J. Cell Sci. 33, 17-36 (1978) I 7

Printed in Great Britain © Company of Biologists Limited

A COLLISION ANALYSIS OF LYMPHOID

CELL AGGREGATION

CLIVE W. EVANS* AND JOHN PROCTORDepartments of Cell Biology and Statistics, The University,Glasgow G12 8QQ, Scotland

SUMMARY

We have obtained data on the frequency of aggregates of different size classes and on themaximum aggregate diameter of lymphoid cells subjected to aggregation in the laminar shearfield of a Couette viscometer. Maximum aggregate diameter reached a plateau level duringaggregation. This plateau is considered to be the result of a balance between the hydrodynamicshear forces tending to resist the formation of aggregates, and the adhesive forces of theaggregated cells tending to resist dissociation. Stepwise increases in the shear rate producedaggregates of progressively smaller maximum diameter until the limiting (i.e. control equivalent)diameter was reached. Equations governing these stepwise changes in aggregate diameter wereobtained by regression analysis, and an estimate of the force of dissociation (FB) was made fromderived values of the critical shear rate. Thymocytes (Fv = 254 x io"0 N m-2/cell) werefound to be more adhesive than lymphocytes (FB = 205 x io~G N m~2/cell), in agreement withcurrent concepts.

The observed data on aggregate frequency were seen to be of poor fit with a model ofaggregation derived by collision analysis of the aggregation process. This led us to consider thepossibility that all cells may not share the same probability of forming an adhesion. We thusderived further models of aggregation in which some fraction of the total cells was consideredto have enhanced possibilities of a collision producing an adhesion. Of the models we considered,a ' 5 % preferred fraction offered best agreement with the experimental observations. Wetherefore conclude that the populations of cells studied in this report are not 'homogenous'in that some cells are more adhesive than others. Alterations in the percentage of the preferredfraction of cells will lead to different aggregate-frequency indices. Such changes might beexpected to occur during the initial stages of carcinoma development.

INTRODUCTION

The continuous circulation of lymphocytes between the blood and the lymph, theiraberrant migration to non-lymphoid sites during some diseases, and the movement oflymphoblasts to sites of inflammation presumably reflect changes in a number of cellproperties. In all of those examples of lymphocyte traffic, the circulating cell mustrecognize a relevant stimulus and then respond by adhering to endothelial cells liningthe blood vessels before migration into a specific organ or an inflammatory site canoccur. From these examples we can thus derive a simple model of cell traffic in whichat least 2 cell properties - the recognition of a stimulus and the subsequent adhesiveresponse - must be involved (Davies, Haston, Evans & Curtis, 1977). Although thecontrol mechanisms of cell traffic are probably multifactorial, their analysis requires

* Correspondence to: Dr Clive W. Evans, Department of Anatomy and ExperimentalPathology, St Andrews University, St Andrews, Fife, KY16 9TS, Scotland.

18 C.W. Evans and J. Proctor

a simple experimental model in which the effects of changes in cell adhesiveness inresponse to stimulation can be examined. In pursuance of such a study we haveadopted the cell-aggregation technique in which changes of this type can be bothinduced and quantified.

Two questions immediately arise when the cell aggregation technique is employedin the measurement of adhesiveness: (i) What type of adhesion is being measured?(2) Does the technique actually establish differences in the strength of adhesion? Theanswers to both of these questions are by no means clear. We consider the possibilitythat there is a temporal aspect in the establishment of cell adhesiveness leading toessentially 2 types of adhesion which are not necessarily mutually exclusive. The firsttype of adhesion arises when 2 cells approach each other at least to within the secondaryminimum and is considered by us to be the initial form of adhesion during aggregation.This initial type of adhesion may have a stabilizing effect permitting the subsequentdevelopment of adhesions in the primary minimum. The London-Dispersion forcepossibly represents an example of the first type of adhesion, while certain cell junctionsmay be examples of the second type. There is some evidence suggesting the involve-ment of long-range Van der Waal's forces in the adhesion of erythrocytes (Gingell,Todd & Parsegian, 1977), and cell junctions have been detected between many cellsincluding lymphocytes maintained in vitro (Mclntyre, Pierce & Karnovsky, 1976).Uncertainties in the nature of the collision between 2 cells during aggregation leadto difficulties in obtaining precise values of contact time, but since available datasuggest that junctional complexes take some seconds to develop (Heaysman &Pegrum, 1973) then it is nevertheless possible to conclude that only the first type ofadhesion may be of consequence in initial aggregate formation.

