Abstract
I derive formulas for the electrostatic potential of a charge in or near a membrane modeled as one or more dielectric slabs lying between two semi-infinite dielectrics. One can use these formulas in Monte Carlo codes to compute the distribution of ions near cell membranes more accurately than by using Poisson-Boltzmann theory or its linearized version. Here I use them to discuss the electric field of a uniformly charged membrane, the image charges of an ion, the distribution of salt ions near a charged membrane, the energy of a zwitterion near a lipid slab, and the effect of including the phosphate head groups as thin layers of high electric permittivity.
Abstract
The plasma membrane of an animal cell and the membranes of the endoplasmic reticulum, the Golgi apparatus, the endosomes, and other membrane-enclosed organelles are lipid bilayers about 5-nm thick studded with proteins. The lipid constituents are mainly phospholipids, sterols, and glycolipids.
Abstract
Of the four main phospholipids in membranes, three-phosphatidylethanolamine (PE), phosphatidylcholine (PC), and sphingomyelin (SM)-are neutral, and one, phosphatidylserine (PS), is negatively charged. In a living cell, PE and PS are mostly in the cytosolic layer of the plasma membrane; PC and SM are mostly in the outer layer [1] ; and the electrostatic potential of the cytosol is 20 to 120 mV lower than that of the extracellular environment.
Abstract
After pioneering work by Gouy, Chapman, and Wagner [2] , and by Onsager and Samaras [3], many scientists have studied the electrical properties of cell membranes [4, 5] . This paper presents the exact electrostatic potential due to a charge in or near a membrane in the continuum limit in which the membrane is taken to be one or more dielectric slabs lying between two different infinite dielectric media. Because of the superposition principle, this monopole potential also gives the multipole potential due to any array of charges in or near a membrane.
Abstract
One can use these formulas to simulate the interactions of ions with other ions and with fixed charges near membranes while modeling water and other neutral molecules as bulk media. For instance, one can use them in Monte Carlo simulations to compute the behavior of salt ions and protons in water near neutral or charged membranes even in the presence of fixed charges of arbitrary geometry. This method is more accurate than the PoissonBoltzmann mean-field approximation and much more accurate than its linearized version [6] . These formulas also provide a context for and a check on all-atom computer * david38@example.net simulations [6, 7] .
Abstract
As early as 1924, Wagner [2] noted that an ion in water near a lipid slab induces image charges that repel the ion. No mean-field theory can describe this simple effect. But work-arounds are available for the PoissonBoltzmann theory [5] .
Abstract
Formulas for the electrostatic potential of a charge in or near a cell membrane modeled as a single slab are derived in section II. As pedagogical illustrations of their utility, I use them to compute the electric field of a charged membrane in section III and the response of bound charge to an ion in section IV. In section V, I use them to simulate the distribution of salt ions near a charged membrane. I discuss the Debye layer in section VI and the energy of a zwitterion near a lipid slab in section VII. In section VIII, I calculate the potential of a charge near a membrane modeled as several dielectric layers of different permittivities between two different semi-infinite dielectrics. I use this analysis in section IX to model a phospholipid bilayer as a lipid layer bounded by two layers of head groups of high electric permittivity. The phosphate head groups cause the membrane to attract rather than to repel ions. I summarize the paper in section X.
Abstract
In electrostatic problems, Maxwell's equations reduce to Gauss's law ∇ · D = ρ f which relates the divergence of the electric displacement D to the free-charge density ρ f (not including the polarization of the medium), and the static form of Faraday's law ∇×E = 0 which implies that the electric field E is the gradient of an electrostatic potential E = −∇V .
Abstract
Across an interface with normal vectorn between two dielectrics, the tangential component of the electric field is continuousn
Abstract
while the normal component of the electric displacement

The plasma membrane of an animal cell and the membranes of the endoplasmic reticulum, the Golgi apparatus, the endosomes, and other membrane-enclosed organelles are lipid bilayers about 5-nm thick studded with proteins. The lipid constituents are mainly phospholipids, sterols, and glycolipids.

