Relevance of Hydrodynamic Effects for the Calculation of Outer Surface Potential of Biological Membrane Using Electrophoretic Data

In this paper, we present the results of a study on the infl uence of hydrodynamic effects on the surface potentials of the erythrocyte membrane, comparing two different models formulated to simulate the electrophoretic movement of a biological cell: the classical Helmholtz-Smoluchowski model and a model presented by Hsu et al. (1996). This model considers hydrodynamic effects to describe the distribution of the fl uid velocity. The electric potential equation was obtained from the non-linear Poisson–Boltzmann equation, considering the spatial distribution of electrical charges fi xed in glycocalyx and cytoplasmic proteins, as well as electrolyte charges and ones fi xed on the surfaces of lipidic bilayer. Our results show that the Helmholtz-Smoluchowski model is not able to refl ect the real forces responsible to the electrophoretic behavior of cell, because it does not take account the hydrodynamic effects of glycocalyx. This charged network that covers cellular surface constitutes a complex physical system whose electromechanical characteristics cannot be neglected. Then, supporting the hypothesis of other authors, we suggest that, in electrophoretic motion analyses of cells, the classical model represents a limiting case of models that take into account hydrodynamic effects to describe the velocity distribution of fl uid.


INTRODUCTION
The electric surface potential is an important control parameter of the cellular metabolic process (Cortez et al. 2008, Genet et al. 2001, Bruner 1986).The activity of most enzymes is affected by this potential, since it is able to alter the passive transport rates and local concentrations of substrates and anionic ligands (Cortez and Bisch 1993, Ahrens 1983, Nørby and Essmann 1997, Dong et al. 2007).It also directly infl uences on the morphological and mechanical properties of erythrocytes, as remarked by Cortez andBisch (1995, 1996), interfering in their movement through blood capillaries.
The decrease of the surface electric charge in erythrocytes can induce important structural molecular alterations, which may cause a reduction in the membrane elasticity and changes in shape and size of cells, IZAN M. SILVA JUNIOR, MARIA CLÍCIA S. CASTRO, DILSON SILVA and CÉLIA M. CORTEZ contributing to impaired physiological function and ageing process (Chen et al. 2007).The biconcave form of the erythrocyte is attributed to the appropriate balance among hydrostatic pressure, electrostatic forces and membrane tension (Katnik andWaugh 1990, Lopez et al. 1968); thus, the evaluation of cell surface electrical properties constitutes a basic condition to the understanding of its hydrodynamics and physiology.
Electrochemical aspects of the erythrocyte membrane have been studied based on diverse models.It is well known that, like other cells of mammals, erythrocytes have a net negative surface charge at physiological pH.Classically, the Helmholtz-Smoluchowski equation (HS-equation) is consider valid to estimate the electrical surface potential and charge density of cells from electrophoretic data (Eylar et al. 1962), but now its validity scale has been examined.Searching a more realistic mathematical interpretation for experimentally determined electrophoretic mobility (μ), several models have been formulated.Many of them take into account the structure of the glycocalyx and its infl uence on the cell peripheral regions (Chen et al. 2007, Mehrishi and Bauer 2002, Hsu et al. 1993, Levine et al. 1983, Sharp and Brooks 1985, Donath and Lerche 1980).
The study of electric characteristics of cell surface and the electric potential profi le across the membrane using electrophoretic data can effectively contribute to the study of alterations in diffuse electrical double layers in function of changes at the lipid bilayer surface, and vice versa (Pinto et al. 2014, Cruz et al. 2000, Heinrich et al. 1982).The diffuse double layer occurs in the electrolytic solution very near the lipid bilayer surfaces.This accumulation of electric charges on both sides of membrane affects its mechanical properties (Ziebert and Lacoste 2010), infl uencing in the affi nity, adhesivity, aggregation and endocytosis, as well as the cellular mobility and immunity (Chen et al. 2007, Pöckel-Fernandes et al. 2011, Nagura et al. 1973).
In this paper, we are presenting results of a study on the infl uence of hydrodynamic effects in the surface potentials of erythrocyte membrane.For this, we built a membrane model considering the spatial charge distribution fi xed on the glycocalyx and cytoplasmatic proteins, as well as electrolytic charges (mono and divalent electrolytes) and ones fi xed at lipidic bilayer.
Then, effects of electrophoretic mobility, ionic strength, and surface charges on the surface potentials and potential profi le were verifi ed based on two different models formulate to simulate the electrophoretic motion of a biological cell: the classical Helmholtz-Smoluchowski model and a model presented by Hsu et al. (1996).For the numerical simulations, we used values of eletrophoretic mobility given by Bateman and Zellmer (1956) for four different ionic strength values.

