A rate equation for blood platelet aggregation

J. theor. Biol. (1987) 129, 257-261

A Rate Equation for Blood Platelet Aggregation MOIDEEN P. JAMALUDDIN AND LISSY K. KRISHNAN

Thrombosis Research Unit, Sree Chitra Tirunal Institute for Medical Sciences and Technology, Biomedical Technology Wing, Trivandrum--695012, India (Received 27 July 1987) A kinetic scheme for agonist-induced aggregation of blood platelets was formulated in terms of the kinetics of agonist interaction with the nonaggregable, discoid platelets, formation of aggregable forms by shape-change reactions, and interactions among shape-changed forms. Taking into account the relative magnitudes of the rate constants of the different steps and assuming aggregation to be by hydrophobic forces, an equation similar in form to the Michaelis-Menten equation was derived to characterize aggregation kinetics. The kinetic formulation could account for several empirical observations and may be used to interpret kinetic effects of antiplatelet drugs more informatively than at present.

1. Introduction Platelets are the smallest among the formed elements of blood. They play crucial roles in the physiological process of haemostasis and the pathological process of thrombosis (Zucker, 1980). Their participation in both processes is contingent upon their adequate activation, by various external stimuli or agonists. Fully activated normal platelets change shape, become sticky, aggregate and secrete the contents of their storage granules, the secretions causing activation of passing platelets (Zucker, 1980). This is potentially a thrombotic event. Mechanisms are necessary, therefore, to delimit or modulate the aggregatory reactions of platelets if they are to meet the varying demands and exigencies of different situations in the vascular system without endangering life. Unravelling such mechanisms kinetically, employing isolated platelets, forms an important aspect of haemostasis and thrombosis research. But the interpretation of the kinetics, perforce, remains tentative in the absence of a rational formulation of a rate law of aggregation. We give here a reasonable kinetic scheme and derive a rate equation for platelet aggregation.

2. Formulation of a Kinetic Scheme of Platelet Aggregation Aggregatory reactions of platelets comprise the following reversible steps: Occupation of a free, nonaggregable, discoid platelet, P, by the agonist A; shape-change of the platelets into at least two other, not necessarily spheroidal, forms: one, P* having ruffled surfaces or membranous protrusions (Hantgan, 1984; Nachmias, 1983) and another, P**, derived from P*, not necessarily in one step, and having pseudopods (Deranleau et al., 1982; Gear, 1984; Hantgan, 1984); and 257 0022-5193/87/220257+5 $03.00/0

© 1987 Academic Press Limited

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JAMALUDDIN

AND

L.

K.

KRISHNAN

aggregation. So the following kinetic equation may be written: kt

P + A.

k2

k- I

k3

• P* ~

• PA. k- 2

k4

P** :~

k- 3

~ Aggregation.

(1)

k- 4

3. Derivation of a Rate Equation of Aggregation Aggregation of platelets increases with the degree of their activation (Beival & Hellums, 1984) and pseudopod formation is essential for aggregation (Milton & Frojmovic, 1984) there being a direct relationship between pseudopod number and aggregation (Macmillan & Oliver, 1965). Adrenaline, an activator of ADP-induced platelet aggregation (Mills & Roberts, 1967) accelerates ADP-induced formation of a platelet species analogous to P* (Gear, 1984). For these reasons and because membranous irregularities and pseudopods facilitate intercellular interactions (Rogers, 1979), we assume that platelet aggregation is initiated by the interaction of pseudopods of P** with P*. We assume, further, that this interaction is by the operation of the long-range and strong hydrophobic forces (Pashley et al., 1985) in the manner enunciated by Chothia & Janin (1975) for specific protein-protein recognition. According to Chothia and Janin, the Gibbs free energy change (AG) favouring protein-protein association is derived from the decrease in the protein surface area accessible to water molecules, that occurs on their association. Envisaging platelet aggregation as such a type of hydrophobic interaction between (structurally and electronically) complementary molecular surfaces of P* and P** species, its rate, d ( - A G ) / d t , may be described as the rate of decrease of their accessible surface area. This rate will be proportional to the pseudopod concentration and so to the number concentration of P**. The initial rate of aggregation may then be written as ro = k4(P**).

