Regulation of Aquaporin Z osmotic permeability in ABA tri-block copolymer

Wenyuan Xie; Jason Wei Jun Low; Arunmozhiarasi Armugam; Kandiah Jeyaseelan; Yen Wah Tong; Wenyuan Xie; Jason Wei Jun Low; Arunmozhiarasi Armugam; Kandiah Jeyaseelan; Yen Wah Tong

doi:10.3934/biophy.2015.3.381

AIMS Biophysics

2015, Volume 2, Issue 3: 381-397. doi: 10.3934/biophy.2015.3.381

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Research article Special Issues

Regulation of Aquaporin Z osmotic permeability in ABA tri-block copolymer

1.
NUS Environmental Research Institute (NERI), National University of Singapore #02-01, T-Lab Building 5A Engineering Drive 1, Singapore 117411;
2.
Department of Biochemistry, Yong Loo Lin School of Medicine, National University of Singapore, 8 Medical Drive, Singapore 117597;
3.
Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117585

Received: 15 May 2015 Accepted: 20 August 2015 Published: 31 August 2015

Aquaporins are transmembrane water channel proteins present in biological plasma membranes that aid in biological water filtration processes by transporting water molecules through at high speeds, while selectively blocking out other kinds of solutes. Aquaporin Z incorporated biomimetic membranes are envisaged to overcome the problem of high pressure needed, and holds great potential for use in water purification processes, giving high flux while keeping energy consumption low. The functionality of aquaporin Z in terms of osmotic permeability might be regulated by factors such as pH, temperature, crosslinking and hydrophobic thickness of the reconstituted bilayers. Hence, we reconstituted aquaporin Z into vesicles that are made from a series of amphiphilic block copolymers PMOXA-PDMS-PMOXAs with various hydrophobic molecular weights. The osmotic permeability of aquaporin Z in these vesicles was determined through a stopped-flow spectroscopy. In addition, the temperature and pH value of the vesicle solutions were adjusted within wide ranges to investigate the regulation of osmotic permeability of aquaporin Z through external conditions. Our results show that aquaporin Z permeability was enhanced by hydrophobic mismatch. In addition, the water filtration mechanism of aquaporin Z is significantly affected by the concentration of H⁺ and OH^- ions.

Keywords:

Citation: Wenyuan Xie, Jason Wei Jun Low, Arunmozhiarasi Armugam, Kandiah Jeyaseelan, Yen Wah Tong. Regulation of Aquaporin Z osmotic permeability in ABA tri-block copolymer[J]. AIMS Biophysics, 2015, 2(3): 381-397. doi: 10.3934/biophy.2015.3.381

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Abstract

1. Introduction

Fractional calculus is a main branch of mathematics that can be considered as the generalisation of integration and differentiation to arbitrary orders. This hypothesis begins with the assumptions of L. Euler (1730) and G. W. Leibniz (1695). Fractional differential equations (FDEs) have lately gained attention and publicity due to their realistic and accurate computations ^{[1,2,3,4,5,6,7]}. There are various types of fractional derivatives, including Riemann–Liouville, Caputo, Grü nwald–Letnikov, Weyl, Marchaud, and Atangana. This topic's history can be found in ^[8,9,10,11]. Undoubtedly, fractional calculus applies to mathematical models of different phenomena, sometimes more effectively than ordinary calculus ^[12,13]. As a result, it can illustrate a wide range of dynamical and engineering models with greater precision. Applications have been developed and investigated in a variety of scientific and engineering fields over the last few decades, including bioengineering ^[14], mechanics ^[15], optics ^[16], physics ^[17], mathematical biology, electrical power systems ^[18,19,20] and signal processing ^[21,22,23].

One of the definitions of fractional derivatives is Caputo-Fabrizo, which adds a new dimension in the study of FDEs. The new derivative's feature is that it has a nonsingular kernel, which is made from a combination of an ordinary derivative with an exponential function, but it has the same supplementary motivating properties with various scales as in the Riemann-Liouville fractional derivatives and Caputo. The Caputo-Fabrizio fractional derivative has been used to solve real-world problems in numerous areas of mathematical modelling for example, numerical solutions for groundwater pollution, the movement of waves on the surface of shallow water modelling ^[24], RLC circuit modelling ^[25], and heat transfer modelling ^[26,27] were discussed.

Rach (1987), Bellomo and Sarafyan (1987) first compared the Adomian Decomposition method (ADM) ^{[28,29,30,31,32]} to the Picard method on a variety of examples. These methods have many benefits: they effectively work with various types of linear and nonlinear equations and also provide an analytic solution for all of these equations with no linearization or discretization. These methods are more realistic compared with other numerical methods as each technique is used to solve a specific type of equations, on the other hand ADM and Picard are useful for many types of equations. In the numerical examples provided, we compare ADM and Picard solutions of multidimentional fractional order equations with Caputo-Fabrizio.

The fractional derivative of Caputo-Fabrizio for the function $ x\left(t\right) $ is defined as ^[33]

$\begin{equation} ^{CF}D_{0}^{\alpha }x\left( t\right) = \frac{B\left( \alpha \right) }{ 1-\alpha }\int_{0}^{t}\frac{d}{ds}\left( x\left( s\right) \right) \text{ } e^{-\frac{\alpha }{1-\alpha }(t-s)}ds, \end{equation}$ $

(1.1)

and its corresponding fractional integral is

$\begin{equation} ^{CF}I^{\alpha }x\left( t\right) = \frac{1-\alpha }{B\left( \alpha \right) } x\left( t\right) +\frac{\alpha }{B\left( \alpha \right) }\int_{0}^{t}x^{ \text{ }}\left( s\right) ds,{ \ \ \ \ }0 < \alpha < 1, \end{equation}$ $

