1. Introduction
The cell membrane is a biological membrane that separates the interior of all cells from the outside environment which protects the cell from its environment. Ion channels are large proteins embedded in cell membranes that have holes open to the inside and the outside of cells. Ion channel opening gives rise to a passageway through which charged ions can cross the cell membrane. It is now well-known that migration of charges for ionic flow through ion channels can be described mathematically by the Poisson-Nernst-Planck model [1] [2].
A stationary one-dimensional Poisson-Nernst-Planck model [3] [4] [5] is
(1.1)
where
is the electric potential,
is the concentration for the ith ion species,
is the valence,
is the permanent charge of the channel,
is the electrochemical potential,
is the area of the cross-section of the channel,
is the flux density,
is the diffusion coefficient,
is the relative dielectric coefficient,
is the vacuum permittivity, k is the Boltzmann constant, T is the absolute temperature, and e is the elementary charge.
The boundary conditions are, for
,
(1.2)
in the classical Poisson-Nernst-Planck model takes the following form
(1.3)
with
is a constant.
The Poisson-Nernst-Planck model (1.1) can be viewed as a simplified model which is derived from the Maxwell-Boltzmann equations [6] [7] and the Langevin-Poisson equations [8] [9]. More sophisticated Poisson-Nernst-Planck model has been also developed and analyzed [10] [11]. The dynamics of the classical model (1.1) has been analyzed [12] [13] [14] [15] to a great extent. Especially, the existence and uniqueness of solutions for the boundary value problems (1.1) and (1.2) has been obtained in [16] under the assumption that
. In [17], under the assumption that
is a piecewise constant function, the general dynamical system framework for studying the boundary value problems (1.1) and (1.2) has been developed by employing the geometric singular perturbation theory [18] [19] [20]. As we know, under the assumption that
is a piecewise constant function, it is basically difficult to obtain the explicit formula for individual flux with respect to permanent charges, so it is also not easy to analyze the effects of permanent charges on individual flux. In this paper, the effects of permanent charges on ionic flows through ion channels are investigated under the following assumptions.
(A1)
and
.
(A2)
for
,
for
and
for
, where Q is a constant and Q will be set to be small in the later analysis.
By re-scaling,
The model (1.1) is reduced to a standard singularly perturbed system of the following
(1.4)
with the boundary condition, for
.
(1.5)
Under the assumptions (A1) and (A2), the existence of the solutions of (1.4) and (1.5) has been studied in [21]. In this paper, it is additionally assumed that the constant Q is small, then by expanding the solutions of (1.4) and (1.5) with respect to small Q, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated. As
in (1.4) and (1.5), namely, only one positively charged ion and one negatively charged ion are involved in the Poisson-Nernst-Planck model, the effects of small permanent charges on individual flux has been analyzed in [22]. On the other hand, assuming that the constant Q is large, the effects of large permanent charges on individual flux have been also analyzed in [23].
2. Brief Reviews of Relevant Results in [21]
Let
,
. System (1.4) becomes
(2.6)
By using the rescaling
, one has
(2.7)
Define
(2.8)
Then a solution to Equations (1.4) and (1.5) is to finding an orbit of Equation (2.6) or (2.7) from
to
.
Due to the fact that
is a piecewise constant function, so we analyze the limiting fast and limiting slow orbits of Equations (2.6) and (2.7) on three intervals
,
and
respectively.
Let
,
,
,
, where
,
,
,
are unknowns to be determined. Let
Let
,
,
,
, where
,
,
,
are unknowns to be determined. Let
Then an singular orbit of Equation (2.6) or (2.7) from
to
consists of three parts: that is, a singular orbit over the interval
connecting orbit from
to
, a singular orbit over the interval
connecting orbit from
to
, and a singular orbit over the interval
connecting orbit from
to
.
Based on [21], an singular orbit of Equation (2.6) or (2.7) from
to
is equivalent to solving the following algebraic equations:
(2.9)
and
(2.10)
where
(2.11)
3. Taylor Expansions of (2.9)-(2.11) with Respect to Small
In this section, it is assumed that
is small. All unknown quantities in (2.9)-(2.11) are expanded in Q as follows
(3.12)
Let
(3.13)
Inserting the formulae (3.12) into (2.9)-(2.11) and expanding the algebraic equations (2.9)-(2.11) in Q, then by comparing the terms of like-powers in Q, one has
Proposition 3.1. Zeroth order solution in Q of (2.9)-(2.11) is given by
(3.14)
Corollary 3.2. Under electroneutrality boundary conditions
and
, one has
,
,
,
, and
(3.15)
Proposition 3.3. First order terms of the solution in Q of (2.9)-(2.11) are given by
(3.16)
and
(3.17)
where
(3.18)
4. Effects of Small Permanent Charge
In this section, the effects of small permanent charges on individual fluxes are analyzed under electroneutrality conditions
and
.
For
small, the individual flux
of the ith ion species are
From Proposition 3.3, it follows that
(4.1)
where, in terms of
defined in (3.13), A and B defined in (3.18) become
(4.2)
Remark 4.1. Note that
, it means that
is not a zero point of
, also, it can be easily seen that there are only two values
and
such that
.
Remark 4.2. Note that
,
, where the sign of
has been analyzed by in [22].
Remark 4.3.
in (3.3) is exactly similar to
in [22], whose properties have been analyzed in [22].
Let
denote the larger value between
and
,
denote the smaller value between
and
.
Theorem 4.4. (i) If
, for
or
, then
; for
, then
.
(ii) If
, for
or
, then
; for
, then
.
Proof. If
, based on Remark 4.2, then it follows that
By Remark 4.1, there are only two values
and
such that
therefore the statement (i) can be obtained. Similarly, the statement (ii) can be also proved.
Theorem 4.5. If
, for
, then
; for
, then
.
If
, for
, then
; for
, then
.
Equivalently, for
and
, small positive Q strengthens the individual flux
; for
and
, small positive Q reduces the individual flux
.
For
and
, small positive Q reduces the individual flux
; for
and
, small positive Q strengthens the individual flux
.
Proof. Based on Corollary 3.2 and Equation (4.1), one has
(4.3)
From (4.3), the statement can be obtained. □
5. Conclusion
In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that
positively charged ion species have the same valence and the permanent charge is small. By expanding an singular orbit of Poisson-Nernst-Planck model (1.1) in small
, the explicit formulae for
and
are obtained. The signs of
are discussed in Theorem 4.4, which indicates that as
is sufficiently large, fixing the other parameters,
behaves like
. The effects of small permanent charges on individual flux are investigated in Theorem 4.5, which means that small Q can strengthen or reduce the individual flux under suitable conditions. However, for
that is not small, the regular perturbation analysis does not work, so it seems not easy to analyze the effects of permanent charges on individual flux by directly using (2.9)-(2.11).
Acknowledgements
The author was supported by the NNSFC 11971477.