A Flux Ratio and a Universal Property of Permanent Charges Effects on Fluxes

Abstract In this work, we consider ionic flow through ion channels for an ionic mixture of a cation species (positively charged ions) and an anion species (negatively charged ions), and examine effects of a positive permanent charge on fluxes of the cation species and the anion species. For an ion species, and for any given boundary conditions and channel geometry,we introduce a ratio _(Q) = J(Q)/J(0) between the flux J(Q) of the ion species associated with a permanent charge Q and the flux J(0) associated with zero permanent charge. The flux ratio _(Q) is a suitable quantity for measuring an effect of the permanent charge Q: if _(Q) > 1, then the flux is enhanced by Q; if _ < 1, then the flux is reduced by Q. Based on analysis of Poisson-Nernst-Planck models for ionic flows, a universal property of permanent charge effects is obtained: for a positive permanent charge Q, if _1(Q) is the flux ratio for the cation species and _2(Q) is the flux ratio for the anion species, then _1(Q) < _2(Q), independent of boundary conditions and channel geometry. The statement is sharp in the sense that, at least for a given small positive Q, depending on boundary conditions and channel geometry, each of the followings indeed occurs: (i) _1(Q) < 1 < _2(Q); (ii) 1 < _1(Q) < _2(Q); (iii) _1(Q) < _2(Q) < 1. Analogous statements hold true for negative permanent charges with the inequalities reversed. It is also shown that the quantity _(Q) = |J(Q) − J(0)| may not be suitable for comparing the effects of permanent charges on cation flux and on anion flux. More precisely, for some positive permanent charge Q, if _1(Q) is associated with the cation species and _2(Q) is associated with the anion species, then, depending on boundary conditions and channel geometry, each of the followings is possible: (a) _1(Q) > _2(Q); (b) _1(Q) < _2(Q).


Introduction
A central topic of physiology concerns functions of ion channels. Ion channels are large proteins embedded in cell membranes that have "holes" open to inside and outside of charge Q, if δ 1 (Q) is associated with the cation species and δ 2 (Q) is associated with the anion species, then, depending on boundary conditions and channel shapes, each of the followings is possible: (a) δ 1 (Q) > δ 2 (Q); (b) δ 1 (Q) < δ 2 (Q).
The rest of the paper is organized as follows. In Section 2, we recall the threedimensional and a quasi-one-dimensional PNP type models for ionic flows and a dimensionless form of the quasi-one-dimensional model. A statement of our result is provided. In Section 3, some relevant results from [31] are recalled that also serve as a part of motivations for our study in this paper and the above stated claim on the quantity δ(Q) is established. Section 4 contains proofs of our main results on λ(Q). In Section 5, concluding remarks and further related problems are provided.
2 Poisson-Nernst-Planck type models and the main result Permanent charges and channel shapes are the key structures of ion channels. Their effects on ionic flows are the main concern of ion channel functions. We will study the effects based on analysis of PNP type models.

Three-dimensional and quasi-one-dimensional PNP models
Taking the structural characteristics into considerations, PNP type systems are primitive models for ionic flows that treat the aqueous medium (in which salts are dissolved to free ions and ions are migrating) as dielectric continuum. PNP systems can be derived as reduced continuum models from molecular dynamic Langevin models ( [55]), from Boltzmann equations ( [4]), and from variational principles ( [26,27,28,60]), etc.
For an ionic mixture with n ion species, PNP reads ∇ · ε r (r)ε 0 ∇Φ = −e 0 n s=1 z s C s + Q(r) , ∇ · J k = 0, − J k = 1 k B T D k (r)C k ∇µ k , k = 1, 2, · · · , n (2.1) where r ∈ Ω with Ω being a three-dimensional cylindrical-like domain representing the channel, Q(r) is the permanent charge density, ε r (r) is the relative dielectric coefficient, ε 0 is the vacuum permittivity, e 0 is the elementary charge, k B is the Boltzmann constant, T is the absolute temperature; Φ is the electric potential, and, for the kth ion species, C k is the concentration, z k is the valence (the number of charges per particle), µ k is the electrochemical potential depending on Φ and {C j }, J k is the flux density vector, and D k (r) is the diffusion coefficient. Reduction of three-dimensional PNP systems (2.1) to quasi-one-dimensional models was first proposed in [48] based on the fact that ion channels have narrow cross-sections relative to their lengths, and was partially justified in [45] for special cases. A quasione-dimensional PNP model is where X ∈ [0, l] is the coordinate along the axis of the channel, A(X) is the area of cross-section of the channel over the location X. Equipped with system (2.2), we impose the following boundary conditions (see, [16] for a reasoning), for k = 1, 2, · · · , n, For a solution of BVP (2.2) and (2.3), the current I is For fixed L k 's and R k 's, J k 's depend on V only and formula (2.4) defines the I-V (current-voltage) relation -an important characteristic of an ion channel. The electrochemical potential µ k = µ id k (X) + µ ex k (X) consists of the ideal component µ id k (X) and the excess component µ ex k (X). The ideal component µ id k (X), given by with a fixed reference concentration C 0 , reflects the point-charge component of ions.
We point out that BVP (2.2) and (2.3) is generally well-posed if the excess potentials µ ex k (X) are local models, that is, for any X, µ ex k (X) depends on {C j (X)} at the given location X. In general, the excess potentials µ ex k (X) are nonlocal and, in this case, the boundary value problem with boundary conditions (2.3) is severely under determined. We refer the readers to [58] for more detailed discussion and for a correct formulation of the boundary conditions for PNP with nonlocal excess potentials.

