Proc. NatL Acad. Sci. USA Vol. 78, No. 3, pp. 1634-1637, March 1981 Biophysics Aggregation of colloidal particles modeled as a dynamical process (mass action Idnetics/phospholipid vesicles/Na' binding/membranes/intermolecular forces) J. BENTZ AND S. NIR Department of Experimental Pathology, Roswell Park Memorial Institute, 666 Elm Street, Buffalo, New York 14263 Communicated by David Harker, November 19, 1980 ABSTRACT Aggregation kinetics of sonicated phosphatidyl- the process of Na'-induced aggregation of PS vesicles is re- serine (PtdSer) vesicles in NaCl indicate that the process is fully versible to both NaCl dilution and temperature changes. A dra- reversible and dynamical, involving the rapid formation and dis- matic illustration (8) has been that the dispersal of aggregates persal of aggregates. Accordingly, the general mass action kinetic equations are analyzed with respect to the equilibrium state and upon dilution of NaCl is a fast process, which means that the the formation of higher order aggregates. For a general class of backward rates (D.. in Eq. 1 below) can have a significant effect systems, the values for the mass average aggregate size at equi- on the outcome of an analysis of the kinetics of aggregation. librium are obtained from simple closed-form expressions. It is Hence, a consideration of the equilibrium distribution has to shown that an analysis of the aggregation equilibrium will yield supplement the analysis of aggregation kinetics. We will elab- estimates for the potential energy well that holds the aggregates orate on this point in discussing the question of rates and extent together. A fit to the experimental data for kinetics of Na -in- duced aggregation of the vesicles has been achieved by employing of aggregation. mass action kinetic equations that include the dissociation reac- tions. The threshold of NaCl concentration required for aggre- MASS ACTION MODEL gation involves the clear distinction between the rate and extent of aggregation. Consider a system of essentially identical primary particles, denoted X1. The aggregation process is modeled by the general Analysis of the kinetics and equilibrium of particle aggregation mass action reaction is one of the central problems of colloidal science and is a pri- CY. mary source for estimating interaction energies between these Xi + Xj = Xi+j, [1] colloidal particles (1, 2). The most useful approximate solutions Dij to the kinetics of aggregation were obtained by Smoluchowski where Xe is an e-mer aggregate (composed of e primary par- (3, 4), who treated the aggregation as an irreversible diffusional ticles), C.- is the rate constant for the rate at which i-mers and process. Similarly, very useful approximations to the equilib- j-mers form (i + j)-mers, and DU is the rate constant for the re- rium aggregate size distribution were obtained by Mukerjee verse reaction. (5, 6). In both cases, assumptions about the reaction rate con- Let [Xi(t)] denote the concentration of i-mers at time t. Typ- stants were made in order to obtain tractable analytical solutions ically the system is initially monodisperse-i.e., [Xl(O)] = [X,] to the kinetic and equilibrium equations. We have extended this and [X,(O)] = 0 for i - 2; for simplicity, we will assume that this analysis to a more general class of aggregating systems. For is the case. aggregation that follows the mass action model (7), we have found effective algorithms for obtaining both the time devel- Equilibrium distribution classification opment of aggregation and the equilibrium distribution of At equilibrium the principle of detailed balance (7, 9) requires aggregates. for each reaction of Eq. 1 Our interest has been to couple this general theoretical CY = framework to the study of aggregation of sonicated phospha- , [2] tidylserine (PS) vesicles (which have a net negative surface Do mlo.-] charge density under the conditions considered in this work). where [Xi] denotes the equilibrium concentration of i-mers. Eq. In particular, the Na'-induced aggregation of PS vesicles is bf inductive to show that biological interest and is a system well-characterized with re- 2 is used in an argument and + spect to both the magnitude of the negative surface charge Ci_ F(i j) [3] the size of the vesicle. We were led to this work by the obser- F(i)F(j) vation that the rate of increase of light scattered from aggre- Dy gating vesicles (in NaCl) would decrease as the temperature was where raised (8). This implied that the overall rates ofaggregation were likewise decreasing, even though the frequency of collisions t-1 Clk F(e) =- fl F(1) = 1 [4] between the aggregates was increasing. This result could be I Dlk readily explained by assuming that the aggregation is dynami- k cal-i.e., that the dissociation reactions became significant as and the temperature was increased. This gave the impetus to extend the analysis of the kinetic [Xe] = F(C)[Xl]t. [5] equations for aggregation in a manner that would account for the dissociation of aggregates. The results (8) demonstrated that Hence, the concentration of monomers at equilibrium defines the equilibrium concentration of higher order aggregates. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- Abbreviations: PtdSer, phosphatidylserine; DLVO theory, Derja- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. guin-Landau-Verwey-Overbeek theory. 1634 Downloaded by guest on September 24, 2021 Biophysics: Bentz and Nir Proc. NatL Acad. Sci. USA 78 (1981) 1635 At equilibrium all possible aggregate sizes must be consid- Analysis of kinetics ered. A conservation of the total concentration of primary par- These considerations of the equilibrium allow a straightforward ticles, [X<], gives analysis of the rates of aggregation. A useful parameter of this [X0] = analysis is the average aggregate size at time t, (((t)), defined E [Xij [6] equivalently by Eq. 7 with [X,(t)] in the place of [Xi,]. In the early stages of the aggregation of small particles that are Rayleigh The mass average aggregate size at equilibrium may be de- scatterers, the relative increase in turbidity is given by (t(t)) fined by (11, 14). The mass action kinetic equations corresponding to the re- - Zoej,2[XRj/[XO], [7] actions of Eq. 1 may be written which is explained by noting that i[5i]/[X0] is the probability that d[Xe(t)] - - Mef(I + e)Cje[X)][Xe(t)] - De[X(i+e)(t)]} a randomly chosen particle will be found in an aggregate of size dt i. When the system is composed exclusively of j-mers, then + Ef'21{Ci(e.i)[X1(t)][XY(e.i(t)] - Dj(f-jXe(t)]} [X.] = [Xo]/j and (?) = j. Mukerjee (5, 6) considered this problem in its simplest form, [14] which assumes that the equilibrium constants CldDlk are all equal-i. e., where 3,e = 1 when i = eand 0 otherwise, and {(/2} denotes the largest integer s (/2. For the purposes of numerical in- a C111DI, = Cld/Dlk [8] tegration, it is necessary to fix a maximum-allowed aggregate size, denoted M. In practice M may be set sufficiently large Applying Eq. 8 to Eqs. 6 and 7 yields (20-30 in most cases that we treated) such that it does not ar- a[XI] []+ 1 - 9V tificially constrain the evolution of the solutions to the kinetic equations. Eq. 14 may be understood by noting that the first summation accounts for all reactions of the type XY + Xe - X+e and the factor (1 + Si,) accounts for the stoichiometry of the and reaction. The second summation accounts for all reactions of the type X, + X-i - Xe. These are the only two reaction types (e) = (1 + 4a[Xo])'2. [10] which affect the concentration of the species Xe. Hence, the average aggregate size (?) increases linearly with The numerical integrations of Eq. 14 have been performed [X0] when 4a[Xo] << 1, and it increases as [XO]12 when 4a[XO] by using several techniques, all of which gave essentially iden- >> 1. tical results (11). A straightforward Taylor series solution to Eq. In order to classify the behavior of other types of aggregating 14 is sufficient to accurately yield the concentration of each systems, we have chosen an empirical prescription for the species as a function of time from the initial condition t = 0 to higher order equilibrium constants. Suppose now that the ag- the final equilibrium state; the values obtained as t -- 00 are in gregation system may be represented by complete agreement with those calculated directly from the equilibrium distribution. These solutions also account for a sit- uation in which the degree of aggregation is reduced from its CldDlk= a( + k) [11] initial value by electrolyte.dilution or temperature changes. for some value of T, where a C111/D1. Setting 7 = 0 gives back the case of equal equilibrium constants. The format of Eq. THEORY OF DIMERIZATION AND THE 11 arose from Fuoss' (10) equilibrium analysis of cation-anion POTENTIAL ENERGY OF INTERACTION pairs; however, this extension to the argument is not rigorous, For two spherical vesicles of radius R (cm) which are held to- and we prefer to treat Eq. 11 as empirical. For more details on gether by a potential energy well of depth u (where u must be the results presented below, see refs. 11 and 12. Use of Eq. 11 negative if measurable dimerization is to occur), the equilib- in either Eqs. 6 or 7 requires that Ti . 0 for the series to con- rium dimerization constant C11/D11 is related to u (10, 15) by verge. For the case of T = 1, we obtain C11 a = NA x 10-3 [15] (?) = 1 + 2a[X1] [12] D11 -4(2R)33 exp{-u/kT} M'1, where a[FL] is given by solving where NA is Avogadro's number, kT is Boltzmann's constant a[X0] = a[XI] exp {2a[Xl]}.