\documentclass[aps,prl%
%,preprint%
,twocolumn%
,showpacs%
,superscriptaddress%
]{revtex4}

\usepackage{graphicx}
\usepackage{times}


\begin{document}

\setlength{\textfloatsep}{1.2\parindent plus 2pt minus 2pt} %%% DELETE %%%
\setlength{\arraycolsep}{1pt}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\f}{\frac}
\newcommand{\p}{\partial}
\newcommand{\no}{\nonumber}
\newcommand{\kT}{k_{\rm B}T}
\newcommand{\e}{{\rm e}}
\newcommand{\dd}{{\rm d}}

\newcommand{\eg}{{\it e.g.}}
\newcommand{\ie}{{\it i.e.}}
\newcommand{\pc}{p_{\rm c}}

\newcommand{\kav}{\left<k\right>}

\newcommand{\affila}{
 Department of Biological Physics, E\"otv\"os University,
 P\'azm\'any P.\ stny.\ 1A, H-1117 Budapest, Hungary
}
\newcommand{\affilb}{
 Biological Physics Research Group of HAS,
 P\'azm\'any P.\ stny.\ 1A, H-1117 Budapest, Hungary
}

\title{
 Phase transition in the collective migration of tissue cells: experiments and
 models
}

\author{B. Szab\'o}
 %\email{}
 \affiliation{\affila}
\author{G. Sz\H{o}l\H{o}si}
 %\email{}
\author{B. G\"onci}
 %\email{}
 \affiliation{\affila}
 \author{Zs. Jur\'anyi}
 %\email{}
 \affiliation{\affila}
 \author{D. Selmeczi}
 %\email{}
 \affiliation{\affila}
\author{Tam\'as Vicsek}
 %\email{vicsek@angel.elte.hu}
 \affiliation{\affila}
 \affiliation{\affilb}

\date[]{\protect\today}
%\date[]{submitted to where..., when...}


\begin{abstract}
We have been recording the swarming-like collective migration of a
large number of keratocytes (tissue cells obtained from the scales
of gold fish) using long-term videomicroscopy. By increasing the
overall density of the migrating cells, we have been able to
demonstrate experimentally a kinetic phase transition from a
disordered into an ordered state. Near the critical density a
complex picture emerges with interacting clusters of cells moving
in groups. A novel flocking model with no assumption about the
knowledge of the direction of motion of the neighbors is
introduced and used to interpret our experimental results.
\end{abstract}




\pacs{87.17.Aa, 87.17.Jj, 87.64.Rr}

\maketitle

The collective motion of organisms is a spectacular phenomenon
sometimes involving huge schools of fish, thousands of birds
exhibiting complex areal displays \cite{parrish} herds of
quadrupeds and even bacteria producing fractal colonies
\cite{czirokpre} or amoeba \cite{levine}. In addition to being a
common mechanism by which organisms self-organize, a deeper
understanding of the simultaneous adjustment of the velocities of
many moving objects has important potential applications ranging
from the swarming of distributed robots exploring new territories
\cite{jadbabaie} to healing of wounds through the coherent
migration of skin cells
\cite{collective_epithelial_cells_on_collagen}.

Although widely observed in nature, collective motion is less
accessible for experimental investigations under laboratory
conditions. In particular, a well controlled series of experiments
aimed at characterizing the nature of the transition from a
disordered to an ordered phase in the velocity space is expected
to be useful for both providing a quantitative reference for
further experimental studies and prompting more realistic models
for group behavior. While the related laboratory observations are
scarce, a number of models based on assuming self-propelled
particles adopting a common direction have been recently developed
(see, e.g., Refs. \cite{vicsek,gregoire} to account for the above
ordering transition.

In this paper we present our experimental results concerning the
collective migration of a large number of tissue cells
(keratocytes) using long-term videomicroscopy. As the overall
density of the migrating cells is increased, we observe a kinetic
phase transition from a disordered (low density) into an ordered
(at larger densities) state in which most of the cells move in a
direction approximately agreeing with their average direction of
motion. Just below the transition a complex picture emerges with
interacting groups of cells moving in random directions. A novel
flocking model with no assumption about the knowledge of the
direction of motion of the neighbors is introduced and used to
interpret our experimental results. Our model assumes repulsive
and adhesive forces only and leads to results being in good
agreement with the observations.

As we next describe, our experimental setup allowed us to make
long-term videomicroscopy of keratocytes together with a
quantitative analysis of their motion.

