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Rate process in the final stage of complement hemolysis
lmmunochemistry, 1977, Vol. 14, pp. 421~428. Pergamon Press. Printed in Great Britain
RATE PROCESSES IN THE FINAL STAGE OF COMPLEMENT HEMOLYSISt C O N A N K. N. LI~ and R, P. L E V I N E The Biological Laboratories, Harvard University, Cambridge, MA 02138, U.S.A.
(Received 9 November 1976) A b s t r a c t - - W e define S* as the complex resulting just after the final complement components (C5-C9)
have adsorbed to the membrane at one site. An erythrocyte bearing one or more S*'s is called an E*,
The kinetics of lysis of E*'s made from sheep red cells and human complement is examined. The temperature dependence of the initial rate of lysis gives an Arrhenius plot showing an activation energy of 13.7 kcal/mole for lysis. This rules out free diffusion of cellular components, which has a derived activation energy of 4.6 kcal/mole, as the rate-limiting step in the lytic process. The maximum extent of E* lysis is also found to vary with temperature. This observation could be explained by a scheme which postulates that the final lytic site, S*, becomes inactive with time. A dependence of the extent of E* lysis on the volume of fluid surrounding the cells implies that this inactivation could be caused by desorption of some component of S* from the membrane. By assuming that the distribution of S*'s on the complement-treated cells is initially described by a Poisson distribution function, an expression for the time dependence of lysis is derived in terms of the rate constants of lysis and desorption and the initial average number of S*'s per cell. This expression fits the data very well and yields an activation energy for desorption of less than 4 kcal/mole. On the basis of the observations, speculations are made on the nature of the final control point in complement lysis. Associates, Bethesda, MD, U.S.A.) was dissolved in 5 ml distilled water and stored in 0.l-ml aliquots at -80°C. Isotonic veronal-buffered glucose with gelatin, Ca 2 + and Mg z+ (DGVB z+) was prepared as in Rapp and Borsos (1970). DGVB 2- refers to the same buffer without divalent cations. Ethylene diamine tetraacetic acid (EDTA), 0.09 M, pH 7.4, was prepared containing 0.1~ (w/v) gelatin.
INTRODUCTION
The end result of complement treatment of sensitized cells is the formation of a site on the m e m b r a n e capable of lysing the cell. In view of the physiological importance of the lytic process, it is unfortunate that there have been few attempts to clarify the site's mode of action. At present, while it is difficult to devise meaningful experiments to deduce the lytic mechanism on the molecular level, the kinetics of lysis can be measured with ease. In this context, we examined the dependence of lysis kinetics on temperature, hoping that the kinetic behavior would permit postulation of the final events in lysis while the observed activation energy of lysis would narrow down the possible rate limiting steps in the lyric process.
Preparation of sensitized erythrocytes (EA's) Sheep. erythrocytes were washed by suspending in DGVB 2- at about 1 x 10s cells/ml and centrifuging at 1500g for 8 rain. The supernatant was discarded, mad the pellet washed twice more. The cells were then suspended at 1 x 109/ml in DGVB 2-, mixed wiih an equal volume of diluted hemolysin, and incubated 15 min at 37c'C. EA's so prepared were kept at 0-2°C and used within two days. (The extent of hemolysin dilution varied from batch to batch. Hemolysin was diluted in DGVB 2- to the weakest concentration that yielded 100% lysis when EA's prepared as above were incubated with an equal volume of excess complement for 15 hr at 10°C.)
EXPERIMENTAL
Reagents and solutions Sheep erythrocytes were drawn from the same animal for all experiments. The cells were stored in Alsever's solution at 4°C and used between 5 days and 20 days after drawing. Human serum was used as the source of complement. Whole blood was drawn from a volunteer and clotted for 45rain at room temperature, then 20hr at 6°C. The clot was then removed and the serum centrifuged at 17,000 g for 5 rain. The clear supernatant was removed and centrifuged again; the second supernatant was divided into 0.5-ml aliquots and stored at -80°C. For hemolysin, rabbit anti-sheep erythrocyte stromata serum (Microbiological t This work was supported by grants from the National Science Foundation (BMS 75-09919) and the National Institutes of Health (AI 12885). :~ Pre-doctoral Trainee supported by NIH Training grant No. 5T01GM00782.
Determination of the optimal time for E* formation E*'s were prepared by mixing EA's, complement, and DGVB ~+ in the volume ratio 8:1:7 followed by shaking at 10°C. At 30-sec intervals after mixing, 0.l ml of the reaction was sampled into 2 ml 0.09 M EDTA at 0c'C. These tubes were centrifuged for 15rain at 1500g in the cold, then the supernatants were carefully poured off and the tubes wiped. The pellets were resuspended in 0.5ml DGVB z- and incubated at 35°C. After 7 hr the per cent lysis in each tube was determined: an aliquot of the sample was delivered into 20 vol of 0.09 M EDTA and centrifuged at 1500g for 15 min. The absorbance of the supernatant was read at 414nm on a Zeiss Spectrophotometer PM2 DL, and the A,~14 of an aliquot mixed in 20vol of 0.1% NazCO 3 was taken as 100% lysis. The per cent lysis of the E* samples as a function of the time of reaction at 10°C is shown in Fig. I. The optimal time for E* formation
421
422
('()NAN K. N. L1 and R. P. LEVINE
,oo1
~ ~ ,
~o-~60
0
I
2
I
4
i'
6 rain
t,
I
8
12
is that at which lysis of the E*'s just reached 1000,,. At shorter times the E* formation is submaximal; at longer times the E*'s become coated too heavily with complement and the effect of desorptive inactivation of the sites would be unnoticeable compared to lysis (see Discussion). Therefore, an incubation of 10 min at [0~C was used for forming E*'s.
Kitterics 0[ E* lysis Short term. E* formation was stopped by mixing the whole reaction in 20 vol of cold 0.9 M EDTA. The cells were washed three times in 0.09 M EDTA (3000 0, 10 rain centrifugations); just before the last centrifugation the E* suspension was divided into six equal parts. Each of the six final pellets was suspended (at about 1 x 10~ cells/ml) in DGVB z- which had been pre-equilibrated at one of the six temperatures (10.0, 14.3', 19.7, 23.9', 28.4. and 31.3 C). The tubes were kept in a shaking bath at the respective temperatures. For 48 rain after the pellets were suspended, an aliquot was drawn every 4 rain from each tube and delivered immediately into 20vol of 0.09 M EDTA at 0°C. The samples were centrifuged at 1500.q for 15 rain in the cold and the A , ~ of the supernatants taken. Upon dividing by the A,u4 of an aliquot delivered into 20 vo} of 0.1°;', Na2COs, the per cent lysis of the samples were obtained. After shaking for 6 hr at their respective temperatures, the tubes were sampled again for per cent f
O
,
O
O
C 14.3
5
×~X,~,"
-
,/J9
x ~ X~
7
t
23.9
8° ~;I
/×i"
y~"
U 101 sl .~.,.xl'"'
J
/
~s!
Six /31
?
20
_
/ x 20
/
t-,
J 36
I
300
40O
rnin
lysis: this percentage was taken as the maximum E* lysis since there was no further lysis after 6 h r (see Fig, 3). The short term kinetics of E* lysis at six temperatures is shown in Fig. 2. (Here, A4z,, is plotted,) The initial rate of I\sis as well as the maximum % lysis (at 6 hr) varied ~ith temperature. For the six temperatures in increasing order the maximum fractions lysed were 0,30, &39, 0.43, 0.53, 0.60 and 0,67, respectively. Long term. In an experiment similar to the above, aliquots were drawn at various times up to 347rain. The kinetics of E* lysis over a longer time range at two temperatures is shown in Fig. 3. We have only reported our studies of the terminal reaction in DGVB 2- However, similar results were obtained for the reaction in GVB 2- (see Rapp & Borsos, 1970, for buffer description).
Volume dependence o[ E* lysis E* tk~rmation was stopped by mixing the reaction into 20vol of 0.09 M EDTA at 0°C, The E*'s (about 2.5 × 10~,iml) were washed in 0.09 M EDTA twice, suspended m cold DGVB-'-, divided into 8 equal parts, and pelleted at 30000 for 2rain. After removing the supernatants, four of the pellets were suspended separately in 0.05 ml DGVB 2-, the other four in 8 ml DGVB 2-. The eight tubes were shaken at 10°C for 12hr; then the first four tubes were brought to 8 ml with 0.09 M EDTA and the o lysis in the eight samples determined. There was a definite if slight dependence of maximum E* lysis upon \,plume of buffer added to the E* pellets (see Table 1 for results) In another experiment, similar to the above, the E*'s ~erc divided into 4 equal parts instead of 8. To the final Table I. Dependence of maximum lysis on volume ol buffer
Mean __+ S.D.
x { 24
2O0
Fig. 3. Kinetics of E* lysis. The curves are theoretical fits [see equation (12) of text].
x/x/
x I 12
I
tO0
Maximum fraction lysed Group A Group B
x
15
I
t,
/'/"
~o511 IO
g
7a
Fig. 1. Kinetics of E* formation. EA's and complement were shaken in a 1 0 C bath and sampled at 30-sec intervals after mixing. The samples were washed of complement and allowed to lyse at 35"C for 7 hr to determine the per cent E* formation.
~
4o
I
IO
t 48
rain
Pig. 2. The initial rate of change in absorbance ot' an E* sample at six temperatures.
