Journal of Computational Physics 419 (2020) 109627

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Model reduction-based initialization methods for solving the

Poisson-Nernst-Plank equations in three-dimensional ion

channel simulations

Qianru Zhang a,b, Sheng Gui a,b, Hongliang Li c,d,∗, Benzhuo Lu a,b,∗∗a State Key Laboratory of Scientific and Engineering Computing, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, Chinab School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, Chinac The Department of Mathematics, Sichuan Normal University, Chengdu 610066, Chinad The Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621900, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 October 2019Received in revised form 8 April 2020Accepted 30 May 2020Available online 26 June 2020

Keywords:Reduced modelPoisson-Nernst-PlanckSmoluchowski-Poisson-BoltzmannInitializationLinear approximation of PNP modelFinite element method

The Poisson-Nernst-Planck (PNP) model is frequently-used in simulating ion transport through ion channel systems. Due to the nonlinearity and coupling of PNP model, choosing appropriate initial values is crucial to obtaining a convergent result or accelerating the convergence rate of finite element solution, especially for large channel systems. Continuation is an effective and commonly adopted strategy to provide good initial guesses for the solution procedure. However, this method needs multiple times to solve the whole system at different conditions. We utilize a reduced model describing a near-or partial-equilibrium state as an approximation of the original PNP system (describing a non-equilibrium process in general). Based on the reduced model, we design three initialization methods for the solution of PNP equations under general conditions. These methods provide the initial guess of the PNP system by solving a specifically designed Poisson-Boltzmann-like model, Smoluchowski-Poisson-Boltzmann-like model, and linear approximation model. Simulations of potassium channels 1BL8 and 2JK4 demonstrate that these methods can effectively reduce the number of Gummel iteration steps and the total CPU time in the solution of the PNP equations, and especially do not need the continuation approach anymore. The reason is that these initial guesses can approximate the PNP solution well in the channel region. Besides, our numerical experiments demonstrate that as one of the initialization methods, the linear approximation method can even produce very close results such as current-voltage curves to that from the PNP model when the membrane potential is not high.

© 2020 Elsevier Inc. All rights reserved.

* Corresponding author at: The Department of Mathematics, Sichuan Normal University, Chengdu 610066, China.

** Corresponding author at: State Key Laboratory of Scientific and Engineering Computing, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

E-mail addresses: qrzhang@lsec.cc.ac.cn (Q. Zhang), shenggui@lsec.cc.ac.cn (S. Gui), lihongliang@lsec.cc.ac.cn (H. Li), bzlu@lsec.cc.ac.cn (B. Lu).

https://doi.org/10.1016/j.jcp.2020.1096270021-9991/© 2020 Elsevier Inc. All rights reserved.

2 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

1. Introduction

Ion channels are pores surrounded by a class of proteins, which provide pathways for charged mobile ions to pass through the otherwise impermeable cell membranes [1]. The directional transport process of ions within a channel is closely related to fundamental biological functionals, such as the maintenance of salt and water balance, the information transfer and processing in nervous systems, the coordination of muscle contraction, and the regulation of secretion of hormones [2]. The Poisson-Nernst-Planck model (PNP for brevity) [3–12], which couples the electrostatic potential equation with the convection-diffusion equations, is one of the most successful approaches to characterizing the electro-diffusion process of ions in an electrolyte solution.

Since the PNP model is a nonlinear coupled model, the nonlinear iteration is indispensable to the whole algorithm, no matter what numerical schemes we employ for ion channel simulation. And then an initial point is crucial to ensure the convergence of nonlinear iteration, especially for the case with high biases (such as bulk concentration, permanent charge, and membrane potentials). Continuation method is an effective and widely used strategy to improve the robustness of simulation, which has been introduced to calculate current-voltage characteristics of semiconductor devices [13–15] and cope with large permanent charges of ion channel proteins. For the continuation method, the process of loading biases is regarded as a quasi-static process, i.e., the biases are divided into numerous parts and gradually increase step by step. A drawback of the classical continuation method is its expensive computation.

We propose a novel method to guess the starting point of iteration, in which a reduced model is constructed, and its solution serves as the initial starting point of the iterative procedure. A “good” reduced model should be close to the original PNP model and can be solved with a low computation cost. We propose the reduced model by utilizing a near-equilibrium state to approximate the general non-equilibrium state governed by the PNP model. The Poisson-Boltzmann (PB for brevity) -like model and Smoluchowski-Poisson-Boltzmann (SPB for brevity) -like model are tested because their original forms, the PB model and the SPB model [16], are special cases of the PNP model. These two near-equilibrium models are specifically designed to enforce the compatibility of boundary conditions (BCs). And as to be shown later, both of them can often capture well the main properties of ion density distributions inside the channel region, while the electrostatic potential has a large deviation. A linear and uncoupled perturbation model is introduced to overcome the poor approximation accuracy of electrostatic potential, which is derived by expansion based on the PB-like model. The solution of this reduced model is composed of the PB-like solution and solution to the system of linear equations obtained by decomposing PNP equations based on the PB-like equation and taking out of the nonlinear terms after decomposition. Simulations of both channel proteins 1BL8 and 2JK4 indicate that the solution of the perturbation model is a very good initial iteration point. Overall, the novel initialization methods enable us to solve the PNP model with high membrane potential directly, while the classical continuation method repeats the Gummel iteration procedure step by step from zero potential. Especially, for many cases, we are more interested in current-voltage characteristic in the nonlinear response region, i.e., the calculation should start from a non-zero potential, Vm > 0.

It is worth mentioning that the standard finite element method (FEM for brevity) is employed for space discretization of the PNP model in this paper. The robust numerical results imply that convection dominance is not severe for the channels 1BL8 and 2JK4 in K Cl electrolyte using our meshes under our simulation conditions (i.e., membrane potential and bulk concentration). For a convection dominated problem, the stabilization method should be considered in numerical schemes. We refer to [17–22] and the references therein and [23,24] and references therein for recent progress for ion channel simulations, respectively. The remaining parts of this paper are organized as follows. In the next part, we review the PNP model and associated finite element schemes. The novel initialization method is introduced detailedly in Sect. 3. And the simulation of actual channels can be found in Sect. 4.

2. Model and numerical methods

2.1. Steady-state PNP equations

Computational region is an ion channel embedded in membrane proteins; See Fig. 1. The solvent region �s is a mixed solution with diffusive ion species. The solute region �m is composed of membrane proteins. The whole computational domain is denoted with � = �s ∪ �m . The �m and �s are separated by the molecular surface �m , which is highlighted by red line in Fig. 1. The �D = �t ∪ �b and �N are with the Dirichlet boundary conditions and the Neumann boundary conditions, respectively.

The PNP model is to describe the electrostatics, density distributions, and electrodiffusion processes of mobile ions in solvated biomolecular systems under non-equilibrium conditions. In this paper, the steady-state PNP model couples the Nernst-Planck (NP) equations{−∇ · J i = 0, in �s, i = 1,2, ..., K ,

J i = −Di(∇ci + βqi∇φci),(1)

and the electrostatic Poisson equation with the internal interface �m:

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 3

Fig. 1. A two-dimensional (2D) schematic view of an ion channel embedded in membrane proteins. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)⎧⎨

⎩−∇ · ε∇φ = ρ f + λ

∑Ki=1 qici, in �,

φm = φs, on �m,

εmε0∂φm∂�n = εsε0

∂φs∂�n on �m.

