DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS …€¦ · DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS FOR MEMBRANE SUBSTANCE TRANSPORT M. Jarzyns ka Technical High

DERIVATION OF PRACTICAL KEDEM -KATCHALSKY EQUATIONS FOR

MEMBRANE SUBSTANCE TRANSPORT

M. Jarzynska

Technical High School of Environment Developing,

Piotrkow Trybunalski, Broniewskiego 16, Poland,

e-mail: [email protected]

M. Pietruszka

Faculty of Biology and Environmental Protection,

University of Silesia Katowice, ul. Jagiellonska 28,

Poland,


(Received 7 November 2007; accepted 5 December 2007)

Abstract

The following paper includes a derivation of practical Kedem-Katchalsky (K-K) equations for the volume flow Jν and thesolute flow Js for non-electrolytes. This derivation makes theequations clearer and consequently their interpretation alsobecomes easier. The equations have been derived on the basisof the analysis of the membrane transport generated by simul-taneous action of two thermodynamic stimuli: the hydrostatic

Concepts of Physics, Vol. V, No. 3 (2008)DOI: 10.2478/v10005-007-0041-8

459

pressure difference ∆p and the osmotic pressure difference ∆Π.Furthermore, the derivation of the solute permeability coeffi-cient ω, which results from K-K equations, is also presented inthis paper. The formulas for coefficients characterizing mem-brane permeability, ωd and ωk, as well as their derivation andphysical interpretation are also presented below. Finally, aformula for the membrane coefficient LD, which represents adiffusional mobility, is derived.

460 Concepts of Physics, Vol. V, No. 3 (2008)

Kedem-Katchalsky equations for membrane substance transport

1 IntroductionMembrane transport for binary non-electrolyte solutions, gener-

ated by the hydrostatic pressure difference ∆p and the osmotic pres-sure difference ∆Π, can be described by practical KedemKatchalskyequations. The K-K equations have been derived from the principlesof linear thermodynamics of irreversible processes. Such membranetransport is described by the equations for the volume flow Jν andthe solute flow Js [1-4]:

Jν = Lp∆P − Lpσ∆Π, (1)

Js = ω∆Π + c(1− σ)Jν , (2)

where c stands for mean concentration, c ≈ 12 (c1 + c2) and (Lp, σ, ω)

are coefficients of filtration, reflection and permeation, respectively.The above equations have widely been used in research on sub-

stance permeability through artificial and biological membranes [6,11]. This paper includes a complete analysis and interpretation of theK-K equations. Some preliminary results in this field are presentedin Ref. [15].

In the following paper the transport K-K equations have been de-rived on the basis of phenomenological equations. Consequently, theK-K equations have become more comprehensible and we can inter-pret them thoroughly. It is important from biological point of viewsince the effects of substance transport through biological and artifi-cial membranes can be understood easily, whereas the application ofthe K-K equations in their classical version is limited to membranesystems with two-component solutions, sufficiently diluted and wellstirred [7, 9,16]. The presented way of deriving these equations leadsto the derivation of membrane permeability parameters ωd, ωk aswell as LD parameter. The physical meaning of these parameters ispresented further in the paper.

2 Scheme of the solute fluxes across a membraneFig. 1 depicts a cell, which consist of two compartments with a

solution separated by the membrane in isothermal conditions. Sub-stance transport is generated by the simultaneous action of two stim-uli: the osmotic pressure difference and the mechanical pressure dif-ference. The solute fluxes transport through the membrane is de-scribed in the text.

Concepts of Physics, Vol. V, No. 3 (2008) 461

M. Jarzynska and M. Pietruszka

The above K-K equations, (1) and (2), are derived on the basisof linear thermodynamics of irreversible processes. Membrane trans-port is generated by two thermodynamic stimuli: ∆p and ∆Π. Inthe method developed here, the analysis of the membrane transportis presented for the case of a homogeneous membrane, which sepa-rates two aqueous well-mixed solutions of a non-electrolyte with two-component solutions. There are the following conditions inside thecompartments: c1 > c2 and p1 > p2. The osmotic pressure difference∆Π, on the basis of vant Hoffs formula, is assumed as ∆Π = RT∆cwhere ∆c = c1 − c2 is the concentration difference [5, 14] while Rand T are the gas constant and thermodynamic temperature, respec-tively. The hydrostatic pressure difference ∆p = p1 − p2. Accordingto the isothermal conditions, the temperature gradient equals 0.