0.573 0.591 0.599 0.589 0,588 ± 0.011
0.556 0.555 0.530 0.544 0.546 + 0.012
F'our E* pellets were suspended in 0.05 ml DGVB-' Igroup At, while four other pellets were suspended in 8.0 ml DGVB z- (group B). After 12 hr at 10°C, samples in group A were brought to 8 ml with cold 0.09 M EDTA and the fraction of E*'s lysed determined for all samples.
Final Stage of Complement Hemolysis
423
transform into the inactive state, S* (see Fig. 5). We shall attempt to account for the experimental results on the basis that the rate constant for each process is dependent upon temperature.
L
54
The Poisson distribution of sites on the E*'s
._a 1J 52
[
5O
0
I 2
I
I
4
I
6 VoL,
8
I0
mL
Fig. 4. Volume dependence of E* lysis. Different volumes of buffer were added to identical E* pellets and percentage lysis determined after 10 hr at 10°C. Curve is theoretical fit (see text). pellets the following amounts of DGVB 2- were added: 0.l ml, 0.5ml, 2ml and 10ml. After shaking for 10hr at 10°C the lysis percentages were determined. See Fig. 4 for results.
Viscosity of DGVB2- as a function of temperature Viscosity of DGVB 2 -, r/, was measured with a CannonFenske viscometer (Curtin Matheson Scientific) immersed in thermoregulated bath water. A timer, accurate to 0.1 sec, was used to clock buffer flows. The viscometer was calibrated using water, whose viscosity is known over the temperatures studied. Over a range from l0 ° to 33°C the plot of In (T/q) against l/Tis linear and gives a slope of 4.6 kcal/ mole (not shown). DISCUSSION
Evidence for the inactivation of S* The observation that the maximum extent of lysis of an E* population is temperature-dependent is not explainable by published hypotheses. Frank et al. (1965) suggest that pre-heating E*'s causes a subset of them to lyse with a different rate constant. Valet and Opferkuch (1975) propose that E*'s lyse by way of two consecutive steps, the first of which depends on temperature. Although both models predict that the initial rate of E* lysis is a function of temperature, neither model can explain why the maximal per cent lysis of a sample of E*'s is temperature-dependent, as is observed in the experiments presented here. In fact, it is difficult to see how such a variation in lysis for like E* populations could arise from any model which postulates that all final complement sites produce lytic damage. We therefore consider the case in which, besides converting into a functional lesion (L) that lyses the cell, the final complement site S* can irreversibly kl S 'm
P
L
Fig. 5. Proposed mode of action of the final complementgenerated site, S*. Besides converting into a functional lesion, L, which lyses the cell, S* can irreversibly transform into the inactive form, S*.
Before a model describing the behavior of sites can be used to interpret the mechanism of cytolysis, we must know the distribution of sites on the cells. When EA's and complement interact, the formation of S*'s on a cell surface is, to a large extent, a random process independent of other S*'s on the surface. Therefore, in a sample of E*'s it is logical to assume that the initial distribution of S*'s on the cells is described by the Poisson distribution function. That is,
P(n, z) = e -~ z"/n.
(1)
is the proportion of cells having n S*'s, where z is the average number of sites per cell.
How lysis affects the distribution Assume the probability that a site will transform into a functional lesion in the span of 1 min is k~. The probability that a cell bearing n sites will lyse in that time interval is nkl (Hoffmann, 1969). Hoffmann has shown that if the distribution of S*'s on the cells is Poisson to start with, then at any time thereafter the S*'s left upon the surviving cells will be Poisson distributed on those cells. Consider the subpopulation of cells having n sites; initially they comprise a fraction P(n, Zo) of all the cells. Cells in this group disappear at the rate nkl (per min). Thus at time t the number of surviving cells in this subpopulation, divided by the total population at time zero, is P(n, Zo)e-"k' ,. (2) However, by time t the number of cells in the total population has been cut down, due to lysis in all the subpopulations, to a fraction of the original number. This fraction is
~ P ( n , zo)e -"k''= e x p [ - z o ( 1 - e-k't)].
(3)
n=0
Dividing equations (2) by (3) gives the fraction of surviving cells bearing n sites: [exp( - z0 e -
k lt)]
(Z0 e - k,,), n!
(4)
The expression (4) is equivalent to equation (1) with z = Zoe-k"; thus, the distribution of sites on the unlysed cells is Poisson at all times.
How inactivation of sites affects distribution Consider next the inactivation of S*'s in a situation where lesion formation does not occur. Given that the distribution of sites on cells is initially Poisson, if inactivation occurs randomly and independently throughout the population of active sites then at any time the remaining active sites will be Poisson distributed on the cells. The proof of this, kindly provided by Prof. H. Chernoff, appears in the Appendix.
The kinetics of E* lysis We see from the preceding sections that in the case where either the creation of lesions or inactivation
424
CONAN K. N. LI and R. P. LFVINI!
of sites occurs separately, the distribution of active S*'s upon unlysed cells is Poisson at all times, given that it is initially Poisson. We infer that whenever S* possesses both lysing and inactivating properties. then the distribution will also take the form of a Poisson at all times. This will be needed in the following development in which we solve for the time behavior of an E* population. Let us consider the group of all active sites borne by the unlysed cells. The total number of such sites equals Mz where M is the number of intact red cells and z the average number of S*'s per intact cell. The change in time of this total number of sites has two contributions. Firstly, S*'s become inactive at a rate proportional to the number of active sites remaining. Assuming that N(n) is the number of cells with n S*'s and that k2 is the inactivation rate constant per site, then the rate of S* loss from the cell population is Z, Lok2nN(n ). Secondly, when a cell lyses, all the S*'s on that cell have left the group. Each cell bearing , S*'s lyses at the rate nk~, where k~ is the rate constant for conversion of an S* into a lesion. For each such cell lysed n S*'s are lost; hence the rate of S* loss from all cells due to this contribution is Z,:okd~2N(n). In view of the above, the change in total S* number with time is given by d ~ ~tt(Mz) = - ~ k 2 n N ( n ) - ~ k, neN(n). (5) n=O
n=O
Since cells bearing n S*'s lyse at the rate nk~ N(n), the change in the total number of intact E*'s with time is dM dt ~ nkl N(n). (6) n=0
We can solve for the time change of the average number of S*'s per cell, dz/dt, by writing d
dz dM (M~) = M a T + : d~
(7)
Substituting equations (5) and (6) into (7), then rearranging, we obtain dz
m dt
z ~ klnN(n) n=O
- ~ k2nN(n)-
kln2N(n),
n-O
(8)
n-0
which, upon recognizing that
M = Z N(n)
and
z = y" nN(n)/M,
(9)
n=O
n=O
becomes z:
dz__ k~z 2 _ k 2 z - kt ~ n2p(n) ' dt
(10)
,:0
where .L
P(n) = N(n)/ ~ N(,) n-O
is the fraction of cells with n S*'s. Although the results thus far are valid regardless of how the S*'s are distributed on the cells it is not possible to evaluate the sum in equation (10) without knowing the distribution.
We have already concluded that the distribuiion is Poisson at all times. We therefure evaluate the sum by using equation qll: kl ~', n-'P(n)= k~c = \ n =o
~,
S'i,
I =
k I e
I)'.
1
......
(:'c:
4
i
zc:)
- k l - : F k~_-. Substituting the above in equation (1 O) gives dz,/dt =
(k I ~ k, ):_
- = zoc ,I,,-k.1¢
{ll)
From equations (6), (9), and (I I) we can now obtain the number of unlysed cells as a function of time.
dM/dt = -=Mk~. dM/M = -k~ :o[exp
Il,~t":"+1~(e
M/Mo = exp[
(k~ + kz)t ] dr,
]
*'+~'" - I)~.
(12)
That equation (12) accurately describes E* lysis is readily verified experimentally. In Fig. 3 two examples are shown. Two E* samples from the same preparation were suspended in buffers of 25.8{C and 30.1°C and lysis at those temperatures was monitored with time. The curves are theoretically calculated from the relation P = 1 M.'M~.
The actitation energy q! complement lysis We can apply the approximation that e ~ ' = 1 - xt, whenever t is small, to obtain, from equation (12), the initial rate of lysis: dP
d
di = di(l
.... :,~t .~,t,) = - k , :0.
(13)
Since Zo is the same for E* samples from a single preparation, the initial rate of lysis of those samples at different temperatures would provide the variation of k, with temperature (see Fig. 2). From the theory of rate processes we C~Ul write /, t = gl e r, Rr where R is the gas constant, E~ is the activation energy of the lytic process, and ~ is a constant invariant with temperature. Taking the initial rates of lysis from Fig. 2, we plot ln(kl:0) against I."T in Fig. 6 to obtain E~ = 13.7 kcal/molc. One result of this is that free diffusion of cellular components, such as hemoglobin, cannot be the ratelimiting step in lysis. The diffusion coefficient, D, is proportional to T/')1 where ~1 is the viscosity of the solvent; the proportionality factor is dependent on the geometry of the diffusing molecule but independent of T (Gosting, 1956). Hence, the activation energy of diffusion in DGVB 2 is - R ? } n D = _ R ? In (T/,1) : 4.6 kcal/mole.
