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Directed cell movement and cluster formation : physical principles
Hans Gruler, Anne de Boisfleury-Chevance
To cite this version:
Hans Gruler, Anne de Boisfleury-Chevance. Directed cell movement and cluster formation : physical principles. Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.1085-1105. �10.1051/jp1:1994186�.
�jpa-00246966�
Classification Fhy.çic.ç Absn.aciç
61.30 64.70H 87.10
Directed cell movement and cluster formation : physical principles (*)
Hans Gruler (**) and Anne de Boisfleury-Chevance
Centre d'Ecologie Cellulaire and INSERM U313, Hôpital de la Salpétnère, 47 Bd de l'Hôpital.
75013 Paris, France
(Receiied 4 Noi>embei /993, roi>1,çed 3 Marc.fi J994, ac.c.epied 6 Apii/ J994)
Abstract. The cell-cell interaction of migrating human leukocytes (granulocytes) was investiga-
ted. We have found that the attractive pair interaction of granulocytes can be switched-off at high
calcium concentrations and switched-on at low calcium concentrations. Through this experiment
we established that the cells attracted each other and formed clusters containing actively moving cells :(1) the cluster formation was a function of the mean cell density, <ii) where no clusters were observed for small cell densities, pj ~150-300 cells/mm~ ; threshold behaviour). <iii) and the
mean cluster size was a function of the mean cell density. <iv) In addition we established that the dynamic process of the cluster formation was a function of the mean cell density, and (v) the migrating cells were oriented towards the cluster's center. The cluster formation is discussed in the framework of a droplet model where two dynamic processes are observed :<a~ the cells in a cluster
attract further cells and <b~ the cells at the boundary of the cluster have the possibility to move
away. The droplet model in connection with the cell conservation law is confirmed by the expenments. The analogy between the liquid-gas transition of interacting molecules and the condensation of interacting cells is shown. The migrating and onented cells of a cluster are in an orientational liquid crystal state of polar symmetry. The polar order is discussed in the framework of a polar mean field.
1. Introduction.
The process by which particles like molecules, biological cells, etc, exchange information constitutes one of the most intnguing areas of biology and of physics [1, 2]. The physical
processes of reducing a gas or vapor with freely moving molecules to a liquid, liquid-
crystalline or sohd form are known to a large extent. Likewise the interaction processes of
reducing the state of freely migrating cells to a condensated state are less known. The
following processes discussed in the formai hterature [4] is as follows : (1) adhesion or cohesion the molecular forces in the area of contact between mi grating cells act to hold them
together; (ii) chemotaxis or galvanotaxis: every cell transmits signais and guides its
(~) In Memory of Marcel Bessis.
l'fi Feimaneiii addiess Biophysics Department. University of Ulm. Ulm, Germany.
1086 JOURNAL DE PHYSIQUE I N° 7
movement according to the received signais (chemotaxis for chemical signais and galvanotaxis
for electrical signais). The migrating cells con form a condensated state in case of attractive
signais.
The cell-cell communication between migrating human leukocytes (granulocytes) is
investigated. The cell-cell communication can be altered reversibly by means of the bivalent
ions hke calcium and magnesium. At high calcium concentrations <2.5 mmolel), the migration behaviour of a single granulocyte was net affected by other granulocytes except in direct cell-
cell contact <steric exclusion). But at low calcium concentrations (~ nmolel ), cell behaviour is
altered granulocytes attracted each other and formed clusters as demonstrated in figure1.
Migrating granulocytes are ideal test cells for the investigation of cell-cell interaction because the behaviour of granulocytes exposed to extracellular guiding fields is known to a large extend [5, 6].
<IX
fl. ~ù'/à Î~É,
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ll~ljms SQ
~
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~ ?M*~ *
~
fp ~ f
~ ~~
~3
~.&
_'
« ~
~f
~
W
a)
b)
j R
~* ( . f~ '
~
, i L,j
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~t~~f'?
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~
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d)
Fig. l
times per ml blood plasma). Cell density 1000cells/mm2. The bar
carre;ponds 20 ~m. a) two minutes after starting the expenment. The
monocyte in
clu~ter acted as a nucleation center. b) After 2 min and 50 s. c) After 4 min
~quized Dut) and )14 min and 30 s.