3 Derivation of Kedem-Katchalsky equations formembrane transport from phenomenological equa-tions

Dissipation function Φ for the membrane of thickness ∆x is givenby the following formula (Katchalsky and Curran, 1965):∫ ∆x

0

Φdx = Φ =n∑

i=1

Ji∆µi = Jw∆µw + Js∆µs, (3)

where Ji is the volume flow, and µi is the chemical potential of so-lution components. Equation (3) describes n different flows in termsof n forces. The subscripts w and s denote the solvent and the so-lute, respectively. Provided the chemical potentials at the membranesurfaces are identical to the corresponding chemical potentials in thesolutions, Φ can be expressed as [2]

Φ = (JwVw + JsVs)∆p +(

Js

cs

)∆Π, (4)

where Vw and Vs are the partial molar volume of solvent and solute;Jw and Js are flows of solvent and solute, respectively. We obtainthe transformed dissipation function in the following form

Φ = Jν∆p + JD∆Π. (5)

In the conditions of two thermodynamic stimuli ∆p and ∆Π, thephenomenological K-K equations for the volume flow (Jν) and thediffusion flow (JD) through the membrane and the forces defined byEq. (5) can be written as

Jν = Lp∆p + LpD∆Π, (6)

JD = LDp∆p + LD∆Π, (7)

where Lp, LDp, Lpd and LD are coefficients of the membrane (fil-tration, ultrafiltration, osmotic and diffusional, respectively ). If∆p = ∆Pi, then Lp = −LpD = LD; JD is the diffusional flow.



3.1 Equation for the volume flow Jν

The overall volume flow Jν is equal to the sum of the volumeflux caused by ∆p and the volume flux caused by ∆Pi. This overallvolume flow Jν can be given by Eq. (6).

Making use of the reflection coefficient σ, introduced originally byStaverman (1951), we can write [2]

σ = −LpD

Lp. (8)

As a result, equation (9), written below, has exactly the same formas Kedem-Katchalsky Eq. (1):

Jν = Lp∆P − Lpσ∆Π. (9)

The relation among transport parameters Lp and σ is expressed inthe following definition:

Lp =(

Jν

∆p

)∆Π=0

. (10)

The parameter Lp is the hydraulic conductivity or the mechanicalfiltration coefficient of a given membrane. This coefficient has thecharacter of mobility and represents the velocity of fluid per unit ofpressure difference [7]. It expresses the overall filtration properties ofall pores within a given membrane.

The parameter σ is the reflection coefficient. If σ = 1, the mem-brane is semi-permeable, what means that it is permeable only forthe solvent but not for the solute molecules, which are reflected. Ifσ = 0, the membrane is not selective, so every pore of the membraneis permeable for the solvent as well as for the solute. If 0 < σ < 1,the membrane is selective, which means that it has pores with differ-entiated diameters [6, 8].

3.2 Equation for the solute flow (Js)

The overall solute flow Js through the membrane, generated by∆p and ∆Π, can be written in the following form [2]

Js = c(Jν + JD). (11)



For such a membrane, the Onsager reciprocal relation (ORR) is givenby

LpD = LDp. (12)

Using Eqs. (6), (7) and (12), we can write Eq. (11) for the totalsolute flow as follows:

Js = c (Lp∆p + LpD∆Π + LDp∆p + LD∆Π) (13)= cLD∆Π + cLpD∆Π + c (Lp + LDp) ∆p,

= c (LD + LpD) ∆Π + c (Lp + LDp) ∆p,

where Lp∆p is the hydraulic term; LpD∆Π is the osmotic term;LDp∆p is the ultrafiltration term and LD∆Π is the diffusional term.

Next, according to Eq. (8), Eq. (13) can be expressed as:

Js = ωd∆Π + c(1− σ)Lp∆p. (14)

After taking Eq. (6) into account and after some transformations,Eq.(14) can be written as:

Js = cLD∆Π + cLpD∆Π + c(1− σ)(Jν − LpD∆Π) (15)= cLD∆Π + cσLpD∆Π + c (1− σ) Jν .

Using Eq. (8), we can express Eq. (15) in the following form:

Js = c

(LDLp − L2

pD

Lp

)∆Π + c(1− σJν). (16)

If in Eq. (16) we insert the coefficient [2]:

ω =(

Js

∆ΠJν=0

)= c

(LDLp − L2

pD

Lp

), (17)

then Eq. (16) for the solute flow Js has the same form as Kedem-Katchalsky Eq. (2):

Js = ω∆Π + c(1− σ)Jv. (18)

The phenomenological K-K equations can also be written in the fol-lowing form [12, 13]:

Jv = Lp∆p− LpD∆Π, (19)

JD = −LDp∆p + LD∆Π. (20)



3.3 Equation for the volume flow (Jν)

Taking Eqs. (19) and (20) into account, we can rewrite the Staver-mann relation, given by Eq. (8), in the following form [6, 12, 13]:

σ =LpD

Lp. (21)

According to Eq. (21), Eq. (19) for the volume flow Jν can beexpressed in the same way as Kedem-Katchalsky Eq. (1):

Jν = Lp∆p− Lpσ∆Π. (22)

3.4 Equation for the solute flow (Js)

According to Eqs. (12), (19) and (20), equation (11) for the totalsolute flow Js can be written as follows

Js = c(Lp∆p− LpD∆Π− LDp∆p + LD∆Π) (23)= c(Lp − LDp)∆p + cLD∆Π− cLpD∆Π.