Of the four main phospholipids in membranes, three-phosphatidylethanolamine (PE), phosphatidylcholine (PC), and sphingomyelin (SM)-are neutral, and one, phosphatidylserine (PS), is negatively charged. In a living cell, PE and PS are mostly in the cytosolic layer of the plasma membrane; PC and SM are mostly in the outer layer [1] ; and the electrostatic potential of the cytosol is 20 to 120 mV lower than that of the extracellular environment.

After pioneering work by Gouy, Chapman, and Wagner [2] , and by Onsager and Samaras [3] , many scientists have studied the electrical properties of cell membranes [4, 5] . This paper presents the exact electrostatic potential due to a charge in or near a membrane in the continuum limit in which the membrane is taken to be one or more dielectric slabs lying between two different infinite dielectric media. Because of the superposition principle, this monopole potential also gives the multipole potential due to any array of charges in or near a membrane.

One can use these formulas to simulate the interactions of ions with other ions and with fixed charges near membranes while modeling water and other neutral molecules as bulk media. For instance, one can use them in Monte Carlo simulations to compute the behavior of salt ions and protons in water near neutral or charged membranes even in the presence of fixed charges of arbitrary geometry. This method is more accurate than the PoissonBoltzmann mean-field approximation and much more accurate than its linearized version [6] . These formulas also provide a context for and a check on all-atom computer * david38@example.net simulations [6, 7] .

As early as 1924, Wagner [2] noted that an ion in water near a lipid slab induces image charges that repel the ion. No mean-field theory can describe this simple effect. But work-arounds are available for the PoissonBoltzmann theory [5] .

Formulas for the electrostatic potential of a charge in or near a cell membrane modeled as a single slab are derived in section II. As pedagogical illustrations of their utility, I use them to compute the electric field of a charged membrane in section III and the response of bound charge to an ion in section IV. In section V, I use them to simulate the distribution of salt ions near a charged membrane. I discuss the Debye layer in section VI and the energy of a zwitterion near a lipid slab in section VII. In section VIII, I calculate the potential of a charge near a membrane modeled as several dielectric layers of different permittivities between two different semi-infinite dielectrics. I use this analysis in section IX to model a phospholipid bilayer as a lipid layer bounded by two layers of head groups of high electric permittivity. The phosphate head groups cause the membrane to attract rather than to repel ions. I summarize the paper in section X.
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
In electrostatic problems, Maxwell's equations reduce to Gauss's law ∇ · D = ρ f which relates the divergence of the electric displacement D to the free-charge density ρ f (not including the polarization of the medium), and the static form of Faraday's law ∇×E = 0 which implies that the electric field E is the gradient of an electrostatic potential E = −∇V .
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
Across an interface with normal vectorn between two dielectrics, the tangential component of the electric field is continuousn
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
while the normal component of the electric displacement jumps by the surface density σ of free chargê
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
In a linear dielectric, the electric displacement D is the electric field scaled by the permittivity ǫ of the material
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The lipid bilayer is taken to be flat, extending to infinity in the x-y plane, and of a thickness t ≈ 5 nm. The interface between the extracellular salty water and the lipid bilayer is at z = 0. The permittivity ǫ ℓ of the lipid bilayer is about twice that of the vacuum ǫ ℓ ≈ 2ǫ 0 ; those of the extracellular environment ǫ w and of the cytosol ǫ c are about 80 times ǫ 0 .
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The potential of a charge q at a point (0, 0, h) on the z-axis is cylindrically symmetric, and so Bessel functions are useful here. In cylindrical coordinates with ρ = x 2 + y 2 , the functions J n (kρ) e inφ e ±kz form a complete set of solutions of Laplace's equation, but due to the azimuthal symmetry, we only need the n = 0 functions J 0 (kρ) e ±kz . We will use them and the relation [8] 1
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
to represent the potential of point charge at (0, 0, h).
