A Flux Ratio Independent of the Permanent Charge in PNP Models ()
1. Introduction
The cell membrane is a biological membrane that separates the interior of all cells from the outside environment and protects the cell from its environment. The cell membrane consists of a lipid bilayer that is semipermeable. It regulates the transport of materials entering and exiting the cell. Ion channels are large proteins embedded in cell membranes that have holes open to the inside and the outside of cells. The charged ions flow through the open channels and represent an electric current. These currents alter the distribution of charge and the voltage across the membrane is changed. 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)
which
is a constant.
The Poisson-Nernst-Planck model (1.1) is actually a simplified model which is derived from the Maxwell-Boltzmann equations [6] [7] and the Langevin-Poisson equations [8] [9] by capturing key features. Recently, the Poisson-Nernst-Planck model (1.1) has been studied [10] - [17] greatly. In [18], under the assumption
is a piecewise constant function, the boundary value problems (1.1) and (1.2) have been analyzed based on the geometric singular perturbation theory [19] [20] [21]. However, due to the lack of the explicit formula for individual flux, it is difficult to analyze the properties of individual flux. In this paper, a property of individual flux, that is, a flux ratio is independent of the permanent charge, is identified under the following assumptions.
(A1)
and
.
(A2) For
, let
for
;
for
;
for
;
for
;
for
;
for
;
for
; where
, are constants and m is an arbitrary positive integer.
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 assumption that the permanent charge
is small, the effects of small permanent charges on individual flux are investigated in [22]. On the other hand, under the assumption that the permanent charge
is large, the effects of large permanent charges on individual flux have been also analyzed in [23] [24]. Actually, due to the assumption that the permanent charge
is small or large, the solutions of (1.4) and (1.5) can be expanded with respect to
, therefore, 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 or large permanent charges on individual flux can be analyzed in [22] [23] [24].
In this paper, under the assumptions A1 and A2 and without the assumption the permanent charge
is small or large, although it seems that there is no methods to get the explicit formula for
, but it still can be verified that a flux ratio is independent of
, that is,
(1.6)
The rest of this paper is organized as follows. In Section 2, limiting fast and slow orbits for (1.4) and (1.5) are constructed. In Section 3, limiting fast and slow orbits for (1.4) and (1.5) in Section 2 are matched, which results in a series of very complicated algebraic equations. The main results in this paper are presented in Section 4. Some conclusions are given in Section 5.
2. Limiting Fast and Slow Orbits for (4)-(5) over [0, 1]
Let
,
, system (1.4) becomes
(2.7)
By using the rescaling
, one has
(2.8)
Define
(2.9)
Then a solution to Equations (1.4) and (1.5) is to finding an orbit of Equation (2.7) or (2.8) from
to
.
By letting
, we analyze the limiting fast and limiting slow orbits of Equations (2.7) and (2.8) on intervals
,
,
,
,
,
and
respectively due to the fact that
is a piecewise constant function.
Let
,
,
,
, where
,
,
,
are unknowns to be determined. Let
Let
,
,
,
, where
,
,
,
are unknowns to be determined. Let
Then limiting fast and slow orbits of Equation (2.7) or (2.8) from
to
will consists of several parts: limiting fast and slow orbits over the interval
connecting orbit from
to
, limiting fast and slow orbits over the interval
connecting orbit from
to
, limiting fast and slow orbits over the interval
connecting orbit from
to
, limiting fast and slow orbits over the interval
connecting orbit from
to
, limiting fast and slow orbits over the interval
connecting orbit from
to
, limiting fast and slow orbits over the interval
connecting orbit from
to
, and limiting fast and slow orbits over the interval
connecting orbit from
to
.
For convenience, let
.
2.1. Limiting Fast and Slow Orbits on
Where
In this section, we will construct limiting fast and slow orbits that connects
to
by letting
in Equations (2.7) and (2.8). As shown in [25], limiting fast and slow orbits that connect
to
are satisfied by:
(2.10)
where
and
(2.11)
2.2. Limiting Fast and Slow Orbits on
Where
In this section, we will construct limiting fast and slow orbits that connects
to
by letting
in Equations (2.7) and (2.8). Limiting fast and slow orbits that connects
to
are satisfied by:
(2.12)
where
,
and
(2.13)
2.3. Limiting Fast and Slow Orbits on
Where
In this section, we will construct limiting fast and slow orbits that connects
to
by letting
in Equations (2.7) and (2.8). Limiting fast and slow orbits that connect
to
are satisfied by:
(2.14)
where
,
and
(2.15)
2.4. Limiting Fast and Slow Orbits on
Where
In this section, we will construct limiting fast and slow orbits that connects
to
by letting
in Equations (2.7) and (2.8). Limiting fast and slow orbits that connect
to
are satisfied by:
(2.16)
where
and
(2.17)
3. Matching Limiting Fast and Slow Orbits on
Based on Sections 2.1-2.4, to obtain limiting fast and slow orbits from
to
, the following algebraic equations should holds simultaneously:
(3.18)
and
(3.19)
Note that the total number of the unknown parameters
and
is
. Also, the total number of Equations (3.18) and (3.19) is exactly
, which matches the total number of the unknown parameters.
It can be seen that Equations (3.18) and (3.19) are very complicated nonlinear algebraic equations, which are intricately difficult to be solved, however, algebraic Equations (3.19) can be solved in the next section.
4. Main Results
In this section, the main results of this paper, that is, the ratio of
to
is independent of the permanent charge
, will be proved. As shown in the following, to justify the main results, it is sufficient that only Equations (3.19) are used.
Theorem 4.1. Under the assumptions A1 and A2, let
in Equations (2.7) and (2.8), then one has
(4.20)
where
, which indicates that the ratio of
to
is independent of the permanent charge
.
Proof. Substitute the formulae for
in Equations (2.11), (2.13), (2.15) and (2.17) into algebraic Equations (3.19), then direct calculations show that
(4.21)
where
and
.
Again, substitute the formulae for
in Equations (4.21) into algebraic Equations (3.19), then the statement in Theorem 4.1 can be obtained. □
Remark 4.2. The remain parameters
,
,
,
,
,
,
,
, are determined by nonlinear algebraic Equations (3.18), and it seems that it is extremely difficult to get the explicit formulae for these parameters.
5. Conclusion
In this paper, PNP models with an arbitrary number of positively charged ion species and one negatively charged ion species are investigated under the assumptions A1 and A2. By using the geometric singular perturbation theory and solving Equations (3.19), it is proved that the ratio of
to
is independent of the permanent charge
. Also, it can be seen that although Equations (3.18) are not used in the proof of Theorem 4.1, the number of solutions to the boundary value problems (1.1) and (1.2) is determined by Equations (3.18). However, due to the fact that Equations (3.18) are very sophisticated algebraic equations, it is very challenging to solve Equations (3.18).
Acknowledgements
The author was supported by the NNSFC 11971477.