Coupled Interactions among Solute Diffusions, Membrane Surface

By itself, such a membrane charge effect cannot lead to active transport phenomena. However, the combined effects between two opposite enzymic reactio...
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J. Phys. Chem. B 2001, 105, 9623-9630

9623

Coupled Interactions among Solute Diffusions, Membrane Surface Potentials, and Opposing Enzyme Reactions as a Mechanism for Active Transports Performed with Biomimetic Membranes† Bernard Maı1sterrena‡ Laboratoire de Biochimie applique´ e, IUTA, UniVersite´ Claude Bernard-LYON I, 43 bouleVard du 11 NoVembre 1918, 69622 Villeurbanne Cedex, France ReceiVed: February 16, 2001; In Final Form: June 9, 2001

The present theoretical analysis demonstrates that the membrane surface potential effect of a uniformly charged artificial membrane can be substituted by an electrical resistance term which balances the passive membrane resistance term. Depending on the sign of this electrical resistance term, the global membrane resistance for a charged solute may be either enhanced or lowered as compared to the passive membrane resistance term. By itself, such a membrane charge effect cannot lead to active transport phenomena. However, the combined effects between two opposite enzymic reactions acting on both parts of a uniformly charged porous membrane and in unstirred layers with the membrane surface potentials lead to the concept of biomimetic membranes able to specifically transport a molecule against its chemical gradient.

1. Introduction Although a majority of theoretical analyses dealing with active transport phenomena are presently devoted to the concept of a mobile carrier,1,2 we have recently proposed a new kinetic model for active transport processes by coupling reaction/diffusion/ membrane charge effects. Our approach has led us to prove experimentally that two membrane-bound reversed enzymes (i.e., catalyzing two opposite reactions such as a phosphatase/ kinase reactional sequence) acting in unstirred layers (USLs) and on both parts of a charged artificial porous membrane (i.e., principally an organic membrane) behave as a transporter.3-6 However, in these previous studies, the membrane’s charge effect was experimentally determined and its value was just introduced in the corresponding mathematical models as an additive electrical resistance term rψ.3,4 A better understanding of the interrelations occurring among membrane-bound enzyme reactions, solute diffusions, and membrane surface potentials may be considered as an obligatory step for the development of biomimetic membranes able to perform a specific separation and purification of small hydrophilic molecules. Therefore, the aim of the present study was to substitute the indeterminate electrical membrane resistance term rψ introduced in our previous theoretical analysis4-6 by a term including the membrane surface potentials. The substitution of rψ by a precise membrane resistance term now allows a more general mechanism to be proposed for active transports performed with artificial biomimetic membranes. 2. Theory In recent studies we have presented new membrane topographies capable of performing an active transport of a small hydrophilic molecule.3,4 These topographies, which involve a † Abbreviations: ATP, adenosine triphosphate; CR, concentration ratio; USLs, unstirred layers. ‡ Fax: + 33 (0) 4 72 20 50. E-mail: [email protected].

uniformly charged porous membrane bearing two reversed enzymic activities acting on both parts of the membrane and inside USLs, were analyzed.5,6 However, in the corresponding mathematical analysis, the membrane electrical effect was just introduced as a global electrical resistance term, that is, rψ. The goal of the following analysis is to substitute this global membrane electrical term by the surface potentials present on both sides of this biomimetic membrane. Before presenting such a general theory, we will first recall what is known concerning the movement of a charged molecule submitted both to the influence of a concentration gradient and to a constant electrical field.7 The coupled interactions between solute diffusions and membrane surface potentials issued from this first section will serve to assess precisely the coupled interactions among reversed enzymic reactions, solute diffusions, and membrane surface potentials occurring in the microenvironment of these biomimetic membranes. 2.1. Diffusion of a Negatively Charged Substrate through a Negatively Charged Porous Membrane. Let us consider the topography reported in Figure 1. A uniformly negatively charged porous membrane of thickness d separates a feed compartment, II (outside), and a receiving compartment, I (inside). We assume that, on both parts of the membrane, the fixed negative charges covering the membrane’s surfaces generate two thin diffuse double layers (i.e., Stern layers) of thicknesses DL1 and DL2, respectively, in which constant electrical fields exist. It is also assumed that these thin diffuse double layers are included inside two USLs of global thicknesses ∆1 and ∆2, respectively. Therefore, each USL is composed of a passive diffusion zone of thickness δ and a diffuse double layer of thickness DL. It is to be noted that, whatever the theoretical treatment of the relationship between the potential and the surface charges (i.e., the Gouy-Chapmann theory, the Helmotz theory, or the Stern theory), the thickness of this double layer is thin and does not exceed a few nanometers even under a low ionic strength.8 Let us now consider the transport of a negatively charged substrate S from the bulk solution of compartment II (outside)

10.1021/jp010625z CCC: $20.00 © 2001 American Chemical Society Published on Web 09/07/2001

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Maı¨sterrena

JD ) -A(Ds)(dS/dx)

(2)

On the other hand, the electrical field leads to an electrical flux (JE) which can be expressed as follows:

B2 JE ) AS(ωs)(Zs)eE

(3)

where ωs is the substrate mobility, Zs the charge number of S, and e the elementary charge. The total flux of S through DL2 (Jb) corresponds to the summation of the two driving forces given by eqs 2 and 3. Assuming a constant electrical field (that is, a linear potential variation inside DL2) and substituting Ds by (ωs)kT (Einstein relation) in eq 2 and the electrical field by -∆U2/(DL2) in eq 3, we obtain the following equation:

