Mathematical modeling of molecular recognition by an ion-gatingmembrane oscillator

Taichi Ito a, Hidenori Ohashi b, Takanori Tamaki b, Takeo Yamaguchi b,n

a Center for Disease Biology and Integrative Medicine, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japanb Chemical Resources Laboratory, Tokyo Institute of Technology, Nagatsutacho 4259, Midori-ku, Yokohama, Kanagawa 226-8503, Japan

a r t i c l e i n f o

Article history:Received 13 February 2013Received in revised form30 June 2013Accepted 2 August 2013Available online 13 August 2013

Keywords:Molecular recognitionMembrane oscillatorNonequilibrium thermodynamicsMathematical modeling

a b s t r a c t

A mathematical model to explain the mechanism of molecular recognition by an ion-gating membraneoscillator has been proposed, in which the hydrostatic pressure-driven flow (pore-open state) andosmotic flow (pore-closed state) are switched alternatively in response to a specific ion signal. The modelwas based on transport equations derived from nonequilibrium thermodynamics, where the dependencyof the reflection coefficient and the solvent hydrostatic permeability on the signal ion concentration wereassumed to follow the Hill equation, based on experimental data. A time-delay effect was also introducedinto the permeation parameters by first derivation. As a result, the model reproduced the characteristicparameters of the oscillator, such as the period and amplitude. It also clarified that an autocatalyticprocess, the key for nonlinear oscillation, was generated for an osmotic period by a slow response of thereflection coefficient and a fast response of the solvent hydrostatic permeability, i.e., the osmotic flowremoves the signal ions out of the membrane to the ion-fed side, which increases the osmotic flow, andthen more ions are removed from the membrane on opening the pores from the solvent side of themembrane.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Nonlinear phenomena are observed in various fields, frombiosystems [1], such as organs [2] and cells [3–5], to artificialengineering systems, such as BZ reactions [6–9], chemical reac-tions on catalysts [10,11] or electrodes [12], and hydrogels [13–16].They can create unique patterns and/or rhythms [17], which confera high degree of functionality to each system. In particular, variousmembrane systems [18–23] are an interesting research field fornonlinear science, because they are an interface that separates twodifferent spaces having different chemical potentials. Mass trans-port can occur through them, and sometimes reactions arecoupled to these transport phenomena.

As one of such membrane system, a new nonlinear oscillatorshown in Fig. 1 [24] was fabricated, which is called an ion-gatingmembrane oscillator for molecular recognition, inspired by theTeorell oscillator [25,26]. Utilizing the swelling and shrinking of agrafted copolymer of N-isopropylacrylamide and Benzo-18-crown-6-acrylamide [27] inside the pores, the gating membrane couldcontrol the osmotic pressure [28] and hydrostatic pressure-drivenflow [29] by opening and closing its pores in response to theconcentration of a signal ion, such as Ba2þ . This gating membrane

was placed between a solvent chamber and a solution chamber.A capillary was attached to the solution chamber, which was filledwith an aqueous ion solution. Thus, the resulting osmotic inflowfrom the solvent chamber increased the water level in thecapillary. In the case of Ba2þ , the Ba2þ ions entered the membraneunder hydrostatic flow, and then the pores closed and stoppedthe hydrostatic flow. The water level in the capillary decreased atthe same time. After the pores closed, an osmotic flow generatedby the Donnan potential. Then the osmotic flow passed throughthe membrane in the opposite direction to the hydrostatic flow.This osmotic flow eliminated any complexed ions from themembrane. At the same time, the water level in the capillarygradually increased. When all the captured ions were eliminated,the pores of the membrane then reopened, and a hydrostatic flowreentered the membrane. This cycle could be repeated withoutany external stimuli, and therefore a nonlinear self-excited oscilla-tion occurred [24]. For comparison, other ions, such as Ca2þ , didnot form complexes with the membrane. Consequently, in thosecircumstances the membrane did not generate an osmotic flow,and no oscillation occurred.

