# The anomalous tango of hemocyte migration in _Drosophila melanogaster_
embryos

Nickolay Korabel  Department of Mathematics, The University of Manchester, M13
9PL, UK<EMAIL_ADDRESS>Giuliana D. Clemente  School of
Biochemistry, University of Bristol, BS8 1TD, UK  Daniel Han  Department of
Mathematics, The University of Manchester, M13 9PL, UK  Faculty of Biology,
Medicine and Health, The University of Manchester, M13 9PT, UK  Biological
Physics, Department of Physics and Astronomy, The University of Manchester,
M13 9PL, UK  Felix Feldman  Biological Physics, Department of Physics and
Astronomy, The University of Manchester, M13 9PL, UK  Tom H. Millard  Faculty
of Biology, Medicine and Health, The University of Manchester, M13 9PT, UK
<EMAIL_ADDRESS>Thomas Andrew Waigh  Biological Physics,
Department of Physics and Astronomy, The University of Manchester, M13 9PL, UK
Photon Science Institute, The University of Manchester, M13 9PL, UK
<EMAIL_ADDRESS>
###### Abstract

_Drosophila melanogaster_ hemocytes are highly motile cells that are crucial
for successful embryogenesis and have important roles in the organism’s
immunological response. Hemocyte motion was measured using selective plane
illumination microscopy. Every hemocyte cell in one half of an embryo was
tracked during embryogenesis and analysed using a deep learning neural
network. The anomalous transport of the cells was well described by fractional
Brownian motion that was heterogeneous in both time and space. Hemocyte motion
became less persistent over time. _LanB1_ and _SCAR_ mutants disrupted the
collective cellular motion and reduced its persistence due to the modification
of viscoelasticity and actin-based motility respectively. The anomalous
motility of the hemocytes oscillated in time with alternating epoques of
varying persistent motion. Touching hemocytes experience synchronised contact
inhibition of locomotion; an anomalous tango. A quantitative statistical
framework is presented for hemocyte motility which provides new biological
insights.

## Introduction

Cellular motility is a crucial process in biology. It is essential for
morphogenesis, immune responses, cancer metastasis and wound healing. Although
some progress has been made in understanding the motion of isolated eukaryotic
cells on artificial substrates [1], less is known about their collective
behaviour _in vivo_. A diverse selection of experimental results suggest that
single cell movements can range from simple persistent random motion [2] and
heterogeneous persistent random motion [3, 4] to Lévy-like movement patterns
[5] and Lévy walks [6, 7]. Our quantitative understanding of cell motility
inside organisms is much more patchy [8]. We are not aware of any previous
detailed evidence for the anomalous diffusion of cells inside embryos. Single
cell tracking has been previously performed in embryos (e.g. Zebra fish,
_Drosophila_ and mice), but the resultant data was predominantly analyzed in
terms of cell destinations to understand morphogenesis, combined with
relatively qualitative measures of cell motility with small numbers of cells
e.g. instantaneous velocities and apparent diffusion coefficients that do not
accurately describe the signatures of anomalous transport of the whole
ensemble [11, 9, 10]. Some work has been performed on glassy dynamics in
jammed tissue from zebra fish embryos [12, 13, 14], but this is predominantly
from a mechanical perspective. Here we focus on cell motility in _Drosophila_
embryos using advanced optical microscopy techniques.

A major barrier in cell motility research has been experimental, since
cellular aggregates are fragile, often opaque and it is challenging to image
them _in vivo_ with sufficient resolution without causing damage. We used
selective plane illumination microscopy (SPIM) to image the migration of
hemocyte cells in developing _Drosophila melanogaster_ embryos during early
stages of their morphogenesis [15]. The SPIM experiments allowed hemocytes to
be non-invasively imaged for 6 hours. The nuclei of hemocytes were made
fluorescent using genetically encoded histone tags and this allowed the
motility of every hemocyte to be imaged and subsequently tracked in three
dimensions across half an embryo. This would not be possible with conventional
confocal microscopy techniques due to radiation damage and has not before been
achieved with hemocytes. Previous live analyses of hemocyte migration have
focused on relatively few cells migrating in a small and defined region of the
embryo, most frequently the narrow space between the ventral nerve cord and
epidermis. Previously we developed a deep learning neural network analysis
tool to analyze two dimensional tracks of intracellular endosomes that allowed
automatic dynamic segmentation of the tracks in terms of fractional Brownian
motion (FBM) with anomalous diffusion exponents and generalized diffusion
coefficients that are heterogeneous in both space and time [16, 17]. A new
theory for the heterogeneous anomalous transport was developed which was made
possible by the order of magnitude improvement in sensitivity provided by the
neural network [18]. In the current study, we extend our neural network
methodology for anomalous transport to the motion of hemocyte cells in three
dimensions inside embryos and present a new model for their motility based on
new SPIM data.

It has been observed in many experiments that due to viscoelasticity and
heterogeneity of the environment, obstructions, spatio-temporal correlations
and non-specific interactions; intracellular transport and cell motility do
not follow Brownian diffusion [19, 20, 21, 22, 23, 24, 25] or simple
persistent random walks [3, 4]. Single cell tracking of Dictyostelium
discoideum cells revealed anomalous diffusion and power-law tails for the
velocity distributions which were explained by a generalized Langevin model
[26]. Anomalous diffusion was also found in trajectories of single wild-type
and mutated epithelial (transformed Madin–Darby canine kidney) cells [27]. The
super-diffusive increase of the mean squared displacement, non-Gaussian
displacement probability distributions, and power-law decays of the velocity
autocorrelations were interpreted using a fractional Klein–Kramers equation.
Single metastatic cancer cells migrating on linear microtracks were shown to
follow Lévy-like patterns of movement [5]. Even more complex motility was
observed in tracks of single CD8+ T cell migration, which were found to be
consistent with generalized Lévy walks [6]. The simultaneous motility of
multiple cells is expected to impact the forms of anomalous transport
demonstrated by single cells e.g. via jamming, contact inhibition,
intercellular signalling and complex three dimensional geometries [28, 29]. We
therefore made a quantitative study of the anomalous transport of hemocyte
cells in _Drosophila_ embryos.

