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## Brazilian Journal of Physics

##
*On-line version* ISSN 1678-4448

### Braz. J. Phys. vol.30 no.2 São Paulo June 2000

#### http://dx.doi.org/10.1590/S0103-97332000000200023

**Solution of non-linear Poisson-Boltzmann equation for erythrocyte membrane **

Frederico A. O. Cruz, Fernando S. D. S. Vilhena, and Célia M. Cortez

*Universidade do Estado do Rio de Janeiro*

*Departamento de Ciências Fisiológicas*

*Av. Prof Manuel de Abreu, 48 *

*20550-170, Rio de Janeiro, RJ - Brazil*

**Received 25 October, 1999 **

The space-dependent electric potential equations were determined by solution of non-linear Poisson-Boltzmann equation, which decribes an erythrocyte membrane model which takes into account the charges of molecules associated with lipidic bilayer (glycocalyx molecules, outer side, and spectrin, inner side). It was assumed that on both sides of the membrane the charges are homogeneously distributed. Therefore, the potential was dependent on the z-coordinate only. Thus, we obtained the potential profile curve. The results are compared to previous ones obtained from more simple membrane model (Bioelectroch. Bioenerg. 1993,32:305-315).

**I Introduction**

It is well known that the passive transport of charged particles across biologic membranes is strongly affected by electrostatic forces and that the flux rates are dependent on intramembrane field forces [1-5]. But realistic values of membrane potential from experimental searches are obtained only for transmembrane potential (i.e., difference of potential between two points distant from outer and inner surfaces) [6]. The determination of potential profile, which describes the potential behaviour on all adjacent points along perpendicular axis to membrane, is performed based on theoretical models [6, 7, 8]. This is a way to estimate the potential in the vicinity of the membrane. However, informations from experimental searches are essential to formulate such mathematical models which allows the calculation of the membrane potential in function of space. Thus, several models have been suggested for biologic membrane and several electrostatic theories have been developped [1, 2, 6, 9]. Some modern concepts, such as interionic correlations and surface conductivity, were already introduced within of the Gouy-Chapman theory, being its range of validity still discussed [10]. According to Cevc [6], the effects due to surface irregularities and to interfacial hydration decrease the significance of interionic correlation effects in several membranes.

In previous work [1], we associated some recent knowledge on membrane electrostatics with classical Gouy-Chapmann model to describe erythrocyte membrane. For this model we performed a systematic study on the effects of ionic strength change on the electric potential profile. In the present work, we determine the potential profile for a complex membrane model, which takes into account electric charges of glycocalyx and spectrin layers. Donath and Patushenko [11] already have taken into account space charges of the glycocalyx for calculations of the outer surface potential. Heinrich et al [2] used space distributions of charges of glycocalyx, spectrin layer and hemoglobin for numerical calculation of potential profiles. Here, we determined the analytical solution for non-linear Poisson-Boltzmann equation, which includes parameters due to fixed charges on glycocalyx and spectrin. The results are compared with previous results obtained for a simple model [1]. Analytical solution of the electric potential in neighboring of the membrane is of special importance for studies involving more complex mathematics, such as the study of mechanical stability of membrane by hydrodynamic analysis [12-16].

**II Adopted model **

As already known, the erythrocyte membrane is constituted by a double bilayer of phospholipids, cholesterol and proteins [17], being asymmetric by distribution of phospholipids between the two lipid monolayers [18]. Molecules of phosphatidilserine and phosphatidylethanolalanine are preferentially located in the inner monolayer and phosphatidylcholine and sphingomyelin are founded in the outer monolayer [19]. In the outer surface there is an additional layer rich in polysacharides, which are associated with the bilayer molecules, this is called the membrane glycocalyx layer. On the inner surface there is a protein trap that supports the membrane, the spectrin [20]. In a solution of NaCl at pH 7.0, erythrocytes exhibit a negative electrophoretic mobility. This is especially due to charged molecules of the glycocalyx layer [6]. In the same form, spectrin has a net negative charge at physiological pH [21]. In addition, all membranes are embedded in diffuse inorganic salt double layers [6].

Based on these knowledge, we adopted for our study the model shown in Fig. 1(a) and (b). The system is formed by five regions: extracellular bulk region (phase 1), glycocalyx layer (phase g), lipidic bilayer (phase f), spectrin layer (phase p) and cytoplasmatic region (phase 2). Within glycocalyx and spectrin layer there are space charge distributions denoted respectively by r_{g} and r_{p}. The charge densities of these two later layers are considered almost constant in the z direction. The lipidic bilayer has dielectric constant e_{f} and thickness h (extends from z=-h/2 to z=h/2), and it is limited by surfaces *S*_{1} and *S*_{2}, as can be seen in Fig. 1(b). At these points the glycocalyx/bilayer and spectrin/bilayer interfaces are located, respectively. Similar to the Heinrich model [1], the inner surface charges are due only to the head group of phospholipids. So, there are electric charge distributions on S1 and S2 and their densities are denoted by *Q*_{S1} and *Q*_{S2} (Fig. 1(b)). The origin of the Cartesian coordinate axis is located at the bilayer centre. The electric potential depends only of the z coordinate, because charge distribution is considered constant along x and y axis. In aqueous phases, the electrochemical potentials, m_{i}, of each solute ion g are considered constant. The charge density r in the bilayer film is taken as zero (r_{i}(+) = r _{i(-)} = 0). *S*_{g} is the phase 1/glycocalyx interface and *S*_{p} is spectrin/phase 2 interface.