(2)

In the permanent charge distribution

ρ f =N∑

j=1

q jδ(�r −�r j),

δ(�r − �r j) is the Dirac delta distribution at point �r j , N is the number of particles with singular atomic charges q j located at �r j inside biological membrane proteins. Moreover, φ is the electrostatic potential, ci is the concentration of the ith ion species carrying charge qi = ziec , where zi is the valence of the ith ion species and ec is the elementary charge. K is the number of diffusive ion species, Di denotes the spatial-dependent diffusion coefficient. The constant β = 1/(kB T ) is the inverse Boltzmann energy with the Boltamann constant kB and the absolute temperature T . The characteristic function λ is equal to 0 in �m and 1 in �s , which implies that the solute region is impenetrable to mobile ions. The dielectric coefficient ε is defined as ε = εmε0 in �m and ε = εsε0 in �s , where εm (or εs) usually takes 2 (or 80) and ε0 is the vacuum dielectric constant.

This PNP system is augmented with the following boundary conditions:⎧⎨⎩

φ = 0 on �t , φ = Vm on �b,

ci = cbulkit on �t , ci = cbulk

ib on �b,∂φ

∂�n = 0 on �N , J i · �n = 0 on �m ∪ �N ,

(3)

where �n is the unit normal vector at �m or �N .

2.2. Finite element method

Regularization. The permanent charge distribution ρ f implies that the electrostatic potential φ is not with full regularity, i.e., φ ∈ H1−ε(�) for ε > 0. A regularization scheme [9,25] can remove the singular component of the potential such that the remaining component can be solved with standard FEM. The potential is decomposed into a singular component, a harmonic component, and a regular component, that is

φ = G + H + ϕ.

The singular component G and the harmonic component H are only confined in the solute region. The analytic form of Gis as follows,

G =N∑

j=1

q j

4πεmε0∣∣�r −�r j

∣∣ .Moreover, H satisfies the Laplace equation,

−�H = 0 in �m,

H = −G on �m.

Then we can get the regular component ϕ from

4 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

−∇ · (ε∇ϕ) = λ

K∑i=1

qici in �, (4)

which satisfies interface conditions on �m:{ϕm = ϕs,

εsε0∂ϕs∂�n − εmε0

∂ϕm∂�n = εmε0

(∂G∂�n + ∂ H

∂�n).

(5)

It is worth noting that there is no deposition of the electrostatic potential in the solvent region, that is φ = ϕ, in �s .For convenience, we let u = ecβϕ , which is dimensionless.Finite element discretization. Let H1

b(�) = {v ∈ H1(�)|v = 0 on �D

}, and H1(�) is a Sobolev space of weakly differ-

entiable functions. The weak form of the Poisson equation (4) is to find u ∈ H1(�) satisfying the Dirichlet BCs (3)1 such that ∫

�

ε∇u · ∇vd� = e2c β

∫�

λ

K∑i=1

zici vd� −∫�m

ecβεmε0

(∂G

∂�n + ∂ H

∂�n)

vd�, ∀v ∈ H1b(�). (6)

Moreover, the weak form of the ith (i = 1, 2, ..., K ) ion species’ NP equation (1) is to find ci ∈ H1(�s) satisfying the Dirichlet BCs (3)2 such that∫

�s

(Di∇ci · ∇w + Di zici∇u · ∇w)d�s = 0, ∀w ∈ H1b(�s). (7)

Readers can refer to [26] for more details of the finite element discretization.Gummel iteration. The numerical discrete system of the PNP model can be either solved as a whole nonlinear coupled

system using Newton method [27] or solved through a decoupled approach using Gummel iteration [28]. Since the decou-pled method only needs to solve a set of relatively “small” sub-problems in each iteration, it becomes a commonly used method in ion channel simulations. We employ the Gummel iteration combined with the under-relaxation strategy (the relaxation factor α ∈ (0, 1)) to solve the PNP model (6)-(7) in our previous works [11,29]. Firstly, we need to provide the initial ion concentrations c0

i , i = 1, 2, ..., K , and the initial electrostatic potential u0. Then, we obtain predictor of electro-static potential u1 by solving (6) with ci = c0

i and reconstruct u1 = αu1 + (1 − α)u0 with relaxation parameter α. Similarly, the concentration ci

1 is also obtained from eq. (7) with u = u1, and c1i = αc1

i + (1 − α)c0i (i = 1, 2, ..., K ). We set u1 and c1

ias initial values and repeat the iteration procedure until the tolerance is reached.

Continuation method. It has been widely recognized that both Newton iteration and Gummel iteration have a difficulty to converge even with moderate BCs. An appropriate initial value is significant to obtain a convergent result or accelerate the solution strategy’s convergence rate, especially for large channel protein systems. Continuation is an effective and commonly adopted strategy to provide a good initial starting point for the iteration procedure. For the sake of simplicity, we let � =(ρ f , Vm, cbulk

it , cbulkib , · · · ) be the parameters of the simulation, such as the permanent charge inside the channel protein, the

membrane potential, bulk concentrations, and so on, and let C = (c1, c2, · · · , cK ) be ion concentrations. Then the numerical discrete system of the PNP model can be rewritten as

L(�, u, C) = 0. (8)

Details of the procedure are as follows; See Algorithm 1. Actually, for a large and complex ion channel system, we shall increase only one or a few parameters in one step to enforce convergence of simulation. It means more expensive compu-tation costs. To accelerate the whole solution procedure, “int_tol” is usually set larger than the final convergent tolerance

Algorithm 1Given M , int_tol, and tol (int_tol and tol are convergence criteria of the nonlinear iteration procedure in solving the PNP system);Given an initial value S = (u, C)�;for k=1,2,...,M-1 do

Let �k = kM �;

Solve L(�k, u, C) = 0 with the initial value S and the convergent criterion int_tol, and get a new result S = (u, C)�;S ← S;

end forSolve L(�, u, C) = 0 with the initial value S and the convergent criterion tol, and get the final PNP solution S = (u, C)�;

“tol”. And the parameter “M” is set according to different conditions. This traditional initialization method is denoted with “M0”.

Stability condition. In semiconductor device simulations, the NP equation usually serves as a convection dominated problem, which promotes the development of discretization methods and stabilization schemes [17,19,21,22] for PNP equa-tions. And some of these numerical techniques are also employed for ion channel simulations [23,24]. In many previous

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 5

Table 1Checking the stability condition (9).

|qK + β�φ|in 1BL8

|qK + β�φ|in 2JK4

cbulkib = cbulk

it = 0.1M , i = K +, Cl−

1.527∼1.618 1.876∼2.263

cbulkib = cbulk

it = 2.0Mi = K +, Cl−

1.126∼1.288 1.649∼1.909

cbulkib = 2.0M cbulk

it = 0.1Mi = K +, Cl−

1.497∼1.611 1.679∼1.798

biological simulations [7,26,30–32], the standard finite element or finite difference discretization without stabilization can also obtain convergent results and agree well with the experimental data because the convection dominance is not severe in these cases. Here, referring to [23], we can estimate the stability condition as follows,

|qiβ(φ(xl+1) − φ(xl))| ≤ 2, ∀l = 1,2, ..., Mi, i = K +, Cl−, (9)

which is also the famous Scharfetter-Gummel stability condition [17]. Here, xl and xl+1 are adjacent grid points. We check the stability condition (9) of the standard finite discretization for PNP equations under all Dirichlet BCs considered in Numerical simulations on ion channel systems, 1BL8 and 2JK4. Since |qK + | = |qCl−|, we choose the NP equation for K +as a representative, and the corresponding results are shown in Table 1, where |qiβ�φ| denotes the range of the maxi-mum absolute difference of qiβφ between all adjacent pairs of grid points in three-dimension with membrane potentials Vm ∈ [−0.5V , 0.5V ] under fixed Dirichlet BCs of concentrations. The checking results show that convection dominance is not severe for these two typical channel systems in KCl electrolyte using our meshes under our simulation conditions (i.e., membrane potential and bulk concentration). This explains why some previous methods in ion channel simulation still worked without employing stabilization techniques. But for a really convection dominated situation, the stabilization meth-ods should be considered in numerical schemes. In the remainder of this paper, we will mainly discuss the initialization methods for PNP equations.

