A theoretical basis for self-electrophoresis

J. theor. Biol. (1981) 88, 599-630

A Theoretical RAIMA

LARTER~

Department (Received

Basis for Self-electrophoresis AND

PETER

ORTOLEVA

of Chemistry, Indiana University, Indiana 47405, U.S.A.

29 November

Bloomington,

1979, and in revised form 3 April

1980)

Electrical effects are known to have a profound influence on morphogenesis. Experiments on the egg of MUCUS have pointed to the establishment of a transcellular electrical current, “self-electrophoresis,” as an important step in the intracellular localization processes which seem to control asymmetric differentiation of the egg. We treat the transformation to the self-electrophoretic state as an example of spontaneous pattern formation brought about by a symmetry breaking instability in the state of symmetrically distributed membrane potential. We show that these selfsustaining gradients can exist in systems without inherent asymmetry, and are proposed as an example of Turing’s physico-chemical morphogenesis hypothesis. Starting from the continuity and Poisson’s equations and using a multiple scaling technique we derive a set of differential equations for the description of macroscopic biological phenomena. These equations comprise a theory that provides the first self-consistent means of predicting a selfelectrophoretic state. The differential equations are shown to possess solutions typical of non-linear systems; for example, asymmetric solutions in a simple symmetric model system. The existence of self-electrophoretic (asymmetric) states is demonstrated for physiologically reasonable transport coefficients, concentrations and cellular dimensions.

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lations and chemical waves, see Eyring (1978). Nicolis & Prigogine ( 1977 1 and Prigogine & Rice (1978). Theoretical studies of cell models have demonstrated that the non-linear properties of chemical reaction and membrane transport may lead to multiple steady states and cellular oscillations (Rashevsky, 1960; Hahn, Ortoleva & Ross, 1973). These model calculations are valuable because they demonstrate that a variety of biological phenomena can have mechanisms other than direct gene control. The problem of asymmetric cell differentiation has been considered using a simple two-box model to simulate intracellular transport and reaction (Ortoleva & Ross, 1973). It was shown using a model reaction mechanism that the symmetric state (both boxes having the same concentrations) could become unstable to asymmetric concentration perturbations, and evolve to a new asymmetric state (two boxes sustained at different concentrations) in an external environment without imposed gradients. In the above studies only neutral species were considered. Here we derive basic equations of reaction and transport which allow for the full electrochemical nature of cellular phenomena. Certainly, the presence of ionic species and membrane potentials is expected to play a non-trivial role in cell dynamics. The importance of membrane potentials is obvious in neuromuscular physiological systems, but, in fact, membrane potentials have been measured in a large variety of cells (from single-ceil organisms to fully differentiated mammalian cells). The presence of membrane potentials, as will be shown later, imposes a subtle complicating factor on the derivation of the descriptive equations of a biological system. A particularly interesting case is the asymmetric differentiation of the spherical egg of the seaweed FUCUS (Jaffe, Robinson & Nuccitelli, 1974). Careful experiments have shown that, at a critical stage in development, the egg makes a transition from a spherically symmetric state of the distribution of membrane potential to a state of overall polarization with a net transcellular current leaving one pole and entering the other (see Fig. 1). This phenomenon has been termed self-electrophoresis by Jaffe (1969). The transcellular potential gradient is believed to be essential in the development of asymmetry which leads to dramatically different rhizoid and thallus cells after the first division of the egg. It might be argued that the Fucrts phenomenon is a singular case since in most instances of egg development there is a high degree of compositional segregation (asymmetry) at the outset. We believe that this argument misses the important point that the pre-existence of asymmetry is not necessary for the eventual development of asymmetric growth. As we will show, the appearance of order and pattern in a cell can be due to the presence of a

THEORETICAL

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601

SELF-ELECTROPHORESIS A

FIG. 1. Polarization of Fucus egg. The initially spherical egg with membrane potential spontaneously develops a transcellular flux of fertilization. Arrows indicate the direction of flux of calcium ions. elongate in the direction of the flux and undergo asymmetric division defined by the autonomous “self-electrophoretic” field.

symmetrically ions within an The egg will perpendicular

distributed hour after eventually to the axis

common feedback mechanism in a completely symmetric system. For a system with inherent asymmetry, the direction of polarity might indeed be pre-determined: however, this fact provides no explanation of the driving force towards ever more complicated and highly patterned states of the growing system. It is in this sense that we focus on the Fucus problem seeking an explanation for self-electrophoresis, which is a phenomenon that does not, at first glance, seem possible in a system that lacks an inherent asymmetry. There have been a number of earlier attempts to develop a theoretical basis for self-electrophoresis (Jaffe, 1969, 1977; Jaffe et al., 1974). In contrast with the present work these attempts are incomplete and do not explain self-electrophoresis in a self-consistent way. A self-consistent calculation is one in which asymmetric concentration and voltage profiles are predicted to arise spontaneously in an initially symmetric system. All earlier attempts at developing a theory of self-electrophoresis have assumed the existence of one gradient-say, a concentration gradient-and then analyzed its effect on some other property-say, the voltage profile. Thus, these earlier attempts do not explain how the Fucus system can develop and sustain an autonomous ionic current. The present work provides a completely self-consistent prediction of the self-electrophoresis phenomenon. (B)

JUSTIFICATION

OF

ASSUMPTIONS

In order to construct a model consistent with the known aspects of Fucus, let us briefly review several basic experimental facts about this system.