Rescaling of the quasi-one-dimensional model problem
We make a dimensionless rescaling following [19]. Let C 0 be a characteristic concentration of the problems, for example, In terms of the new variables, BVP (2.2) and (2.3) becomes with boundary conditions at x = 0 and x = 1 Our analysis is based on the scaled quasi-one-dimensional BVP (2.7) and (2.8). All results can be transformed easily back to BVP (2.2) and (2.3). In the following, we assume ε > 0 is small and examine BVP (2.7) and (2.8) as a singularly perturbed BVP. We comment that if the distance between the boundary points is l = 2.5(nm) and the characteristic concentration is C 0 = 10(M ), then ε is of order 10 −3 (see, for example, [17]).
For fixed boundary conditions, we denote any solution of BVP (2.7) and (2.8) by and often its zeroth order approximation in ε by We indicate the dependence of solutions on Q but, of course, all these quantities depend on other parameters of the problem too such as the boundary conditions, the diffusion coefficients and the channel geometry. We will also use the notation D k (x; Q) to indicate the dependence of diffusion coefficients on the environment with the presence of Q(x).

Statement of results for
A simple but important observation was made explicitly in [18], that is, the Nernst-Planck equation in (2.7) for the flux J k gives An immediate consequence is that the sign of J k (Q, ε) is determined solely by the transmembrane electrochemical potentialμ k (0)−μ k (1): under the same boundary conditions, the sign of J k (Q, ε) is the same as that of J k (0, ε), independent of the permanent charge Q(x). However, the permanent charge Q(x) influences magnitudes of J k (Q, ε)'s. To measure this effect for kth ion species, we introduce a flux ratio In this case, one would say that the permanent charge has no effect on the flux and attempt to set λ k (Q; ε) = 1 as a convention. A closer examination provides a correct meaning and definition of λ k (Q; ε) for this case. In fact, in this case, λ k (Q; ε) is given by a ratio of 0/0-type -a case occurs in calculus where one uses L'Hoptial rule to determine if the ratio is well-defined as a limit. Indeed, treating the ratio as a limit aŝ µ δ k =μ k (0) −μ k (1) → 0, it follows from (2.10) that The latter ratio is well-defined, independent of whether or notμ δ k = 0. In general, thought, the ratio does not equal to 1 (see Remark 3.1, for example). In case that λ k (Q; ε) = 1, say λ k (Q; ε) > 1, it is still consistent with the statement that the permanent charge enhances the flux J k (Q; ε) = λ k (Q; ε)J k (0, ε) but the enhancement is annihilated by J k (0, ε) = 0.
Thus, the definition of λ k (Q; ε) in the case thatμ δ k = 0 should be understood as that in (2.12). It is worthwhile to mention a technical reason for this definition. The flux ratio λ k (Q; ε) depends also on (V 0 , L, R). If one uses the convention that λ k = 1 forμ δ k = 0, then, viewing λ k as a function of (V 0 , L, R, Q), it would have (removable) discontinuity at parameters whereμ δ k = 0. The definition of λ k in (2.12) for the casê µ δ k = 0 removes the discontinuity, and hence, is of advantage. 3 We consider ionic flows of ionic mixtures consisting of two ion species (n = 2), one cation species with valence z 1 > 0 and one anion species with valence z 2 < 0. We will show that (Theorem 4.1): If Q(x) ≥ 0, Q(x) = 0, then λ 1 (Q, ε) < λ 2 (Q, ε) when ε is small enough. (2.13) Furthermore, combining with results from [31] (recalled in Section 3 for reader's convenience), this result is optimal or sharp.
In general, the quantity δ k (Q) = |J k (Q; ε) − J k (0; ε)| is not suitable for comparing the effects of permanent charges on cation flux and anion flux. This is discussed in Proposition 3.3 in Section 3.
3 Relevant results from [31] and the quantity δ k (Q)