3. New initialization methods based on reduced models

Our new initialization method provides an initial guess (u∗, C∗) for the numerical discrete PNP system (8) through solving the numerical discrete of a reduced model

L(�, u∗,C∗) = 0.

This reduced model should be close to the original PNP model and can be solved with a low computation cost. In the first two initialization methods, our specifically designed PB- or SPB-like equation(s) serves as the reduced model. In the third initialization method, the reduced model is proposed by further utilizing a sort of linear perturbation of the PNP system with respect to the PB-like model.

3.1. Initialization through solving a PB-like equation

In this method, the initial value for the PNP solution is provided through solving a designed PB-like equation whose BCs are carefully set to be consistent with that of the PNP system as much as possible. At an equilibrium state, the equivalence between the PNP solution and the standard PB solution [10,33] is as follows: If PNP equations (1)-(2) satisfy the Dirichlet BCs φ = 0 on �D and ci(�r) = cbulk

i , i = 1, 2, ..., K on �D , where cbulki is the bulk concentration of the ith ion species, and

the reflective condition for each ion species on the interface �m to enforce zero-flux across the interface J i = 0 on �m , then the solution of the PNP equations is equivalent to that of the standard PB equation with the boundary condition φ = 0 on �D . The reason for the above equivalence has been explained in the subsection “How to get SMPB results from the solution of the SMPNPEs” of [10]. Therefore, we expect the initial guess obtained with the PB-like equation can approximate the PNP system well, especially inside the channel region.

The equilibrium (PB) solution can only exist under special conditions in the PNP system (e.g., the boundary values of ci are equal to a constant concentration cbulk

i , and φ is also equal to a constant potential on all boundary patches of �D ). And the general BCs in the PNP system do not guarantee the existence of the equilibrium state. However, we can design a PB-like equation with BCs consistent with the PNP’s BCs. This can be considered as a reduced model of a “near-equilibrium state”, which is useful, and its rationality is to be verified in later ion channel simulations. This PB-like model will serve as our first initialization method.

In this initialization method, the initial guess of electrostatic potential is the PB-like equation’s solution φ, and the initial values of ion concentrations are calculated with ci = ci exp{−βqiφ} (i = 1, 2, .., K ). In other words, all the required initial values for potential and ion concentrations are obtained from the PB-like equation,

6 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

⎧⎪⎪⎨⎪⎪⎩

−∇ · ε∇φ = ρ f + λ∑K

i=1 qicie−βqiφ in �,

φ = 0 on �D ,

φm = φs, εmε0∂φm∂�n = εsε0

∂φs∂�n on �m

∂φ

∂�n = 0 on �N .

(10)

Here, the ith ion species’ “bulk concentration” ci is a spatial-dependent function, whereas it is a constant in the traditional PB equation. If cbulk

ib = cbulkit , ci can be set to cbulk

it (= cbulkib ), and eq. (10)1 reduces to the traditional PB equation. Otherwise,

it is defined as a linear interpolation function, which is equal to cbulkit on �t and cbulk

ib on �b . In this PB-like equation, the ion concentration ci = ci exp{−βqiφ} (i = 1, 2, .., K ) satisfies “a local Boltzmann distribution”, which is different from the standard PB equation.

In fact, we have tried to make the BCs of φ in this PB-like equation in accordance with that in the PNP model, i.e. φ =0 on �t and φ = Vm on �b , but the initial guess acquired from the PB-like equation with these BCs of φ cannot approximate the PNP system well. In the following of this paper, we only consider the PB-like equation (and the following SPB-like equations as well) that satisfies the ion concentrations’ BCs of the PNP model and takes a zero potential on boundaries �t

and �b . To give an initial value including the influence of the Dirichlet BCs of the potential, we propose another initialization method “M3” that will be introduced later.

The singular charge distribution ρ f in eq. (10)1 can also be disposed with the regularization scheme, and the regular component ϕ of the electrostatic potential can be solved from

−∇ · (ε∇ϕ) = λ

K∑i=1

qicie−βqiϕ in �, (11)

which satisfies the interface conditions (5) on �m . Then the weak form of eq. (11) is to find u ∈ H1b(�) such that

∫�

ε∇u · ∇vd� − e2c β

∫�

λ

K∑i=1

(zicie

−zi u)

vd� +∫�m

ecβεmε0

(∂G

∂�n + ∂ H

∂�n)

vd� = 0, ∀v ∈ H1b(�). (12)

A backtracking line search Newton-like method is adopted to solve the above nonlinear equation (12). Let uk be the finite element approximation of u at the kth Newton iteration, which could be expressed with finite element bases {v1, v2, ..., v M}. Referring to eq. (12), we define a nonlinear function F (u), whose jth component is given as

F (u) j =∫�

ε∇u · ∇v jd� − e2c β

∫�

λ

K∑i=1

(zicie

−zi u)

v jd� +∫�m

ecβεmε0

(∂G

∂�n + ∂ H

∂�n)

v jd�, j = 1,2, ..., M.

The Jacobian matrix of F (u) is denoted with F ′(u), and the j, kth element of which is described as

(F ′(u)

)j,k =

∫�

ε∇vk · ∇v jd� + e2c β

∫�

λ

K∑i=1

(z2

i cie−zi u

)vk v jd�, j,k = 1,2, ..., M.

Moreover, the merit function of F (u) is the sum of squares, which is defined by

f (u) = 1

2‖F (u)‖2

2 = 1

2

M∑j=1

(F (u) j)2.

The framework of the backtracking line search Newton-like method [34] is stated in Algorithm 2.In this paper, we set h = 10−5, ρ = 0.618, γinit = 1.0, and γ = 0.05. Because the solution of eq. (10)1 serves just as an

initial guess, “tol_PB” is generally set to 10−2, which can reduce the time of the initialization part. As for the initial value u0, we find that u0 = 0 sometimes is not a proper choice. Therefore we utilize the following technique to get it. Firstly, we get p by solving the following Newton equations,

F ′(0)p = −F (0).

Then we take u0 = θ p, where θ is chosen according to different systems. In general, θ is set to 1.This initialization method is denoted with “M1”.

