INTRODUCTION

Top Abstract INTRODUCTION BASIC MODEL ASSUMPTIONS PHENOMENOLOGICAL FORCES OF THE... PHENOMENOLOGICAL COEFFICIENTS... SIMULATIONS DISCUSSION References

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., 1973; 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., 1988; Schroeder et al., 1991), 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., 1993; Heinrich et al., 1997) 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. (1993), 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. (1997) 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 ek, 1989; Seifert et al., 1991; Heinrich et al., 1993). 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

Top Abstract INTRODUCTION BASIC MODEL ASSUMPTIONS PHENOMENOLOGICAL FORCES OF THE... PHENOMENOLOGICAL COEFFICIENTS... SIMULATIONS DISCUSSION References

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 N_i^c and N_i^e. They are related to the differences n_i = N_i^c  N_i^e and to the total amounts of lipids N_i = N_i^c + N_i^e as follows:

(1)

The time-dependent changes of the composition of the monolayers are governed by the differential equations

(2)

where J_i^act and J_i^pass 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 N_i 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 J_i^act we use an ATP-dependent carrier mechanism as described previously (Heinrich et al., 1997; 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,

(3)

where X_j denotes thermodynamic forces. The coefficients L_ij 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 X_j 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 N_i^c or N_i^e 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

Top Abstract INTRODUCTION BASIC MODEL ASSUMPTIONS PHENOMENOLOGICAL FORCES OF THE... PHENOMENOLOGICAL COEFFICIENTS... SIMULATIONS DISCUSSION References

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 = k_BT ln Z, where k_B 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 _i^c = L_AN_i^c and _i^e = L_AN_i^e of species i on either monolayer, e or c, respectively. L_A denotes Avogadro's number.

The configurational partition function of the bilayer reads

(4)

Z takes into account the number of microscopic distributions of _i molecules of each lipid species among the monolayers with a lipid number _i^c and _i^e on the two layers.

The entropic contribution to the total free energy (in Joules) of a single membrane reads

(6)

Using Stirling's formula, ln x!  x ln x  x for x 1, one derives from Eqs. 4-6

(7)

It is easy to see that at variations of _j^c and _j^e under the constraint _j = _j^c + _j^e = 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

(8)

Nonequilibrium states are characterized by the variables

(9)

which are related to the differences n_i = N_i^c  N_i^e by n_i = 2y_i. Defining entropic forces in units of J/mol as

(10)

one derives

(11)

This formula for the entropic force shows some correspondence to an expression used in a previous study (Heinrich et al., 1997). 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 (n_i N_i) the entropic forces may be expressed in a linear approximation as

(12)

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

(13)

where a_i and _i^c,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. _i 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 _i^c,e of areas will depend on the deviations n_i 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,

(14)

where A₀^c,e are the total equilibrium surfaces of the monolayers. In nonequilibrium states the two areas may be expressed as

(15)

The term a_j(1 + _j^c,e) represents the compressed ( < 0) or expanded ( > 0) area of lipid j in layer c and e, respectively. From Eq. 15 it follows with condition 14 that

(16)

In this equation it is assumed that A₀^c = A₀^e = 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., 1997) relation 16 was used to calculate the relative area changes under the assumption _j^c = _j^e = , that is, they are identical for all lipids. Such a simplification may not be justified for lipid species of different equilibrium areas a_i and of different force constants _i. 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 v^c and v^e. With

(17)

the lateral elastic energy of the bilayer has an extremum if

(18a,b)

From Eqs. 13, 15, 17, and 18 one obtains

(19)

which under consideration of Eq. 1 yields

(20)

By taking into account the constraints in 16 in Eq. 20, one derives

(21)

For sufficiently small deviations from equilibrium, the linear approximation

(22)

is valid. The total mechanical energy E = E^c + E^e, obtained by introducing relation 22 into Eq. 13, reads

(23)

where terms higher than second order are neglected.

The mechanical forces are defined as

(24)

(see the definition of the entropic forces in Eq. 10). From Eqs. 23 and 24 one obtains with n_i = 2y_i

(25)

It is worth mentioning that in the special case of unspecific parameters (_i = , a_i = a), Eqs. 22 and 25 reduce to previously derived expressions (cf. Heinrich et al., 1997). 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  value. Equation 25 reveals that forces are related pairwise by X_i^mech/X_j^mech = a_i/a_j. Furthermore, all forces are proportional to

(26)

which may be considered as the area difference of uncoupled monolayers.

In the following we combine the entropic force and the mechanical force as the total force of passive translocation:

(27)

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 (n_i = 0).

	PHENOMENOLOGICAL COEFFICIENTS FOR PASSIVE TRANSLOCATIONS

Top Abstract INTRODUCTION BASIC MODEL ASSUMPTIONS PHENOMENOLOGICAL FORCES OF THE... PHENOMENOLOGICAL COEFFICIENTS... SIMULATIONS DISCUSSION References

Parameterization with respect to lipid amounts

In Eq. 3 the phenomenological coefficients L_ij are unknown. A certain knowledge concerning the dependence of L_ij 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 N_a and N_b and the deviations from equilibrium n_a and n_b are related to the corresponding quantities of species k as follows:

(28a,b)

(28c,d)

The parameter  that quantifies the decomposition of species k is confined by 1/2    1/2. It follows directly from Eqs. 1, 12, and 25 that the forces of species a and b are the same as the force of species k:

(29)

For fixed k, the linear flux-force relations in Eq. 3 may be rewritten for the original system as