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
If the charge q is at (0, 0, h) in the water above the membrane (h > 0), then we may write the potentials in the extracellular water V 
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
Imposing the constraints (1 & 2) , writing u(k) as u, m(k) as m, and so forth, and setting β ≡ qe −kh /4πǫ w and y = e 2kt , we get the four equations
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
In terms of the abbreviations
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
their solutions are
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
Inserting these solutions into the Bessel expansions (4) for the potentials, expanding their denominators
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
and using the integral (3), we find that the potential V w w (ρ, z) in the extracellular water due to a charge q at (0, 0, h) in that water is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
in which r = ρ 2 + (z − h) 2 is the distance to the charge q, and ρ 2 + (z + h) 2 is the distance to the principal image charge pq. Similarly, the potential V w ℓ in the lipid bilayer is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
and the potential V w c in the cytosol is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
If the charge q is in the lipid bilayer at (0, 0, h) with −t < h < 0, then the Bessel representations of the potentials are
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
With γ = qe kh /4πǫ ℓ and x = e −2kh , the Maxwell constraints (1 & 2) give us the equations
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
whose solutions are
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
After putting these solutions into the Bessel expansions (12) and using the integral (3) and the denominator sum (8) , one finds that the potential V ℓ w in the extracellular water due to a charge q at (0, 0, h) in the lipid bilayer is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The potential V ℓ ℓ in the lipid bilayer is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
Finally and somewhat redundantly, we turn to the case of a charge q in the cytosol at (0, 0, h) with h < −t. Now the Bessel expansions of the potentials are
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The continuity conditions (1 & 2) give us the four equations
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
whose solutions are
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
Thus using (3 & 8) in (18), we find that the potential V c w in the extracellular water due to a charge q at (0, 0, h) in the cytosol is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The potential V c ℓ in the lipid bilayer is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The potential V c c in the cytosol is
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
in which r is the distance to the charge and ρ 2 + (z + h + 2t) 2 is the distance to the principal image charge p ′ q.
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
wℓ , the series (9) (10) (11) for the potentials V w ℓ (ρ, z), V w w (ρ, z), and V w c (ρ, z) agree with those derived by the method of image charges [9] . The first eight terms of the infinite series (9-11, 15-17, & 21-23) give the potentials to within a percent. They are fast.
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
The first 1000 terms of the series (9, 10, & 11) for the potentials V (15, 16, & 17) (V) for ρ = 1 as a function of the height z above the phospholipid bilayer for a unit charge q = |e| in the phospholipid bilayer at (ρ, z) = (0, −4.5) (left curve, blue), (0, −2.5) (middle curve, magenta), and (0, −0.5) (right curve, red). Distances and permittivities are as in Fig. 1 . q = |e| in the extracellular medium at (ρ, z) = (0, 1) nm (right curve, red) and in the cytosol at (ρ, z) = (0, − 6) nm (left curve, blue). The errors due to the truncation of these series at 1000 terms are less than 1 part in 10
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
15 . The first 1000 terms of the series (15, 16, & 17) Figs. 1 & 2 show that the potentials fall off sharply as they cross the lipid bilayer. The reason for this effect is a build-up of bound charge in the water near the lipid bilayer due to the high electric permittivities of the extracellular environment ǫ w and of the cytosol ǫ c .
II. THE POTENTIAL OF A CHARGE IN OR NEAR A LIPID BILAYER
In sections VIII & IX, I will extend this derivation to the case of several dielectric slabs. This generalization will allow us to add two layers of high-permittivity dielectric that will represent the head groups of the phospholipids.
III. A SURFACE CHARGE ON A MEMBRANE
As a pedagogical application of the formulas of the preceding section, let us consider a uniform charge density σ on the surface of a lipid bilayer of thickness t. To avoid minus signs, I will put the charge density on the extracellular leaflet. After doing the computation, I will translate the result to the case of phosphatidylserine on the cytosolic leaflet.
III. A SURFACE CHARGE ON A MEMBRANE
Maxwell's jump equation (2) tells us that the electric displacement D w in the water differs by σ from its value D ℓ in the lipid, which in turn is the same as its value D c in the cytosol. So we have two equations D w = D ℓ + σ and D ℓ = D c for three unknowns. We can use the electrostatic potentials of section II to resolve this ambiguity.
