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Dissipative spinodal condensation of membrane proteins
Volume 154, number 2
DISSIPATIVE
CHEMICAL
SPINODAL
PHYSICS LETTERS
CONDENSATION
13 January
OF MEMBRANE
1989
PROTEINS
Peter FROMHERZ Abteilung Biophysik der Universitdt Urn, D- 7900 Urn-Eselsberg, Federal Republic of Germany Received
1 August
1988; in final form 3 November
1988
A model of dissipative self-organisation of mobile charged channel proteins in membranes - far from electrochemical equilibrium - is presented. It is formulated by analogy with the condensation of a real gas and the decomposition of an alloy on the basis of a dissipative intermolecular potential. For the linear geometry of a cable the spinodal and the growth constants for periodic patterns are derived in terms of pertinent parameters for biological membranes.
1. Introduction
2. Dissipative
The “fluid mosaic model” - a fluid bimolecular layer of lipid with floating protein molecules - is a basic physical concept in modelling the structure of biological membranes [ I 1. The mobile protein molecules may form specific ion channels [Z] and may bear electrical charges [ 31. The electric current through a given distribution of channels affects the electric potential along the membrane. Gradients of the membrane potential affect the distribution of proteins by lateral electrophoresis. The coupled dynamics of the distribution of proteins and membrane potentials has been investigated [ 41. It has been shown that a structural instability of the homogeneous fluid mosaic of mobile charged channels may occur far from equilibrium. In the present paper dissipative self-organization of membrane protein is treated by the thermodynamical approach used to describe first-order conservative phase transitions. The lateral spread of ionic current around each molecular channel gives rise to a local electrical potential gradient. Other charged molecules thus experience a potential energy. On the basis of a pair potential of dissipative origin the threshold and the linear dynamics for self-organization are derived by analogy with the condensation of a real gas and the spinodal decomposition of an alloy. 146
intermolecular
potential
Consider a cylindrical cable of a membrane embedded in an electrolyte. The electrical features per unit length are described by an equivalent circuit (fig. 1) composed of a capacitance C, of the coat, an unspecific conductivity G, of the coat and a resistance R, of the core (neglecting the resistance of the bath). Instead of C,, GNI and RIit is convenient to use the time constant TM= CM/GM, the length constant &,= (R,GM)-"2and the input resistance
R,=(R,/G,)"'[5]. The electrical potential
hi
kH
vM(x, t) across the mem-
%H
Fig. 1. Structural and electrical features of a cylindrical cable comprising a coat of a lipid membrane with mobile charged channel proteins embedded in an electrolyte. Left: The protein molecules are characterized by the diffusion coefficient DC-, the conductance &, and the electrical charge qcH exposed to the core of the cable. Right; Each length element of the coat is characterized by an equivalent circuit. The top of the circuit refers to the core, the bottom to the bath. V, is the membrane potential. The components are the capacitance CM of the coat, the leak conductlvity Gw of the coat, the conductivity nCHNcH of the channels of density Ncn, the resistance R, of the core and the reversal potential E& of the channels. The conductivity A&V,-, is variable due to the lateral rearrangement of the channels.
0 009-2614/89/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 154, number 2
CHEMICAL PHYSICS LETTERS
brane as a function of space x and time t is governed by the balance of current densities according to Kelvin’s cable equation c
ah M--~!!$L+G~V~=Zinj. at ,
3. Dissipative condensation
) = Ee”“C”exp(-Ix-x’I/~~). ZGhlCM
(2)
The exponential polarization V$” is caused by the current flowing through the channel, spreading along the core and flowing back through the leaky coat. Its range is defined by the length constant &,+ A channel moving along the cable with trajectory x’ =x’ (t) is surrounded by a local polarization Vc,” (x-x’ (t)) if the motion is slow as compared to the electrical relaxation. The diffusive motion is characterized by the time constant rCu= &h /DC” with diffusion coefficient &u. The assumption of adiabatic polarization around a floating channel is valid for r,,.,< rcCH. Let each protein bear an electrical charge qCH exposed to the core of the cable. Every charged protein molecule at a distance r along the cable from another conductive protein molecule has a potential energy ticu(r)=(lcHv&H (r) as expressed by
%&
characterized by parameters such as the conductance AcZHand the charge qCH. (“actio-reactio” remains valid, of course, as the matrix is involved implicitly.)