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 7

Algorithm 2Given h ∈ (0, 12 ), ρ ∈ (0, 1), γinit > 0, γ > 0, and tol_PB;Choose u0;for k=0,1,2,... do

Calculate a solution pk from the Newton equations

F ′(uk)pk = −F (uk);Set γ ← γinit ;while f (uk + γ pk) > f (uk) + hγ ∇ f (uk)� pk and γ > γ do

γ ← ργ ;end whileγk ← γ ;uk+1 ← uk + γk pk ;if ||uk+1 − uk||2/||uk+1||2 < tol_PB then

break;end if

end for

3.2. Initialization through solving the SPB-like equations

The above initialization method “M1” provides initial values for both electrostatic potential and ion concentrations. A possibly improved consideration is just using the PB-like equation to provide an initial guess φ for electrostatic potential, then using this potential’s initial guess φ as an external field and solve the drift-diffusion equations (Smoluchowski or NP equations) to provide initial guesses ci, i = 1, 2, ..., K for ion concentrations. This reduced model, SPB-like equations, de-scribes a partial-equilibrium state with the electrostatic potential following a PB-like equation (10)1 and the concentrations following Smoluchowski equations that are described as

⎧⎪⎪⎨⎪⎪⎩

−∇ · J i = 0, in �s, i = 1,2, ..., K ,

J i = −Di(∇ci + βqi∇φci),

ci = ci on �D ,

J i · �n = 0 on �m ∪ �N .

(13)

Here, ci is the same as that in the PB-like equation (10)1. Due to the uncoupling nature of the SPB-like equations, eq. (13)1−2with the same weak form and finite element discretization as eq. (1) just needs to be solved once after obtaining φ from eq. (10)1.

We use “M2” to denote this initialization method.

3.3. Initialization through a linear perturbation solution of the PNP system with respect to the PB-like model

If a nonzero voltage difference exists in the system, the solution of the PB-like equation doesn’t satisfy the Dirichlet boundary condition of the electrostatic potential on �b in the PNP model. In this case, we can explore an initialization method to get a better approximation of the PNP solution. As the PB-like model works well for the case of zero voltage difference, we can deploy this component and use a perturbation idea to decompose the PNP solution with the PB-like solution as a reference.

Firstly, we let

{φ = φp1 + φp2

ci = cp1i + cp2

i , i = 1,2, . . . , K .(14)

Here, φp1 satisfies the PB-like equation (10)1, and cp1i = ci exp{−βqiφ

p1}, i = 1, 2, ..., K . Then the flux J i is divided into two parts J p1

i and J p2i ,

⎧⎪⎨⎪⎩

J i = J p1i + J p2

i ,

J p1i = −Di(∇cp1

i + βqi∇φp1cp1i ) = −Di∇ cie−βqiφ

p1,

J p2i = −Di

(∇cp2

i + βqi∇φp1cp2i + βqi∇φp2cp1

i + βqi∇φp2cp2i

).

(15)

Lastly, through taking eqs. (14) and (15) into PNP equations, φp2 and cp2(i = 1, 2, .., K ) satisfy the following equations:

i

8 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∇ · ε∇φp2 = λ∑K

i=1 qicp2i in �,

−∇ · J p2i = −∇Di · ∇ cie−βqiφ

p1 + Diβqi∇φp1 · ∇ cie−βqiφp1

in �s, i = 1,2, ..., K ,

J p2i = −Di(∇cp2

i + βqi∇φp1cp2i + βqi∇φp2cp1

i + βqi∇φp2cp2i )

φp2 = 0 on �t , φp2 = Vm on �b

cp2i = 0 on �D ,

φp2m = φ

p2s , εmε0

∂φp2m

∂�n = εsε0∂φ

p2s

∂�n on �m,

J p2i · �n = 0 on �m ∪ �N ,

∂φp2

∂�n = 0 on �N .

(16)

In this initialization method, the initial value for the PNP solution is given as φ = φp1 + φp2 and ci = cp1i + cp2

i , i =1, 2, ..., K .

It is observed that the equation system (16) is again a nonlinear coupled system, which is not our goal. To reduce the computational expenses, we take out of the nonlinear term ∇φp2cp2

i in J p2i . Then the equation system (16) becomes a linear

approximation form of the original ones, which can be solved as a whole linear system. The weak form of (16) without the nonlinear term ∇φp2cp2

i is finding (up2, cp21 , cp2

2 , ..., cp2K ) ∈ H1(�) × H1

b(�s) × ... × H1b(�s)︸ ︷︷ ︸

K

, where up2 satisfies the Dirichlet

BCs (16)4 - (16)5, such that⎧⎪⎪⎪⎨⎪⎪⎪⎩

∫�

ε∇up2 · ∇vd� − λe2c β∫�

∑Ki=1 zic

p2i vd� = 0, ∀v ∈ H1

b(�),

∫�s

Di(∇cp2i + zi∇up1cp2

i + zi∇up2cp1i ) · ∇wid�s = ∫

�m∪�NDie−zi u

p1∇ ci · �nwid�

− ∫�s

∇Di · ∇ cie−zi up1

wid�s + ∫�s

zi Die−zi up1∇ ci · ∇up1 wid�s, ∀wi ∈ H1

b(�s), i = 1,2, ..., K ,

(17)

and up2 = ecβφp2 is dimensionless. Furthermore, up2 and cp2i , i = 1, 2, ..., K can be respectively expressed with finite

element bases {

v1, v2, ..., v M0

}and

{wi

1, wi2, ..., wi

Mi

}, i.e. up2 =∑M0

l=1 ul vl and cp2i =∑Mi

l=1 cil wi

l . A set of column vectors

{λi}Ki=0 are utilized to represent the freedoms of up2 and cp2

i :

(λ0)l = ul, l = 1,2, ..., M0,

(λi)l = cil , l = 1,2, ..., Mi, i = 1,2, ..., K .

Then the linear system (17) can be rewrote as⎡⎢⎢⎢⎢⎢⎣

A B1 B2 · · · B K

C1 D1

C2 D2

.... . .

C K D K

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

λ0

λ1

λ2

...

λK

⎤⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎣

0F 1

F 2

...

F K

⎤⎥⎥⎥⎥⎥⎦ , (18)

where A is a M0 square matrix, Bi is a M0 × Mi matrix, C i is a Mi × M0 matrix, Di is a Mi square matrix, and Fi is a Mi

dimensional column vector. We utilize (Q )l,m to represent the l, mth element of the matrix Q , then the concrete expression of every block matrix in the linear system (18) is given as

(A)l,m =∫�

ε∇vl · ∇vmd�,

(Bi)

l,m = −∫�

λe2c βzi vl w

imd�,

(C i)

l,m =∫�s

zi Dicp1i ∇wi

l · ∇vmd�s,

(Di)

l,m =∫�s

Di

(∇wi

l · ∇wim + zi wi

m∇up1 · ∇wil

)d�s,

(F i)

l =∫

�m∪�N

Die−zi up1∇ ci · �nwi

l d� −∫�s

∇Di · ∇ cie−zi up1

wil d�s +

∫�s

zi Die−zi up1∇ ci · ∇up1 wi

l d�s.

This initialization method is denoted with “M3-1”, and it will have a simplified form as introduced below.