There are four phases of development for _Drosophila melanogaster_. The
embryonic phase is the first of these four stages and it is the focus of the
current study. During embryogenesis, hemocytes are formed at a single site in
the developing head and undergo a process called dispersal in which they
migrate away from their origin over the course of several hours until they are
spread throughout the embryo. The hemocytes are involved in phagocytosis
(immune responses), encapsulation, melanization and extracellular matrix
production [30]. In general, cell migration can be either passive or active.
Cells can be passively carried by fluid flow of the circulatory system or the
movement of the surrounding tissue e.g. during germ band retraction. Active
migration involves a crawling motion of the cells, driven by self-assembly of
cytoskeletal proteins inside the cells. The motility is modulated by adhesion
with the surrounding cells and the extracellular matrix (ECM), viscoelasticity
and confinement effects (there are well defined pathways through the embryo
and haemocytes experience contact inhibition) [30]. In addition the cells’
motion is directed by extracellular signals [30]. These can attract the cells
to migrate along specific routes and cause them to become oriented.
Chemoattractants for hemocyte motion include Pvf (important for guiding
migration) and factors involved in contact inhibition of locomotion (CIL) in
which hemocyte cells repel one another [32, 31, 40]. Pvf and CIL are thought
to be the main mechanisms that guide hemocyte migration during dispersal.

In the current study, two separate mutations were examined in addition to wild
type _Drosophila_ , called _LanB1_ and _SCAR_. _LanB1_ has the beta subunit of
laminin deleted, which affects the extracellular matrix (ECM) and thus the
localized viscoelasticity experienced by the hemocytes, which in turn affects
their migration [33]. In _SCAR_ mutant embryos, SCAR, a protein that promotes
nucleation of actin filaments at the leading edge of the migrating cells in
absent [34]. Thus it was possible to separate the effects of ECM
viscoelasticity and internal cellular dynamics on the anomalous transport of
the hemocytes.

## Results

### Quantitative characterisation of temporal heterogeneity in hemocyte
migration

We quantified the heterogeneity of hemocyte motion via the analysis of their
local dynamics in wild-type embryos and embryos with loss of function
mutations in genes encoding _SCAR_ and _LanB1_. We used a neural network
method [18] and estimated the local Hurst exponents $H(t)$ of individual
hemocyte trajectories. The spatio-temporal dynamics were then classified as
persistent $H(t)>0.5$ or anti-persistent $H(t)<0.5$. Trajectories in three
dimensions displayed different proportions of persistent and anti-persistent
movement for the control, LanB1 and SCAR embryos (Fig. 1). All the
instantaneous values of the Hurst exponent can be plotted together for an
embryo and anatomical details of the _Drosophila_ embryos are observed. The
instantaneous Hurst exponents illustrate distinct changes in their spatial
arrangements for the control and the two mutants (LanB1 and SCAR) (Fig. 1).
Trajectories in LanB1 and SCAR experiments display progressively greater
amounts of anti-persistent motion compared to the control.

Figure 1: Hemocytes randomly transition between persistent and anti-persistent
motion in different proportions while exploring control, _LanB1_ and _SCAR_
embryos. (a, b, c) Three dimensional hemocyte trajectories with colours
corresponding to local persistent ($H(t)>0.5$) (violet) and anti-persistent
($H(t)<0.5$) (yellow) motion. Here $H(t)$ is the local Hurst exponent
estimated using the neural network. Trajectories in _SCAR_ experiment have a
greater amount of anti-persistent motion compared to the control and LanB1.
(d, e, f) Two dimensional projections of all the tracks reveal large scale
dynamical patterns and anatomical features of the embryos become visible.

### Heterogeneous anomalous diffusion of hemocytes: non-Gaussian displacement
distributions, super-diffusive MSDs and power-law decaying VACFs

The probability distributions of hemocyte displacements scaled by their
standard deviation were plotted at the shortest time interval probed (Fig.
2(a)). The distributions are clearly non-Gaussian and possess exponential
tails (Laplace-like distributions). Over the range of time scales probed, the
rescaled distributions for the control, LanB1 and _SCAR_ are almost
indistinguishable. Fig. 2(b) shows the mean square displacement (MSD) and time
averaged mean squared displacement (TMSD) of the cells as a function of time
interval. The control, LanB1 and _SCAR_ MSDs are all superdiffusive
($\alpha>1$) and the amplitude of the motion decreases in the order
control$>$LanB1$>$_SCAR_. Thus the mutations are not noticeably affecting the
form of the statistics of the displacement distribution of the motility (Fig.
2(a)), rather it is the changes in displacement magnitude (Fig. 2(b)) and
direction (persistence, Fig. 1) that are having a major impact. The velocity
autocorrelation functions (VACF) as a function of time interval (Fig. 2(c))
show similar decays for the control and LanB1 sample, whereas the _SCAR_
sample is much slower to decay.