Essentially 2 parameters involving the measurement of maximum aggregate size(Moscona, 1961) or the rate of disappearance of single cells into aggregates (Curtis &Greaves, 1965) have been utilized in the determination of adhesiveness from cellaggregation data. Neither of those parameters provides a direct measure of the forceof adhesion, and recent data suggest that in fact they may measure different aspectsof the aggregation phenomenon (Ede & Agerbak, 1968; Gershman, 1970; Morris,1976). One criticism of these studies is that measurement of the maximum aggregatesize and the rate of aggregation were made over different time intervals and we havealready drawn attention to changes possibly associated with the temporal aspects ofaggregation. Clearly there is a need to measure the actual force of adhesion betweenspecific cells, particularly if we assume that it is this force which contributes to in vivocell interactions. An approximation of the force of adhesion (FA) can be obtained fromdetachment studies (Coman, 1944; McKeever, 1974), providing the assumption ismade that the force holding 2 particles together (FA) is equivalent to the force requiredto pull them apart (FD). Although the detachment technique may have the advantageover other methods in that it provides some estimate of FB, its basic assumption maynot be valid. The force of dissociation, for example, is dependent upon the mode ofseparation: a peeling mode requires much less force to separate 2 bodies than a tensilemode. This is seen very clearly when an attempt is made to separate 2 coverslipsheld together by a liquid smear. The easier way to separate them is to slide them over

Lymphoid cell aggregation analysis 19

each other (i.e. peeling mode) rather than to pull them apart (i.e. tensile mode).Analysis by detachment fails to distinguish between separation by membrane ruptureand separation in which the membranes are kept intact, and also does not take intoaccount the effects of connective tissue substances such as collagen fibres or the effectsof cell deformability, both of which may resist separation. We have attempted toovercome some of these criticisms by measuring FD from shear spectrum analyses onaggregating cell suspensions. Since at low shear rates larger aggregates form than athigher shear rates, then, as the shear is increased in a continuous experiment, the largeraggregates forming initially will subsequently break up to form smaller aggregates whichsatisfy the requirement of the higher shear conditions. Break-up has been consideredby Albers & Overbeek (i960), and presents an analytical problem which might lead todeviation from the expected aggregate frequency index. Its effect can be reduced,however, by considering discrete (stepwise) rather than continuous experiments ina shear spectrum analysis. The stepwise experimental method was employed in thisstudy, and we make the assumption that supra-optimal-sized aggregates cannot developsince -FA < FD, thus leading to reduced break-up. Break-up of suboptimal-sizedaggregates cannot be accounted for, and no estimate can be made of its contribution.

Unlike an earlier study (Brooks, Millars, Seaman & Vassar, 1967) we have notmeasured the force required to bring about dissociation of preformed aggregates orsmall tissue clumps, but rather the force required just to prevent association of singlecells into couplets. Although the data obtained using such a stepwise failure-to-adhere technique are necessarily approximate and not precisely equivalent to FA, asa preliminary step in aggregation analysis the method offers a means by which quan-titative differences in adhesion can be compared and assessed.

One further problem concerning aggregation technique warrants some elaboration.In any aggregating system the degree of aggregation is proportional to the number ofcollisions induced by shear in the medium and is thus related to the period of aggre-gation, the shear rate, initial particle concentration, and volume fraction of aggregatingparticles. If a comparison is to be made between different experiments or betweenvarious methods of assessing adhesiveness then clearly these parameters must eitherbe known or kept constant. In this study we have employed the principles of Couetteviscometry, a technique originally introduced into cell aggregation studies by Curtis(1969) in order to include the above parameters, quantify their effects, and measureaggregation. We thus report on shear spectrum analyses in which maximum aggre-gate diameter and aggregate frequency have been determined, and use these data toderive a measure of cell adhesiveness.

Most aggregation studies assume an homogeneous cell population in which allparticles are considered equivalent in that they are assigned an equal value for theiradhesiveness. This assumption is probably not correct in the majority of cases, andwe have presented evidence elsewhere that in the thymus, for example, at least 2subpopulations of cells exist which differ in their degrees of mutual adhesiveness(Evans & Davies, 1977). Based on collision analysis it is possible to derive a model ofaggregation which, under a set of denned conditions, reflects the frequency ofaggregates of different size classes. We extend such a model to include situations in

20 C. W. Evans and J. Proctor

which a varying fraction of cells is considered to be more adhesive than the remainingcells of the initial population, and subsequently examine the derived models for good-ness of fit with observed aggregate-frequency data.

MATERIALS AND METHODS

Preparation of cells

Single-cell suspensions were obtained from the thymuses or combined lymph nodes (inguinal,mesenteric and axillary) of female CBA/ca mice (6-8 weeks old) by teasing the organs apartin RPMI 1640 medium (Gibco-Biocult) supplemented with 0-04% (w/v) NaHCO3 andbuffered to pH 7-4 with io~2 M HEPES (A7-2-hydroxyethylpiperazine-iV'-2-ethanesulphonicacid). After teasing, the suspensions were washed free of cell clumps by filtration through sterileglass wool columns, and subsequently diluted with supplemented RPMI 1640 medium asrequired. Cell viability was assessed by the 02 % trypan blue exclusion technique both beforeand after aggregation, and was judged to be > 90 % in all cases.