2. Material and methods.
2.1 CELL PREPARATION. -The cells were prepared as previously described [8]. In short
human blood from healthy donors was taken in Calciparin <Choay, Paris, 10 units/ml) as an
anticoagulant in the case of the central expenment and in ethylene diamine tetraacetic acid
(EDTA) (Sigma, ? %w/v 1drap per ml blood 1drap contains il mg EDTA) as an
anticoagulant in the case of experiments at low calcium concentrations. EDTA is a chelator
which binds bivalent ions like calcium (pKc~
=
10.7 ) and magnesium (pK~~ = 8.7). The
blood cells are allowed to sediment by gravity for approximately 2 heurs. For thÉ granulocytes studies, the buffy coat was suspended in plasma and used as stock suspension (typical 5 x 10~ cells/ml). Similar cell preparations were made with ethyleneglycol-(aminoethyl ether)
tetraacetic acid (EGTA) (pKc~
=
II and pK~~
=
5.2) as an anticoagulant. For the monocytes studies, the buffy coat was suspended in 2.5 ml of autologous plasma and deposited
on top of a
2.5 ml lymphoprep (Nyegaard, Oslo, Norway). The tube was centrifuged at 400 g for 30 min.
The mononuclear cells were enriched in the layer between the lymphoprep and plasma. This
layer was washed three times in the plasma of the same donor (first wash : 200 g for 10 min, second and third washes 100 g for 10 mini resulting in the final stock solution of monocytes.
For lymphocyte studies, the upper part of plasma above the buff» coat wa~ removed and
centrifuged at 100 g for 10 min in order to increase the lymphocytes number in the stock solution.
To study the interaction of EDTA with leukocytes the different stock solutions wer~
enriched with different high doses EDTA (addition of EDTA : 1drap/ml, 2 draps/ml, and
3 draps/ml at higher EDTA doses, the cell's appearance was never healthy). A drap of the
granulocyte suspension was deposited between a glass slide and a glass coverslip. The preparation was then sealed with parafin and made ready for the use in the phase contrast microscope. For monocytes, the glass shdes were covered with glycol methacrylate
(Polysciences Inc, Warrington) and allowed to dry for a week at room temperature before use.
For lymphocytes, the slides were replaced by plastic coverslips <plexiglas).
Ail the expenments were performed with blood plasma because <1) the cell; look healthy and (ii) migrate well. If, however, the blood plasma is exchanged by an artificial medium, then the
cellular speed is reduced by a factor of two or more.
2.2 MicRoscoPlc OBSERVATION. The cell movement was observed by means of a phase-
contrast light microscope equipped with a vider camera. The vider signal was recorded on a
VHS vider tape (time lapse). Ail the experiments were performed at 37 °C. Different
objectives were used : 10X, to have many but poorely resolved cells in the viewing field. ?5X and 40X, to have a few, but highly resolved cells in the viewing field.
The vider signal of the low magnification pictures was digitized for further computenzed picture analysis. The center of gravity of the cells is approximated by the center of area of the
cell coutour fine. The interval between consecutive images was generally 20 s.
2.3 DATA ANALYSIS.
Ramdam walk and mean-squai"ed displacement.-The mean-squared displacement was
perforrned to determine (1) the charactenstic time quantifying the locomotory machinery and (ii) the diffusion coefficient quantifying the random walk activity.
The mean-squared displacement was determined from the trajectones of the migrating cells.
The displacement between an arbitrary chosen time, t
= 0 and the time, t, was determined AJ =,i(t) ->"(0) and ày = y(t) y(0). The procedure was repeated for different starting
positions and different trajectones. The final result was the mean-squared displacements,
(AJ~), and, (ày~), as
a function of time.