In general, mathematical modeling is an effective approach toclarify the essential mechanism of nonlinear phenomena [30]. Forexample, the Hodgkin–Huxley model [31,32] is a representativemodel for showing the dynamic mechanism of the excitation of anerve system due to the active potential through a cellular membrane,which is the driving force of the oscillation. Another typical example

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0376-7388/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.memsci.2013.08.001

n Corresponding author. Tel.: þ81 45 924 5254; fax: þ81 45 924 5253.E-mail address: yamag@res.titech.ac.jp (T. Yamaguchi).

Journal of Membrane Science 448 (2013) 231–239

is the van der Pol equation [30,33]. This equation can be used todescribe many systems, such as a tunnel diode oscillator. Theequation contains a so-called negative resistance, which alsogenerates an autocatalytic process. These models are ordinaldifferential equation models, which can simplify nonlinear phe-nomena and clarify an essential character of the oscillationdynamics, such as an autocatalytic or feedback process.

Nonequilibrium thermodynamics are useful for modelingmembrane oscillators [34–36]. In addition, it is very convenientto reduce equations to an ordinal differential system, such as thevan der Pol equation or the Hodgkin–Huxley model, to clarify anoscillation mechanism. In the case of the Teorell oscillator, this wasspecifically based on nonequilibrium thermodynamics [25,26].Assuming that the electrical resistance is a function of the solventflow and has a first-order time-delay effect, Teorell reduced thethree simultaneous equations of the solvent flow, the solute flow,and the electric current to an equation of the van der Pol type.Although some inaccurate points about the modeling werepointed out, and improved and/or modified models have beensuggested [37,38], the Teorell model shows that an approach basedon nonequilibrium thermodynamics is still efficient today, espe-cially when the physicochemical details are unknown [39,40].

In this study, a molecular recognition by an ion-gating mem-brane oscillator is modeled based on nonequilibrium thermody-namics, the nonlinear relationships between the permeabilityparameters of the steady state, the concentration of a signal ionderived from an experimental approach, and any time-delayeffects introduced by the response rate of the permeabilityparameters. The mechanism by which an autocatalytic processoccurs is discussed based on the model.

2. Experimental

2.1. Membrane preparation

The membranes were prepared by a graft polymerization methodusing a peroxide plasma, as described in an earlier publication [28].Briefly, a porous high-density polyethylene film (pore size¼0.2 μm,

thickness¼100 μm) was used as a substrate, and the plasma treat-ment power and time were 30W and 1 min, respectively. Themembrane used, which had a grafted density of 1.01 mg/cm2, wasthe same as the one used in the previously reported research [24,29].

2.2. Osmotic pressure and osmotic flow measurements

The osmotic pressure was measured using an osmometer cell at43 1C, as described in an earlier publication [29]. The osmometerhad a solution chamber and a solvent chamber. The membranewas placed between these two chambers. The solution chamberwas filled with an ion solution and was closed using a valve toprevent the inflow of solvent. The increase in pressure inside thesolution chamber was detected by a pressure sensor (Keyence, AP-12A, Japan). The initial condition of the gating membrane waschanged by immersing the membrane in aqueous BaCl2 solutionsof various concentrations. The osmotic pressure was alwaysmeasured when the initial condition of the gating membranewas 0 mM, as described in the previous publications. The osmoticflow was measured using the same apparatus at 43 1C withoutclosing the valve, and the flux of poured solution was measured.

3. Model

3.1. Equations of membrane transport based on nonequilibriumthermodynamics

Katchalsky derived the equations of membrane transport basedon nonequilibrium thermodynamics as follows [34]:

Jv ¼ LPðΔP�sΔπÞ; ð1Þand

Js ¼ PðCm�CpÞþð1�sÞcs Jv; ð2Þwhere Jv and Js are the flux of the solvent and the solute, respectively.The terms LP and s are the permeability of pure water and thereflection coefficient, respectively. The terms P, Δπ, ΔP, Cm, Cp, andcs are the solute permeability, the difference in osmotic pressure,

Fig. 1. A schematic drawing of the operation of molecular recognition by a gating membrane oscillator. The hydrostatic flow and the osmotic flow are generated alternativelyby the opening and closing of the pores.

