the (j+1)thturn for the data from all 17
trajectories. There are 2023 data points in the second and fourth
quadrant, 1218 in the first and third, and thus the (j+1)thturn is
biased by the jthturn by a factor of 1.7. (B) Autocorrelation
function for the turn directions. Blue: Experimental values and
standard errors. Black: Theoretical expectation value for a Markov
process with probabilities taken from panel A. Insert: Verification
that turn-correlations are real and not an artifact of the turn-
Polysphondylium motion on an agar surface. (A) The jth
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. (C) Histogram of turn amplitudes. These data are
well fitted by an exponential distribution (characteristic an-
gle=0.72 rad). Lower left panel: Histogram of a. Upper right
panel: Autocorrelation function for turn amplitudes, no correlation
was observed. (D) Histogram of time intervals between detected
turns. These data are well fitted by an exponential distribution
(characteristic time=0.98 min). Data is from all 17 trajectories.
Lower left panel: Same histogram but on linear scale. The smallest
detected value for tj+12tjis 1 min, 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. (E1) Experimental power
spectral density of Q is well fitted by theory. Two time-scales
were returned by the fit: (f0)21=4.360.2 min and 1 rad2/
Dh=7.960.5 min. (E2) Experimental autocorrelation function
for DQ (blue) is consistent with the theoretical expectation (black)
calculated from a simulation. Qs were calculated for t=50 s in E1
and E2.
Found at: doi:10.1371/journal.pone.0002093.s004 (4.14 MB TIF)
Acknowledgments
We thank S. Schvartsman, I. Golding, H. Flyvbjerg, R. Austin, S. Sawai, J.
Shaevitz, and F. Ju ¨licher for discussions, and all members of the Cox
laboratory for their help and advice. This paper benefited greatly from the
comments of an anonymous referee.
Author Contributions
Conceived and designed the experiments: EC LL. Performed the
experiments: LL. Analyzed the data: LL SN. Contributed reagents/
materials/analysis tools: EC LL. Wrote the paper: EC LL SN.
References
1. Viswanathan GM, Buldyrev SV, Havlin S, da Luz MG, Raposo EP, et al. (1999)
Optimizing the success of random searches. Nature 401: 911–914.
2. Bartumeus F, Catalan J, Fulco UL, Lyra ML, Viswanathan GM (2002)
Optimizing the encounter rate in biological interactions: Levy versus Brownian
strategies. Phys Rev Lett 88: 097901.
3. Raposo EP, Buldyrev SV, da Luz MGE, Santos MC, Stanley HE, et al. (2003)
Dynamical robustness of Levy search strategies. Phys Rev Lett 91: 240601.
4. Santos MC, Viswanathan GM, Raposo EP, da Luz MGE (2005) Optimization
of random searches on regular lattices. Phys Rev E 72: 046143.
5. Edwards AM, Phillips RA, Watkins NW, Freeman MP, Murphy EJ, et al. (2007)
Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and
deer. Nature 449: 1044–1048.
6. Boyer D, Miramontes O, Ramos-Ferna ´ndezGabriel (2008) Evidence for
biological Le ´vy flights stands. arXiv:08021762v1 [q-bioPE]; http://arxivorg/
abs/08021762.
7. Benichou O, Coppey M, Moreau M, Suet PH, Voituriez R (2005) Optimal
search strategies for hidden targets. Phys Rev Lett 94: 198101.
8. Benichou O, Loverdo C, Moreau M, Voituriez R (2006) Two-dimensional
intermittent search processes: An alternative to Levy flight strategies. Phys Rev E
74: 020102.
9. Shlesinger MF (2006) Mathematical physics - Search research. Nature 443:
281–282.
10. Kramer DL, McLaughlin RL (2001) The behavioral ecology of intermittent
locomotion. Am Zool 41: 137–153.
11. Obrien WJ, Browman HI, Evans BI (1990) Search Strategies of Foraging
Animals. Am Sci 78: 152–160.
12. Flanagan JG (2006) Neural map specification by gradients. Curr Opin Neurobiol
16: 59–66.
13. Willard SS, Devreotes PN (2006) Signaling pathways mediating chemotaxis in
the social amoeba, Dictyostelium discoideum. Eur J Cell Biol 85: 897–904.
14. Arrieumerlou C, Meyer T (2005) A local coupling model and compass
parameter for eukaryotic chemotaxis. Dev Cell 8: 215–227.
15. Rubinstein M, Colby RH (2003) Polymer physics. Oxford New York: Oxford
University Press. 440 p.
16. Uhlenbeck GE, Ornstein LS (1930) On the theory of the Brownian motion. Phys
Rev 36: 0823–0841.
17. Schaap P, Winckler T, Nelson M, Alvarez-Curto E, Elgie B, et al. (2006)
Molecular phylogeny and evolution of morphology in the social amoebas.
Science 314: 661–663.
18. Erdmann U, Ebeling W, Schimansky-Geier L, Schweitzer F (2000) Brownian
particles far from equilibrium. Eur Phys J B 15: 105–113.
19. Xu J, Van Keymeulen A, Wakida NM, Carlton P, Berns MW, et al. (2007)
Polarity reveals intrinsic cell chirality. Proc Natl Acad Sci U S A 104:
9296–9300.
20. Czirok A, Schlett K, Madarasz E, Vicsek T (1998) Exponential distribution of
locomotion activity in cell cultures. Phys Rev Lett 81: 3038–3041.
21. Andrew N, Insall RH (2007) Chemotaxis in shallow gradients is mediated
independently of PtdIns 3-kinase by biased choices between random protrusions.
Nat Cell Biol 9: 193–200.
22. Potel MJ, Mackay SA (1979) Pre-Aggregative Cell Motion in Dictyostelium. J Cell
Sci 36: 281–309.
23. Hartman RS, Lau K, Chou W, Coates TD (1994) The fundamental motor of
the human neutrophil is not random: evidence for local non-Markov movement
in neutrophils. Biophys J 67: 2535–2545.
24. Garcia R, Moss F, Nihongi A, Strickler JR, Goller S, et al. (2007) Optimal
foraging by zooplankton within patches: the case of Daphnia. Math Biosci 207:
165–188.
25. Cox EC, Vocke CD, Walter S, Gregg KY, Bain ES (1990) Electrophoretic
karyotype for Dictyostelium discoideum. Proc Natl Acad Sci U S A 87: 8247–8251.
26. Berg-Sorensen K, Flyvbjerg H (2004) Power spectrum analysis for optical
tweezers. Rev Sci Instrum 75: 594–612.
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