Jb ) -A(ωs)kT(dS/dx) - AS(ωs)(Zs)e[∆U2/(DL2)] (4) Figure 1. Scheme describing a compartmentalized system involving a negatively charged porous membrane of thickness d separating a feed compartment, II, and a receiving compartment, I. The membrane is supposed to be surrounded by two global USLs of thicknesses ∆1 and ∆2 adjacent to compartments I and II, respectively. Each global USL ∆ is composed of a diffuse layer of thickness DL (in which acts a constant electrical field) and a passive diffusion zone of thickness δ. The diffusion analysis of a negatively charged substrate from compartment II toward compartment I takes into account five different steps numbered from a to e. (For the symbols used, see the text.)

to the bulk solution of compartment I (inside). For the sake of clarity, we have adopted a general nomenclature which can be found useful to describe more complex topographies. Concentration (S or P) localizations have been identified by a superscript and two subscripts: the superscript indicates the localization of the substrate, i.e., in the bulk (b), at the δ/DL interface (δ), and at the membrane level (m). The first subscript specifies the compartment (I or II) while the second subscript is relative to time (0, t). Therefore, the substrate localizations reported in Figure 1 are as follows: SII,0b as the initial substrate concentration in the bulk of comparment II; SII,tδ at the δ2/DL2 interface; SII,tm at the membrane level facing compartment II; SI,tm at the membrane level adjacent to compartment I; SI,tδ at the δ1/DL1 interface; SI,tb in the bulk solution of compartment I. To cross this global barrier in the 0x direction, the substrate first diffuses through the passive diffusion zone δ2 (step a). Then it crosses DL2 to reach the membrane surface (step b) and diffuses through the membrane structure (step c). On the compartment I side, the substrate crosses DL1 (step d) and last diffuses through the passive diffusion zone δ1 to reach the bulk solution of compartment I (step e). These five consecutive steps can be expressed by the following mass transfer equations. In step a, the substrate diffusion inside δ2 can be expressed by Fick’s first law, the integration of which leads to the following:9,10

Ja ) [1/(RδsII)](SII,0b - SII,tδ)

(1)

where RδsII is the passive diffusion resistance of δ2 for S, that is, δ2/[A(Ds)], A and Ds being the total membrane area and the diffusion coefficient of S, respectively. (For the symbols used, see Table 1). In step b, the negatively charged substrate moves both on the influence of a concentration gradient and on the influence of an electrical field E B2 which is oriented in the 0x direction. Considering only a unidirectional transport in the 0x direction, the concentration gradient generates a diffusion flux (JD) which may be described by Fick’s first law:

where k is the Boltzmann constant, T the absolute temperature, and ∆U2 the potential difference between the potential existing at the δ2/DL2 interface and the negative potential present at the membrane surface level. Equation 4 may be easily integrated in the interval dx corresponding to DL2, and we find the following:

Jb ) [[SII,tδ exp(Ψs2) - SII,tm]/ [exp(Ψs2) - 1]]A(ωs)(Zs)e[∆U2/(DL2)] (5) where Ψs2 is the Boltzmann factor for S inside DL2, that is, (Zs)e(∆U2)/(kT). If we substitute ωs by (Ds)/kT in eq 5, we can write this equation in the following simplified form:

Jb ) [(Ψs2)/ [(RDLs2)(exp(Ψs2) - 1)]][SII,tδ exp(Ψs2) - SII,tm] (6) where RDLs2 is the diffuse double layer resistance for S, that is, (DL2)/[A(Ds)]. Equation 6, already presented,7,11 implies that ∆U2 varies linearly with the distance in the DL2 interval. This in turn means that the variation of S inside DL2 cannot be a linear function of the distance x. It is to be noted that when equilibrium is reached, Jb ) 0 and therefore SII,tδ exp(Ψs2) ) SII,tm, that is, the wellknown Nernst-Planck relation. In our example, ∆U2 is positive whereas Zs is negative. Therefore, Ψs2 is negative and the membrane surface potential repels the substrate inside DL2 and in the 0x direction. In step c and in the absence of a membrane potential (the membrane is supposed to be uniformly charged, and thus the difference between the potentials on the two surfaces and across the membrane is zero), S passively diffuses through the membrane pore structures to reach the membrane surface adjacent to compartment I. The corresponding flux Jc can be expressed by Fick’s first law, that is:

Jc ) [1/(rms)](SII,tm - SI,tm)

(7)

where rms is the passive membrane resistance for S, that is, d/[Aw(Ds)], d and Aw being the membrane thickness and the effective membrane pore area, respectively. In step d, the substrate moves inside DL1 under both the influence of a concentration gradient and the influence of an electrical field B E1 directed in the opposite direction of 0x. The resulting flux of S inside DL1 may therefore be expressed by an equation similar to eq 6, that is:

Active Transports Performed with Biomimetic Membranes

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9625

TABLE 1: Symbols and Nomenclature symbol

definition

units

A Aw d DL1 DL2 Ds Dp e Km1, Km2 k k1, k2 P RDLs1 RDLs2 RDLp1 RDLp2 RδsI RδsII RδpI RδpII Rms Rmp rms rmp S t T ∆U1 ∆U2 V Vm1, Vm2 Zs, Zp

total membrane area effective membrane pore area membrane thickness diffuse layer thickness adjacent to compartment I diffuse layer thickness adjacent to compartment II diffusivity of the substrate S diffusivity of the product P elementary charge (1.6 × 10-19 C) Michaelis constant for the membrane-bound enzymes I and II Boltzmann constant (1.38 × 10-23 J‚K‚mol-1) first-order rate constants relative to enzymes I and II product concentration diffuse layer resistance for S in compartment I diffuse layer resistance for S in compartment II diffuse layer resistance for P in compartment I diffuse layer resistance for P in compartment II unstirred layer resistance for S in compartment I unstirred layer resistance for S in compartment II unstirred layer resistance for P in compartment I unstirred layer resistance for P in compartment II global membrane resistance for S global membrane resistance for P passive membrane resistance for S passive membrane resistance for P substrate concentration time absolute temperature (273.15 + °C) potential difference inside DL1 potential difference inside DL2 compartment volume maximal velocity for the membrane-bound enzymes I and II charge number of S and P

cm2 cm2 cm cm cm cm2‚min-1 cm2‚min-1 C mol‚cm-3 J‚K‚mol-1 cm3‚min-1 mol‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 min‚cm-3 mol‚cm-3 min K V V cm3 mol‚min-1 dimensionless

∆1, ∆2 δ1, δ2 Ψs1 Ψs2 Ψp1 Ψp2

Greek Symbols global boundary layer thickness (including the diffuse layers) in compartments I and II diffusion layer thickness (excluding the diffuse layers) in compartments I and II Boltzmann factor for S in the diffuse layer DL1 Boltzmann factor for S in the diffuse layer DL2 Boltzmann factor for P in the diffuse layer DL1 Boltzmann factor for P in the diffuse layer DL2

cm cm dimensionless dimensionless dimensionless dimensionless

Jd ) [(Ψs1)/ [(RDLs1)(exp(Ψs1) - 1)]][SI,tm exp(Ψs1) - SI,tδ] (8) where Ψs1 is the Boltzmann factor for S inside DL1, that is, (Zs)e(∆U1)/(kT). It is to be noted that on this membrane side ∆U1 is negative, leading to a positive Boltzmann factor value, which implies the fact that the negative membrane charges repel the negatively charged substrate in the 0x direction. In step e, the substrate passively diffuses through δ1 to reach the bulk solution of compartment I. The resulting flux can be expressed by Fick’s first law:

Je ) [1/(RδsI)](SI,t - SI,t ) δ

b

(9)

where RδsI is the resistance of δ1 for S, that is, δ1/[A(Ds)]. Because the five steps a, b, c, d, and e are consecutive, at steady state, the corresponding fluxes are identical, that is, Ja ) Jb ) Jc ) Jd ) Je ) J. Therefore, the global flux of S from compartment II toward compartment I can be expressed by the following simplified relationship:

J ) (1/R)(SII,0b - SI,tb)

(10)

where R is a global resistance for S including the membrane barrier and the two global USL resistances. To simplify the R determination, we assume that the membrane is symmetrical in terms of the charge density so that ∆U2 ) -∆U1. Therefore, Ψs2 ) -Ψs1, DL2 ) DL1 ) DL,

and consequently RDLs2 ) RDLs1 ) RDLs. We also assume that the hydrodynamic conditions are identical on both parts of the membrane so that δ2 ) δ1 ) δ, leading to RδsII ) RδsI ) Rδs. Under these conditions, eqs 1, 6, 7, 8, and 9 can be expressed in the following simplified forms, respectively:

J(Rδs) ) (SII,0b - SII,tδ)

(11)

J[[(RDLs)(exp(Ψs2) - 1)]/ (Ψs2)] ) [SII,tδ exp(Ψs2) - SII,tm] (12) J(rms) ) (SII,tm - SI,tm)

(13)

J[[(RDLs)(exp(-Ψs2) - 1)]/ (-Ψs2)] ) [SI,tm exp(-Ψs2) - SI,tδ] (14) J(Rδs) ) (SII,tb - SI,tb)

(15)

The summation of these five equations allows SII,tδ, SII,tm, SI,tm, and SI,tδ to be eliminated and thus an expression of the global flux of S to be obtained, that is

J ) [[(Ψs2) exp(Ψs2)]/[2(Rδs)(Ψs2) exp(Ψs2) + (rms)(Ψs2) + 2(RDLs)(exp(Ψs2) - 1)]](SII,0b - SI,tb) (16) If the term including the diffuse double layer resistances RDLs for S is small compared to the terms including rms and

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Rδs, then eq 16 may be simplified as follows:

J ) [exp(Ψs2)/[2(Rδs) exp(Ψs2) + rms]](SII,0b - SI,tb) (17) Such a simplification may be justified by the fact that the thickness of a diffuse layer is very thin compared to the thickness of an artificial membrane. On the other hand, eq 17 takes into account the USL resistances, which often have significant values.12,13 If we now define the intrinsic membrane resistance as the combination only of the two diffuse layers’ resistances and the passive membrane resistance, the use of eqs 12, 13, and 14 allows, with the same reasoning, the intrinsic membrane flux of S to be determined, that is:

J ) [[(Ψs2) exp(Ψs2)]/[(rms)(Ψs2) + 2(RDLs)(exp(Ψs2) - 1)]](SII,tδ - SI,tδ) (18) Equation 18 describes the flux between SII,tδ at the δ2/DL2 interface and SI,tδ at the δ1/DL1 interface. By neglecting the diffuse layer resistance term as compared to the membrane resistance term, we obtain the following:

J ) [exp(Ψs2)/(rms)](SII,tδ - SI,tδ)

(19)

Therefore, under such restricted (but justified) conditions, the global intrinsic membrane resistance for S designed by Rms (see Table 1 for the symbols used) can be substituted by the following fundamental relation:

Rms ) (rms) exp(-Ψs2)

(20)

To conclude this section, it can be strongly underlined that, in the example chosen and reported in Figure 1, Ψs2 takes a negative value and therefore the global intrinsic membrane resistance Rms becomes superior to the passive membrane resistance rms: the negative charges covering the membrane surface repel the negatively charged substrate. On the other hand, if we consider now either the transport of a positively charged substrate through a negatively charged membrane or the transport of a negatively charged substrate through a positively charged membrane, the global intrinsic membrane resistance term Rms will be lower than rms: in these latter two cases the membrane’s charges favor the substrate transport. However, it must be underlined that, whatever the situation considered and in all cases, no active transport may occur (i.e., a transport against its chemical gradient). 2.2. Coupling of Diffusion/Enzymic Reactions/Membrane Electrical Effects. The topography illustrated in Figure 2a is proposed as a general scenario allowing the active transport of a small hydrophilic substrate to be performed. This general topography mainly consists of a uniformly charged porous membrane (positive or negative) which separates a feed compartment, II (outside), and a receiving compartment, I (inside). It is assumed that hydrodynamic conditions allow the creation of USLs at the membrane/solution interfaces of global thicknesses ∆1 and ∆2 in compartments I and II, respectively. We also assume that each global USL is composed of a passive diffusion zone of thickness δ and a thin diffuse layer of thickness DL, close to the membrane, in which a constant potential exists. We suppose that enzyme E2 is bound to the membrane facing compartmentI II while enzyme E1 is bound to the membrane facing compartment I. E2 and E1 are two reversed enzymes, that is two enzymes catalyzing two opposite reactions. Such reversed enzymes are

Figure 2. Scheme describing the general topography of a biomimetic membrane. In (a), a charged porous membrane separates a feed compartment, II, and a receiving compartment, I. This membrane bears two reversed enzymes, i.e., E2 and E1, facing comparments II and I, respectively. The bienzymic membrane is supposed to be surrounded by two global USLs of thicknesses ∆1 and ∆2 adjacent to compartments I and II, respectively. Each global USL ∆ is composed of a diffuse layer of thickness DL (in which acts a constant electrical field) and a passive diffusion zone of thickness δ. The substrate is symbolized by a closed circle, while the intermediary product is symbolized by an open circle. For the sake of clarity the different steps occurring in the microenvironment of the biomimetic membrane are numbered and reported in (b). The general nomenclature used for a precise localization of both the substrate and the intermediary product is reported in (c). (For the symbols used, see the text.)

plentiful in living cells, and a phosphatase associated with a kinase may be considered as the best example of reversed enzymes. Therefore, a phosphatase/kinase enzymic reactional sequence (or the contrary) has been selected for the design of artificial biomimetic membranes. Consequently, in our proposed scenario, if E2 is identified as a phosphatase, then E1 is necessarily a kinase and conversely. To be as close as possible to our previous studies,3-6 we suppose that both enzymes act outside the thin diffuse layers. (This peculiar hypothesis will be discussed in the Conclusion.) In this chosen topography, we consider the transport of a substrate (symbolyzed in Figure 2a by a closed circle) from compartment II toward compartment I. This substrate may be either negatively charged or uncharged. If it is negatively charged, we assume that it is a phosphorylated substrate. For the sake of clarity, the different steps occurring during this transport process are reported in Figure 2b and numbered as follows: In step 1, the substrate initially present in the bulk solution of compartment II may cross δ2. In step 2, this substrate may cross DL2, the membrane, and DL1. In step 3, the substrate may diffuse through δ1 to reach the bulk of compartment I. In step 4, the substrate may also react with E2 at the δ2/DL2 interface, thus generating a product (symbolyzed in Figure 2a by an open circle). In step 5, this generated product may cross DL2, the membrane, and DL1. In step 6, the product may react

Active Transports Performed with Biomimetic Membranes with E1 at the δ1/DL1 interface, thus generating the substrate on the compartment I side. Step 7 illustrates the diffusion of the product (generated by E2) through δ2 in the direction of compartment II (that is, in the direction opposite 0x). Step 8 corresponds to the product diffusion through δ1. These eight steps may be described by eight equations. To be as precise as possible, Figure 2c illustrates the nomenclature, defined in the previous section, which has been used to identify precisely the substrate and the product localizations. In step 1, the passive diffusion of SII,0b inside δ2 can be expressed by Fick’s first law:

J1 )

[1/(RδsII)](SII,0b

- SII,t ) δ

(22)

Step 3 corresponds to the passive diffusion of SI,tδ through δ1 and can be expressed as follows:

J3 ) [1/(RδsI)](SI,tδ - SI,tb)

(23)