III. A SURFACE CHARGE ON A MEMBRANE
The electrostatic potential V w w (ρ, z) due to a point charge at (0, 0, h) as given by (9) is equal to the electrostatic potential V w w (0, z) due to a point charge at (ρ, h). Thus setting the height h in (9) equal to zero, and differentiating with respect to z, we find for the z-component of the electric field at (0, 0, z) due to a charge q at (ρ, 0)
III. A SURFACE CHARGE ON A MEMBRANE
or
III. A SURFACE CHARGE ON A MEMBRANE
in which r = ρ 2 + z 2 and z ≥ 0. Replacing the charge q by σ 2πρdρ and integrating over ρ from ρ = 0 to ρ = ∞, we have
III. A SURFACE CHARGE ON A MEMBRANE
The dependence upon the variables z and t has dropped out. Doing the sum and using the definitions (6) of p and p ′ , we get
III. A SURFACE CHARGE ON A MEMBRANE
The field points in theẑ direction. In the limiting case in which the electric permittivity of the extracellular medium is the same as that of the cytosol, ǫ w = ǫ c , the electric field is E w (σ) = σ/2ǫ w .
III. A SURFACE CHARGE ON A MEMBRANE
We may apply similar reasoning to the formula (11) for the potential V w c (ρ, z) in the cytosol due to a charge q in the extracellular water at a height h above the membrane. If we keep in mind that the quantity z − 2nt − h is negative, then we find for the electric field in the cytosol of a surface-charge density σ at h = 0
III. A SURFACE CHARGE ON A MEMBRANE
Since there is no surface charge between the cytosol and the membrane, it follows from Maxwell's jump equation (2) that D ℓ = ǫ ℓ E ℓ = D c = ǫ c E c , and so that the electric field in the membrane is proportional to that in the cytosol
III. A SURFACE CHARGE ON A MEMBRANE
Our formula (28) for E c (σ) now gives E ℓ (σ) as
III. A SURFACE CHARGE ON A MEMBRANE
The jump in the displacement D across the layer of surface charge is
III. A SURFACE CHARGE ON A MEMBRANE
in agreement with Maxwell's equation (2) . If the layer of surface charge of density, like that of phosphatidylserine, lies on the cytosolic leaflet at z = −t, then the electric fields are
III. A SURFACE CHARGE ON A MEMBRANE
E c (σ) = − σ ǫ w + ǫ c .
IV. IMAGE CHARGES
As noted by Wagner [2] , a charge q in or near a membrane polarizes the membrane and the surrounding water. The potential formulas (9) (10) (11) , (15) (16) (17) , and (21-23) represent these bound charges as infinitely many mirror charges. The mirror charges affect the behavior of ions near an interface between two dielectrics in ways that mean-field theories can't describe.
IV. IMAGE CHARGES
For instance, a charge q in the lipid bilayer induces mirror charges in the cytosol and in the extracellular environment. These induced charges are of opposite sign, and they attract the charge q in the lipid membrane. We can be more precise about this attraction if in the formula (16) 
IV. IMAGE CHARGES
Keeping only the first term in each sum and using C for the constant log term, we recognize two image charges
IV. IMAGE CHARGES
familiar from freshman physics. They attract the charge q no matter what its sign. Water is better than lipid at attracting charges. Similarly, a charge q in the extracellular water induces a mirror charge in the lipid and others in the cytosol. The mirror charge in the lipid is of the same sign and, being closer, repels the charge q. We can describe this repulsion in terms of the formula (9) 
IV. IMAGE CHARGES
The first term is the potential of the textbook mirror charge
IV. IMAGE CHARGES
An ion of charge q in this potential has an energy proportional to q 2 p, which is positive for both cations and anions. A lipid membrane therefore repels both cations and anions; the water attracts the ion more than the lipid does. In a mean-field theory, such as unpatched PoissonBoltzmann theory, every particle responds to the same potential V (x), so the force qE(x) = −q∇V (x) is proportional to the charge q of the ion and therefore must be opposite for cations and anions. Mean-field theories can't describe why a lipid membrane repels both cations and anions. 
V. IONS NEAR A CHARGED MEMBRANE
One can use the formulas of Sec. II in a Monte Carlo code to compute the distribution of salt ions near a charged membrane. Here I present the result of such a simulation of the distributions of potassium and chloride ions near a membrane of a vesicle whose inner leaflet contains phosphatidylserine (PS) at a level of 4 percent, which is about that of the plasma membrane of a liver cell.