ELECTROPHORETIC MOBILITY AND THE CHARGES ON CELL SURFACE
It is well known that, under physiological pH and ionic strength conditions, erythrocytes of many animal species exhibit a negative electrophoretic mobility, especially due to charged groups present in the glycocalyx layer (Jitendra et al. 2002, Jan andChien 1973).There are strong evidences that the carboxyl group of sialic acid is mainly responsible for that negativity rather than phosphate groups of phospholipids (Eylar et al. 1962, Kabanov et al. 2008).
The discrepancy between the electrophoretic mobility of erythrocyte as experimentally determined and that estimated by HS-equation has motivated many studies (Chen et al. 2007, Mehrishi and Bauer 2002, Hsu et al. 1993, Levine et al. 1983, Sharp and Brooks 1985, Donath and Lerche 1980, Hsu et al. 1996, Wunderlich 1982).Levine et al. (1983) developed a model to study the electrophoresis of human erythrocyte which provided mobility values signifi cantly lower than those estimated by HS-equation.The model considered the glycocalyx as a permeable layer of polielectrolytic polymer molecules anchored to the cell membrane, and containing uniformly distributed charges.Stokes friction forces generated by segments of an idealized polymer were taken as dominant for the fl uid fl ow in the glycocalyx.Thus, the authors obtained an expression for μ in function of the glycocalyx thickness and the mean radius of polymer segment, concluding that μ for a cell is closely related to the properties of the membrane phase.
In 1993, Hsu et al. extended the mathematical treatment applied to a particle coated with a layer of charged polymer to the case of a continuous distribution of fi xed charge.Then, Hsu and Kuo (1994) examined the electrophoretic mobility for an ion-penetrate cell, considering the case of a linear or exponential distribution of fi xed charges.Hsu and Fan, in 1995, analysed the case when the permittivity of the membrane phase was position-dependent for low potential values and/or symmetrical electrolytes.As the models did not consider a nonlinear Poisson-Boltzmann equation, they did not conduce to an explicit form for the electrical potential distribution, and the electrophoretic mobility expressions derived contained an implicit integration.In 1996, Hsu et al. introduced in the model of planar particle covered by a membrane the case of asymmetric electrolyte solutions, examining the effects of the parameters characterizing the nature of the liquid phase on the electrophoretic mobility.

ADOPTED MODEL AND METHOD
Figure 1 shows the adopted model in our study.It is constituted by 4 different regions: electrolytic phase surrounding the cell or bulk phase (region 1), glycocalyx region (region g), lipidic bilayer (region b) and cytoplasmatic region (region 2).As schematized of this fi gure, proteins are generic proteins, including channel ones.
The extracellular bulk region is constituted by extracellular electrolytic solution, where the spatial charge density is denoted by 1 ρ .Homogeneous spatial distributions of electric charges are found in glycocalyx and cytoplasm regions, denoted respectively by the lipid bilayer, the glycocalyx extends from z=-h/2 to z=-h g , and the membrane surface is coincident with the glycocalyx surface (S g ), i.e., S g is the interface bulk phase/glycocalyx, where the surface electric charge density is denoted by g S Q .
The lipidic bilayer (region b) has dielectric constant f ε and thickness h, being delimitated by surfaces S 1 and S 2 , which correspond to interfaces glycocalyx/bilayer (z=-h/2) and cytoplasm/bilayer (z=h/2), respectively.We named 1 S Q and 2 S Q the surface electric charge densities on S 1 and S 2 , respectively.These charges are due to charged groups of phospholipid heads, being the phospholipid distribution in the two monolayers asymmetric (Gurtovenko andVattulainen 2007, Hall andLatorre 1976).It is known that in the erythrocyte membrane, phosphatidylserine and phosphatidylethanolamine are preferentially located in the inner monolayer, whereas phosphatidylcholine and sphingomyelin are found in the outer monolayer.Phosphatidylserine is pointed as mainly responsible to the negative charge on the inner membrane surface (Heinrich et al. 1982).
Our model considered that all charge distributions, spatial and surface, were homogeneous, and that the charge density Q S1 was negligible, because of the dominant presence of neutral phospholipids in the outer layer of bilayer.In addition, due to the high hydrophobic degree in the lipidic bilayer, b ρ was taken as being vanish ( b ρ =0) (Cortez et al. 2008).