(1)
Ilnj(X, t) is an injected current density. Consider an ion channel at a location x’ with Ohmic conductance A The injected current density is Ii”,(X) S;-, (P CH- V,) 6(x-x’ ). E&., is the reversal potential due to the ion concentration gradient. The stationary distribution of the membrane potential V&” (x-x’ ) around the channel is obtained from eq. ( 1). If the input resistance of the cable is low as compared to the resistance of the channel with R, -z=zAcA this local polarization is vc,” (x-x’
13 January 1989
ewd-r/L).
(3)
The potential energy of a molecule i in the field of molecule j is equal to the potential energy of molecule j in the field of molecule i as the molecules involved bear the same charge and conductance. It is an important aspect of this kind of “dissipative pair potential”, however, that this reciprocity does not hold automatically_ The mutual potential energy may be unequal for unequal proteins, as the creation of the electric field and the detection of the field are
Let us consider a “fluid mosaic” of mobile charged channels in a cable as an ensemble of particles interacting via the “pair potential” of eq. (3). Heuristically let us describe the ensemble thermodynamically by defining a coarse-grained free energy fcH per unit length. The average energy a,-u of a single channel in the field of all other channels with an average density NT,, is z&, =NcHJCLXuCH(x). The energy z+u with respect to one selected neighbour may be expressed by eq. (3 ) if the electrical features of the cable are not perturbed significantly by the presence of the channels, i.e. if the number &u of channels is low (.4 eHNeH c GM). Integration yields ti,-u = Ncuqcu xE&&,/G,. Assigning an energy per unit length of ~tZcHflcH [ 61 and an ideal entropy contribution (regular solution) [ 71 the free energy fCHis expressed up to a constant by fCH=kBT(&
ln&,+B,,N~,)
kBT is the thermal coefficient, BcH=~c~E~H~cHI~~sTGw.
.
(4)
energy, BCH is the second virial
(5)
The spinodal is defined by the condition a2fcH/ a%, i 0. According to eq. (4) the stability of the fluid mosaic is controlled by the mean energy per channel &-,,/k,T=2B,&,. The condensation condition is 2&&,-u i - 1 as expressed explicitly by
Condensation requires a negative virial coefficient, i.e. an opposite sign for the charge q,-_, and the reversal potential E$u .
147
4. Dissipative
13 January 1989
CHEMICALPHYSICS LETTERS
Volume 154, number 2 spinodal patterns
a4NCH ---2GHaX4
Heuristically let us describe an inhomogeneous fluid mosaic using the concept of spinodal decomposition [ 8- 10 1. The total free energy FCHdepends on the local free energy&,(x) per unit length as determined by the local channel density NcH = NCH(x) in analogy to eq. (4) and the contribution of the channel density gradients weighted by the coefficient rccH according to (7) The mean energy of a channel at location x is ticH(x)=JdYNcH(~‘) ucH(x--x’). Expansion of NcH (x’ ) around x to second order leads to a-H = 2kB TBcH the total free
(8) Comparison with eq. (7) - by partial integration defines the coefficient KCHof the energy gradient according to
-keTBci&
.
