Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 76, Issue 3, 1293-1309, 1 March 1999
doi:10.1016/S0006-3495(99)77292-9
Biophysical Theory and Modeling
Kinetic and Thermodynamic Aspects of Lipid Translocation in Biological Membranes
Stephan Frickenhaus and Reinhart Heinrich
, 
Humboldt University Berlin, Institute of Biology and Theoretical Biophysics, D-10115 Berlin, Germany
Address reprint requests to Reinhart Heinrich, Institute of Biology and Theoretical Biophysics, Humboldt University Berlin, Invalidenstrasse 42, 10115 Berlin, Germany. Tel.: 49-30-20938698; Fax: 49-30-20938813.
Abstract
A theoretical analysis of the lipid translocation in cellular bilayer membranes is presented. We focus on an integrative model of active and passive transport processes determining the asymmetrical distribution of the major lipid components between the monolayers. The active translocation of the aminophospholipids phosphatidylserine and phosphatidylethanolamine is mathematically described by kinetic equations resulting from a realistic ATP-dependent transport mechanism. Concerning the passive transport of the aminophospholipids as well as of phosphatidylcholine, sphingomyelin, and cholesterol, two different approaches are used. The first treatment makes use of thermodynamic flux-force relationships. Relevant forces are transversal concentration differences of the lipids as well as differences in the mechanical states of the monolayers due to lateral compressions. Both forces, originating primarily from the operation of an aminophospholipid translocase, are expressed as functions of the lipid compositions of the two monolayers. In the case of mechanical forces, lipid-specific parameters such as different molecular surface areas and compression force constants are taken into account. Using invariance principles, it is shown how the phenomenological coefficients depend on the total lipid amounts. In a second approach, passive transport is analyzed in terms of kinetic mechanisms of carrier-mediated translocation, where mechanical effects are incorporated into the translocation rate constants. The thermodynamic as well as the kinetic approach are applied to simulate the time-dependent redistribution of the lipid components in human red blood cells. In the thermodynamic model the steady-state asymmetrical lipid distribution of erythrocyte membranes is simulated well under certain parameter restrictions: 1) the time scales of uncoupled passive transbilayer movement must be different among the lipid species; 2) positive cross-couplings of the passive lipid fluxes are needed, which, however, may be chosen lipid-unspecifically. A comparison of the thermodynamic and the kinetic approaches reveals that antiport mechanisms for passive lipid movements may be excluded. Simulations with kinetic symport mechanisms are in qualitative agreement with experimental data but show discrepancies in the asymmetrical distribution for sphingomyelin.
Introduction
Plasma membranes of eukaryotic cells show a pronounced asymmetry with respect to the distributions of the major lipid components among the two monolayers. The aminophospholipids phosphatidylserine (PS) and phosphatidylethanolamine (PE) are predominantly located on the cytoplasmic leaflet, whereas the phospholipids phosphatidylcholine (PC) and sphingomyelin (SM) are mainly found on the external leaflet (Bretscher, 1972,Verkleij et al,Gordesky and Marinetti, 1973; cf. Devaux, 1991,Zachowski, 1993). Evidence of the distribution of cholesterol (Ch) as another membrane component is still contradictory. Some authors found for cholesterol a preference for the cytoplasmic layer of the red blood cell membrane (Brasaemle et al,Schroeder et al), whereas other results indicate a rather symmetrical distribution (Blau and Bittman, 1978,Lange and Slayton, 1982).
As has been demonstrated by Seigneuret and Devaux, 1984, the asymmetrical distribution of the aminophospholipids may be understood by an ATP-dependent translocation of these components from the external to the cytoplasmic layer. The response of the membrane to this directed transport will concern not only the counter-directed movement of PS and PE, but also a redistribution of PC, SM, and Ch. Furthermore, a change in membrane curvature may occur because of geometrical restrictions and corresponding mechanical forces caused by the coupling of the monolayer surfaces. This reasoning shows that the membrane asymmetry is determined by a multitude of processes, depending on 1) the metabolic state of the cell, 2) the mechanism of active translocation, 3) the transmembrane concentration differences of lipids, and 4) mechanical forces.
Obviously, the interaction of different translocation processes may be adequately described only on the basis of biophysical models allowing quantitative estimates for different experimental situations under time-independent and time-dependent conditions. Mathematical models of molecular mechanisms of lipid translocation are still rare. Previous investigations (Brumen et al,Heinrich et al) indicated that an explanation of the asymmetrical molecular composition of bilayers needs to consider free energy contributions from mixing entropy as well as from mechanical effects of lateral compressions. However, in the work of Brumen et al, the expressions for fluxes resulting from mechanical stress, the so-called compensatory fluxes, are not well defined in their physical meaning. A drawback of the paper of Heinrich et al is that the use of monolayer mixing entropy is not fully justified for two coupled monolayers. Furthermore, the latter analysis assumes that the mechanical properties of the lipids are species independent, which is an oversimplification, at least for cholesterol.