Cell aggregation

Aggregation of single-cell suspensions was performed in Couette viscometers which maintaina laminar shear flow under known conditions (Van Wazer, Lyons, Kim & Colwell, 1963;Curtis, 1969). Briefly, a Couette viscometer consists of 2 cylinders one of which rotates relativeto the other. The cylinders are separated by a narrow gap containing the cell suspension, andthe shear rate can be determined from the geometry of this gap and the speed of rotation.Aggregation was performed under varying conditions, but different populations of cells wereadjusted to a constant volume fraction (18-63 x I 0~2%)- After aggregation a sample was takenfrom the gap, placed on a haemocytometer, and the following parameters of aggregation weremeasured: (1) Maximum aggregate diameter (£>max). This term refers to the mean maximumdiameter of aggregates obtained from multiple samples of a number of experiments withoutregard to orientation or aggregate shape. And (2) Aggregate-frequency index. The numbers ofcell couplets, triplets, quadruplets, and aggregates containing more than 4 cells were countedin each sample and expressed as percentages of the total aggregates counted.

Measurement of the critical force of dissociation (FD)

The critical force of dissociation (Fv) was derived by Albers & Overbeek (i960) from con-sideration of the requirements to separate a lyophobic couplet under the influence of a shearingforce. For redispersal of couplets, FD is derived from the following equation:

^D = i-57r'/K2»- + ̂ )Gcritsin(2acrit) (1)

Where •»/ is the continuous-phase viscosity, r the cell radius, H the cell separation, Gcrn theshear rate at which redispersal occurs, and acr,t the angle between a line drawn connecting thecentres of each cell and the plane defined by the direction of flow and the direction of shearwhen separation occurs (Fig. 1). In the present analysis Gcrit is determined as that shear rateat which 2 cells just fail to form a couplet and is derived from the shear-spectrum analysespresented in Fig. 5 (p. 23).

RESULTS

Shear-spectrum analysis

Preparations of thymocytes and lymphocytes were subjected to aggregation inCouette viscometers. Experiments were carried out at different shear rates, but withconstant volume fraction conditions and over the same time period. The results ofsuch shear spectrum analyses are shown in Fig. 2, from which it can be deduced that

Lymphoid cell aggregation analysis 21

under the relevant conditions the largest aggregates for both experimental cellpopulations are obtained at a shear rate of about ioo s~x, but that thymocytes formlarger cell aggregates than lymphocytes. Final aggregate size is considered to bea function of the force of adhesion, and is dependent upon the collision conditionsemployed in aggregation. Since the number of collisions is related to the duration of

Cell 1

Cell 2

Flow

Fig. i . Two adhering cells under the influence of a shearing field. Two cells of radiusr are separated by a gap H. The shearing force acting on cell 2 (F x = P ^ sin a) is theresult of hydrodynamic flow, and varies as the couplet rotates from a = 0° to a = 90°with a maximum at a = 450. At some critical angle, when Fx = F D > FA, separationoccurs. Since Fv varies as sin 2acrit the precise value of acrlt is not too important forangles greater than about 25° (Albers & Overbeek, 1960).

10 15 20 25

102><G. s- '

Fig. 2. Shear spectrum analysis. Thymocytes (broken line) and lymphocytes (solidline) were subjected to varying shear rates but under constant fraction conditions(f = 60 min). Each point represents the mean of a series of discrete experiments.

22 C. W. Evans and J. Proctor

aggregation, then at shear rates less than ioo s"1 the maximum observed aggregatesize must be limited by time. At shear rates greater than ioo s"1, and under theconditions employed in this analysis, maximum aggregate size is believed to belimited by the dissociating force acting on the aggregates.

E

50

40

30

20

10

r T

1

11111

T/

/ I

T/'

-

1 1 1

T1

i

11

i

T

i

10 20 30 60 120 180Time, min

Fig. 3. Thymocyte aggregation - appearance of a size plateau. Thymocytes wereaggregated under constant conditions (G = 75 s"1, «10 = 1-75 x io6 cm"1) for up to3 h in a continuous experiment. Samples were taken after various intervals and themean D,mx calculated (+ standard error).

Establishment of a plateau

A test was made to determine whether Dmax reaches a plateau value or not bysubjecting thymocytes to analysis at a constant shear rate and for an extended periodof time (Fig. 3). Under the conditions employed in this study (G = 75 s~\ n10 (theinitial number of single cells) = 1-75 x io6 cm"3) a plateau was established near/ = 30 min.