akn
according to generalized diffusion. With the conservation of protein molecules as expressed by aNcH/ at= - u&ax, the linear dynamics of the channel distribution is described by 148
(%,I
)’ (11)
-
In the limit of long waves (K-0) spontaneous growth sets in at 2B.-,&,< - 1. This threshold agrees with the spinodal as derived by the static approach (eq. (6 ) ). Above the threshold a finite band of modes grows spontaneously. The wavenumber of fastest growth Km”” and its growth constant kr are determined by
(12)
up to linear terms [ 9 1. The protein current along the cable is &H&H k,TF
+~BCH%H(K~M)~
~&-I&H)
(9)
The gradient of the chemical potential along the cable governs the dynamics of phase separation [ 9, lo]. The chemical potential L(CH(x) is obtained from FCH by variation as
-&H=-
NcH is the average channel density. The curvature aZfcH/SV& and the weight ~~~ of the energy gradient have to be substituted (cf. eqs. (4) and (9)). Let us consider the dynamics of the normal modes of NCH(x, t) with respect to the average flcH as NCH-~cH=N~H(t)~~~k5 with wavenumbers K and amplitudes N&. These modes may refer to a finite cable of length IM with tight ends, i.e. with aNcH /ax=0 at x=0, 1,. In that case the allowed modes are K=nn/l, with n= 1, 2, ... _Insertion into eq. (10) leads to linear rate equations dN&/dt = kKN& for the amplitudes. The growth constants kK are defined by kKtcH = - ( I +
With an ideal local entropy contribution energy FCH is expressed by
KCH=
(10)
kyvcH
= -2&,&H(Km”“~M)4
_
(13)
At a strongly negative value of 2BcHNcH, i.e. at a high density of channels and an effective attraction, a fast assembly of highly structured patterns is promoted. Applied to modes in a finite cable eqs. ( 11 )- ( 13) are approximations which are strictly valid only for (IM/rc&,,)2>~ 1 as the local polarization around channels within a range CMfrom the ends differs from the exponential law of eq. (2 ) . In a finite cable the threshold refers to growth of the lowest mode of wavenumber K=x/l,. The effective spinodal is controlled by the length constant CM,i.e. by the length resistance RI according to
Volume 154, number 2
CHEMICAL PHYSICS LETTERS
5. Discussion A fluid mosaic of charged channel proteins in a membrane exhibits an instability with respect to spatial patterns. The instability is driven by the salt concentration gradient across the membrane as expressed by the reversal potential of the channels. The instability occurs far from electrochemical equilibrium. The features of this dissipative process correspond in all details to the features of a first-order conservative phase transition such as in a real gas or a metallic alloy. Two levels of the dissipative process are to be distinguished: (i) The flow of ions across the membrane which gives rise to attractive intermolecular potential energies. (ii) The assembly of proteins in the membrane which is amenable to thermodynamics. The spinodal and growth constants of intraspinodal patterns are in agreement with the results of the phenomenological dynamics of channel density and membrane potential within the limits of the present approach, i.e. in the linear regime at low channel density in a long cable [ 41. This agreement justifies the heuristic thermodynamic treatment of a dissipative system a posteriori. The particles involved are defined by their conductance &,, their charge qcH and their diffusion coefficient DCH. These parameters are commonly used to characterize membrane proteins [ 2,3]. Pattern formation is controlled at constant temperature: On the dissipative level by the reversal potential E&, on the thermodynamic level by the average density ii& of charged channels and by the length &,, of the cable. These parameters play a preeminant role in living cells: (i) Gradients of salt concentration which determine the reversal potential may be adjusted by metabolism through ion pumps. (ii ) The density of charged channels may be modulated by chemical agents creating and annihilating conductance or charge. (iii) The size of cells and organelles may change by growth. Consider a numerical example. The proteins are defined by .4 ,-- = 10 pS [ 2 1, qCH= 5~ (e. is the eiementary charge) [ll], Do=1 pm*/s [3] and E& = - 100 mV. The cable is defined by C,=O.l pF/um, GM= 1 nS/pm and RI= 0.1 Ma/urn, i.e. by rM=O.l ms, &=lOO pm and R-=10 MR. Such
I3 January 1989
conditions may be realized e.g. by a cable of radius 2 pm at a specific conductivity of 1 uF/cm’ and a specific conductivity of 0.0 1 S/cm2 for the coat and a specific resistance of r, = 100 R cm for the core [ 6 1. The ratio .4,&/R, = 1O4 is sufficiently large such that the approximation of eq. (2 ) for the local polarization is valid. The ratio rccn/rM= lo8 is sufficiently large such that the concept of an adiabatic intermolecular potential is justified. The amplitude of local depolarization by a single channel (eq. (2) > is 5 uV leading to an amplitude for the intermolecular potential (eq. (3) ) of the order of 1O- 3kBT. The low energy justifies the mean field approach [ 71. Using these numbers the second virial coefficient (es. (5)) is Be”=-0.1 Pm. The threshold of the spinodal 2BCHNcH= - I (es. (7) ) is reached at a channel density of Nc, = 5 urn- ‘. It may be surprising that a condensation process is triggered at all in a system described by an intermolecular energy which is as low as k,T/ 1000. However, in the second virial coefficient the weak amplitude of interaction is compensated by the extreme range &,, of interaction. The cooperativity in the system goes far beyond a next-neighbour interaction. In the case of a finite cable of lM= 100 urn the threshold of the lowest mode is enhanced up to NcH=50um-’ (eq. (lS)).Inacableofradius2um this value corresponds to a density of 5 molecules/ urn*. Such a value is at the lower limit of channel densities observed [ 21. A dissipative spinodal condensation of channels is within the capabilities of a living cell.