{\em Cell Culture:} 2--4 fish scales were collected from living
5--15 cm long goldfish {\it (Carassius auratus)} with tweezers,
and placed external side up in a 35-mm Petri dish (Greiner)
similarly to Ref.\cite{Borisy_keratocytes}. After a few seconds
scales were flood with 1 ml culture medium: RPMI 1640 supplemented
with 10\% fetal calf serum (Gibco), 40 $\mu$g/ml Gentamycin
(Gibco), 0.5 $\mu$g/ml Amphotericin B (Sigma). Scales were kept at
room temperature in 5\% CO$_2$ atmosphere overnight to let
epidermal keratocyte cells migrate out from the scales. Before
time-lapse microscopy scales were removed, and cultures were
flushed with fresh medium. To gain low density cultures we applied
phosphate saline buffer (PBS) for 5--10 min and/or 0.25\% Trypsin
(Gibco) for 1--2 min. PBS and trypsin weaken cell-cell and
cell-substrate connections reversibly, a controllable percentage
of cells could be removed in this way.  {\em Time-lapse
microscopy:}   We used a home developed fully computer-controlled
time-lapse microscope \cite{jojo}, which enabled us to monitor
cell motility up to four cultures in several fields of view with a
frequency of 1 shot per min. Cultures were positioned over the 10x
objective of a Leica DM IRB inverted phase contrast microscope by
a 2D motorized stage (M\"arzh\"auser). Either an Olympus C4040-z
digital camera or a DP70 fluorescent camera captured the images.
During the typically 24--h long time-lapse microscopy cells were
kept in a room temperature custom made mini-incubator to maintain
5\% CO$_2$ atmosphere and avoid steaming up. {\em Data processing
and analysis}: In each time-lapse frame 20--30 representative
cells were randomly selected from the population. Positions of the
cells were determined manually along 100--200 frames by using a
custom written software. This process resulted in the $ r_{ij}$
positon vectors of cells, where $i$ denotes the index of the cell,
$j$*$\Delta t$ is the elapsed time from the beginning of the
cell's trajectory. $\Delta t$ stands for the time difference
between the frames. When calculating the order parameter and its
standard deviation -- to avoid noise and reduce error from manual
position determination -- we took into consideration only one
frame in every $10$ minutes.


\begin{figure*}
\centerline{
\mbox{\includegraphics[angle=0,width=0.75\textwidth]{fig1}}}
\caption{Typical behavior of cells in case of three different
densities.} \label{fig1}
\end{figure*}

Our {\em experimental results} are based on monitoring the cell
migration from very low cell densities to confluent (nearly full
coverage) cultures. A relatively sharp transition could be
observed from the random motility in sparse cultures to the
ordered collective migration of dense islands of cells. Fig.\
\ref{fig1} shows the typical behavior of cells in case of three
different densities. Single cell migration is usually described as
a persistent random walk \cite{Selmeczi}. It has recently been
demonstrated that some tissue cells can also move in dense sheets
\cite{cell_sheet_cryptic_lamellipodia,
collective_epithelial_cells_on_collagen}. To measure the level of
coherence of the migration pattern we used an order parameter
which was defined as the mean direction of motion of the
individual cells. Unit vectors of the cellular velocity were
calculated for each analyzed cell. Length of the mean vector,
i.e., the {\it r} order parameter was calculated as follows:

\begin{eqnarray}
\nonumber r&=&\frac{\sqrt{\left(\sum_{i=1}^N\sum_{j=1}^{M_i}x_{ij}\right)^2+
              \left(\sum_{i=1}^N\sum_{j=1}^{M_i}y_{ij}\right)^2}}
    {\sum_{i=1}^{N} M_i}\\
&=&\sqrt{\left<x\right>^2+\left<y\right>^2}\enspace,\\ \nonumber
x_{ij}^2+y_{ij}^2&=&1
\enspace \forall i,j \enspace, \text{which infers that } 0\leq r \leq1
\end{eqnarray}
where x$_{ij}$ is the x coordinate of the unit tangent vector of the i$^{th}$
evaluated cell at the j$^{th}$ point of its trajectory. M$_i$ is the number of
points along the trajectory of the i$^{th}$ cell: the i$^{th}$ cell was
followed for a time of M$_i$*$\Delta t$. N: number of evaluated cells.

\begin{figure}
\centerline{
\includegraphics[angle=0,width=1.0\columnwidth]{fig2}}
\caption{Order parameter as a function of cell density.}
\label{fig2}
\end{figure}

Fig.\  \ref{fig2}  displays the order parameter as a function of cell density.
A sharp step can be observed around 5*10$^{-4}$ cell/$\mu$m$^2$.