(1.2)

where $ x\left(t\right) $ be continuous and differentiable on [0, T]. Also, in the above definition, the function $ B\left(\alpha \right) > 0 $ is a normalized function which satisfy the condition $ B\left(0\right) = B\left(1\right) = 0. $ The relation between the Caputo–Fabrizio fractional derivate and its corresponding integral is given by

$\begin{equation} \left( ^{CF}I_{0}^{\alpha }\right) \left( ^{CF}D_{0}^{\alpha }f\left( t\right) \right) = f\left( t\right) -f\left( a\right) . \end{equation}$ $

(1.3)

2. Construction of the algorithm

In this section, we will introduce a multidimentional FDE subject to the initial condition. Let $ \alpha \in \; (0, 1] $, $ 0 < \alpha _{1} < \alpha _{2} < ..., \alpha _{m} < 1, $ and $ m $ is integer real number,

$\begin{eqnarray} ^{CF}Dx & = &f(t,x,^{CF}D^{\alpha _{1}}x,^{CF}D^{\alpha _{2}}x,...,^{CF}D^{\alpha _{m}}x,)\text{ }, \\ x\left( 0\right) & = &c_{0}, \end{eqnarray}$ $

(2.1)

where $ x = x\left(t\right), t\in J = \left[ 0, T\right], T\in R^{+}, x\in C\left(J\right) $.

To facilitate the equation and make it easy for the calculation, we let $ x\left(t\right) = c_{0}+X\left(t\right) $ so Eq (2.1) can be witten as

$\begin{eqnarray} ^{CF}D^{\alpha }X & = &f(t,c_{0}+X,^{CF}D^{\alpha _{1}}X,^{CF}D^{\alpha _{2}}X,...,^{CF}D^{\alpha _{m}}X), \\ \text{ }X\left( 0\right) & = &0. \end{eqnarray}$ $

(2.2)

the algorithm depends on converting the initial condition from a constant $ c_{0} $ to 0.

Let $ ^{CF}D^{\alpha }X = y\left(t\right) $ then $ X = \; ^{CF}I^{\alpha }y, $ so we have

$\begin{equation} ^{CF}D^{\alpha i}X = \text{ }^{CF}I^{\alpha -\alpha i{ \ }CF}D^{\alpha }X = \text{ }^{CF}I^{\alpha -\alpha i}y,{ \ \ }i = 1,2,...,m. \end{equation}$ $

(2.3)

Substituting in Eq (2.2) we obtain

$\begin{equation} y = f(t,c_{0}+\text{ }^{CF}I^{\alpha }y,\text{ }^{CF}I^{\alpha -\alpha _{1}}y,...,\text{ }^{CF}I^{\alpha -\alpha _{m}}y). \end{equation}$ $

(2.4)

Assume $ f $ satisfies Lipschtiz condition with Lipschtiz constant$ \ L $ given by,

$\begin{equation} \left\vert f(t,y_{0},y_{1},...,y_{m})\right\vert -\left\vert f(t,z_{0},z_{1},...,z_{m})\right\vert \leq L\sum\limits_{i = 0}^{m}\left\vert y_{i}-z_{i}\right\vert , \end{equation}$ $

(2.5)

which implies

$\begin{eqnarray} && \begin{array}{c} \left\vert f(t,c_{0}+^{CF}I^{\alpha }y,^{CF}I^{\alpha -\alpha _{1}}y,..,^{CF}I^{\alpha -\alpha _{m}}y)\right. \\ \left. -f(t,c_{0}+^{CF}I^{\alpha }z,^{CF}I^{\alpha -\alpha _{1}}z,..,^{CF}I^{\alpha -\alpha _{m}}z)\right\vert \end{array} \\ &\leq &L\sum\limits_{i = 0}^{m}\left\vert \text{ }^{CF}I^{\alpha -\alpha _{i}}y-\text{ }^{CF}I^{\alpha -\alpha _{i}}z\right\vert . \end{eqnarray}$ $

(2.6)

The solution algorithm of Eq (2.4) using ADM is,

$\begin{eqnarray} y_{0}\left( t\right) & = &a\left( t\right) \\ y_{n+1}\left( t\right) & = &A_{n}\left( t\right) ,{ \ }j\geqslant 0. \end{eqnarray}$ $

(2.7)

where $ a\left(t\right) $ pocesses all free terms in Eq (2.4) and $ A_{n} $ are the Adomian polynomials of the nonlinear term which takes the form ^[34]

$\begin{equation} A_{n} = f\left( S_{n}\right) -\sum\limits_{i = 0}^{n-1}A_{i}, \end{equation}$ $

(2.8)

where $ f\left(S_{n}\right) = \sum_{i = 0}^{n}A_{i} $. Later, this accelerated formula of Adomian polynomial will be used in convergence analysis and error estimation. The solution of Eq (2.4) can be written in the form,

$\begin{equation} y\left( t\right) = \sum\limits_{i = 0}^{\infty }y_{i}\left( t\right) . \end{equation}$ $

(2.9)

lastly, the solution of the Eq (2.4) takes the form

$\begin{equation} x\left( t\right) = c_{0}+X\left( t\right) = c_{0}+\text{ }^{CF}I^{\alpha }y\left( t\right) . \end{equation}$ $

(2.10)

At which we convert the parameter to the initial form $ y $ to $ x $ in Eq (2.10), so we have the solution of the original Eq (2.1).

3. Convergence analysis

3.1. Existence and uniqueness

Define a mapping $ F:E\rightarrow E $ where $ E = \left(C\left[ J\right], \left\Vert \cdot \right\Vert \right) $ is a Banach space of all continuous functions on $ J $ with the norm $ \left\Vert x\right\Vert = \underset{t\epsilon J}{\text{ }\max\limits } \; x\left(t\right) $.

Theorem 3.1. Equation (2.4) has a unique solution whenever $ 0 < \phi < 1 $ where $ \phi = L\left(\sum_{i = 0}^{m}\frac{\left[ \left(\alpha-\alpha _{i}\right) \left(T-1\right) \right] +1}{B\left(\alpha -\alpha_{i}\right) }\right) $.