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(1) The axis of polarization appears to be randomly oriented in dark grown cultures but may be fixed with the imposition of unidirectional light, strong gravitational fields, composition gradients and other eggs growing within a few cell diameters of a given egg (for a review see Jaffe & Nuccitelli, 1977). This, in light of the apparent high degree of spherical symmetry of the egg, seems to suggest that the axis of polarization is not determined by an inherent asymmetry imposed early in the egg’s development. (2) Cytoplasmic streaming appears to have no role in the determination of the axis of polarization since eggs stratified under high speed centrifugation can develop a polar axis while the resultant cell component stratification is still well defined (Whitaker, 1940). In these experiments the axis is found to be along the centrifugal direction (either parallel or antiparallel depending on pH) except when external gradients are also applied, in which case the direction of polarization is determined by the interplay of the centrifugal and gradient effects. (3) The process of rhizoid formation and the accompanying localization of macromolecules and particles have been shown to be separate events from the determination of the polar axis fixation (Quatrano, 1973). Using inhibitors, the following were shown: (i) cycloheximide inhibits protein synthesis by 90%, but does not affect axis determination; (ii) sucrose appears (by a simple osmotic effect) to prevent water uptake, but not axis determination; and (iii) colchicine, known to block microtubulin assembly, was used to show that the spindle apparatus and cell division were not necessary to establish and maintain a stable polarity. In light of these observations we propose a theory for self-electrophoretic effects in Focus-like systems. We assume that the important features of these phenomena can be described by a model which does not involve the nucleus, protein synthesis, cytoplasmic streaming and cleavage or distortion of the spherical shape of the cell. The cell interior and the environment will be treated as reaction-transport continua with isotropic rate and transport laws. In an ionic system there is a very strong tendency toward charge neutrality. Thus the motion of all the ions is co-ordinated by the charge neutrality constraint. As one might imagine, only the most naive optimist would believe that the proper equations of electrophysiology would have a structure that is simple. It is one of the goals of the present report to set forth the proper starting equations for the application of Turing’s biomorphogenetic hypothesis in a way which properly handles the important electrical aspects of the pJoblem. As we shall discuss at greater length later, our starting equations are quite different from the reaction diffusion equations for uncharged species used in all earlier studies of the Turing hypothesis. Thus

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most earlier work in this area must be regarded as suggestive but clearly misses a main aspect of a biologically relevant theory. A brief overview of the derivation of the electrophysiological equations is given in section 2. The mathematical details of the analysis are enumerated in the Appendix at the end of the paper. The results of section 2 are applied, in section 3, to a simple model, and the presence of a symmetric membrane non-linearity is shown to lead to asymmetric self-electrophoretic states. This detailed model study shows that self-sustained polarity is the result of biologically reasonable parameter values, and, hence, serves as a feasibility study for the applicability of the theory. Section 4 is a discussion of some of the many possible applications of the theory of electrical effects in biological systems embodied in these equations. A detailed model of Fucus early development, using the general formalism set forth here, is presented in Larter & Ortoleva, 1980a,b and Larter, Strickholm & Ortoleva, 1980. 2. Derivation

of Bioelectric (A)

BASIC

Equations

for Morphogenesis

EQUATIONS

A logical starting point for the derivation of macroscopic electrostatic equations is the full Maxwell equations. For the slow processes of interest here (on ms time scales or longer) it has been shown (Plonsey, 1969) that these equations reduce to Poisson’s equation pjf-!a9

---z*v F

(1)

where V is the voltage, 9 is Faraday’s constant, E is the dielectric coefficient, assumed to be constant in a given phase, W is a column vector of concentrations v={%?~,%&.

..,%‘,},

(2)

z is a row vector of valences Zi for species i = 1, 2, . . . , s and we introduce the notation A*B=

1 Ai&

i=l

(3)

for any vectors A and B. s is the number of chemical species in the system. W obeys the continuity equation which, assuming a linear relation for electro-diffusive transport, is the reaction-diffusion equation for charged species (Fitts, 1960; DeGroot & Mazur, 1963)

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where C@is a matrix of diffusion coefficients, .M a matrix of mobilities (including valence factors) and %( %‘) a column vector of chemical reaction rate terms. Implicit in (4) is the assumption that we may describe the system as homogeneous-i.e. there are no special features such that 62, ,M, or !g explicitly depend on ?, the spatial variable. The system consists of cell plus external medium and we may allow for differences between the cell interior and the environment, but neglect the effects of heterogeneous reaction at internal sites such as the nucleus and membraneous structures in the cell interior. The nucleus occupies less than a few per cent of the cell volume and its impeding effect on transcellular fluxes, produced by gradients on a length scale of the order of a cell diameter, is a small correction which is known to vanish as the ratio of the volume of the obstructing particles to the total system volume (Batchelor, 1975). Furthermore, other membraneous structures will only affect the transcellular ion fluxes by effectively reducing the diffusion and mobility coefficients of the migrating ions. Finally the high speed centrifugation experiments of Whitaker (1940) suggest that a distribution of local sites is not important to axis determination. If we find a mechanism that can break the symmetry of the system without such heterogeneities, it will not be difficult to include the effects of local sites of reaction other than the plasma membrane. The Debye length specifies the extent of the electrical double layer near the membrane. For biological systems, it is on the order of 10 A. This is much shorter than the length scale over which the transcellular field is measured (on the order of the cell diameter, about 10’ A) but it is, nevertheless, important in determining the membrane potential. Rather than neglect variations on this length scale, we have developed a multiplescale treatment whereby these short length scale effects may be incorporated in a description which preserves the essential role of the membrane potential. The quantities of interest in (1) and (4), such as L??,%‘, r, etc., are scaled with characteristic values for biological systems (see the Appendix for details). This introduces dimensionless quantities D, c, t, etc., and brings in a small parameter (Y,defined by a = (XL) -’ <<1, where x is the inverse Debye length and L is the characteristic pattern length (a cell diameter). After scaling, the continuity equation becomes cx

=~.(D~c+M&V)

(6)

THEORETICAL

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FOR

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605

and Poisson’s equation takes the form

v2v=-z*c. Equations dynamics.