?(1/T)
711 T )
Since the rate-limiting step has a much higher activation energy, i.e. 13.7 kcale/mole, it must be due to a process besides free diffusion. Our value for E~ compares well with an activation energy of 15 + 2 kcal/mole for immune hemolysis
Final Stage of Complement Hemolysis reported by Shanin et al. (1968). Shanin et al. (1968) incubated sheep erythrocytes at 6 x 10S/ml with guinea-pig serum (final concentration 1:600) and two concentrations of rabbit anti-sheep serum. Monitoring lysis by change in absorbance with time, they noted the time required for 50~o lysis. The experiment was performed at seven temperatures in the range 20-40°C. From the plot of ln(1/tso~) against I/T Shanin et al. (1968) obtained an activation energy of 15 kcal/mole for the overall process, including adsorption of antibody and all the complement components. This value agrees with the activation energy we obtained for lysis of cells already in the EACI-9 state, suggesting that the final step may be the slowest stage in the overall complement lysis reaction.
6
't= E o_.
2
Nb I 0.8 I 06 I o4
I
•
Explanation for the variation of maximum E* lysis with temperature After incubation of E*'s has proceeded for a long time, the per cent lysis saturates at a maximum value, P,,. From equation (12) Pm = 1 - M/Mo = 1 - e x p [ - k l z o / ( k l + kz)].
(14)
Thus, the variation of P,, with temperature is wholly accountable for by a non-zero rate constant of S* self-inactivation, k2. We can obtain an estimate of E2, the activation energy of the inactivation process, kt + k: can be derived from the measured quantities dP/dt and P,, by eliminating - k l Z o from (13) and (14). The plot of -Rln(k~ + k2) against 1/Tis nearly linear (see Fig. 7), implying that in the temperature range studied, one of the rate constants is dominant over the other, with the slope indicating approximately the activation energy of the dominant process. Since the slope, 4 kcal/mole, is far from the known value of 13.7 kcal/mole for El, k2 must be much greater than kz ~> kl.
(15)
Therefore, E2 is approximately 4kcal/mole. This value is an upper limit for Ez since the slope of the Arrhenius plot of ka + k2 must lie between Et and E2. Evidence that the inactivation of S* occurs through desorption of some component of S* If S* became inactive through the loss of one of its component parts via desorption from the cell membrane, then if that component were re-adsorbed, one would expect the maximum per cent E* lysis to be a function of the volume of solution in which the E*'s were suspended. Alternatively, if the inactivation of S* occurred on the membrane then the final percentage of E* lysed should be independent of the volume of suspension. In fact, all experiments in which identical E* samples were suspended in different amounts of DGVB z- for lysis showed a small but distinct increase in the final extent of E* lysis with increasing E* concentration. The results of such an experiment are shown in Table 1. Four of eight identical E*-con-
IMM. 14/6 c
I
3.3
J
3.4 I/T,
3.5
10-3deg -~
Fig. 6. Arrhenius plot of klz0 to determine El, the t~ctivation energy of lysis. taining pellets were suspended in 0.05 ml DGVB z-, the other four in 8.0 ml DGVB 2-. After 12 hr at 10°C the mean lysis in the first group was 0.588 + 0.01l and that in the second group 0.546 + 0.012. The probability that the two groups of,samples in reality have the same mean per cent lysis, based on Student's t-test, is less than 0.5~o. In what follows, we see whether the re-adsorption hypothesis can provide a fit to experiment. The proposed scheme may be written k,
S* --# B + C ,
(16)
where B is a membrane-bound portion of S* and C is the residual component, which desorbed into solution. (In the next section of the Discussion we show that C9 does desorb from S* and hence imply that C includes C9.) Neither B nor C alone are capable of lyric action but the adsorption of C to B creates an active S*. In the scheme (16), (S*) and (B) are in units of molecules,t while (C) is in molecules/ml. This is because the net rate of desorption from and adsorption to a membrane-bound species depends
E o
4.¢-
3.2
t It is understood that S*, B, and perhaps C are macromolecular entities, but the unit 'molecules' is apptied for the sake of convenience.
425
I
L
3.3
3.4. I/T,
3.,5
3.6
lO'3de~ -~
Fig. 7, Arrhenius plot of kx + kz. The slope corresponds to 4 kcal/mole.
426
CONAN K. N. LI and R. P. LEVINE
only on the absolute quanti D of that species, irrespective of its concentration in solution: however, the adsorption rate does depend upon the concentration in solution of the adsorbing component. At any instant in an E* lysis experiment, the effeclive or observed rate of desorption, k;_(S*L equals the actual desorption rate less the contribution due to adsorption :
k~(S*) = k2(S*) - k3(B)(C), k2 =
k2 -
(17)
k3{B)(C)/'(S*),
Naturally, the effective desorption rate constant, k'2. will be time-dependent by virtue of the changes in (B), (C), and (S*) from moment to moment, but we may obtain an average k~ by substituting, in equation (17), time-averaged values of (B}, (C), and (S*). To a first approximation the time-averaged quantity equals half the sum of the initial value and the equilibrium, or final value. Initially, let the amount of (S*) be S~. (B) = (C) = 0 at the beginning. After the experiment has proceeded to equilibrium let the value of (C) be Qq. Then the equilibrium values of (B) and IS*) are. respectively, Q q I / a n d (S* - C~q V)/2, where V is the volume of extracellular buffer.+ Hence, (CJa , -- C~q/2, (BL, = QqV/2, (S*),,, = (2S~ - C , V)/2.
(l 8l
Examination of the data reveals that the change in thc final amount of E* lysed with variations in volume, though distinct, is slight. Thus, the contribution of adsorption is not very great and, to a first approximation, the equilibrium amount of material desorbed, C,.4 K is the same in all experiments. In particular, when the volume is very large practically all the S*'s eventually desorb, so C~qV= S*.
(19)
Eliminating C,. from equations (18) with the relation (19) and s u b s t i t u t i n g (17), we obtain
the t i m e - a v e r a g e d
values
k 2 -~ k 2 - k~S*,"2l':
into
(20)
To explain how the maximal E* lysis depends on I we make use of equations (14) and (15), recognizing that the effective desorption rate constant, k'2, replaces k2 in those equations when considering the effect of volume. We obtain k'2 = - k I zo/ln(l - P,.).
(21)
We can eliminate k~ from equations (20) and (21). Then. noticing that S* = zoMo and rearranging,
P,, = I - e x p [ - k j z o / l k : - k3zoMo/2V).
(22)
By setting k3Mo/2k~ = 0.02 (molecule/ml)-' and k:"k,zo = 1.38 molecule -~, we see that a plausible fit to experiment is obtained (see Fig. 4).
Evidence that C9 desorbs fi'om S* Kolb and Miiller-Eberhard (1974) examined the kinetics of adsorption of human C9 to EACI-8 made + We shall ignore the interstitial buffer trapped in the E* pellet, whereupon Vbecomes the volume of DGVB 2added to the pellet,
from sheep cells and human complement. We wish to see whether their data are consistent with desorption of a component from the terminal complement site. Kolb and Miiller-Eberhard (19741 mixed 1.2 x 109 EAC1-8's with 80rig or 16ng radiolabelled (?9 in a final volume of 20 mL At various time intervals samples were withdrawn and the cells pelleted. The bound radioactivity indicated C9 uptake by the cells. The upper and lower binding experiments (filled circles, Fig. 5 in Kolb & Miiller-Eberhard,. 1974) differ only in the number of C9 molecules. Hadding and Miiller-Eberhard (19691 found the mol. wt of C9 to be 79,000; thus, there were 6 × 10 ~ and 1.2 × 10 ~ C9 molecules per 20 ml reaction volume in the two experiments. The initial number of SACI-8 sites on all the cells, which is the same for both experiments, is not specifically given. However, based on values reported for other experiments in the same paper, there were about 2(~3 sites per cell or 2.4 10 ~J SACI-8's initially. The experiment cited can be described by tt~c scheme SAC1-8 + C9 ~
SACI:L
I231
where we included reversibility of (;9 adsorption for generality. When starting with only SACl-8 and free C9, the amount of C9 bound as a function of time, from the solution of (23), is II-
(SACI-9) . . . . .
F
~ +
2k~
//r/k3
i - ,;,r[i, + .i,i,;ii, ~ ,.i]" 1=4t
where l~ -- k3[(C9) o + {SACI-8)o] + k~. y = [u-' - 4k3(C9)o(SAC1-8)o] I -' and (C9)o and (SACI-SL are the starting amounts, measured in the respective units of [(') and (BI in the preceding section. F r o m these equations and the above experimental values we compute the curves in Fig. 8 for the generation of SACI-9 (bound C9~ as a function of time. (It was necessary to assume the lower experiment began al t -:: l m i n to accommodate the otherwise inexplicable lag.) The fit yields adsorption and desorption rate constants of, respectively, k 3 = 2 . 5 x 10-12rain ~ (molecule/ml) ~ and k, = 0.144rain-1. ,'3
~6 g "6 E o O -41
2
S
1
~I
I0
20 t-,
I
]
50
40
rain
Fig. 8, Kinetics of SACI-9 formation from (.'9 and SACI-8. Circles are experimental data from Kolb and Mbller-Eberhard 11974), Curves are calculated fits based on scheme of reversible adsorption of (79 to SA('I-~ (sce text).