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239232

the difference in pressure, the concentration of solute on the fed sideof the membrane, the concentration of solute on the permeated sideof the membrane, and the average concentration of solute inside themembrane, respectively.

Experimentally, the diffusive flux of the ions was much smallerthan the advective flux in Eq. (2), and thus, the first term,PðCm�CpÞ, can be neglected:

Jsffið1�sÞcs Jv ¼ ð1�sÞcsLPðΔP�sΔπÞ: ð3ÞThe inside of the membrane is assumed to be a continuously

stirring tank reactor, and neglected the ion concentration distribu-tion inside the membrane. In this case, the concentration of theion inside the gating membrane can be represented by the averageion concentration inside the membrane, c. Then, the mass balanceof the ions in the membrane can be represented as follows:

Vdcdt

¼ AmðJs_in�Js_outÞ; ð4Þ

where V and Am are the pore volume of the membrane and thearea of the membrane, respectively, and Js_in and Js_out are the ionfluxes of the inflow and the outflow.

In the case of Js_in, the value of cs is equal to the ionconcentration of the inflow solution from outside the membrane(term cin in Eq. (3)). The direction of the flow switched alterna-tively between the osmotic period and the hydrostatic period. Inthe osmotic period, the value of Js_in was negative and theion concentration of the inflow solution, cin, was equal to the fedion concentration in the feed chamber, cf . On the other hand, inthe hydrostatic period, the value of Js_in was positive, and the ionconcentration of the inflow solution, cin, was equal to zero in thesolvent chamber. Thus, Eq. (3) is reduced to

Js_in ¼ ð1�sÞcinLP jΔP�sΔπj; ð5Þwhere

at ΔP�sΔπ40 : cin ¼ cfat ΔP�sΔπr0 : cin ¼ 0

In the case of Js_out , the value of cs is equal to the ionconcentration inside the membrane as follows:

Js_out ¼ ð1�sÞc LP jΔP�sΔπj: ð6ÞIn Eqs. (2) and (3), the value of cs is constant, because the

equation depicts the steady state. In this study, the ion concentra-tion inside the gating membrane changed, and thus cs wasreplaced by c. Combining Eqs. (4)–(6), the equation of ion trans-port was derived as follows:

Vdcdt

¼ Amð1�sÞðcin�cÞLP U ΔP�sΔπ :j�� ð7Þ

The mass balance of the solvent also needs to be considered.The change in solvent volume in the capillary is equal to the inflowor outflow of the solvent through the membrane.

Ac

gdðΔPÞdt

¼�Am LpðΔP�sΔπÞ; ð8Þ

where Ac and gare the cross-sectional area of the capillary and theacceleration due to gravity, respectively.

3.2. Nonlinear relationship between the permeability parametersand the ion concentration

The terms LP and s are the transport coefficients of themembrane. These are usually constants in the case of separationmembranes, such as ultrafiltration membranes, because the mem-brane properties, such as the pore size and the density of fixedcharges, are constant. However, in the case of a gating membrane,

these change in response to c, the ion concentration inside themembrane. The value of LP decreases with an increase in c, whilethe value of s increases with an increase in c. A nonlinearrelationship existed between LP , s, and c, is assumed based onHill's equation in the steady state, by reference to the experi-mental data. Hill's equation is known to describe the cooperativebinding effect of hemoglobins to oxygen in biochemistry field. It isgenerally used to determine nonlinear relationship based onexperimental data in various fields such as muscle mechanics[42]. In the present research, the experimental data shown inFigs. 2 and 5 are well fitted by Hill's equations as mentionedbelow. Thus, Hill's equation is introduced in Eqs. (11) and (12). It isan important future work to clarify the relationship between ionconcentration, swelling and shrinking of the grafted copolymer

Fig. 2. Determination of LP1 as a function of c by fitting Hill's equation. The opensquares denote the experimental values previously reported in Ref. [24]. The solidline is the determined function.

Fig. 3. The dependency of the osmotic pressure on the initial ion concentrationinside the gating membrane. The ion concentrations of the feed ion and solventchamber were 100 and 0 mM, respectively. The initial ion concentration inside thegating membrane was 0 mM (A), 40 mM (B), 100 mM (C), and 150 mM (D).

