INTRODUCTION

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Most biomembranes are charged. These charges arise from charged headgroups of phospholipids, adsorbed ions, and proteins. Phospholipids, the basic structural component of membranes, are charged due to the dissociation of protons. Depending on the charges of additional groups that may be bound to the phosphate group, phospholipids in water can have a valency between 2 and +1, and also neutral groups are possible (Cooper, 2000). The state of charge of a phospholipid is not a fixed quantity, but depends on the pH and the ionic composition of the adjacent electrolyte solution. For this reason, a specific phospholipid group is best characterized by a chemical-binding constant rather than by a fixed charge.

Biomembranes are usually in a fluid state in which individual membrane components are free to move in lateral directions, i.e., within the plane of the membrane, whereas their normal movements are highly restricted (Almeida and Vaz, 1995). Depending on their specific biological function, membranes are composed of mixtures of many different lipids and amphiphilic proteins, and it is, in particular, the proteins that are decisive for their specific function. However, if more general properties of membranes are concerned, it often makes sense to neglect this diversity (and especially the proteins), and to study a model membrane solely made of phospholipids (Sackman and Lipowsky, 1995).

In this article, we study such a model membrane. It is assumed to be a collection of surface groups, specified not other than that they can become charged and that they are mobile in the membrane plane. The membrane shape changes are neglected. Different types of groups are allowed for, each type being characterized by a chemical dissociation constant rather than a charge. With such a model, we take account of three basic properties of a lipid bilayer: that it may be composed of different types of phospholipids, that the state of charge of each surface group is controlled by a pH-dependent chemical reaction, and that the surface groups can diffuse laterally.

Specifically, this article addresses the question of how such a model membrane interacts electrostatically with other charged objects in an electrolyte solution. The interaction between charged macroscopic objects in an electrolyte solution is, in fact, an "effective" one (Löwen and Hansen, 2000), meaning that, in addition to the direct Coulomb interaction between both objects, there is a contribution to the interaction energy coming from the distance-dependent density distribution of the electrolyte ions around both objects. More precisely, the effective interaction can be viewed as the free energy of the whole system (composed of both macroions and microions) as a function of the distance between the macroions. In a mean-field approach, the essential input to calculate this free energy, and thus the effective interaction, is the electrostatic mean-field potential; it can be obtained from a Poisson-Boltzmann (PB) (Barrat and Joanny, 1996; Andelman, 1995) boundary value problem (BVP), where the boundaries are the surfaces of the two objects carrying the fixed charges. Important here is the choice of the boundary conditions, which must be made on physical grounds. Besides the constant-charge and constant-potential boundary condition, fixing either the potential or its derivative at the boundary, a third boundary condition is well established, the charge-regulation boundary condition, where the surface charge is assumed not to be fixed, but to result from ionization of discrete surface sites (Ninham and Parsegian, 1971; Healy and White, 1978; Healy et al., 1980; Chan et al., 1976). The surface-charge density distribution is then a result, not a parameter, of the calculation; input parameters are rather the set of acid dissociation constants and the pH value.

This charge-regulation boundary condition is based on the assumption that the ionizable groups are locally fixed, and is thus not adequate for our case of a model membrane composed of mobile groups. This brings us to the major point of this paper. We derive a boundary condition for a PB BVP that goes beyond the traditional charge-regulation boundary condition by taking explicit account of surface group mobility. Once this point is clarified, the calculation of effective interactions isthough technically involvedconceptually simple. We then calculate the effective interactions between a charged rod and a charged membrane. Here we think of a DNA molecule interacting with a lipid membrane, which we see as a potential field of application of our results.