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 9

We find that the initial guess provided by “M3-1” can even be utilized to calculate current-voltage curves, which are very close to that from the complete PNP solution when ci = cbulk

ib = cbulkit and the charge on the surface of the ion channel

or the membrane potential is not high. In other words, the initial guess obtained by “M3-1” can be an already good approx-imation of the PNP solution in these situations (see section Numerical results). We call this scheme a “linear approximation method”. In this method, the approximated electric current along the ion channel is

I = −∫S

K∑i=1

qi Di

(∇cp2

i + βqi∇φp1cp2i + βqi∇φp2cp1

i + βqi∇φp2cp2i

)· �ndS, (19)

where S is a cut plane inside the ion channel. And �n is the unit normal vector at S . Due to the conservation of the total flux, S can be any cut plane located either inside or outside of the ion channel. For convenience, we take S as the top surface �t of our computational region.

In fact, we can further simplify the equation system (16) to make it solved more easily. Because cp2i is equal to zero on

�D , we expect that it is relatively small compared with ci (e.g. the L2 norm of cp2i is smaller than ten percent of the L2 norm

of ci ), cp2i can be set to zero. Then the simplified initialization method can utilize φ = φp1 + φp2 and ci = cp1

i , i = 1, 2, .., Kto initialize the PNP solution, where φp1 satisfies the PB-like equation, and cp1

i = ci exp{−βqiφp1}, i = 1, 2, ..., K . As for φp2,

it can be solved from⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∇ · ε∇φp2 = 0 in �,

φp2 = 0 on �t , φp2 = Vm on �b,

φp2m = φ

p2s , εmε0

∂φp2m

∂�n = εsε0∂φ

p2s

∂�n on �m,∂φp2

∂�n = 0 on �N .

(20)

But eq. (20)1 doesn’t count for the effect of cp1i on φp2. To avoid the redefinition, φp2 in �m is defined as zero and φp2 in

�s satisfies the following equations:⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∇ · J p2i = −∇Di · ∇ cie−βqiφ

p1 + Diβqi∇φp1 · ∇ cie−βqiφp1

in �s, i = 1,2, ..., K ,

J p2i = −Di(βqi∇φp2cp1

i )

φp2 = 0 on �t, φp2 = Vm on �b,

J p2i · �n = 0 on �m ∪ �N .

(21)

The weak form of eq. (21)1−2 is to find up2 ∈ H1(�s) satisfying the Dirichlet BCs (21)3 such that∫�s

Di zicp1i ∇up2 · ∇vd� = −

∫�s

∇Di · ∇ cie−zi u

p1vd�s +

∫�s

zi Die−zi u

p1∇ ci · ∇up1 vd�s,

∀v ∈ H1b(�s), i = 1,2, ..., K . (22)

For every ion species, φp2 can be gotten from eq. (21)1−2, which is denoted with φp2i when eq. (21)1−2 is relevant to the

ith ion species. However, they may be different due to the ignored cp2i and numerical deviations. To obtain a unique value

of φp2, it can be set to the average of all φp2i , i = 1, 2, .., K , or the average of φp2

i , i ∈ {cations or anions}.We use “M3-2” to denote this simplified initialization method. Our later numerical experiments show that the initial

guess obtained from “M3-2” performs as well as that provided by “M3-1” when ci = cbulkib = cbulk

it on �t and �b , but it only needs to solve a simple equation after solving the PB-like equation. Although “M3-1” and “M3-2” are two different initial-ization methods in the form, they originate from the same idea. Therefore, these two initialization methods are collectively denoted as “M3” in this paper. In later numerical tests, “M3-1” is adopted to initialize the PNP solution when cbulk

ib �= cbulkit ,

and “M3-2” is employed when cbulkib = cbulk

it .From descriptions of these initialization methods, we find that initial values provided by “M1” and “M2” are not affected

by the electrostatic potential’s boundary condition (3)1. In other words, if we are willing to solve a series of PNP models only with different boundary values of the potential on �b , “M1” or “M2” need to be carried out just once. If we want to get the initial value with “M3”, after solving the PB-like equation, solving a simple linear system for each boundary value of the potential is enough. This is an advantage that the continuation method does not possess because in the continuation method, some/all coefficients and/or certain boundary condition(s) of the PNP model need to change progressively during the solution procedure.

3.4. Software implementations

Equations mentioned in this paper are all solved with the parallel FEM, and these initialization methods are implemented with the open-source finite element toolbox Parallel Hierarchical Grid (PHG) [35]. There exist several well-developed PB

10 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

Fig. 2. The unstructed tetrahedral volume mesh and triangular surface mesh of 1BL8.

solvers, such as APBS [36], AFMPB [37], DelPhi [38], MEAD [39], and so on. In this work, we adopt our PB solver that is developed for solving the PB-like equation satisfying general BCs based on the PHG. The main considerations that we didn’t employ other existing PB solvers are as follows: (1) Those existing PB solvers cannot directly solve the PB-like equation under non-equilibrium BCs without code modifications. (2) Different solvers use different program architecture, different meshes, and output formats, which always cause inconveniences in coding and usage. Based on the same FEM library PHG, we can call the PB solver inside instead of outside from the PNP solver, and the PB solver can run in a massively parallel environment for large channel protein systems. (3) The PB solution is much cheap relative to the solution of PNP equations. FEM in three dimensions needs tetrahedral meshes. Firstly, we use our biomolecule surface meshing software TMSmesh [40] to produce a molecular surface conforming mesh, which is capable of generating a manifold surface mesh for arbitrarily large molecular systems. Secondly, we smooth the mesh employing the program SMOPT [41] to improve the quality of our surface mesh. Then a volume mesh is generated based on the surface mesh using TetGen [42]. At last, we adopt a walk-and-detect algorithm [43] to identify the ion channel region and embed it in a membrane correctly.

4. Numerical simulations

At first, in subsection 4.1, we will evaluate the effectiveness of our model reduction-based initialization methods by applying them to a realistic potassium channel: 1BL8 [Protein Data Bank (PDB) entry] with +8e net charge and a voltage-dependent anion channel (VDAC): 2JK4 [PDB entry] with +4.987e net charge. Then in subsection 4.2, we will investigate the conditions under which the current-voltage curves obtained with the “linear approximation method” can be good approximations of that calculated with the original PNP equations also on ion channels 1BL8 and 2JK4. Because these two ion channels are both large channel protein systems, we also choose another small ion channel protein system 1MAG [PDB entry] to investigate the “linear approximation method” more comprehensively.

In numerical experiments, we always consider the electrolyte solution as K Cl. In the bulk region, the diffusion coefficients for K + and Cl− are respectively set to their experimental values: Dbulk

K = 0.196 Å/ps, DbulkCl = 0.203 Å/ps.

All numerical experiments were carried out on the cluster Lenovo DeepComp 8800 of the State Key Laboratory of Scien-tific and Engineering Computing of Chinese Academy of Sciences, which consists of dual Intel Gold 6140 CPU nodes (2 × 18cores, 2.30 GHz) and 100 Gbps EDR Infiniband network.

4.1. Apply initialization methods to PNP solution for 1BL8 and 2JK4

For simplicity, the diffusion coefficients inside the ion channels are set to be their values in the bulk region when making comparisons among these initialization methods.

The potassium channel 1BL8 is composed of 5,892 atoms. In normal conditions, the potassium ions highly concentrate inside the pore region, which is essentially the difficulty in the numerical solution. The unstructured tetrahedral volume mesh and triangular surface mesh of 1BL8 are shown in Fig. 2, which have 102,572 vertices and 643,832 tetrahedra over the whole domain.