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239 233

inside the pores of the membrane, the pore size of the membrane,and the flux values. In addition, as discussed in the followingsection in detail, LP and s are assumed to have a first-order time-delay effect. Thus, the steady-state values of LP and s, LP1 and s1,respectively, were determined from a Hill's equation-type rela-tionship. The values of LP and s were determined according to thefollowing equations:

dsdt

¼ 1τsðs1�sÞ; ð9Þ

and

dLpdt

¼ 1τLp

ðLp1�LpÞ; ð10Þ

where τLp and τs are the characteristic times of the hydrostaticperiod and the osmotic period.

s1 ¼ s0ððc�c3Þ=c2Þm

bþððc�c3Þ=c2Þm; ð11Þ

and

Lp1 ¼ L0 1� ðc=c1Þnaþðc=c1Þn

� �; ð12Þ

where L0, s0, c1, c2, c3, a, b, n, and m are the fitting parameters.

3.3. Equations of state of the oscillator and numerical calculations

Simultaneous equations comprised of Eqs. (7)–(12) representmolecular recognition by the ion-gating membrane oscillator. Therepresentative parameter set is shown in Table 1. The previousoscillations experiments were performed from the pore-openingstate with no ions inside the gating membrane. Thus, the initialvalues of ΔP, c, Lp, and s were 1.5 kPa, 0 mM, 1.1�10�6 m3 m�2

s�1(kPa)�1, and 0, respectively. The equations were solvednumerically using the Runge–Kutta method employing the Cþþsoftware package. The effect and sensitivity of parameters such ascf , τLp , τs, and m on the computations were studied.

4. Results and discussion

4.1. Determination of LP1 as a function of c

The value of LP1 needed to be determined as a function of c forthe modeling. The experimental values of LP1 at various ionconcentrations have already been reported, as shown in Fig. 2[28]. Eq. (12) is fitted to the data shown in Fig. 2, and the fittingparameters are determined, as shown in Table 1.

4.2. Dependency of osmotic pressure on the initial ion concentrationinside the gating membrane

Fig. 3 shows the time course of the osmotic pressure fordifferent initial concentrations of Ba2þ ions inside the membrane.The initial concentrations of curves (A), (B), (C), and (D) in Fig. 3 of

Fig. 4. Osmotic flux for various feed ion concentrations between 0 and 100 mM.The initial ion concentration was fixed at 0 mM for all conditions.

Fig. 5. Determination of s1 as a function of c by fitting Hill's equation. The opensquares are experimental values calculated from data in Fig. 4 using Eq. (17). Thesolid line is the determined function for m¼4. The dashed–dotted line and thedotted line are the cases for m¼2 and 10, respectively, without changing the otherparameters of Hill's equation.

Table 1Parameters used in the numerical simulations.

Parameter Unit Value

Predetermined parametersAc m2 1.5�10�5

Am m2 1.4�10�3

cf mM 40Δπ kPa 105

V m3 8.2�10�8

Parameters estimated by auxiliary experimentsa – 6.0�10�3

b – 1c1 mM 30c2 mM 30c3 mM 5L0 m3 m�2 s�1(kPa)�1 1.1�10�6

m – 4n – 5s0 – 9.0�10�1

Fitting parameters for oscillation calculationτs (osmotic period) s 3600τs (hydrostatic period) s 3600τLp (osmotic period) s 900τLp (hydrostatic period) s 1

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239234

the gating membrane are 0, 40, 100, and 150 mM, respectively. Thefeed ion concentration was 100 mM for all conditions, and so thedriving force of the osmotic pressure generation had the samevalue. Thus, the osmotic pressure for a period of several hoursafter starting a measurement was constant for all conditions.

However, the osmotic pressure in the early stages of themeasurements was different for each condition. The osmoticpressure increased rapidly when the initial ion concentrationinside the gating membrane was lower than the feed ion concen-tration of 100 mM in the cases of curves (A) and (B). The pores of asolvent-fed side can open, which would make it possible for arapid diffusion of water through the membrane. On the otherhand, the pores can close on the ion-fed side immediately aftermeasurements are commenced, because of the ions diffusing intothe gating membrane. In general, the existence of a very thin layerthat can reject the diffusion of a solute is adequate to generate anosmotic pressure.