Using Eq. (21) and Eq. (19) we obtain:

Js = c(1− σ)Lp∆p + cLD∆Π− cLpD∆Π (24)= c(1− σ)(Jv + LpD∆Π) + cLD∆Π− cLpD∆Π= c(LD − σLpD)∆Π + c(1− σ)Jν

= c

(LDLp − L2

pD

Lp

)∆Π + c(1− σ)Jν .

If in Eq. (24) we insert Eq. (17) for the coefficient ω, then theequation Eq. (24) for the solute flow Js has the same form as Eq.(2):

Js = ω∆Π + c(1− σ)Jν , (25)

where ω is the coefficient of a solute permeability at Jν = 0. Itexpresses the permeation of all pores of a given membrane with si-multaneous action of both stimuli, ∆Π and ∆p.



4 Derivation of membrane transport parameters:ωd, ωk and LD

4.1 Membrane permeation coefficient ωd

Parameter ωd is a coefficient expressing permeation of all poresof a given membrane for the solute flow due to the presence of ∆Πstimulus, when ∆p = 0. For the sake of its derivation, we define thecoefficient ωd in the following way

ωd =(

Js

∆Π

)∆p=0

. (26)

Using Eqs. (9) and (18), we obtain the following relation:

Js = ω∆Π + c(1− σ)(Lp∆p− Lpσ∆Π) (27)= [ω − c(1− σ)Lpσ]∆Π + c(1− σ)Lp∆p.

When ∆p = 0, Eq. (27) takes on the form:

(Js)∆p=0 = (ω − c(1− σ)Lpσ) ∆Π. (28)

The obtained equation, Eq. (28), allows one to calculate the ωd

coefficient due to its definition given by Eq. (26). Therefore, ωd canbe expressed as:

ωd = ω − (1− σ)cLpσ. (29)

Next, inserting Eq. (17) and Eq. (8) into Eq. (30), we get the finalform of the ωd coefficient [11]:

ωd = c(LD + σLpD − Lpσ + σLpσ) (30)= c(LD + LpD).

The coefficient ωd, expressed by Eq. (30), can be interpreted moreeasily than the one expressed by Eq. (29). Moreover, the form ofEq. (30) makes membrane transport, described by practical Kedem-Katchalsky equations, more comprehensible.



4.2 Parameter LD :

The above equations (30) and (29) can be written as follows:

c(LD + LpD) = ω − (1− σ)cLpσ. (31)

The final formula describing transport parameter LD has the follow-ing form [11]:

LD =ω

c+ Lpσ

2. (32)

The diffusional flow JD, caused by the osmotic pressure difference,is characterized by the coefficient LD. This coefficient representsdiffusional mobility per unit of osmotic pressure [2]. Therefore, thecoefficient LD can be defined as:

LD =(

JD

∆Π

)∆p=0

. (33)

4.3 Transport parameter ωk

Parameter ωk is a permeation coefficient of all pores of a givenmembrane for the solute flow due to the action of the hydrostaticpressure difference ∆p, when ∆Π = 0 The coefficient ωk can be de-fined by means of the following formula

ωk =(

Js

∆p

)∆Π=0

. (34)

After taking Eqs. (18) and (9) into account, we obtain Eq. (33) inthe following form

ωk = c(1− σ)Lp. (35)

Inserting Eq. (8) into Eq. (34), we can write eventually the formulafor the coefficient ωk as:

ωk = c(Lp + LDp). (36)



5 Conclusions

The analysis of the expressions for particular fluxes and other rela-tions derived here, representing membrane permeation coefficients ωd,ωk, ω and parameter LD, together with their physical interpretation,make practical K-K equations more straightforward and comprehen-sible. Hence, the physical interpretation of these equations seems tobe easier now, while it has been considered as difficult especially inthe case of the equation for the solute flow Js.

The method presented here, which has resulted in the derivationof practical Kedem-Katchalsky equations for the volume flow Jν andthe solute flow Js, allows one to carry out a more detailed and com-prehensible analysis of results of the research on transport throughbiological and artificial membranes.

References

[1] O. Kedem, A. Katchalsky: Thermodynamics analysis of the per-meability of biological membranes to non-electrolytes, Biochim.Biophys. Acta (1958), 27, 229- 246.

[2] A. Katchalsky, P. F. Curran, Non-equilibrium Thermodynamicsin Biophysics, Harvard University Press, Cambridge, MA, 1965.