Final
Stage of Complement
Note that it is impossible to fit the given conditions with a model that assumes irreversible adsorption of C9 (i.e. scheme (23) with k2 = 0), for this would imply that C9 binding should saturate at 100 and 500 molecules per cell for the lower and upper curves, which is not observed. Alternatively, if the number of SACl-8 sites were limiting, then C9 binding should saturate to the same value in both experiments. Therefore, the scheme postulating reversible adsorption of C9 to SACl-8 is the simplest model satisfying the given conditions and fitting the experimental data. This demonstrates that C9 must desorb from SACl-9 complexes on the cell. It therefore seems plausible that C9 is part of the component which desorbs from S* to cause its inactivity. Eoents in the jinal stage of complement lysis Lysis does not take place instantaneously after the five terminal components, C5-C9, have complexed on the cell surface. A further step, inhibitable by 0.09 M EDTA, must occur before lysis is effected. This step may consist of a conformational change or a molecular rearrangement in either the complex or the membrane: there is as yet no evidence clarifying the mechanistic details. The step proceeds with an activation energy of 13.7 kcal/mole and results in impairment of the membrane’s selective permeability. Prior to the execution of this final step it is possible for molecules of the membrane-bound complex to desorb from the membrane back into the solution. Although CSb6 in itself is loosely bound to the membrane (Tamura & Baba, 1976) the reaction of C7 creates a heat-stable, cell-bound C5b7 complex which does not dissociate for several hours at 37°C (Lachmann, 1973). Hammer et al. (1975) observed that C5b which had been incorporated into membranebound C5b-7 was more resistant to trypsin digestion than C5b bound to the membrane as C5b-6. C7 in C5b-7 was also trypsin-resistant. Hammer et al. (1975) also found that C5b could not be eluted from cell-bound CSb7 under conditions which easily eluted C5b6. These phenomena imply that C5b7 is firmly inserted into the lipid bilayer (Hammer et al., 1975). From the above considerations it seems unlikely that either C5b, C6, or C7, once incorporated into the C5b7 complex, would desorb from the membrane. It is plausible to assume that these components will be no more inclined to desorb from the membrane site after attachment of C8 and C9. It has been proposed that polypeptide chains of C8 are inserted into the bilayer to form a transmembrane channel (Kolb & Miiller-Eberhard, 1976; Miiller-Eberhard, 1975). Also, classical pathwaygenerated and membrane extracted C5b9 complexes were shown to be stable (Kolb & Miiller-Eberhard, 1973; Bhadki et al., 1975). These statements cannot be used to argue against desorption of C8 or C9 since they concern post-lytic C5b-9. Desorption from C5b-9 occurs prior to the intracomplex rearrangement which causes lysis. That C9 was found to associate with only ‘C8 (Kolb rt u!., 1973) and not with C5, C6, C7 or C5, 6, 7 (trimolecular aggregate of C5, C6 and C7) suggests that C9 binds only to C8 in the C5b8 cell-bound complex. If this is so then desorption of C8 from C5bF-9 would cause concomitant loss of C9. We have
421
Hemolysis
already presented arguments suggesting that C9 does desorb, and in doing so abolishes the lytic potential of the complex. That C8 associates reversibly with C5 and C5, 6, 7 (Kolb et al., 1973) at least introduces the likelihood of C8 desorption, though whether or not desorption of C8 from actual C5b8 or C5b9 occurs cannot be deduced from the available evidence. It might be argued that due to the lytic tendency of the C5b-8 complex (Stolfi, 1968; Tamura et al., 1972) C8 must somehow become inactive; otherwise, the E* samples at different temperatures would approach the same extent of lysis through C5b8 action. However, under the present experimental conditions the contribution of C5b8 lysis would be below detection. The desorption of C9 and possibly of C8 during the final S* rearrangement preceding lysis apparently provides the last of many control points in the complement lysis process. CONCLUSIONS
The final stage of complement lysis. the transformation of S* to a functional lesion, has an activation energy of 13.7 kcal/mole. Free diffusion of cellular molecules occurs with an activation energy of 4.6 kcal/mole and is thus ruled out as the rate-limiting step in lysis. Membrane bound S*‘s can also become inactive by desorbing a component into solution. Apparently, C9 desorbs from S*, and may therefore be part of this component. Acknowledgements-The Chernoff, L. Hoffmann, advice and interest.
authors wish to thank M. Mayer, and S. Ruddy
Drs. H. for their
REFERENCES
Bhadki S., Bjerrum 0. J., Rother U., Kniifermann H. & Wallach D. F. H. (1975) Biochim. hiophys. Acta 406, 21. Frank M. M., Rapp H. J. & Borsos T. (1965) J. Immun. 94, 295. Gosting L. J. (1956) Adti. Prof. Chem. II, 429. Hadding U. & Miiller-Eberhard H. J. (1969) Immunology 16, 719. Hammer C. H., Nicholson A. & Mayer M. M. (1975) Proc. mtn. Acad. Sci., U.S.A. 12, 5076. Hoffmann L. G. (1969) Immunochemistry 6, 309. Kolb W. P., Haxby J. A., Arroyave C. M. & Miiller-Eberhard H. J. (1973) J. exp. Med. 138, 428. Kolb W. P. & Miller-Eberhard H. J. (1973) J. exp. Med. 138, 438. Kolb W. P. & Miiller-Eberhard H. J. (1974) J. Immun. 113, 479. Kolb W. P. & Miiller-Eberhard H. J. (1976) J. exp. Med. 143, 1131. Lachmann P. J. (1973) Defence and Recognition, Biochemistry Series One (Edited by Porter R. R.) Vol. IO, p, 370. Butterworths, London. Miiller-Eberhard H. J. (1975) A. Rec. Biochem. 44, 697. Rapp H. J. & Borsos T. (1970) Molecular Basis of Complemew Artiou. pp. 75-77. Appleton-Century-Crofts, New York. Shanin S. S., Gostev B. S. & Kriger Y. A. (1968) Vop. Med. Khim. 14, 145. Stolfi R. I. (1968) J. Immun. 100, 46. Tamura N. & Baba A. S. (1976) Immunology 31, 151. Tamura N., Shimada A. & Chang S. (1972) Immunology 22, 131. Valet G. & Opferkuch W. (1975) J. Immun. 115, 1028.
428
CONAN
K+ N. LI and R. P. L E V I N E
APPENDIX
Demonstratiou that a Poisson distribution (?]' sites (m cells will remain Poisson under random Kite desorptiou (due to Prof. H. Chernoff. Mathematics Department, M.1.T.)
The demonstration consists of two parts. First, wc introduce the multi-nomia[ distribution as a description of sites on cells and show it is invariant under r a n d o m site desorption (or inactivation). Then we show h o w the Poisson distribution function is a case of the multi-nomial. Consider a total of N c o m p l e m e n t sites distributed a m o n g s t a population of M cells. Pick a possible arrangement of sites on the cells: the first cell has .\]' sites, the second has x* sites, and so on, with the Mth cell having +\*M sites where x* + x*~ + . . . + .x~ * = N. Then the probability that this arrangement will occur, that is, the n u m b e r of ways to create this distribution divided by the total n u m b e r of possible distributions, is
Y,! (t /ill ~ P O l l - ">. . . . . . . . . . . --~l!x,!..
" ](I
:"
+ (I,,'M) +~.
2 .....
g~l
(All
1
Let us say that, starting with N sites distributed on M cells, one of the N sites chosen at r a n d o m desorbed from its cell. W h a t is the probability that we will n o w find a distribution P'(x~,x 2. . . . . . VM)'? The site could have desorbed from cell 1. The chance of this equals the probability of that particular pre-desorption distribution occurring, P(.\-I + 1,x2 . . . . , xM), times the chance that a site desorbing at r a n d o m left cell I. (x~ + 1)/'N. Hence, the probability that the new distribution resulted from a site leaving celt I is N! { + + (.'c, + ')+.'c;+ . . . . '++,,+ (I,'.+ l ,+ :': ':t. .N. . l ] ( N - 1)! ( I , . M ) '~ •\h ! .\z ! . . . . ~M !
(A2)
( ,,I. + ')'
N! p(n)=iiiiN~ii('14)"
'.
v,'...~u!
t-
I
The s u m m a n d is simply' the probability that a particular distribution of .\' u sites on M -- I cells will occur, by analogy to equation (A1), and since the sum is over all possiblc distributions the sum must equal unity. Therelore. wc arrive at N!
(N -
I)t ( I / ' M } N+. 1
.+ .v ~! x 2 ! . . . . vM!
I M)"(
1
"
for the probability that a cell bears H sites whenever a total of N sites are available for distribution, and the chance is I / M that any single site will find its wa_,, onto a particular cell (i.e. there are M cells). Normally. in the case of c o m p l e m e n t adsorption to cells, M and N arc large and comparable. In this case the above equation, k n o ~ n as the binomial distribution, approaches the Poisson distribution, POt) =
Likewise. the probability that the site had desorbed from one of the other cells is given by (A21: therefore, the probability of finding the new distribution, P'. is the sum of probability (A2) over all M cells, which is P'(x I . x 2 . . . . . . "%+) =
\,+t!"
where the sum is over all "~2, v:~ . . . . . * xt " N. [ h i s bccomc~
N~
pi +* ., • ,-'(t - \_~. . . . . . "%a) =
P~
or simply (A l l with N replaced by A ; fhis sil~,~> th:t r a n d o m site desorption will not allkect the mttlti-mm+i,ll distribution, {All, except to drop the total n u m b e r o! >ilc, by I. Say we wished to consider o1115 the Ilunlbcl t~i ,,itc:, o~: one cell• The probability that cell I has pl situs is the smi~ over all probabilities (All in which v+ ,1:
C
: 2 Iz
where : N / M is the average n u m b e r of sitcs per cell. (That the Poisson distribution a p p r o x i m a t e s the binomial distribution under the stated conditions is shown in most probability texts. See, for example, Fry T, C. (I928) Prohability and its Engineerin,q l'ses, p. 214. D. Van Nostrand, New York.}
RATE PROCESSES IN THE FINAL STAGE OF COMPLEMENT HEMOLYSISt C O N A N K. N. LI~ and R, P. L E V I N E The Biological Laboratories, Harvard University, Cambridge, MA 02138, U.S.A.