Calculation of FD

We next considered the 2 profiles of the shear spectrum analysis separately byplotting the maximum aggregate diameter either side of the nodal point (aboutG = 100 s"1) against the logarithm of the shear rate (Figs. 4, 5). The transformed datawere then submitted to a linear regression analysis and tested for goodness of fit. Thevalue of Dmax at G = o s"1 (see Fig. 2) does not correspond with the maximum diameterof a single cell. There are 2 possible interpretations of this observation: either a puresingle-cell suspension cannot be obtained experimentally or (because optical limi-tations restrict their resolution) random scatter results in 2 cells lying close to eachother being scored as a couplet even though they are not adhering. Even at extremely

Lymphoid cell aggregation analysis

50

40

i 30

20

10

10

Log,o

2 0

Fig. 4. Transformed aggregation profile (low shear). Lines fitted by regression analysis.For thymocytes (broken line), y = 21-53;*; + io-66 with P < 0001. For'lymphocytes(solid line), y = 15-10:*:+1154 with P < 0001.

60

f50 r

40

§ 30

20

10

2 0 3 0

Log10 shear

4 0

Fig. 5. Transformed aggregation profile (high shear). Lines fitted by regressionanalysis. For thymocytes (broken line), y — 21-oo;e +88-36 with P < 0001. Forlymphocytes (solid line), y = — i7-io« + 72-o.6 withP < o-ooi. The derivation of GcM(arrows) is explained in the text.

24 C. W. Evans and J. Proctor

high shear rates (about 5 x io3 s-1) apparent couplets remain detectable. Consequentlywe derive Gcrlt from the substitution of the minimum observed i ) m a x into the linear equa-tions determined by regression analysis, and assume that GCTlt is that shear which justprevents association of single cells into couplets. For thymocytes GCTit = 3-7 x io 3 s - 1 ;for lymphocytes GcrU = 2-4 x io 3 s-1.

Now when r |> H, equation (1) becomes

FD = 37n?r26?crltsin(2acrlt). (2)

Substituting i f = i x io~3 N s m~2, ac r i t = 300, and observed values for r (3-25 /tmfor lymphocytes, 2-9 /tm for thymocytes) and Gcnt we obtain

FD = 2-54 x io~6 N m~2/cell for thymocytesand

FD = 2-05 x io~6 N m~2/ceh* f° r lymphocytes.

Collision analysis

The rate of build up of aggregates of different cell number reflects the number ofcollisions resulting in adhesions. In Table 1 we present aggregate-frequency data froma number of experiments in which conditions other than the shear rate have been keptconstant. We now pose the following question: how do the observed frequency datacompare with that predicted by a model obtained from the theoretical treatment ofaggregation ?

Table 1. Aggregate-frequency data

Shear, s"1

1 0

5°1 0 0

5 0 0

1000

Doublets, %

2-0610-342-37+0-243-91+0-364-27 ± 1-073-16 ±0-30

Triplets, %

o-33±o-n0-27 ±0060-90 + 0 1 3

1-06 ±0-570-671015

Quadruplets, %

0-15 +006

0-09 + 0-07

049 + 010

0-30 ±0-17

0-04 ±0-03

Conditions of experiment: n10 thymocytes = 1-75 x io6 cm"3; t = 60 min. The nos. ofaggregates of different-size classes are expressed as percentages of the total particles (aggregates+ single cells) ±s.E.

Derivation of the model

Equations for particle adhesiotis. Suppose at time zero there are n10 single cells perunit volume and no aggregates, and at time t there are nx single cells, w2 doublets, n3

triplets and so on per unit volume. If there are ciinini collisions between z-tuplets and/-tuplets per unit time per unit volume, and a proportion p of all collisions result inadhesions, then we obtain the following set of differential equations governing thevariation of the (wj) with time.

drc. <°'5i> °°-77= S pci, i_j«jwi- j- S pcnnini-pcnn\ (i = 1,2, . . . ) . (3)

The first summation arises from adhesions of smaller aggregates that producez'-tuplets and the second from adhesions of z-tuplets with other aggregates. The final

Lymphoid cell aggregation analysis 25

term— pclKn\ reflects the fact that when 2 z'-tuplets adhere together then both are lostproducing a depletion — 2pcnnf in the number of j-tuplets.

For any finite time of experiment the number of aggregates over a certain size willbe negligible. If we thus neglect aggregates of larger than m particles, we can considersystem (3) as a finite system of differential equations

dn, f0;^) '"-1

— 2 Pci\nin\{~Pcnnt if i < 0*5 m) (i = 1, 2, ..., m). (4)3 = 1

o - ^ - u+Gscos 0

scos 0

u

Fig. 6. Collision path of two cells in a uniform shear field. The terms are referred toin the text.

We need to know the coefficients c{i in order to solve equation (4), and even though itstill does not yield to analytical evaluation an adequate numerical solution can bederived.