Step 4 corresponds to the enzymic reaction of E2 at the δ2/ DL2 interface. Assuming first-order kinetic conditions (i.e., a low substrate concentration compared to the Km value of E2), the enzymic rate V2 can be expressed as follows:

V2 ) k2SII,tδ

(24)

where k2 is the first-order rate constant of E2, (i.e., (Vm2)/(Km2)). It must be stressed that if E2 is the kinase, the assumption that the enzyme functions under first-order conditions implies that ATP is not rate-limiting. In step 5, the product generated by E2 may diffuse through DL2, the membrane, and DL1. This product being possibly charged and still neglecting the diffuse layers’ resistances as compared to the membrane resistance, this step may be illustrated by the following simplified equation:

J5 ) [exp(Ψp2)/(rmp)](PII,tδ - PI,tδ)

(25)

where rmp is the membrane resistance for P, that is, d/[Aw(Dp)], Dp being the diffusion coefficient of P. Ψp2 is the Boltzmann factor for P inside DL2, that is, (Zp)e(∆U2)/(kT), Zp being the charge number of P (for the symbols used, see Table 1). Step 6 corresponds to the enzymic reaction of E1 occurring at the δ1/DL1 interface on the compartment I side. Assuming always first-order kinetic conditions, this enzymic rate V1 can be expressed as follows:

V1 ) k1PI,tδ

(26)

where k1 is the first-order rate constant of E1 (i.e., (Vm1)/(Km1)). Step 7 describes the product diffusion of PII,tδ (generated by E2) through δ2 and in the direction of the bulk solution of compartment II. This step may be expressed by Fick’s first law, that is:

J7 ) [1/(RδpII)](PII,tδ - PII,tb)

where RδpII is the δ2 layer resistance for P, that is, δ2/[A(Dp)]. Step 8 corresponds to the passive diffusion of the product through δ1, that is:

J8 ) [1/(RδpI)](PI,tδ - PI,tb)

(27)

(28)

where RδpI is the δ1 layer resistance for P, that is, δ1/[A(Dp)]. Using the equations corresponding to these eight steps, at steady state, we can define the following relationships among these steps:

k2SII,tδ + [exp(Ψs2)/(rms)](SII,tδ - SI,tδ) ) [1/(RδsII)](SII,0b - SII,tδ) (29)

(21)

Step 2 represents the substrate diffusion through DL2, the membrane, and DL1. Assuming a possible charged substrate (the membrane being considered always charged) and neglecting the DL resistances, this substrate mass transfer can be described by an equation similar to eq 19, that is

J2 ) [exp(Ψs2)/(rms)](SII,tδ - SI,tδ)

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9627

k2SII,tδ ) [1/(RδpII)](PII,tδ - PII,tb) + [exp(Ψp2)/(rmp)](PII,tδ - PI,tδ) (30) k1PI,tδ + [exp(Ψs2)/(rms)](SII,tδ - SI,tδ) ) [1/(RδsI)](SI,tδ - SI,tb) (31) [exp(Ψp2)/(rmp)](PII,tδ - PI,tδ) ) k1PI,tδ + [1/(RδpI)](PI,tδ - PI,tb) (32) On the other hand, substrate and product appearance inside compartment I may also be expressed by the following two equations, respectively:

dSI,tb/dt ) [1/VI(RδsI)](SI,tδ - SI,tb)

(33)

dPI,tb/dt ) [1/VI(RδpI)](PI,tδ - PI,tb)

(34)

where VI is the volume of compartment I. Assuming constant membrane-bound enzyme activities and a large compartment II (outside) in which the initial substrate concentration remains constant (i.e., SII,0b ) SII,tb), the system of eqs 29-34 can be solved as previously reported4 so as to obtain the expressions of SI,tb and PI,tb as a function of time. The general solutions of the system are the following:

SI,tb ) G exp(r1t) + H exp(r2t) + R18/R17

(35)

PI,tb ) R19G exp(r1t) + R20H exp(r2t) + R21

(36)

The constant Rn values and the roots of the system (r1 and r2) appearing in eqs 35 and 36 are reported in Table 2. To simplify the writing of these Rn constants, it must be underlined that the global membrane resistances for S and P have been introduced as Rms and Rmp, respectively. In fact Rms and Rmp correspond to (rms) exp(-Ψs2) and to (rmp) exp(-Ψp2), respectively. Therefore, in the presence of an uncharged substrate the global membrane resistance Rms becomes equal to the passive membrane resistance rms. On the other hand, if the intermediary product is uncharged, then the global membrane resistance Rmp becomes equal to the passive membrane resistance rmp. This general model may be found useful to predict either the active transport of a phosphorylated substrate or the active transport of a neutral substrate performed with uniformly charged biomimetic membranes. 3. Simulated Transport Rates 3.1. Active Transport of a Phosphorylated Substrate. In this first case, we consider a uniformly negatively charged

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TABLE 2: Constants Used in the Mathematical Model Rms R1 ) S b RδsII II,0

(

R2 ) (Rms) k2 + R3 )

R2R5R9 k2R3 R10 - R4 R2R5 R 2R 4 R14 ) R3 k2 k2(Rmp) R13 )

(

)

1 +1 RδsII RδpI

k1(Rmp)(RδpI) + Rmp + RδpI

R15 )

VI(RδsI) R14 -(R13 - 1) RδsI (R10 - 1) R14 RδpI R14

R4 )

