polymer-mediated electrostatic interactions between charged lipid assemblies and electrolyte...

J. theor. Biol. (1991) 149, 1-20

Polymer-mediated Electrostatic Interactions between Charged Lipid Assemblies and Electrolyte Solutions: A Tentative Model

of the Polyethylene Glycol-induced Cell Fusion

ANTONIO RAUDINO AND PATRIZIA BIANCIARDI

Dipartimento di Scienze Chimiche, Universit~ di Catania, Viale, A.Doria 6-95125, Catania, Italy

(Received on 26 February 1990, Accepted in revised form on 7 August 1990)

We developed a theoretical model to investigate the interaction between charged lipid aggregates and a water solution containing ions and uncharged polymers. The local concentration of ions and polymer chains around the lipid aggregate have been treated as variational parameters which can be found by minimizing the total energy of the system. We divided the energy into the following main contributions:

(a) Solvation energy of the ions. This depends on the local polymer concentration through the variation of the solvent dielectric properties.

(b) Ions-lipid aggregate interactions. These depend on the local concentrations both of the ion cloud and polymer chains.

(c) Conformational energy of the polymer. This term is related to the inhomogeneous spatial density of the polymer segments.

Any direct interaction between the charged lipid surface and the polymer coils has been intentionally neglected. The minimization procedure leads to a non-linear Poisson-Boitzmann equation coupled with a non-linear algebraic equation describ- ing the polymer distribution. The solution of the above system allows one to calculate the ions and polymer spatial distribution around the lipid aggregate. The knowledge of such parameters is useful to predict the effect of non-ionic polymers on the structure and properties of lipid assemblies such as the mean area per lipid molecule, the aggregation number, the critical micellar concentration and the formation of immiscibility gaps in mixed lipid systems. A possible involvement of these parameters into the fusion process between lipid vesicles is discussed.

1. Introduction

In last few years several theoretical studies have been devoted to the problem of the ;nteraction between water-soluble polymers (or proteins) and lipid aggregates. Many of these investigations depict the membrane surface as a fiat, rigid boundary interacting with the polymeric coils through solvent-screened electrostatic interactions or other short-ranged forces (Wiegel, 1977; Eisentiegel, 1985; Scheutjens & Fleer, 1979, 1980; Koopal & Ralston, 1986; Koopal et al., 1988; Gaylord & Zhang, 1987).

In these studies the solvent is treated as a homogeneous continuum medium which basically acts by screening the electrostatic interactions between the surface and the polymer. This could well-illustrate how the charged membranes interact with poly- electrolytes or proteins bearing several ionic residues. The role of the solvent becomes

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0022-5193/91/050001 +20 $03.00/0 © 1991 Academic Press Limited

2 A. R A U D 1 N O A N D P. B I A N C I A R D I

more intriguing when the polymer hydrophilicity decreases. When this situation takes place, the role of the solvent is two-fold. On the one hand it still screens the local density of the polymeric coils which contract themselves in order to reduce their contact with water. These "drop of oil" models have been used to rationalize the adsorption data of slightly soluble proteins on solid substrates (Andrade & Hlady, 1986; Lundstrom et al., 1987; Lee & Ruckenstein, 1988) and, with some caution, they are useful also for lipid-polymer interactions (Raudino et al., 1990) [very hydrophobic protein domains could penetrate inside the lipid bilayer hydrophobic core (Boggs, 1983) making the interpretation of the results more complex]:

In these years some interesting and largely unexplained phenomena have been observed for instance when the interaction between lipid assemblies and non-ionic but hydrophilic polymers is investigated. In this case neither strong surface-polymer forces, nor hydrophobic effects occur; nevertheless, noticeable variations in the lipid assemblies properties have been observed. Among the non-ionic hydrophilic polymers, one of the most popular is the poly(ethylene glycol) (PEG) which is widely used to induce fusion between cells (Davidson & Gerald, 1977; Lucy, 1978; Westerwoudt, 1980; Hui et al., 1985) or lipid vesicles (Arnold et al., 1985; Boni et al., 1981, 1984a) Aldwinkle et al., 1982; MacDonald, 1985; Rupert et al., 1988). Several explanations of PEG fusogenic properties have been suggested in the literature. Some of them are based on the very high hydrophilicity of the polymer chains which compete for water with the lipid head groups. They become partially dehydrated allowing a better contact between two adjacent bilayers (Morgan et al., 1983; Boni et al., 1984b; Parente & Lentz, 1986; MacDonald, 1985; Rupert et al., 1988).

Moreover, by reducing hydration of the head groups, a tighter lipid packing should be observed. Indeed, several measurements showed a decreasing of the membrane fluidity as a consequence of PEG addition. This has been confirmed by differential scanning calorimetry (DSC) (Tilcock & Fisher, 1979; Boni et al., 1984b), spin labels (Herrmann et al., 1983; Surewicz, 1983) and NMR measurements (Ohno et al., 1981; Boni et al., 1984a). However, other data suggest a less direct interaction between PEG and lipid bilayers. For example, it has been shown that the PEG fusogenic properties are maintained even if the polymer containing the solution and the lipid vesicles suspension are separated by a water-permeable membrane (MacDonald, 1985), emphasizing the role of the osmotic pressure related phenomena in polymer induced fusion. Similar conclusions have been reached in recent adhesion measurements between giant unilamellar vesicles (Evans & Needham, 1988a, b).

