T H E KINETICS O F AGGLUTINATION OF RED BLOOD CELL SUSPENSIONS* BY JEAN OLIVER A S D PEARL SMITH
In a study of the action of electrolytes on the electric charges and stability of red blood cell suspensions Oliver and Barnardl have shown that a t a certain concentration of trivalent salts agglutination of the cells proceeds at a maximum speed. It was also observed that at this concentration the electric charge of the cells was completely abolished, whereas in other concentrations
where t,he charge had been only depressed below the critical potential, agglutination, although it occurred, was much slower (Fig. I). It was therefore suggested that the conditions were analogous to those described by Zsigmondy in the coagulation of colloidal solutions by electrolytes. In his experiments with colloidal gold Zsigmondy2 found that the time of color change a t first decreased rapidly and then became constant as the con*From the Department of Pathology of the Medical School of Stanford University, 8an Francisco, California. J. Gen. Physiol. 7, 99 (1924). Z. physik. Chem. 92, 600 (1917).
JEAN OLIVER AND PEARL SMITH
2
centration of the coagulating electrolyte was increased and this range of concentration in which the color change was constant he called the zone of “rapid” coagulation. By comparing his measurements with those of Galecki’ on the cataphoresis of similar particles he assumed that this zone of rapid coagulation lay in the immediate neighborhood of the isoelectric point, and was the result of the complete neutralization of the charge of the particles. These experiments of Zsigmondy formed the starting point for the development of a theory of coagulation. He assum.ed that in the rapid coagulation of spherical particles of equal size, any particle which comes within a certain distance of another particle becomes permanently fixed to it since it possesses no electric charge to cause repulsion. This distance he defined as the radius of the attraction sphere of the particle. The mathematical treatment of the problem was developed by v. Smoluchowski2who calculated formulae for the number of single, double, treble and larger aggregates in a suspension from ( I ) the original number of particles (Vo), (2) the radius of the attraction sphere (R), and (3) the velocity constant of Brownian movement (D) . Von Smoluchowski first calculated the probability that any particle will enter the attraction sphere of a given particle by applying the equation for Brownian movement and also the probability that within the time T no particle will enter such an attraction sphere. From these probabilities he calculated the decrease in number of the single particles. By making corrections for the Brownian movements of the particles around which coagulation is supposed to take place, and from the fact that the aggregates already formed act as coagulating nuclei he finally arrived a t the following system of equations : VI
+ vz + + . . . . . . . . 2, = + t/T V O
v3
I
VK =
V, (t/T)K-l (I t/T)K+l
+
(1)
(4)
where 2 , = total number of particles; V1 = number of single particles; VZ = number of double particles; T = 1/4rDRVO Von Smoluchowski’stheory has been verified by a number of experiments. Zsigmondp first tested the equation V1 = Vo / ( I t/T)2 by measuring the decrease in the number of single particles in a gold sol. He was able with the ultramicroscope t o distinguish between the single particles and aggregates
+
Z. anorg. Chem. 74, 199 (1912). Z. physik. Chem. 92, 129 (1917). * Z. physik. Chem. 92,600 (1917). 2
AGGLUTINATION O F RED BLOOD CELL SUSPENSIONS
3
because the former emit green and the latter yellow light. By substituting the actual values in the equation he obtained a value for I/T which was fairly constant for all but ,the shorter periods of time. With the average value for I/T he calculated the theorectical values of VI for the various time intervals, and found that they agreed approximately with the experimental figures. With the same data, he obtained values between 2 . 0 and 3.0 for the ratio between the radius of the attraction sphere and the radius of the particle by applying the formula T = 1/4nDRV0 in which the value of D was substituted, giving R/r = N 37/H82V0 r/T,* Westgren and Reitstotter‘ studied the change in the total number of particles in gold sols and found that their results were in accord with the equation Zy = V0/(r t / T ) , where H = 8.31 X 107, gas constant; N = 6.06 X 1 0 ~ ~Avogadro’s ; constant; 7 = oog, viscosity of medium; 8 = absolute temperature Luers2 has found v. Smoluchowski’s formulae applicable to the change of color of Congo red hydrosols when mixed with an electrolyte. Somewhat divergent results were obtained by Kruyt and van ArkeP in their experiments with selenium hydrosols. In several of their experiments, the total number of particles was found to decrease in a way not consistent with v. Smoluchowski’s formula. When an agreement with the theory did occur, R/r varied from 0.5 to 1.0. The formula 2” = Vo/ ( I t/T) was found by Ehringhaus and Wintgen4 to apply in experiments with a very different sort of material, In a flux of borax and gold chloride, the particles of gold were found to decrease according to the formula when heated a t a temperature of 92 j C. for I to 500 minutes. In all but one of their series the average value of R/r was 2.39; in one series it was 13.7 In the present investigation an attempt was made to apply v. Smoluchowski’s formulae to the “rapid agglutination” of red blood corpuscles. 3
+
+
Methods In all of our experiments a 1.5 per cent. suspension of red blood cells in 8.5 per cent. sucrose solution was used. In the preparation of the cells fresh defibrinated rabbit blood was washed three times with 0.9 per cent. NaCl solution and three times with 8.5 per cent. sucrose solution. To highly charge and stabilize the suspension sufficient KaOH was added t o made a M/5oo solution of the hydroxide, and enough A12CI8 was then added to this mixture to make the concentration of the electrolyte approximately M/4000. This concentration previous experimentation had shown brings the cells of such a suspension to the isoelectric point and produces “rapid agglutination”. (Fig. I). I Z . physik. Chem. 92, 750 (1917). 2 Kolloid-Z., 27, 123 (1920). aRev. Trav. chim., 39, 656; 40,916 (1920). 4 Z. physik. Chem. 104, 301 (1923).