The issue of mobility of surface groups in an electrostatic context has been addressed before by Guttman and Andelman (1993) and Fogden and Ninham (1991), who investigated the interplay of a spontaneous curvature of a single membrane and the spatial modulation of the surface-charge density of mobile and immobile ions (Andelman, 1995). The effect of mobile surface charges has also been investigated treating the surface charges and counterions as strongly correlated two-dimensional (2D) liquids, which is a valid approximation at very large coupling parameters (i.e. low temperature or multivalent counterions) (Nguyen et al., 2000). Motivated by the recent interest in the DNA-cationic liposome complexes observed by Rädler et al. (1997) and Salditt et al. (1997), a sequence of theoretical papers appeared in which a periodic array of charged rods is considered that is adsorbed onto an oppositely charged surface with mobile charged groups (Menes et al., 1998; Dan, 1997; Bruinsma and Mashl, 1998; Harries et al., 2000; Wagner et al., 2000; Mashl et al., 1999; Mashl and Gronbech-Jensen, 1998). In the work of Harries et al. (2000), the appropriate boundary condition is derived by minimizing a free-energy functional. Quite recently, May et al. (2000a) considered the adsorption of charged proteins on membranes, taking explicit into account surface-group mobility. However, in all these works, the equilibrium between dissociated and associated surface groups was not considered. The case of a membrane consisting of equal amounts of negative and positive mobile lipids has been of special interest. The effective interaction between two fluid membranes is, in this case, solely due to correlation of in-plane charge fluctuations of mobile surface groups (Attard et al., 1988a; Pincus and Safran, 1998). The effect of such lateral charge fluctuations on the elastic properties of a membrane has been considered by Lau and Pincus (1998), and the effective interaction with test charges has been calculated using a generalized Green's formalism (Netz, 1999).

The outline of this paper is as follows. In the section Formulation of the Problem, we formulate the theoretical problem and present the results to make clear the underlying physics. The Theory section contains the formal solution that is derived from the grand-canonical partition function, a somewhat technical analysis that, however, is unnecessary to an understanding of the main result. In the Discussion, various simple limiting cases are considered to make the result more transparent and intuitively understandable. The next section is devoted to a typical application of our theory; we set up a PB BVP and calculate numerically the interaction of a charged cylinder approaching an oppositely charged wall consisting of mobile surface groups.

	FORMULATION OF THE PROBLEM

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We consider a charged surface S embedded in an aqueous electrolyte solution. In addition to the mobile electrolyte ions, there are ions on the surface that we assume to result from a dissociation of ionizable groups. We assume that there are M different types of such groups, each denoted by the symbol A_iH_vi (i = 1, ... , M). In water, these groups dissociate according to the reaction formula,

(1)

where v_i are the stoichiometric coefficients of the reaction and A denote the negatively charged ions that remain at the surface. The valency of the ion of type i, q_i, is therefore v_i. Each of the M different reactions in Eq. 1 is characterized by a dissociation constant K given by the law of mass action. For the moment, only simple acid reactions are allowed for, but generalization to basic groups is straightforward. Neutral surface groups are also included in the scheme, and can be realized by setting the corresponding dissociation constant equal to zero. We assume each surface group to cover some small area a² of the surface, which we assume to be the same area for every surface group type i. We can then regard the surface as being entirely composed of such groups. Every point on the surface belongs to one specific surface group. This leads to the idea of a regular lattice of site area a² being superposed on the surface, with each lattice site being occupied by one and only one surface group.

Inside the electrolyte solution and close to the surface, there is a charged object, which, for the moment, we need not specify further. Essential is that, in a mean-field description, the reduced electrostatic mean-field potential (r)that is, the potential multiplied by e with e being the elementary charge and  = 1/kT, the inverse temperatureis now a function of all three spatial dimensions. Because of the presence of the charged object, there is a variation of (r) directly on the surface. Let us denote the position vector on the surface, r  S, by r_S. Far away from the object, the surface potential (r_S) approaches the constant value . Note that this implies that the perturbation of the system due to the presence of the charged object is local.

What we calculate here is the partial surface density _i(r_S) of the ion type A for 1) a given surface potential (r_S), 2) a given pH value of the electrolyte solution, and 3) a given set of dissociation constants K (i = 1, ... , M). This we want to do under the additional assumption that the surface groups are free to move in the surface. To set the stage, let us briefly consider the opposite case of immobile surface groups, where our task is easily solved. In case (r_S) = , _i is a constant, , and the law of mass action reads

(2)

with exp(pH ln 10  ) the concentration of H⁺ ions at the surface, and c_i the number of surface ionizable groups of type i per area. Note that the concentration of water molecules is adsorbed into the definition of K. Hence,

(3)

where pK = ln K/ln 10. In the following, we refer to the ratio /c_i as the degree of dissociation _i. For neutral surface groups (K = 0), the degree of dissociation becomes zero. If (r_S) is now a function slowly varying on a length scale that is large compared to the lattice constant a of our regular lattice, then, Eqs. 2 and 3 should be valid for every single lattice cell and _i(r_S)/c_i results from simply replacing e^q_i by e^q_i(r_S) in Eq. 3. Expressing the resulting formula in terms of the degree of dissociation _i defined in Eq. 3, one obtains