V. IONS NEAR A CHARGED MEMBRANE
In the simulation, I let the potassium and chloride ions move according to a Metropolis algorithm within a box whose width and length were 50 nm and whose height was 10 nm. I took the potassium concentration to be 150 mM so as to allow for a 10 mM concentration of sodium ions. The box contained 2258 K + ions. The bottom of the box was covered by a uniform negative surface charge density whose total charge was − 143 |e| corresponding to 143 phosphatidylserines at a molar density of 4 %. I used 2115 Cl − ions to make the whole system neutral; these chloride ions played the role of the whole ensemble of anionic cell constituents.
V. IONS NEAR A CHARGED MEMBRANE
To mitigate edge effects, I surrounded the box with eight identical boxes into which I mirrored all 4373 ions. So there were 39,357 ions in nine identical boxes. All the boxes had the same uniform surface-charge density due to the presence of the PSs at a level of 4 %. To strictly enforce periodic boundary conditions, one should use Ewald sums [10] , but I did not do this because they would have slowed the code down and because liquids are not crystals. Since the maximum step size in the zdirection was only 1Å; the error due to using only eight boxes of mirrored ions altered the energy difference ∆E of a Monte Carlo move by less than 0.064 kT , usually about 0.016 kT , far less than the thermal noise.
V. IONS NEAR A CHARGED MEMBRANE
In my Monte Carlo code [11] , I used the constant (27) for the electric field E z (σ) of the surface charge of phosphatidylserines, the series (35) for the self-potential V w w (z) arising from the response of the bound charge to an ion of charge q, and the sum (9) for the electrostatic potential V w w (ρ, z) due to a charge in the extracellular water near a membrane. In this way, I took exact account of the fields of the charges of the problem while treating the neutral molecules of the extracellular environment, the membrane, and the cytosol as bulk dielectric media. The simulations consisted of eight separate runs in which 23,000 sweeps were allowed for thermalization. Four of the runs collected data for an additional 50,000 sweeps; the other four for an additional 9,000 sweeps.
V. IONS NEAR A CHARGED MEMBRANE
In Poisson-Boltzmann theory [12] , all charges respond to a common potential V that obeys Poisson's equation with a charge density that respects the Boltzmann dis- (35) and that Vσ(z) = − σ |z|/(ǫw + ǫc) due to the electric field (32) of a lipid bilayer whose cytosolic leaflet is charged to a phosphatidylserine mole percent of 4% (red, solid curve) or 20% (blue, dot-long-dash curve) is plotted against the distance |z| (nm) from the leaflet. The uncorrected linear potentials Vσ(z) of the two surface-charge densities, 4% (blue dots) and 20% (red dashes), appear as straight lines.
V. IONS NEAR A CHARGED MEMBRANE
Here V is the electrostatic potential, ρ pf is a prescribed distribution of free charge, q i is the charge of species i and ρ f 0i is its free-charge density where V vanishes [12] . This non-linear equation is hard to solve except in onedimensional problems where the Gouy-Chapman solution [2] is available [12] . In the present context, that solution for the potential V (z) is [12] V (z) = − 2kT e ln 1 + e
V. IONS NEAR A CHARGED MEMBRANE
in which λ = 1/ √ 8πℓ B c ∞ , the Bjerrum length is ℓ B = e 2 /4πǫ w kT , and c ∞ is the bulk ion concentration (taken to be the same for potassium and chloride). If σ is half the absolute value of the surface-charge density of the phosphatidylserines, then the offset is
V. IONS NEAR A CHARGED MEMBRANE
The Gouy-Chapman formulas for the concentrations of the potassium and chloride ions (normalized to unity where V = 0) are then
V. IONS NEAR A CHARGED MEMBRANE
and Cl GC (z) = e eV (z)/kT . (40) In Fig. 3, I have plotted my Monte Carlo predictions for the relative concentrations of potassium ions ρ K (z)/ρ K (red solid curve) and of chloride ions ρ Cl (z)/ρ Cl (blue dashed curve) as functions of the distance z from the charged membrane. The for the normalized potassium K GC (z) (red, dotdash) and chloride Cl GC (z) (blue, dots) concentrations also are plotted there. The solid K + and dashed Cl − Monte Carlo concentrations correctly drop sharply for respectively z < 1 and z < 2 nm due to the repulsion by the induced image charges as discussed in Section IV. These concentrations are much lower than the GouyChapman predictions. The Gouy-Chapman-PoissonBoltzmann potassium concentration actually rises monotonically when the K + is less than 4 nm from the membrane. (The behavior of both ρ K (z)/ρ K and ρ Cl (z)/ρ Cl for z > 9 nm is an artifact due to the absence of ions at z > 10 nm in the simulation.)