Figure 2: Hemocytes in control, _LanB1_ and _SCAR_ embryos move via
heterogeneous anomalous diffusion. (a) Distributions of displacements,
$dx=x(t+\tau)-x(t)$, calculated at $\tau=17.6$ s are non-Gaussian and follow
the Laplace distribution (the dashed line),
$PDF(\xi)=\exp\left(-|\xi|/b\right)/2b$ with $\xi=dx/\sigma_{x}$ and $b=0.71$.
(b) MSDs and TMSDs as a function of time interval grow superdiffusively with
anomalous exponents $\alpha=1.3\pm 0.1$ in the control and $\alpha=1.1\pm 0.1$
for LanB1 embryo. LanB1 and _SCAR_ embryos have progressively more anti-
persistent motion on the shortest time scales while their long time behaviour
remains superdiffusive. At longer time scales the MSD of Scar increases with
anomalous exponents $\alpha=1.5\pm 0.1$ (see Fig. S2). The dashed line
represents the power-law functions $t^{1.3}$ for the control as a guide. (c)
Velocity auto-correlation functions as a function of time interval are
positive and decay to zero as power-laws, $(t/\delta)^{\alpha-2}$, with
$\alpha$ determined from the MSDs. (d) Distributions of incremental
displacements $dR=\sqrt{dx^{2}+dy^{2}+dz^{2}}$ calculated for 17.6 s time
intervals decay exponentially. The dashed line represents an exponential
function (the dashed line). (e) PDFs of incremental angles $\theta$ for 17.6
time intervals calculated using the dot product of consecutive position
vectors, $\bf{r_{1}}=(x_{1},y_{1},z_{1})$ and
$\bf{r_{2}}=(x_{2},y_{2},z_{2})$, as
$\theta=\arccos(\bf{r_{1}}\cdot\bf{r_{2}}/|\bf{r_{1}}||\bf{r_{2}}|)$.
Statistical quantities in this figure were averaged over $5$ data sets of
control, $2$ LanB1, $4$ SCAR embryos. The error bars correspond to standard
error of the mean.

The ensemble averaged mean squared displacements (MSD) (Fig. 2(b)) demonstrate
a clear super-diffusive growth ($\alpha>1$) in the control, _LanB1_ and _SCAR_
embryos, indicating actively driven motion. Ensemble MSDs and time averaged
TMSDs have near identical behaviour (see Fig. S2) which suggests ergodic
motility for the hemocytes [35]. At short time scales, the slope of the MSDs
is smaller for LanB1 and _SCAR_ in agreement with the increased amount of
anti-persistent motion observed in Fig. 1 (a,b,c). The velocity auto-
correlation functions (VACF) (Fig. 2(c)) are positive for all the embryos
studied and decay to zero as power-laws, $t^{\alpha-2}$, with $\alpha$ in
agreement with that from the MSDs. The power-law behaviour of the MSDs and
VACFs together with ergodic behaviour of the MSDs (Fig. 2(b,c)) suggest that
hemocytes are described by fractional Brownian motion (FBM) or the fractional
Langevin equation, rather than non-ergodic continuous time random walks
(CTRW). These three models (FBM, FLE and CTRW) are the main possibilities used
to describe anomalous transport in the literature i.e. to motivate fractional
values of $\alpha$. This supports our decision to train the neural network
using FBM trajectories [18]. The distributions of incremental displacements
$dR=\sqrt{dx^{2}+dy^{2}+dz^{2}}$ with $dx=x(t+\tau)-x(t)$, $dy=y(t+\tau)-y(t)$
and $dz=z(t+\tau)-z(t)$ also decay exponentially (Fig. 2(d)) similar to the
distributions of displacements (Fig. 2(a)). However, they show that hemocytes
in the _SCAR_ embryos are characterized by smaller incremental displacements
compared to the control and LanB1 mutants. From Fig. 2(e) we also calculated
the mean cell speed $\bar{v}=0.066\pm 0.046$ $\mu$m/s for control,
$\bar{v}=0.074\pm 0.05$ $\mu$m/s for _LanB1_ and $\bar{v}=0.045\pm 0.033$
$\mu$m/s for _SCAR_. Hemocytes in control embryos move persistently compared
to _LanB1_ and _SCAR_ embryos. Corresponding distributions of incremental
angles are shown in Fig. 2(e). This is consistent with our observation that
the amount of persistent motion decreases in the order
control$>$LanB1$>$_SCAR_ (Fig. 2(a,b,c)). The non-Gaussian form of the
distributions of the displacements (Fig. 2(a)) suggests that the FBM-like
motility of hemocytes is heterogeneous. Below we explore the heterogeneity of
hemocyte motion in detail by studying their local dynamics.

### Distributions of hemocytes’ local anomalous exponents and local
generalized diffusion coefficients

To study the heterogeneity of hemocyte motion, we first calculated time
dependent Hurst exponents $H(t)$ for individual hemocyte trajectories using
the neural network [18] for the control, LanB1 and _SCAR_ embryos. For
individual trajectories we also calculated local time averaged MSDs and
extracted time dependent local anomalous exponents $\alpha_{L}(t)$ and local
generalized diffusion coefficients $D_{\alpha_{L}}(t)$. Time dependent Hurst
exponents $H(t)$ and time dependent local anomalous exponents $\alpha_{L}(t)$
quantify similar information (for standard FBM $\alpha=2H$), but $H(t)$ is
determined more accurately by the neural network [18]. We did both methods of
analysis to provide an accurate test of the neural network analysis, bench-
marked against a more traditional methodology. The typical behaviour of
$\alpha_{L}(t)$ and $D_{\alpha_{L}}(t)$ calculated for sample trajectories in
the LanB1 embryo (Fig. 3(a,b)) reveals that $\alpha_{L}(t)$ and
$D_{\alpha_{L}}(t)$ display oscillatory behaviour and are anti-correlated. A
Fast Fourier Transform of $\alpha_{L}(t)$ (Fig. 3(c)) (should be SI Fig.3)
reveals two periods of oscillation of approximately $3$ and $8$ minutes for
the control, which vary for LanB1 and SCAR. Time dependent Hurst exponents
$H(t)$ for the control, LanB1 and _SCAR_ embryos have broad distributions with
a decreasing amount of persistent motion, following control$>$LanB1$>$_SCAR_
and an increasing amount of anti-persistent movement with
control$<$LanB1$<$_SCAR_ (Fig. 3(d)). The distributions of local generalized
diffusion coefficients $D_{\alpha_{L}}(t)$ were found to decay with a Weibull
distribution form (Fig. 3(e)) and _SCAR_ showed large numbers of small values
of $D_{\alpha_{L}}(t)$.