(Received 9 November 1976) A b s t r a c t - - W e define S* as the complex resulting just after the final complement components (C5-C9)
have adsorbed to the membrane at one site. An erythrocyte bearing one or more S*'s is called an E*,
The kinetics of lysis of E*'s made from sheep red cells and human complement is examined. The temperature dependence of the initial rate of lysis gives an Arrhenius plot showing an activation energy of 13.7 kcal/mole for lysis. This rules out free diffusion of cellular components, which has a derived activation energy of 4.6 kcal/mole, as the rate-limiting step in the lytic process. The maximum extent of E* lysis is also found to vary with temperature. This observation could be explained by a scheme which postulates that the final lytic site, S*, becomes inactive with time. A dependence of the extent of E* lysis on the volume of fluid surrounding the cells implies that this inactivation could be caused by desorption of some component of S* from the membrane. By assuming that the distribution of S*'s on the complement-treated cells is initially described by a Poisson distribution function, an expression for the time dependence of lysis is derived in terms of the rate constants of lysis and desorption and the initial average number of S*'s per cell. This expression fits the data very well and yields an activation energy for desorption of less than 4 kcal/mole. On the basis of the observations, speculations are made on the nature of the final control point in complement lysis. Associates, Bethesda, MD, U.S.A.) was dissolved in 5 ml distilled water and stored in 0.l-ml aliquots at -80°C. Isotonic veronal-buffered glucose with gelatin, Ca 2 + and Mg z+ (DGVB z+) was prepared as in Rapp and Borsos (1970). DGVB 2- refers to the same buffer without divalent cations. Ethylene diamine tetraacetic acid (EDTA), 0.09 M, pH 7.4, was prepared containing 0.1~ (w/v) gelatin.
INTRODUCTION
The end result of complement treatment of sensitized cells is the formation of a site on the m e m b r a n e capable of lysing the cell. In view of the physiological importance of the lytic process, it is unfortunate that there have been few attempts to clarify the site's mode of action. At present, while it is difficult to devise meaningful experiments to deduce the lytic mechanism on the molecular level, the kinetics of lysis can be measured with ease. In this context, we examined the dependence of lysis kinetics on temperature, hoping that the kinetic behavior would permit postulation of the final events in lysis while the observed activation energy of lysis would narrow down the possible rate limiting steps in the lyric process.
Preparation of sensitized erythrocytes (EA's) Sheep. erythrocytes were washed by suspending in DGVB 2- at about 1 x 10s cells/ml and centrifuging at 1500g for 8 rain. The supernatant was discarded, mad the pellet washed twice more. The cells were then suspended at 1 x 109/ml in DGVB 2-, mixed wiih an equal volume of diluted hemolysin, and incubated 15 min at 37c'C. EA's so prepared were kept at 0-2°C and used within two days. (The extent of hemolysin dilution varied from batch to batch. Hemolysin was diluted in DGVB 2- to the weakest concentration that yielded 100% lysis when EA's prepared as above were incubated with an equal volume of excess complement for 15 hr at 10°C.)
EXPERIMENTAL
Reagents and solutions Sheep erythrocytes were drawn from the same animal for all experiments. The cells were stored in Alsever's solution at 4°C and used between 5 days and 20 days after drawing. Human serum was used as the source of complement. Whole blood was drawn from a volunteer and clotted for 45rain at room temperature, then 20hr at 6°C. The clot was then removed and the serum centrifuged at 17,000 g for 5 rain. The clear supernatant was removed and centrifuged again; the second supernatant was divided into 0.5-ml aliquots and stored at -80°C. For hemolysin, rabbit anti-sheep erythrocyte stromata serum (Microbiological t This work was supported by grants from the National Science Foundation (BMS 75-09919) and the National Institutes of Health (AI 12885). :~ Pre-doctoral Trainee supported by NIH Training grant No. 5T01GM00782.
Determination of the optimal time for E* formation E*'s were prepared by mixing EA's, complement, and DGVB ~+ in the volume ratio 8:1:7 followed by shaking at 10°C. At 30-sec intervals after mixing, 0.l ml of the reaction was sampled into 2 ml 0.09 M EDTA at 0c'C. These tubes were centrifuged for 15rain at 1500g in the cold, then the supernatants were carefully poured off and the tubes wiped. The pellets were resuspended in 0.5ml DGVB z- and incubated at 35°C. After 7 hr the per cent lysis in each tube was determined: an aliquot of the sample was delivered into 20 vol of 0.09 M EDTA and centrifuged at 1500g for 15 min. The absorbance of the supernatant was read at 414nm on a Zeiss Spectrophotometer PM2 DL, and the A,~14 of an aliquot mixed in 20vol of 0.1% NazCO 3 was taken as 100% lysis. The per cent lysis of the E* samples as a function of the time of reaction at 10°C is shown in Fig. I. The optimal time for E* formation
421
422
('()NAN K. N. L1 and R. P. LEVINE
,oo1
~ ~ ,
~o-~60
0
I
2
I
4
i'
6 rain
t,
I
8
12
is that at which lysis of the E*'s just reached 1000,,. At shorter times the E* formation is submaximal; at longer times the E*'s become coated too heavily with complement and the effect of desorptive inactivation of the sites would be unnoticeable compared to lysis (see Discussion). Therefore, an incubation of 10 min at [0~C was used for forming E*'s.
Kitterics 0[ E* lysis Short term. E* formation was stopped by mixing the whole reaction in 20 vol of cold 0.9 M EDTA. The cells were washed three times in 0.09 M EDTA (3000 0, 10 rain centrifugations); just before the last centrifugation the E* suspension was divided into six equal parts. Each of the six final pellets was suspended (at about 1 x 10~ cells/ml) in DGVB z- which had been pre-equilibrated at one of the six temperatures (10.0, 14.3', 19.7, 23.9', 28.4. and 31.3 C). The tubes were kept in a shaking bath at the respective temperatures. For 48 rain after the pellets were suspended, an aliquot was drawn every 4 rain from each tube and delivered immediately into 20vol of 0.09 M EDTA at 0°C. The samples were centrifuged at 1500.q for 15 rain in the cold and the A , ~ of the supernatants taken. Upon dividing by the A,u4 of an aliquot delivered into 20 vo} of 0.1°;', Na2COs, the per cent lysis of the samples were obtained. After shaking for 6 hr at their respective temperatures, the tubes were sampled again for per cent f
O
,
O
O
C 14.3
5
×~X,~,"
-
,/J9
x ~ X~
7
t
23.9
8° ~;I
/×i"
y~"
U 101 sl .~.,.xl'"'
J
/
~s!
Six /31
?
20
_
/ x 20
/
t-,
J 36
I
300
40O
rnin
lysis: this percentage was taken as the maximum E* lysis since there was no further lysis after 6 h r (see Fig, 3). The short term kinetics of E* lysis at six temperatures is shown in Fig. 2. (Here, A4z,, is plotted,) The initial rate of I\sis as well as the maximum % lysis (at 6 hr) varied ~ith temperature. For the six temperatures in increasing order the maximum fractions lysed were 0,30, &39, 0.43, 0.53, 0.60 and 0,67, respectively. Long term. In an experiment similar to the above, aliquots were drawn at various times up to 347rain. The kinetics of E* lysis over a longer time range at two temperatures is shown in Fig. 3. We have only reported our studies of the terminal reaction in DGVB 2- However, similar results were obtained for the reaction in GVB 2- (see Rapp & Borsos, 1970, for buffer description).
Volume dependence o[ E* lysis E* tk~rmation was stopped by mixing the reaction into 20vol of 0.09 M EDTA at 0°C, The E*'s (about 2.5 × 10~,iml) were washed in 0.09 M EDTA twice, suspended m cold DGVB-'-, divided into 8 equal parts, and pelleted at 30000 for 2rain. After removing the supernatants, four of the pellets were suspended separately in 0.05 ml DGVB 2-, the other four in 8 ml DGVB 2-. The eight tubes were shaken at 10°C for 12hr; then the first four tubes were brought to 8 ml with 0.09 M EDTA and the o lysis in the eight samples determined. There was a definite if slight dependence of maximum E* lysis upon \,plume of buffer added to the E* pellets (see Table 1 for results) In another experiment, similar to the above, the E*'s ~erc divided into 4 equal parts instead of 8. To the final Table I. Dependence of maximum lysis on volume ol buffer
Mean __+ S.D.
x { 24
2O0
Fig. 3. Kinetics of E* lysis. The curves are theoretical fits [see equation (12) of text].
x/x/
x I 12
I
tO0
Maximum fraction lysed Group A Group B
x
15
I
t,
/'/"
~o511 IO
g
7a
Fig. 1. Kinetics of E* formation. EA's and complement were shaken in a 1 0 C bath and sampled at 30-sec intervals after mixing. The samples were washed of complement and allowed to lyse at 35"C for 7 hr to determine the per cent E* formation.
~
4o
I
IO
t 48
rain
Pig. 2. The initial rate of change in absorbance ot' an E* sample at six temperatures.