(4)

with (r_S) = (r_S)  . In the same way, one obtains the surface density of the associated (A) groups A_iH_vi of type i, which we denote by (r_S),

(5)

Obviously,

(6)

for all points r_S on the surface. Eq. 4 then is the partial surface density _i(r_S) for given values of pH and K and a given surface potential caused by the presence of the charged object in the vicinity of the membrane. The main message of the last three equations is that the degree to which a certain ionizable group dissociates, now depends on its position on the surface. As a result of such a spatial dependence of the degree of dissociation, a 2D surface-charge distribution forms. Eq. 6 states, in essence, that the surface groups are immobile; a group at r_S can dissociate or not, but it can never leave its position, so that the surface density of the dissociated and associated species must everywhere add up to c_i.

Things are different if the surface groups can freely move in the interfacial plane. There are now two possibilities for the surface groups to respond to the surface potential (r_S). The first is the old one, the charge-regulation mechanism of adjusting the degree of dissociation to (r_S), which is still effective, as in the case of immobile ions. However, in addition, the free energy of the system can now be lowered further by allowing the surface charges to move to their most favorable position in the 2D surface potential (r_S).

The quantity that governs the movement of the surface groups is the set of chemical potentials µ_i for all types of surface groups. They regulate the exchange of surface groups with a reservoir. A change of sites between two groups of type i and j at lattice positions r_i and r_j is then to be understood as a process consisting of four steps: transferring particle at r_i to the reservoir (energy change µ_i), putting ion of type j from the reservoir to site r_i (+µ_j), removing particle at r_j (µ_j) to the reservoir and inserting particle of type i at r_j (µ_i). The net energy change for a site change of two groups is thus zero, which is why we say that the groups can move freely. If, however, there is a r_S dependence of the surface potential, an exchange of sites can cause a change of energy, because it is now the r_S-dependent electrochemical potential µ_i  q_i(r_S) rather than the chemical potential that regulates the exchange of sites.

With these few remarks, it should have become clear that the case of mobile surface groups is not simply a straightforward generalization of the results obtained for immobile ions, but that another charge-regulating mechanism is allowed for, and that more input parameters, as the chemical potentials of all groups, must now be incorporated into the theory. Starting from the grand-canonical partition function, we derive, in the next section, the following for the partial surface density of mobile ions of type i,

(7)

which is the pendant of Eq. 4, now for the case of mobile surface groups. We will also show that Eq. 5, for the case of mobile ions, becomes

(8)

and one can recognize already that Eq. 6 is no longer valid, a feature that best shows the basic difference between the case of mobile and immobile ions. We continue this discussion after having derived Eqs. 7 and 8.

Once we know _i(r_S) for all group types i, we can calculate the total surface charge density distribution _c(r_S),

(9)

which, via Eq. 7, still depends on the 2D surface potential (r_S). So far, we have assumed this surface potential to be a quantity known a priori. In practice, the spatially dependent electrostatic potential (r), and with it (r_S), must be calculated in a self-consistent way from the PB BVP in which _c(r_S) (and thus (r_S)) enter as boundary condition (see section DNA Near an Oppositely Charged Planar Membrane).

	THEORY

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We start with the grand partition function for a multicomponent electrolyte consisting of Q different types of ions, free to move in the three-dimensional configuration space GG*, where G is the configuration space for the whole system and G* = S  C. S is a 2D smooth manifold embedded in G, and C is the region occupied by an additional arbitrary distribution of fixed charges, denoted by (r), located somewhere in GS. On S, we define an regular lattice, i.e., the area per site is constant. Each site is occupied by one out of M different surface groups. The area per site can be understood as the size of the surface group; all surface groups are thus assumed to be of the same size. A surface group on site i can be in one of two possible states (associated/dissociated), which yields in total 2M possible states per site. We label each site n with a state variable S_n similar to the spin variable in the Ising model. S_n can be any integer between 1 and 2M. We introduce the particle density for the mobile electrolyte ions of type j in GG*,

(10)

where r denotes the position vector of particle k of species j, and N_j the total number of particles of type j. Similarly, we write for the density of surface groups of type i in S,

(11)

Here, P is the number of lattice sites and r_n is the position vector of lattice site n. All together, we have three different sorts of ions, mobile electrolyte ions (density (r)) in GG*, fixed ions in GS (density (r)) and charged/uncharged surface groups (density _i(r)) in S, and the total charge density reads accordingly,

(12)

with q_i (q_j) being the valency of the surface groups (bulk ions) (q_i = 0 for an uncharged group). These charges interact via the Coulomb interaction, (r, r'), so that the Hamiltonian of our system takes the simple form