JEAN OLIVER AND PEARL SMITH
4
At varying intervals the process of agglutination was interrupted by pipetting 2 ccm. of the mixture into a test tube containing 1.0 cm. of a 1.0 per cent. gelatin solution. The progress of agglutination was followed by making counts of single cells or aggregates of cells from each tube. In each case a t least three counts were made on the undiluted suspension, using an ordinary blood counting chamber.
SERIESI I1
Table I-Exp. Secs. VI
o 30 60 90
x
@
10-6
VI
Obs
x
10-8
Calc
60.00 3.15 1.30
-
-
,1131 ,0973
3.42
.so
.1115
.53
Av.
r
Secs. 0
30 60 90
1.10
598.6
.1071
120
I11
Table 11-Exp. -
0
45.15
30 60 90
5.50
.0618 6 . 7 8
2.55
,0534 2 . 6 0 .0470 1 . 3 7
120
1.50
120
1.65 .90
,0506
90
180
.40
,0527
.40
240
.25
.0605
.25
.os27
Table 111-Exp. 0
37.40
30
12.25
60 90 120
I50
7.45 6.60 4.35 2.45
X 10-6 Obs. 33.20
VI
@
VI
-
X
10-6
Calc. -
r
2.85(.08041) 1 . 6 0 .os925 .OS291
1.89 .99 .75 .o4711 .61 Av. .Os309
1.00
Table V-Exp.
-
o 30 60 90
Av ,
XI1
Table IV-Exp.
46.40
1.60
-
542.5
XI -
7.45 .0498 8.52 .0518 3 . 4 5
2.75
2.35
,0382 .0365
1.85 1.16
. 7 5 .0458 .77 Av. ,0445
325.3
396.0
XI11
-
-
(.0249)
-
.ozo7 8.71
.0153 5 . 4 9 ,0165 3 . 7 8 ,0194 2.76 Av. ,0179 163 . o
The Experiments I n Series I single cells only were counted. Fig. 2 shows the manner in which the number of single cells decreases with time; the logarithms of the number of cells per ccm. are used as ordinates and time in seconds as abscissae. I n the Tables I-V the constant I/T or p was calculated from the formula ,6 =
(vg
I/T
- I) .
The calculated values for the number of single
+
cells were obtained by applying the formula VI = VJ(I pt)z, In each case the p used is the average for the experiment with the exclusion of those
AGGLUTINATION O F RED BLOOD CELL SUSPENSIONS
5
enclosed by brackets. These are omitted as obviously too large, since it is impossible to stop the progress of agglutination promptly at this point. The same difficulty was observed by Zsigmondy and others who worked with colloidal solutions. The p for each experiment is fairly constant and the calculated numbers of single cells agree approximately with the actual counts.
60
nM€ IN
It0
S€CONDS
FIG.3
FIG. 2
SERIESI1 Table VI-Exp. IX Sees. 0
30 60 90 I20
zv
x
IO”.
Obs.
22.60 4.70 3.80 2.65 2.60
P
I20 150
.90
30 60 90
x
10-5
Calc.