Specifics of flux ratios for small positive permanent charges
In [31], the authors considered a setting of classical PNP models with n = 2,ε r = 1, constant diffusion coefficients D k (x)'s, and Q(x) as where Q 0 is a constant with |Q 0 | small relative to the boundary concentrations l k 's and r k 's. For the zeroth order approximation in ε of BVP (2.7) and (2.8), one can write For kth ion species, denote the difference of its boundary electrochemical potentials bŷ Applying the methods in [16,44] to the study of BVP (2.7) and (2.8), the following results were obtained in [31] (see [31] for details and for other results).
Under the electroneutrality boundary conditions z 1 l 1 = −z 2 l 2 = L and z 1 r 1 = −z 2 r 2 = R, it is derived in [31] that and where, (3.5) Therefore, for |Q 0 | small, Remark 3.1. We provide a discussion directly related to Remark 2.1. Consider the case, sayμ δ 1 = z 1 V 0 + ln L − ln R = 0 but L = R so V 0 = 0. Then, for small |Q 0 | and up to O(Q 0 ), If the coefficient of Q 0 on the right-hand side of above is not zero, then λ k (Q; 0) = 1 for small |Q 0 | = 0. On the other hand, the coefficient of Q 0 equals to zero if either A = 0 The assumption that L = R implies that A = 0. Thus, The latter does not hold in general, and hence, λ 1 (Q; 0) = 1 even ifμ δ 1 = 0. 3 A complete classification of properties of J k1 's as consequences of interplay between boundary concentrations, boundary potential, channel geometry, and permanent charges Q(x) with small |Q 0 | is obtained. The following is a summary of Theorems 4.7 and 4.8 in [31] in terms of the flux ratios λ k (Q, ε). Theorem 3.2. Suppose ε > 0 is small enough. Then, depending on the boundary condition (V 0 , L, R) and the characteristic (α, β) of the channel geometry, a small positive Q 0 can (i) reduce the flux of cations and enhance that of anions: λ 1 (Q, ε) < 1 < λ 2 (Q, ε); (ii) enhance the fluxes of both cations and anions: λ 1 (Q, ε) > 1 and λ 2 (Q, ε) > 1; (iii) reduce the fluxes of both cations and anions: λ 1 (Q, ε) < 1 and λ 2 (Q, ε) < 1; (iv) but cannot enhance the flux of cations while reduce that of anions.
To complement the above results, we comment that, in [67], it is shown that, for values of (V 0 , L, R) in a bounded region, if Q 0 > 0 is large enough, then λ 1 (Q, ε) < 1; but, either λ 2 (Q, ε) < 1 or λ 2 (Q, ε) > 1 may occur, depending on the specifics of (V 0 , L, R). For moderate values of Q 0 , a numerical study in [66], focusing on parameter values of (V 0 , Q 0 ) for which λ k = 1, reveals a rich behavior of the flux ratios.

On the quantities δ k (Q; ε)
To end this section, we present a result to indicate that the quantity δ k (Q, ε) = |J k (Q, ε) − J k (0, ε)| is not a suitable quantity for comparing the effects of permanent charges on fluxes of cations and anions. Proposition 3.3. Assume the setup in Subsection 3.1 with z 1 = 1 and z 2 = −1. For Q 0 > 0 small and ε > 0 small, depending on specifics of (V 0 , L, R, α, β), each of the following inequalities is possible Proof. It suffices to establish the statement for δ k (Q; 0). Note that, for 0 )|, and hence, inequality (a) is equivalent to |J 11 |/|J 21 | > 1 and inequality (b) is equivalent to |J 11 |/|J 21 | < 1. It follows from (3.4) that Consider L > R. It has been shown that one can choose (α, β), depending on L and R but independent of V 0 , so that B > 1 (Case (i) in Lemma 4.6 of [31]). Assume this is the case. Note that, in the above expression inside the absolute value sign, the numerator If we choose V 0 so that then N > 0 and D > 0, and N − D = 2BV 0 ln(L/R) < 0, and hence, |J 11 | < |J 21 |, which implies inequality (b).