Evaluation of cn. The following derivation of c n was discovered by VonSmoluchowski (1916). Consider spherical cells of effective radius r and concentration«t in a uniform field of shear G. Suppose a cell C is moving along Ox with velocity u(Fig. 6). Cells which collide with C will be those whose path of centre is not more than2r from that of C in plane Oyz. Those whose path of centre is a distance s away ina direction making an angle 6 with Oz will have a velocity u + Gs cos 0, i.e. they willhave a velocity relative to C of Gs cos 0. Thus in the sector of plane Oyz boundedby (s, s + ds), (0, 0 + dO) where ds and AO are small there will be n-^Gs cos 0\ sAsAOcollisions per unit time involving C. Hence the total number of collisions of C withother single cells per unit time will be

r f niJ s =0 J 6 = 0

As\ cosO I AO = (5)

But there are n± cells like C per unit volume suggesting \2- Gnfr3 as the total numberof collisions. However, this counts every collision twice so the correct value is

yielding^ Gn\r3

= is Gr3 = \GV

where V = f nr3, the volume of single cell.

26 C. W. Evans andj. Proctor

Evaluation of higher c^. The problem of evaluating the collision rate between largeraggregates is complicated by the fact that the number of collisions on any particularaggregate depends not just on its size, but also on the configuration of its constituentcells and on its orientation in the shear field. Experience has suggested no strongpreference for any particular orientation or configuration during aggregation, so thefollowing model was developed to simulate aggregate formation.

A single cell was placed at the origin. A second cell selected a point of the first onwhich to adhere according to a spherical uniform distribution, i.e. all points wereequally likely to be selected. This association constitutes a random couplet, and largeraggregates can be formed in a similar manner providing subsequent attachment pointsdo not involve intersection of the preformed particle. If an intersecting point waschosen, the selection process commenced again.

Now given an z-tuplet with its centre of mass on Ox and occupying an area A whenprojected on to Oyz then, as in the derivation of cn, the number of collisions per unittime with /-tuplets will be

jJA^GzdA, (6)where z replaces s|cos 0\ in equation (5), dA expresses sdsdO, and Aj* is that areawithin which the projection of the mass centre of a/-tuplet would need to lie in orderfor any point of that/-tuplet to project on to A. Now A^* depends not only on A andj but also on the orientation and configuration of the/-tuplet in question, which arerandom variables. We have made an attempt to account for this randomness by thefollowing Monte-Carlo method for evaluating equation (6).

Construct a random «-tupIet and translate it so that its mass centre lies on Ox.Also construct a random/-tuplet and translate it so that its mass centre projects on toa point in Oyz which has been randomly selected from a uniform distribution in thesquare of side 2r(i+j) with centre at O and sides parallel to Oy and Oz. If the two willcollide when placed in a shear field, or equivalently if the projections of their cross-sectional areas intersect when projected on to Oyz, then score

(2r(i+j))\Gz

where (zr(i+j))z is the area of the square, Gz is the relative velocity of the/-tupletto the z'-tuplet where z is the z-coordinate of the/-tuplet's mass centre, and n;- is thevolume density of/-tuplets. We can now repeat with k random /-tuplets and averagetheir scores which, if k is large, should be an accurate estimate of the integral (6).The procedure can be repeated for a number of different z-tuplets and an overallestimate for cy can be obtained. Table 2 presents estimates of cit (± standard error)in which all figures are in multiples of GV/n.

Solution of the system. We now looked at a truncated system as in equation (4), takingm = 6 and considering the particular values G = o-i s"1, V = 106-48 x io~12 cm3,M10 = i-75>< io6 cm"3, p = 1. The system was solved numerically for a 30-minperiod, and the results are presented in Fig. 7. The 3 lines plotted indicate thepercentage of aggregates that were couplets, triplets, and quadruplets evolvingthrough time. The number of larger aggregates was considerably less than the numberof quadruplets justifying the truncation at m = 6.

Lytnphoid cell aggregation analysis

Table 2. Estimates of c^

27

j

i

value

value i

2

345

i

8-oo*14-86 ±00922-50 ±0222931 +0263716 ±029

14-862464*

2

±0-0910-27

35-3010-504631 10-68

3

22-50 ±035-3010

47-75*10

• 2 2

•5°•61

29

46•31•31

4

± 0±.0

• 2 6

•68

5

371610-20

GVc,j values and their standard errors are expressed as multiples of .When i = j * , the estimate should be halved since each collision is counted twice. We include

the doubled figures in the Table to maintain regularity. The quantity cn, which was calculatedanalytically, served to confirm the validity of the Monte-Carlo method, which estimated it as8-05 ±0-07.