For a system with the given BCs (Vm = 0.2V and cbulkit = cbulk

ib = 0.1M, i = K +, Cl−), we compare the absolute initial deviations (the absolute value of the deviation between the PNP solution and its corresponding initial value obtained with an initialization method) of the electrostatic potential and ion concentrations in our model reduction-based initialization methods. In these comparison experiments, the initialization method “M3” actually refers to the method “M3-2” due to cbulk

it = cbulkib . The cross-section views of the absolute initial deviations for different initialization methods are presented in

Figs. 4–6, respectively. And cross-section views of the electrostatic potential and ion concentrations solved from the PNP model are shown in Fig. 3.

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 11

Fig. 3. Cross-section views of the PNP solution in the whole domain of 1BL8: (a) electrostatic potential (V ), (b) K + concentration (M), (c) Cl− concentration (M).

Fig. 4. Cross-section views of the absolute initial deviations in the whole domain of 1BL8 in the initialization method “M1”: (a) electrostatic potential (V ), (b) K + concentration (M), (c) Cl− concentration (M).

Fig. 5. Cross-section views of the absolute initial deviations in the whole domain of 1BL8 in the initialization method “M2”: (a) electrostatic potential (V ), (b) K + concentration (M), (c) Cl− concentration (M).

Fig. 6. Cross-section views of the absolute initial deviations in the whole domain of 1BL8 in the initialization method “M3” (“M3-2”): (a) electrostatic potential (V ), (b) K + concentration (M), (c) Cl− concentration (M).

12 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

Table 2Relative deviations (in L2 norm) of the electrostatic potential and ion concentrations inside the channel pore obtained with our model reduction-based initialization methods to their corre-sponding part of the PNP solution for the ion channel 1BL8 case.

M1 M2 M3 (M3-2)

Electrostatic potential 95.75% 95.75% 0.4%K + ion concentration 5.34% 4.55% 4.09%Cl− ion concentration 6.85% 4.09% 6.27%

Table 3The number of Gummel iteration steps and the total CPU time of the whole process for solving PNP equations with cbulk

ib = cbulkit =

0.1M, i = K +, Cl− on 1BL8.

Vm(V ) M0 M1 M2 M3-2

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

-0.5 167 141.01 140 113.67 139 119.39 146 124.29-0.2 113 114.32 77 82.83 77 68.86 63 55.96-0.1 138 135.69 77 75.83 77 80.59 66 59.590 142 127.19 68 68.04 91 84.44 68 59.710.1 116 98.76 77 62.78 78 65.05 60 51.120.2 139 131.30 77 76.88 84 70.29 64 52.240.5 138 109.15 77 58.35 77 58.55 66 53.06

These cross-section views show that the initial values of the potential obtained from “M1” and “M2” are similar. And “M3” (“M3-2”) produces a better approximation of the potential compared with “M1” and “M2”. As for concentrations, initial values provided by these initialization methods are overall good approximations through comparing Figs. 4–6 with Fig. 3. Especially, the “trouble-making” K + concentrations inside the crucial region, channel pore, obtained with reduced models only have relatively slight deviations from those of the original PNP model. This implies that there are only relatively small differences between the non-equilibrium charge density distribution (governed by the PNP system) and the approximate quasi-equilibrium Boltzmann distribution determined in our model reduction-based initialization methods inside the chan-nel region, and demonstrates that these initial guesses can be close to the PNP solution well in the channel region. To further illustrate this, relative initial deviations in L2 norm in the channel region are separately listed in Table 2. And in Table 2, it could be noticed that the relative electrostatic potential deviation is large for “M1” and “M2”. This is because these two methods only utilize the solution of the PB-like equation that doesn’t satisfy the Dirichlet boundary condition of the potential on �b in the PNP model. In other words, the absolute potential deviation between the PNP solution and the initial guess provided by “M1” or “M2” becomes larger and larger in the region close to �b (refer to Fig. 4 (a) and Fig. 5(a)). In “M3”, we count for all BCs of the potential, which enables “M3” to provide an initial guess with a small relative deviation.

In the following numerical experiments, we will investigate the effectiveness of initialization methods for two ion chan-nel systems, 1BL8 and 2JK4, when the Dirichlet BCs are different. In the continuation method “M0”, we set M = 10. For convenience, we make Gummel iteration always conducts 5 steps during the first M − 1 times of increases of all coefficients in � = (ρ f , Vm, cbulk

it , cbulkib ). In the procedure of solving PNP equations, the relaxation factor α is generally set to 0.2. But

Gummel iteration may not converge under some BCs, so we make α smaller until Gummel iteration can converge.We first consider a system with low bulk concentration conditions (cbulk

ib = cbulkit = 0.1M, i = K +, Cl−) in the potassium

channel 1BL8. The number of the Gummel iteration steps and the total CPU time of the whole process of solving the PNP model with different initialization methods are shown in Table 3. From the best of our experience, if the initial values are provided by “M1”, “M2”, and “M3-2”, we take the relaxation factor α = 0.2. If the PNP model is initialized with “M0”, we set α = 0.15 when Vm = −0.1, 0, and 0.5V , and α = 0.2 when Vm takes the other values. In the backtracking line search Newton-like method, we set θ = 0.3, then the number of Newton iteration steps for solving the PB-like equation is 9. For this potassium channel, we also test the high bulk concentration conditions (cbulk

ib = cbulkit = 2.0M, i = K +, Cl−). The

corresponding numerical results are presented in Table 4. In this case, we always set α = 0.1 and θ = 1.0, then the number of Newton iteration steps for solving the PB-like equation is 10.

Then, we apply these initialization methods on systems with low bulk concentration (cbulkib = cbulk

it = 0.1M, i = K +, Cl−) and high bulk concentration (cbulk

ib = cbulkit = 2.0M, i = K +, Cl−) conditions in another ion channel 2JK4 that is composed of

4,393 atoms [44]. The unstructured tetrahedral volume mesh and triangular surface mesh of 2JK4 are presented in Fig. 7, which have 55,178 vertices and 343,973 tetrahedra over the whole domain. We set α = 0.2 in the low bulk concentration case and α = 0.1 in the high bulk concentration case. In these two cases, we always set θ = 1.0, then the number of Newton iteration steps for solving the PB-like equation is separatively equal to 7 (for the low bulk concentration case) and 5 (for the high bulk concentration case). The corresponding numerical results are respectively shown in Table 5 and Table 6.

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 13

Table 4The number of Gummel iteration steps and the total CPU time of the whole process in solution for solving PNP equations with cbulk

ib =cbulk

it = 2.0M, i = K +, Cl− on 1BL8.

Vm(V ) M0 M1 M2 M3-2

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

-0.5 602 508.38 592 476.48 583 402.48 661 450.88-0.2 187 141.73 154 124.35 154 132.50 141 100.57-0.1 186 137.95 155 107.75 155 109.40 143 100.900 197 159.86 143 112.22 155 120.72 143 98.360.1 186 120.72 155 107.75 155 109.40 131 90.040.2 187 131.39 154 121.04 154 119.47 126 86.240.5 186 148.47 154 111.95 154 101.89 124 85.51

Fig. 7. The unstructured tetrahedral volume mesh and triangular surface mesh of 2JK4.