On the other hand, the osmotic pressure increased graduallywhen the initial ion concentration inside the gating membranewas higher than the feed ion concentration of 100 mM in the caseof curves (C) and (D). The pores of the gating membrane wereinitially closed at a given point in the membrane. Thus, waterdiffusion could be slower when compared with the case of curves(A) and (B), which would result in a slow increase in the osmoticpressure.

The period of this oscillation as reported in the previousresearch was between 10 and 25 min [24]. Thus, the ion concen-tration inside the gating membrane may have a strong effect onthe oscillation. The ion concentration profile throughout thedirectional thickness of the membrane reflects the pore size profilethroughout that thickness. In addition, the pore size profile shouldbe considered as the key factor influencing the oscillations.

4.3. Determination of s1 as a function of c by measuring theosmotic flux

Fig. 4 shows the osmotic flux for a feed ion concentrationbetween 0 and 100 mM. The osmotic flux depended on the feedion concentration, because of the difference in the driving force forthe generation of the osmotic pressure; this was generated for afeed ion concentration above 20 mM, which agrees well with theprevious osmotic pressure data [24].

The initial ion concentration inside the gating membrane wasfixed at 0 mM for all conditions, because the concentration insidethe membrane, which is defined as the value of c in Eq. (7), mustdecrease and the pores of the gating membrane must open eachtime the osmotic period switched into the hydrostatic period inthe oscillator. This means that osmotic flux when the ion concen-tration inside the gating membrane was 0 mM is an importantparameter. The osmotic flux decreased with time and becamealmost constant 4 h after commencing the measurements shownin Fig. 4. This coincided well with the osmotic pressure of curve(A) in Fig. 3.

Based on these considerations, the osmotic flux was defined asthe value measured 10 min after the commencement of themeasurements, because the period of this oscillation as reportedin the previous research was between 10 and 25 min [24]. Theequations of membrane transport, Eqs. (13) and (14), can bederived from linear response theory as follows [34]:

Jv ¼ LPΔPþLPDΔπ; ð13Þand

JD ¼ LDPΔPþLDΔπ; ð14Þwhere Jv and JD are the hydrostatic pressure-driven flow and thediffusive flow, and LP , LPD, LDP , and LD are proportionality coefficients.

The value of LPD was determined from the measured osmotic fluxusing Eq. (15), because ΔP ¼ 0 in Eq. (13):

LPD ¼ΔπJv

ð15Þ

Katchalsky defined s as follows:

s¼�LPDLP

ð16Þ

Thus, the following relationship holds for the research:

s1 ¼�LPDLP1

ð17Þ

The value of LP1 was reported in the earlier publication [28],and is shown in Fig. 2. Hence, s1 was calculated for each feed ionconcentration using Eq. (17), as shown in Fig. 5. In the modelingcalculations, Eq. (11) was fitted to the data, and the parameterswere determined, as shown in Table 1. The parameter m is animportant parameter, and a value of m¼4 fitted the data well. Inaddition, in Fig. 5, the lines for other values of m are drawn,without changing the other parameters in Eq. (17), for use infurther discussion.

4.4. Reproduction of the capillary water pressure of the oscillatorusing numerical calculations

Numerical simulations were performed using all the deter-mined parameters, as summarized in Table 1. The parameters ofthe membrane and capillary, V , Am, and Ac , were determined fromcalculations using the experimental data shown in Table 1. Thevalue of cf at which oscillations were observed experimentally was40 mM, as reported in the previous publication [24]. The osmoticpressure, Δπ, was calculated using van't Hoff's law from the valueof cf . The characteristic times of the hydrostatic period and theosmotic period, τLp and τs, were assigned for both the osmotic andhydrostatic periods without obtaining experimental values, asshown in Table 1. The value of τLp and τs were significantlydifferent, where τs is much higher than τLp . The response rate ofthe membrane to change the hydrostatic pressure-driven flux inresponse to the BaCl2 concentration was extremely fast as reportedin the previous study. In fact, the pores closed below 30 s [41].Partially because Ba2þ was fed to the membrane by pressure-driven flux. On the other hand, the characteristic time of thegeneration of osmotic pressure was ca. 60 min as shown inFigs. 3 and 4. Therefore, τLp was assumed as 1 s in the hydrostaticperiod or 900 s in the osmotic period, while τs was assumed as3600 s.