Furthermore, it has been observed by dielectric measurements that the solvent polarity is largely reduced by adding increasing amounts of PEG (Heri'mann et al., 1983; Arnold et al., 1985; Zaslawsky et al., 1989). According to the well-known theories on the stability of lipid aggregates based on the so-called capacitance model of the surface electrostatic interactions (Israelachvili et al., 1976; Mitchell & Ninham, 1981), it can be easily seen that a decreasing of the medium dielectric constant should increase the electrostatic repulsion, leading to less packed lipid aggregates and to a decreasing of the melting temperature. Similar conclusions have been reached recently by Nagarajan (1989) who developed an interesting model which takes into account the lipid-polymer interactions.

P O L Y E T H Y L E N E G L Y C O L - I N D U C E D C E L L F U S I O N 3

In our opinion, some of the previous conflicting results could be reconciled if we take into account in an explicit way the role of the ions in these complex systems. Indeed, most of the experimental results have been obtained in the presence of alkaline ions, generally added to stabilize the lipid aggregates, or divalent ions (mainly Ca 2÷) used to accelerate the fusion process (Rupert et al., 1988; Boniet al., 1984a; Hoekstra, 1982).

Moreover, any realistic model of the lipid-proteins interactions cannot disregard the role played by the ions because of their large concentration or the high binding constants of some divalent ions (e.g. Ca 2+-, Mg 2÷) with lipid or proteins charged domains.

Finally, it must be recalled that the combined effect of ions and non-ionic polymers like PEG has interesting effects on other charged biomolecules; for example, PEG induces aggregation of nucleotides (Yevdokimov et al., 1988) and precipitation of proteins (van Oss & Good, 1984).

In this paper we develop quite a simple theoretical model by focusing our attention on the polymer-mediated electrostatic interaction between a charged lipid aggregate and the surrounding ion atmosphere. Intentionally, any direct lipid-polymer interaction will be neglected.

2. Theory

A schematic drawing of the system investigated by us is reported in Fig. 1. The sphere of radius R describes a charged micelle (or a closed bilayer) surrounded by an ion cloud containing both positive and negative point-like charges. In the surrounding aqueous medium there are some polymeric chains each of them containing m monomeric units.

FIG. 1. Schematic representation of the polymer-vesicle-electrolyte solution association structure.

4 A. R A U D I N O A N D P. B I A N C I A R D I

Both the ions concentration and that of the polymer segments vary with the distance r from the micelle centre reaching a constant value at large distances. Moreover, when t->o o, the electroneutrality condition of the saline solution must be fulfilled.

In order to calculate the ion and polymer distribution around the lipid aggregate we have to calculate the minimum energy configuration of the whole system and this can be done once the total energy has been partitioned into different main contributions. It is useful to consider one at the time the ions and polymer energies as follows.

2.1. I O N S A N D L I P I D S E N E R G I E S

2.1.1. Ions configurational entropy

Let Ni, N2 . . . . . N~ be the number of particles and N,o~ = Nj + N2 + . . . N, be the number of sites, the configurational entropy is given by the well-known relationship (Reichl, 1980):

S mi~=-k log ( N1 IN2 ! . . . N~ ! N , + N 2 + . . . N~)I' (1)

where k is the Boltzmann constant. In the present case ~ot = N~.+ inNv+~, s N~s, where Nw is the number of water molecules, Np that of the polymer chains each of them consisting of m identical segments and N o that of the generic j th dissolved ion. By introducing the definition Xi -~ N~/N, ot and by applying the Stifling formula we can easily evaluate eqn (1). Since the ions distribution is not homogeneous, the above formula must be generalized. Assuming constant the ion concentration X,s(r) within a generic layer lying between r and r + d r , we may calculate the free energy associated to the configurational entropy of the ions by integrating over the whole volume occupied by the electrolyte solution

GTiX = -S~'iXT= kT" - ~ n X°(r) l °gX°(r )

+ ( 1 - ~ X i j ( r ) ) l o g ( 1 - ~ X o ( r ) ) ] r 2 d r , (2)

a 3 being the averaged volume of each site.

2.1.2. Ions self-energy

The self-energy of the dissolved ions can be divided into two contributions. The first one, say E°s, depends only on the ion electronic structure. The knowledge of its numerical value is not necessary since all the constant terms cancel in the next calculations. The latter one, say o o, depends on the interactions with the surrounding medium. Within a homogeneous electrolyte solution this term is constant and it is discarded in the subsequent mathematical handling. This is not true for inhomogeneous media where the solvation energy depends on the ion position. In the present system containing charged lipid aggregates, ions and non-ionic polymers


the polarity spatial variations can be very large. In fact, it is likely to suppose that the ions concentrate themselves near the lipid surface, both to better interact with the charged amphiphile head groups and to escape from the polymer-rich bulk solution which is a poor solvent for the ions. This latter effect can be very large for hydrophilic solvents exhibiting moderately low dielectric constants, a situation occuring in many cases (the PEG behaves in such a way, see, e.g. Herrmann et al., 1983; Arnold et al., 1985; Zaslawski et al., 1989). Furthermore, if the polymer does not contain groups strongly interacting with the aggregate surface, it will not compete with the ions for the lipid charged groups, and the only force opposing to the polymer removal from the interface region is the entropic one. Also this second perequisite is fulfilled by non-ionic molecules like PEG.

The simplest way to calculate the ion self-energy is to employ the Born equation (Bottcher, 1973), even if more refined formulas have been reported in the literature (Abe, 1986; Gersten & Sapse, 1985):

o o ( r ) - - - z~e2[ 1 - e-~r) ] 1 , (3) Ehj

where zje is the ion charge, hj its radius and e(r) the local dielectric constant of the surrounding medium. In this system e(r) depends on the local concentration ~ ( r ) of polymer segments and, with a good approximation, we may write:

e(r) ~- epdP(r) + ew[1 - qb(r)], (4)

ew and ep being the pure water and pure polymer dielectric constants, respectively. The validity of eqn (4) has been tested on many substances and proved on the basis of theoretical considerations (Chelkowski, 1980). Also the PEG satisfies eqn (4) with a fair accuracy in a wide range of concentrations (Arnold et al., 1985; Zaslawsky et al., 1989).