6. Summary The concepts of thermodynamics and spinodal decomposition are applied to describe the dissipative self-organization of proteins in a membrane, introducing the notion of a dissipative pair potential. Threshold and growth constants are derived for the one-dimensional geometry of a cable. Generalization to two-dimensional systems is straightforward. Spatial order by dissipative long range interaction in a fluid mosaic can be distinguished from a structural transition in a compact lattice of proteins by conservative nearest-neighbour interaction [ 12 1. 149
Volume 154,number 2
CHEMICALPHYSICSLETTERS
A particular feature of a dissipative intermolecular energy is its dependence on two “charges” coupling the particles and the mediating field: The conductance ACK gives rise to an electrical field, the charge qCHdetects the field. In a one-Component system the macroscopic effects are similar to those in conservative systems as shown in the present paper. In a twocomponent system, however, the forces between different dissipative particles are no longer reciprocal. Novel phases may appear as oscillatory and solitary waves [ 131.
References [ 1] S.J. Singer and G.L. Nicolson, Science 175 ( 1972) 720.
150
13January 1989
[2] B. HiIle, Ionic channels of exitable membranes
(Sinauer, Sunderland, 1984). [ 31 M. McCloskey and M. Poo, Int. Rev. Cytol. 87 ( 1985 ) 19. [4] P. Fromherz, Proc. Natl. Acad. Sci. US 85 (1988) 6353. [ 51 B. Katz, Nerve, muscle and synapse (McGraw-Hill, New
York, 1966). [6] F. Reif, Fundamentals of statistics and thermal physics (McGraw-Hill,New York, 1965). [7] C. Wagner, Thermodynamic of alloys (Addison-Wesley, Reading, 1952). [8] J.W. Cahn andJ.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [9] J.W. Cahn, ActaMet. 9 (1961) 795. [lo] J.W. Cahn, Trans. Metal. Sot. AIME 242 (1968) 166. [ 111 T.A. Ryan, I. Myers, D. Holowka, B. Bairds and W.W. Webb, Science 289 (1988) 61. [ 121 J.P. Changeux, J. Thi&y, Y. Tung and C. Kittel, Proc. Natl. Acad. Sci. US 57 (1967) 335. [ 131 P. Fromherz, Biochim. Biophys. Acta 944 (1988) 108.
DISSIPATIVE
CHEMICAL
SPINODAL
PHYSICS LETTERS
CONDENSATION
13 January
OF MEMBRANE
1989
PROTEINS
Peter FROMHERZ Abteilung Biophysik der Universitdt Urn, D- 7900 Urn-Eselsberg, Federal Republic of Germany Received
1 August
1988; in final form 3 November
1988
A model of dissipative self-organisation of mobile charged channel proteins in membranes - far from electrochemical equilibrium - is presented. It is formulated by analogy with the condensation of a real gas and the decomposition of an alloy on the basis of a dissipative intermolecular potential. For the linear geometry of a cable the spinodal and the growth constants for periodic patterns are derived in terms of pertinent parameters for biological membranes.