0.573 0.591 0.599 0.589 0,588 ± 0.011
0.556 0.555 0.530 0.544 0.546 + 0.012
F'our E* pellets were suspended in 0.05 ml DGVB-' Igroup At, while four other pellets were suspended in 8.0 ml DGVB z- (group B). After 12 hr at 10°C, samples in group A were brought to 8 ml with cold 0.09 M EDTA and the fraction of E*'s lysed determined for all samples.
Final Stage of Complement Hemolysis
423
transform into the inactive state, S* (see Fig. 5). We shall attempt to account for the experimental results on the basis that the rate constant for each process is dependent upon temperature.
L
54
The Poisson distribution of sites on the E*'s
._a 1J 52
[
5O
0
I 2
I
I
4
I
6 VoL,
8
I0
mL
Fig. 4. Volume dependence of E* lysis. Different volumes of buffer were added to identical E* pellets and percentage lysis determined after 10 hr at 10°C. Curve is theoretical fit (see text). pellets the following amounts of DGVB 2- were added: 0.l ml, 0.5ml, 2ml and 10ml. After shaking for 10hr at 10°C the lysis percentages were determined. See Fig. 4 for results.
Viscosity of DGVB2- as a function of temperature Viscosity of DGVB 2 -, r/, was measured with a CannonFenske viscometer (Curtin Matheson Scientific) immersed in thermoregulated bath water. A timer, accurate to 0.1 sec, was used to clock buffer flows. The viscometer was calibrated using water, whose viscosity is known over the temperatures studied. Over a range from l0 ° to 33°C the plot of In (T/q) against l/Tis linear and gives a slope of 4.6 kcal/ mole (not shown). DISCUSSION
Evidence for the inactivation of S* The observation that the maximum extent of lysis of an E* population is temperature-dependent is not explainable by published hypotheses. Frank et al. (1965) suggest that pre-heating E*'s causes a subset of them to lyse with a different rate constant. Valet and Opferkuch (1975) propose that E*'s lyse by way of two consecutive steps, the first of which depends on temperature. Although both models predict that the initial rate of E* lysis is a function of temperature, neither model can explain why the maximal per cent lysis of a sample of E*'s is temperature-dependent, as is observed in the experiments presented here. In fact, it is difficult to see how such a variation in lysis for like E* populations could arise from any model which postulates that all final complement sites produce lytic damage. We therefore consider the case in which, besides converting into a functional lesion (L) that lyses the cell, the final complement site S* can irreversibly kl S 'm
P
L
Fig. 5. Proposed mode of action of the final complementgenerated site, S*. Besides converting into a functional lesion, L, which lyses the cell, S* can irreversibly transform into the inactive form, S*.
Before a model describing the behavior of sites can be used to interpret the mechanism of cytolysis, we must know the distribution of sites on the cells. When EA's and complement interact, the formation of S*'s on a cell surface is, to a large extent, a random process independent of other S*'s on the surface. Therefore, in a sample of E*'s it is logical to assume that the initial distribution of S*'s on the cells is described by the Poisson distribution function. That is,
P(n, z) = e -~ z"/n.
(1)
is the proportion of cells having n S*'s, where z is the average number of sites per cell.
How lysis affects the distribution Assume the probability that a site will transform into a functional lesion in the span of 1 min is k~. The probability that a cell bearing n sites will lyse in that time interval is nkl (Hoffmann, 1969). Hoffmann has shown that if the distribution of S*'s on the cells is Poisson to start with, then at any time thereafter the S*'s left upon the surviving cells will be Poisson distributed on those cells. Consider the subpopulation of cells having n sites; initially they comprise a fraction P(n, Zo) of all the cells. Cells in this group disappear at the rate nkl (per min). Thus at time t the number of surviving cells in this subpopulation, divided by the total population at time zero, is P(n, Zo)e-"k' ,. (2) However, by time t the number of cells in the total population has been cut down, due to lysis in all the subpopulations, to a fraction of the original number. This fraction is
~ P ( n , zo)e -"k''= e x p [ - z o ( 1 - e-k't)].
(3)
n=0
Dividing equations (2) by (3) gives the fraction of surviving cells bearing n sites: [exp( - z0 e -
k lt)]
(Z0 e - k,,), n!
(4)
The expression (4) is equivalent to equation (1) with z = Zoe-k"; thus, the distribution of sites on the unlysed cells is Poisson at all times.
How inactivation of sites affects distribution Consider next the inactivation of S*'s in a situation where lesion formation does not occur. Given that the distribution of sites on cells is initially Poisson, if inactivation occurs randomly and independently throughout the population of active sites then at any time the remaining active sites will be Poisson distributed on the cells. The proof of this, kindly provided by Prof. H. Chernoff, appears in the Appendix.
The kinetics of E* lysis We see from the preceding sections that in the case where either the creation of lesions or inactivation
424
CONAN K. N. LI and R. P. LFVINI!
of sites occurs separately, the distribution of active S*'s upon unlysed cells is Poisson at all times, given that it is initially Poisson. We infer that whenever S* possesses both lysing and inactivating properties. then the distribution will also take the form of a Poisson at all times. This will be needed in the following development in which we solve for the time behavior of an E* population. Let us consider the group of all active sites borne by the unlysed cells. The total number of such sites equals Mz where M is the number of intact red cells and z the average number of S*'s per intact cell. The change in time of this total number of sites has two contributions. Firstly, S*'s become inactive at a rate proportional to the number of active sites remaining. Assuming that N(n) is the number of cells with n S*'s and that k2 is the inactivation rate constant per site, then the rate of S* loss from the cell population is Z, Lok2nN(n ). Secondly, when a cell lyses, all the S*'s on that cell have left the group. Each cell bearing , S*'s lyses at the rate nk~, where k~ is the rate constant for conversion of an S* into a lesion. For each such cell lysed n S*'s are lost; hence the rate of S* loss from all cells due to this contribution is Z,:okd~2N(n). In view of the above, the change in total S* number with time is given by d ~ ~tt(Mz) = - ~ k 2 n N ( n ) - ~ k, neN(n). (5) n=O
n=O
Since cells bearing n S*'s lyse at the rate nk~ N(n), the change in the total number of intact E*'s with time is dM dt ~ nkl N(n). (6) n=0
We can solve for the time change of the average number of S*'s per cell, dz/dt, by writing d
dz dM (M~) = M a T + : d~
(7)
Substituting equations (5) and (6) into (7), then rearranging, we obtain dz
m dt
z ~ klnN(n) n=O
- ~ k2nN(n)-
kln2N(n),
n-O
(8)
n-0
which, upon recognizing that
M = Z N(n)
and
z = y" nN(n)/M,
(9)
n=O
n=O
becomes z:
dz__ k~z 2 _ k 2 z - kt ~ n2p(n) ' dt
(10)
,:0
where .L
P(n) = N(n)/ ~ N(,) n-O
is the fraction of cells with n S*'s. Although the results thus far are valid regardless of how the S*'s are distributed on the cells it is not possible to evaluate the sum in equation (10) without knowing the distribution.
We have already concluded that the distribuiion is Poisson at all times. We therefure evaluate the sum by using equation qll: kl ~', n-'P(n)= k~c = \ n =o
~,
S'i,
I =
k I e
I)'.
1
......
(:'c:
4
i
zc:)
- k l - : F k~_-. Substituting the above in equation (1 O) gives dz,/dt =
(k I ~ k, ):_
- = zoc ,I,,-k.1¢
{ll)
From equations (6), (9), and (I I) we can now obtain the number of unlysed cells as a function of time.
dM/dt = -=Mk~. dM/M = -k~ :o[exp
Il,~t":"+1~(e
M/Mo = exp[
(k~ + kz)t ] dr,
]
*'+~'" - I)~.
(12)
That equation (12) accurately describes E* lysis is readily verified experimentally. In Fig. 3 two examples are shown. Two E* samples from the same preparation were suspended in buffers of 25.8{C and 30.1°C and lysis at those temperatures was monitored with time. The curves are theoretically calculated from the relation P = 1 M.'M~.
The actitation energy q! complement lysis We can apply the approximation that e ~ ' = 1 - xt, whenever t is small, to obtain, from equation (12), the initial rate of lysis: dP
d
di = di(l
.... :,~t .~,t,) = - k , :0.
(13)
Since Zo is the same for E* samples from a single preparation, the initial rate of lysis of those samples at different temperatures would provide the variation of k, with temperature (see Fig. 2). From the theory of rate processes we C~Ul write /, t = gl e r, Rr where R is the gas constant, E~ is the activation energy of the lytic process, and ~ is a constant invariant with temperature. Taking the initial rates of lysis from Fig. 2, we plot ln(kl:0) against I."T in Fig. 6 to obtain E~ = 13.7 kcal/molc. One result of this is that free diffusion of cellular components, such as hemoglobin, cannot be the ratelimiting step in lysis. The diffusion coefficient, D, is proportional to T/')1 where ~1 is the viscosity of the solvent; the proportionality factor is dependent on the geometry of the diffusing molecule but independent of T (Gosting, 1956). Hence, the activation energy of diffusion in DGVB 2 is - R ? } n D = _ R ? In (T/,1) : 4.6 kcal/mole.