Since the ion solvation energy o o depends on the local polymer concentration qb(r) through the local dielectric constant e(r), it contains a coupling term between the ions and polymer distributions, and, in order to stress this peculiar role, we will use the notation o 0 = vj [¢ '(r)] .

Integrating eqn (3) over all the space we obtain the total ion self-energy:

4~ I °:' G ~ e = ' ~ R Xtj(r)oj[dP(r)]r2 dr" (5)

where Xo(r) is the local concentration of the generic j th ion which has to be determined.

2.1.3. Ions-tipids interaction

The solvent-screened electrostatic potential ~(r ) originating from the central lipid aggregate interacts with the surrounding ions.

This energy (per ion) is +zje~O(r), and integrating over all the ions we find:

f2 G~"'e'-4rrE zje Xl j ( r )~b(r )r2 dr. (6) a 3 j

6 A . R A U D I N O A N D P. B I A N C I A R D I

The analytical expression of ~b(r) is still unknown and it will be obtained shortly later by applying a generalized Poisson-Boltzmann equation.

2,2. P O L Y M E R E N E R G Y

2.2.1. Configurational entropy

Let ~(r ) be the local polymer fraction and 1 -d~(r) that of the electrolyte solution, the polymer configurational entropy per unit volume can be obtained by the Flory's formula (Flory, 1978):

- k [ ~ log ~ + (1 - ~ ) log (1 - ~ ) ]

m being the number of monomeric units forming the chain. When the distribution around the lipid particle is not uniform (see the previous discussion) • is a function of r. Integrating the above expression over the whole volume surrounding the spherical aggregate we get:

mix 4~r f ~ G~ ix = - S p T = k T . - ~ _ R

OP( r) + (1-Og( r) ) i°g (1-dP( r) ) ] r2 (7)

2.2.2. Polymer-solvent interactions

Let aep, ass and asp be the monomer-monomer, solvent-solvent and monomer- solvent energies per unit volume, respectively, and let Npp, N~s and Nsp be the number of pairs of closest neighbours, we may write the mean polymer-solvent interaction per unit volume as: Npvapp+Nssas~+Np~a,,. In writing the above equations we assumed only short-ranged intermolecular interactions which is a very good approximation for non-ionic polymers where only van der Waals, steric or other rapidly falling interactions take place. Let z be the averaged co-ordination number (the same value has been assumed for the solvent and monomeric units) we have: zN, = 2N, s + Nsp and: zNp = 2Npp + Nsp (Prigogine, 1957), recalling that: NpocO, Nsoz l -qb and Nspoczd9(1-O), we can combine the above formulas yielding:

El.toe z [apv 0 + ass(1 - 0) ] + wO(1 - 0) , P 2 (8)

where: w =-z[asv- ½(ass+ app)]. The parameter w (which, in general, is positive and depends on the temperature) contains all the polymer-polymer, water-polymer and water-water interactions, but not the polymer-ions interactions since the most relevant contributions to this term has been described in eqn (3).


When the polymer shows a inhomogeneous spatial distribution, eqn (8) must be generalized as:

= a3 (appO(r)+ct,5(1-O(r)))+wO(r)(1-O(r)) r 2 dr. (9) R

Adding together eqns (2), (5-7) and (9) we obtain the expression for the total energy of the system. The explicit expressions for the local ion [X o (r)] and polymer [O(r)] concentrations can be obtained by minimizing the total energy with respect to these variational parameters. By differentiating and rearranging we obtain after simple algebra:

OEto t _4~ f°~[EOj_uj(O(r))+zjeO(r)+kTIog Xlj(r) .]r2 dr OXo(r) a3 JR 1 -~j Xo(r)J

--0 (10a)

aE~o, 4~r I°~ E - avj (O(r))+ z, 3 X , , ( r )

- 2wCP(r) + 1 log O(r) - l o g (1 - O(r))[ r 2 dr = 0, (10b) m m d

where oj (O(r)) has been defined in eqn (3). The above expressions can be simplified noticing that at low ion concentration ~.j Xo(r)- 1. Moreover, the unknown ions and polymer self-energies [E°j and (z/2)(ctpp - ass)] can be eliminated by investigating the behaviour of eqn (10) at r--)cc. Recalling that 0(r) --) . . . . 0, (I)(r) - - ) , _ ~ and X o ( r ) ~ , ~ )(;j, we have:

E°j - vj (O) +kT log 2 o =0 ( l la)

- E )(o ooj (~) ~z ( ~ _ ~ ) -2w4) j OO 2

1 + - - l o g - - - l o g (1 -(P) =0, ( l ib)

m rR

and eqn (10) can be rearranged yielding:

X,j(r)=Xoexp[~T(Oj(O(r))-oj(O)-zjeO(r)) ] (12a)

1 qb(r) 1-qb(r) 2w dp r - - ~[ ()-'~] m l°g - -~ - - l ° g 1-~)

Ooj(O(r)) Ooj(O)] 1 ~ Xo(r ) c)dP(r) X,j --~ j = 0 (12b)

In order to solve the system of eqn (12) we need to calculate the electrostatic potential O(r) within the electrolyte solution surrounding the lipid aggregate. This can be done making use of the generalized Poisson equation valid for inhomogeneous dielectrics (Conway, 1970):

grad [e(r) div O(r)] = -4rrep(r), (13a)

8 A. R A U D I N O AND P. B I A N C I A R D I

where ep(r) is the total electric density of the free moving charges (ions) in the dielectric medium:

ep(r)=-ff3 ~ zjXo(r), (13b)

the analytical expressions for X~j(r) are given in eqn (12b). e(r) is the spatially varying dielectric permittivity of the medium which is related to the local polymer concentration by eqn (4).