1. Introduction
2. Dissipative
The “fluid mosaic model” - a fluid bimolecular layer of lipid with floating protein molecules - is a basic physical concept in modelling the structure of biological membranes [ I 1. The mobile protein molecules may form specific ion channels [Z] and may bear electrical charges [ 31. The electric current through a given distribution of channels affects the electric potential along the membrane. Gradients of the membrane potential affect the distribution of proteins by lateral electrophoresis. The coupled dynamics of the distribution of proteins and membrane potentials has been investigated [ 41. It has been shown that a structural instability of the homogeneous fluid mosaic of mobile charged channels may occur far from equilibrium. In the present paper dissipative self-organization of membrane protein is treated by the thermodynamical approach used to describe first-order conservative phase transitions. The lateral spread of ionic current around each molecular channel gives rise to a local electrical potential gradient. Other charged molecules thus experience a potential energy. On the basis of a pair potential of dissipative origin the threshold and the linear dynamics for self-organization are derived by analogy with the condensation of a real gas and the spinodal decomposition of an alloy. 146
intermolecular
potential
Consider a cylindrical cable of a membrane embedded in an electrolyte. The electrical features per unit length are described by an equivalent circuit (fig. 1) composed of a capacitance C, of the coat, an unspecific conductivity G, of the coat and a resistance R, of the core (neglecting the resistance of the bath). Instead of C,, GNI and RIit is convenient to use the time constant TM= CM/GM, the length constant &,= (R,GM)-"2and the input resistance
R,=(R,/G,)"'[5]. The electrical potential
hi
kH
vM(x, t) across the mem-
%H
Fig. 1. Structural and electrical features of a cylindrical cable comprising a coat of a lipid membrane with mobile charged channel proteins embedded in an electrolyte. Left: The protein molecules are characterized by the diffusion coefficient DC-, the conductance &, and the electrical charge qcH exposed to the core of the cable. Right; Each length element of the coat is characterized by an equivalent circuit. The top of the circuit refers to the core, the bottom to the bath. V, is the membrane potential. The components are the capacitance CM of the coat, the leak conductlvity Gw of the coat, the conductivity nCHNcH of the channels of density Ncn, the resistance R, of the core and the reversal potential E& of the channels. The conductivity A&V,-, is variable due to the lateral rearrangement of the channels.
0 009-2614/89/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 154, number 2
CHEMICAL PHYSICS LETTERS
brane as a function of space x and time t is governed by the balance of current densities according to Kelvin’s cable equation c
ah M--~!!$L+G~V~=Zinj. at ,
3. Dissipative condensation
) = Ee”“C”exp(-Ix-x’I/~~). ZGhlCM
(2)
The exponential polarization V$” is caused by the current flowing through the channel, spreading along the core and flowing back through the leaky coat. Its range is defined by the length constant &,+ A channel moving along the cable with trajectory x’ =x’ (t) is surrounded by a local polarization Vc,” (x-x’ (t)) if the motion is slow as compared to the electrical relaxation. The diffusive motion is characterized by the time constant rCu= &h /DC” with diffusion coefficient &u. The assumption of adiabatic polarization around a floating channel is valid for r,,.,< rcCH. Let each protein bear an electrical charge qCH exposed to the core of the cable. Every charged protein molecule at a distance r along the cable from another conductive protein molecule has a potential energy ticu(r)=(lcHv&H (r) as expressed by
%&
characterized by parameters such as the conductance AcZHand the charge qCH. (“actio-reactio” remains valid, of course, as the matrix is involved implicitly.)
(1)
Ilnj(X, t) is an injected current density. Consider an ion channel at a location x’ with Ohmic conductance A The injected current density is Ii”,(X) S;-, (P CH- V,) 6(x-x’ ). E&., is the reversal potential due to the ion concentration gradient. The stationary distribution of the membrane potential V&” (x-x’ ) around the channel is obtained from eq. ( 1). If the input resistance of the cable is low as compared to the resistance of the channel with R, -z=zAcA this local polarization is vc,” (x-x’
13 January 1989
ewd-r/L).
(3)
The potential energy of a molecule i in the field of molecule j is equal to the potential energy of molecule j in the field of molecule i as the molecules involved bear the same charge and conductance. It is an important aspect of this kind of “dissipative pair potential”, however, that this reciprocity does not hold automatically_ The mutual potential energy may be unequal for unequal proteins, as the creation of the electric field and the detection of the field are
Let us consider a “fluid mosaic” of mobile charged channels in a cable as an ensemble of particles interacting via the “pair potential” of eq. (3). Heuristically let us describe the ensemble thermodynamically by defining a coarse-grained free energy fcH per unit length. The average energy a,-u of a single channel in the field of all other channels with an average density NT,, is z&, =NcHJCLXuCH(x). The energy z+u with respect to one selected neighbour may be expressed by eq. (3 ) if the electrical features of the cable are not perturbed significantly by the presence of the channels, i.e. if the number &u of channels is low (.4 eHNeH c GM). Integration yields ti,-u = Ncuqcu xE&&,/G,. Assigning an energy per unit length of ~tZcHflcH [ 61 and an ideal entropy contribution (regular solution) [ 71 the free energy fCHis expressed up to a constant by fCH=kBT(&
ln&,+B,,N~,)
kBT is the thermal coefficient, BcH=~c~E~H~cHI~~sTGw.