?(1/T)
711 T )
Since the rate-limiting step has a much higher activation energy, i.e. 13.7 kcale/mole, it must be due to a process besides free diffusion. Our value for E~ compares well with an activation energy of 15 + 2 kcal/mole for immune hemolysis
Final Stage of Complement Hemolysis reported by Shanin et al. (1968). Shanin et al. (1968) incubated sheep erythrocytes at 6 x 10S/ml with guinea-pig serum (final concentration 1:600) and two concentrations of rabbit anti-sheep serum. Monitoring lysis by change in absorbance with time, they noted the time required for 50~o lysis. The experiment was performed at seven temperatures in the range 20-40°C. From the plot of ln(1/tso~) against I/T Shanin et al. (1968) obtained an activation energy of 15 kcal/mole for the overall process, including adsorption of antibody and all the complement components. This value agrees with the activation energy we obtained for lysis of cells already in the EACI-9 state, suggesting that the final step may be the slowest stage in the overall complement lysis reaction.
6
't= E o_.
2
Nb I 0.8 I 06 I o4
I
•
Explanation for the variation of maximum E* lysis with temperature After incubation of E*'s has proceeded for a long time, the per cent lysis saturates at a maximum value, P,,. From equation (12) Pm = 1 - M/Mo = 1 - e x p [ - k l z o / ( k l + kz)].
(14)
Thus, the variation of P,, with temperature is wholly accountable for by a non-zero rate constant of S* self-inactivation, k2. We can obtain an estimate of E2, the activation energy of the inactivation process, kt + k: can be derived from the measured quantities dP/dt and P,, by eliminating - k l Z o from (13) and (14). The plot of -Rln(k~ + k2) against 1/Tis nearly linear (see Fig. 7), implying that in the temperature range studied, one of the rate constants is dominant over the other, with the slope indicating approximately the activation energy of the dominant process. Since the slope, 4 kcal/mole, is far from the known value of 13.7 kcal/mole for El, k2 must be much greater than kz ~> kl.
(15)
Therefore, E2 is approximately 4kcal/mole. This value is an upper limit for Ez since the slope of the Arrhenius plot of ka + k2 must lie between Et and E2. Evidence that the inactivation of S* occurs through desorption of some component of S* If S* became inactive through the loss of one of its component parts via desorption from the cell membrane, then if that component were re-adsorbed, one would expect the maximum per cent E* lysis to be a function of the volume of solution in which the E*'s were suspended. Alternatively, if the inactivation of S* occurred on the membrane then the final percentage of E* lysed should be independent of the volume of suspension. In fact, all experiments in which identical E* samples were suspended in different amounts of DGVB z- for lysis showed a small but distinct increase in the final extent of E* lysis with increasing E* concentration. The results of such an experiment are shown in Table 1. Four of eight identical E*-con-
IMM. 14/6 c
I
3.3
J
3.4 I/T,
3.5
10-3deg -~
Fig. 6. Arrhenius plot of klz0 to determine El, the t~ctivation energy of lysis. taining pellets were suspended in 0.05 ml DGVB z-, the other four in 8.0 ml DGVB 2-. After 12 hr at 10°C the mean lysis in the first group was 0.588 + 0.01l and that in the second group 0.546 + 0.012. The probability that the two groups of,samples in reality have the same mean per cent lysis, based on Student's t-test, is less than 0.5~o. In what follows, we see whether the re-adsorption hypothesis can provide a fit to experiment. The proposed scheme may be written k,
S* --# B + C ,
(16)
where B is a membrane-bound portion of S* and C is the residual component, which desorbed into solution. (In the next section of the Discussion we show that C9 does desorb from S* and hence imply that C includes C9.) Neither B nor C alone are capable of lyric action but the adsorption of C to B creates an active S*. In the scheme (16), (S*) and (B) are in units of molecules,t while (C) is in molecules/ml. This is because the net rate of desorption from and adsorption to a membrane-bound species depends
E o
4.¢-
3.2
t It is understood that S*, B, and perhaps C are macromolecular entities, but the unit 'molecules' is apptied for the sake of convenience.
425
I
L
3.3
3.4. I/T,
3.,5
3.6
lO'3de~ -~
Fig. 7, Arrhenius plot of kx + kz. The slope corresponds to 4 kcal/mole.
426
CONAN K. N. LI and R. P. LEVINE
only on the absolute quanti D of that species, irrespective of its concentration in solution: however, the adsorption rate does depend upon the concentration in solution of the adsorbing component. At any instant in an E* lysis experiment, the effeclive or observed rate of desorption, k;_(S*L equals the actual desorption rate less the contribution due to adsorption :
k~(S*) = k2(S*) - k3(B)(C), k2 =
k2 -
(17)
k3{B)(C)/'(S*),
Naturally, the effective desorption rate constant, k'2. will be time-dependent by virtue of the changes in (B), (C), and (S*) from moment to moment, but we may obtain an average k~ by substituting, in equation (17), time-averaged values of (B}, (C), and (S*). To a first approximation the time-averaged quantity equals half the sum of the initial value and the equilibrium, or final value. Initially, let the amount of (S*) be S~. (B) = (C) = 0 at the beginning. After the experiment has proceeded to equilibrium let the value of (C) be Qq. Then the equilibrium values of (B) and IS*) are. respectively, Q q I / a n d (S* - C~q V)/2, where V is the volume of extracellular buffer.+ Hence, (CJa , -- C~q/2, (BL, = QqV/2, (S*),,, = (2S~ - C , V)/2.
(l 8l
Examination of the data reveals that the change in thc final amount of E* lysed with variations in volume, though distinct, is slight. Thus, the contribution of adsorption is not very great and, to a first approximation, the equilibrium amount of material desorbed, C,.4 K is the same in all experiments. In particular, when the volume is very large practically all the S*'s eventually desorb, so C~qV= S*.
(19)
Eliminating C,. from equations (18) with the relation (19) and s u b s t i t u t i n g (17), we obtain
the t i m e - a v e r a g e d
values
k 2 -~ k 2 - k~S*,"2l':
into
(20)
To explain how the maximal E* lysis depends on I we make use of equations (14) and (15), recognizing that the effective desorption rate constant, k'2, replaces k2 in those equations when considering the effect of volume. We obtain k'2 = - k I zo/ln(l - P,.).
(21)
We can eliminate k~ from equations (20) and (21). Then. noticing that S* = zoMo and rearranging,
P,, = I - e x p [ - k j z o / l k : - k3zoMo/2V).
(22)
By setting k3Mo/2k~ = 0.02 (molecule/ml)-' and k:"k,zo = 1.38 molecule -~, we see that a plausible fit to experiment is obtained (see Fig. 4).
Evidence that C9 desorbs fi'om S* Kolb and Miiller-Eberhard (1974) examined the kinetics of adsorption of human C9 to EACI-8 made + We shall ignore the interstitial buffer trapped in the E* pellet, whereupon Vbecomes the volume of DGVB 2added to the pellet,
from sheep cells and human complement. We wish to see whether their data are consistent with desorption of a component from the terminal complement site. Kolb and Miiller-Eberhard (19741 mixed 1.2 x 109 EAC1-8's with 80rig or 16ng radiolabelled (?9 in a final volume of 20 mL At various time intervals samples were withdrawn and the cells pelleted. The bound radioactivity indicated C9 uptake by the cells. The upper and lower binding experiments (filled circles, Fig. 5 in Kolb & Miiller-Eberhard,. 1974) differ only in the number of C9 molecules. Hadding and Miiller-Eberhard (19691 found the mol. wt of C9 to be 79,000; thus, there were 6 × 10 ~ and 1.2 × 10 ~ C9 molecules per 20 ml reaction volume in the two experiments. The initial number of SACI-8 sites on all the cells, which is the same for both experiments, is not specifically given. However, based on values reported for other experiments in the same paper, there were about 2(~3 sites per cell or 2.4 10 ~J SACI-8's initially. The experiment cited can be described by tt~c scheme SAC1-8 + C9 ~
SACI:L
I231
where we included reversibility of (;9 adsorption for generality. When starting with only SACl-8 and free C9, the amount of C9 bound as a function of time, from the solution of (23), is II-
(SACI-9) . . . . .
F
~ +
2k~
//r/k3
i - ,;,r[i, + .i,i,;ii, ~ ,.i]" 1=4t
where l~ -- k3[(C9) o + {SACI-8)o] + k~. y = [u-' - 4k3(C9)o(SAC1-8)o] I -' and (C9)o and (SACI-SL are the starting amounts, measured in the respective units of [(') and (BI in the preceding section. F r o m these equations and the above experimental values we compute the curves in Fig. 8 for the generation of SACI-9 (bound C9~ as a function of time. (It was necessary to assume the lower experiment began al t -:: l m i n to accommodate the otherwise inexplicable lag.) The fit yields adsorption and desorption rate constants of, respectively, k 3 = 2 . 5 x 10-12rain ~ (molecule/ml) ~ and k, = 0.144rain-1. ,'3
~6 g "6 E o O -41
2
S
1
~I
I0
20 t-,
I
]
50
40
rain
Fig. 8, Kinetics of SACI-9 formation from (.'9 and SACI-8. Circles are experimental data from Kolb and Mbller-Eberhard 11974), Curves are calculated fits based on scheme of reversible adsorption of (79 to SA('I-~ (sce text).