Once the analytical expressions for e(r) and ep(r) have been obtained, the non-linear differential eqn (13a) coupled with the non-linear algebraic eqn (12b) can be solved and explicit expressions for the polymer and ions concentrations can be obtained. This problem can be solved only at numerical level, however, interesting analytical relationships may be obtained in some limiting cases through a series expansion.

At moderately low surface charge densities of the lipid aggregates, we may assume that the total ions concentration (positive+ negative ions) is not very far from that of bulk solution. Since the polymer concentration depends on the total ions concentration, in the zeroth-order approximation we may assume O(°)(r)= ~. Within this approximation eqn (13a) reduces to the well-known Poisson-Boltzmann equation (PB), which, according to the previous assumption of moderately low surface charge density, can be further simplified by the usual expansion of ep(r) in power series of ~b(r). The solution of the linearized PB equation subject to the proper boundary conditions gives the zeroth-order potential ff(°)(r). Inserting this solution into eqn (12a) we may calculate the zeroth-order ions concentrations Xt°)/r), which, in turn, are inserted back into eqn (12b) giving the first-order tj correction to the polymer concentration qb(~)(r). This procedure can be iterated several times until a good convergence is reached. Therefore, putting:

qb(r) -- (~ + -qqb (] I(1) + -q2cI)(2)(r) + -r/3(I~ (3)(r) + . . . (14a)

~b(r) = const. + r/~bt°~(r) + ~72~btl)(r) + r/3~t2)(r) + . . . , (14b)

we can transform eqn (12) in a simpler sets of coupled linear algebraic and differential equations which can be easily solved by analytical methods. ~7 is a disposable parameter put equal to 1 at the end of the calculations. After tedious but straightforward algebra we obtain quite cumbersome equations which, in the case of uni- univalent symmetrical electrolytes reduce to very simple expressions. In particular, eqn (12b) transforms as:

qb~l~(r) =0 (15a)

qb(2,(r) _ __J___l e 2 a........~ ep-ewK~¢(o,2(r ) (15b) 167rA kT a2Ep g

a~ b2 4,

1 e-" a 3 ep-- ew Kg~btOl(r)~b(I)(r) ' (15c)

aqb- 6


where g = ep(~)+ ew(1 - ~ ) is the macroscopic dielectric permittivity of the solution and the analytical expression of O'-Ep/O02]oo is

kT l + O ( m - 1 ) m (~(1 - O ) 2w,

while Ko is the so-called Debye length defined as:

4rre 2 Ko = a3gkT ()(+ + ) ( - ) •

In deriving eqn (15) and the following equations we assumed the same radius for anions and cations which has been put equal to the mean value ~.. As we shall see shortly later, ~ ( l l ( r )=0 , then: (I)t3)(r) =0. By combining eqn (12a) with eqn (13)

2m -- and making use of the relationships: ~i z~ "+1 )(u = 0 and ~j z~ X,j = )(÷ +.~_ valid for symmetrical univalent electrolytes we find: const. = 0 [see eqn (14b)] and

V~Ot°)(r) - Ko@')(r) = 0 (16a)

V ~0t'l(r) - Ko2O(' )(r) = 0 (16b)

V 2r @"l(r) - Ko0(2'(r)

O0(°)(r) O0(21(r) a30(o).,( = a , - - - - ~ - a 2 O m ) ( r ) O t 2 1 ( r ) + _ r) (16c)

Or Or

where:

E'p - - E w a l ~ - - - -

, _ . , ( 2 ) a2= g K8 2XTkT 1

a 3 --= ~,K8

In spherical co-ordinates the differential operator V 2r is 02/0r2+ (2/ r ) O/Or. Equation (16a) is the linearized PB equation valid for homogeneous dielectrics. As we can see, in the zeroth-order approximation the only effect of the polymer is to affect the numerical value of the dielectric constant which changes from ew to g. Since eqns (16a) and (16b) are identical, we have q,('~(r) = 0, therefore the effect of the polymer inhomogeneous distribution on the ionic atmosphere is described by the coupled eqns (15b) and (16c), the solution of which will be discussed shortly later.

Interesting results are obtained when we consider asymmetrical electrolytes containing divalent cations. By repeating the above procedure yields:

O(,)(r ) = 3e a 3 ep - ew , )?++ 4rcA O2E. g K8 g+~_ @(°)(r) (17)

002

10

and:

A . R A U D I N O A N D P. B I A N C I A R D I

V 2r 0 (° ' (r ) - Ko20(°)(r) = 0 (18a)

a#°>(r) aO"~(r) V2~b(')(r) - Ko0~')(r) = b, - - Or Or

+b2df°)(r)~P(X)(r)+b3~Pt2)(r)+b4~bt°l:(r) (18b)

where: bl = al , b2 = a2 [see eqn (16)] and

bs -= 3 e e p - e w , X++

b - e 2 X++ 4 = - 3 K o - .

In deriving eqns (17) and (18) we neglected higher order terms in the concentration of the divalent ions as well as their influence on the Debye length Kg ~, and the following straightforward identity ~j z~ J?o = 6X÷÷ has been used. These approxima- tions are permissible because in biological media the divalent ions concentration is much lower than that of the monvalent ions (10-4-10 .6 and 10 -1, respectively). Therefore, the overall effect of the electrolytes can be approximated by the sum of the mono- and divalent ions contributions.