Proof. First, we define the mapping $ F:E\rightarrow E $ as

$\begin{equation*} Fy = f(t,c_{0}+\text{ }^{CF}I^{\alpha }y,\text{ }^{CF}I^{\alpha -\alpha _{1}}y,...,\text{ }^{CF}I^{\alpha -\alpha _{m}}y). \end{equation*}$ $

Let $ y $ and $ z\in E $ are two different solutions of Eq (2.4). Then

$\begin{equation*} Fy-Fz = f(t,c_{0}+^{CF}I^{\alpha }y,^{CF}I^{\alpha -\alpha _{1}}y,..,^{CF}I^{\alpha -\alpha _{m}}y)-f(t,c_{0}+^{CF}I^{\alpha }z,^{CF}I^{\alpha -\alpha _{1}}z,...,^{CF}I^{\alpha -\alpha _{m}}z) \end{equation*}$ $

which implies that

$\begin{eqnarray*} \left\vert Fy-Fz\right\vert & = &\left\vert f(t,c_{0}+\text{ }^{CF}I^{\alpha }y,\text{ }^{CF}I^{\alpha -\alpha _{1}}y,...,\text{ }^{CF}I^{\alpha -\alpha _{m}}y)\right. \\ &&-\left. f(t,c_{0}+\text{ }^{CF}I^{\alpha }z,\text{ }^{CF}I^{\alpha -\alpha _{1}}z,...,\text{ }^{CF}I^{\alpha -\alpha _{m}}z)\right\vert \\ &\leq &L\sum\limits_{i = 0}^{m}\left\vert \text{ }^{CF}I^{\alpha -\alpha _{i}}y-\text{ }^{CF}I^{\alpha -\alpha _{i}}z\right\vert \\ &\leq &L\sum\limits_{i = 0}^{m}\left\vert \frac{1-\left( \alpha -\alpha _{i}\right) }{ B\left( \alpha -\alpha _{i}\right) }\left( y-z\right) +\frac{\alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }\int_{0}^{t}\left( y-z\right) ds\right\vert \\ \left\Vert Fy-Fz\right\Vert &\leq &L\sum\limits_{i = 0}^{m}\frac{1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }\underset{ t\epsilon J}{\max }\left\vert y-z\right\vert +\frac{\alpha -\alpha _{i}}{ B\left( \alpha -\alpha _{i}\right) }\underset{t\epsilon J}{\max }\left\vert y-z\right\vert \int_{0}^{t}ds \\ &\leq &L\sum\limits_{i = 0}^{m}\frac{1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }\left\Vert y-z\right\Vert +\frac{\alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }\left\Vert y-z\right\Vert T \\ &\leq &L\left\Vert y-z\right\Vert \left( \sum\limits_{i = 0}^{m}\frac{1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }+\frac{\alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }T\right) \\ &\leq &L\left\Vert y-z\right\Vert \left( \sum\limits_{i = 0}^{m}\frac{\left[ \left( \alpha -\alpha _{i}\right) \left( T-1\right) \right] +1}{B\left( \alpha -\alpha _{i}\right) }\right) \\ &\leq &\phi \left\Vert y-z\right\Vert . \end{eqnarray*}$ $

under the condition $ 0 < \phi < 1, $ the mapping $ F $ is contraction and hence there exists a unique solution $ y\in C\left[ J\right] $ for the problem Eq (2.4) and this completes the proof.

3.2. Proof of convergence

Theorem 3.2. The series solution of the problem Eq (2.4)converges if $ \left\vert y_{1}\left(t\right) \right\vert < c $ and $ c $ isfinite.

Proof. Define a sequence $ \left\{ S_{p}\right\} $ such that $ S_{p} = \sum_{i = 0}^{p}y_{i}\left(t\right) $ is the sequence of partial sums from the series solution $ \sum_{i = 0}^{\infty }y_{i}\left(t\right), $ we have

$\begin{equation*} f(t,c_{0}+\text{ }^{CF}I^{\alpha }y,\text{ }^{CF}I^{\alpha -\alpha _{1}}y,...,\text{ }^{CF}I^{\alpha -\alpha _{m}}y) = \sum\limits_{i = 0}^{\infty }A_{i}, \end{equation*}$ $

$\begin{equation*} f(t,c_{0}+\text{ }^{CF}I^{\alpha }S_{p},\text{ }^{CF}I^{\alpha -\alpha _{1}}S_{p},...,\text{ }^{CF}I^{\alpha -\alpha _{m}}S_{p}) = \sum\limits_{i = 0}^{p}A_{i}, \end{equation*}$ $

From Eq (2.7) we have

$\begin{equation*} \sum\limits_{i = 0}^{\infty }y_{i}\left( t\right) = a\left( t\right) +\sum\limits_{i = 0}^{\infty }A_{i-1} \end{equation*}$ $

let $ S_{p}, S_{q} $ be two arbitrary sums with $ p\geqslant q $. Now, we are going to prove that $ \left\{ S_{p}\right\} $ is a Caushy sequence in this Banach space. We have

$\begin{eqnarray*} S_{p} & = &\sum\limits_{i = 0}^{p}y_{i}\left( t\right) = a\left( t\right) +\sum\limits_{i = 0}^{p}A_{i-1,} \\ S_{q} & = &\sum\limits_{i = 0}^{q}y_{i}\left( t\right) = a\left( t\right) +\sum\limits_{i = 0}^{q}A_{i-1.} \end{eqnarray*}$ $