(6) and (7) provide

(B)

a complete,

MULTIPLE

(7) macroscopic

picture of cell

SCALING

Equations (6) and (7) are valid in all regions of the cell and its environment-including the thin double layer region near the membranes. The double layer region is entirely different in character than the bulk of the system, being electrically charged and having concentration gradients that vary rapidly over a relatively short distance. As mentioned before, though many important processes take place in, on and near the membranes and it is not consistent to ignore the small double layer when considering a theory of electrical phenomena in biological systems. The smallness of the parameter (Y, (5), is exploited to convert (6) and (7) into sets of equations that are valid either in the bulk or near the membrane, but not both, and to correctly match the sets in the region where both are applicable (at a distance slightly greater than X-’ from the membrane). The essential difference between the two regions of interest is the length scale over which significant changes occur. Techniques to expand the double-layer region mathematically are well known in hydrodynamic boundary layer theory (see Van Dyke, 1975). These techniques permit a correct description of processes on the short length scale (x-l) and the long length scale (L) by defining a sequence of distance variables scaled with some function of a small parameter characteristic of the system. These techniques have been used in a variety of physical systems (Nayfeh, 1973) and, of particular relevance here, for the description of the effect of electrical fields on chemical waves (Schmidt & Ortoleva, 1977, 1979). As shown in the Appendix we take the cell membrane to be a sphere (see Fig. 2) although the final results are independent of geometry. A sequence of scaled variables is defined [equation (A8)]; the shortest is the distance variable appropriate for the boundary layer, the next is the length scale appropriate for the bulk of the system and all the remaining longer ones are not important for this particular problem. To complete the multiple scale development, the descriptive variables c (concentration) and V (voltage) are expanded as power series in (Y, (A9). Upon substitution of (A8) and (A9) into the general equations (6) and (7) we find one set of equations which characterize the double layer [(A17), (18)] and another set appropriate for the bulk of the system [(A30), (31)].

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m

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2. Spherical co-ordinate system. The cell membrane is taken to be a sphere of radius at the origin. The radial coordinate is scaled by a factor of x-’ and the scaled coordinate, [ = x-l (Ir/ --;A), IS such that 4’< 0 for the cell interior and 5 > 0 for the environment. 0 and 4 are the usual, unscaled angular variables. FIG.

A/2 centered

Our theory shows that a simplistic approach to setting forth a set of electrophysiological equations for biomorphogenesis is not only inaccurate but fundamentally inconsistent. In the simplistic treatment one neglects charge and hence Poisson’s equation reduces to Laplace’s equation v2v=o.

(8)

This theory is then completed by solving (8) and inserting solution V into the continuity equation (6) and solving for c. But evolution in (6) is not constrained so that charge neutrality is rapidly violated even if it was satisfied initially. Thus the initial assumption leading to (8) is violated and the simplistic theory breaks down. In contrast to the simplistic approach, our theory results in an evolution equation for the concentrations which forces the concentrations to always obey charge neutrality if they do so initially. This is because very small charge densities, neglected in the simplistic approach, actually have a very large effect on the motion of the ions. It is this vanishingly small charge density that causes an electric field just sufficient to force the ionic distribution to stay charge neutral at high ionic strength, cy+ 0. Thus a consistent theory of biomorphogenesis can only be formulated, if the ionic nature of most biologically relevant molecules is to be treated in a meaningful way, by the multiple scaling approach used in the present work. The central results of our theory are (A30) (A31). For a complete solution to a set of differential equations such as these we must have the correct

THEORETICAL

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conditions on the fluxes at the boundary. Here, too, the simplistic approach (which amounts to setting the charge in the double layer to zero) seems too naive to be right. However, as can be seen by working through the derivation in the appendix, the simplistic approach surprisingly gives the correct boundary conditions by a very subtle cancellation of terms. The key is that, as mentioned above, the concentrations vary quite rapidly on a length scale x.-l at the membrane; as (Y (and hence X-‘) + 0 we might expect enormous contributions to the flux at the boundary. However, these large contributions exactly cancel out and we are left with the usual boundary conditions described in the next section. (Cl

BOUNDARY

CONDITIONS

In this section we summarize the boundary conditions discussed above. For a single cell isolated in a large bath solution we must specify the conditions far from the surface. Since the concentration of the bath solution is at the control of the investigator we may take the asymptotic boundary conditions to be

c - cot, a constant li~+m

(91 v-

v”=o, li~-bcc’

where we have defined the voltage to be zero at a distance far from the cell. For a system of cells in close proximity, asymptotic boundary conditions are not appropriate, but we must impose a similar condition-namely, that the total concentration is a known constant,

- = d3rc(?). CTV I0 c,p is the total number of moles of each species in a system of finite volume P. The most interesting boundary conditions occur at the membrane surface. Let e denote the solution surrounding the cell and i denote the cell interior. Let ti be a unit vector normal to the outer surface, pointing in the e direction. The boundary conditions at the membrane are

where 3”

is the flux within the membrane,

$‘.’

are the fluxes in the

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indicated regions, W”” are surface reaction rate terms and all quantities arc evaluated at the indicated side of the membrane. 3“’ can have components parallel to the membrane (see Poo, 1977), which may be included in our formalism, but we shall neglect them here. Hence, we set 3” = -,#&‘n*.

ir+a/z

0 K-’

FIG. 3. Qualitative behavior of E and V near the membrane. The membrane surface acts like an electrode in solution and the ordering of ions near the membrane creates a so-called “double layer” which has a thickness, for biological systems, of K -’ 5 10 A. In this region, c and V are rapidly varying functions of Iii.