As expected, the zeroth-order eqs (16a) and (18a) are identical in both cases --2 (apart from small terms proportional to X++) while the first-order terms do not

vanish when mixtures of ions beating different charges are present in solution. This fact further evidences the effectiveness of divalent ions like Ca 2+ or Mg 2+ in modulating membrane phenomena.

Coming back to the system of eqns (15) and (16), it is easy to see that it can be solved by standard methods, provided the proper boundary conditions are imposed. Recalling that the potential inside the spherical lipid aggregate is the sum of the bare potential ~o(r) of the charged head groups and of the polarization potential 0~(r) induced at the sharp lipid-water interface, the boundary conditions are (Bottcher, 1973):

[0o(r) + iPi(r)] [R = 0~(r)lR (19a)

eiO[O°(r)+~bi(r)][R=[e(r)O~b~(r)][Or J R (19b)

lim ~be(r) = 0 (19c)

lim 0~(r) = finite, (19d) r~O

where ~ ( r ) and O~(r) are the internal and external to the spherical aggregate potentials, whereas e~ and e(r) are the corresponding dielectric constants. In most of the cavity models e~ = 1. Since inside the sphere there are not free moving charges, ~b~ (r) must obey the Laplace's equation V 2r ~/'~ (r) = 0. Assuming that the electric charge


density or(r) is uniformly smeared over the amphi_phile-water interface [i.e. or(r)= ( Q / 4 ~ r R 2 ) 6 ( r - R ) , Q being the net charge of the sphere of radius R], it is easy to prove that inside the sphere the only solution satisfying the condition (19d) is 0~(r) = Ai, whereas, at the interface we have 0o(r)ln = Q / R . Inserting these results into eqn (19) and making use of eqns (15) and (16) we find:

and

07'(r)lR = AI °' + -Q-Q (20a) R

°G°'(r) R Q Or = - R-- 5 (20b)

0~2'(r)lR = A~ 2' (21a)

R O0~°J(r) =0, (215) gO~2~(r)or +(%-ew)O~2'(r)ln Or

where the relationship: Oe/O¢14, = %-e,v has been used. The solution of eqn (16a) with the boundary conditions (20a, b) gives:

¢,7'(r) 1 Q 1 exp [_Ko(r_ R)]. (22) I+KoR r

From which, making use of eqn (20a), the zeroth-order term of the electrostatic potential inside the micelle can be calculated. Its knowledge allows one to calculate the electrostatic solvation energy by the relationship (Bottcher, 1973):

Eso,v = ½ f ? or(r')$,(~) d~ 3. (23)

To the contribution calculated by eqn (23) we must add the repulsion between the charged heads, which, assuming the above expression for or(r) can be easily calculated yielding:

E~p = +Q2/ 2R . (24)

Combining the previous equations we have in the zeroth-order approximation:

0 2 ( 1 1 ) Q2 E~,- 2R 1 g l + ~ o R +2---R

1 Q2 - - + O(Ko2R-4).

g 2KoR 2 (25)

The approximation 1 + KoR = KoR, which will be used several times in the following calculations, is valid since at physiological conditions we have Ko-0.1 A-~ and R = 200-600/~ for lipid vesicles.

We procede further by calculating the first non-vanishing correction to the zeroth- order solution. Inserting eqn (22) into eqn (16d) and combining this with eqn (15b),


we obtain a inhomogeneous differential equation which can be solved exactly by standard procedure giving:

e-Kor e+Kar G2'(r) = C, + C 2 - -

r r

-I- e-K°r d r ] 1 [~-~fr f (r ) e - ' r r d r - r I f ( r ) e+K°rr (26)

2Ko J where f ( r ) =- 3'1 e -3K°'/r~ + 2 Y2(1 + Kor) 2 e -3K°r/r-~ with 3'1 and 3'2 numerical constants.

C1 and C2 are two integration constants to be determined by applying the boundary conditions. From eqn (19c) it follows: C2=0. Making use of the above formulas and mathematical handbook (Gradshteyn & Ryzhik, 1980) we may easily calculate the above integrals obtaining:

~(e2)(r) C1 e-K°r " "~ e-K°r = + (3'1 - ~K~3'2) ~'(Kor)

r r

e-3Kor

+ 3'2 3r 3 , (27)

where: ~'(Kor)-= -E~(2Kor)+2 e+2K°rE~(4Kor). E~(x) being the so-called exponen- tial integral defined as: ~ e-Y/y dy (Abramowitz & Stegun, 1972). Since KoR >> 1, we may use (at least for vesicles) the following asymptotic expansion valid for large x: (Abramowitz & Stegun, 1972):

e-X( ' 2 ) E 1 ( x ) ~ - - x 1 - - + x - ' 5 - " " " x ( 2 8 )

Inserting this result into eqn (27) and making use of the boundary condition (21b) we may calculate exactly the integration constant Cl:

Ci = -- R----T- 3'1 •

Combining eqns (27) and (29) and recalling the condition (21a) we obtain:

@~e21(r)lR =A~ 2J R 3 3" ,

which, with the aid of eqn (23), eventually can be used to calculate the correction term to the solvation energy caused by the polymer inhomogeneous distribution. Adding this latter result to the zeroth-order result [reported it, eqn (25)], we get:

Z, 9 ~ 2 4 ~ 4 6 1 z~,e-At, z h A t , e N Eet=277 gKo-~ 47"r3 gZK3(kT)2A3

a 1 (% ~__Lqew'-~ kTK~ (1 + _e2)] • I-t 8rri ~ 0 " 2 4, 2AgkT]]" (31a)