V. IONS NEAR A CHARGED MEMBRANE
In Fig. 4 , I have plotted my Monte Carlo predictions for the average total potential that a K + ion (solid, red) or a Cl − ion (dashed, blue) feels due to the surface charge of the phosphatidylserines, to the other ions, and to its polarization of the three dielectrics. The induced bound charge sharply raises the potential felt by the potassium ion for z < 1 nm and lowers that felt by a chloride ion for z < 2 nm. The dot-dash magenta curve represents the potential due to the surface-charge layer of PSs and to all the ions (including the bound charges they induce) but without the image charges of equation (35). These three potentials differ significantly over the whole range in which the Gouy-Chapman concentrations differ from their bulk values. (The dip in the three potentials for z > 9 nm is an artifact due to the absence of ions at z > 10 nm.)
V. IONS NEAR A CHARGED MEMBRANE
In the related papers [9, 13] , I neglected the selfpotential.
VI. VALIDITY OF THE DEBYE LAYER
We have seen in the last two sections that mean-field theory cannot account for the behavior of ions near an interface between two dielectrics with two very different permittivities. Does this mean that the usual interpretation of the Debye layer is incorrect?
VI. VALIDITY OF THE DEBYE LAYER
The answer depends upon the difference between the two permittivities and upon the magnitude of the surface-charge density. An image charge is proportional to the ratio of the difference of the two permittivities to their sum. So if the dielectrics have similar permittivities, then the induced charges will be weak, and the image-charge correction to a mean-field Debye layer will be small. Similarly, a high surface-charge density will dominate the field due to the induced bound charges un-less the ions are very close to the interface. Thus the validity of the Debye layer depends on the relative magnitudes of the self-potential V w w (z) ≈ qp/8πǫ w |z| due to the image-charge correction (35) and the potential V σ (z) = −σ |z|/(ǫ w + ǫ c ) due to the electric field (32) of the phosphatidylserines. In Fig. 5, I have plotted for a K + ion both their sum V w w (z) + V σ (z) and the potential V σ (z). For the surface-charge density σ of a 4 mole-percent concentration of phosphatidylserine, as on the cytosolic leaflet of a liver cell, the sum V w w (z) + V σ (z) (red solid curve) differs from the surface-charge potential V σ (z) (red dashed straight line) for z < 2 nm. But for a higher mole percent of 20%, the relative difference between the sum V w w (z) + V σ (z) (blue dot-long-dash curve) and V σ (z) (blue dotted straight line) is somewhat less.
VI. VALIDITY OF THE DEBYE LAYER
The importance of the image-charge correction (35) rises as the surface-charge density falls. It is therefore particularly important in the case of an uncharged membrane, such as the outer leaflet of the plasma membrane.
VII. A ZWITTERION
Let us consider a simple model of a zwitterionic molecule in salty water above an uncharged lipid bilayer. The toy zwitterion is just a point charge q ′ at r ′ and another q ′′ at r ′′ . The charges are separated by s = r ′ − r ′′ which makes an angle θ with the verticalẑ so that z · s = s cos θ where s is the distance between the charges s = √ s 2 . The square of the horizontal distance between the charges is ρ
VII. A ZWITTERION
The midpoint of the molecule is r = (r ′ + r ′′ )/2 and its mean height is z =ẑ · r. The heights of the charges are z ′ = z + 
VII. A ZWITTERION
Similarly, the function u(z) gives the energy of the interaction of the other point charge q ′′ with the polarization it induces as
VII. A ZWITTERION
2 s cos θ). To find the energy E(r ′ , r ′′ ) of the charge q ′ in the full potential of the charge q ′′ and the energy E(r ′′ , r ′ ) of the charge q ′′ in the full potential of the charge q ′ , we use our formula (9) for the potential V w w (ρ, z). To compute E(r ′ , r ′′ ), we imagine the charge q ′′ to be at (x ′′ , y ′′ , z ′′ ) = (0, 0, h) and the charge q ′ to be at This interaction is unchanged when we interchange the locations of the two charges, and z
VII. A ZWITTERION
The total electrostatic energy of the molecule is then
VII. A ZWITTERION
In Fig. 6 , I plot the energy (44 ) of the model zwitterion when it is z nm away from a plasma membrane considered as a lipid slab with salty water on both sides. The four lower, solid curves are for a point charge q ′ = |e| separated by 0.5 nm from a charge q ′′ = −|e|; the four upper, dashed curves are for a point charge q ′ = 2|e| separated by 2 nm from a charge q ′′ = −|e|. The lipid slab repels the molecules.