Figure 3: Local characterization of hemocytes’ heterogeneous anomalous
diffusion in control, _LanB1_ and _SCAR_ embryos. (a), (b), (c) Local
anomalous exponents $\alpha_{L}(t)$ for representative hemocyte trajectories
in a control, LanB1 and SCAR embryo. $\alpha_{L}(t)$ display oscillatory
behaviour. (d) Distributions of time dependent Hurst exponents $H(t)$
estimated using the neural network. (e) Distributions of local generalized
diffusion coefficients $D_{\alpha_{L}}$. The dashed line represents a fit of
the distribution tail with the Weibull density function, $PDF(\eta)\sim
c(\eta^{c-1}/\eta_{0}^{c})\exp(-(\eta/\eta_{0})^{c})$ with $c=0.39$ and
$\eta_{0}=0.008$. The distributions were averaged over $5$ data sets of
control, $2$ LanB1, $4$ SCAR embryos. The error bars correspond to standard
error of the mean.

To quantify the changes in proportion of persistent and anti-persistent
motion, we used Gaussian mixture models to fit the distributions of local
Hurst exponents $H(t)$. Four separate Gaussian modes were chosen to describe
the data whose number was determined from the BIC criteria (SI, Fig. S3). The
individual Gaussian components in the mixture models show a significant
difference in modes of persistent ($H(t)>0.5$) motion between the control,
_SCAR_ and LanB1 embryos. The Hurst exponents of modes 1, 2 and 4 decrease
with time (they become less persistent), whereas 3 is relatively constant.

The distributions of generalized diffusion coefficients $D_{\alpha_{L}}(t)$
obtained by calculating the local MSDs of individual hemocyte trajectories and
fitting them with power-law functions, have Weibull distribution tails (Fig.
3(e)) for the control, LanB1 and _SCAR_ embryos. Numerical simulations of
heterogeneous FBM with a Weibull distribution of generalized diffusion
coefficients (Fig. 4) are in good agreement with the Laplace form of the
displacements observed in the control, LanB1 and _SCAR_ embryos (Fig. 2(a)).

Figure 4: Heterogeneous FBM (hFBM) with the Weibull distribution of
generalized diffusion coefficients reproduces displacements distributions in
control, _LanB1_ and _SCAR_ embryos. For clarity only the distribution of
displacements in LanB1 is shown.

### Oscillatory movement of hemocytes

Strikingly, we found that the persistence of the motion of single cells is
oscillatory with reasonably well defined time periods of $3\pm 0.5$ and $8\pm
0.5$ minutes for the wild type and $8\pm 0.5$ and $14\pm 0.5$ minutes (with
small remnants of the 3 minute periodicity) for _LanB1_ and _SCAR_ (Fig.
3(a,b,c)). This is three orders of magnitude slower that the heart beat of
_Drosophila melanogaster_ which is in the range 4-6 beats/s and the heart does
not start to beat until later stages of development. The motion of the
control, _LanB1_ and _SCAR_ were all found to be oscillatory. The absolute
values of the anomalous exponents of single cell motility varied in space and
time, which could be driven by varying viscoelasticities, confinement,
adhesion, jamming or intracellular signalling (probably a combination of all
these factors). It is expected the time period is internally driven by the
cell, whereas the absolute values of the persistence of the motion are
modulated by the external environment.

The oscillatory behaviour of local anomalous exponents was synchronised for
the local motion of non-contacting hemocytes. We found several nearby
trajectories (Fig. 5(b)) which are similar in space and their local anomalous
exponents display coherent oscillations in time (Fig. 5(a)). The slight time
delays between the oscillations allow us to discount extrinsic experimental
factors, such as instrument vibrations (the displacement amplitudes are also
much larger than the static error in the tracking analysis). Furthermore, the
neighbouring trajectories in Fig. 5(b) have displacement correlation
coefficients approaching 1, whereas more distant cells are found to have much
lower values e.g. some are negatively correlated (SI Fig. S5). This suggests
that local anomalous exponents are a valuable tool to detect synchronised
motion of migrating cells in embryos.

Upon their contact, the hemocyte motion was also synchronised, although their
clear sense of directional motion was lost (Fig. 6). Contact of the cells
inhibits their motion (a process of contact inhibitory locomotion [36]) and
their anomalous transport continues to be synchronised in terms of the
$\alpha_{L}(t)$, $D_{\alpha_{L}}(t)$, displacements and the velocities (Fig. 6
a,b, c and d). An anomalous tango is thus observed between the cells when they
touch (Fig. 6 e).