Correct theoretical investigations of the membrane on a microscopic level are crucial for understanding macroscopic cellular phenomena such as shapes of cells. In the latter field of research much theoretical work has been done using methods of elasticity theory (see Svetina and Žekš, 1989,Seifert et al,Heinrich et al). Membrane properties such as spontaneous curvature and relative area changes of the monolayers, which enter the shape-determining energy functional, should be directly related to the asymmetrical composition of the bilayer.
This study is intended to gain a more complete understanding of the phenomena of transbilayer lipid movement by finding an appropriate phenomenological description of the lipid fluxes. The steady-state asymmetrical lipid distribution is governed by dynamics equations. A reference simulation with a minimal set of phenomenological parameters yields qualitative restrictions to the many possible translocation mechanisms. Relations between phenomenological and kinetic model parameters serve, furthermore, as guidelines for the selection of kinetic constants in a quantitative way. It is shown how the mechanical driving forces of the phenomenological model are to be incorporated into a kinetic model of lipid translocation.
Basic model assumptions
Let us consider a bilayer membrane of one cell composed of s different lipids, which are subject to translocation processes between the cytoplasmic monolayer c and the external monolayer e. The amounts, in units of moles per cell, of the lipids i, i=1, …, s, on the monolayers are denoted by Nic and Nie. They are related to the differences ni=Nic−Nie and to the total amounts of lipids Ni=Nic+Nie as follows:
The time-dependent changes of the composition of the monolayers are governed by the differential equations
where
Jiact and
Jipass denote the fluxes of active and passive transport, respectively. Fluxes have positive sign if lipid amounts on the cytoplasmic side are increased. In this equation it is assumed that the total amount
Ni is constant, that is, the model does neither include insertions of the lipids into the membrane or extractions of the lipids from the membrane. Lipids are considered to be distributed homogeneously in lateral directions. Furthermore, there is no intermediate state at the transport of the lipids from one leaflet to the other.
For the fluxes Jiact we use an ATP-dependent carrier mechanism as described previously (Heinrich et al; cf. also Simulations). Passive fluxes are described first in the framework of linear irreversible thermodynamics, and second on the basis of kinetic translocation models.
In the thermodynamic approach we apply linear flux-force relationships,
where
Xj denotes thermodynamic forces. The coefficients
Lij are referred to as phenomenological coefficients and are assumed to be state independent, i.e., they have constant values in time. They may depend, however, on system parameters such as total lipid amounts and lipid-specific molecular parameters. For the forces
Xj we analyze in the following sections entropic effects and mechanical effects within lipid bilayers. Both types of forces may be expressed as functions of the variables
Nic or
Nie and of lipid-specific parameters. Gradients in temperature are neglected, that is, the solvent on both sides of the membrane acts as a heat bath.
In the kinetic approach it is assumed that passive diffusion fluxes of lipids are mediated by a protein carrier. Two different mechanisms are analyzed: first, an antiport mechanism, and second, a symport mechanism. Near equilibrium the linearized kinetic equations may be directly compared to the phenomenological equations of the thermodynamic approach. In this way the coupling coefficients may be expressed in terms of the kinetic parameters of the carrier.
Phenomenological forces of the lateral ideally mixing bilayer
Entropic forces
The free energy F of the bilayer can be derived from combinatorial considerations under the following assumptions: 1) ideal mixing of the lipids within both monolayers (i.e., non-lipid-specific interactions), 2) negligible molecular transbilayer interactions, and 3) no internal degrees of freedom for the conformation of the lipid molecules. Hence F is related to a configurational partition function Z by F=−kBT ln Z, where kB is Boltzmann's constant. The partition function may be calculated from the macroscopic configurations of a bilayer, which are characterized by the numbers of lipid molecules Λic=LANic and Λie=LANie of species i on either monolayer, e or c, respectively. LA denotes Avogadro's number.
The configurational partition function of the bilayer reads
Z takes into account the number of microscopic distributions of Λ
i molecules of each lipid species among the monolayers with a lipid number Λ
ic and Λ
ie on the two layers.
The entropic contribution to the total free energy (in Joules) of a single membrane reads
Using Stirling's formula, ln
x!≈
x ln
x−
x for
x≫1, one derives from Eqs.
(4)It is easy to see that at variations of Λ
jc and Λ
je under the constraint Λ
j=Λ
jc+Λ
je=constant, the free energy attains its minimum for an equal distribution of each lipid species among the two monolayers. In terms of molar amounts the equilibrium state is, therefore, characterized by
Nonequilibrium states are characterized by the variables
which are related to the differences
ni=
Nic−
Nie by
ni=2
yi. Defining entropic forces in units of J/mol as
one derives
This formula for the entropic force shows some correspondence to an expression used in a previous study (
Heinrich et al). However, in the latter work entropic forces have been expressed from differences of lateral mixing entropies between the monolayers, which would be a correct treatment if the monolayers were allowed to uncouple in their surface areas.