9 and W in (11) are functions of % and -V, the unscaled concentration and voltage. We scale these quantities with characteristic values, as in section 2.~; again, LYis made to appear explicitly,

ac

D-+MC35

av ‘,.I=a(rRe.'+Jbf),

(121

ad- >

we have used the linear flux law for $e,’ and JM is now the dimensionless, normal, transmembrane flux. Using our multiple scale-expansion procedure on (12) we find, as shown in the Appendix, (A34), the first-order scaled boundary condition. Equation (A34) depends on both the bulk solutions, denoted co and Vo, as well as on the double layer corrections, cl and VI. One might expect that the double layer corrections should appear in the boundary conditions because, even though cl and VI are small, they are rapidly varying near the boundary. As shown in the Appendix, however, we find that the terms due to cl and VI just cancel, yielding (A39). Equation (A39) is an important result; it states that the bulk concentration and voltage, co and Vo, obey the full boundary conditions with respect to their long length scale variation, and that c, and VI do not enter. Since we now have a closed system in CO, Vo, we henceforth drop the “0” subscript for simplicity.

THEORETICAL

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The results of this section-a summary of the multiple scale development of the Appendix-provide a macroscopic framework in which to study electrical phenomena in biological systems. To get an idea of the validity of these equations we note that, for physiological systems, x-l = 10 8, and the pattern length L is around a cell diameter, L = lo6 A. Therefore, CY= lo-‘. Hence, the lowest order theory should agree well with experimental measurements. 3. One-dimensional

Morphogenesis

It is important to show that self-electrophoresis can occur in a system with physically reasonable cellular dimensions, transport coefficients, rate laws and chemical concentrations. The purpose of the present section is to show the feasibility of self-electrophoresis in such a system. A specific model of Fucus is studied elsewhere (Larter & Ortoleva, 1980a,b : Larter, Strickholm & Ortoleva, 1980). (A)

A MODEL

SYSTEM

A simple, one-dimensional model with a membrane-localized nonlinearity is sufficient to demonstrate the existence of self-electrophoretic solutions to our equations. A one-dimensional model contains most of the interesting features, since we are looking for the development of asymmetry along a polar axis, as in Fucus. This one-dimensional model can also be applied to a field of cells which exhibit an ionic current during morphogenesis. More will be said, in section 4, about this latter application.

FIG. 4. One-dimensional array of self-electrophoretic one end of the cell to the other is essentially a breaking potential and sets in before elongation starts.

Fucus eggs. The onset of ion flow from of left-right symmetry of the membrane

Early experiments on the Fucus phenomenon (Jaffe, 1966) were done using the configuration of eggs shown schematically in Fig. 4. We construct a model of this configuration, consisting of parallel, planar membranes (see Fig. 5) arranged in a one-dimensional array along the x-axis. This can also be thought of as a model of an isolated Fucus embryo, in which case the solutions must obey periodic boundary conditions, i.e. the value of a quantity at x must equal that at x + L + A. For the isolated egg, L represents

610

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-(zL+A) -(L+A) -L FIG. 5. Parallel, planar Model of egg configuration

I'.

ORTOI.E\

0

membranes, separating shown in Fig. 4.

4 regions

A

L+4

of cytoplasm,

1+2A

i, and bath solution,

8.

a typical distance that current must travel in the culture medium; hence, L -A, and for simplicity, we assume L = A. We consider a model consisting of a charged species Szs in the presence of a background electrolyte, such as KCl, in high concentration. Let

c={S,K, C}

(13)

where S, K and C are the concentrations of Szs, K’ and Cl-, respectively. The membrane flux is taken to be a sum of passive transport and active (9) pumping terms: JM (c, V)

= [H(C'

- c' ) +

m( V’ - V’)] + P’(c’, cl, V’ - V’ )

(14)

where ceViand V"' are evaluated at the surface of the membrane, H is a 3 x 3 diagonal matrix of membrane transport coefficients, m is a three-component vector of valence-dependent membrane electrical mobility coefficients and B is a non-linear pumping term. We take a Jacob-Monod type non-linearity for 9’,;

cP)K=cPc=o W/SC)’ y>. 9s = -kS’ 1 + (s’,sc)”

(15)

i.e. K and C are only transported passively through the membrane, but S is, in addition, actively pumped out of the cell at a rate proportional to the concentration of S just outside the cell membrane, when the concentration of S inside the cell at the membrane exceeds the critical value S,. Equation (15) for y+ co is shown schematically in Fig. 6(a), along with a curve corresponding to a more realistic form for P’s [Fig. 6(b), y =.2]. The step-function shown in curve (a) of Fig. 6 will be used here in order to find an analytical solution to the equations. Later, we comment on the effect of using a function such as curve (b) for 9s. The pump is a strictly nonequilibrium term in that, at equilibrium (i.e. a dead cell), 9 must vanish since it takes metabolic energy to run the pump.