Rmp k1(Rmp)(RδpI) + Rmp + RδpI

R16 )

R5 )

R3 1 1 + RδpII Rmp Rmp

R17 )

R6 ) R2 R7 ) R 1

( (

R9 ) -

) )

R6 1 1 P b + RδsI Rms k2(RδpII) II,0

R7 R8

r1 )

R6R5 k2R3

(R13 - 1)(R10 - 1)

-R16 + (R162 - 4R15R17)1/2 2R15

-R16 - (R162 - 4R15R17)1/2 2R15 VI(RδsI)r1 - R13 + 1 R19 ) R14 r2 )

1 (RδsI)R8

R10 ) R11 )

R9 VI(RδpI)R14 VI(RδpI) R12(R10 - 1) R11 R18 ) VI(RδpI) VI(RδpI)R14

1 1 1 + RδsI Rms Rms

R8 ) k 1 -

(

R4R6 1 R4R5R6 + R8 k2R3 k2(Rmp)

)

R20 )

VI(RδsI)r2 - R13 + 1 R14

R21 ) -

R2R5R11 + R12 ) R1 k2R3 k2(RδpII) R2PII,tb

membrane bearing a phosphatase on the compartment II side (E2) and a kinase on the compartment I side (E1). It is to be noted that an alkaline phosphatase dephosphorylates a phosphorylated substrate according to the reaction:

(P-OPO3)

)( )

2-

(H2O)

98 P-H + (HPO4)

2-

(37)

while a kinase specifically phosphorylates a substrate by transferring a phosphate group from ATP as follows:

P-H + (ATP/Mg)2- f (P-OPO3)2- + (ADP/Mg)- + H+ (38) In the case considered, (P-OPO3)2- corresponds to the substrate and P-H to the product. Therefore, the substrate is negatively charged, while the intermediary product generated by the phosphatase is uncharged. In this case, the membrane surface potential only acts on the substrate movement. The above general model may be used by substituting the global membrane resistance Rms (appearing in the Rn constants reported in Table 2) by (rms) exp(-Ψs2). On the other hand, the intermediary product being uncharged, the global membrane resistance Rmp becomes equal to rmp. In the case considered, Ψs2 is negative (Zs is negative and ∆U2 is positive), leading to a exp(-Ψs2) value >1. Therefore, the global membrane resistance Rms appears superior to the passive membrane resistance rms. It must be underlined that the mathematical model relative to this case previously presented (and corroborated by experiments performed with artificial membranes)4,14 was inaccurate in terms of the membrane’s charge effect.4 As a matter of fact, in this previous model dealing with the active transport of a phosphorylated substrate,4,5 the global membrane resistance Rms for S was introduced as Rms ) rms + rψs, where rψs was an indeterminate electrical resistance term. The identification of Rms as (rms) exp(-Ψs2) (see section 2.1) leads to the following

[

]

(R13 - 1)R18 + R12R17 R17R14

relation:

rψs ) (rms)[exp(-Ψs2) - 1]

(39)

Therefore, our previous theoretical analysis may still stay valid5 and may be generalized (for the case considered) only by substituting rψs by (rms)[exp(-Ψs2) - 1]. To illustrate this case, simulated transport rates are provided and reported in Figure 3. These simulations have been conducted in a very plausible configuration, that is, in the diffusion cell configuration used in our last experimental studies.14 In this configuration, the parameters used were as follows: k1 ) 55 cm3 × min-1; k2 ) 10 cm3‚min-1; d ) 1.2 × 10-2 cm; A ) 39.2 cm2; Aw ) 6.66 cm2; δ1 ) 1 × 10-2 cm; δ2 ) 1.8 × 10-2 cm; Ds ) 4.09 cm2‚min-1; Dp ) 5.05 cm2‚min-1;VI ) 4 cm3. Curves a, b, and c reported in Figure 3 were calculated assuming a linear potential gradient inside DL2 of 10, 20, and 50 mV, respectively, while the reference curve was calculated without any potential effect. Figure 3 clearly shows that the concentration ratio CR, that is, SI,tb/SII,0b, depends on the membrane potential value. It is to be noted that without a membrane surface potential (reference curve) or in the absence of USLs or without enzyme activities no transport occurs, i.e., CR ) 1. To explain such an active transport phenomenon, it must be stressed that when considering the transport of a phosphorylated substrate which bears two negative charges at an alkaline pH, a credible potential difference inside DL2 of 50 mV leads (at 25 °C) to a global membrane resistance Rms equal to 48.9(rms). On the other hand, the uncharged intermediary product generated close to the membrane surface encountered a membrane resistance equal to rmp, that is, a resistance nearly 50 times lower than Rms. Therefore, the entire transport phenomenon is based on this important difference between the membrane resistance for the substrate and for the intermediary product. This difference, which is a function of the membrane

Active Transports Performed with Biomimetic Membranes

Figure 3. Simulation of glycerol-3-phosphate active transport. The glycerol-3-phosphate concentration ratio (CR) versus time was calculated for different membrane surface potentials by using eq 35. The parameters determined in the diffusion cell used in a previous experimental study14 were as follows: k1 ) 55 cm3‚min-1; k2 ) 10 cm3‚min-1; d ) 1.2 × 10-2 cm; A ) 39.2 cm2; Aw ) 6.66 cm2; δ1 ) 1 × 10-2 cm; δ2 ) 1.8 × 10-2 cm; Ds ) 4.09 cm2‚min-1; Dp ) 5.05 cm2‚min-1;VI ) 4 cm3. Assuming a uniformly negatively charged artificial membrane, curves a, b, and c were calculated with a linear potential gradient inside DL2 of 10 mV ([), 20 mV (9), and 50 mV (]), respectively. The reference curve (0) was calculated in the absence of a membrane surface potential.