VII. A ZWITTERION
One may find the energy of a more realistic zwitterion by integrating the two-charge formula (44) over a suitable charge distribution.
VIII. SEVERAL DIELECTRIC LAYERS
We may extend our derivation of the electric potential of a charge in a material of three dielectrics to the case of several dielectrics. One biological application is to a phospholipid bilayer considered as a layer of lipids bounded by two thin layers of head groups. These three layers with the extracellular water and the cytosol pose a five-dielectric problem.
VIII. SEVERAL DIELECTRIC LAYERS
Let us first consider the case of four dielectric layers with the charge q in the first layer of permittivity ǫ w in the region z > 0. Instead of the three potentials (4), we have four
VIII. SEVERAL DIELECTRIC LAYERS
in which the first internal layer of permittivity ǫ 1 fills the region −t 1 < z < 0 while the second internal layer of permittivity ǫ 2 fills the region −t 1 − t 2 < z < −t 1 . The constraints (1 & 2) give six equations
VIII. SEVERAL DIELECTRIC LAYERS
in which y 1 (k) = exp(2kt 1 ) and y 2 (k) = exp(2k(t 1 + t 2 )), while as in (5) the parameter β(k) = q exp(−kh)/4πǫ w represents the charge at z = h > 0. The functions m 1 (k) and f 1 (k) determine m 2 (k) and f 2 (k) as
VIII. SEVERAL DIELECTRIC LAYERS
We now address the problem of a charge q in a semiinfinite region of permittivity ǫ w at a height h above n internal layers of permittivity ǫ i and thickness t i which in turn are above a semi-infinite region of permittivity ǫ c . In the ith internal layer, the potential is
VIII. SEVERAL DIELECTRIC LAYERS
while the first and fourth equations of the set (45) describe the potentials in the two semi-infinite regions. The functions m i (k) and f i (k) determine those m i+1 (k) and f i+1 (k) of the next layer by matrix multiplication. If
VIII. SEVERAL DIELECTRIC LAYERS
in which y 0 = 1 and y i = exp[2k(t 1 + · · · + t i )] for i > 0. Let us set m 0 (k) = β(k) and f 0 (k) = u(k) as well as m n+1 (k) = d(k) and f n+1 (k) = 0. In these formulas, ǫ n+1 is ǫ c , and ǫ 0 is ǫ w , not the permittivity of the vacuum. If E ℓ is the product of ℓ + 1 of the matrices (50)
VIII. SEVERAL DIELECTRIC LAYERS
then we have
VIII. SEVERAL DIELECTRIC LAYERS
Setting i = n gives us
IX. THREE DIELECTRIC LAYERS
A phospholipid bilayer consists of a layer of phosphate head groups, a (double) layer of lipids, and a second layer of phosphate head groups. In this section, we will apply the formulas of section VIII to the problem of a charge q at a height h above such a bilayer. There are now three slabs and two semi-infinite regions. We must compute E (3) . I get
IX. THREE DIELECTRIC LAYERS
and E
IX. THREE DIELECTRIC LAYERS
Stern and Feller [14] , Nymeyer and Zhou [15] , and Baker [16] have estimated the relative electric permittivity of phospholipid membranes as being 1 from 0 to 10Å from the center, 4 from 10 to 15Å, 180 from 15 to 20Å, 210 from 20 to 25Å, and like bulk water beyond 25 A. I will approximate their results by using 2 from 0 to 15Å and 195 from 15 to 25Å and will take the relative permittivities of the cytosol and of the extra-cellular environment to be 80. These approximations greatly simplify ) at the point (ρ, z) due to a charge q = |e| on the z-axis at (0, h) above a three-slab phospholipid bilayer that is neutral (lowest, solid, blue) or has a 4 mole-percent layer of phosphatidylserine on its cytosolic leaflet (upper, solid, red) . Both ρ and h are 1 nm. The potential (9) of a charge above a single neutral lipid slab without head groups is plotted as a dashed magenta curve to illustrate the effect of the layers of head groups.