[3] O. Kedem, A. Katchalsky, Permeability of composite mem-branes, Trans. Faraday Soc. 59 (Part 1-3) (1963).

[4] O. Kedem, A. Katchalsky, A physical interpretation of thephenomenological coefficient of membrane permeability, J Gen.Physiol. 45 (1961) 143-179.

[5] K. Guminski, Thermodynamics of Irreversible Processes, PWN,Warsaw, 1962 (in Polish) 57-63.

[6] M. Kargol, A. Kargol: Mechanistic equations for membranesubstance transport and their identity with Kedem-Katchalskyequations, Biophys. Chem. (2003), 103, 117-127.

[7] A. Narebska, W. Kujawski and S. Koter: Irreversible thermody-namics of transport across charged membranes, J. Membr. Sci.,25 (1985) 153-170.



[8] B. Z. Ginzburg, A. Katchalsky: The frictional coefficients of theflows on nonelectrolytes through artificial membranes. J. Gen.Phisiol. (1963), 47, 403-418.

[9] S. Koter: The Kedem-Katchalsky equations and the sieve mech-anism of membrane transport, Journal of Membrane Science(2005), 246, 109-111.

[10] M. Jarzynska: Mechanistic equations for membrane substancetransport are consistent with Kedem Katchalsky equations, J.Membr. Sci. 263 (2005) 162-163.

[11] A. Kargol, M. Przestalski, M. Kargol: A study of porous struc-ture of cellular membranes in human erythrocytes, Cryobiology50 (2005) 332-337.

[12] G.Suchanek : On the derivation of the Kargols mechanistictransport equations from the Kedem-Katchalsky phenomenolog-ical equations. Gen. Physiol. Biophys. (2005) 24, 247-258.

[13] G. Suchanek: Mechanistic Equations for Membrane Transportof Multicomponent Solutions. Gen. Physiol. Biophys. (2006), 25,53-63.

[14] K. Do lowy, A. Szewczyk, S. Piku la: Biological membranes, page109, Scientific Publisher “Silesia” , Katowice-Warszawa 2003 (in Polish).

[15] M. Jarzynska: New method of derivation of practical Kedem-Katchalsky) membrane transport equations, Polymers inMedicine, 35 (2005) pp. 19-24 (in Polish).

[16] G. Monticelli: Some remarks about a mechanistic model oftransport processes in porous membranes, J. Membr. Sci. (2003),214, 331-333.

[17] M. Jarzynska: The application of practical Kedem-Katchalskyequations in membrane transport, Central European Journal ofPhysics, accepted 14 July 2006.

[18] B. Z. Ginzburg, A. Katchalsky: The frictional coefficients of theflows on nonelectrolytes through artificial membranes. J. Gen.Phisiol. (1963), 47, 403-418.



Nomenclature

Js, Js1, Js2, Js3, Js4 solute fluxesJν volume flowνs, νw velocity of solute and solventµi chemical potential of the th component∆µi difference of the chemical potentials of

the i-th componentVw Vs partial molar volume of the solvent

and the solute∆p ∆Π hydrostatic and osmotic pressure differences

(Pa)Lp, LD, LpD, LDp hydraulic coefficientsω, ωk, ωd solute permeability coefficientsσ reflection coefficient (-)c mean concentrationc1, c2 solutions concentrations∆c concentration differenceR gas constantT absolute temperature


Comment

Comment onDERIVATION OF PRACTICAL KEDEM -

KATCHALSKY EQUATIONS FORMEMBRANE SUBSTANCE TRANSPORT

Jacek WasikLaboratory of Surface Physics

Jan Długosz AcademyCzęstochowa, Poland


This paper is aimed to give more physical sense to the Kedem-Katchalsky (K-K) equations when considering the case of solvent andsolute transport under the action of hydrostatic pressure and osmoticpressure gradient.It is a good way to clarify some chapters of the non-equilibrium ther-modynamics.

This derivation, in particular makes the equation for solute flow(Js) and the solute permeability coefficient (ω) clearer and conse-quently their interpretation also becomes easier. Furthermore, thispaper makes some new contribution to membrane expressed by theso called K-K equations. It includes a transformation of K-K equa-tion on solute flow to its new form: Js = ωd∆Π + c(1 − σ)Lp∆p ascoefficient derivations which characterize the membrane ( ωd - thecoefficient of diffusive solute permeation and LD - conductivity for


Comment

volume flow ).It should be emphasized that the K-K equation on solute flow

transformed to a new form is more comprehensible. Together withthe derivated parameters, it gives new opportunities for the researchconcerning solute permeation across biological and artificial mem-branes.


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DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS …€¦ · DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS FOR MEMBRANE SUBSTANCE TRANSPORT M. Jarzyns ka Technical High