The same procedure can be repeated for the case of an asymmetrical electrolyte. When we consider a solution containing monovalent ions containing also a small amount of divalent cations, a further contribution has to be added to eqn (31a), it is:

z3.~3e4 [ -~++ e 2 .X++ (t~p-Sw)2( + 3e2 "~] 87'/'2 g2K2kTA2 )~++..g_ + ~. a2Ep g3 1 4hgkT " (31b)

In deriving eqn (31) we used the following relationships between the lipid head groups area A, the radius R and the net charge Q:

- e

Q = zhXh -~ 4zrR 2 (32a)

NA = 4zrR 2, (32b)

where )(h is the fraction of charged lipids with charge equal to Zh e and N is the aggregation number.

Once the polymer-perturbed solvation energy has been calculated, one may gain informations on the structure of the lipid assembly. This goal can be reached starting from a complete description of the lipid aggregate energy contributions and by minimizing the total energy with respect to the relevant parameters of the aggregate. For the sake of simplicity, we adopt, with only minor modifications, a very simple model developed some years ago by Israelachvili et al. (1976).

According to this picture, we may partition the energy into two contributions, the first one arises from the hydrophobic forces acting at the hydrocarbon-water interface. Let 3' be the interfacial free energy per unit area and A the averaged lipid head groups area, this term can be written as: +3'A. The presence of ions and polymer molecules in solution may affect the numerical value of 3'. Usually the two effects are opposite, however, at large non-ionic polymer concentrations an overall decreasing of the interfacial tension is likely. The latter contribution includes the electrostatic, short-range repulsion forces between the heads and any specific interaction. The purely electrostatic contribution has been calculated by eqn (31) and it contains also the ions and polymer effects. The remaining contributions can be taken into account as in the two-dimensional van der Waals equation and equated to +Co~A, Co being a positive constant. When the lipid aggregate is built-up from different species, we have to consider two further terms: the mixing entropy contribution and the preferential interaction between specific pairs. Their analytical expressions have been already obtained in calculating the ions and polymer energies [see eqns (1) and (8)]. Adding together these terms we get:

1 to, ~ C o + C i ) ( ~ D N W~gs~ - TA + A A 3 ,~-4 + zlip ~ ~7- - - -~- ta,-~h + a_,(1 - X h ) ]

+ W.vP~h (1 - )~h) + kT[)fh log .~h + (1 - )(h) log (1 - Xh)], (33)

14

where:

and:

A . R A U D I N O A N D P. B I A N C I A R D I

2 2 Ct ~- 27.r zhe .

z e6[a3kT p'w'2 +e21 D ~ 4 ¢ r 3 ~3Ko3(kT)2 ~-~ 8zrX O2Ep g2 Ko 1 2X~kT .

By differentiating eqn (33) with respect to A and equating it to zero we obtain the searched expression for the polymer-perturbed area per lipid molecule:

-G / ,i J

~(Co+ C'"~'1'/2 r 3 D'4 ], -; / L 1- ico+c,g ) 2 *

A(o) where ..~q = (Co/y)~/2 is the lipid area at zero surface charge density. Equation (34) shows interesting features. In fact, the first term in the right-hand

side leads to an increasing of the lipid area on raising the polymer concentration because of the lowering of the electrostatic screening. Likely, this effect should be very small. In fact, even in the limiting situation where all the repulsion between the lipid head groups is due to the electrostatic interactions (i.e. Co = 0), we have: AeqO C ~-1/4 On the contrary, the latter term leads to a decreasing of the lipid area because o f the stronger lipid-electrolyte interaction induced by the polymer. The enhanced ion binding is consistent with early observations which showed the PEG ability in decreasing the surface potential of lipid vesicles (Maggio et al., 1976; Maggio & Lucy, 1978).

Other parameters can be easily calculated. For example, for a spherical micelle and a bilayer vesicle the aggregation numbers Nm and Nv are related to the equilibrium lipid area Aeq [-calculated by eqn (34)] by the relationships (Nagarajan, 1989):

9 v~ N,, -- 3 6 r r - T- (35a)

Aeq

(16rrl3°'~ (A'ql°-l)-' (35b) Nv = \-~-Vo / \ Vo

where Vo is the volume of a lipid molecule and 1o is the bilayer thickness, clearly showing an inverse relationship between the aggregation number and the surface area.

Making use of the above equations we may calculate the polymer influence on the critical micellar concentration (CMC), the general expression of which is given

POLYETHYLENE GLYCOL-INDUCED CELL FUSION 15

by the well-known formula (see, e.g. Nagarajan, 1989):

[1o 1 log XcMc Wag~rleq - Wtl ° = _ , (36)

kT

where (l/N)Wt~tgrl~q is the total energy of a lipid molecule forming the aggregate whereas Wt~ °t is the corresponding energy of the isolated molecule dissolved in the same medium. Inserting eqn (34) into eqn (33) we may easily calculate the energy of a lipid molecule at the minimum energy (equilibrium) configuration. The major contribution to the energy of the isolated amphiphile is the hydrophobic energy, which can be estimated to be: +ySo, So being the lipid surface in contact with the solvent. Inserting these results into eqn (36) we find:

1 "~ 4 8 N Waggr[eq 2 y l / 2 ( C o + C , . X 2 h ) ' / 2 - T 3/2 D X 4 __ tot = 4- O ( D - X h ) . (37) (Co+ c ,g~) ~:-

Since for most of the phospholipids the CMC in water solution is very small [~10 -~° M (Tanford, 1980)], it follows that the leading term in eqn (37) is the last one (W~°t). Therefore, owing to the decreasing of the interface tension y, the main effect of PEG at high concentrations should be the solubilization of the lipid aggregates. At low polymer concentrations (or for amphiphiles with a low CMC) a more complex situation could take place because of the PEG-induced stabilization of the aggregates. In the next section these conclusion will be compared with some experimental results.