Figure 5: Coherent oscillations of local anomalous exponents reveal
synchronised movement of hemocytes. (a) Oscillating local anomalous exponents
of sample trajectories of hemocytes in LanB1. (b) Corresponding trajectories
of hemocytes which are moving in a synchronised manner. We quantified
correlations between trajectories (see SM Fig. S5). Pairs of trajectories in
(b) have correlation coefficients close to $1$.  Figure 6: Contact inhibition
leads to synchronization of hemocyte movement. Oscillations of local anomalous
exponents (a), local generalized diffusion coefficients (b) and velocity (c)
synchronize when hemocytes experience contact inhibition. Hemocytes during the
contact remain in close proximity to each other performing a tango of
anomalous diffusion. (d) Shows the relative distance between contacted
hemocytes depicted in (e). Correlation coefficients for trajectories shown in
(e) calculated by Corr$(\lVert\bf{r_{i}}\rVert,\lVert\bf{r_{j}}\rVert)$ where
$\lVert\bf{r_{i}}\rVert$ and $\lVert\bf{r_{j}}\rVert$ are distances vectors of
trajectories $i$ and $j$. The correlation coefficients of three pairs of
trajectories were $-0.08$, $0.52$ and $0.86$ indicating that trajectories
remain correlated.

### Truncated power-law survival times in persistent and anti-persistent
hemocyte migration

Hemocytes constantly switch between persistent and anti-persistent motion
(Fig. 1). To quantify the amount of persistent and anti-persistent motion in
the control and the two mutant embryos, we calculated the survival functions
of the times the hemocytes spent in persistent and anti-persistent states
(Fig. 7) using the non-parametric Kaplan–Meier estimator [37].

Figure 7: Survival functions of the times hemocytes move in persistent and
anti-persistent motion in the control, LanB1 and SCAR embryos are truncated
power-laws. Survival functions $\Psi(t)$ of persistent motion (solid curves)
and anti-persistent motion (dashed curves) for control (blue), LanB1 (red) and
SCAR (yellow) embryos. Survival functions were averaged over $5$ data sets of
control, $2$ LanB1, $4$ SCAR embryos. The error bars correspond to standard
error of the mean. The dashed-doted and dotted curves are truncated power-law
fits given by $(a/(a+t))^{b}e^{-ct}$ with $a=300$, $b=3.3$, $c=0.0016$ and
$a=405$, $b=0.33$, $c=0.002$, respectively.

Survival analysis revealed that the majority of hemocytes spent longer in
persistent than anti-persistent states. The duration of anti-persistent motion
increases in embryos from the control to _SCAR_ in the order
control$<$LanB1$<$_SCAR_. This agrees with the visual inspection of
trajectories in Fig. 1 (a,b,c). The reduction of persistence of the movement
of the hemocytes in LanB1 and _SCAR_ embryos compared to the control is also
corroborated by the MSDs (Fig.2 (b)). Furthermore, the largest difference of
survival times between persistent and anti-persistent states is exhibited by
the control embryo hemocytes. The LanB1 mutation reduces the survival times of
the persistent states of motion while keeping anti-persistent states the same
as the control. In contrast, the _SCAR_ mutation increases the survival times
of the anti-persistent states of motion while keeping persistent states the
same as the control. Thus, LanB1 and _SCAR_ mutations produce significant
differences in hemocyte migration. This suggests that actin polymerisation is
responsible for regulating the anti-persistent states of motion, whereas
laminin is responsible for persistent states of motion. This is the first time
evidence has been provided that external and internal cellular mechanisms
contribute to different aspects of cellular migration in a population of cells
_in vivo_ in a distinct and quantifiable manner. Finally, this evidence
confirms that the incremental displacement steps and incremental angles
between steps shown in Figs. 2d and e are connected to specific mutations
within the hemocytes.

## Discussion

We present the first quantitative analysis of hemocyte migration in three
dimensions inside live _Drosophila_ embryos and provide strong evidence that
hemocyte motility must be considered within the framework of anomalous
transport [38]. The average Hurst exponents for all the cells decreased with
time as the embryos aged and the cell motility became both less motile and
less persistent, with many cells becoming trapped in localized regions. The
control, _LanB1_ and _SCAR_ embryos had distinct patterns of anomalous
motility. The control hemocytes quickly dispersed to perform their diverse
biological functions. In _LanB1_ the process of dispersal was significantly
slowed and the embryos failed to hatch into larvae. In _SCAR_ hemocytes
dispersion was even more significantly disrupted and the mutation also made
the embryos unviable.

A broad spectrum of Hurst exponents was found for the control, _LanB1_ and
_SCAR_ embryos (Fig. 3(d)). The distributions of Hurst exponents were
characteristic of the type of mutant considered. The hemocytes are switching
between different levels of persistent motility rather than there being
distinct subpopulations of cells that have constant amounts of persistent
motility (four populations of gait were identified by a Gaussian mixture
model, SI Fig. S3).

Detailed analysis of the statistics of motion showed distinct changes with the
type of embryo considered: control, _LanB1_ and _SCAR_. These could be
explained in terms of the known biological functions of the mutants. _LanB1_
affected laminin in the extracellular matrix and thus the local
viscoelasticity. The _LanB1_ mutant (deduced to increase the local
viscoelasticity) reduced the amplitude of the motion (observed via the MSDs),
decreased the persistence of motion (observed via Hurst exponents and angular
distributions), but did not change the gait of the motility (observed in the
velocity autocorrelations). _SCAR_ directly affected the motility by
perturbing intracellular actin polymerisation. This caused the largest
reduction in the amplitude of the motion observed (via the MSDs), decreased
the persistence of motion (observed via Hurst exponents and angular
distributions) and had a significant change to the gait of the motility
(observed in the velocity autocorrelations).

The survival probabilities for the durations of persistent motion followed
truncated power law distributions (Fig. 7). The distributions of displacement
increments were non-Gaussian with exponential tails (Laplace-like) in common
with other previous studies of single cell motility [3]. In terms of their
spatial distribution, hemocytes began densely packed in a small area (the head
mesoderm) and rapidly spread around the embryo. In _LanB1_ embryos, anti-
persistent motion dominates from an earlier stage than with the control and
_SCAR_ , highlighting the influence of the local viscoelasticity on the
trapping of cells in antipersistent states.