Table 5The number of Gummel iteration steps and the total CPU time of the whole process for solving PNP equations with cbulk

ib = cbulkit =

0.1M, i = K +, Cl− on 2JK4.

Vm(V ) M0 M1 M2 M3-2

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

-0.5 113 65.61 77 49.77 77 50.83 68 43.11-0.2 114 76.43 77 51.17 77 49.79 70 41.82-0.1 114 68.50 77 50.93 77 55.15 68 41.000 119 73.11 71 47.28 78 46.12 71 41.480.1 114 72.23 77 45.63 77 46.05 62 36.670.2 113 65.82 77 44.39 77 47.92 63 40.650.5 113 64.62 77 51.77 77 44.16 62 32.61

Table 6The number of Gummel iteration steps and the total CPU time of the whole process for solving PNP equations with cbulk

ib = cbulkit =

2.0M, i = K +, Cl− on 2JK4.

Vm(V ) M0 M1 M2 M3-2

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

total time (s)

Gummel iteration steps

Total time (s)

-0.5 186 93.51 155 72.01 155 65.81 119 50.89-0.2 187 102.09 155 81.16 155 77.05 114 58.39-0.1 186 94.65 154 75.45 154 79.85 117 57.720 200 102.49 146 75.80 152 66.10 146 64.530.1 186 102.50 154 78.99 154 78.58 124 62.610.2 186 104.61 155 77.82 155 76.67 122 58.950.5 186 96.00 154 72.90 154 68.42 121 53.70

Referring to these numerical results, it is observed that the continuation method “M0” can be replaced with our model reduction-based initialization methods “M1”, “M2”, and “M3-2” when cbulk

it = cbulkib . And they are more effective in reducing

the number of Gummel iteration steps and/or the total time of the whole process for solving the PNP equations than “M0”. These numerical results also show that the effectiveness of “M1” is similar to that of “M2”. Because the numbers of Gummel iteration steps with initial values provided by “M1” and “M2” are almost the same and their corresponding CPU time of the whole process is also similar in most cases. By comparing these initialization methods, it is found that “M3-2” has evident advantages than the other methods when the membrane potential is not very large (−0.2V ≤ Vm ≤ 0.2V, this is the normal range of the membrane potential in physiological conditions).

14 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

Table 7The number of Gummel iteration steps and the total CPU time of the whole process for solving PNP equations with cbulk

ib =2.0M, cbulk

it = 0.1M, i = K +, Cl− on 1BL8.

Vm(V ) M0 M1 M2 M3-1

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

-0.5 411 313.12 377 266.97 444 321.22 384 276.61-0.2 139 107.11 103 81.30 156 125.20 91 73.35-0.1 143 110.54 103 89.05 305 243.10 91 72.170 148 130.74 105 93.43 321 245.68 105 82.560.1 138 106.37 104 77.81 155 117.60 92 73.870.2 138 104.56 104 97.09 301 222.84 87 73.700.5 138 101.96 103 86.20 303 229.35 83 62.20

Table 8The number of Gummel iteration steps and the total CPU time of the whole process for solving the PNP equations with cbulk

ib =2.0M, cbulk

it = 0.1M, i = K +, Cl− on 2JK4.

Vm(V ) M0 M1 M2 M3-1

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

Gummel iteration steps

Total time (s)

-0.5 186 107.02 154 79.37 155 76.67 126 66.20-0.2 186 94.08 154 83.42 154 84.20 127 65.92-0.1 187 103.48 155 88.86 154 88.05 131 76.880 198 110.33 159 85.27 163 79.76 156 75.150.1 187 109.31 155 84.31 154 84.80 132 71.040.2 186 103.01 155 88.73 154 84.30 124 69.270.5 186 96.69 155 76.25 154 83.41 123 61.88

At last, we test our initialization methods with more general boundary conditions. We apply the initialization methods to PNP simulations on 1BL8 and 2JK4 systems under unsymmetric bulk concentration conditions (cbulk

ib = 2.0M, cbulkit =

0.1M, i = K +, Cl−). The number of Gummel iteration steps and the total CPU time of the whole process in the solution of the PNP equations with different initialization methods are presented in Table 7 and Table 8, respectively. For the ion channel 1BL8, we set the relaxation factor α = 0.15, if the initial values are provided by “M0”, “M1”, and “M3-1”. If “M2” is utilized to initialize the PNP solution, α = 0.1 when Vm = −0.5, −0.2, and 0.1V , and α = 0.05 when Vm takes the other values. In the backtracking line search Newton-like method, we set θ = 1.0, then the number of Newton iteration steps for solving the PB-like equation is 12. For the convergence of “M2”, tol_PB is set to 10−4 when Vm = 0.5V , then the number of Newton iteration steps is 14. For the ion channel 2JK4 case, we always set α = 0.1 and θ = 1.0, then the number of Newton iteration steps for solving the PB-like equation is 5.

From Table 7 and Table 8, it is found that the continuation method “M0” can also be replaced with our model reduction-based initialization methods “M1”, “M2”, and “M3” when cbulk

it �= cbulkib . Similar to the above situation where cbulk

it = cbulkib ,

“M1” and “M3-1” perform better in reducing the number of Gummel iteration steps and/or the total CPU time of the whole process for solving the PNP equations than “M0”. Moreover, “M3-1” is still the most effective method among these initialization methods when the membrane potential is not very large (−0.2V ≤ Vm ≤ 0.2V ). However, it is observed that “M2” performs worse than “M0” on the 1BL8 system. This implies that the SPB-like solution may not approximate the PNP solution well when cbulk

it �= cbulkib . In fact, the ion concentration ci (i = 1, 2, ..., K ) is assumed to satisfy “the local Boltzmann

distribution” in the PB-like equation (10)1. If cbulkit �= cbulk

ib , the flux J i = −Di∇ ci exp{−βqiφ} is not equal to zero and the divergence of it ∇ · J i = −∇Di · ∇ ci exp{−βqiφ} + Diβqi∇φ · ∇ ci(�r) exp{−βqiφ} is also nonzero, which does not coincide with the Smoluchowski (NP) equation (13)1−2.

According to the above numerical results, we conclude that the continuation method “M0” can be replaced with our model reduction-based initialization methods “M1”, “M2”, and “M3”. Furthermore, “M1” and “M3” are always more effective in reducing the number of Gummel iteration steps and/or the total CPU time of the whole process for solving the PNP equations than “M0”. And “M3” is always the most effective one among these initialization methods for not very large membrane potential conditions, i.e. −0.2V ≤ Vm ≤ 0.2V , which is usually the normal range of the membrane potential in physiological conditions. As for “M2”, its effectiveness is similar to that of “M1” when cbulk

it = cbulkib , but it performs worse

than “M0” when cbulkit �= cbulk

ib .In addition, we here also present a comparison between our initialization method and the traditional voltage scanning

continuation method (as seen in [23]), which is popular in current-voltage curve calculations but different from the gen-erally effective continuation method “M0” in ion channel simulations as described in this work. To further illustrate the effectiveness of our initialization methods, we compare our most effective method “M3” with the voltage scanning contin-uation method under all Dirichlet BCs mentioned above on ion channel systems, 1BL8 and 2JK4. In Table 9 and Table 10, we separately list the Gummel iteration step numbers of every incremental voltage in the process of solving PNP equa-

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 15

Table 9The Gummel iteration step numbers of every incremental voltage in the process of solving of PNP equations on 1BL8.