For small deviations from the equilibrium state (ni ≪ Ni) the entropic forces may be expressed in a linear approximation as
Mechanical forces
Changes in the lipid compositions of the monolayers caused by passive or active translocation will affect the mechanical energy of the membrane because of changes in lateral distances between the molecules. For small deviations from equilibrium, the lateral mechanical energies of the two monolayers c and e may be expressed in a harmonic approximation as
where
ai and ξ
ic,e denote, for lipid species
i, the equilibrium membrane surface area per molecule, and the area change relative to the equilibrium area on the two layers, respectively.
γi represents the force constant per unit area. In Eq.
(13) the relative area changes are assumed to be equal for all molecules of one species
i, but different in each monolayer, that is, the tension in the monolayer is distributed homogeneously over the molecules of each species. Furthermore, equilibrium areas and force constants are considered to be layer independent.
Because the two monolayers are coupled in such a way that there is a common closed contact surface within the membrane, the surface area of the cytoplasmic layer and that of the external layer cannot vary independently (cf. the bilayer couple hypothesis of Sheetz and Singer, 1974). In particular, one expects that the surface areas of lipids located in the monolayer with increased amounts will be compressed, whereas the areas of lipids within the other monolayer will be expanded. Thus the relative changes ξic,e of areas will depend on the deviations ni characterizing the nonequilibrium state of the membrane. To derive the corresponding relation, one may assume in a first approximation that, under deviation from equilibrium, the total surface areas of each layer remain unchanged, that is,
where
A0c,e are the total equilibrium surfaces of the monolayers. In nonequilibrium states the two areas may be expressed as
The term
aj(1+ξ
jc,e) represents the compressed (ξ<0) or expanded (ξ>0) area of lipid
j in layer
c and
e, respectively. From Eq.
(15) it follows with condition 14 that
In this equation it is assumed that
A0c=
A0e=
A0, i.e., the membrane is symmetrical at equilibrium (for a more general ansatz see the Discussion). In a previous study (cf.
Heinrich et al) relation 16 was used to calculate the relative area changes under the assumption ξ
jc=−ξ
je=ξ, that is, they are identical for all lipids. Such a simplification may not be justified for lipid species of different equilibrium areas
ai and of different force constants
γi. Accordingly, further relations must be taken into account besides Eq.
(16) to calculate the individual area changes. Such relations may be obtained by considering the elastic energy of each monolayer to be at minimum. This assumption is supported by the fact that, after perturbation of the equilibrium state, relaxation of the mechanical stress on both monolayers will occur on a much shorter time scale than transversal redistribution of lipids. This mechanical relaxation will be supported, for example, by the fast lateral redistribution of lipids. Minimization of this energy under the constraints in Eq.
(16) can be performed with two Lagrange multipliers
vc and
ve. With
the lateral elastic energy of the bilayer has an extremum if
From Eqs.
(13) one obtains
which under consideration of Eq.
(1) yields
By taking into account the constraints in 16 in Eq.
(20), one derives
For sufficiently small deviations from equilibrium, the linear approximation
is valid. The total mechanical energy
E=
Ec+
Ee, obtained by introducing relation 22 into Eq.
(13), reads
where terms higher than second order are neglected.
The mechanical forces are defined as
(see the definition of the entropic forces in Eq.
(10)). From Eqs.
(23) one obtains with
ni=2
yiIt is worth mentioning that in the special case of unspecific parameters (
γi=
γ,
ai=
a), Eqs.
(22) reduce to previously derived expressions (cf.
Heinrich et al). As expected, all mechanical forces are reduced in the presence of lipid species that are soft with respect to lateral area changes, i.e., species that have a low
γ value. Equation
(25) reveals that forces are related pairwise by
Ximech/
Xjmech=
ai/
aj. Furthermore, all forces are proportional to
which may be considered as the area difference of
uncoupledmonolayers.
In the following we combine the entropic force and the mechanical force as the total force of passive translocation:
The equilibrium state where the total energy
F+
E has its minimum is characterized by a symmetrical distribution of each lipid component among the monolayers (
ni=0).
Phenomenological coefficients for passive translocations
Parameterization with respect to lipid amounts
In Eq. (3) the phenomenological coefficients Lij are unknown. A certain knowledge concerning the dependence of Lij on molecular membrane parameters may be obtained on the basis of special kinetic models (see below). Furthermore, these coefficients may be fitted to experimental data. More generally, one may apply invariance principles to derive various restrictions for the structure of these coefficients, particularly concerning their dependencies on the total amounts of lipids. We use the principle that macroscopically the behavior of the system should be independent of a decomposition of a certain lipid species into two identical subspecies.