.
(4)
energy, BCH is the second virial
(5)
The spinodal is defined by the condition a2fcH/ a%, i 0. According to eq. (4) the stability of the fluid mosaic is controlled by the mean energy per channel &-,,/k,T=2B,&,. The condensation condition is 2&&,-u i - 1 as expressed explicitly by
Condensation requires a negative virial coefficient, i.e. an opposite sign for the charge q,-_, and the reversal potential E$u .
147
4. Dissipative
13 January 1989
CHEMICALPHYSICS LETTERS
Volume 154, number 2 spinodal patterns
a4NCH ---2GHaX4
Heuristically let us describe an inhomogeneous fluid mosaic using the concept of spinodal decomposition [ 8- 10 1. The total free energy FCHdepends on the local free energy&,(x) per unit length as determined by the local channel density NcH = NCH(x) in analogy to eq. (4) and the contribution of the channel density gradients weighted by the coefficient rccH according to (7) The mean energy of a channel at location x is ticH(x)=JdYNcH(~‘) ucH(x--x’). Expansion of NcH (x’ ) around x to second order leads to a-H = 2kB TBcH the total free
(8) Comparison with eq. (7) - by partial integration defines the coefficient KCHof the energy gradient according to
-keTBci&
.
akn
according to generalized diffusion. With the conservation of protein molecules as expressed by aNcH/ at= - u&ax, the linear dynamics of the channel distribution is described by 148
(%,I
)’ (11)
-
In the limit of long waves (K-0) spontaneous growth sets in at 2B.-,&,< - 1. This threshold agrees with the spinodal as derived by the static approach (eq. (6 ) ). Above the threshold a finite band of modes grows spontaneously. The wavenumber of fastest growth Km”” and its growth constant kr are determined by
(12)
up to linear terms [ 9 1. The protein current along the cable is &H&H k,TF
+~BCH%H(K~M)~
~&-I&H)
(9)
The gradient of the chemical potential along the cable governs the dynamics of phase separation [ 9, lo]. The chemical potential L(CH(x) is obtained from FCH by variation as
-&H=-
NcH is the average channel density. The curvature aZfcH/SV& and the weight ~~~ of the energy gradient have to be substituted (cf. eqs. (4) and (9)). Let us consider the dynamics of the normal modes of NCH(x, t) with respect to the average flcH as NCH-~cH=N~H(t)~~~k5 with wavenumbers K and amplitudes N&. These modes may refer to a finite cable of length IM with tight ends, i.e. with aNcH /ax=0 at x=0, 1,. In that case the allowed modes are K=nn/l, with n= 1, 2, ... _Insertion into eq. (10) leads to linear rate equations dN&/dt = kKN& for the amplitudes. The growth constants kK are defined by kKtcH = - ( I +
With an ideal local entropy contribution energy FCH is expressed by
KCH=
(10)
kyvcH
= -2&,&H(Km”“~M)4
_
(13)
At a strongly negative value of 2BcHNcH, i.e. at a high density of channels and an effective attraction, a fast assembly of highly structured patterns is promoted. Applied to modes in a finite cable eqs. ( 11 )- ( 13) are approximations which are strictly valid only for (IM/rc&,,)2>~ 1 as the local polarization around channels within a range CMfrom the ends differs from the exponential law of eq. (2 ) . In a finite cable the threshold refers to growth of the lowest mode of wavenumber K=x/l,. The effective spinodal is controlled by the length constant CM,i.e. by the length resistance RI according to
Volume 154, number 2
CHEMICAL PHYSICS LETTERS
5. Discussion A fluid mosaic of charged channel proteins in a membrane exhibits an instability with respect to spatial patterns. The instability is driven by the salt concentration gradient across the membrane as expressed by the reversal potential of the channels. The instability occurs far from electrochemical equilibrium. The features of this dissipative process correspond in all details to the features of a first-order conservative phase transition such as in a real gas or a metallic alloy. Two levels of the dissipative process are to be distinguished: (i) The flow of ions across the membrane which gives rise to attractive intermolecular potential energies. (ii) The assembly of proteins in the membrane which is amenable to thermodynamics. The spinodal and growth constants of intraspinodal patterns are in agreement with the results of the phenomenological dynamics of channel density and membrane potential within the limits of the present approach, i.e. in the linear regime