,0826 .0838 .0627 Av. .0769
r
4.03 2.86 2.21
1152.0
X
-
-
(. 803) .631 .68r
2.17
.55I
1.11
,630
.g9
Av. , 6 2 3
R -
-
( . 1273)
Table VII-Exp. 84. IO 3.35 2.45 1.35 1.25
0
zv
-
1.47
2513.0
i80
6
JEAN OLIVER AND PEARL SMITH
Table VIII-Exp. 29.10 4.50
0
30 60 90
VI
-
-
.1820
5.24 2.88
I20
1.50
.2040 .1670 . I530
150
1.70
. I073
I80
1.60
2.20
1.85
2.03 1.52
1.23 1.03
,0955 Av. .1516
1832. o
In Series I1 the total number of cells anc, aggregates was countec. and similar calculations made with the formulae
p
=
I/t
V,/Z, -
and
I
+
2 , = V,/I Pt Tables VI-VI11 and the upper curve in Fig. 3 illustrate experiments of this type. Agreement between the theoretical and observed values is not as close as observed in Series I, but is approximate.
SERIESI11 Table IX-Exp.
90
e30
120
.20
.oo
30 1.60 .70 60 4 0 90 120 .40 150 . 2 0 Av.
30 60 go I20
150
r 0
30 60 90
0.0
I20
VZ X
10-6
Obs. .oo
.85 .60 .35 .35
-
.I27
2.20
.IIO
.82 .42
.ogo .077 .092 .Og9
0
30
.26
.I7
723.8
X -
(.44 .253 .31 . ~ j .246 . ~ .IO .228 .og ,265 .06 .os Av. .248
p
0.45 0.30
j
1001.0
.oo
VZ X
(.120)
.076
5 r
.62 .33
.072
.074 Av. -074
-
10-6
Calc.
-
Table XIII-Exp.
XI
-
Table XI-Exp. 0
Secs.
1032 . o
.IOI
Table X-Exp. 0
_R
.099 .294 .og4 .178
Av.
IX
Table XII-Exp.
XI1
Secs. VZ X 10% p VZ X 10Obs. Calc. - 0 .oo 30 .80 (.162) 60 . s o .I10 ,571
.21
1108.0
XI11
(.095) -
1.90 60 1.30 .063 1.64 go 1.00 .93 120 .65 .os0 .60 150 .50 ,047 .42 Av. .os3 478.9
7
AGGLUTINATION O F RED BLOOD CELL SUSPENSIONS
TABLE XIV Valves of /3 obtained in same experiment by different methods. Exp.
VI11 IX X XI XI1 XI11
From VI R/r
From Z V R/r
P ,046 .0769 .623
P ,0269
345.2 1152.0 2513.0
206.3
-
I
-
,0445 .053I .0179
325.3 542.5 163.0
From Vz
P
R/r
.074 .246 ,099
1108.0
-
.IOI
.0528
1001.0
723.8 1032.0 478.9
+ FIG.
4
I n Series 111, two-celled aggregates were counted and calculations of the values of /3 and of VZmade by the use of the equation,
vz
=
v, @ / ( I
+ PtY.
The data obtained are shown in Tables IX-XIII, and the type of curve is illustrated by the lower curve in Fig. 3. I n Series IV two different kinds of counts were made from each tube to see if the values for /3 would be tjhe same for the two counts. There is considerable variation in the /3 of some of the experiments as is shown by Table XIV. I n others the agreement is as satisfactory as can be expected in such a rigorous test of the theory. Two methods have been devised to give a graphic comparison between the theoretical and experimental values. Von Smoluchowski constructed curves showing the theoretical progress of coagulation based on equations (I) to (4) using v/vo as ordinates and t/T as abscissae. Such a set of theoretical curves, t,ogether with the experimental data is shown in Fig. 4.
8
JEAN OLIVER AND PEARL SMITH
The second method in which the theoretical values are represented by straight lines was used by Ehringhaus and Wintgenl. By the rearrangement
of the equation Zv
=
V O ___
I
+ t/T
is obtained and since I/V,
the equation
=
I/&
+
I/V,
I /T 4-is constant for a given experiment, plotting
VO
against t should give a straight line. The results are shown in Figs. 5-7.
I/&
$k fxp. m
L I
I V
I
r
I
30
I
I
60
90
1 lZC
Tine m Seconds Fro. 6
t
I
30
I 90
I
60
I
I20 30
pine in Seconds FIG.
5
6
=
I
+( 5 ) t
Tim? m Seconds
FIG.
Similarly the equation VI = and
90
BO
/ZO
0
7
V O
E (I
+ t/T)2was rearranged, giving .
,
I
plotted against t. The results are illus-
trated bv - Figs. - 8-10. Considering- the difficulties of the experiment the agreements between the theoretical and observed values are quite satisfactory.