(13)

We introduce the fugacities _j = e^µ_j/ and chemical potentials µ_j for the Q different types of bulk ions (_t the thermal wave length), and the fugacities and chemical potentials for the 2M different types of surface groups, _i = e^µ_i/ (i = 1, ... , 2M). The grand partition function of this system can then be written in the from,

(14)

With Eq. 14, we have brought our problem into a form well suited for applying standard field-theoretical methods. The details of what follows now are not specific to this calculation, and has been described elsewhere; we refer the reader, for example, to Netz and Orland (1999, 2000) Netz (1999, 2000), and continue with a more condensed description of the calculation. After renormalizing the fugacities to get rid of diagonal terms, a Hubbard-Stratonovich transformation leads us to

(15)

where

(16)

with  being a fluctuating field and (r) a dielectric field defined on G. To be able to calculate later the expectation values of the charge density operators, we introduce at this point, the generating fields h_i(r) and h_j(r), which couple to the densities _i(r) and (r), respectively. Resolving our abbreviations in Eq. 16, Eqs. 10, 11, and 12, performing the sums and making use of the series expansion of the exponential function, we can bring the partition function into the form,
with the abbreviation

(17)

This can be further simplified to

(18)

If the physical properties of the system vary on a much larger scale than the size of a lattice site, we can avoid the sum over a discrete lattice. Introducing the functional,

(19)

we can rewrite Eq. 18 as

(20)

We approximate the integral over all possible configurations by the configuration for which the partition function is stationary (saddle-point approximation),

(21)

where the mean-field potential _SP results from,

(22)

From the mean-field partition function, Eq. 21, we can now derive all quantities needed for the following. We start with the densities of the electrolyte ions; it can be obtained with the help of the functions h_j(r),

(23)

which yields

(24)

where we have introduced := i_SP. The bulk ion fugacities _j may be determined from the ion densities far way from the surface S and the fixed charge distribution  where one may safely assume that (r) = c with c being the concentration of electrolyte ions of type j ( q_jc = 0). This leads to _j = c. The densities of the surface groups in mean-field approximation can be calculated from (r_S = r  S)

(25)

resulting in the expression

(26)

Again the fugacities need to be determined. Henceforth, we denote the density of the associated species by , the fugacity of the associated species by _i, and that of the dissociated one by _i (i  {1, ... , M},  = e^µ). Furthermore, we set the valencies of the neutral surface groups to zero. For r_S far away from any fixed charge distribution (r) we expect a homogeneous density,

(27)

and hence,

(28)

This is an eigenvalue equation for the fugacities for the eigenvalue 1 with the eigenvector,

(29)

We determine the by means of the mass action law, Eqs. 2 and 3. At infinity, the ratio of and  = c_i   must be equal to _i/(1  _i) as defined in Eq. 3. In contrast, Eq. 26 yields / = e^q_i so that

(30)

Inserting the expression for _i and in Eq. 26 leads us directly to the main result of this paper, Eqs. 7 and 8.

The mean-field partition function provides us also with the grand potential ,

(31)

It is important to realize that this equation is only valid if we use the mean-field potential defined by Eq. 22 in H_G and H_S. Using Eqs. 17 and 19, we obtain for the grand potential

(32)

An interesting property of the system is that the partition function factorizes due to the mean-field description,

(33)

with Z_G[] := exp{H_G[/i]} and Z_S[] := exp{H_S[/i]}. Therefore, it is easy to extend our model to several independent lattice systems. Let us denote the kth of these lattices by S_k. The partition function for each lattice factorizes itself and is just the product of the partition functions of each single lattice site as can be seen in Eq. 18. Allowing on S_k, 2M_k different states on each site, we get for the partition sum of this sub-system Z_Sk

(34)

which, for a slowly varying field , can be approximated by

(35)

It is not needed that the lattices are spatially distinct. Due to this property, we are capable of describing a system of several interpenetrating lattices and thus modeling a surface with various immobile surface groups. The partition function for a system with L different lattices hence reads,

(36)

If we determine the fugacities for each lattice similar to the procedure for one lattice done above, we arrive at the following expression for the grand potential:

(37)

where G* now becomes S_k  C. If we specialize to one ionizable surface group on each lattice, i.e., two different states on each lattice site (M_k = 1), we readily arrive at the densities given by Eqs. 4 and 5 above.