Fig. 6 shows the time course of a simulated capillary waterpressure, where the value of ΔP in Eq. (8) was equal to the waterlevel of the attached capillary. The value of ΔP showed a nonlinearoscillation with an amplitude of about 0.1 kPa for a period of about20 min. The simulated amplitude and period of ΔP were com-mensurate with the experimental values. (The experimental data[24] are available in the Supplementary information.) The oscilla-tion was finally dampened at ΔP¼18.7 kPa in the model, which issimilar to the experimental value, because the oscillation appearedas a transitional phenomenon. The wave forms obtained fordifferent values of ΔP are also shown in Fig. 6(B) and (C) frommagnified sections of Fig. 6(A). These showed a slow decrease inΔP in the hydrostatic period for low values of ΔP. However,they showed a sharp decrease in ΔP in the hydrostatic period forhigh values of ΔP. Similar change in the waveforms with increas-ing values of ΔP was also observed in the experiments.The proposed model captures well the essential mechanism ofmolecular recognition in an ion-gating membrane oscillator, basedon the above comparison between the experiment data available

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239 235

in the Supplementary information and the modeling resultsshown in Fig. 6.

4.5. Clarification of autocatalytic process in the osmotic period basedon the model calculations

In general, an oscillator requires both an autocatalytic and afeedback process. The autocatalytic process occurred during theincrease of water level in the case of the present oscillator.Feedback occurred via the hydrostatic flow, which rapidly closedthe pores of the gating membrane by feeding ions into themembrane. However, the autocatalytic process mechanism wasunclear, even qualitatively. The model makes it possible to suggesta mechanism of the autocatalytic process.

Fig. 7(A) shows the time course of the ion concentration insidethe gating membrane, c. When the pores were open, c increased

sharply owing to the inflow of the ion solution from the hydro-static flow in the hydrostatic period. This increase in c rapidlyclosed the pores because of the small value of τLp . If the auto-catalytic process did not exist, then the pores would have beenkept in the intermediate state between open and closed, and nooscillation would have occurred. However, the value of c began togradually decrease because the osmotic flow pushed out the ionsinside the gating membrane. Once the value of c decreased to agiven value, the value of LP began to increase sharply, as shown inFig. 7(B). This increase in the value of LP brought about an increasein the osmotic flow, because the osmotic flow, Jv, obeys Eq. (1). Thestronger the osmotic flow became, the more efficiently theosmotic flow pushed the ions out from the gating membrane.The larger the decrease in c became, the stronger the osmotic flowbecame. This autocatalytic process brought about the reopening ofthe pores of the gating membrane. Once the pores were reopened,the same phenomena occurred, and so the oscillation continued. Itshould be noted that the change in the value of c depended on theion concentration of the feed solution, cf . When the value of cf waslow, as shown in curves (C) and (D) in Fig. 7(A), no oscillationswere observed.

A schematic diagram of this autocatalytic process in theosmotic period is shown in Fig. 8 based on the modeling calcula-tion results. The model suggests that LP increases and the value of

Fig. 6. Reproduction of the capillary water pressure of the oscillator by thenumerical calculations. The parameters used for the numerical calculations areshown in Table 1. (A) An overall view. (B) Partially magnified at the time pointsbetween 0.5 and 3.0 h. (C) Partially magnified at the time points between 45.0 and47.0 h.