Final
Stage of Complement
Note that it is impossible to fit the given conditions with a model that assumes irreversible adsorption of C9 (i.e. scheme (23) with k2 = 0), for this would imply that C9 binding should saturate at 100 and 500 molecules per cell for the lower and upper curves, which is not observed. Alternatively, if the number of SACl-8 sites were limiting, then C9 binding should saturate to the same value in both experiments. Therefore, the scheme postulating reversible adsorption of C9 to SACl-8 is the simplest model satisfying the given conditions and fitting the experimental data. This demonstrates that C9 must desorb from SACl-9 complexes on the cell. It therefore seems plausible that C9 is part of the component which desorbs from S* to cause its inactivity. Eoents in the jinal stage of complement lysis Lysis does not take place instantaneously after the five terminal components, C5-C9, have complexed on the cell surface. A further step, inhibitable by 0.09 M EDTA, must occur before lysis is effected. This step may consist of a conformational change or a molecular rearrangement in either the complex or the membrane: there is as yet no evidence clarifying the mechanistic details. The step proceeds with an activation energy of 13.7 kcal/mole and results in impairment of the membrane’s selective permeability. Prior to the execution of this final step it is possible for molecules of the membrane-bound complex to desorb from the membrane back into the solution. Although CSb6 in itself is loosely bound to the membrane (Tamura & Baba, 1976) the reaction of C7 creates a heat-stable, cell-bound C5b7 complex which does not dissociate for several hours at 37°C (Lachmann, 1973). Hammer et al. (1975) observed that C5b which had been incorporated into membranebound C5b-7 was more resistant to trypsin digestion than C5b bound to the membrane as C5b-6. C7 in C5b-7 was also trypsin-resistant. Hammer et al. (1975) also found that C5b could not be eluted from cell-bound CSb7 under conditions which easily eluted C5b6. These phenomena imply that C5b7 is firmly inserted into the lipid bilayer (Hammer et al., 1975). From the above considerations it seems unlikely that either C5b, C6, or C7, once incorporated into the C5b7 complex, would desorb from the membrane. It is plausible to assume that these components will be no more inclined to desorb from the membrane site after attachment of C8 and C9. It has been proposed that polypeptide chains of C8 are inserted into the bilayer to form a transmembrane channel (Kolb & Miiller-Eberhard, 1976; Miiller-Eberhard, 1975). Also, classical pathwaygenerated and membrane extracted C5b9 complexes were shown to be stable (Kolb & Miiller-Eberhard, 1973; Bhadki et al., 1975). These statements cannot be used to argue against desorption of C8 or C9 since they concern post-lytic C5b-9. Desorption from C5b-9 occurs prior to the intracomplex rearrangement which causes lysis. That C9 was found to associate with only ‘C8 (Kolb rt u!., 1973) and not with C5, C6, C7 or C5, 6, 7 (trimolecular aggregate of C5, C6 and C7) suggests that C9 binds only to C8 in the C5b8 cell-bound complex. If this is so then desorption of C8 from C5bF-9 would cause concomitant loss of C9. We have
421
Hemolysis
already presented arguments suggesting that C9 does desorb, and in doing so abolishes the lytic potential of the complex. That C8 associates reversibly with C5 and C5, 6, 7 (Kolb et al., 1973) at least introduces the likelihood of C8 desorption, though whether or not desorption of C8 from actual C5b8 or C5b9 occurs cannot be deduced from the available evidence. It might be argued that due to the lytic tendency of the C5b-8 complex (Stolfi, 1968; Tamura et al., 1972) C8 must somehow become inactive; otherwise, the E* samples at different temperatures would approach the same extent of lysis through C5b8 action. However, under the present experimental conditions the contribution of C5b8 lysis would be below detection. The desorption of C9 and possibly of C8 during the final S* rearrangement preceding lysis apparently provides the last of many control points in the complement lysis process. CONCLUSIONS
The final stage of complement lysis. the transformation of S* to a functional lesion, has an activation energy of 13.7 kcal/mole. Free diffusion of cellular molecules occurs with an activation energy of 4.6 kcal/mole and is thus ruled out as the rate-limiting step in lysis. Membrane bound S*‘s can also become inactive by desorbing a component into solution. Apparently, C9 desorbs from S*, and may therefore be part of this component. Acknowledgements-The Chernoff, L. Hoffmann, advice and interest.
authors wish to thank M. Mayer, and S. Ruddy
Drs. H. for their
REFERENCES
Bhadki S., Bjerrum 0. J., Rother U., Kniifermann H. & Wallach D. F. H. (1975) Biochim. hiophys. Acta 406, 21. Frank M. M., Rapp H. J. & Borsos T. (1965) J. Immun. 94, 295. Gosting L. J. (1956) Adti. Prof. Chem. II, 429. Hadding U. & Miiller-Eberhard H. J. (1969) Immunology 16, 719. Hammer C. H., Nicholson A. & Mayer M. M. (1975) Proc. mtn. Acad. Sci., U.S.A. 12, 5076. Hoffmann L. G. (1969) Immunochemistry 6, 309. Kolb W. P., Haxby J. A., Arroyave C. M. & Miiller-Eberhard H. J. (1973) J. exp. Med. 138, 428. Kolb W. P. & Miller-Eberhard H. J. (1973) J. exp. Med. 138, 438. Kolb W. P. & Miiller-Eberhard H. J. (1974) J. Immun. 113, 479. Kolb W. P. & Miiller-Eberhard H. J. (1976) J. exp. Med. 143, 1131. Lachmann P. J. (1973) Defence and Recognition, Biochemistry Series One (Edited by Porter R. R.) Vol. IO, p, 370. Butterworths, London. Miiller-Eberhard H. J. (1975) A. Rec. Biochem. 44, 697. Rapp H. J. & Borsos T. (1970) Molecular Basis of Complemew Artiou. pp. 75-77. Appleton-Century-Crofts, New York. Shanin S. S., Gostev B. S. & Kriger Y. A. (1968) Vop. Med. Khim. 14, 145. Stolfi R. I. (1968) J. Immun. 100, 46. Tamura N. & Baba A. S. (1976) Immunology 31, 151. Tamura N., Shimada A. & Chang S. (1972) Immunology 22, 131. Valet G. & Opferkuch W. (1975) J. Immun. 115, 1028.
428
CONAN
K+ N. LI and R. P. L E V I N E
APPENDIX
Demonstratiou that a Poisson distribution (?]' sites (m cells will remain Poisson under random Kite desorptiou (due to Prof. H. Chernoff. Mathematics Department, M.1.T.)
The demonstration consists of two parts. First, wc introduce the multi-nomia[ distribution as a description of sites on cells and show it is invariant under r a n d o m site desorption (or inactivation). Then we show h o w the Poisson distribution function is a case of the multi-nomial. Consider a total of N c o m p l e m e n t sites distributed a m o n g s t a population of M cells. Pick a possible arrangement of sites on the cells: the first cell has .\]' sites, the second has x* sites, and so on, with the Mth cell having +\*M sites where x* + x*~ + . . . + .x~ * = N. Then the probability that this arrangement will occur, that is, the n u m b e r of ways to create this distribution divided by the total n u m b e r of possible distributions, is
Y,! (t /ill ~ P O l l - ">. . . . . . . . . . . --~l!x,!..
" ](I
:"
+ (I,,'M) +~.
2 .....
g~l
(All
1
Let us say that, starting with N sites distributed on M cells, one of the N sites chosen at r a n d o m desorbed from its cell. W h a t is the probability that we will n o w find a distribution P'(x~,x 2. . . . . . VM)'? The site could have desorbed from cell 1. The chance of this equals the probability of that particular pre-desorption distribution occurring, P(.\-I + 1,x2 . . . . , xM), times the chance that a site desorbing at r a n d o m left cell I. (x~ + 1)/'N. Hence, the probability that the new distribution resulted from a site leaving celt I is N! { + + (.'c, + ')+.'c;+ . . . . '++,,+ (I,'.+ l ,+ :': ':t. .N. . l ] ( N - 1)! ( I , . M ) '~ •\h ! .\z ! . . . . ~M !
(A2)
( ,,I. + ')'
N! p(n)=iiiiN~ii('14)"
'.
v,'...~u!
t-
I
The s u m m a n d is simply' the probability that a particular distribution of .\' u sites on M -- I cells will occur, by analogy to equation (A1), and since the sum is over all possiblc distributions the sum must equal unity. Therelore. wc arrive at N!
(N -
I)t ( I / ' M } N+. 1
.+ .v ~! x 2 ! . . . . vM!
I M)"(
1
"
for the probability that a cell bears H sites whenever a total of N sites are available for distribution, and the chance is I / M that any single site will find its wa_,, onto a particular cell (i.e. there are M cells). Normally. in the case of c o m p l e m e n t adsorption to cells, M and N arc large and comparable. In this case the above equation, k n o ~ n as the binomial distribution, approaches the Poisson distribution, POt) =
Likewise. the probability that the site had desorbed from one of the other cells is given by (A21: therefore, the probability of finding the new distribution, P'. is the sum of probability (A2) over all M cells, which is P'(x I . x 2 . . . . . . "%+) =
\,+t!"
where the sum is over all "~2, v:~ . . . . . * xt " N. [ h i s bccomc~
N~
pi +* ., • ,-'(t - \_~. . . . . . "%a) =
P~
or simply (A l l with N replaced by A ; fhis sil~,~> th:t r a n d o m site desorption will not allkect the mttlti-mm+i,ll distribution, {All, except to drop the total n u m b e r o! >ilc, by I. Say we wished to consider o1115 the Ilunlbcl t~i ,,itc:, o~: one cell• The probability that cell I has pl situs is the smi~ over all probabilities (All in which v+ ,1:
C
: 2 Iz
where : N / M is the average n u m b e r of sitcs per cell. (That the Poisson distribution a p p r o x i m a t e s the binomial distribution under the stated conditions is shown in most probability texts. See, for example, Fry T, C. (I928) Prohability and its Engineerin,q l'ses, p. 214. D. Van Nostrand, New York.}