{\em Model} To interpret the above phase transition-like ordering
phenomenon we designed a model taking into account the specific
features of the experimental system. Obviously, when forming
coherently migrating groups, tissue cells are unable to explicitly
adjust their direction of motion to the average velocity of their
neighbors, this behavioral pattern emerges as a result of direct
cell to cell interactions (forces) only. In particular, the
locomotion in vertebrate cells typically involves the extension of
a broad veil-like lamella or  pseudopod at the cell front, using
which cells can move forward by crawling\cite{Lodish}. A complex
interplay of dynamic cytoskeletal growth and shrinkage, motor
protein facilitated contraction and the action of a plethora of
adhesive and regulatory proteins results in sustained well
directed cell motion\cite{Lodish}.

When constructing a model we first consider single cells,  moving
forward in a well defined
 --highly persistent-- direction, with relatively
constant speeds. Individual cells $i \in \{1,N\}$ are assumed to
move with velocity $v_0$ in the direction of a unit vector ${\bf
n}_i.$ This movement occurs at length scales of $L \approx 100 \mu
m$, speeds of $v_0 \approx 0.1 \frac{\mu m }{s}$ and with net
forces in the range of tens of piconewtons. From these values it
is clear that  the dynamics of cell locomotion is overdamped with
a Reynold's number of around $R \approx 10^{-3}$. The motion of
the center of mass of such a cell can be well approximated by an
equation for a particle subject to a constant self propelling
force
  $v_0 {\bf p}$ and total external force ${\bf F}$ (expressed in units
of velocity):
\begin{equation}
{\bf v}_i =  v_0 {\bf n}_i + {\bf F}, \label{eom1}
\end{equation}
where $ {\bf v}_i = \dot {\bf r}_i$ is the velocity of particle
$i$. The direction of motion of individual cells ${\bf n}_i$ is,
however not constant; it slowly changes with time\cite{Selmeczi},
the trajectory of a cell has a finite persistence length  and the
internal processes of the cell which generate cell motion are
noisy. Introducing the angle $\theta_i^{{\bf n}_i}$ describing the
direction of the self propelling force $v_0 {\bf n}_i$ we take
this into consideration by including a noise term $\xi \in
[\eta/2, \eta/2 ]$ into the dynamics of the self propelling force:
\begin{equation}
% \bf{v}_i &=  \dot {\bf r}_i &=  v_0 {\bf n}_i \nonumber   ,\\
\omega_i^{\bf n} =  \xi, \label{eom2}
\end{equation}
where $\omega_i^{\bf n} = \dot \theta_i^{\bf n}$ is the angular
velocity of the vector ${\bf n}_i$.

To describe the collective motion of cells we must introduce
forces acting between the $N$ cells. We consider pairwise
inter-cellular forces which are a function of the distance between
the two centers of mass only:
\begin{equation}
{\bf F}_{i j} = \left({\bf r}_i-{\bf r}_j \right)  \big(
F_{\mathrm{rep.}}(d_{ij}) + F_{\mathrm{int.}}(d_{ij}) +
F_{\mathrm{adh.}}(d_{ij}) \big) \label{force}
\end{equation}
%\left({\bf r}_i-{\bf r}_j\right) \left( v_r
%  \exp(-\frac{d-R_0)}{R_r}]) + v_i \left( d-\frac{R_i}{2}\right)
%    +\Theta(d-R_a)v_a \left( \frac{d-R_a}{R_i-R_a}^p\right)\right)
,where $d_{i j} = || {\bf r}_i-{\bf r}_j ||$. The force acting
between two cells is a result of three main contributions: i)
Cells which come into contact form a variety of intracellular
bonds aided by dedicated membrane proteins  (these bonds act to
keep them together), and resulting in an effective adhesive
interaction $F_{\mathrm{adh.}}$. ii) Incompressibility of the
intracellular material and rigidness of the cytoskeleton  together
act as a repulsive force $F_\mathrm{rep.}$ that prevent cells from
''crawling into'' each other. iii) In between these two regimes
the cells experience a intermediate force $F_\mathrm{int.}$ due to
deformation of the membrane and cytoskeleton.  In our simulations
we used:
\begin{eqnarray}
 F_{\mathrm{rep.}}  &=& v_r \mathrm{e}^{(-\frac{d_{ij}-R_0}{R_r} )},\quad F_{\mathrm{int.}}  =  v_i \left( d_{ij}-\frac{R_0}{2}\right),\nonumber\\
F_{\mathrm{adh.}}  &=& \Theta(d_{ij}-R_a) v_a\left(
\frac{d_{ij}-R_a}{R_0-R_a}\right)^p,
\end{eqnarray}
with the values $v_0=1$, $v_r=40$, $R_0=1$, $R_r=0.3$, $v_i=0.2$,
$v_a=0.5$,$R_a=0.8$, $p=6$ , ($\Theta(x)$ is the Heavside step
function). However, the results we obtained were not sensitive to
the particular choice for the parameter values.