From the perspective of theoretical modelling, the equivalence of the MSD and
TAMSD implies the cellular motion is ergodic (Fig. 2(b)). This allows CTRW
models to be discounted and implies FBM or fractional Langevin equation (FLE)
models may be relevant. It also allow glassy dynamics to be discounted,
observed previously in jammed embryonic cells [13], since it would again be
non-ergodic. The major difference between the FBM and the FLE models is that
the FLE fulfills the fluctuation-dissipation theorem while the FBM does not.
Since most cellular and intracellular processes consume energy (e.g. ATP),
they break the conditions for thermal equilibrium needed for the fluctuation-
dissipation theorem. Therefore, the FBM model seems preferable to describe
cell motility.

In general, heterogeneous FBM appears to be a very flexible model to describe
anomalous transport in biology provided its assumptions are fulfilled i.e.
ergodicity, bounded Hurst exponents in the range $[0,1]$ etc. A previous study
showed hFBM could be used to describe endosomal transport in human cells [18,
16, 17]. Here we use it to describe the motion of whole cells on much larger
time and length scales. Although both hemocyte and endosomal motility were
found to be heterogeneous in time and space, crucial differences are seen in
the statistics. Hemocyte displacements are found to be Laplace-like, whereas
endosomes follow power law distributions. The reasons for these non-Gaussian
distributions of displacements were found to be in the power law distributions
of endosome diffusion coefficients [16, 17] and Weibull-like distributions of
diffusion coefficients of hemocytes (see Figs. 3). Curiously, the generalised
diffusion coefficients for hemocytes are anticorrelated, whereas those for
endosomes are correlated. A loose analogy would be to a driver of a vehicle
hitting the brake and accelerator pedals at the same time with hemocytes and
hemocyte motility must be under careful control by a feedback loop [39].
Furthermore the motion of neighbouring hemocytes are found to be correlated,
whereas endosomes are not. The hFBM model accurately predicts the
distributions of displacements given the distribution of generalized diffusion
coefficients of the hemocyte cells in embryos from the experiments (Fig. 4)
and it is consistent with all the other statistical data extracted. This is a
major success for the hFBM model in this context.

It is fascinating to observe the synchronisation of the motion of
neighbouring, but non-contacting, hemocytes in embryos. The exact biological
origin of this synchronisation would require further research, but chemotaxis
in response to a common small molecule signal (e.g. Pvf [40]) is a primary
candidate to explain the motion of non-contacting cells (Fig. 5). Upon contact
the anomalous transport of hemocytes continues to be synchronised, but the
directions become randomized due to cell scattering. Contact inhibition of
locomotion (CIL) is hypothesized to make a major contribution to the dispersal
of hemocytes during development [32, 31]. Classically Abercrombie defines two
types of CIL [41]. In type I CIL cells stop following one another after
contact, whereas in type II CIL cells repolarise after contact and migrate
away from one another. In contrast, our data indicates a more nuanced model
for hemocyte motion upon contact, i.e. anomalous CIL, in which the amplitude,
scaling and persistence of the anomalous transport becomes synchronised upon
contact (an anomalous tango of hemocytes). This moves beyond simple notions of
cells in pure stationary, diffusive or ballistic states. Coherent motion due
to large scale embryonic remodelling (e.g. germ band retraction and nerve cord
condensation) is also expected to contribute, but there would be no time
delays in the oscillations in this case.

Some _in vitro_ biophysical work has previously been performed on CIL with
human cancer cells in microfluidic geometries [42] and machine learning
techniques were used to classify the processes involved in cell scattering.
Here, we extend the biophysical analysis of CIL and provide clear evidence
that anomalous cellular transport is important with hemocytes i.e. anomalous
CIL occurs, characterised by synchronised fractional kinetics _in vivo_.

Oscillations in the motion of hemocytes on the time scale of $>$2 mins have
been previously described in the biological literature due to oscillations of
hemocyte protrusions [40]. The oscillations of hemocyte protrusions were
characterised by increases in the polarity of cells i.e. the contact area of
the cells oscillated along the directions of motion. We expect these
oscillations in hemocyte polarity also explain the oscillations in anomalous
motility and anomalous CIL, although we have not measured the polarity
directly (only the nuclei were labelled in our experiments, not the external
cellular membrane).

A huge amount of biologically relevant information is available from cell
tracking experiments in embryos. Future research on anomalous transport could
investigate other cells beyond hemocytes, later stages of morphogenesis and
other types of embryos e.g. zebra fish or murine. Heterogeneous fractional
Brownian motion informed by neural networks is a powerful new tool to
automatically segment the dynamics of cell motility in time and space. Future
studies could link the generalized diffusion coefficients and anomalous
exponents with specific molecular processes in the cells [39]. This would
provide an accurate quantitative model to connect the molecular biology with
the cell motility over the complete duration of the embryonic stage of
development. For example, a current challenge is to explain the anti-
correlation of the generalized diffusion coefficients and the anomalous
exponents in oscillatory hemocyte motility, which has not previously been
observed in the field of anomalous transport. The feed forward neural network
used provides state of the art dynamic segmentation of the tracks, but it is
possible that other architectures could improve on this performance e.g. the
Wavenet convolutional neural network with causality constraints and a wider
range of dynamic sensitivity [43]. Lattice light sheet microscopy could allow
the tracking of smaller features inside cells with higher resolution [11],
although it is expected SPIM will still be competitive for the analysis of
whole cell motions.