surface potential value, may also be expressed by the following kinetic conditions: steps 4, 5, and 6 (reported in Figure 2b) become faster than step 2, leading to an accumulation of the phosphorylated substrate in the bulk of the receiving compartment I. Therefore, the active transport of a phosphorylated substrate can be performed by using a negatively charged biomimetic membrane and involving a phosphatase/kinase reactive sequence acting on both parts of the membrane and necessarily inside USLs adjacent to the membrane surfaces. To conclude this section, the events occurring in the case considered can be summarized as follows: the dephosphorylation/phosphorylation reactive sequence occurring at the membrane pore entrance/exit structure and necessarily in unstirred layers, by removing/adding a phosphate group on a small hydrophilic molecule, leads to an electrical coupling effect between the charged substrate and the membrane’s charges, while rendering the uncharged intermediary product free of any potential influence. In this case, our general model can be used to predict the active transport of the phosphorylated substrate only by substituting Rms by (rms) exp(-Ψs2) and Rmp by rmp. The role of the other parameters on this active transport phenomenon may be found in our previous work.5 3.2. Active Transport of a Neutral Substrate. We consider now the active transport of a neutral substrate through a uniformly positively charged membrane bearing a kinase on the compartment II side and a phosphatase on the compartment I

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9629

Figure 4. Simulation of glycerol active transport. The glycerol concentration ratio (CR) versus time was calculated for different membrane surface potentials by using eq 35. Assuming a uniformly positively charged artificial membrane, curves a, b, and c were calculated with a linear potential gradient inside DL2 of -10 mV ([), -20 mV (9), and -50 mV (]), respectively. The reference curve (0) was calculated in the absence of a membrane surface potential. Computation was conducted with the numerical values described in Figure 3.

side. In the proposed scenario, the uncharged substrate is phosphorylated by the kinase on the compartment II side, thus generating an intermediary phosphorylated product. In this case, Ψs2 ) 0 (Zs ) 0) and the global membrane resistance Rms corresponds to the passive diffusion membrane resistance rms (i.e., Rms ) rms). On the other hand, Ψp2 is positive (Zp < 0; ∆U2 < 0), leading to a global membrane resistance for the product, that is, Rmp ) (rmp) exp(-Ψp2). It is to be noted that, in this case, Rmp appears lower than rmp and the membrane’s charges favor the intermediary product mass transfer through the membrane. It must be underlined that the theoretical analysis relative to this case and previously presented6 was inaccurate in terms of the membrane’s charge effect. As a matter of fact, in this previous model dealing with the active transport of a neutral substrate,6 the global membrane resistance for P, i.e., Rmp, was introduced as Rmp ) rmp - rψp, where rψp was an indeterminate electrical resistance term lowering the passive membrane resistance for P. The identification of Rmp as (rmp) exp(-Ψp2) leads to the following relation:

rψp ) (rmp)[1 - exp(-Ψp2)]

(40)

Therefore, the previous analysis relative to this case may still stay valid6 and the corresponding model may be generalized only by substituting rψp by (rmp)[1 - exp(-Ψp2)]. To illustrate this case, simulated transport rates are provided and reported in Figure 4. These simulations have been conducted by using the numerical values of the parameters reported in the previous section. Curves a, b, and c reported in Figure 4 were

9630 J. Phys. Chem. B, Vol. 105, No. 39, 2001 calculated with a linear potential gradient inside DL2 of -10, -20, and -50 mV, respectively, while the reference curve was calculated without any potential effect. Figure 4 clearly shows that the concentration ratio CR, that is, SI,tb/SII,0b, depends on the membrane surface potential value. Once again, it is to be noted that without a membrane surface potential (reference curve) or in the absence of USLs or without enzyme activities no transport occurs, i.e., CR ) 1. To explain the case considered here, the active transport phenomenon results in the difference of the membrane resistance for the uncharged substrate and for the intermediary phosphorylated product. As an example, a potential difference inside DL2 of -50 mV leads to a global membrane resistance for the negatively charged intermediary product Rmp equal to 0.020 (rmp). In this case and in our topography (see Figure 2), the facilitated diffusion of P through the membrane structure leads to the active transport of S. This can also be expressed by the following kinetic conditions: steps 4, 5, and 6 (see Figure 2b) becomes faster than step 2, explaining why the uncharged substrate can accumulate in the receiving compartment I. In conclusion, regarding the active transport of an uncharged substrate, the biomimetic membrane must be positively charged and must involve a kinase/phosphatase reactive sequence acting in USLs. The events occurring in this case can be summarized as follows: the phosphorylation/dephosphorylation reactive sequence occurring at the membrane pore entrance/exit structure and necessarily in unstirred layers, by adding/removing a phosphate group on a small hydrophilic molecule, leads to an electrical coupling effect between the charged intermediary product and the membrane’s charges, while rendering the uncharged substrate free of any electrical influence. In this case, our general model can be used to predict the active transport of the uncharged substrate only by substituting Rms by rms and Rmp by (rmp) exp(-Ψp2). It is to be noted that the role of the other parameters on this active transport phenomenon may be found in our previous work.6 4. Conclusions Recent experimental scale-up studies have clearly demonstrated that the specific active transport of small hydrophilic molecules by using artificial biomimetic membranes could be markedly improved.14 The technological development of biomimetic membranes also depends on a better understanding of the main forces acting in the microenvironment of these membranes and their interrelations to each other. The present analysis demonstrates that the global membrane resistance of a porous and uniformly charged membrane for a charged solute, i.e., Rm, can be substituted by a very simple relationship, i.e., (rm) exp(-Ψs), where rm is the passive membrane resistance and Ψs is the Boltzmann factor for the solute. Depending on the sign of Ψs, which is related to the solute charge number and the sign of the membrane surface potential, Rm can be either enhanced (if Ψs is negative) or lowered (if Ψs is positive) as compared to the passive membrane resistance term rm. Obviously, if the solute is uncharged, then Rm becomes equal to rm. However, by itself, such a membrane surface potential effect cannot permit a substrate active transport phenomenon. On the other hand, as previously reported,4-6 the interrelations of such a membrane charge effect with membranebound reversed enzymic reactions and the diffusional movements of S and P occurring in USLs can lead to active transport phenomena. This model, which can be related to the Goddard model,15 is an example of secondary active transport in which concentration gradients provide the driving forces. Since this is gradient