(30a)

(30b)

Characterizing the quantities of the decomposed system by the superscript * gives

(31a)

(31b)

Obviously, the macroscopic properties of the membrane, particularly the passive translocation fluxes, should be invariant with respect to the decomposition in Eq. 28, that is,

(32a,b)

The deviations n_i from equilibrium and, therefore, the forces X_i may vary independently. Thus, taking into account Eqs. 30 and 31 in Eq. 32, and using Onsager's reciprocity relation L_ij = L_ji, one derives that the phenomenological coefficients have to fulfill the relations

(33a)

(33b)

(33c)

To derive dependencies of the phenomenological coefficients on the lipid amounts, the following ansatz is used:

(34a)

(34b)

The parameters _ij and ^*_ij, which must be components of a symmetrical matrix, do not contain any further factors of N_i and N_j, but possibly factors N_l with l  i, j. Constraints on further relations for the dependencies of the parameters _ij 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,b into Eq. 33b gives, together with the decomposition 28a,b,

(35)

Because species a, b, and k have the same physical properties, their parameters  will be identical:

(36)

Furthermore, Eq. 35 must be valid for any decomposition of species k, that is, the left-hand side should be independent of . For x = y, Eq. 35 reads, considering Eq. 36,

(37a)

where z = x = y. This equation is independent of  only for z = 1. For x  y two separate conditions are obtained:

(37b,c)

since Eq. 35 holds for arbitrary values of both amounts N_i and N_k. Obviously these two equations (37b,c) cannot be fulfilled simultaneously for x  y and arbitrary . 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

(38)

To find expressions for the diagonal elements L_kk, we use relation 34a,b in Eq. 33c and take into account Eq. 38. This yields

(39)

Because ^*_aa = ^*_bb = _kk, Eq. 39 gives

(40)

where z = x + y. For z there are two solutions such that Eq. 40 is independent of . The first solution reads z = 1, with ^*_ab = 0, and arbitrary values of _kk denoted in the following _k. The second solution reads z = 2, for which Eq. 40 holds independent of  only with _kk = ^*_ab. Linear combination of these two solutions yields for the diagonal elements

(41)

This expression shows that the diagonal elements L_kk are composed of a diffusion term _kN_k and a self-coupling term _kkN_k².

Combining relations 38 and 41, one obtains the following general expression for the phenomenological coefficients:

(42)

Concerning the parameters entering the phenomenological coefficients in Eq. 42, we use the notation diffusion parameters _i, self-coupling parameters _ii, and cross-coupling parameters _ij for i  j.

Under the assumption of a lateral homogeneous membrane, two general properties of the diffusion parameters _j and coupling parameters _ij that enter Eq. 42 can be derived. For any subsection of the bilayer characterized by N'_k = N_k and n'_k = n_k for k = 1, ... , s with a scaling parameter , confined to 0 <   1, the following relations should be fulfilled:

(43a)

(43b)

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 _i is homogeneous of degree zero and _ij of degree 1, respectively. Accordingly, one derives from Euler's theorem on homogeneous functions the following conditions:

(44a,b)

In the most simple case, Eq. 44a is fulfilled if all diffusion parameters are independent of the lipid amounts, whereas Eq. 44b 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

(45)

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 _ij in Eq. 42 may be related to the equilibrium surface areas a_i 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

(46a,b)

Equation 46a entails that the entropic force in Eq. 12 of species k vanishes, whereas Eq. 46b 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 and 12, in the vicinity of equilibrium,

(47)

Taking into account the general structure for the nondiagonal elements L_kj given in Eq. 42, one derives from Eq. 47 the following condition for the cross-coupling parameters:

(48a)

Equation 48a states that a vector ^k = (_k1 ··· _{k,k1 k,k+1} ··· _ks)^T is orthogonal to any vector of perturbations ^k = (n₁ ··· n_k1 n_k+1 ··· n_s)^T, which in vector notation reads

(48b)

The space of perturbations ^k restricted by Eqs. 46a,b is of dimension s  2. Accordingly, this space is spanned by s  2 vectors ^(i,k) (i = 1, ... , s  2). In the case k = s, for example, a special choice of these vectors reads

(49)

Generally, the elements of the vector ^(i,s) are given by b_j^(i,s) = 0 for 1  j  i  1 and i + 2  j  s  1, and b_i^(i,s) = a_i+1, b_i+1^(i,s) = a_i. Similar representations are obtained for k  s.

The condition 48b of orthogonality is fulfilled if ^k is orthogonal to all corresponding vectors ^(i,k), which yields

(50)

It can be shown that there is one parameter v_k, such that Eq. 50 is fulfilled with

(51)

For k = s this result may be easily verified by using Eq. 49. The symmetry relations _kj = _jk require v_ka_j = v_ja_k, which leads to

(52)

The combination of Eqs. 42 and 52 gives

(53)

According to this equation, the s(s + 1)/2 phenomenological coefficients L_ij are fixed by only s + 1 phenomenological parameters v and _j. Relation 53 may be used also for the case of equal lipid area parameters, that is, for a_j = a.

The requirement for the matrix L_ij 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 L_ii are positive. Thus there is a lower limit for v depending on the values of _i > 0 and a_i > 0.

Phenomenological coefficients as derived from kinetic models

On the phenomenological level of description, the parameters _j and _ij 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

(54)

(55)

(56)

(57a,b)

(58a,b)

(59)

(60)

(61)

(62)

(63a,b)

(64)

(65)