Let us consider an arbitrary decomposition of a certain species k into two subspecies a and b, such that their total amounts Na and Nb and the deviations from equilibrium na and nb are related to the corresponding quantities of species k as follows:
The parameter ϵ that quantifies the decomposition of species
k is confined by −1/2≤ϵ≤1/2. It follows directly from Eqs.
(1) that the forces of species
a and
b are the same as the force of species
k:For fixed
k, the linear flux-force relations in Eq.
(3) may be rewritten for the original system as
Characterizing the quantities of the decomposed system by the superscript * gives
Obviously, the macroscopic properties of the membrane, particularly the passive translocation fluxes, should be invariant with respect to the decomposition in Eq.
(28a,b), that is,
The deviations
ni from equilibrium and, therefore, the forces
Xi may vary independently. Thus, taking into account Eqs.
(30a) in Eq.
(32a,b), and using Onsager's reciprocity relation
Lij=
Lji, one derives that the phenomenological coefficients have to fulfill the relations
To derive dependencies of the phenomenological coefficients on the lipid amounts, the following ansatz is used:
The parameters
λij and
λ*ij, which must be components of a symmetrical matrix, do not contain any further factors of
Ni and
Nj, but possibly factors
Nl with
l≠
i,
j. Constraints on further relations for the dependencies of the parameters
λij on the total lipid amounts are given below (see Eq.
(44a,b)). The values of the exponents
x and
y have to be determined from the invariance properties mentioned above. Inserting Eqs.
(34a) into Eq.
(33b) gives, together with the decomposition 28a,b,
Because species
a,
b, and
k have the same physical properties, their parameters
λ will be identical:
Furthermore, Eq.
(35) must be valid for any decomposition of species
k, that is, the left-hand side should be independent of ϵ. For
x=
y, Eq.
(35) reads, considering Eq.
(36),
where
z=
x=
y. This equation is independent of ϵ only for
z=1. For
x≠
y two separate conditions are obtained:
since Eq.
(35) holds for arbitrary values of both amounts
Ni and
Nk. Obviously these two equations
(37b,c) cannot be fulfilled simultaneously for
x≠
y and arbitrary ϵ. Thus we are left with the only possibility,
x=1 and
y=1. Using this result, we obtain from Eq.
(34a) for the cross-coupling coefficients
To find expressions for the diagonal elements Lkk, we use relation 34a,b in Eq. (33c) and take into account Eq. (38). This yields
Because
λ*aa=
λ*bb=
λkk, Eq.
(39) gives
where
z=
x+
y. For
z there are two solutions such that Eq.
(40) is independent of ϵ. The first solution reads
z=1, with
λ*ab=0, and arbitrary values of
λkk denoted in the following
κk. The second solution reads
z=2, for which Eq.
(40) holds independent of ϵ only with
λkk=
λ*ab. Linear combination of these two solutions yields for the diagonal elements
This expression shows that the diagonal elements
Lkk are composed of a diffusion term
κkNk and a self-coupling term
λkkNk2.
Combining relations 38 and 41, one obtains the following general expression for the phenomenological coefficients:
Concerning the parameters entering the phenomenological coefficients in Eq.
(42), we use the notation diffusion parameters
κi, self-coupling parameters
λii, and cross-coupling parameters
λij for
i≠j.
Under the assumption of a lateral homogeneous membrane, two general properties of the diffusion parameters κj and coupling parameters λij that enter Eq. (42) can be derived. For any subsection of the bilayer characterized by N′k=αNk and n′k=αnk for k=1, … , s with a scaling parameter α, confined to 0<α≤1, the following relations should be fulfilled:
that is, the fluxes and forces are homogeneous functions in the amounts of lipids of first degree and degree zero, respectively. Applying the linear flux-force relations to these subsections of the membrane, one obtains that the phenomenological coefficients are homogeneous functions of first degree in the lipid amounts. Taking into account expression 42, one obtains that
κi is homogeneous of degree zero and
λij of degree −1, respectively. Accordingly, one derives from Euler's theorem on homogeneous functions the following conditions:
In the most simple case, Eq.
(44a,b) is fulfilled if all diffusion parameters are independent of the lipid amounts, whereas Eq.
(44a,b) is fulfilled if all coupling parameters are proportional to the inverse of the total lipid amount.
Taking into account result 42, the passive translocation fluxes, defined in Eq. (3), read
For the simulation of lipid translocation this flux equation may be used in Eq.
(3).
Parameterization with respect to membrane surface areas of lipids
In this subsection we show that the coupling parameters λij in Eq. (42) may be related to the equilibrium surface areas ai of the lipids, which, besides the lipid amounts, are the relevant parameters of the present model. Let us consider a certain lipid species, say k, and a perturbation from the equilibrium state such that
Equation 46a entails that the entropic force in Eq.
(12) of species
k vanishes, whereas Eq.
(46a,b) implies that no mechanical forces occur (see Eq.