at low channel density in a long cable [ 41. This agreement justifies the heuristic thermodynamic treatment of a dissipative system a posteriori. The particles involved are defined by their conductance &,, their charge qcH and their diffusion coefficient DCH. These parameters are commonly used to characterize membrane proteins [ 2,3]. Pattern formation is controlled at constant temperature: On the dissipative level by the reversal potential E&, on the thermodynamic level by the average density ii& of charged channels and by the length &,, of the cable. These parameters play a preeminant role in living cells: (i) Gradients of salt concentration which determine the reversal potential may be adjusted by metabolism through ion pumps. (ii ) The density of charged channels may be modulated by chemical agents creating and annihilating conductance or charge. (iii) The size of cells and organelles may change by growth. Consider a numerical example. The proteins are defined by .4 ,-- = 10 pS [ 2 1, qCH= 5~ (e. is the eiementary charge) [ll], Do=1 pm*/s [3] and E& = - 100 mV. The cable is defined by C,=O.l pF/um, GM= 1 nS/pm and RI= 0.1 Ma/urn, i.e. by rM=O.l ms, &=lOO pm and R-=10 MR. Such
I3 January 1989
conditions may be realized e.g. by a cable of radius 2 pm at a specific conductivity of 1 uF/cm’ and a specific conductivity of 0.0 1 S/cm2 for the coat and a specific resistance of r, = 100 R cm for the core [ 6 1. The ratio .4,&/R, = 1O4 is sufficiently large such that the approximation of eq. (2 ) for the local polarization is valid. The ratio rccn/rM= lo8 is sufficiently large such that the concept of an adiabatic intermolecular potential is justified. The amplitude of local depolarization by a single channel (eq. (2) > is 5 uV leading to an amplitude for the intermolecular potential (eq. (3) ) of the order of 1O- 3kBT. The low energy justifies the mean field approach [ 71. Using these numbers the second virial coefficient (es. (5)) is Be”=-0.1 Pm. The threshold of the spinodal 2BCHNcH= - I (es. (7) ) is reached at a channel density of Nc, = 5 urn- ‘. It may be surprising that a condensation process is triggered at all in a system described by an intermolecular energy which is as low as k,T/ 1000. However, in the second virial coefficient the weak amplitude of interaction is compensated by the extreme range &,, of interaction. The cooperativity in the system goes far beyond a next-neighbour interaction. In the case of a finite cable of lM= 100 urn the threshold of the lowest mode is enhanced up to NcH=50um-’ (eq. (lS)).Inacableofradius2um this value corresponds to a density of 5 molecules/ urn*. Such a value is at the lower limit of channel densities observed [ 21. A dissipative spinodal condensation of channels is within the capabilities of a living cell.
6. Summary The concepts of thermodynamics and spinodal decomposition are applied to describe the dissipative self-organization of proteins in a membrane, introducing the notion of a dissipative pair potential. Threshold and growth constants are derived for the one-dimensional geometry of a cable. Generalization to two-dimensional systems is straightforward. Spatial order by dissipative long range interaction in a fluid mosaic can be distinguished from a structural transition in a compact lattice of proteins by conservative nearest-neighbour interaction [ 12 1. 149
Volume 154,number 2
CHEMICALPHYSICSLETTERS
A particular feature of a dissipative intermolecular energy is its dependence on two “charges” coupling the particles and the mediating field: The conductance ACK gives rise to an electrical field, the charge qCHdetects the field. In a one-Component system the macroscopic effects are similar to those in conservative systems as shown in the present paper. In a twocomponent system, however, the forces between different dissipative particles are no longer reciprocal. Novel phases may appear as oscillatory and solitary waves [ 131.
References [ 1] S.J. Singer and G.L. Nicolson, Science 175 ( 1972) 720.
150
13January 1989
[2] B. HiIle, Ionic channels of exitable membranes
(Sinauer, Sunderland, 1984). [ 31 M. McCloskey and M. Poo, Int. Rev. Cytol. 87 ( 1985 ) 19. [4] P. Fromherz, Proc. Natl. Acad. Sci. US 85 (1988) 6353. [ 51 B. Katz, Nerve, muscle and synapse (McGraw-Hill, New
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