A universal property of permanent charge effects
We will establish our main result on the flux ratios for cations and anions.

Basic assumptions
For our result, we will consider n = 2 with z 1 > 0 > z 2 and make the following general assumptions for PNP type system (2.7).
(A3) The diffusion coefficients satisfy for a constant σ > 0 independent of x and Q(x).
We comment that (A3) and (A4) are assumed to be true also for Q = 0. Before a statement of our main claim, we briefly comment on assumptions (A3) and (A4). In general, the nature of diffusion coefficients D k (x; Q)'s is not completely understood. They are though not constants and vary from environment to environment. In (A3), we do not assume specifics on individual diffusion coefficients but assume a relation among them. (A3) roughly means that, as the environment varies from location to location, its influences on the two diffusion coefficients at the same location x are assumed to be the same; that is, the two diffusion coefficients vary from one common environment to another common environment in a way so that their ratio is independent of locations. This is clearly not a justification of this assumption but only an explanation of what it reflects.
The identity in (A4) is nothing but, up to the zeroth order approximation in ε, the pointwise-electroneutrality condition. Global electroneutrality is widely accepted while, in general, it is reasonable to assume only approximate pointwise-electroneutrality. Our assumption of "approximate" is specific: pointwise-electroneutrality is assumed to be exact only at the zeroth order approximation in ε. Mathematically, the zeroth order pointwise-electroneutrality condition has been rigorously justified for classical PNP models ( [16,44]) and PNP with hard-sphere potentials ( [30,38]).
We emphasize that, in the following, we do not assume PNP system (2.7) to be classical; that is, the electrochemical potential µ k can include any local excess components to account for the finite sizes of ions. Also, from the proof of Theorem 4.1 below, we only need to assume the excess components to be local at x = 0 and x = 1. Away from these two points, the models for the excess potentials could be nonlocal.
A numerical study on flux ratios λ k (Q) was recently conducted in [66] based on the classical PNP model and a PNP model with a Hard-Sphere component (PNP-HS). The permanent charge Q(x) was taken as in display (3.1) and the quantity Q 0 was increased from 0 to a large value. The numerical results for both the classical PNP and the PNP-HS models verify the property λ 1 (Q; ε) < λ 2 (Q; ε) in Theorem 4.1.
Furthermore, for small Q 0 , the numerical results on the boundary conditions for each of the cases (i), (ii) and (iii) in Theorem 3.2 agree perfectly with the analytical results in [31].

Concluding remarks and further related problems
In this work, we discover a universal property of positive permanent charge effects on fluxes of cations and anions. Combining with the results in [31], a fairly global picture is obtained: A positive permanent charge can enhance the fluxes of both cations and anions, can reduce the fluxes of both cations and anions, can reduce the flux of cations while enhance that of anions, but cannot enhance the flux of cations while reduce that of anions. Furthermore, in case both fluxes are enhanced, the relative amount of flux increased for cations is smaller than that for anions; in case both fluxes are reduced, the relative amount of flux decreased for cations is greater than that for anions. It is straightforward to state the results of the effects of negative permanent charges on fluxes of cations and anions. The assumptions (A1) and (A4) are not independent: Without (A1), the pointwise electroneutrality in (A4) may not be a good approximation. It would be interesting to know if the claim of Theorem 4.1 is still true without assuming ε to be particularly small. Other problems that are worthwhile to examine include (P1) the effect of a positive permanent charge on fluxes of three and more ionic species; (P2) the effect of sign changing permanent charges (e.g. piecewise constant with different signs) on fluxes of ionic mixtures.
We have used a quasi-one-dimensional PNP model for the study of the present topic. For three-dimensional channels, the permanent charge may not be ring-like so its effective one-dimensional version is not clear. This is an interesting problem to study. Of course, it is important to study the topics discussed in this paper and the above mentioned related problems for three-dimensional PNP type models.