Fig. 7. (A) The time course of the ion concentration inside the gating membrane, c,from numerical calculations. The parameters used for the numerical calculationsare shown in Table 1. The concentration of the fed ion, cf , was 100 mM (A), 40 mM(B), 20 mM (C), and 10 mM (D). Only (A) and (B) generated an oscillation of the ionconcentration inside the gating membrane. (B) The time course of LP and s fromnumerical calculations. The parameters used for the numerical calculations areshown in Table 1. The concentration of the fed ion, cf , was 40 mM. An autocatalyticincrease in LP was observed during the osmotic period.

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239236

s is kept high during the autocatalytic process. In general, inmembrane science, the value of LP depends on the membranethickness of a dense separation layer. However, the value of s doesnot depend on the membrane thickness of a dense separationlayer. Hence, separation membranes have evolved from a symme-trical membrane structure into an asymmetrical membrane struc-ture, which has a thin separation layer and a thick supportivelayer. An asymmetric membrane can achieve both a high selectiv-ity and a high permeation flux, because of the compatibility of ahigh value of LP and a high value of s. Based on this knowledgewell known in membrane science, the changes in LP and s in theosmotic period are interpreted to be related to the pore size profilethroughout the directional thickness of the membrane, as shownin Fig. 8. Owing to the osmotic flow, the pores began opening onthe pure water side of the gating membrane, which increased thevalue of LP more and more. At the same time, the existence of avery thin layer of closed pores on the feed ion side was enough tomaintain a high value of s. Thus, the membrane properties, such asthe amplitude and period of the oscillator, are considered to beclosely related to the thickness of the gating membrane.

The following point should be noted. The fluctuation of s wasvery small; however, oscillations were not observed without themodulation of s(data not shown). This means that little change ofs is crucially important for the oscillations. That is to say, thedecrease of s in the osmotic period contributed to the increase ofJs_out in Eq. (6) as well as the change of LP .

4.6. The relationship between the gating effect of the membrane andthe generation of oscillations

The characteristic times, τLp and τs, are important, because nooscillations would occur if the time-delay effect was not assumedin the model. Fig. 9(A) and (B) show the calculated values ofΔP fordifferent values of τLp and τs. Oscillations occurred over a widerange of τLp and τs. In addition, the period and amplitude bothdepended on the values of τLp and τs.

A limitation of the model was the assumption of τLp and τswithout experimental proof of the values. Thus, the qualitativerelationship between the membrane structure and the oscillationproperties are difficult to be discussed. A physical meaning ofthese characteristic times is the time-delay effect on LP and s,which arises from the pore size profile throughout the membranethickness, as shown in Fig. 8. Thus, a large value of τs in the

osmotic period corresponds to a large membrane thickness.A normal membrane thickness is thought to exist for the oscillator.

The effect of s1 on the oscillations is also investigated. Thecurvature of the relationship between s1 and c depends onthe parameter m in Eq. (12), as shown in Fig. 5. Fig. 10 showsthe calculated value of ΔP for various values of m. The oscillationswere sensitive to the value of m. The pores closed at an ionconcentration inside the gating membrane of 20 mM. When m islarge, the pores of the gating membrane close rapidly and aresensitive to the change in ion concentration around 20 mM. On theother hand, the pores of the gating membrane close slowly inresponse to a change in the ion concentration when the value of mis small. The gating membrane did not generate oscillations bothfor large values of m (m¼10) and small values of m (m¼2). Thismeans that the pores of the gating membrane needed to closecontinuously in response to a change in the ion concentration. Inaddition, the pores needed to close rapidly to some degree inresponse to the change in ion concentration inside the membrane.

In the previous publication, the oscillations were generated byexperimental trial and error [24]. Through investigating the effectof parameters such as τLp , τs, and m, the structure and propertiesof the gating membrane used in the previous research [24] wereoptimized by chance for generating the oscillations. The structuralfeatures of the gating membrane, such as the thickness, pore size,and porosity of the substrate used, and amount and profile of thegrafting copolymer, determine the functions of the gating mem-brane that control the hydrostatic flow and the osmotic flow. Onlywhen these functions are properly combined are oscillationsobserved.