Electric Potential Equations.
For the adopted membrane model, the Poisson's equation is written as whose solution gives the electric potential φ(x,y,z) in any point of region i (i=1,g,b,2), and i ρ is the volumetric charge density due to electrolytes in solution, and As electric charges were homogeneously distributed on all surfaces, the electric potential in Eq. 1 was only depended on coordinate z.In bilayer, this equation assumes the form ( ) where j S φ is the electric potential on S j ( j=g,1,2), is molar concentration of γ at surface S j .Eq. 2 is the Boltzmann distribution of charge due to ion γ in phases adjacent to the bilayer.
Substituting Eq. 2 in Eq. 1, and considering the electroneutrality condition for solutions ( ), we have and considering only mono and divalent ions in the model, we have that where β = e/KT, γ = and .Taking into account the boundary conditions mentioned above and that sinh (2βu) = 2sinh (βu) .cosh (βu), the integration of Eq. 4 gives which is the equation for glycocalyx (i=g and j=1) and cytoplasmic regions (i=p and j=2), where For bulk regions (i=1 and j=g), where 0 i f ρ = , Eq. 5 becomes Outer Surface Charges.In the classical Helmholtz-Smoluchowski theory, μ is defi ned as μ = U/E, being U the steady-state velocity of the charged particles moving in the electrophoretic fi eld, and E is the electric fi eld strength, and it can be written as where ( ) φ the cell surface potential and 0 i φ is the outer bulk potential (potential at the steady-state region, far from the membrane surface), which is normally considered vanish; ν is the medium viscosity.Eq. 7 is the HS-equation.It is normally used in problems involving electrophoresis of rigid particles, and was considered as a good approach to biological cell in the past.However, this equation neglects both, IZAN M. SILVA JUNIOR, MARIA CLÍCIA S. CASTRO, DILSON SILVA and CÉLIA M. CORTEZ the dimensions of glycocalyx, where there is a spatial distribution of electric charges, and the radius of cellular curvature.
In the model of Hsu et al. (1996), μ is estimated considering the cellular movement in the electrophoretic fl uid, and the distribution of fl uid velocity is described by the Navier-Stokes equation: where ν is the medium viscosity, ζ denotes the membrane friction coeffi cient, and E is the electric fi eld strength.For convenience, they assumed that the coordinate moved with the cell, and considered the following boundary conditions: and / x 0, when x 0, where U is constant.
The mobility μ of the cell, considering that μ = U/E, can be estimated by where is dimensionless potential, d ′ is the adimensional glycocalyx thickness and X the adimensional distance.In this case, , and X x