Figure 1. (a) Adopetd membrane model. The erythrocyte membrane is constituted by a lipidprotein bilayer, a outer layer rich in polysacharides, the glycocalyx, and a inner layer of spectrin. The bilayer surfaces located at positions z = ±h/2. The glicocalyx extends from -h/2 to -h_{g} and the spectrin layer extends from h/2 to h_{p}. (b) Phases 1 and 2 correspond to extracellular medium and cytoplasm medium, respectively. f_{1}(z) represents electric potential along z axis into phase 1, being f_{01} the potential for z ® -¥ and f_{Sg} the potential on z = -h_{g} (on S_{g} interface). In phase 2, we have f_{2}(z) as the electric potential along z, f_{02} is the potential for z ® ¥ and f_{Sp} is the potential on z = -h_{p} (on S_{p} interface). Phases g and p correspond glycocalyx and spectrin layer, respectively, being f_{g}(z) and f_{p}(z) the electric potentials along z axis in these phases. S_{1} and S_{2} are limit surface of bilayer and f_{S1} and f_{S2} are potentials on these interfaces. Q_{S1} and Q_{S2} are outer and inner surface charge density of bilayer. |

**III Calculation of potential profile **

The Poisson equation applied for the five considered phases is

and it solution give us the electric potential f(*x*,* y*, *z*) at any point in each one. The considered boundary are as follows (see Fig. 1 (b)):

- (I) When
*z*® -¥ and*z*® ¥, (a) the electric potential tends to limiting values f_{01}and f_{02}, respectively, and (b) the corresponding ionic concentration tends to limiting values h_{g01}and h_{g02}. In these regions one has the electroneutrality conditions: S_{g}*Z*_{g}*e*h_{g01}= 0 and S_{g}*Z*_{g}*e*h_{g02}= 0. - (II) At
*z*= -*h*_{g},*z*= ±*h*/2 and*z*=*h*_{p}, the electric potential is continuous through the surfaces and takes the values f_{Sg}at the surfaces*S*_{g}and f_{Sp}at surface*S*_{p}, f_{S1}and f_{S2}at*S*_{2}. - (III) In these surfaces the condition of discontinuity of the electric displacement vector is valid.

Solution within phases 1 and 2. The potential equation for 1 and 2 phases already was calculated in previous paper [1]. In this regions we have only aqueous phase. Using the Boltzmann equilibrium, symmetric electrolytes and electroneutrality condition in both phases we obtained,

*k*_{i} is related to the of Debye constant, f_{Sj} the surface potential on *S*_{j} (*j* = *g*, *p*), f_{oi} is the bulk potential in phase *i*, b = *Ze*/*KT*, *K* is Boltzmann constant, *Ze* is molar charge. The reference state is assumed to be stationary.

**IV Solution within glycocalyx and spectrin layers **

Within the Glycocalyx and spectrin layers the Poisson equation is non-linear due to addition of the term related to fixed charges on these molecular nets [2].

where r_{j} is the electrolytic charge density and r_{fj} refer to fixed charges into glycocalyx (r_{fg}) or spectrin (r_{fp}).

Using a Boltzmann distribution for the ions in aqueous phases and considering charge homogeneity on directions *x* and *y*, we have for each type of ions:

where *Z*_{0}*e* is the molar electric charge h_{0j} the corresponding molar concentration of ions in solution within the phase *j*, and e_{j} dielectric constant. In simplified form we have,

where

The first order solution of equation 6 give us

Applying the boundary conditions II on *S*_{g} e *S*_{p}, we have

where *E*_{Sj} = ¶f_{j}/¶*z*, at *S*_{j}.

Thus, we can write

being

The solution of equation (13) is not trivial, hence simplifications was required. Using Taylor's expansion, we obtained that

which is a infinite series, where *a*_{j} = (*l*/c_{j})^{1/2}.

Taking into account experimental data from literature for erythrocyte parameter values, and previous results [1], we studied the series terms. We found that, for *u*_{j} < 200mV, the terms decrease significantly as the *u*_{j}-exponent increases. So the higher order terms do not contribute appreciably and the equation can be reduced to

where *b*_{j} = (*l*/2) *g*_{j}*a*_{j}^{3}.