(25)). If there are no lipid-specific interactions, one may conclude that all states fulfilling Eq.
(46a,b) are characterized by a vanishing passive flux of species
k, which yields, using Eqs.
(3), in the vicinity of equilibrium,
Taking into account the general structure for the nondiagonal elements
Lkj given in Eq.
(42), one derives from Eq.
(47) the following condition for the cross-coupling parameters:
Equation
(48a) states that a vector
= (
λk1 ···
λk,k−1λk,k+1 ···
λks)
T is orthogonal to any vector of perturbations
= (
n1 ···
nk−1nk+1 ···
ns)
T, which in vector notation reads
The space of perturbations
restricted by Eqs.
(46a,b) is of dimension
s−2. Accordingly, this space is spanned by
s−2 vectors
(
i=1, … ,
s−2). In the case
k=
s, for example, a special choice of these vectors reads
Generally, the elements of the vector 
are given by bj(i,s)=0 for 1≤j≤i−1 and i+2≤j≤s−1, and bi(i,s)=ai+1, bi+1(i,s)=−ai. Similar representations are obtained for k≠s.
The condition 48b of orthogonality is fulfilled if 
is orthogonal to all corresponding vectors 
, which yields
It can be shown that there is one parameter
vk, such that Eq.
(50) is fulfilled with
For
k=
s this result may be easily verified by using Eq.
(49). The symmetry relations
λkj=
λjk require
vkaj=
vjak, which leads to
The combination of Eqs.
(42) gives
According to this equation, the
s(
s+1)/2 phenomenological coefficients
Lij are fixed by only
s+1 phenomenological parameters
v and
κj. Relation 53 may be used also for the case of equal lipid area parameters, that is, for
aj=
a.
The requirement for the matrix Lij to be positive definite (see de Groot and Mazur, 1962) sets a certain limit for the unknown parameter v. For example, any choice of v must ensure that the diagonal elements Lii are positive. Thus there is a lower limit for v depending on the values of κi>0 and ai>0.
Phenomenological coefficients as derived from kinetic models
On the phenomenological level of description, the parameters κj and λij in Eq. (42) and, similarly, the parameter v in Eq. (52) have no mechanistic interpretation. In this section we consider the case in which lipid translocation is mediated by a carrier protein. This allows us to express the phenomenological parameters in terms of kinetic constants.
Kinetic models without mechanical effects
Antiport mechanism. Let us assume that the translocation carrier has only one binding site, to which the lipids bind in a competitive way. If this binding is fast compared to translocation, the following equilibrium relations hold:
where
Pic,e and
Pc,e denote the amounts of the loaded forms and of the unloaded forms, respectively. In this equation
Nic,e are the amounts of free lipids, that is, lipids not bound to the carrier. We assume that the total amount of loaded carrier forms is much smaller compared to the individual lipid amounts such that
Ni=
Nic+
Nie holds true. Conservation of the total amount
P of carrier molecules leads with Eq.
(54) to the relation
A quasi-steady-state approximation for the distribution of the carrier among the two layers yields
where
li+ and
li− denote the translocation rate constants of the loaded carrier forms, and
k+ and
k− denote the rate constants of movement of the unloaded carrier forms. One obtains with the help of Eqs.
(54)with
Because the time-dependent changes of the lipid amounts are characterized by
one obtains from Eqs.
(54)where again
Pic,e≪
Nic,e is assumed (see
Schultz, 1980).
To express the coupling parameters in terms of kinetic constants, Eq. (60) has to be applied for states near equilibrium. In the case characterized by Kic=Kie=Ki, li+=li−=li and k+=k−=k (symmetrical carrier), the equilibrium amounts are Nic=Nie=Ni/2 (symmetrical membrane). Near equilibrium a linear expansion in ni of Eq. (60) yields, after some algebra,
A and B correspond to the quantities defined in Eq. (58a,b) by inserting the equilibrium amounts of the lipids. To calculate the phenomenological parameters, we relate the right-hand side of Eq. (61) to the expression
The latter formula follows from linearization of Eq.
(44a,b) as well as from neglecting the mechanical forces. One obtains for the phenomenological parameters
where
keff denotes an effective rate constant of the carrier defined as follows:
P0 and
Pi denote the total equilibrium amounts of the unloaded carrier form and loaded carrier forms, respectively, that is,
P0=
P/
A and
Pi=
P0KiNi/2.
Combining Eq. (42) with Eq. (63a,b) yields for the phenomenological coefficients
Equations
(63a,b) show that the phenomenological coefficients resulting from a special kinetic equation are in accordance with the general structure given in Eq.
(42).