## Methods

### Drosophila husbandry

Drosophila stocks were raised on iberian food according to standard
procedures. Flies were maintained in incubators at 25∘C and controlled
humidity. For embryo collection, flies were placed in cages and left to lay
eggs on apple juice plates prepared in house. The following Drosophila stocks
were used: _SCAR_ delta37 [34], _LanB1_ def [33] and w1118(BL$\\#$5905) served
as wild-type control. Hemocyte nuclei were labelled by expression of
Histone2A-mCherry under the direct control of the serpert promoter [34].
Embryos of the appropriated genotype were identified based on the absence of
balancer chromosomes carrying fluorescent markers.

### Preparation of embryos for imaging and Selective plane illumination
microscopy (SPIM)

Embryos of an overnight collection were dechorionated in 50% bleach for 1 min
and extensively washed in water. Embryos at stage 12 of development (440-580
min AEL) were selected for SPIM imaging and mounted on a cylinder of 1% low
melting point agarose (Sigma Aldrich).

Embryos were imaged using a Carl Zeiss Z1 lightsheet microscope equipped with
a 20x/1.0NA Plan Aprochomat objective. 1200 frame movies of the embryos in
three dimensions were created. The time interval between consecutive frames
was 17.6 seconds. The excitation wavelength was 561 nm. Cells were detected
and tracked using the IMARIS software package (version 8.2, Oxford
Instruments, United Kingdom).

### Neural networks

A feedforward neural network (FNN) architecture was used to dynamically
segment tracks for hemocyte cells that was trained on fractional Brownian
motion [18]. The neural network takes the incremental displacements of a
trajectory and returns the time-dependent Hurst exponent $0<H(t)<1$ which is a
measure of persistence of the motion. For a standard fractional Brownian
motion, the Hurst exponent is constant and the mean squared displacement grow
as $MSD(t)\simeq t^{2H}$. Our previous FNN study used two dimensional tracks
from intracellular endosomes, but the extension to the three dimensional
motion of whole cells (specifically the nuclei of whole cells) was
straightforward.

### Statistical analysis of experimental hemocyte’s trajectories

The ensemble-averaged mean squared displacement (MSD) of 3D experimental
hemocyte’s trajectories $\\{x_{i},y_{i},z_{i}\\}$ is defined as:

$\mbox{MSD}(t)=\frac{1}{l^{2}}\left<(x_{i}(t)-x_{i}(0))^{2}+(y_{i}(t)-y_{i}(0))^{2}+(z_{i}(t)-z_{i}(0))^{2}\right>=6D_{\alpha}\left(\frac{t}{\tau}\right)^{\alpha}.$
(1)

The angled brackets denotes averaging over an ensemble of trajectories,
$\left<A\right>=\sum_{i=1}^{N}A_{i}/N$, where $N$ is the number of
trajectories in the ensemble. We set the time scale $\tau=1$ sec and the
length scale $l=1$ $\mu$m in order to make the generalized diffusion
coefficient $D_{\alpha}$ dimensionless. The MSD is fitted to the power law
function to extract the anomalous exponent $\alpha$ and the generalized
diffusion coefficient $D_{\alpha}$. Notice that $\alpha$ and $D_{\alpha}$ are
constants which characterize averaged transport properties of the ensemble of
trajectories.

The time-averaged mean squared displacement (TMSD) of a individual trajectory
of a duration $T$:

$\mbox{TMSD}_{i}(t)=\frac{1}{l^{2}}\frac{1}{T-t}\int_{0}^{T-t}\left(x_{i}(t^{\prime}+t)-x_{i}(t^{\prime}))^{2}+(y_{i}(t^{\prime}+t)-y_{i}(t^{\prime}))^{2}+(z_{i}(t^{\prime}+t)-z_{i}(t^{\prime}))^{2}\right)dt^{\prime},$
(2)

where $l$ is the length scale which we choose to be $l=1$ $\mu$m. TMSDs of
individual trajectories can be averaged further over the ensemble of
trajectories:

$\mbox{TMSD}(t)=\left<\mbox{TMSD}_{i}(t)\right>.$ (3)

We calculated the local TMSD (L-TMSD) e.g. TMSDi in a time window
$(t-W/2,t+W/2)$ which is shifted along a trajectory and L-TMSD is calculated
in each window. By fitting the first $10$ L-TMSDs points to power law
functions, we extracted the time dependent local anomalous exponents
$\alpha_{L}(t)$ and the time dependent local generalized diffusion
coefficients $D_{\alpha_{L}}(t)$:

$\mbox{L-TMSD}(t)=6D_{\alpha_{L}}\left(\frac{t}{\tau}\right)^{\alpha_{L}}.$
(4)

Again, the time scale $\tau=1$ sec and the length scale $l=1$ $\mu$m are
introduced in order to make the local generalized diffusion coefficients
dimensionless and facilitate comparison.

The time dependence of the MSD determine the character of the movement of the
ensemble of hemocytes as normal diffusion for $\alpha=1$, sub-diffusion for
$\alpha<1$ and super-diffusion for $\alpha>1$. Similarly the time dependence
of the L-TMSD determine the character of local movement of a single hemocyte
trajectory.

### Velocity auto-correlation function of experimental trajectories

The velocity auto-correlation function (VACF) is defined as:

$\mbox{VACF}(t)=\frac{\int_{0}^{T-t-\delta}\vec{v}(t^{\prime}+t)\vec{v}(t^{\prime})dt^{\prime}}{T-t-\delta}.$
(5)

where $\vec{v}=\frac{\vec{r}(t+\delta)-\vec{r}(t)}{\delta}$ and $\delta=17.6$
s. To improve statistics, the VACF is further averaged over the ensemble of
all the trajectories.