Voltage scanning/M3

Vm(V )

-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5

cbulkib = cbulk

it = 0.1M , i = K +, Cl−(relaxation factor α = 0.2)

120/146 151/175 147/169 74/63 77/66 77/60 74/64 72/61 71/61 70/66

cbulkib = cbulk

it = 2.0Mi = K +, Cl−(relaxation factor α = 0.1)

523/661 343/425 194/240 148/141 155/143 155/131 148/126 144/124 141/124 139/124

cbulkib = 2.0M cbulk

it = 0.1Mi = K +, Cl−(relaxation factor α = 0.15)

303/384 95/109 96/95 99/91 103/91 104/92 99/87 96/87 95/84 93/83

Table 10The Gummel iteration step numbers of every incremental voltage in the process of solving of PNP equations on 2JK4.

Voltage scanning/M3 Vm(V )

-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5

cbulkib = cbulk

it = 0.1M , i = K +, Cl−(relaxation factor α = 0.2)

70/68 71/69 72/70 74/70 77/68 77/62 74/63 72/63 71/62 70/62

cbulkib = cbulk

it = 2.0Mi = K +, Cl−(relaxation factor α = 0.1)

139/119 141/116 144/113 148/114 154/117 154/125 148/122 144/115 141/120 139/121

cbulkib = 2.0M cbulk

it = 0.1Mi = K +, Cl−(relaxation factor α = 0.1)

139/126 142/124 144/123 148/127 155/131 155/132 148/124 144/123 141/124 139/123

tions. Through comparing these numerical results, it is observed that “M3” always works better than the voltage scanning continuation method except Vm ≤ −0.3V on 1BL8. Moreover, if we want to get the solution at a certain voltage (such as Vm = 0.5V ) or calculate the current-voltage curves starting from a nonzero voltage, adopting our initialization method “M3” is more economic because the traditional voltage scanning continuation method always starts from zero voltage and obtaining solutions at unnecessary voltages still needs a lot of computations.

4.2. Apply the “linear approximation method” to current-voltage characteristic calculations

In this subsection, we will study under what conditions the “linear approximation method” described in “M3-1” can work with respect to the current-voltage characteristics. The 1BL8 and 2JK4 systems are still taken as examples. Because these two ion channels are both large channel protein systems, we also choose a small ion channel system, 1MAG, as the third example for investigating the “linear approximation method” more comprehensively. Although there is no experimen-tal data available for diffusion coefficients inside the ion channel, it is known that diffusion coefficients in the bulk region and the channel pore region should be different [45]. Therefore, we make the diffusion coefficients continuously change inside the channel region when calculating the current-voltage curves. A diffusion coefficient function is described as [46]

D(�r) =⎧⎨⎩

Dbulk, �r ∈ bulk region,

Dchan + (Dchan − Dbulk) f (�r), �r ∈ buffering region,

Dchan, �r ∈ channel region,

where the function f (�r) is given by

f (�r) = n

( |�r −�rchan||�rbulk −�rchan|

)n+1

− (n + 1)

( |�r −�rchan||�rbulk −�rchan|

)n

,

and n is set to 9. Here, �rchan is the boundary coordinate of the channel region, and �rbulk is the boundary coordinate of the bulk region. The buffering region locates between the bulk region and the channel region, whose length is equal to |�rbulk −�rchan|. In this paper, |�rbulk −�rchan| = 5 Å and Dchan = ratio × Dbulk . We set ratio = 0.4 in 2JK4 [26], ratio = 0.1 in 1BL8[30], and ratio = 0.0725 in 1MAG to match the experimental data in [47].

The ion channel 1MAG is composed of 552 atoms [48]. Fig. 8 shows the unstructured tetrahedral volume mesh and triangular surface mesh of it, which have a total of 17,872 vertices and 92,480 tetrahedra over the whole domain.

16 Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627

Fig. 8. The unstructed tetrahedral volume mesh and triangular surface mesh of 1MAG.

Table 11Summary of initialization methods’ effect.

cbulkit = cbulk

ib cbulkit �= cbulk

ib

Help get a convergent solution

Reduce total time (compare with “M0”)

Provide very close PNP I-V curves

Help get a convergent solution

Reduce total time (compare with “M0”)

Provide very close PNP I-V curves

M0 � −− No � −− No

M1 � � No � � No

M2 � � No � No No

M3 M3-1 � �(most effective method when −0.2V ≤ Vm ≤ 0.2V )

�(the charge on the surface of the ion channel is small or −50mV ≤ Vm ≤ 50mV )

−− −− −−

M3-2 −− −− −− � �(most effective method when −0.2V ≤ Vm ≤ 0.2V )

No

Fig. 9 shows the current-voltage curves computed with the “linear approximation method” and the original PNP equa-tions separately on ion channel systems 1BL8, 2JK4, and 1MAG. The ionic salt solution KCl is studied, and the bulk concentration conditions are cbulk

ib = cbulkit = 0.1M, i = K +, Cl− . By comparing the current-voltage curves in Fig. 9, we can

draw the following conclusions. If the charge on the surface of the ion channel is small and its current-voltage curve is nearly linear, such as 1MAG, the electric current calculated with the “linear approximation method” is a suitable approx-imation to that calculated with the original PNP model. But if the charge on the surface of the ion channel is large, such as 2JK4 and 1BL8, the real electric current can only be approximated well by the “linear approximation method” at a not large membrane voltage ranging from -50 mV to 50 mV. This is reasonable because the “linear approximation method” ignores the nonlinear term in the linear perturbation scheme of the PNP solution (see the initialization method “M3-1”).

5. Conclusion

In this paper, we propose three initialization methods based in a class of reduced models describing a near- or partial-equilibrium state of the original PNP system. These initialization methods have shown improvements on the traditional continuation method for the solution of the PNP equations for real ion channel systems. Two of the initialization methods are obtained by directly solving a specifically designed PB- or SPB-like equation(s) whose boundary conditions of ion con-centrations are consistent with that of the PNP model. Another initialization approach is studied by decomposing the PNP solution with one component satisfying the PB-like equation. The backtracking line search Newton-like method is adopted to solve the nonlinear PB-like equation. The underlying fact for these methods is that the deviations of the approximate quasi-equilibrium Boltzmann distributions from the non-equilibrium charge density distributions governed by PNP equa-tions are insignificant inside the ion channel when cbulk

it = cbulkib and the membrane potential is not very large. Table 11

summarizes the effects and main conclusions of the three model reduction-based initialization methods proposed in this work.

Declaration of competing interest

The authors declare no competing interests that could have appeared to influence the work reported in this paper.

Q. Zhang et al. / Journal of Computational Physics 419 (2020) 109627 17

Fig. 9. Current-voltage curves computed with the “linear approximation method” and the original PNP model on (a) 1BL8, (b) 2JK4, (c) 1MAG.

Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments, and Xuejiao Liu for her help in algorithm implementation.

Funding: This work was supported by the Science Challenge Program under Grant TZ2016003-1; the National Key Re-search and Development Program of Ministry of Science and Technology under Grant 2016YFB0201304; and the China NSF under Grants 21573274 and 11771435.

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