The following properties of the coupling parameters of the antiport mechanism are worth mentioning: 1) The phenomenological parameters in Eqs. (63a,b) are proportional to the total amount of the carrier P. 2) The diffusion parameters κi as well as the cross-coupling parameters λij (i≠j) and the self-coupling parameters λii are proportional to the respective rate constants li and lj of the translocation of the carrier. 3) All coupling parameters λij are negative, which reflects that a driving force Xi giving rise to a diffusion flux of species i is accompanied by a cross-coupled flux as well as a self-coupled flux in the opposite direction. 4) The quantity A in the denominators of Eqs. (63a,b) reflects a saturation of the carrier with lipids. For very low affinities of the lipids for the carrier KiNi≪1, the quantity A tends to unity, that is, saturation effects are negligible. 5) The coupling parameters λij are inversely related to the effective rate constant keff of the carrier. In the case that the unloaded form of the carrier moves much faster than the loaded forms (k≫li), the λij tend to zero, that is, the fluxes become uncoupled. In the opposite limit k→0, it follows from Eq. (65) that Lii=−Σj≠iLij. In this case one row of the matrix of phenomenological coefficients becomes linearly dependent on the other rows. Such a matrix is no longer positive definite, because at least one of its eigenvalues vanishes. Because principles of irreversible thermodynamics require that the matrix Lij is positive definite, vanishing unloaded flip-flop (k=0) is excluded. For example, in the special case of one component (s=1), it is easily verified that diffusion and self-coupling coefficients cancel out in the limit k→0. Because any nonequilibrium state is accompanied by entropy production, such a situation of vanishing flux is forbidden.
Similar to the scaling properties in Eq. (44a,b) derived for the phenomenological parameters of the thermodynamic model, the parameters 63a,b of the kinetic models must be homogeneous functions of all quantities involving units of amounts. Rescaling of the system by a factor a necessitates in the case of carrier mechanisms not only a change of lipid amounts Nic,e→αNic,e, but also of the total amount P3aP of carrier molecules as well as of the binding constants Ki→α−1Ki.
Symport mechanism. A carrier with two identical lipidbinding sites will be able to translocate single molecules of the various lipid species as well as different pairs of molecules. Making assumptions analogous to those in the case of an antiport, one derives as a kinetic equation of such a symport mechanism:
The quantities A and B are defined in analogy to Eq. (58a,b), that is,
In the derivation of these equations, it has been assumed that the two binding sites of the carrier have identical properties. Furthermore, the binding constants are assumed to be layer independent, that is,
Kic=
Kie=
Ki. The factors 2 on the right-hand side of Eq.
(66) arise for combinatorial reasons. In the first term this factor reflects that each lipid
i may bind to the unloaded carrier at two different binding sites. The factor 2 in the second term corresponds, for
i≠
j, to the two different forms of the carrier loaded with lipid
i and lipid
j, whereas for
i=
j this factor takes into account that two molecules of lipid
i are transported in one translocation step. In Eq.
(67a,b) the exponent 2 occurs because the two identical binding sites are assumed to be independent. Near equilibrium a linear approximation of Eqs.
(66) leads for a symmetrical parameter choice
li+,−=
li,
lij+,−=
lij, and
k+,
−=
k to an equation that may be directly compared with Eq.
(62). In this way the coupling parameters are identified as
where
A and
B are obtained from Eqs.
(67a,b) by inserting the equilibrium amounts of the lipids.
P0,
Pk, and
Pkm denote the total equilibrium amounts of the unloaded carrier form, the forms loaded with one lipid of species
j, and the double loaded forms of the carrier, respectively.
Compared to the parameters of the antiport, the following features of the phenomenological parameters of the symport are remarkable: 1) A coupling parameter λij may become positive in case in which the corresponding double-loaded carrier is translocated (lij≠0). The sign of the coupling parameters depends crucially on the value of the effective rate constant keff defined in Eq. (69). If keff is much higher than all translocation rate constants li, lj, and lij, then one obtains λij>0 for all i, j. This can be achieved, for example, if the translocation rate constant k of the unloaded carrier is high enough. 2) The negative term in λij in Eq. (68b) is closely related to the diffusion parameters κi and κj, because it is proportional to their product.
Inclusion of mechanical effects
A translocation of lipid molecules will generally be accompanied by a change in the mechanical energy of both monolayers. This energy change will affect the free energy barriers determining the rate constants of translocation of the loaded carrier forms. As an example, we consider the kinetic equation (Eq. (60)) of the antiport mechanism. The energy change ΔEi for the translocation of Δyi moles of species i, leading to a change of ni by 2Δyi, may be approximated by (∂E/∂yi)Δyi. For 1 mole of translocated lipids, this expression reduces, according to Eq. (24), to
Assuming a linear profile of the mechanical energy along the translocation coordinate as well as a symmetrical energy barrier in the absence of mechanical stresses, the rate constants obey the following relation:
Taking into account these expressions in the rate equation (Eq.
(60)) as well as in the terms
Bc,e defined in Eq.
(58a,b), the mechanical forces are included in the kinetic model.