$\begin{eqnarray*} S_{p}-S_{q} & = &\sum\limits_{i = 0}^{p}A_{i-1}-\sum\limits_{i = 0}^{q}A_{i-1} = \sum\limits_{i = q+1}^{p}A_{i-1} = \sum\limits_{i = q}^{p-1}A_{i-1} \\ & = &f(t,c_{0}+\text{ }^{CF}I^{\alpha }S_{p-1},\text{ }^{CF}I^{\alpha -\alpha _{1}}S_{p-1},...,\text{ }^{CF}I^{\alpha -\alpha _{m}}S_{p-1})- \\ &&f(t,c_{0}+\text{ }^{CF}I^{\alpha }S_{q-1},\text{ }^{CF}I^{\alpha -\alpha _{1}}S_{q-1},...,\text{ }^{CF}I^{\alpha -\alpha _{m}}S_{q-1}) \end{eqnarray*}$ $

$\begin{eqnarray*} \left\vert S_{p}-S_{q}\right\vert & = &\left\vert f(t,c_{0}+\text{ } ^{CF}I^{\alpha }S_{p-1},\text{ }^{CF}I^{\alpha -\alpha _{1}}S_{p-1},..., \text{ }^{CF}I^{\alpha -\alpha _{m}}S_{p-1})-\right. \\ &&\left. f(t,c_{0}+\text{ }^{CF}I^{\alpha }S_{q-1},\text{ }^{CF}I^{\alpha -\alpha _{1}}S_{q-1},...,\text{ }^{CF}I^{\alpha -\alpha _{m}}S_{q-1})\right\vert \\ &\leq &L\sum\limits_{i = 0}^{m}\left\vert \text{ }^{CF}I^{\alpha -\alpha _{i}}S_{p-1}- \text{ }^{CF}I^{\alpha -\alpha _{i}}S_{q-1}\right\vert \\ &\leq &L\sum\limits_{i = 0}^{m}\left\vert \frac{1-\left( \alpha -\alpha _{i}\right) }{ B\left( \alpha -\alpha _{i}\right) }\left( S_{p-1}-S_{q-1}\right) +\frac{ \alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }\int_{0}^{t}\left( S_{p-1}-S_{q-1}\right) ds\right\vert \\ &\leq &L\sum\limits_{i = 0}^{m}\frac{1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }\left\vert S_{p-1}-S_{q-1}\right\vert +\frac{ \alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }\int_{0}^{t}\left \vert S_{p-1}-S_{q-1}\right\vert ds \end{eqnarray*}$ $

$\begin{eqnarray*} \left\Vert S_{p}-S_{q}\right\Vert &\leq &L\sum\limits_{i = 0}^{m}\frac{1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }\underset{ t\epsilon J}{\max }\left\vert S_{p-1}-S_{q-1}\right\vert +\frac{\alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }\underset{t\epsilon J}{ \max }\left\vert S_{p-1}-S_{q-1}\right\vert \int_{0}^{t}ds \\ &\leq &L\left\Vert S_{p}-S_{q}\right\Vert \sum\limits_{i = 0}^{m}\left( \frac{ 1-\left( \alpha -\alpha _{i}\right) }{B\left( \alpha -\alpha _{i}\right) }+ \frac{\alpha -\alpha _{i}}{B\left( \alpha -\alpha _{i}\right) }T\right) \\ &\leq &L\left\Vert S_{p}-S_{q}\right\Vert \left( \sum\limits_{i = 0}^{m}\frac{\left[ \left( \alpha -\alpha _{i}\right) \left( T-1\right) \right] +1}{B\left( \alpha -\alpha _{i}\right) }\right) \\ &\leq &\phi \left\Vert S_{p}-S_{q}\right\Vert \end{eqnarray*}$ $

let $ p = q+1 $ then,

$\begin{equation*} \left\Vert S_{q+1}-S_{q}\right\Vert \leq \phi \left\Vert S_{q}-S_{q-1}\right\Vert \leq \phi ^{2}\left\Vert S_{q-1}-S_{q-2}\right\Vert \leq ...\leq \phi ^{q}\left\Vert S_{1}-S_{0}\right\Vert \end{equation*}$ $

From the triangle inequality we have

$\begin{eqnarray*} \left\Vert S_{p}-S_{q}\right\Vert &\leq &\left\Vert S_{q+1}-S_{q}\right\Vert +\left\Vert S_{q+2}-S_{q+1}\right\Vert +...\left\Vert S_{p}-S_{p-1}\right\Vert \\ &\leq &\left[ \phi ^{q}+\phi ^{q+1}+...+\phi ^{p-1}\right] \left\Vert S_{1}-S_{0}\right\Vert \\ &\leq &\phi ^{q}\left[ 1+\phi +...+\phi ^{p-q+1}\right] \left\Vert S_{1}-S_{0}\right\Vert \\ &\leq &\phi ^{q}\left[ \frac{1-\phi ^{p-q}}{1-\phi }\right] \left\Vert y_{1}\left( t\right) \right\Vert \end{eqnarray*}$ $

Since $ 0 < \phi < 1, p\geqslant q $ then $ \left(1-\phi ^{p-q}\right) \leq 1 $. Consequently

$\begin{equation} \left\Vert S_{p}-S_{q}\right\Vert \leq \frac{\phi ^{q}}{1-\phi }\left\Vert y_{1}\left( t\right) \right\Vert \leq \frac{\phi ^{q}}{1-\phi }\underset{ \forall t\epsilon J}{\max }\left\vert y_{1}\left( t\right) \right\vert \end{equation}$ $

(3.1)

but $ \left\vert y_{1}\left(t\right) \right\vert < \infty $ and as $ q\rightarrow \infty $ then, $ \left\Vert S_{p}-S_{q}\right\Vert \rightarrow 0 $ and hence, $ \left\{ S_{p}\right\} $ is a Caushy sequence in this Banach space then the proof is complete.