Discussion Certain differences between the mechanism of chemical reactions and of coagulation have been pointed out by v. Smoluchowski. In chemical union 1
Z. physik. Chem. 104, 301 (1923).
AGGLUTINATION O F RED BLOOD CELL SUSPENSIONS
9
only a definite number of atoms or atom groups unite to form a molecule while in coagulation successive complexes are formed whose size has theoretically no limit. In Figures 11 and 2 the usual kinetic formulae of chemical reactions are applied to the observed rate of agglutinations as expressed by the decrease in the number of single particles, the concentration V1 being plotted as log VI (Fig. 2 ) ) and l/V1 and 1/Vl2 (Fig. 11). It is seen that the process of agglutination does not follow the course of a mono-, bi- or trimolecular reaction. The experimental data on the rate of rapid agglutination of red blood cells are therefore in fairly good agreement with the reported results with colloids and with v. Smoluchowski’s theory. But when we come to the calculation of R/r i.e.,the ratio of the radius of the cell to its sphere of attraction, a striking difference appears. mhereas the largest values
I
Time in Seconds FIG.8
30
Emme
I
60 90 in 5econds
120
FIG. 9
heretofore obtained is 13.7 (Ehringhaus and Wintgen), our values range from 142.3 to 2 513, the average of allexperimentsbeing6 53.0. In theequationR/r= B the quantity 2/3 H8/Nv is practically a constant for ordinary temperatures, so that the value of R/r depends almost wholly on the ratio P/V,. In all of our experiments the value of IT, is small as compared to the concentrations of colloids used by other investigators. Our values for P, on the contrary, do not, except in a few instances differ markedly from those obtained by others. This follows from the fact that the absolute rate of the agglutination of the weak cell suspension is almost as rapid as that of the coagulation of the concentrated colloidal solution. I n order for agglutination of the cells to proceed as rapidly as coagulation of the particles there are at least two differences in the suspensions which must be compensated for if their units are to come in contact and form aggregates at equal rates. I n the first place, the red blood cells are farther apart. Comparing one of Westgren and Reitstotter’s experiments with our experiment IX, their particles are 7.9 X IO-^ cm. apart whereas the red blood cells are 35.4 x IO-^ cm. apart. This gives a ratio,
JEAN OLIVER AND PEARL SMITH
IO
I
30
60
90
/BO
I50
120
h e in Seconds
I
Z/O
i!%
FIG.I O
Distance between cells = 4.4 Distance between particles The second factor is that the Brownian movement of red blood cells is relatively slight. From the formula Az = RT/N 1/3wqr . t, where A = average distance; R = 8.31 X 10’ (gas constant); N = 6.06 X 1oZ3(Avogadro’s constant); T = absolute temperature; t = time in seconds; q = coefficient of viscosity; r = radius of particle. The Brownian movement of the gold particles in the experiment cited above which have a radius of 96pp is 19.2 X
30-
20
-
10
I
I
I
120
150
BO
Time in Seconds
FIG.11
AGGLCTINATION O F RED BLOOD CELL SUSPENSIONS
I1
IO-^ cm. per second. The Brownian movement of the red blood cells used, which have a radius of 2 6 0 0 p p is theoretically 4.05 x 10-5 cm. per cc. This gives a ratio Brownian movement of cells = 0.21 Brownian movement of particles and the ratio Distance ratio = 20.95 Brownian movement ratio In actual fact this ratio of discrepancy between the two types of suspensions is certainly much larger, as the red cells have even less Brownian movement b
ExpZ
v
$=598.6
I
60
I
120
I
180
1
240
I
3CQ
I
36c
T , ~ Pin Seconds F I G . 12
than the theory would indicate. Practically speaking they have none when one considers the distance which separates them in our suspensions. It is therefore evident that some factor must operate to enable the red blood cells to form, as they do, aggregates as rapidly as colloidal particles. This unknown factor may well be included in what is considered a larger attration sphere. This necessity for a large attraction sphere is shown graphically in Fig. 1 2 , where the upper curve illustrates what the course of agglutination in experiment I1 would be if R/r = 2.0, and the lower curve illustrates the actual data where R/r = 598.6. In other words the maximum range of effective attraction between the cells in this experiment was 0.15 cm! We have at present no explanation to offer for this remarkable difference between the cell suspensions and the colloidal solutions in regard to the “radii of attraction” of their particles. The cells and particles themselves differ in at least two ways. The cells are relatively enormous, their volume being approximately 50,000 times that of the gold particles. They also bear a different relation to the surrounding medium than does the gold particle as they are permeable to water and certain ions. Mass effects and surface energies may therefore differ considerably in the two cases and these may account for the relatively large “sphere of attraction” that is observed in the rapid agghtination of the red cells. An investigation of these problems is in progres.