The question arises whether this kind of kinetic description is consistent with the thermodynamic flux-force relationships used to obtain Eq. (45). For the comparison of the two approaches, the kinetic rate equation, obtained by combination of Eqs. (60), as well as the phenomenological flux equation (Eq. (45)) must be linearized in the deviations ni. The linearized kinetic equation reads
Linearization of Eq.
(45) yields
which is an extension of Eq.
(62) by inclusion of mechanical forces. A comparison of these two equations shows that near equilibrium the kinetic model that includes mechanical forces may be characterized by the same phenomenological parameters as a model that does not contain any mechanical effects (see Eq.
(63a,b)).
These results support our previous assumption that in the framework of thermodynamic flux-force relations, a unique set of coupling parameters exists, that is, that the entropic and mechanical forces are additive. The incorporation of mechanical forces in the rate constants of the kinetic model may give, therefore, a good representation also of the nonlinear kinetic equation system in Eq. (60). The given procedure has been applied also for the symport mechanism discussed above, with the result that the coupling parameters given in Eqs. (68a) remain unchanged if the mechanical forces are included in the rate constants (Frickenhaus and Heinrich, unpublished results).
Simulations
General remarks
The theory outlined in the previous sections is applied to the transversal lipid distribution of the human red blood cell membrane. Time-dependent changes as well as steady states are calculated by means of numerical integration of the differential equations resulting from the thermodynamic and from the kinetic approach for passive fluxes. We take into account the five abundant lipids, phosphatidylserine, phosphatidylethanolamine, phosphatidylcholine, sphingomyelin, and cholesterol (see Zachowski, 1993). The lipid specifying indices i=1, … , s with s=5 refer to PS, PE, PC, SM, and Ch, respectively.
For the simulations we use the system equations 2. Jipass is represented either by Eq. (45) (thermodynamic model) or by a kinetic equation resulting from a carrier mechanism (Eq. (60) or 66). An expression for the active fluxes Jiact, which also appears in Eq. (2), is derived below. Furthermore, we have to specify the units of the model quantities.
For the active transport we refer to the result of Seigneuret and Devaux, 1984, that there exists in plasma membranes of eukaryotic cells an ATP-dependent carrier that specifically translocates PS and PE from the external to the cytoplasmic leaflet. A simple kinetic expression for this process is obtained under the following assumptions: 1) fast and competitive binding of PS and PE to the translocase on both sides of the membrane, 2) unidirectional transport of PS and PE to the cytoplasmic layer under consumption of one ATP per transported aminophospholipid (Beleznay et al), and 3) bidirectional flip-flop of the unloaded carrier form. The kinetic equations for the active transport of PS and PE read
where
In these equations
T denotes the total translocase concentration,
KPSc,e and
KPEc,e are the binding constants of the lipids to the translocase,
k+,− are the flip-flop rate constants of the unloaded translocase, and
lPS+ and
lPE+ are the effective rate constants of the unidirectional translocations for a fixed ATP concentration. We leave out a detailed derivation of Eq.
(74a,b), because it is easily done by the procedure explained above for the passive carrier mechanisms. In fact, Eq.
(74a,b) is formally a special case of the kinetic expression in Eq.
(60). However, in the case of active transport, the principle of detailed balance, which would exclude irreversible steps within a translocation cycle, is not relevant because of the coupling to ATP consumption. For a derivation of Eq.
(74a,b), see also
Heinrich et al.
Rate constants in the kinetic equations of passive and active carrier mechanisms have units of min−1. Binding constants, which also enter the kinetic models, are usually given in units of mM−1. To meet this choice, the lipid amounts must be expressed in concentration units (mM). For that, we normalize the lipid amounts with respect to a fixed reference volume. A convenient choice is the volume of the red cell as a reference, because it is independent of the cell suspension volume. Of course, the fixed reference volume could be chosen differently, e.g., the bilayer membrane volume, which would rescale only the numerical values of the concentration-dependent system parameters. Furthermore, we use concentration units for the quantities P and T of two of the types of carriers; thus the fluxes are expressed in mM min−1. Concerning the thermodynamic model equations, rescaling of concentrations will affect the unit of the flux, but not the unit of the thermodynamic forces (cf. Eqs. (12)). The diffusion parameters κi and coupling parameters λij, which enter the phenomenological coefficients, have, according to their original definitions, units of min−1 J−1 and min−1 J−1 mol−1, respectively. Both parameters may be rescaled in such a way that they have the dimension of rate constants. This leads to κ′i=4RTκi for the diffusion parameters and to λ′ij=4NRTλij for the coupling parameters, where N denotes the total concentration of lipids, i.e., N=ΣiNi. In both expressions the factor 4 arises from the linearization of the system equations (Eq. (45)) in the vicinity of equilibrium.