(2)

The magnitude of k~ in eqn (1) may be in the range of -105 M-1 sec -1, found for the interaction of a proteinaceous ligand with a cell-surface bound receptor (Wank et al., 1983), or more. The magnitude of k2 is much less, of the order of 0.16-0-5 sec -~ (Deranleau et al., 1982; Hantgan, 1984), and that of k3 may be smaller still (Hantgan, 1984). From the experimental data of Gear (1982) and Frojmovic et al. (1983), it appears that k4 is small. In agreement with this, Hantgan (1984) found that estimates of k2 and k3, under aggregating as well as nonaggregating conditions, did not differ significantly. This suggests that k4 may be rate limiting. The magnitude of k_l may vary widely (Bell, 1978) but the very nature of the reactions of the other reverse rate constants would make it certain that they are <
- 0 = kl ( P ) ( A ) + k _ 2 ( P * ) - (k_~ + k ~ ) ( P A )

(3)

RATE

EQUATION

FOR

PLATELET

AGGREGATION

d(P*) - 0 = k2(PA) + k-s(P**) - (k_2 + k3)(P*) dt d(P**) = 0 = ks(P*) - k_3(P**), dt

259 (4) (5)

together with the conservation equation (Pr) -- ( P A ) + ( P ) + ( P * ) + (P**),

(6)

where (Pr) is the total number concentration of platelets. Substituting solutions of eqns (3)-(5) for (P), ( P A ) and (P*), respectively, into eqn (6), solving for (P**), substituting its value into eqn (2), and rearranging: ro =

[k4(Pr)(A)]/(1 + K~+ K2K~) • • r _}_ r r KI[K2K3/(1 K;+K2Ks)]+(A)'

(7)

where K~, K~, and K~ are the reverse equilibrium constants of the first, second and third steps, respectively, of eqn (1). Equation (7) may be written as ro -

R(A) So.5+(A)"

(8)

R is the maximum value ro can attain, at a given concentration of platelets, and (A) >>S0.s. So.5 is, operationally, the value of (A) at which r0=0.5R, or its halfmaximal saturation concentration. Equation (8) is similar in form to the MichaelisMenten equation and may be written in a more general form analogous to the empirical Hill equation (Hill, 1910), to take care of co-operative effects: R ( A / So.5) h ro - 1 + ( A / So.5) h

(9)

where h is the Hill coefficient. It is obtained as the slope of the linear transformation of eqn (9): log

ro

R-ro

=hlog(A)-hlogSo.5.

(10)

The left-hand side of eqn (10) is plotted against log (A). The plot also yields So-5 as the value of (A) at l o g [ r o / ( R - t o ) ] =0. 4. Discussion

The need for platelets to undergo shape-change reactions before they can aggregate is now well recognized (Gear, 1984; Macmillan & Oliver, 1965; Milton & Frojmovic, 1984; White & Gerrard, 1978). This information, together with the quantitative aspects of the kinetics of shape-change reaction (Deranleau et al., 1982; Hantgan, 1984) and aggregation (Gear, 1982; Frojmovic et al., 1983), suggested to us the kinetic scheme of eqn (1). In order to derive a rate equation, we have invoked hydrophobic interactions between specific shape-changed platelet forms in analogy with the postulates of

260

M.

P. J A M A L U D D I N

AND

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K.