3.3. Error estimate

Theorem 3.3. The maximum absolute truncated error Eq (2.4)is estimated to be $ \underset{t\epsilon J}{\max }\left\vert y\left(t\right)-\sum_{i = 0}^{q}y_{i}\left(t\right) \right\vert \leq \frac{\phi ^{q}}{1-\phi }\underset{t\epsilon J}{\max }\left\vert y_{1}\left(t\right) \right\vert $

Proof. From the convergence theorm inequality (Eq 3.1) we have

$\begin{equation*} \left\Vert S_{p}-S_{q}\right\Vert \leq \frac{\phi ^{q}}{1-\phi }\underset{ t\epsilon J}{\max }\left\vert y_{1}\left( t\right) \right\vert \end{equation*}$ $

but, $ S_{p} = \sum_{i = 0}^{p}y_{i}\left(t\right) $ as $ p\rightarrow \infty $ then, $ S_{p}\rightarrow y\left(t\right) $ so,

$\begin{equation*} \left\Vert y\left( t\right) -S_{q}\right\Vert \leq \frac{\phi ^{q}}{1-\phi } \underset{t\epsilon J}{\max }\left\vert y_{1}\left( t\right) \right\vert \end{equation*}$ $

so, the maximum absolute truncated error in the interval $ J $ is,

$\begin{equation} \underset{t\epsilon J}{\max }\left\vert y\left( t\right) -\sum\limits_{i = 0}^{q}y_{i}\left( t\right) \right\vert \leq \frac{\phi ^{q}}{1-\phi }\underset{t\epsilon J}{\max }\left\vert y_{1}\left( t\right) \right\vert \end{equation}$ $

(3.2)

and this completes the proof.

4. Numerical examples

In this part, we introduce several numerical examples with unkown exact solution and we will use inequality (Eq 3.2) to estimate the maximum absolute truncated error.

Example 4.1. Application of linear FDE

$\begin{equation} ^{CF}Dx\left( t\right) +2a^{CF}D^{1/2}x\left( t\right) +bx\left( t\right) = 0, { \ \ \ \ \ \ \ }x\left( 0\right) = 1. \end{equation}$ $

(4.1)

A Basset problem in fluid dynamics is a classical problem which is used to study the unsteady movement of an accelerating particle in a viscous fluid under the action of the gravity ^[36]

Set

$\begin{equation*} X\left( t\right) = x\left( t\right) -1 \end{equation*}$ $

Equation (4.1) will be

$\begin{equation} ^{CF}DX\left( t\right) +2a^{CF}D^{1/2}X\left( t\right) +bX\left( t\right) = 0, { \ \ \ \ \ \ \ }X\left( 0\right) = 0. \end{equation}$ $

(4.2)

Appling Eq (2.3) to Eq (4.2), and using initial condition, also we take a = 1, b = 1/2,

$\begin{equation} y = -\frac{1}{2}-2I^{1/2}y-\frac{1}{2}I\text{ }y \end{equation}$ $

(4.3)

Appling ADM to Eq (4.3), we find the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &-\frac{1}{2}, \\ y_{i}\left( t\right) & = &-2\text{ }^{CF}I^{1/2}y_{i-1}-\frac{1}{2}\text{ } ^{CF}I\text{ }y_{i-1},\ \ \ \ \ i\geq 1. \end{eqnarray}$ $

(4.4)

Appling Picard solution to Eq (4.2), we find the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &-\frac{1}{2}, \\ y_{i}\left( t\right) & = &-\frac{1}{2}-2\text{ }^{CF}I^{1/2}y_{i-1}-\frac{1}{2} \text{ }^{CF}I\text{ }y_{i-1},\ \ \ \ \ i\geq 1. \end{eqnarray}$ $

(4.5)

From Eq (4.4), the solution using ADM is given by $ y\left(t\right) = \underset{q\rightarrow \infty }{Lim}{_{i = 0}^{q} y_{i}} \left(t\right) $ while from Eq (4.5), the solution using Picard technique is given by $ y\left(t\right) = \; \underset{i\rightarrow \infty }{ Lim} \; y_{i}\left(t\right) $. Lately, the solution of the original problem Eq (4.2), is

$\begin{equation*} x\left( t\right) = 1+\text{ }^{CF}I\text{ }y\left( t\right) . \end{equation*}$ $

One the same processor (q = 20), the time consumed using ADM is 0.037 seconds, while the time consumed using Picard is 7.955 seconds.

Figure 1 gives a comparison between ADM and Picard solution of Ex. 4.1.

Figure 1. ADM and Picard solution of Ex. 4.1.

DownLoad: Full-Size Img PowerPoint

Example 4.2. Consider the following nonlinear FDE ^[35]

$\begin{eqnarray} ^{CF}D^{1/2}x & = &\frac{8t^{3/2}}{3\sqrt{\pi }}-\frac{t^{7/4}}{4\Gamma \left( \frac{11}{4}\right) }-\frac{t^{4}}{4}+\frac{1}{8}\text{ }^{CF}D^{1/4}x+\frac{ 1}{4}x^{2},\text{ } \\ x\left( 0\right) & = &0. \end{eqnarray}$ $

(4.6)

Appling Eq (2.3) to Eq (4.6), and using initial condition,

$\begin{equation} y = \frac{8t^{3/2}}{3\sqrt{\pi }}-\frac{t^{7/4}}{4\Gamma \left( \frac{11}{4} \right) }-\frac{t^{4}}{4}+\frac{1}{8}\text{ }^{CF}I^{1/4}y+\frac{1}{4}\left( ^{CF}I^{1/2}y\right) ^{2}. \end{equation}$ $

(4.7)

Appling ADM to Eq (4.7), we find the solution algorithm will be become

$\begin{eqnarray} y_{0}\left( t\right) & = &\frac{8t^{3/2}}{3\sqrt{\pi }}-\frac{t^{7/4}}{4\Gamma \left( \frac{11}{4}\right) }-\frac{t^{4}}{4}, \\ y_{i}\left( t\right) & = &\frac{1}{8}\text{ }^{CF}I^{1/4}y_{i-1}+\frac{1}{4} \left( A_{i-1}\right) ,\ \ \ \ \ i\geq 1. \end{eqnarray}$ $