### Heterogeneous Fractional Brownian Motion

Heterogeneous FBM trajectories were simulated using the overdamped Langevin
equation with generalized diffusion coefficient $D$ and Hurst exponent $H$:

$\frac{d\mbox{r}(t)}{dt}=\sqrt{2D}\mbox{f}^{fGn},$ (6)

where $\mbox{f}^{fGn}=\\{f^{fGn}_{x},f^{fGn}_{y},f^{fGn}_{z}\\}$ is the vector
of fractional Gaussian noise with correlation function

$\left<\mbox{f}^{fGn}(t_{1})\mbox{f}^{fGn}(t_{2})\right>\simeq
H(2H-1)|t_{1}-t_{2}|^{2H-2}.$ (7)

We generated $\mbox{f}^{fGn}$ in MATLAB (The MathWorks, Nantick, MA) using
ffgn function. In the simulations shown in Fig. 4, the Hurst exponents were
constant ($H=0.65$) for all trajectories. The generalized diffusion
coefficients were assigned from the Weibull distribution
$f(D)=c(D^{c-1}/D_{0}^{c})\exp(-(D/D_{0})^{c})$ with $c=0.33$ and
$D_{0}=0.0018$ which were determined from haemocyte trajectories (see Fig.
3c). The PDF of $D$ was truncated at $D_{min}=0.1$ and $D_{max}=100$.

### Survival analysis

We used instantaneous Hurst exponents $H(t)$ to segment haemocyte trajectories
into persistent ($H(t)>0.5$) and anti-persistent ($H(t)<0.5$) states of
motion. The survival time in a persistent or anti-persistent states is than
defined as a time the cell remains in the same state for time $t$. Kaplan-
Meier non-parametric estimator of the time dependent survival probability that
an event has not occurred at a time $t_{i}$ is defined as
$\Psi(t_{i})=\left(1-\frac{d_{i}}{n_{i}}\right)\Psi(t_{i-1})$, where $d_{i}$
is the number of events that occur at time $t_{i}$ and $n_{i}$ is the number
of events survived up to time $t_{i}$. By definition, $\Psi(t_{0})=1$.

## Acknowledgements

DH acknowledges financial support from the Wellcome Trust Grant No.
215189/Z/19/Z. NK acknowledges financial support from EPSRC Grant No.
EP/V008641/1. TM acknowledges financial support from the Wellcome Trust Grant
no. 201958/Z/16/Z. We thanks the MPhys students for their preliminary
contributions: Tom Hughes-Jordan, Jack Forrester, Nicole Rolph and Charlotte
Payne. Viki Allan and Sergei Fedotov are thanked for many useful
conversations. We thank the University of Manchester Fly and Bioimaging
Facilities, in particular David Spiller, for assistance with microscopy and
image analysis.

## Author contributions statement

## Competing financial interests

The authors declare no competing interests.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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## Supplementary Material

Figure S1: An example of estimation of instantaneous Hurst exponent $H(t)$
using the neutral network. See the main text for details. (a) shows the
displacement and estimated instantaneous Hurst exponent $H(t)$ (b) of a sample
experimental trajectory (c). Green color in (c) corresponds to persistent
motion $H(t)>0.5$ and purple to anti-persistent motion $H(t)<0.5$. Figure S2:
Ergodic behavior of hemocytes. The EMSDs and ETMSDS for (a) control, (b) LanB1
and (c) SCAR embryos have similar behaviour as functions of time. The dashed
line are fits to power laws with exponents $\alpha=1.3\pm 0.1$ for control,
$\alpha=1.1\pm 0.1$ for LanB1 and $\alpha=1.5\pm 0.1$ for SCAR embryos. Figure
S3: Fitting the distributions of local Hurst exponents $H(t)$ with Gaussian
mixture models. (a) Four separate Gaussian modes were chosen to describe the
data which was determined from the BIC criteria (b). The individual Gaussian
components in the mixture models show a significant difference in modes of
persistent ($H(t)>0.5$) motion between control, SCAR and LanB1 embryos. The
Hurst exponents of modes 1, 2 and 4 decrease with time, whereas 3 is
relatively constant and is thought to be associated with passive cell motility
due to germ band retraction.  Figure S4: Fast Fourier transforms (FFT) of
local anomalous exponents in control, LanB1 and SCAR embryos. FFT of
$\alpha_{L}(t)$ as a function of frequency (f, averaged over $10$
trajectories) for control embryo (blue curve) reveals two oscillation periods
of $3$ and $8$ minutes. Period $3$ minutes might be related to the oscillatory
behaviour of hemocyte polarization during the motion. FFT of $\alpha_{L}(t)$
for LanB1 and SCAR embryos have remnants of the period $3$ minutes and show
periods of $8$ and $14$ minutes instead. Figure S5: Quantification of
correlated movement of hemocytes in LanB1 embryo. Correlation coefficients for
six sample trajectories shown in the figure is calculated by
Corr${}_{1,2}=$Corr$(\lVert\bf{r_{1}}\rVert,\lVert\bf{r_{2}}\rVert)$. Here
$\lVert\bf{r_{1}}\rVert$ and $\lVert\bf{r_{1}}\rVert$ are distances vectors of
trajectories $1$ and $2$ to the origin as functions of time. The correlation
coefficients of pairs of trajectories are Corr${}_{1,2}=0.99$,
Corr${}_{1,3}=0.98$, Corr${}_{1,4}=0.97$, Corr${}_{1,5}=0.83$ and
Corr${}_{1,6}=-0.79$. Trajectories $1,2,3,4$ display strongly correlated
movement while trajectory $6$ is moving anti-correlated relative to them.
Figure S6: Quantification of correlated movement of hemocytes in control,
LanB1 and SCAR embryos. Colours of trajectories represent correlation
coefficients calculated with respect to the first trajectory.