D k =
where is the surface potential, d is the glycocalyx thickness, D k is the inverse of Debye length, is a measure of the asymmetry degree of electrolytes, F and R are the Faraday and ideal gas constants, respectively, and T is the absolute temperature.In addition, , ( ) ( ) where These expressions suggest that the mobility is related to the electric potential at the interface bulk phase-glycocalyx and the electric potential distribution in the glycocalyx region.
From the boundary conditions and Eq. 8, Hsu et al. (1996) shows that where ( ) ( ) Using these equations, it is possible to estimate the potential at glycocalyx surface (cell surface, S g .The analytical solution for Eq. 5 and Eq. 6 is not trivial.Then, the integrations through glycocalyx and cytoplasm regions were performed by means of numerical computation, using the forth order Range-Kutta method.All programs were built in C language and Table I shows some numerical values used in calculations. To estimate g S φ and g S Q , we used Eqs 18 and 19 (Hsu et al. 1996), and Eq. 7 (HS-equation), applying the μ-values to four different ionic strength values taken from Bateman and Zellmer (1956), Table II.In agreement to observations of other authors (Chen et al. 2007, Mehrishi and Bauer 2002, Hsu et al. 1993, Levine et al. 1983, Sharp and Brooks 1985, Donath and Lerche 1980), these results evidence the limitation of the HS-equation to estimate the surface potential for cells.It is a good electrophoretic model for charged and rigid surface particles moving through electric fi eld, but it seems very simple to refl ect forces involved in electrophoretic motion of cells.The Helmholtz-Smoluchowski model do not compute the three-dimensional charged network that covers the biological cells, and the electric fi eld generated by this network is not small, and can be determinant for the calculation of μ.This variation cannot be neglected, and it shows that charges in glycocalyx and on bilayer surfaces contribute to the potential difference responsible for the motion of erythrocytes in an electrophoretic fi eld, although the electrical double layer extends over only about 1 nm from lipidic bilayer surface in physiological conditions.In addition, covering this surface there is a peripheral zone containing a glycoprotein layer, which extends over about 6 nm to the cell limit surface.This layer possesses a spatial distribution of ionogenic groups of sialic acid.(Chen et al. 2007, Mehrishi and Bauer 2002, Hsu et al. 1993, Levine et al. 1983, Sharp and Brooks 1985, Donath and Lerche 1980, Heinrich et al. 1982, Kawahata et al. 1990).
According to Eylar et al. (1962), the treatment of erythrocyte with neuraminidase is able to remove 95-100% of acid sialic from the cell surface, and it reduces μ.Removal by enzyme of 50% of total sialic acid from the outer cell layer can decrease the calculated surface charge of about 66%.It means that the electrophoretic cell motion would be specially produced by charges of glycocalyx.But our modelling evidences that the contribution given by the surface bilayer charges to the surface cell potential ( g S φ ) cannot be neglected.It is also important to consider that the precise distribution of the sialic acid molecules in the cellular surface is not known and this distribution is not independent to surface bilayer charge.
To study the effect of ionic strength, surface charges and transmembrane potential on the potential profi le across membrane, we considered two distinct conditions: (A) the density Q kept fi xed for all ionic strength values.All penetrate ions (Na + , K + and Cl -) maintained into electrochemical equilibrium, being the membrane permeability very high to Cl -, very low to K + and it was practically impermeant to others ions.The cellular electroneutrality was maintained by intra and extracellular chemical potentials due to Cl -and K + .Thus, the transmembrane potential (Δφ m ) signifi cantly changed with the ionic strength variation.To estimate Δφ m for each ionic strength, we used the Nernst equation for chloride, , which is known as a good approximation for erythrocytes, where [ ] in Cl − and [ ] ou Cl − are inner and outer chloride concentrations, respectively.The situation A simulated the experimental condition in which an enzyme (like neuraminidase) is used to remove charged groups from membrane surface, reducing then the surface charge without however altering the electrolytic equilibrium (Coakley andDeeley 1982, Mironov andDolgaya 1985).
In Figure 4, we can see the electric potential profi le for four different values of outer ionic strength: F 1 , 0.5F 1 , 0.25F 1 and 0.1F 1 (Table II), taking into account the condition B. In these last fi gures, values of g S φ and g S Q were also calculated based on the model of Hsu et al. (1996), using electrophoretic data given by Bateman and Zellmer (1956).
An exponential potential fall can be observed across the extracellular medium in Figures 3 and 4, and this fall becomes more pronounced with the increase of negativity on lipid bilayer surface.It is known that, experimentally, variations on the ionic strength and surface charges are interdependent.Any change in the fi rst naturally causes alteration in the ratio between the intracellular and extracellular ionic concentrations (Guyton 1984), as well as modifi es the bilayer and glycocalyx charges, due to the binding or liberation of ions (Chen et al. 2007, Cortez and Bisch 1993, Tatulian et al. 1988), causing variations of both the surface charge and the transmembrane potential.For fi xed surface charges, the reduction of outer ionic strength IZAN M. SILVA JUNIOR, MARIA CLÍCIA S. CASTRO, DILSON SILVA and CÉLIA M. CORTEZ (Figure 4) provoked a relevant deviation on the outer potential profi le, and a considerable change in the intramembrane electric fi eld.
According to Cortez and Bisch (1993), the alteration in the potential profi le is less pronounced when the ionic strength and surface charges vary together.They studied the behavior of the potential profi le of erythrocyte membranes using a simpler model that was described by a linear Poisson-Boltzmann equation, and they applied the HS-equation to estimate outer surface charge densities.The comparison of potential profi les shown in Figure 4 to those calculated by Cortez and Bisch (1993), for the same ionic strength values, evidences the important change in the intracellular potential profi le provoked by the inclusion of the term for representing the spatial charge density due to cytoplasmic protein ionic groups in the Poisson-Boltzmann equation.II), considering the situation B. Potential and charge density at S g were estimated based on the model of Hsu et al. (1996), Eqs.18 and 19, using membrane values shown in Table I, and electrophoretic data given in Table II.I, and electrophoretic data given in Table II.