5 r

(0"o 40)

8 2

1 *

. - - 2

' ' * ' '

* *msikZXZ

-3

10 20 30Time, min

Fig. 7. Standard model of aggregation. The 3 lines (2, 3, 4) plotted in Figs. 7-11indicate the percentage of aggregates that are couplets, triplets, and quadruplets,respectively, as each system evolves through time. In this solution, m = 6, p = 1,G = 01 s"1, V = 10648 x io~12 cm3, nw = 175 x io6 cm"1, t < 30 min. Observeddata are plotted as asterisks by fitting the percentages that are couplets.

Universality of the solution. It can readily be shown that the solution in Fig. 7applies for any values of the parameters G, V, p, n10, the only amendment neededbeing in the time axis (i.e. that the speed of the aggregation process increases inproportion to the product GVpn10).

Comparison of the solution with the data

We next compared the solution as shown in Fig. 7 with data obtained from a shearspectrum analysis of thymocyte aggregation (Table 1), where w10 = 1-75 x io6 cm~3,V = 106-48 x icr12 cm3, and t = 60 min. A cursory comparison suggests that there

28 C. W. Evans andj. Proctor

are far too many triplets and quadruplets observed relative to the number of couplets:clearly a radically different model is required.

Preference models

We now investigated preference models which would predict a higher ratio oftriplets and quadruplets to couplets. One possible way of achieving this higher ratiois obtainable by regarding a fraction/of the cells as preferred or more readily adhesivethan the others. A collision between 2 preferred cells would then result in an adhesionwith probability/)!, a preferred and an ordinary cell with probabilityp2, and 2 ordinarycells with probability/>3 where/)! ^ p2 ^ p3. The standard model takes/)! = p2 = p3

so that/has no effect on the solution. We considered the 2 following special cases tobe worthy of consideration:

preference model (i): p1 = p2 = p, p3 = opreference model (2): p1 = p, p2 = p3 = o

Each special case needs to be investigated for a number of different values of the prefer-ence fraction/. We consider these models in detail below.

Preference model (1). This model considers the experimental cell population toconsist of 2 types of cell. Adhesions between one cell type are considered to beimprobably low, whereas the other cell type has a probability p of adhering to anycell with which it collides. In order to set up a differential equation model for thesystem, we introduced the following notations:

Let xx be the number of ordinary (ord) single cells in the system at time t, x2 bethe number of preferred (pref) single cells in the system at time t, x3 be the number oford/pref couplets in the system at time t, x± be the number of pref/pref couplets inthe system at time t, x5 be the number of ord/pref/pref triplets in the system at time t,x6 be the number of ord/pref/ord triplets in the system at time t, x7 be the number ofpref/ord/pref triplets in the system at time t, x8 be the number of pref/pref/preftriplets in the system at time t, x9 be the number of quadruplets in the system attime t.

Now consider a random triplet with its centre of mass on Ox and observe thecollisions it will have with single cells. Averaged over all random triplets, a fraction Aof these will be collisions with each of the end cells and a fraction 1-2A with thecentre cell of the triplet. Although we could estimate A by Monte-Carlo methods asbefore, as a matter of expediency we chose 0-4 as the value of A since it would seemthat A should exceed one third, i.e. that the end cells should receive more collisionsthan the centre one. We thus obtained the following equation system for this model ofaggregation as far as quadruplets:

\GVi) ~cE = ~^11*1*2~C12#1(0-5 x3 + x4)-c13«1((i - A)x& + (1 - 2A)x6 + 2Ax7 + x8),

I I ~T7~ =—'2

(

Lymphoid cell aggregation analysis 29

I I ~fa — ~C12Xi(Xl + X2)—'- + CnX.2t

77 = 0-5^2*2*3-

I "dl2+ 2c22(x3tf4 + 0-5 xj + f *

2

4

to(fi

roo5 3ro

"o0)

| 2

u>a>oO)

2c iCDO(D

0 -

-

s*////*

.---

— — **"*""_ « - • • - — • • • • —— ™ — • * " "

. . - 3

, - - - " " * 4

20I

40 60Time, mm

Fig. 8. Preference model of aggregation (i). / = io%, G = 05 s"1, t < 60 min,other conditions as before.

These equations can be solved numerically, starting from the boundary condition^IC0) = ^IOC1 ~/)> ^2(0) = niof- Solutions were obtained for/ = 10, 15, 20% and areplotted in Figs. 8-10. Examination suggests that the 15 % preference fraction presentsthe best fit, although the model is not entirely satisfactory. We extrapolated the 15%preference model as shown in Fig. 11, and matched the data from shears 10, 50, ioo,500, 1000 s"""1 to the solution. We can see that the model predicts the aggregatefrequency with reasonable accuracy even at these higher shear rates.