Solving the first order differential equation and considering electric potential continuity (condition II) at surfaces *S*_{1} and *S*_{2},

being

So, we can write: for glycocalyx region:

where

for spectrin region:

where

Thus, from equations 18 and 19 we obtain, respectively,

and

**V Solution within bilayer**

Considering that in lipidic bilayer charge density can be vanished, we have

being the solution of a linear function family.

Taking into account the vector electric field displacement discontinuity (condition III) on the *S*_{1} and *S*_{2}, and making some algebraic manipulations, we have:

being

**VI Results **

Applying the referent values of the erythrocyte membrane (table 1) in results obtained here and using the interpolation method, we could trace the potential profile curve (Fig. 2). Fig. 3 shows the potential profile for the simple model based on the Gouy-Chapman theory of difuse layers, presented in anterior work [1].

Table 1. Numerical values os the parameters used of the calculations |

Figure 2. Electric Potential Profile of the erythrocyte membrane calculated by use of the potential equations from analytical solution of nonlinear Poisson-Boltzmann equation. Numerical values adopted are shown in table 1. (Ionic strength = 172 mM). |

Figure 3. Electric Potential Profile of the erythrocyte membrane calculated by use of the potential equations from analytical solution of the linear Poisson-Boltzmann equation. Q_{S1} = 1.41 × 10^{-3} A.s/m^{2}, Q_{S2} = -9 × 10^{-2} A.s/m^{2}, h = 10 nm. (Ionic strength = 172 mM). |

**VII Discussion and conclusions **

The inclusion of the volumetric charge distribution into Poisson's equation (equation 1) led us to a Poisson-Boltzmann equation one with non linear characteristics and not trivial analytical solution. Thus, after search into possibilities we chose Taylor's series expansion to simplify its second integration (equation 13). However, we observed that the series (equation 15) was an infinit series, and did not converge for any potential value. Thus, it had required a detailed study to find the intervals where it became convergent, and if it was possible into expected interval of potential values, taking into account numerical values of the involved potential equation parameters. So, we investigated the contribution of each series term and we verified convergence for numerical values of erythrocyte parameters. Indeed, the values of the terms decrease significantly with increasing of *u*_{j}-exponent, considering a potential interval between 0-200 mV, or 0 £ (*u*_{i}-*u*_{j}) £ 200 mV. Thus, it became possible to obtain a analytical result for potential, which can be easely used for posterior studies involving more complex calcules. The potential profile curve shown in the Fig. 2 was traced based on these obtained equations.

Using numerical integration, Heinrich et al [2] obtained the potential profile for their membrane model, which considered glycocalyx and spectrin charges hemoglobin charges. They studied the influence of ionic strength on this profile. Their results showed that, for all osmotic states studied, a negative inner surface potential (*S*_{2}) of more than 60 mV was calculated. For a ionic strength of 145 mM the potential on the *S*_{2} was about -75 mV, and on *S*_{1} (outer bilayer surface) was about -10 mV. They also shown that at a decrease of ionic strength causes a increase of negativity of potential on *S*_{2}. Our potential profile (Fig. 2), which was obtained from analitical solution of Poisson-Boltzmann equation, presents the same shape than those shown by Heinrich et al [2]. We considered in our calculation a ionic strength of 172 mM.

In a previous paper [1], we adopted a simple model, which was based on the Gouy-Chapman theory of diffuse layers, in which space distributions of fixed charge of glycocalyx and spectrin layers was neglected. Comparing the curve of Fig. 2 with that of previous results (Fig. 3), we observed that the profile suffers some change when we took into account these additional charges. Within layers where there are fixed charges, the potential varies less pronouncedly than when these space distribution of charges are neglected. In this case (Fig. 3), the observed potential varies exponentially in both two sides, and the linear part of the inner curve presents a pronounced inclination. As can be seen in Fig. 2, the inclination of potential profile in the neighboring of the bilayer is smaller than that shown in Fig. 3 in the same place. In the simple model [1], we adopted a distance of 10 nm between the two parallel planes which limit the membrane region. So, the outer surface was located approximately at half the distance between the bilayer surface and the more extreme point of glycocalyx layer. All charges belonging to this membrane side (bilayer surface and neighbouring fixed charges) were taken into account as surface charge. Thus, in that case, we circumvented space charges by treating them as surface charges becoming trivial the solution of the Poisson-Boltzmann equation by analytical means. By confront of these two figures, it can be observed the importance of glycocalyx charges on potential within the outer membrane surface, as well as the potential within spectrin layer at the inner surface is mainly determined by the charges in this region.

Applying the analytical solution of the electric potential of the membrane in the hydrodynamic analysis, we can study the influence of these charges on the mechanical stability of the membrane. As can be seen in Fig. 2, in spectrin region there is an important electric field, and its effect on transport phenomena through membrane may be imperative, as well as on regulation of other membrane functions.

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