CONCLUSIONS
Our results showed that values of electric potential of the surface cell generated by HS-equation ( HS φ ) do not refl ect the real forces responsible for the electrophoretic cellular behavior.However, the equation may be useful for approximating, since the graph of φ HS in function of μ was found between Associating the expression of Hsu et al. (1996) to our model, we found surface potential values and potential profi les for the erythrocyte membrane very distinct from those previously obtained for a model described by linear Poisson-Boltzmann equation (Cortez and Bisch 1993).The modelling of the cell membrane as a complex structure, including two outer charged interfaces separated by non-neglectful distance (glycocalyx length), evidences the need to review the concept of "cellular surface potential".The glycocalyx confi gures a complex physical system, and the study of cell electrophoresis should include hydrodynamic considerations, in addition to electrostatic data.Thus, our results are in agreement with the suggestion of Hsu et al. (1996) that the HS-model in analysis the electrophoretic motion of cells could be recovered as a limiting case of models which take into account hydrodynamic effects to describe the fl uid velocity distribution.
Figure 1 -Adopted model for erythrocyte membrane.

(
Figure1), whose solution is a family of linear function, and electric fi eld in bilayer ( conditions were: (a) for z→-∞ (i=1) and z→∞ (i=p), bulk regions, the electric potential tended to the limit values 1 , and the ionic concentrations assumed the values 01 Sγ η and 02 Sγ η ; (b) at z=-h g and z=±h/2, the electric potential continuity was considered, assuming the values g S φ on S g , 1 S φ on S 1 and 2 S φ on S 2 ; (c) at these interfaces, the condition of discontinuity of electric displacement fi eld is considered.The density i ρ an be obtained from the summation of concentrations of positive and negative ions,, where Z γ is the valence of ion γ, e is the electron charge, and i γ η is the molar concentration of ion γ.The electrochemical potential for ionic solutes in a diluted solution is given by term due to the ionic concentration, refers to electric potential, and KT are the Boltzmann's constant, and absolute temperature, respectively.Using the electrochemical potential equation and taking into account the homogeneity of charge distribution on bilayer surface and the Boltzmann's electrochemical equilibrium (

FigureφSφ
Figure 2(a) and 2(b) show how, according to our model, potentials g S φ , 1 S φ and 2 S φ vary with the increase of μ and outer ionic strength, when surface charges were kept fi xed.The potential 1 S φ was the most infl uenced by the increase of these parameters.Examining these fi gures, it can be observed that the plot for HS φ , Figure 2(c) evidences the fall in the potentials is important to observe the linearity in the three plots.An increase of 62% in Q S1negativity caused a decrease of about 33% in the g S φ -value (Figure 2(c)).
, all penetrating ions (Na + , K + and Cl -) maintained into electrochemical equilibrium and the transmembrane potential (Δφ m ) was approximately fi xed; (B) the ionic strength varied

Figure 4 -
Figure 4 -Electric potential profi le for four values of ionic strength (F): F 1, 0.5F 1 , 0.25F 1 and 0.1F 1 (F 1 is reference value, TableII), considering the situation B. Potential and charge density at S g were estimated based on the model ofHsu et al. (1996), Eqs.18 and 19, using membrane values shown in TableI, and electrophoretic data given in TableII.
our model equations), and φ HS -values were closer to φ Sg -values than those for 1 S φ .

TABLE II Electrophoretic Mobility versus Ionic Strength.
Bateman and Zellmer (1956).IZAN M. SILVA JUNIOR, MARIA CLÍCIA S. CASTRO, DILSON SILVA and CÉLIA M. CORTEZ Table I shows values for charge densities.