\begin{figure*}
\centerline{\includegraphics[angle=0,width=0.5\textwidth]{fig3}}
\caption{Computer simulations were preformed of
 the system desribed in the text for different values of the density $\rho$,
 these showed a transition to the ordered phase similar to that seen in experiments.
  (a) typical behavior of cells is shown for three different densities.
  (b) as cells moving in different directions come into contact adhesive
  inter-cellular forces act to align  ${\bf n}_i$ and ${\bf n}_i$, resulting
  in an effective averaging of self-driving directions.  (c) The absolute value
  of the average velocity $v_a$ versus the density $\rho$ is shown for $N=1000$
  and $\xi =2.75$. The inset shows the dependence
  of $\ln v_a$ on $\frac{\rho-\rho_c(\eta)}{rho_c}$,
  the slope of the line fitted can be associated with the critical exponent
   $\delta$. The large scaling regime and the similarity of the numerical
   values obtained for the exponents with those found for other models indicates
    the existence of a continuous phase transition in $\rho$.
} \label{modelfig}
\end{figure*}

As two cells with  different direction of motion come into contact
the above inter-cellular forces will prevent the cells from moving
in their prior self-propelled  direction. The force generating
mechanism of the cell must clearly adjust in some manner to the
external forces. Thus, we postulate that it is sufficient to
consider this response as an effective rotation of the cells'
direction of motion ${\bf n}_i$ toward the direction of the total
force acting on them $\sum_{j=1}^N {\bf F}_{i
  j} + v_0 {\bf n}_i$, i.e.,
%Some form of  \emph{positive
%  feedback} acts, such that the cell's locomotive apparatus adjusts
%itself to move in the direction in which it is most ''successful''
%-- the direction the cell is actually moving:
\begin{eqnarray}
{\bf v}_i &=& v_0 {\bf n}_i    + \sum_j {\bf F}_{ij} \nonumber,\\
\omega^{\bf n}_i &=& \frac{1}{\tau} \arcsin(||{\bf n}_i  \times
\frac{{\bf v}_i}{v_i}||) +\xi, \label{eom3}
\end{eqnarray}
where the parameter $\tau$  (a value of $\tau=1$ was used in
simulations) controls the speed of rotation and $v_i$ is the
magnitude of ${\bf v}_i$. If two cells approach each other close
enough, \ref{eom3} leads to a gradual alignment of their direction
of motion (Fig. \ref{modelfig}.b.). Our model is in this sense
similar to other models of systems of self-driven particles
exhibiting emergent collective motion -- ie.\ flocking --
\cite{vicsek}, with the very important difference that the
particles -- the cells -- do not directly use information on the
movement of others around them to determine their own movement.
%they only interact through and respond to inter-cellular forces
%acting between them.

We carried out simulations by solving the system of $3N$
differential equations (\ref{eom3}) with periodic boundary
conditions using an adaptive step size fifth order Runge-Kutta
algorithm based on (\cite{NumRec}), for details see online
supplementary material \cite{online}.

In good agreement with experiments, running our model resulted in
a continuous transition to the ordered phase (where all cells move
in a common direction) as the density of cells $\rho$ is increased
for a fixed value of the noise $\xi$ (Fig. \ref{modelfig}.b.).
While changing the value of $\xi$ was not experimentally feasible,
simulations show that a continuous transition to the ordered phase
also takes place as $\xi$ is decreased while $\rho$ is held fixed
(data not shown, see online material \cite{online}). In both cases
we chose as the order parameter
\begin{equation}
v_a = \frac{1}{N} \left| \sum_{i=1}^N \frac{{\bf v}_i}{v_i}
\right|.
\end{equation}