Now we can estimate ̂ >exP» the probability that a collision between a preferred celland another produces an adhesion by regarding the data in Fig. 11 as being matched

3 c u 33

30 C.W. Evans andjf. Proctor

at the points shown, and then (since the speed of the aggregation process is pro-portional to GVpn10) by equating

(GVpn10)moiel x tmo6el = (GVpnlo)esp x 3600 s.

TOTO

S i3

4

20 40I

60Time, min

Fig. 9. Preference model (1). / = 15 %, G = 0-2 s"1, t < 60 min, other conditionsas before.

. . - * " "

*

%..*

20I

40I

60Time, min

Fig. 10. Preference model (1). / = 20 %, G = 0-2 s-1, t < 60 min, other conditionsas before.

Lymphoid cell aggregation analysis 3

Table 3. Estimates of the probability of a collision between a preferred cell and anyother producing an adhesion

95°11002300

2700

1630

1 0

SO1 0 0

5 0 0

1000

0010560-002440002 56000060

ocoo 18

4 -

3 -

2 -

*/

0/

y'

X/

_..£.-*- *- * •

1 1 (

— - " " " " 3

4

20 30Time, min

40 50 60

Fig. 11. Preference model (1) (Extrapolated). / = 15%, G = 0-4 s"1, t < 60 min,other conditions as before.

That is, the model reaches in time <modei the state the experiment reaches in an hour.If we now substitute the values presented in Fig. 11 (Gmo(lel = 0-4; pmo&c\ = 1;

'model exp, ^lOniodel = nlOexp ) then.

"cxp —"'model,36OO '

°"4

These results for the 15% preference model are shown in Table 3. Interestingly, weshow in Fig. 12 that regression analysis of log Pe x p against log G suggests a linearrelationship (P < o-oi) between these two parameters.

Preference model (2). This model conceives a fraction/of the population behavingas in the standard model, and a fraction 1-/behaving as complete dummies and remain-ing as single cells. In analysis by comparison with experimental data, the number ofsingle cells observed clearly presents no information about the fit of the model. Wehave thus looked at the frequency ratios as the solution evolves through time (Fig. 13).Examination suggests that the fit is not very good, and we therefore abandoned thismodel in favour of preference model (1).

3-2

C. W. Evans and J. Proctor

1 2 3Log,0 G

Fig. 12. Shear-related changes in the estimated probabilities of collisions betweena preferred cell and any other producing an adhesion. A line of the form logi0 Pexp =— 0-8274 log10 G—1-1087 w a s fitted by regression analysis with P < 001.

v

^(0-437, 0-577) asymptotically428, 0-565) at fn,0 GVt=0-6T\

(0-406, 0-536) at fny0 6l/f=0-447

0-417) Gl/f=0-224

#3/#2

Fig. 13. Preference model (2). The ratios of triplets:couplets and quadruplets: tripletshave been plotted as the solution evolves through time. Observed data plotted asasterisks, calculated data as stars.

Lymphoid cell aggregation analysis 33

DISCUSSION

We have presented experimental evidence establishing the existence of a sizeplateau during the aggregation of murine thymocytes. This plateau reflects the factthat at some optimum shear rate the maximum aggregate diameter is limited by theforces of dissociation which act to prevent the adhesion of further cells to the optimum-sized aggregate. The existence of such a plateau is not evidence for the presence ofan equilibrium, i.e. we can say nothing about the relationship between the rate atwhich cells are being added to aggregates and the rate at which cells or smalleraggregates are being returned to the medium by aggregate break up. It is commonlyassumed that such an equilibrium exists, but in the light of our present data we nowprefer to consider supra-optimal aggregates as failing to form rather than as beingformed and subsequently undergoing break up. In an ideal system limited only byshear we envisage the ultimate formation of one or more aggregates of similar size.In a system where such a situation is never achieved, we consider a fraction of thecells to be of such low adhesiveness that mutual collisions never form adhesions.Such a system would manifest itself as a number of aggregates of similar JDmax ina suspension of single cells. Our present data suggest that thymocytes may aggregatein this manner. Analysis of our shear-spectrum studies presents us with the twofollowing possible mechanisms of measuring adhesiveness:

(1) Aggregate size. The rates at which aggregates build up or single cells are removedfrom suspension are not necessarily dependent upon the strength of adhesion.However, at any given shear after plateau formation Dmax will be limited by theforces tending to resist the formation of larger-than-optimum aggregates. Thus inany 2 comparable systems which have reached plateau size, the cell type whichdisplays the larger aggregate size is considered by us to have greater adhesive forcesresisting separation. From Fig. 2, for example, we see that thymocytes are moreadhesive than lymphocytes beyond the plateau point (equivalent to about G = 100 S"1

in this instance). The observations of others (Curtis & De Sousa, 1973, 1975)using different techniques support this conclusion.

