2.4 Regularity of linear elliptic interface problems
3.1.1 Poisson-Boltzmann equation
Let Ω ⊂ Rd, d = 2,3 be a bounded domain with Lipschitz boundary ∂Ω whose outward unit normal vector is denoted byn∂Ω. The domain Ω contains two Lipschitz subdomains, Ωm and Ωs, which denote the molecular region and the solvent region, respectively. It is assumed that Ωm ⊂ Ω, i.e., the molecular region is strictly contained in Ω. Each of the two subdomains Ωm and Ωs is allowed to be a disconnected set, which can be represented as the union of Lipschitz domains. The boundary of Ωm is denoted by Γ, the interface between the molecular region and the solvent region, and its outward unit normal vector is denoted by nΓ. Finally, we can write Ω = Ωm∪Γ∪Ωs. There are several definitions of the molecular surface that have been used in practice, the most common of which is the solvent excluded surface (SES). The SES is formed by the contact points of the Van der Waals surface and a solvent probe sphere that is rolled over it (see [67, 94, 158, 165]). In 1983 Connolly gave an analytic description of the solvent excluded surface and therefore it is also known as the Connolly surface (see [53] for the piecewise analytic definition of this surface).
To model the electrostatic potential in a system of biomolecules with the presence of mov-ing ions the so-called ion exclusion layer (IEL) is introduced. This is a layer around the bio-molecules in which no ions can penetrate and it is defined as the difference between the union of the inflated Van der Waals spheres of the atoms by a counterion radiusRion and
3.1. PROBLEM FORMULATION 29 the molecular region defined by the SES. Alternatively, it is defined as the difference between the region, enclosed by the Connolly surface of the molecule with inflated Van der Waals spheres of the atoms by a counterion radiusRion, and the usual molecular region defined by the Connolly surface. We denote this region by ΩIEL. The part of Ωs without the ion exclusion layer ΩIEL is accessible for ions and we denote it by Ωions. With this notation, it holds Ωs= ΩIEL\Γ
∪Ωions (see Figure 3.1).
Figure 3.1: Computational domain Ω with molecular domain Ωm and solution domain Ωs= ΩIEL\Γ∪Ωions.
The electrostatic potential ϕ is governed by the Poisson equation which is derived from Gauss’s law of electrostatics. In CGS (centimeter-gram-second) units, the Poisson equation reads
−∇ ·(∇ϕ) = 4πρ, (3.1)
whereρ(x) is the charge density at pointxandis the dielectric coefficient, which is assumed to be constant in the molecule region Ωm and Lipschitz continuous in the solvent region Ωs with a possible jump discontinuity across the interface Γ, i.e.,
(x) =
(m, x∈Ωm,
s(x), x∈Ωs. (3.2)
In the molecular region Ωm, there are only fixed partial charges and therefore the charge density is
ρm =
Nm
X
i=1
zie0δxi,
whereNm is the number of fixed partial charges,zi is the valency of the i-th partial charge, xi its position, and e0 = 4.8032424×10−10 esu (= 1.60217662×10−19 Coulombs) is the elementary charge. For all electrostatics units and physical constants that we will be using see Table 3.1 and Table 3.2. The Poisson equation for the potentialϕin Ωm reads
−∇ ·(m∇ϕ) = 4πρm.
In the region ΩIEL there are no fixed partial charges, nor moving ions and therefore the charge density there isρIEL= 0. The Poisson equation for ϕin ΩIEL reads
−∇ ·(s∇ϕ) =ρIEL= 0.
In the region Ωions, there are moving ions whose charge density is assumed to follow Boltzmann distribution and is given by
ρions =
Nions
X
j=1
Mjξje0e−
ξj e0ϕ kB T ,
whereNionsis the number of different ion species in the solvent,ξj is the valency of thej-th ion species,Mj = #ions
cm3 is its average concentration in Ωions,kB= 1.38064852×10−16erg K−1 is the Boltzmann constant, and T is the absolute temperature. Therefore, the Poisson equation for the potentialϕin Ωions reads
−∇ ·(s∇ϕ) = 4πρions.
If we denote the total charge density in Ω by ρ, it holds ρ=ρm+ρIEL+ρions and we can write one equation in the whole computational domain Ω
−∇ ·(∇ϕ)−χΩions4π
Nions
X
j=1
Mjξje0e−
ξj e0ϕ
kB T = 4πe0
Nm
X
i=1
ziδxi =:F in Ω, (3.3a)
[ϕ]Γ = 0, (3.3b)
[∇ϕ·nΓ]Γ = 0, (3.3c)
ϕ = g on ∂Ω, (3.3d)
where [·]Γ denotes the jump across the interface Γ of the enclosed quantity and we have taken into account the continuity condition on the potential and the normal component of the displacement field∇ϕacross the interface Γ. We notice that in fact the physical problem pre-scribes a vanishing potential at infinite distance from the boundary of Ωm, i.e., lim|x|→∞= 0.