1088 jOURNAL DE PHYSIQUE I N° 7
The ability of cells to performe directed movement is due to the existence of a cellular automatic pilot. The rate equation for the angle of migration, ~, is as previously shown [9]
~~
= kpc-j <field sin q~ + r~ ii) <1
~ is the angle between the direction of migration and the guiding field. The first term on the
right side of this equation has the meaning of a deterministic
« torque » which tries to render
the movement parallel to guiding field of the strength, ci (field). The machine coefficient,
kp, characterizes the deterministic' part of the signal/transduction system of the cellular
machinery. The second term, r~, is a 'stochastic torque' which is responsible for the random walk activity of the locomotory machinery.
In case of an isotropic cellular environment there exists no desired direction and no guiding
field <c.j <field)
=
0) and, thus, the angle of migration, q~, is only determined by the stochastic torque, r~ (t ). By knowing the stochastic properties of r~ (t), equation il can be solved. As
previously shown [10], the stochastic term, r~(t), can be approximated by a white noise
source ( (r~ (t ~)
=
0 and (F~(t r~ (t'))
= q~ à (t t~)). q~ is a further machine coefficient
which is related with the mean charactenstic time, T~, of the locomotory machinery
(r~ = 2/q~). T~ is often refered as persistence time.
The mean-squared displacement as a function of time can be obtained from equation (1) since the cellular speed, u~(t), and the angle of migration, q~(t ), are statistically independent
variables [10]
(Aur~) = (ày~)
= 2 D t T~ exp
l.
(?)T~
The persistence time. T~, and the diffusion coefficient, D (= (u))/q~) are obtained by fitting equation(2) to the experimental data. This equation holds only if there is no cell-cell
interaction. Thus in case oi EDTA-treated cells the experiments must be performed at low cell
density.
2.4 DIRECTED MOVEMENT AND POLAR ORDER PARAMETER. In
case of a non-vanishing guiding field, ci (field) # 0, the random movement of granulocytes became directed. The
machine coefficient, kp, of equation(1) can be determined by quantifying the directed
movement. The Fokker-Planck equation for the angle distribution function, f(q~, t), is m case
of a white noise source
~~ÎÎ~ ~~
ô~
~~
~ ' ~~~ ~ ~
Î
ô~ ~~~' ~~' ~~~
One obtains for steady state (ôf/ôt=0) the normalized angle distribution function,
f(~)
2k
exp( ~~' .cos ~)
~~~~ ~~
k ~~~
2 wl~( ~~~
q~
where l~ is a hyperbolic Bessel function. The dimensionless quantity 2 kp cj/q~ measures the
guiding field in natural units. In case of galvanotaxis, the guiding field, cj, is an electric field.
E, and the natural unit is K~ E (with the galvanotaxis coefficient K~
=
2 kp/q~). The polar
order parameter as the average of cas q~ is then
2 kp ci (field Ii
~~°~ ~ ~
2 kp ci Îfield ~~~
'o
q~
The polar order parameter equals ?eio for random movement and one if the cells move parallel
to the guiding field.
Dur model of the automatic controller can be compared with the phenomenological approach
of Patlak [13], Keller and Segel [14]. They assumed that the mean drift velocity,
vii, is proportional to the concentration gradient de/à; (vii = x dc/àr where x is the constant of
proportionality). A similar expression can be used for electric fields. These approximations are only valid for small concentration gradients or small electric field strength. This is evident
since the mean drift velocity approaches for large guiding fields the finite mean track velocity
when the concentration gradient or electric field becomes very large. The model of the
automatic controller is net hmited to small concentration gradients or to small electnc field
strengths. One obtains for small polar order parameter (cas q~) «1) x
= (v~) K~/2.
There exists at least two different ways to determine the average of cas çg. But bath methods lead to the same result. This holds true at least for granulocytes [I Ii Therefore, we do net
distinguish between these two methods here.
Method : the migrating cells drift parallel to the guiding field : the mean displacement in
the direction of the guiding field is the average drift velocity, (vii ), multiplied by the time t.
The mean drift velocity, (vii), is the product of the mean speed, (v~), and the average of
(cosçg). The directed movement can be characterized by the chemotropism index, (vii /(u~), which is identical with the Mccutcheon index (ratio between travel parallel to the guiding field and the actual distance travelled). This method was used to determine the polar
order parameter of cells exposed to an electric gmding field.