Maı¨sterrena pumping, then the accumulations achieved are a consequence of the different resistances of steps 2 and 5, attenuated by the surface potential that modulates the “amplifier” (i.e., the voltage source defined by Goddard in the equivalent electrical circuit). In conclusion, we have succeeded to substitute the indeterminate electrical membrane resistance terms introduced in our previous models4-6 by a precise membrane resistance term which includes the membrane surface potential of an artificial isotropic membrane. Therefore, the different analyses concerning the active transport of a phosphorylated substrate and an uncharged molecule, which were differentiated in first approaches,5,6 may now be analyzed by using only the present general model. Such a contribution to the understanding of biomimetic membrane behavior must not eclipse that this model was elaborated by supposing the reversed enzyme activities acting outside the diffuse layers adjacent to both of the membrane’s faces. As a matter of fact, it should be noted that the kinase requires ATP which is charged and which will respond to the membrane surface potential. The presented model, based on the assumption that the enzyme functions under first-order conditions, implies that ATP is not rate limiting (i.e., at a saturant concentration). The other hypothesis, which supposes enzymic reactions inside the diffuse layers, may lead (in our topography) to more development. In this latter situation, the increase of the membrane potential may strongly affect the local ATP concentration close to the kinase. This will impact the “inside amplifier” differentially to the “outside amplifier”15 and will eventually limit the production of the concentration gradient. Furthermore, we have just presented a simplified analysis dealing with the technological development of uniformly charged biomimetic membranes. Other models, based on the combined effects of solute diffusions, reversed enzymic reactions, and membrane surface potentials, must be analyzed by considering asymmetrical charged membranes and thus by including membrane potentials acting through the pore structures. Even if we are still far from the performances of biomembranes, this new concept of artificial biomimetic membranes, which can specifically concentrate a small hydrophilic molecule against its chemical gradient, appears promising in the field of separation and purification technology. Acknowledgment. I gratefully thank Professor Michel Odin, head of IUTA, for financial support and Be´atrice Riveaux for the correction of this manuscript. References and Notes (1) Cramer, W. A.; Knaff, D. B. Energy Transduction in biological membranes; Springer-Verlag: New York, 1990. (2) Lehninger, A. L.; Nelson, D. L.; Cox, M. M. Principes de biochimie; Flammarion: Paris, 1994. (3) Maı¨sterrena, B. French Patent 96 13554, 1996. (4) Maı¨sterrena, B.; Nigon, C.; Michalon, P.; Couturier, R. J. Membr. Sci. 1997, 134, 85. (5) Nigon, C.; Phalippon, J.; Favre-Bonvin, C.; Maı¨sterrena, B. J. Membr. Sci. 1998, 144, 223. (6) Nigon, C.; Michalon, P.; Perrin, B.; Maı¨sterrena, B. J. Membr. Sci. 1998, 144, 237. (7) Ricard, J.; Noat, G. J. Theor. Biol. 1984, 109, 555. (8) Thellier, M.; Ripoll, C. Bases themodynamiques de la biologie cellulaire; Masson: Paris, 1992. (9) Goldman, R.; Katchalski, E. J. Theor. Biol. 1971, 32, 243. (10) Engasser, J. M.; Horvath, Cs. Applied biochemisry and bioengineering; Academic Press: New York, 1976. (11) Walz, D. Biochim. Biophys. Acta 1990, 1019, 171. (12) Maı¨sterrena, B.; Blum, L. J.; Coulet, P. R. Biochem. J. 1987, 242, 835. (13) Maı¨sterrena, B.; Coulet, P. R. Biochem. J. 1989, 260, 455. (14) Perrin, B.; Couturier, R.; Maı¨sterrena, B. Sep. Purif. Technol. 1999, 17, 195. (15) Goddard, J. D. J. Phys. Chem. 1985, 89, 125.