(2) Force of dissociation. The force opposing dissociation of any 2 cells is related tothe force of adhesion between them, but as we remarked earlier the actual value of theadhesive force obtained from dissociation measurements will vary with the mode ofseparation. The contribution of particular modes of separation to cellular studies ofthe sort undertaken here remains uncertain, and thus although a quantitative valuecan be obtained for FD its precise relationship with FA cannot be deduced. Clearly,however, such values at least provide a minimal measure of forces resisting separation.In the following discussion we assume the contribution of each mode of separationto be similar in the studies of each cell type. The values obtained for FD for thymo-cytes and lymphocytes are again qualitatively similar to results obtained from separatestudies where adhesiveness was measured by the collision-efficiency method (Curtis& De Sousa, 1973, 1975). Direct measurement of FQ has been made by other workers.McKeever (1974), for example, using the micromanipulation method of Coman (1944)to detach adhered cells found values of FD = 1-42 x io~3 N m~2/cell for the adhesion

34 C. W. Evans andj. Proctor

of rabbit alveolar macrophages to glass. Interestingly, McKeever (1974) noted vari-ability in the FB of individual macrophages but could offer no evidence as to whetherthis difference was attributable to cellular or technical variation. Brooks et al. (1967)reported figures of about r j x i o " 4 dynes (1-5 x io~5 N m-2)/cell for mouse liveraggregates, and derived figures for red blood cell adhesive forces of between io~6 andio~8 dynes (io~7 and icr9 N m~2)/cell depending on the data source. These figures,although admittedly from different systems, nevertheless display the expectedqualitative relationship in the observed adhesiveness of macrophages, erythrocytes,lymphocytes and thymocytes. The observation that lymphocytes and macrophagesare more adhesive than erythrocytes may possibly contribute to the fact that the former2 cell types actively adhere to vascular endothelium and migrate into inflammatorysites, whereas erythrocytes are known to migrate only passively.

The models we have obtained reflect our observations during aggregate formationunder known, constant conditions. The standard model represents a statisticalevaluation of the collision frequency of suspended, single cells leading to aggregation.The poor fit of our experimental data with the standard model raised the possibilitythat all the suspended cells cannot be treated as particles of equal adhesiveness. Thisconclusion is not surprising, since variations in adhesion within supposedly homo-geneous cell populations have been reported before (McKeever, 1974; Evans &Davies, 1977). There are many cell properties which might contribute to unequaladhesiveness. These include variations in cell diameter and cell projections, mem-brane perturbations resulting from altered chemical content (such changes certainlyoccur during lymphoblast formation; see Resch, 1976), and changes in charge dis-tribution possibly arising as a consequence of membrane fluidity. We cannot takeinto account the entire adhesive spectrum of any cell population but, after makingcertain basic assumptions, it is possible nevertheless to derive models of collisionfrequency which approximate the observed data. Of the preference models wederive, one including a 15% preference fraction offered the best fit with the data. Inother words, about 15% of the entire thymocyte cell population are ever likely toform active adhesions with other thymocytes. The biological consequences of thisobservation can only be alluded to, but we would envisage a small population of cellsresident in the thymus which display strong mutual adhesion. Such a population maybe represented in part by the persistent phytohaemagglutinin-responsive cells whichare thought to be long-term residents in the thymus (Elliot, 1973), or it may merelyreflect changes in adhesion as thymocytes proceed through maturational steps withinthe thymus (reviewed in Greaves, Owen & Raff, 1973). Interestingly, some 10-20%of thymocytes (the large and medium-sized cells) are actively undergoing mitosis.This raises the possibility that the cell cycle may be involved in adhesion, a theorycurrently under examination in our laboratory.

The derivation of Pexp (the probability of a collision producing an adhesion) is ofinterest as a means of obtaining further insight into collision analysis during aggre-gation. Thus Table 3 indicates that under optimum conditions for thymocyte aggre-g a t i o n ^ , ^ = 100 s~\ t = 60 min) less than 3 in 1000 collisions produce an adhesion.As expected, the number of collisions producing adhesions decreases with increasing

Lymphoid cell aggregation analysis 35

shear rate. This observation supports our assumption that weaker adhesions areunable to form at supra-optimal shear rates.

We have expanded on our models in some detail because an understanding of thecollision processes occurring during cell aggregation is fundamental to the study ofcell adhesion. Our conclusion that not all cells within a supposedly 'homogeneous'population are of similar adhesiveness, and our derivation of models which accountfor this observation, present the possibility of a more detailed analysis of the cellaggregation technique. We envisage our approach as being of direct relevance in thestudy of cancer, for example, where changes in the adhesiveness of an initially smallsubpopulation of cells may ultimately lead to widespread metastasis.

This study was supported by a grant from the Wellcome Trust, C. W. E. was funded bythe Medical Research Council.

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(Received 27 January 1978)