We would like to end the theoretical section by investigating the ability of non-ionic polymers in inducing lateral phase separations of the membrane components in mixed lipid systems. In order to calculate the presence of miscibility gaps in a bilayer containing two lipid species, we need to express the excess energy (calculated at the equilibrium geometry) as a function of their concentration )(h and 1 - ,X 'h .

to t - - w t o t [ ~ - h - tot Recalling that W~xc- ~gg~ h~--XhWagg~(1)--(1---~h) ~ot can W~gg~(0) we calculate the analytical expression of W~°xtc by exploiting the result reported in eqn (37). Multiple minima (i.e. more phases) can be found by differentiating W~°~ with respect to -~h and equating the result to zero:

)fh + q_~.~,, + q3Pf3h + (38) kT log 1 - )~h ql "" "

Where:

q,=-- W L - - ~ + - ~ - ~ + D o ~ e q " - e q

q2---2 WL Aeo - - - 2~

q3 -~ - A~)----~ + D w i t h A ~ ) =

i/2

16 A . R A U D I N O A N D P. B I A N C 1 A R D I

'~I '~ o.o

/ I

0-5

FIG. 2. Graphical solution of the eqn (38).)(h is the stechiometric fraction of the charged lipid, whereas the numbers refer to the function )'2(-~h ) calculated in the absence of polymer ( 1 ), in the presence of low (2) and high (3) polymer concentrations, respectively. The asterisks mark the (multiple) roots of eqn (38).

Equation (38) can be solved graphically as qualitatively sketched in Fig. 2. Here we report yl(.,~h)=--kTlog.~h/(1--Xh) (the S-shaped curve) and y2(-Kh)------ --ql + q2-'~h + q3 .~3 as a function of "~h. The roots (underlined in figure by asteriks) are the intersection points between Y~(Xh) and y2(.~h). In the absence of polymer,

I yE(.,~h) (the curve 1 in figure) cross the straight line y = 0 at -~h > ~_ because of the electrostatic contribution in the rhs o feqn (38). When only weak interactions between like lipids take place, only one root (one phase) is present. The addition of small amount of polymer (curve 2) shifts the function Y_~(PCh) downwards which becomes

--3 slightly concave because of the term proport ional to Xh. Probably, this curvature is not large enough to cause a multiple crossing (more phases) between Y*(PCh) and yE(V~h). At very high polymer concentrations (curve 3) the lower interface tension makes the aggregate looser and the lipid areas increase. The reduced electrostatic repulsion increases the slope of the function y2(PCh) and the existence of three distinct roots (two phases) becomes likely. It would be interesting to compare these qualitative prediction with some scattered observations as it will be done in the next section.

4. C o n c l u s i o n s

Although the present model is very crude, it seems to give a better insight into the intriguing problem of the indirect interactions between charged lipid aggregates suspended in a polymer-containing electrolyte solution. We would like to summarize and discuss further some of the model main predictions.


(1) The presence of a non-ionic polymer strengths the lipids-electrolytes interactions as described by eqn (31). As we can see from a close inspection of the formulae, three factors mainly contribute: (a) the different polarity between solvent (water) and polymer. This is described by the last term in the r.h.s, of eqn (31a) which behaves as -----(ep-ew)Z[ep¢~+ew(1 _~)]-,9/2. It is only partially balanced by the slowly varying first term which behaves as: [epcb+ew(1-c~)] -u2. Comparable behaviour is shown by eqn (31b).

(b) The length of the polymer chains favours stronger lipids-ions interactions. This effect is described by the term (02Ep/O~2[e~) -~ contained in the r.h.s, last term of eqn (31a). The analytical expression of 02Ep/0~214, is reported in eqn (15) and it can be easily proved that it is proportional to the polymer concentration fluctuations (de Gennes, 1979). From its analytical structure we observe that the fluctuations are an increasing function of the polymer length, tending to an asymptotic value for large values of m. For example, putting 2w(T) /kT = 1, (b = 0.2 and m = 50 and assuming unchanged the monomer and polymer dielectric permittivities, we calculated that the polymer-l ipid energy of interaction is about 15 times greater than the monomer- l ip id one. However, by putting m = 100, this energy is only 17.5 times larger. The greater effectiveness of the polymer can be easily understood recalling the strong mixing entropy of the isolated monomers which makes unfavourable the formation of monomer-free regions near the spherical lipid surface. Furthermore, when the coefficient 2w(T)kT becomes comparable with the entropic term, the concentration fluctuations grows tending to the infinity and a phase separation leading to the formation of polymer-rich regions takes place. The PEG shows this transition above the room temperature, the longer chains having a lower transition temperature than the shorter ones, in qualitative agreement with the Flory's theory (Saeki et al., 1976; Boucher & Hines, 1978; Florin et al., 1984; Karlstrom, 1985; Atamar, 1987). These facts show that near the transition temperature (the value of which depends also on the ion concentration), the polymer fluctuations can favour the ion-l ipid interactions.

(c) The absence of any l ipid-polymer direct interaction enhances the ion binding to the lipid aggregate owing to the absence of competition for the lipid head groups.