THEORETICAL

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FIG. 6. The pumping of SzS as a function of its cellular concentration, where S’ is the value of [S’q] at the membrane-i interface and S’ at the membrane--e interface. Cases: (a) y+ CC and (b) y = 2.

Phenomenological relations for transmembrane fluxes have been discussed extensively, showing, for example, how the coupling of reaction and transport within the membrane can lead to facilitated and active transport (Plonsey, 1969; Katchalsky & Spangler, 1968). In a review of this subject Katchalsky & Spangler (1968) discuss some of the thermodynamic and kinetic aspects of membrane transport. These authors point out the importance of making calculations on tractable model systems (regarding Na’-K’ active transport in erythrocytes): “. . . upon mastering the calculational procedures there is no fundamental difficulty in supplementing the scheme with additional data to bring it closer to reality.” Thus, we do not attempt, in the present work, to carefully set forth a detailed model of FUCUS membrane transport. In a more realistic model of Fucus one would have both Na’ and K’ ions and the complications associated with their being pumped out of and into the cell respectively. However we ignore this aspect of the problem in this “feasibility” study. Note, in this regard, thaat the value of [Na’] + [K’] is approximately the same inside and outside the cell so that a simple picture may be reasonable if we envision K’ and Na’ as essentially identical. In any case, it is not hard in principle to add other species if they turn out to be important. (B)

ONE-DIMENSIONAL

EQUATIONS

In this section we will study the steady-state electrophysiological

equa-

tions [(A30) and (A31)] for the model described above. We focus on the

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non-linear effects produced by the pumping term in the membrane flux I 141 & (15) and take G = R”’ = 0 1161 in (A30) and (12). Therefore, obey the following differential D

d2K Kz-

+M

the concentrations equations: dKdV

Kdx

MKK(X)

dx

u

of the three species (13)

F(x)=O,

d2C +M dCdVJfcC(x) -F(x) DC-Tdx Cdx dx r D dZS+M +!!=O ‘dx2 ‘dx dx

17)

= 0,

’

(18) (19)

where

and a=MKK(x)+M&(x)-M&‘(X).

(21)

Note the particularly simple form of these equations, which do not contain any extraneous constants (such as Faraday’s constant, the dielectric constant, etc.) due to our choice of units as described in the Appendix. Equations (17)-( 19) are coupled to Poisson’s equation for the voltage (A3 1) which for our simple model is d=V ~---,dx-

F(x) CT

(22)

The continuity equation for S(x) takes the particularly simple form (19) because S was assumed to be in low concentration, &&S(x) <<(T. From the charge neutrality relation (A16) we have K + z,S = C. Substituting

(23) into (21) and using the fact that S <
Combining

(23)

1 K(MK -MY)’

(24)

(24) with (17), (18) and (22) we arrive at the continuity equations

THEORETICAL

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for K and C dZK 2dx

z&,+1)

-dVdS-= dx dx

2

d2C dx’= and the simplified

0

0

(25) (26)

form of Poisson’s equation: (27)

where we have used Einstein’s relation (M, = z;Di). These equations apply to the bulk of the solution where the gradients of c and V can be expected to be small. With this, we assume that the square of these gradients is negligible: equations (25), (26) and (27) all reduce to Laplace’s equation, (28)

where A = {V, K, C, S}. Equation

(28) has the simple solution

A=b+gx.

(29)

The constants A and dA/dx will be determined tions, as discussed in the next section. (C)

BOUNDARY

from the boundary condi-

CONDITIONS

The solution (29) must satisfy the boundary conditions (A39) with (16) and the definition of J”, (14) and (15). Since the solutions h(x) must be periodic (with period L +A = 2A) we see, from Fig. 5, that the interval from x = 0 to x = L +A = 2A comprises a finite volume in which A(x) can be completely determined. Therefore, the boundary conditions at two membranes (say at x = 0 and x = A) will be sufficient to completely specify the as yet undetermined constants in (29). Each element of A has four of these constants (two each for regions i and e). Therefore, we need 16 conditions to completely determine the solution (29). From (A38) we have, for the one-dimensional model J’ = J”(0), J’ = J’, J”(0)

=

-J”‘(A),

(30)

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where J”(0) and J”‘(A) are (14) evaluated at s = 0 and A, respectively, and are defined to be positive when a flux is directed into “i” (the cell interior). J”’ is given by Jw = Ddc”“ dx

I MC,.Td V“” (” ds (31)

z D dc”” I Mct,P d V”” dx dx ’ where C? is an average concentration in the indicated region. Here we have neglected terms which are proportional to the square of a gradient. Equation (30) is a system of nine simultaneous equations. The remaining seven conditions are as follows: For the system of finite volume 2A we have, from (10) the total mass equations, 2A

dxS(x)

ST=& I

0

(32)

2A

KT=&

I0

dxK(x)

(for K and S only) where Sr and Kr are the (known) total concentrations and equal to the total number of moles in the system divided by the volume 2A. Next, we have the charge neutrality relations (charge neutrality must hold at each of the two surfaces of the two membranes at x = 0, A), z * Cl‘<’= (). (33) Finally, we must specify a zero point for V somewhere to 2A: e.g. we can take the left face of the membrane v(o-)

= 0.

in the interval x = 0 at x = 0 (34)

(30)-(34) is a system of 16 non-linear algebraic equations. The nonlinearities occur in two places: the step-function pumping terms, (14), and the field-dependent term in (31). In order to emphasize the importance of the membrane-localized non-linearity (14) and to obtain an analytical solution we assume that

but that the concentration gradients are non-zero. As will be shown in section 3.~ the analytical solution found here can be generalized to non-zero fields. The restriction (35) puts a constraint on the allowed parameter values available to the system. This point, and the solution of (30)-(34) is discussed in the next section.