(4.8)

at which A$ _{\text{i}} $ are Adomian polynomial of the nonliner term $ \left(^{CF}I^{1/2}y\right) ^{2}. $

Appling Picard solution to Eq (4.7), we find the the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &\frac{8t^{3/2}}{3\sqrt{\pi }}-\frac{t^{7/4}}{4\Gamma \left( \frac{11}{4}\right) }-\frac{t^{4}}{4}, \\ y_{i}\left( t\right) & = &y_{0}\left( t\right) +\frac{1}{8}\text{ } ^{CF}I^{1/4}y_{i-1}+\frac{1}{4}\left( ^{CF}I^{1/2}y_{i-1}\right) ^{2},\ \ \ \ \ i\geq 1. \end{eqnarray}$ $

(4.9)

From Eq (4.8), the solution using ADM is given by $ y\left(t\right) = \underset{q\rightarrow \infty }{Lim}{_{i = 0}^{q}y_{i}} \left(t\right) $ while from Eq (4.9), the solution using Picard technique is given by $ y\left(t\right) = \underset{i\rightarrow \infty }{Lim} y_{i}\left(t\right) $. Finally, the solution of the original problem Eq (4.7), is.

$\begin{equation*} x\left( t\right) = \text{ }^{CF}I^{1/2}y. \end{equation*}$ $

One the same processor (q = 2), the time consumed using ADM is 65.13 seconds, while the time consumed using Picard is 544.787 seconds.

Table 1 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 2):

Table 1. Max. absolute error.

q	max. absolute error
2	0.114548
5	0.099186
10	0.004363

| Show Table

DownLoad: CSV

Figure 2 gives a comparison between ADM and Picard solution of Ex. 4.2.

Figure 2. ADM and Picard solution of Ex. 4.2.

DownLoad: Full-Size Img PowerPoint

Example 4.3. Consider the following nonlinear FDE ^[35]

$\begin{eqnarray} ^{CF}D^{\alpha }x & = &3t^{2}-\frac{128}{125\pi }t^{5}+\frac{1}{10}\left( ^{CF}D^{1/2}x\right) ^{2}, \\ x\left( 0\right) & = &0. \end{eqnarray}$ $

(4.10)

Appling Eq (2.3) to Eq (4.10), and using initial condition,

$\begin{equation} y = 3t^{2}-\frac{128}{125\pi }t^{5}+\frac{1}{10}\left( ^{CF}I^{1/2}y\right) ^{2} \end{equation}$ $

(4.11)

Appling ADM to Eq (4.11), we find the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &3t^{2}-\frac{128}{125\pi }t^{5}, \\ y_{i}\left( t\right) & = &\frac{1}{10}\left( A_{i-1}\right) ,\ \ \ \ \ i\geq 1 \end{eqnarray}$ $

(4.12)

at which A$ _{\text{i}} $ are Adomian polynomial of the nonliner term $ \left(^{CF}I^{1/2}y\right) ^{2}. $

Then appling Picard solution to Eq (4.11), we find the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &3t^{2}-\frac{128}{125\pi }t^{5}, \\ y_{i}\left( t\right) & = &y_{0}\left( t\right) +\frac{1}{10}\left( ^{CF}I^{1/2}y_{i-1}\right) ^{2},\ \ \ \ \ i\geq 1. \end{eqnarray}$ $

(4.13)

From Eq (4.12), the solution using ADM is given by $ y\left(t\right) = \underset{q\rightarrow \infty }{Lim}{_{i = 0}^{q}y_{i}} \left(t\right) $ while from Eq (4.13), the solution is $ y\left(t\right) = \underset{i\rightarrow \infty }{Lim}y_{i}\left(t\right) $. Finally, the solution of the original problem Eq (4.11), is

$\begin{equation*} x\left( t\right) = ^{CF}Iy\left( t\right) . \end{equation*}$ $

One the same processor (q = 4), the time consumed using ADM is 2.09 seconds, while the time consumed using Picard is 44.725 seconds.

Table 2 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 4):

Table 2. Max. absolute error.

q	max. absolute error
2	0.00222433
5	0.0000326908
10	2.88273*10$ ^{-8} $

| Show Table

DownLoad: CSV

Figure 3 gives a comparison between ADM and Picard solution of Ex. 4.3 with $ \alpha = 1 $.

Figure 3. ADM and Picard solution where of Ex. 4.3.

DownLoad: Full-Size Img PowerPoint

Example 4.4. Consider the following nonlinear FDE ^[35]

$\begin{eqnarray} ^{CF}D^{\alpha }x & = &t^{2}+\frac{1}{2}\text{ }^{CF}D^{\alpha _{1}}x+\frac{1}{ 4}\text{ }^{CF}D^{\alpha _{2}}x+\frac{1}{6}\text{ }^{CF}D^{\alpha 3}x+\frac{1 }{8}x^{4}, \\ x\left( 0\right) & = &0. \end{eqnarray}$ $

(4.14)

Appling Eq (2.3) to Eq (4.10), and using initial condition,

$\begin{equation} y = t^{2}+\frac{1}{2}\left( ^{CF}I^{\alpha -\alpha _{1}}y\right) +\frac{1}{4} \left( ^{CF}I^{\alpha -\alpha _{2}}y\right) +\frac{1}{6}\left( ^{CF}I^{\alpha -\alpha 3}y\right) +\frac{1}{8}\left( ^{CF}I^{\alpha }y\right) ^{4}, \end{equation}$ $

(4.15)