IX. THREE DIELECTRIC LAYERS
our formulas (54 & 55) and imply that p 0 = 0.418 = −p 3 and p 1 = −0.98 = −p 2 . The thicknesses of the layers are t 1 = t ≡ 1 nm, t 2 = 3t, and t 3 = t, and so y 1 = e 2kt , y 2 = e 8kt , and y 3 = e 10kt . With these simplifications, our formulas (54 & 55) reduce to
IX. THREE DIELECTRIC LAYERS
and E The key function u(k) then is by (53) the ratio
IX. THREE DIELECTRIC LAYERS
We can use it and our formula (45) to write the potential V w w (ρ, z) at the point (ρ, z) due to a charge q on the z-axis at (0, h) above a three-slab phospholipid bilayer as
IX. THREE DIELECTRIC LAYERS
(59) Figure 7 plots this potential V w w (ρ, z) for ρ = 1 nm and 0 ≤ z ≤ 5 nm for the case of a unit positive charge q = |e| on the z-axis at height h = 1 nm above a three-slab phospholipid bilayer that is neutral (lowest curve, solid, blue) or has a 4 mole-percent layer of phosphatidylserine on its cytosolic leaflet (upper curve, solid, red). The electric field of the PSs is taken from (32) to be E w (σ) = σ/(ǫ w + ǫ c ) in which σ is negative. To illustrate the effect of the layers of head groups with very high electric permittivity, I have replotted the potential (9) of Fig. 1 due to the same charge but above a single naked, neutral lipid slab as a dashed magenta curve. The head-group dipoles lower the potential V w w (ρ, z) in their vicinity. We also can use the ratio u(k) to compute the selfinteraction of a charge q with our three slab model of the phospholipid bilayer. The potential felt by the charge in the salty water due to all the image charges its presence induces in the three slabs and in the cytosol is
IX. THREE DIELECTRIC LAYERS
in which u(k) is the ratio (58) but with the height h of the charge q replaced by z. The potential felt by the charge q at z then is in which the parameter t is one nm. Figure 8 plots the electric potential felt by an ion of charge |e| near a phospholipid bilayer modeled as a lipid slab bounded by two thin polar slabs. The lowest (solid, blue) curve is for a neutral membrane; the upper (solid, red) curve is for a membrane whose cytosolic leaflet is negatively charged by phosphatidylserine at 4 mole percent represented as in Fig. 7 . The two thin layers of phosphate head groups make the potential felt by an ion near a neutral phospholipid bilayer attractive rather than repulsive. Without the outer polar slabs, the potential (35) is purely repulsive (dashed magenta). The head groups attract to the membrane ions that a naked lipid slab would repel. If the membrane has phosphatidylserines on its cytosolic leaflet, then it strongly attracts positive ions that it otherwise would repel. Apparently the phosphate head groups facilitate many physiological processes, such as the docking of ligands and the translocation and endocytosis of positive ions and cell-penetrating peptides.
X. SUMMARY
I derived the electrostatic potential of a charge in or near a lipid bilayer in section II and used it in section III to compute the electric field of a uniformly charged membrane and in section IV to describe the effects of image charges. In section V, I used the results of sections II-IV in a Monte Carlo computation of the distribution of ions near a charged membrane. I discussed the validity of the Debye layer in section VI and computed the energy of a zwitterion near a lipid slab in section VII. In section VIII, I calculated the potential of a charge near a membrane modeled as several dielectric layers of different permittivities between two different semi-infinite dielectrics. I used this analysis in section IX to model a phospholipid bilayer as a lipid layer bounded by two layers of head groups of high electric permittivity. The phosphate head groups cause a neutral membrane to attract rather than to repel ions.

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