In the earlier publication [24], the existence of a trace amountof the additive 18-crown-6 was necessary to induce oscillations. Itshould be noted that the model neglected any effect of this.Besides this, the model suggested some important points forconsideration, such as the mechanism of the autocatalytic processand the relationship between the gating function and the genera-tion of oscillations. The phenomenological approach was useful forthe situation, in which the physical and chemical behavior of thegating membrane was only partially clarified.

5. Conclusions

A molecular recognition by an ion-gating membrane oscillatorhas been mathematically modeled using membrane transport

Fig. 8. A schematic drawing of the autocatalytic process in the osmotic period based on the numerical calculation results shown in Fig. 7. An autocatalytic increase in LP wasgenerated when the pores opened from the solvent chamber side. At the same time, the value of s was kept high during the process because of the thin layer on the solutionside when the pores were closed. Thus, the osmotic flow increased autocatalytically following Eq. (1).

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239 237

equations based on nonequilibrium thermodynamics. A mechan-ism for the oscillator has been proposed based on numericalsolutions of the model. The autocatalytic increase in the purewater permeability, LP , in the osmotic period is considered to bethe key characteristic of the oscillator. In addition, the oscillator isextremely sensitive to the gating function of the membrane. Thesieving coefficient, s1, depends on the ion concentration insidethe membrane. A slight difference in this dependency curvegreatly influences the generation of the oscillations. The resultsof the modeling calculations make a contribution to providing

guidelines on how to design gating membranes for nonlinearoscillation systems.

Appendix A. Supporting information

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.memsci.2013.08.001.

Nomenclature

Ac cross-sectional area of capillary, m2

Am area of membrane, m2

a parameter in Hill equation for Lp1, dimensionlessb parameter in Hill equation for s1, dimensionlessc, cs average concentration of solute inside membrane,

mMc1 parameter in Hill equation for Lp1, mMc2 parameter in Hill equation for s1, mMc3 parameter in Hill equation for s1, mMcf ion concentration in feed chamber, mMcin ion concentration of inflow to the membrane, mMcout ion concentration of outflow from the membrane,

mMCm solute concentration on fed side of membrane, mMCp solute concentration on permeated side of mem-

brane, mMg acceleration due to gravity, m�1 s�2

JD diffusive flux, mol m�2 s�1

Jv hydrostatic pressure-driven flux of solvent,m3 m�2 s�1

Js flux of solute, mol m�2 s�1

Js_in ion flux of inflow to membrane, m3 m�2 s�1

Js_out ion flux of outflow from membrane, m3 m�2 s�1

L0 parameter in Hill equation for Lp1,m3 m�2 s�1(kPa)�1

Lp permeability of pure water, m3 m�2 s�1(kPa)�1

Lp1 steady-state value of Lp, m3 m�2 s�1(kPa)�1

LP proportionality coefficient, m3 m�2 s�1(kPa)�1

LPD proportionality coefficient, m3 m�2 s�1(kPa)�1

LDP proportionality coefficient, mol m�2 s�1(kPa)�1

LD proportionality coefficient, mol m�2 s�1(kPa)�1

m parameter in Hill equation for s1, dimensionless

Fig. 9. Influence of the characteristic times τs and τLp on the oscillation fromnumerical calculations. (A) Capillary water pressure at various characteristic timesin the osmotic period. (B) Capillary water pressure at various characteristic times inthe hydrostatic period.

Fig. 10. Influence of the Hill's equation parameter m on the oscillation fromnumerical calculations. The relevant value of m (m¼4) can generate an oscillationover a long period.

T. Ito et al. / Journal of Membrane Science 448 (2013) 231–239238

n parameter in Hill equation for Lp1, dimensionlessP solute permeability, m3 m�2 s�1

ΔP difference in pressure, kPas0 parameter in Hill equation for s1, dimensionlessV pore volume of membrane, m3

Greek symbols

Δπ difference in osmotic pressure, kPas reflection coefficient , dimensionlesss1 steady-state value of s, dimensionlessτs characteristic time of s in osmotic/hydrostatic period,

sτLp characteristic time of Lp in osmotic/hydrostatic period,

s

Subscripts

c capillaryf feed solutionin inflow to membraneLp permeability of pure waterm membraneout outflow from membranep permeated solutions solutes reflection coefficient

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