Appling ADM to Eq (4.15), we find the solution algorithm become

$\begin{eqnarray} y_{0}\left( t\right) & = &t^{2}, \\ y_{i}\left( t\right) & = &\frac{1}{2}\left( ^{CF}I^{\alpha -\alpha _{1}}y\right) +\frac{1}{4}\left( ^{CF}I^{\alpha -\alpha _{2}}y\right) +\frac{ 1}{6}\left( ^{CF}I^{\alpha -\alpha 3}y\right) +\frac{1}{8}A_{i-1},{ \ \ }i\geq 1 \end{eqnarray}$ $

(4.16)

where A$ _{\text{i}} $ are Adomian polynomial of the nonliner term $ \left(^{CF}I^{\alpha }y\right) ^{4}. $

Then appling Picard solution to Eq (4.15), we find the solution algorithm become

$ y_{0}\left( t\right) = t^{2}, \\ y_{i}\left( t\right) = t^{2}+\frac{1}{2}\left( ^{CF}I^{\alpha -\alpha _{1}}y_{i-1}\right) +\frac{1}{4}\left( ^{CF}I^{\alpha -\alpha _{2}}y_{i-1}\right) \\+\frac{1}{6}\left( ^{CF}I^{\alpha -\alpha 3}y_{i-1}\right) +\frac{1}{8}\left( ^{CF}I^{\alpha }y_{i-1}\right) ^{4}\ \ \ \ \ i\geq 1. $

(4.17)

From Eq (4.16), the solution using ADM is given by $ y\left(t\right) = \underset{q\rightarrow \infty }{Lim}{_{i = 0}^{q}y_{i}} \left(t\right) $ while from Eq (4.17), the solution using Picard technique is $ y\left(t\right) = \underset{i\rightarrow \infty }{Lim} y_{i}\left(t\right) $. Finally, the solution of the original problem Eq (4.14), is

$\begin{equation*} x\left( t\right) = ^{CF}I^{\alpha }y\left( t\right) . \end{equation*}$ $

One the same processor (q = 3), the time consumed using ADM is 0.437 seconds, while the time consumed using Picard is (16.816) seconds. Figure 4 shows a comparison between ADM and Picard solution of Ex. 4.4 $ at \; \alpha = 0.7, \; \alpha _{1} = 0.1, \alpha _{2} = 0.3, \alpha _{3} = 0.5. $

Figure 4. ADM and Picard solution where of Ex. 4.4.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

The Caputo-Fabrizo fractional deivative has a nonsingular kernel, and consequently, this definition is appropriate in solving nonlinear multidimensional FDE ^[37,38]. Since the selected numerical problems have an unkown exact solution, the formula (3.2) can be used to estimate the maximum absolute truncated error. By comparing the time taken on the same processor (i7-2670QM), it was found that the time consumed by ADM is much smaller compared with the Picard technique. Furthermore Picard gives a more accurate solution than ADM at the same interval with the same number of terms.