Note that it is an inherent assumption for our present treatment of the model, that the fixed charge distribution perturbs the surfaces S_k only locally. Only under this condition we can chose the fugacities in the way we did above. Furthermore, we can show that, in the case of local perturbation, the relative number fluctuation of the particle of type i in the surface goes like 1/, where N is the total number of particle in the surface. Thus, in the case of large particle numbers, i.e. large surfaces, our description of the system is equivalent to the case where the particle numbers are fixed.

	DISCUSSION

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Having derived the two expressions, Eqs. 7 and 8, we now want to convey a more intuitive understanding of their meaning. For that purpose, we consider a few simple cases. We re-iterate beforehand that our result relies on the existence of a regular lattice with lattice constant a superposed on the surface, and that it is thus valid only if one can assume that all membrane components are of the same size and arrangeable on such a lattice. In both Eqs. 7 and 8, a² appears in conjunction with surface densities c_i (or _i), and the product c_ia² (_ia²) can be understood just as the surface fraction of species i. Because the surface is closely packed with groups, c_ja² = 1.

The simplest case is that the potential does not depend on the surface position vector r_S, either because the charged surface is well separated from other charged objects in the solution, or because of symmetry reasons (e.g., two parallel planar walls). Then (r_S) = 0 and Eqs. 7 and 8 reduce to _i(r_S)/c_i = _i and (r_S)/c_i = 1  _i. The same result is obtained if the surface groups are immobile (insert (r_S) = 0 in Eqs. 4 and 5). If there is no variation of the potential on the surface, there is no point in distinguishing between mobile and immobile surface groups; the mean-field potential is the same for both cases. This implies, for example, that the effect of mobility of surface groups cannot be studied in problems depending only on the spatial coordinate z on the mean-field level, as, for example, the traditional Gouy-Chapman problem of a single-charged wall bordering to an electrolyte solution.

If (r_S) is now taken to be a spatially varying function, then there is a difference between mobile and immobile surface groups, best to be seen from the following consideration. The total surface density of groups of type i at r_S, _i(r_S) + (r_S), is usually not equal to c_i, as can be seen by adding Eq. 7 to Eq. 8. Because c_i is the total surface density of groups when (r_S) = 0 (i.e., before switching on (r_S)), groups of type i must have moved after (r_S) was switched on, either by disappearing from or by coming to the point r_S. This is in direct contrast to the case of immobile ions where the total surface density of groups of type i remains unaffected by (r_S), and is always equal to c_i, see Eq. 6. Mobility of surface groups, however, does not mean that sites on the surface remain unoccupied:

(38)

which shows that the lattice site at r_S is always occupied by some surface groups, though not necessarily by that specific group it was occupied before (r_S) became nonzero.

We next remark that the distinction between mobile and immobile ions is again pointless if only one type of surface group is present, because the exchange of sites of two identical groups can have no energetic effect, and, indeed, Eqs. 7, 8, and 38 reduce to Eqs. 4, 5, and 6 if M = 1.

Let us now consider a surface composed of two types of mobile surface groups, the first of which can dissociate (degree of dissociation ₁ = ), the second not (₂ = 0). With Eqs. 7 and 9, we then find for the total surface charge density,

(39)

where we have used (c₁ + c₂)a² = 1. Specializing this expression further to the case  = 1, 

(40)

we have the surface charge density resulting only from the mobility of the groups. If, in contrast, we allow for only one type of surface group by setting c₁a² = 1, then Eq. 39 reduces to

(41)

which is identical to the expression one obtains starting from Eq. 4 for immobile groups. Eq. 41 thus represents the case of a surface charge density generated by a r_S-dependent dissociation of one type of ions, no matter if mobile or not. Both Eqs. 40 and 41 appeared in literature before: Eq. 40 has been derived by Harries et al. (2000) (see also (May et al., 2000a)), whereas Eq. 41 is the starting point of the classical paper of Ninham and Parsegian (1971) introducing the concept of the charge-regulation PB boundary condition.

Comparison of Eqs. 40 and 41 reveals the close relationship between dissociation and surface group mobility as the two basic charge-regulating mechanisms: Eq. 40 becomes identical to Eq. 41 if one sets the surface fraction c₁a² in Eq. 40 equal to the degree of dissociation  in Eq. 41. This is not surprising, because  measures the fraction of ions relative to the total number of groups, exactly as c₁a² does in the mixture of neutral and charged groups. Thus, we see that the case of a mixture of neutral and fully dissociated mobile surface groups is equivalent to the case of a surface made of