It is worth noting that points a, b and c are fulfilled at the same time by PEG. This may explain the peculiar role of this polymer in inducing fusion between lipid vesicles. In fact, it is well-known that the fusion between lipid vesicles or cells is triggered by the binding of divalent cations (mainly Ca -'+ ) onto the negatively charged lipid head groups. The binding causes a local destabilization of the bilayer which rearranges itself passing through some ill-defined intermediates, eventually leading to the merging of the two adjacent bilayers (for recent reviews on that matter see, e.g. Presegard & O'Brien, 1987; Hoekstra & Wilschut, 1988). Therefore, any substance like PEG that enhances the lipid aggregate-electrolytes interaction should increase the rate of fusion. This prediction is consistent with experimental results. For example, Hoekstra (1982) observed a substantial fusion of phosphatidylserine (an anionic lipid) containing vesicles treated with 20% of PEG (M =6000) at subfusogenic concentrations of Mg 2+ or Ca z+. Similar results have been obtained in a more recent paper (Rupert et al., 1988). Furthermore, stronger lipids-ions

18 A. R A U D I N O A N D P. B 1 A N C 1 A R D I

interactions should be accompanied by other effects, most of them have been experimentally observed in PEG containing vesicles suspensions: a decreasing of the surface potential (Maggio et al., 1976; Maggio & Lucy, 1978), an increase of the transition temperature associated to the melting of the lipid chains (Tilcock & Fischer, 1979; Boni et al., 1984b), and an increasing of the order parameter within the hydrophobic region of the lipid bilayer (Hermann et al., 1983; Surewicz, 1983; Ohno et al., 1981; Boni et al., 1984a). Recent data concerning the effect of the polymer chain length on the fusion rate showed unambigously that very long chains inhibit the vesicles fusion (Rupert et al., 1988). This result is in apparent disagreement with our predictions. However, as previously discussed, the vesicle-ions interaction grows with the polymer length for short chains and then rapidly reaches an asymptotic value for long chains. On the contrary, the medium microviscosity steadily increases with the polymerization degree (de Gennes, 1979), therefore the vesicles encounter probability sharply decreases.

(2) Our model predicts a depletion of polymer's segments near the charged interface [see eqn (15b), with ep - ew <0 and oEEp/Ocb2I~>O]. Moreover, if we consider a negatively charged aggregate [Ot°~(r)< 0] interacting with an electrolyte solution containing divalent cations a further depletion of the polymer surface layers has to be considered.

(3) A more thorough analysis of the polymer effect on the fusion processes should take into account the role of the so-called osmotic forces. It has been shown a net attraction between two adjacent particles takes place as a consequence of the depleted concentration profiles associated with each surface. The concentration reduction in the gap relative to the exterior bulk solution gives rise to an osmotic effect that acts to draw the surfaces together (Fleer et al., 1984; Evans & Needham, 1988a, b). As we previously discussed, in the presence of charged surfaces facing an electrolyte solution, there is a depletion of polymer segments on the surface even when the two surface are kept away at infinite distance. Therefore, the phenomena described in this paper seem to be sinergistic with the osmotic ones, both favouring the fusion event.

(4) Until now we have disregarded the effect of the polymer on the solution-lipid interface tension y. At higher polymer concentrations y decreases and several properties of the lipid assembly could be modified. For example, our model predicts an increasing of the lipid surface area Aeq [see eqn (34)]; this effect is opposite to that calculated for low polymer concentrations, a decreasing of the aggregation numbers [see eqn [35)] and a raising of the critical micellar concentration [see eqn (37)]. Some indirect evidences of these trends are reported in the literature. For example, Arnold et al. ( 1983, 1986) showed that the partition coefficient of a lipid-like spin probe decreases exponentially with increasing PEG concentration, a result which is consistent with the smaller interface tension of the medium. Moreover, it has been proved that PEG promotes the exchange of lipids between lipid vesicles (Tilcock & Fisher, 1982), once again supporting the above hypothesis.

(5) No direct evidences of the PEG-induced lateral phase separation of the lipid bilayer components exist in the literature. However, there are convincing observations that high PEG concentrations causes the aggregation of intramembrane lipid


p a r t i c l e s ( I M I s ) as r e v e a l e d by f r e e z e - f r a c t u r e e l e c t r o n m i c r o s c o p y s tud ies

( R o b i n s o n et al., 1979; R o o s et al., 1983; H u i et al., 1985). A l t h o u g h o u r resu l t s [ see

e q n (38)] a re b a s e d on the a s s u m p t i o n tha t all t he b i l a y e r c o m p o n e n t s h a v e the

s a m e size a n d tha t o n l y o n e s p e c i e bea r s a c h a r g e , t he a g r e e m e n t b e t w e e n t h e o r e t i c a l

a n d e x p e r i m e n t a l d a t a c o u l d n o t be fo r tu i t ous . N e v e r t h e l e s s , m o r e e x p e r i m e n t s

m a d e on s i m p l e r a n d w e l l - d e f i n e d sys t ems as we l l as t h e o r e t i c a l i m p r o v e m e n t s are

n e e d e d to u n d e r s t a n d t h e i n f l u e n c e o f n o n - i o n i c p o l y m e r s on the i n h o m o g e n e o u s

d i s t r i b u t i o n o f t h e m e m b r a n e c o m p o n e n t s . T h e s a m e c a u t i o n s h o u l d be u s e d in d e a l i n g wi th o t h e r p o l y m e r s effects such as

t he i r ly t ic o r f u s o g e n i c p r o p e r t i e s w h i c h still r e m a i n l a rge ly u n e x p l a i n e d on the

bas is o f t he c u r r e n t t h e o r i e s .

This work has been supported by the Italian Ministero della Pubblica Istruzione.

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