In practice, one uses a bounded computational domain and imposes the boundary condition (3.35d) instead, where the functiong∈C0,1(∂Ω) can usually be calculated accurately enough
by solving a simpler problem, possibly with a known analytical solution.
Under the assumption that there are only two ion species in the solution with the same concentration M1 = M2 = M, which are univalent but with opposite charge, i.e ξj = (−1)j, j = 1,2, we obtain the equation
−∇ ·(∇ϕ) +χΩions8πM e0sinh e0ϕ
kBT
= 4πe0 Nm
X
i=1
ziδxi in Ω. (3.4)
3.1. PROBLEM FORMULATION 31 By introducing the new functionsφ= ke0ϕ
BT and g= e0g
kBT in (3.4) we arrive at the equation for the dimensionless potentialφ
−∇ ·(∇φ) +k2sinh (φ) = 4πe20 kBT
Nm
X
i=1
ziδxi =:F in Ω, (3.5a)
[φ]Γ = 0, (3.5b)
[∇φ·nΓ]Γ = 0, (3.5c)
φ = g on ∂Ω. (3.5d)
The coefficientkis defined by
k2(x) =
0, x∈Ωm∪ΩIEL,
k2ions = 8πNAe20Is
1000kBT , x∈Ωions,
(3.6)
whereNA= 6.022140857×1023 is Avogadro’s number and the ionic strengthIs, measured in moles per liter (molar), is given by
Is = 1 2
2
X
j=1
ciξj2= 1000M NA
withc1 =c2 = 1000MN
A , the average molar concentration of each ion (see [12, 101, 141]).
Equation (3.5) is often referred to as the Poisson-Boltzmann equation [103, 143, 168]. When there are no ions present, Mj = 0, j = 1,2, . . . , Nions, Is = 0, k2 = 0 in Ω, ΩIEL = ∅, Ωions ≡Ωs, and equation (3.5a) becomes the linear Poisson equation of electrostatics.
−∇ ·(∇φ) = F in Ωm∪Ωs, (3.7a)
[φ]Γ = 0, (3.7b)
[∇φ·nΓ]Γ = 0, (3.7c)
φ = g on∂Ω. (3.7d)
The Poisson-Boltzmann equation (3.5a) can be linearized by expanding sinh in Maclaurin series. We obtain the linearized Poisson-Boltzmann equation for the electrostatic potentialφ
−∇ ·(∇φ) +k2φ = F in Ω, (3.8a)
[φ]Γ = 0, (3.8b)
[∇φ·nΓ]Γ = 0, (3.8c)
φ = g on∂Ω. (3.8d)
Remark 3.1
We note that the LPBE (3.8)and the Poisson problem (3.7)are often given for the electrostatic
potential ϕ, and not for the dimensionless potentialφ in order to avoid the scaling with the factor ke0
BT. The LPBE for the potential ϕ with dimension [ϕ] =hcharge length i
reads
−∇ ·(∇ϕ) +k2ϕ = F in Ω, (3.9a)
[ϕ]Γ = 0, (3.9b)
[∇ϕ·nΓ]Γ = 0, (3.9c)
ϕ = g on∂Ω. (3.9d)
We will use this form of the LPBE in Chapter 5.
For the subsequent analysis of the PBE and the LPBE we will need the functionG given by G=
Nm
X
i=1
Gi =− 2e20 mkBT
Nm
X
i=1
ziln|x−xi|, ifd= 2, (3.10)
G=
Nm
X
i=1
Gi = e20 mkBT
Nm
X
i=1
zi
|x−xi|, ifd= 3. (3.11) The function G describes the Coulomb part of the potential due to the partial charges {zie0}Ni=1m in a uniform dielectric medium with a dielectric constantm. It is well known that Gis the distributional solution of the problem
−∇ ·(m∇G) = 4πe20 kBT
Nm
X
i=1
ziδxi =F inRd, d∈ {2,3}. (3.12)
What is meant by (3.12) is that (see, e.g. p.106 in [173])
− Z
Rd
mG∆vdx=hF, vi, for all v∈C0∞(Rd). (3.13)
In particular, (3.13) is valid for all v∈C0∞(Ω). The functionGis weakly differentiable with a weak derivative equal almost everywhere to its classical derivative. Moreover,Gand∇G are in Lp(Ω) for allp < d−1d and thus G∈ T
p<d−1d
W1,p(Ω) (alternatively, we can use the AC characterization of W1,p(Ω) functions - see Theorem 10.35 in [122]). Therefore, by applying the integration by parts formula (see Theorem 2.14), for any v∈C0∞(Ω), we obtain
Z
Ω
mG∆vdx= Z
∂Ω
mG∇v·n∂Ωds
| {z }
=0
− Z
Ω
m∇G· ∇vdx
Z
Ω
m∇G· ∇vdx=hF, vi, for all v∈C0∞(Ω). (3.14)
3.2. LINEARIZED POISSON BOLTZMANN EQUATION 33