Method 2 : the direction of migration can be determined from a single picture [12] since (ii the migrating granulocytes are elongated in the direction of migration and (ii) the leading front and rear end of the cell can morphologically be distinguished. An orientation vector is introduced for every cell and fixed at the center of the cell and, thus, the angle distribution function, f(q~ ), in respect to the guiding field can be determined. The final result is the polar
order parameter as the average of cas q~. This method was used to determine the cellular polar
order in a cluster having a central symmetric guiding field.
Pair coi-i elation function. The cell-cell interaction can be obtained from the pair correlation
function [15]. For more details see Appendix A. The pair correlation function, g(1), of
migrating cells can be obtained from the cell-cell distance distribution evaluated from pictures
made at low magnification. The cell-cell configuration should be in steady state which is reached for granulocytes several minutes after starting the expenment (t w T~).
The cell-cell distance distribution function can be obtained in the following way : an
arbitrary cell is chosen as a center cell (r
=
0). The number of cells. N (1), in the area enclosed
by the circles with radius, r àr, and i, is determined. The largest circle, i~~~, is chosen
accordingly to the desired maximum distance in the cell-cell interaction. An arbitrary chosen cell should be faf enough from the picture's boundary (~ r~~~). The procedure is repeated with ail possible cells as a center. Then the procedure is repeated with different pictures.
1090 JOURNAL DE PHYSIQUE I N° 7
The average number of cells, (N(r)), divided by the enclosed area, AA(r)=
ar(iJ-(r-ài)~), yields the cell density, N~(i"). The normalized pair distribution, n~ = N~jr)IN
j~, (number of counted cell pairs, N
j~ = jj N (r)) is then proportional to the pair correlation function, g~(r). The calibration factor, n, is determined by the asymptotic behaviour of the pair correlation function
lim g~ (r)
= n lim n~(1)
= (6)
- oe - oe
3. Results.
3.1PHENOMENOLOGICAL oBsERvATioNs.-Granulocytes exposed to blood plasma are
stimulated to migrate on a glass substrate. A well accepted picture is that the migration behaviour of single granulocytes is net affected by other granulocytes except at direct contact (steric integration). The cellular behaviour can be altered if the cells were prepared with the
anticoagulant EDTA. The migrating cells attracted each other and formed clusters eut of the
uniforrnly spread cells as can be seen in figure (1 000 cells/mm2 and drap EDTA pet ml blood plasma) and figure 2 (w 400 cells/mm2). The clusters attracted further cells eut of their
vicinity (Fig. 2) and thus the mean cell density close to the cluster became very low. Every
cluster was surrounded by a low cell density ring (see Fig. 2). The movement of the center of
gravity of the whole cluster with its migrating cells was very small. The unification of two clusters was a very seldom event. In the central experiments, where the cells are treated with Calcipanne as anticoagulant, no cluster formation was observed (1 440 and 100 cells/mm2).
first picture 10 pictures (ht = 20s) last picture
~ ~ ~ , ~
~ ~p ~ o , ~~ « ~
o°1]/°(1
° '~j
~
~
~~~ u ~ ~
jo ~Î °Î
Î
'~
/
u~o °~
~ ~
~u~ « ~
Fig. 2. The spatial distribution of EDTA-treated granulocytes (1400 cells/mm2) is shown. The left side picture was taken at the staff of the expenment. The middle side picture is the sum of 10 consecutive taken pictures : a few cells attracted further cells which tried to stay in the cluster. The ride side picture
was taken at the end of the experiment. Note the mean cell density is close to the threshold cell density.
The effect of EDTA on the migrating cells can be demonstrated in a different way first, a sample was prepared with EDTA-treated cells (1 drap per ml blood plasma). The migrating
cells formed clusters as expected. Then, the EDTA-treated cells in the stock solution were
washed. A new sample was prepared and no clusters were observed.
In the next step, the different types of blood cells were investigated. No cellular clusters
were observed with monocytes treated by one, two or three draps of EDTA per ml blood
plasma. Lymphocytes treated by one, two or three draps of EDTA per ml blood plasma
showed no cluster formation. But when a mixture contaming a high concentration of
granulocytes and a low concentration of monocytes, was investigated, then the cellular cluster