THEORETICAL iD)

BASIS BIFURCATION

FOR OF

615

SELF-ELECTROPHORESIS ASYMMETRIC

SOLUTIONS

Let us now consider the solution for constant voltage, i.e. when (35) is valid. With the constraint (35) we find that the system of 16 equations, (30)-(34), reduces to two equations by a process of successive elimination:

- k[2&

-S(A-)]B[S(A-)

x e[S(O+) - SC]- 3k[2&

-S,]

= 0,

- S(A-)le[S(A-,

(36)

- $1 = 0,

(37)

which is a set of coupled equations for S(A ) and S(0’). S(A-) is the concentration of S evaluated inside the cell at the membrane at x = A, S(0’) is the concentration inside the cell at the membrane x = 0 and S is the total concentration defined by (32). For simplicity we have taken all transport coefficients (mK, ms, -mc, h, hs, hc, DK, DC) equal to unity, except Ds, which we set equal to 2. Also recall 9”~ = -kS’O(Si -S,) for y + 00 [where B(x) is the Heaviside step function]. Equations (36) and (37) yield four solutions to the model problem; these correspond, respectively, to pumping through (i) the left membrane, (ii) the right membrane, (iii) both membranes and (iv) neither membrane. S(A’) and S(O+) are given explicitly in Table 1 for each case. These solutions are subject to an additional constraint, -!j[S(A-

) - S(O+)] - 15[&

+:k[2&-S(A-)]t?[S(A-)-S,]=O.

- S(o+)] + yk[2&-

- s(o+)]e[s(o+)

- SC] (38)

This constraint is left over from the reduction of 16 equations to two equations and is due to assumption (35). Solutions (iii) and (iv) satisfy (38) automatically; solutions (i) and (ii), however, when substituted into (38), are found to be valid only for a particular cell length A =A, (= 1 for the parameter choices given in the preceding paragraph). Limits on the other parameters for solutions (i)-(iv) can be determined from the limits on S(0’) and ??(A-) imposed by the step-functions in (36) and (37). We define s as the ratio &-/SC, and find that solution (iii) is valid for the range of values of s and k given in Table 1. Solutions (i) and (ii) are also restricted to a certain range of values of s and k ; these are given in Table 1,

616

R.

1.ARTE.R

AND

P.

ORI‘OI

EL’/4

TABLE 1 Membrane

Current.

potential,

AV

I = z* J 12-4k

S(O+l

12+k 12+2k

SCAmI

(iii) Symmetric pumping

(9, (ii) Self-electrophoretic

Solution Membrane

potential and current

12+k

2kST

k&

12+k 6kST

3-k 0

12+k 12+2k ST? ---ST lZ+k I’-4k ST,

12+k

352k -S, 3-k 3-2k ST

-ST

3-k

livl No pumping (I 0 ST ST

(i), (ii) are the self-electrophoretic solutions corresponding to pumping through the left and right membranes, respectively. They are valid for (12+k)/(12+2k) k >O, A = 1. (iii) is a symmetric solution which corresponds to pumping through both membranes. It is valid for s > i - k +3)/t - 2k +3). k <$, all A. (iv) is another symmetric solution, for no pumping on either side, valid for s < 1 and for all values of A and k.

also. Figure 7 shows the region in the s, k-parameter plane where (i) and (ii) are valid (regions III, IV and VI). In the course of eliminating 14 of the 16 boundary conditions we find an expression for the membrane potential, A V = V(C) - V(O+) = V(A+) - V(A-),

+$k[2&

- S(A-)]fl[S(AP)

-S,].

In Table 1 we list A V for the solutions

(39) (i)-(iv).

I=z*J=Jk-+zJs-JC

The total current is given by (40)

and is also given in Table 1. Notice that solutions (i) and (ii) are the self-electrophoretic solutions and that the current for solution (ii) is directed opposite to that of solution (i). The membrane potential, A V, is non-zero for (i), (ii) and (iii) because the active pump is working in these modes; in addition, AV is equal for the two asymmetric solutions, (i) and (ii), but different from the membrane potential corresponding to the symmetric solution, (iii). AV for solution (iv) is zero because, for this solution, there is no active pumping and, therefore, the cell is at equilibrium with equal concentrations on either side of the membrane. If at least one species did not

THEORETICAL

BASIS

FOR

SELF-ELECTROPHORESIS

617

V I l

Ii--

2

3

k

FIG. 7. s, k-parameter space: asymmetric solutions. Solutions with non-zero current exist for3>k>Oand(k+12)/(2k+12)
permeate the membrane, a non-zero membrane potential would be observed in the passive mode (iv). The current 1, (40), is shown in Fig. 8 as a function of the ratio s. All parameter values are given in the caption to Fig. 8. In particular, we chose k = $. From the limits on s given in Table 1 for (i) and (ii) we see that the self-electrophoretic solutions are only valid for sI
618

R.

LARTER

AND

P.