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	Kumar M, Grzelakowski M, Zilles J, et al. (2007) Highly permeable polymeri membranes based on the incorporation of the functional water channel protein Aquaporin Z. PNAS 104: 20719-20724. doi: 10.1073/pnas.0708762104
[2]	Service RF (2006) Desalination freshens up. Science 313:1088-1090.
[3]	Tal A (2006) Seeking sustainability: Israel's evolving water management strategy. Science 313: 1081-1084.
[4]	Discher BM, Won YY, Ege DS, et al. (1999) Polymersomes: tough vesicles made from diblock copolymers. Science 284:1143-1146. doi: 10.1126/science.284.5417.1143
[5]	Calamita G, Kempf B, Rudd KE, et al. (1997) The aquaporin-Z water channel gene of Escherichia coli: Structure, organization and phylogeny. Biology of the Cell 89: 321-329. doi: 10.1111/j.1768-322X.1997.tb01029.x
[6]	Calamita G, Bishai WR, Preston GM, et al. (1995) Molecular cloning and characterization of AqpZ, a water channel from Escherichia coli. J Biol Chem 270: 29063-29066. doi: 10.1074/jbc.270.49.29063
[7]	Scheuring S, Ringler P, Borgnia M, et al. (1999) High resultion AFM topographs of the Escherichia coli water channel aquaporin Z. EMBO J 18: 4981-4987.
[8]	Gorin MB, Yancey SB, Cline J, et al. (1984) The major intrinsic protein (MIP) of the bovine lens fiber membrane: Characterization and structure based on cDNA cloning. Cell 39: 49-59.
[9]	Ishibashi K, Kuwahara M, Gu Y, et al. (1997) Cloning and functional expression of a new water channel abundantly expressed in the testis permeable to water, glycerol and urea. J Biol Chem 272: 20782-20786. doi: 10.1074/jbc.272.33.20782
[10]	Soupene E, King N, Lee H, et al. (2001) Aquaporin Z of Escherichia coli: Reassessment of Its Regulation and Physiological Role. J Bacter 184: 4304-4307.
[11]	Calamita G, Kempf B, Bonhivers B, et al. (1998) Regulation of the Escherichia coli water channel gene AqpZ. Proc Natl Acad Sci U S A 95: 3627-3631. doi: 10.1073/pnas.95.7.3627
[12]	Borgnia MJ, Kozono D, Calamita G, et al. (1999) Funcation Reconstitution and Characterization of AqpZ, the E. coli Water Channel Protein. J Mol Biol 291: 1169-1179.
[13]	Kozono D, Yasui M, King LS, et al. (2002). Aquaporin water channels: atomic structure molecular dynamics meet clinical medicine. J Clin Inves 109: 1395-1399. doi: 10.1172/JCI0215851
[14]	Nemeth-Cahalan KL, Hall JE (2000) pH and Calcium Regulate the Water Permeability of Aquaporin 0. J Biol Chem 275: 6777-6782. doi: 10.1074/jbc.275.10.6777
[15]	Cahalan K, Kalman K, Hall JE (2004) Molecular Basis of pH and Ca2+ Regulation of Aquaporin Water Permeability. J Gen Physiol 123: 573-580
[16]	Zhou W, Jones SW (1996) The effects of external pH on calcium channel currents in bullfrog sympathetic neurons. Biophys J 70: 1326-1334
[17]	Gonen T, Walz T (2006) The structure of aquaporins. Q Rev Biophys 39: 361-396.
[18]	Chaumont F, Moshelion F, Daniels MJ (2005) Regulation of plant aquaporin activity. Biol Cell 97: 749-764. doi: 10.1042/BC20040133
[19]	Tong J, Canty JT, Briggs MM, et al. (2013) The water permeability of lens aquaporin-0 depends on its lipid bilayer environment. Exp Eye Res 113: 32-40. doi: 10.1016/j.exer.2013.04.022
[20]	Andersen OS, Bruno MJ, Sun H, et al. (2007) Single-molecule methods for monitoring changes in bilayer elastic properties. Meth Mol Biol 400: 543-570 doi: 10.1007/978-1-59745-519-0_37
[21]	Hong H, Tamm LK (2004) Elastic coupling of integral membrane protein stability to lipid bilayer forces. Proc Natl Acad Sci U S A 101: 4065-4070. doi: 10.1073/pnas.0400358101
[22]	Nyholm TK, Ozdirekcan S, Killian JA (2007) How protein transmembrane segments sense the lipid environment. Biochemistry 46: 1457-1465. doi: 10.1021/bi061941c
[23]	Phillips R, Ursell T, Wiggins P, et al. (2009) Emerging roles for lipids in shaping membrane-protein function. Nature 459: 379-385. doi: 10.1038/nature08147
[24]	Yuan C, O'Connell RJ, Jacob RF, et al. (2007) Regulation of the gating of BKCa channel by lipid bilayer thickness. J Biol Chem 282: 7276-7286.
[25]	Dumas F, Tocanne JF, Leblanc G, et al. (2000) Consequences of hydrophobic mismatch between lipids and melibiose permease on melibiose transport. Biochem 39: 4846-4854. doi: 10.1021/bi992634s
[26]	Perozo E, Kloda A, Cortes DM, et al. (2002) Physical principles underlying the transduction of bilayer deformation forces during mechano senditive channel gating. Nat Struct Biol 9: 696-703. doi: 10.1038/nsb827
[27]	Xie W, He F, Wang B, et al. (2013) An aquaporin-based vesicle-embedded polymeric membrane for low energy water filtration. J Mater Chem A 1: 7592-7600. doi: 10.1039/c3ta10731k
[28]	Wang H, Chung TS, Tong YW, et al. (2011) Preparation and characterization of pore-suspending biomimetic membranes embedded with Aquaporin Z on carboxylated polyethylene glycol polymer cushion. Soft Matter 7: 7274-7280.
[29]	Wang H, Chung TS, Tong YW, et al. (2012) Highly permeable and selective pore-spanning biomimetic membrane embedded with aquaporin Z. Small 8: 1185-1190, 1125.
[30]	Duong PHH, Chung TS, Jeyaseelan K, et al. (2012) Planar biomimetic aquaporin-incorporated triblock copolymer membranes on porous alumina supports for nanofiltration. J Membr Sci 409: 34-43.
[31]	Zhong PS, Chung TS, Jeyaseelan K, et al. (2012) Aquaporin-embedded biomimetic membranes for nanofiltration. J Membr Sci 407: 27-33.
[32]	Savage DF, Egea PF, Colmenares YR, et al. (2013) Architecture and selectivity in aquaporins: 2.5A X-Ray Structure of Aquaporin Z. PLoS Biol 1: 334-340.
[33]	Nielsen CH (2009) Biomimetic membranes for sensor and separation applications. Bioanal Chem 395: 697-718. doi: 10.1007/s00216-009-2960-0
[34]	Discher DE, Eisenberg A(2002) Polymer vesicles. Science 297: 967-973.
[35]	Ahmed F, Photos PJ, Discher DE (2006) Polymersomes as viral capsid mimics. Drug Develop Res 67: 4-14. doi: 10.1002/ddr.20062
[36]	Lewis BA, Engelman DM (1983) Lipid Bilayer Thickness Varies Linearly with Acyl Chain Length in Fluid Phosphatidylcholine Vesicles. J Mol Biol 166: 211-217. doi: 10.1016/S0022-2836(83)80007-2
[37]	Dave PC, Tiburu EK, Damodaran K, et al. (2004) Investigating Structural Changes in the Lipid Bilayer upon Insertion of the Transmembrane Domain of the Membrane-Bound Protein Phospholamban Utilizing ³¹P and ²H Solid-State NMR Spectroscopy. Biophys J 86: 1564-1573. doi: 10.1016/S0006-3495(04)74224-1
[38]	Marsh D (2008) Energetics of Hydrophobic Matching in Lipid-Protein Interactions. Biophys J 94: 3996-4013.
[39]	Xu Q, Kim M, David Ho KW, et al. (2008) Membrane Hydrocarbon Thickness Modulates the Dynamics of a Membrane Transport Protein. Biophys J 95: 2849-2858. doi: 10.1529/biophysj.108.133629
[40]	He F, Tong YW (2014) A mechanistic study on amphiphilic block co-polymer poly(butadiene-b-(ethylene oxide)) vesicles reveals the water permeation mechanism through a polymeric bilayer. RSC Adv 4: 15304-15313. doi: 10.1039/c3ra48063a
[41]	Yang B, Verkman AS (1997) Water and Glycerol Permeabilities of Aquaporins 1-5 and MIP Determined Quantitatively by Expression of Epitope-tagged Constructs in Xenopus Oocytes. J Biol Chem 272: 16140-16146. doi: 10.1074/jbc.272.26.16140
[42]	Mehdizadeh H, Dickson JM, Eriksson PK (1989) Temperature effects on the performance of thin-film composite, aromatic polyamide membranes. Ind Eng Chem Res 28: 814-824. doi: 10.1021/ie00090a025

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