Simulations with a reference set of parameters
As a reference model, a set of differential equations (Eq. (2)) with passive fluxes from the thermodynamic model (Eq. (45)) is considered. The whole set of parameters is reduced in such a way that the model is still able to reproduce certain experimental data for human erythrocytes. In Eq. (45) the area parameters ai are assumed to be equal, that is, ai=a. For the phenomenological coefficients Eq. (53) is applied, where the coupling parameters are unspecific (λij=λ=va2). Simulations with a more detailed parameter specification are given in the next sections.
Fig. 1 shows time-dependent changes in the lipid concentration ratios Nic/Ni as well as the fraction of the total lipid concentration Nc/N in the cytoplasmic layer. The curves are obtained by numerical integration of the system equations, using the parameter values listed in Table 1. The initial state at t=0 is symmetrical, that is, Nic/Ni=1/2. For t<10,000min, active translocation for PS and PE takes place, whereas for t>10,000min this translocation is fully inhibited. In the first time interval the lipid concentrations tend toward a steady state, which shows a pronounced asymmetry between the two monolayers. After inhibition of the translocase at t=10,000min, the membrane relaxes to equilibrium, which is characterized by symmetric distributions of its components.
| | |
| Total lipid concentrations* (mM) | |
|---|
| NPS | 0.55 | |
| NPE | 1.12 | |
| NPC | 1.24 | |
| NSM | 1.04 | |
| NCh | 3.5 | |
| Aminophospholipid translocase‡ | | |
| lPS+=lPE+ | 2.0×104min−1 | |
| k+=k− | 4.0×104min−1 | |
| (KPSc)−1=(KPEc)−1 | 1.0×103mM | |
| (KPEe)−1 | 0.5mM | |
| (KPSe)−1 | 5.0mM | |
| T | 5.5×10−6mM | |
| Phenomenological parameters, molecular areas, and mechanical force constants | | |
| 4N RT λij | 1/670min−1§ | |
| 4 RT κPS | 1/600min−1#a | |
| 4 RT κPE | 1/600min−1#a | |
| 4 RT κPC | 1/1000min−1#b | |
| 4 RT κSM | 1/4000min−1# | |
| 4 RT κCh | 1min−1# | |
| γi/LA | 0.25Jm−2¶ | |
| ai | 0.6nm2¶ | |
| | |
The concentration ratios within the asymmetrical steady state and the corresponding experimental data are given in Table 2. It is seen that the model explains rather well the experimentally observed state of the membrane. In particular, it predicts an accumulation of PS and PE on the cytoplasmic layer and a compensating displacement of the other lipids to the external layer. Moreover, the model correctly describes SM as displaced to a greater extent than PC. Taking into account the parameter values from Table 1, it is observed that the slowest diffusing component SM reaches the most pronounced asymmetry, whereas cholesterol shows a nearly symmetrical distribution due to its fast passive translocation.
| | |
| | Lipid Concentration Ratios | |
|---|
| | Experimental Data | | |
|---|
| | a | b | c | d | e | Theoretical (Fig. 1) | |
|---|
| NPSc/NPS | 1.0 | 0.81 | 0.92 | 1.0 | — | 0.98 | |
| NPEc/NPE | 0.95 | — | 0.79 | 0.80 | 0.82 | 0.89 | |
| NPCc/NPC | 0.45 | 0.30 | 0.28 | 0.24 | 0.44 | 0.35 | |
| NSMc/NSM | 0.15 | — | 0.21 | 0.18 | — | 0.12 | |
| NChc/NCh | — | — | — | — | — | 0.47 | |
| | |
Fig. 1 and Table 2 show that within the whole time range, the sum of lipid concentrations is well balanced between the two monolayers (Nc≈Ne), despite strong changes in the concentrations of the individual membrane components. In an initial phase (for ∼100min) the inward translocation of PS and PE is mainly compensated by the displacement of cholesterol. Within the subsequent phase there are also significant displacements of PC and SM.
The results of the simulation depend crucially on the value of the coupling parameter λ. This is demonstrated in Fig. 2, where the normalized steady-state concentrations are plotted as functions of λ. Simulations are possible only for such values of λ, for which Lij is a positive definite matrix, that is, for λ>λc (see text below Eq. (53)). Using the shift λ → λ−λc allows for a logarithmic display of the abscissa in Fig. 2. It is observed that at λ=0 the steady-state ratios Nic/Ni are the same for those species that are translocated only passively. This property can be derived analytically by taking into account that for vanishing cross-couplings the steady-state condition for the species PC, SM, and Ch, that is, for i=3, 4, and 5, simplifies to Xientr+Ximech=0. Because for ai=a the mechanical forces Ximech are independent of i, it follows directly from Eq. (45) that the steady-state ratios Nic/Ni become species independent as well. Crossing the value λ=0 leads to a qualitative change in the lipid distribution. For λ<0 one obtains NChc/NCh<NPCc/NPC<NSMc/NSM, whereas λ>0 leads to the reverse situation, NSMc/NSM<NPCc/NPC<NChc/NCh. The