ORTOI.F;VA

s 8. Current as a function of s = ST/&. Here we have ml, = ms = -mc = hK = hs = hc = DK = DC = z, = S, = 1; Ds = 2. We also choose k = f. The self-electrophoretic branches are valid for g< s f (solution iii). The transcellular current I is given in multiples of S,, i.e. I = *A&s for solutions (i), (ii). FIG.

self-electrophoresis there is no simple relation between the transcellular current and voltage. The membrane potential, AV, is shown in Fig. 9 as a function of the pumping strength, k. All parameters are indicated in the caption to Fig. 8, but, for this figure, we fix s and find limits on the allowed values of k for each

0

1

2 k

FIG. 9. Membrane given in the caption to range of k values O< equation (411, for $$<

potential as a function of gumping strength, k. Parameter values are Fig. 8. Here we choose s = 5. The symmetric solution (iii) is valid for the k <$. The self-electrophoretic solutions (i) and (ii) are valid, from k ~3.

THEORETICAL

BASIS

FOR

619

SELF-ELECTROPHORESIS

solution. Choosing s = 4 we find that the symmetric solution (iii) is valid for 0 < k < +. No net transcellular current or individual ion flux is associated with the type (iii) solution. From Table 1 we find that the asymmetric solutions (i) and (ii) exist for $$< k < 3. Type (i) or (ii) solutions do have an associated net transcellular current, as illustrated in Fig. 8. Again, the asymmetric solutions are seen to bifurcate from the symmetric branch of solutions at a critical parameter value. Since the asymmetric solutions exist only over a welldefined range of parameters, a cell could turn on self-electrophoresis by releasing energy-rich species (e.g. ATP) to run the pump, bringing k into the domain of existence of self-elzctrophoretic solutions, i.e. $$< k < 3, for this example. The sharp corners in Figs 8 and 9 are the result of the step function form of 9,. If gs were to take on a more physically reasonable form, as in curve (b) of Fig. 6, the corners would be rounded off. However, our solutions demonstrate the qualitative behavior of a more physically reasonable pumping function. (E)

THE

GENERAL

A f A,

CASE;

The self-electrophoretic solutions discussed above are valid only for d V/dx = 0 and, hence, only for a particular value of the cell diameter, A, = 1, To find corrections to the particular solution near A = A,, we can write the general asymmetric solution as a power series expansion in A -A,. Let 4 be the column vector of unknowns for the system, ‘I’= {c(O-), c(O+), c(A-),

c(A+),

V(O-),

V(O+), V(A-),

V(A+)}.

(41)

We will study the behavior of the system as a function of the cell size A. Then, (30)-(34) can be written as a column vector of equations, F(O, A) = GN’) + AH(W)

= 0.

We write w as a power series expansion in AA = A -A,, particular value of A for the special case dV/dx = 0. ~=@“‘+@“AA+. .. where wIr”’ is the analytical written as a Taylor series, F[‘@“, A,]+

solution

where A, is the

found for A = A,. Equation

C- aF’“‘,yl.l I+ H’o’ AA + . . . j ’ I d*j

where superscript (0) implies evaluation

at ‘P(O), A,. Since

F[‘PCO’, A,] = 0

(431 (42) is

620

R.

LARTER

AND

P.

ORTOLEVA

we can find the first correction to @’ by solving the matrix equation n@“= -H where,

(461

0, &.

(47)

If one had a general solution to the present problem, then (35) would be two relations among the system parameters. One condition is that A = A,. The solubility conditions for (46) are that either det n is non-zero or, if it is, then H is perpendicular to the null space of $I. Since we find det 0 = 0 as expected [since (35) implies a second restriction among system parameters] our results are strictly valid along a line in the Sk-plane of Fig. 7 although we do not consider this in detail here. From 9”’ we may calculate the pole-to-pole potential drop and the transcellular current as a function of A. The pole-to-pole potential A Vpole= V(A') - V(O-) changes sign as A passes through A,,, but the current does not. Here we have evidence for a phenomenon we call anti-electrophoresis, i.e. a pole-to-pole potential directed opposite to the current. In experimental observations, the pole-to-pole potential is in the same direction as the current. Our result is reasonable in light of the fact that a current due to diffusion exists even for zero pole-to-pole voltage (the analytical solutions of section 3.D). The results derived above for this simple model are only of biological interest if the parameter values chosen are physiologically reasonable. Removing dimensions at the beginning of a derivation sometimes obscures this important point, so we will now justify the chosen parameters. A typical diffusion length L should be on the order of a cell diameter, which for MUCUS is 0.01 cm. A typical ionic diffusion coefficient in biological fluids, fi, is around lo-’ cm’/s. Measured net transcellular fluxes are about .? = lo-l2 mol/cm’-s. From Fick’s law

A&

D

(48)

we may calculate the typical difference in concentration from one side of the cell to the other. For the above choices AC = 10m9 mol/cm3, which is only 1 part in lo5 for a typical concentration of a species like chloride, c = lop4 mol/cm3. However, enzymes can change their activity appreciably over a range of mediator concentration that is of the order of Ai? calculated here. An ion elimination pump that is stimulated by small changes in concentration of the ion which is being eliminated, e.g. Ca’+, would serve as a possible example of this type of feedback species.