Based on the assumption that our model exhibits a kinetic phase
transition in the thermodynamic limit analogous to the continuous
phase transition in equilibrium systems, we proceeded to study
it's critical behavior,  ie.\ ,
\begin{equation}
v_a \propto [\eta_c(\rho)-\eta]^\beta \quad \quad \mathrm{and}
\quad \quad v_a \propto  [\rho-\rho_c(\eta)]^\delta,
\end{equation}
where $\beta$ and $\delta$ are the critical exponents and
$\eta_c(\rho)$ and $\rho_c(\eta)$ are the critical noise and
density, . Analysis of the data yielded the values $\beta=0.45 \pm
0.05$ and $\delta=0.36 \pm 0.01$, strongly suggesting that our
model belongs to the same universality class as the angular noise
model of Vicsek \emph{et al.\ }\cite{vicsek}, for which
$\beta=0.45 \pm 0.07$ and $\delta=0.35 \pm 0.06$.

In this way we have been able to demonstrate how purely adhesive
forces imply an effective alignment term in much simpler models.
We expect that our and similar experiments as well as the
quantitative interpretation of the observations we have provided
for the collective motion of tissue cells will lead to a better
understanding of such vital phenomena as wound healing or
embryogenesis.





\section{Acknowledgments}

We are grateful to Dr. G\'abor Cs\'ucs for his instructive help to
set up a fish keratocyte lab in our institute. We thank Dr.
Zsuzsanna K\"ornyei for her advices in cell culturing. This study
was supported by Hungarian Science Research Funds: NKFP
3A/0005/2002 and OTKA Nos. F49795, T049674 .



\begin{thebibliography}{99}

\bibitem{parrish}
J. K. Parrish,  and W. H. Hapner (eds.) {\em Animal Groups in
Three Dimensions} (Cambridge University Press, New York, 1997).

\bibitem{czirokpre}
A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet and T. Vicsek:
 {\em Phys. Rev. E,} {\bf 54}, 1791 (1996)

 \bibitem{levine}
W.-J. Rappel, A. Nicol, A. Sarkissian, H. Levine, and W. F. Loomis
{\em Phys. Rev. Lett.} {\bf 83}, 1247 (1999)

\bibitem{jadbabaie}
A. Jadbabaie, J. Lin, and A. S. Morse {\em IEEE Transactions on
Automatic Control}, {\bf 48} 988 (2003)

\bibitem{collective_epithelial_cells_on_collagen}
H. Haga, C. Irahara, R. Kobayashi, T. Nakagaski, K. Kawabata,
 Biophys.\ J.\ {\bf 88,} 2250-56 (2005)


\bibitem{vicsek}
T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet {\em
   Phys. Rev. Lett.} {\bf  75}, 1226 (1995)


\bibitem{gregoire}
G. Gregoire, H. Chate, and Y. Tu {\em Physica D} {\bf  181}, 157
(2003)


\bibitem{Borisy_keratocytes} T. M. Svitkina, A. B. Verkhovsky, K. M. McQuade,
 G. M. Borisy, J.\ Cell Biol.\ {\bf 139} (2), 397-415 (1997)

\bibitem{jojo} B. Szab\'o, Zs. K\"ornyei, J. Z\'ach, D. Selmeczi, G. Cs\'ucs,
 A. Czir\'ok and T. Vicsek, Cell Motil.\ Cytoskel.\ {\bf 59} (1), 38 -49 (2004)

\bibitem{Selmeczi} D. Selmeczi, S. Mosler, P. H. Hagedorn, N. B. Larsen,
 H. Flyvbjerg,
Biophys.\ J.\ {\bf 89} (2), 912-31 (2005)

\bibitem{collective_sperms}
 I. H. Riedel, K. Kruse, J. Howard, Science,
 {\bf 309,} 300-3 (2005)

\bibitem{cell_sheet_cryptic_lamellipodia}
R. Farooqui, g. Fenteany,
J.\ Cell Science {\bf 118,} 51-63 (2004)


\bibitem{selforganized_dictyostelium}
W-J. Rappel, A. Nicol, A. Sarkissian, H. Levine,
Phys.\ Rev.\ Lett.\ {\bf 83} (6), 1247-50 (1999)

\bibitem{symmetry_breaking_mammalian_cells}
S. Huang, C. P. Brangwynne, K. K. Parker, D. E. Ingber,
Cell Motil.\ Cytoskel.\ {\bf 61,} 201-13 (2005)

\end{thebibliography}

\end{document}