KR1SHNAN

Chothia & Janin (1975). We believe that specific molecules endowed with topological and electronic complementarity on different shape-changed forms of platelets participate in the aggregatory reactions as in other intercellular interactions (Edwards, 1983) and that molecular bonds are formed as enunciated by Bell (1978). The accessible surface area buried per protein, during protein-protein recognition reactions, 11-17 nm 2, is only a fraction of its total (Chothia & Janin, 1975). The areas involved in platelet aggregation could be larger and more planar. The range and strength of attractive hydrophobic interactions between planar surfaces increase with the area and the hydrophobicity of the surfaces involved, and the force of attraction, in the 8-10-rim range, is 10-100 times stronger than that of long-range attractive van der Waals forces (Pashley et al., 1985). We have not included exogenous fibrinogen-binding interaction in eqn (1) because there appears to be some species-dependent variation in the mechanism of its manifestation (Harfenist et aL, 1985). The inclusion of such rapid reversible steps before aggregation, however, will not alter the general form of e qn (8) but will increase the complexity of So.~ and R. Many workers have assumed a hyperbolic type of rate law for platelet aggregation kinetics (Skoza et al., 1967; Macfarlane & Mills, 1975). Chang & Robertson (1976) found that the kinetic data of ADP-induced aggregation of rabbit platelet-rich plasma yielded linear double-reciprocal plots over a wide range of ADP concentrations. We have found that the initial rate kinetics of ADP-induced aggregation of calf platelet-rich plasma (Jamaluddin & Krishnan, 1987) and gel-filtered calf platelets (Sreedevi et al., 1987) by Brownian motion also followed a hyperbolic rate law. Rossi & Louis (1977) were the first to suggest, without elaboration, that plateletagonist interaction could, in analogy with drug-receptor interactions, lead to a hyperbolic rate law for aggregation. But this is the first time, to our knowledge, that the rates of agonist interaction, shape-change and aggregation have all been explicitly incorporated into a unified whole to formulate a kinetic scheme and derive a rate equation of aggregation. This formulation can be used to account for several empirical observations without having recourse to ad hoc assumptions. Equation (7) predicts proportionality between ro and total platelet concentration. This has indeed been found (Gear, 1982; Jamaluddin & Krishnan, 1987). The collision theory of platelet aggregation, on the other hand, predicts Woportionality to (.or) 2 (Bikhazi & Ayyub, 1978; Davis & Bown, 1983). Equation (7) predicts also that So.5< K~. This, again, has been found; K~ for ADP binding to gel-filtered human platelets is 5.5 ~M (Lips et al., 1980) whereas the So.5 for ADP-induced aggregation is in the 0.5-1.9 tXM range (Gear, 1982; Frojmovic et al., 1983; Skoza et al., 1967; Rossi & Louis, 1977). Since the rate of conversion of P* to P** is slow and since the rate of formation of P* from P is agonist concentration-dependent (Hantgan, 1984), the aggregation reaction, according to the kinetic formulation, will show a lag period inversely related to the agonist concentration. This expectation has also been borne out (Gear, 1982; Skoza et al., 1967). The synergistic effect of adrenaline on ADP-induced aggregation of human platelets can now be explained as being due to accelerated formation of P* in its

RATE E Q U A T I O N FOR PLATELET A G G R E G A T I O N

261

p r e s e n c e , f o u n d b y G e a r (1984), P** f o r m a t i o n f r o m P * b e i n g a g o n i s t c o n c e n t r a t i o n i n d e p e n d e n t ( H a n t g a n , 1984). E q u a t i o n (1) a n d t h e s i m i l a r i t y o f eqn (8) to the M i c h a e l i s - M e n t e n e q u a t i o n also suggest novel ways to i n t e r p r e t effects o f p l a t e l e t a n t a g o n i s t s on d o u b l e r e c i p r o c a l p l o t s o f p l a t e l e t a g g r e g a t i o n kinetics. A n a n t a g o n i s t will e x h i b i t s l o p e effects o n l y i f it a n d t h e a g o n i s t c o m b i n e with the s a m e p l a t e l e t f o r m o r different p l a t e l e t f o r m s t h a t are c o n n e c t e d b y r e v e r s i b l e steps. I n t e r c e p t effects a l o n e will b e o b s e r v e d if the a n t a g o n i s t c o m b i n e s with a p l a t e l e t form different from t h a t w h i c h the a g o n i s t c o m b i n e s with, d e c r e a s i n g , t h e r e b y , the total c o n c e n t r a t i o n o f t h a t f o r m for d i s t r i b u t i o n a m o n g the u s u a l forms. Both s l o p e s a n d i n t e r c e p t s will b e m o d i f i e d b y a g e n t s a c t i n g b y b o t h m e c h a n i s m s . F i n a l l y v a r i a t i o n s o f s l o p e s a n d i n t e r c e p t s as a f u n c t i o n o f a n t a g o n i s t c o n c e n t r a t i o n m a y be u s e d to infer w h e t h e r it can act at m o r e t h a n o n e t y p e o f site o r not. It m a y b e s a i d in c o n c l u s i o n t h a t the k i n e t i c f o r m u l a t i o n s o f this p a p e r m a y p r o v e useful for i n v e s t i g a t i n g v a r i o u s a s p e c t s o f the kinetics o f i n t e r p l a t e l e t i n t e r a c t i o n s and their modulatory mechanisms.

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