THEORETICAL

BASIS

FOR

SELF-ELECTROPHORESIS

621

Characteristic values of k, the pumping strength can be approximated from data on active transport of ions through plant cell membranes. MacRobbie, 1975, reports membrane fluxes, due to pumping, of lo-100 x 10-l’ mol/cm’ . s for Cl-, K’ and Na’. From k”=.~“pZ, and % = lop6 - 10e4 mol/cm3, we find k’= lo-‘teristic value we have chosen is calculated from

10-j cm/s. The charac-

which yields ,6= 10e8 -lo-’ cm/s. Thus the dimensionless parameter k. which is given by k = 6/c takes on typical values in the range k = I- 10J. and our choices of k = 1 - 10 are quite reasonable. 4. Conclusions

and Other Applications

We have rigorously derived a set of macroscopic equations suitable for application to slow electrophysiological phenomena such as morphogenesis. To emphasize the importance of the membrane potential and the accompanying electrical double layer near the membrane we have used a multiplescale technique to reduce the continuity and Poisson’s equations for systems with high ionic strength. We obtain in a self consistent way one set of equations describing the double layer and another describing the variations on the long (morphogenic) length scale. The latter bring in the charge neutrality constraint in a natural way and show the presence of a Planck potential (depending on concentration gradients, only) that guarantees charge neutrality sufficiently far from the membrane. The theory was applied to a simple model system consisting of an ionic species in the presence of high background electrolyte, and possessing a membrane-bound nonlinearity in the form of an active pumping term. This simple biologically suggestive system was sufficient to demonstrate the possibility of a symmetry-breaking instability and the existence of selfelectrophoretic solutions to the equations for a model with dimensions, transport properties and concentrations consistent with physiological data. The model used is not intended to be a conjecture about the mechanism of self-electrophoresis in MUCUS, but rather, is intended to show that selfelectrophoresis is predicted by the electrophysiological theory derived here. A specific, detailed model of the Fucus egg is studied elsewhere (Larter & Ortoleva, 1980a,b; Larter, Strickholm & Ortoleva, 1980). In addition to single cell phenomena, these ideas could be applied to a field of cells as for example the one-dimensional array of Fig. 4. (For such a

622

R.

I.ARTER

AND

I’.

OR

I’01

F\

.A

situation one must, of course, drop the periodic boundary conditions on a one cell length scale used in the previous section.) Borgens, Vanable & Jatie (1977a) have determined that autonomous currents are associated with the regeneration of limbs in newts. The regenerating stump can be modeled by the system described in section 3 where, now, each pair of membranes corresponds to a layer of cells. Borgens, Vanable & Jaffe i 1977h), also found that externally applied currents enhance the rate of regeneration of amphibian limbs and can even cause the stump to degenerate when high enough currents are applied in the opposite direction. A number of examples of the effects of electrical fields on development are found in Brighton, Black & Pollack (1979); application and variations of the theory presented here to uni- and multi-cellular systems is found in Ortoleva, 1980. The self-electrophoretic phenomenon may serve as a mechanism of multicellular or tissue organization and morphogenesis. Indeed, the onedimensional configuration of Fig. 4 could be a cross-section through a developing field of cells. It is seen that such a field could develop a directionality even if no inherent directionality or asymmetry exists. One might be tempted to argue that the self-electrophoretic phenomenon displayed by Fucus is not of general interest since the more typical case is of cells with inherent asymmetry as, for example, in mammalian eggs, where it is known that the point of entry of the sperm can fix axis determination. We believe that this type of argument misses the essential point. The tendency towards self-electrophoretic instability is expected to be commonplace as our simple, minimal model calculation of section 3 serves to indicate. If inherent asymmetry exists, it only serves (in analogy to imposed unidirectional light during Fucus development) to fix the axis of polarity. This does not, however, detract from the ability of the self-electrophoretic state to create or maintain organized motions of ionic species on the cellular or multicellular scale. (We point out that our formalism is equally applicable to asymmetric models, although these would not show the interesting behavior associated with spontaneous pattern formation in an initially symmetric model; see Larter & Ortoleva, 19806 and Ortoleva, 1980.) Because of the rotational invariance of the equations applied here to the Fucus problem, one objection might be that the self-electrophoretic states would only be marginally stable to rotation. Indeed, it is found that the orientation of the polar axis in Fucus may be rotated during the first three hours after polarity appears by changing the direction of, for example, the applied unidirectional light. However, later in development this lability is lost. This eventual “freezing in” of the direction of the polar axis is probably due to secondary processes (microtubule formation?). The key point is that the self-electrophoretic instability provides a mechanism for spontaneous

THEORETICAL

BASIS

FOR

623

SELF-ELECTROPHORESIS

axis formation in the absence of protein synthesis, inherent asymmetry external gradients.

or

We thank Professor A. Strickholm for some very helpful suggestions and Professor L. Jaffe for pointing out some key experimental facts on MUCUSand associated references to the literature. This research Foundation.

was

supported

in

part

by

a grant

from

the

National

Science

REFERENCES BATCHELOR, G. K. (1974). BORGENS, R. B., VANABLE,

Ann. Rev. of Fluid Mech. 6, 227. J. W. & JAFFE, L. F. (1977a). Proc. narn. Acad.

Sci. U.S.A.

74,

4528. BORGENS, BRIGHTON, Cartilage:

R. B., VANABLE, J. W. & JAFFE, L. F. (1977b). .I. exp. Zool. 200,403. C. T., BLACK, J. & POLLACK, S. R. (Eds) (1979). ElectricalPropertiesofBone and Experimental Effects and Clinical Applications. New York: Grune and Stratton. DE GROOT, S. R. & MAZUR, P. (1963). Nonequilibrium Thermodynamics. Amsterdam: North-Holland Pub. Co. DU BOIS-REYMOND, E. (1860), Utersuchungen uber tierische ElektrizitatII.2, Berlin: Reimer. EL-BADEWI, M., LARTER, R. & ORTOLEVA, P., manuscript in preparation. EYRING, D. (Ed.) (1978). Periodicities in Chemistry and Biology, Adv. in Theor. Chem., New York: Academic. FI?~s, D. D. (1960). Non-equilibrium Thermodynamics. New York: McGraw-Hill. FLICKER, M. & Ross, J. (1974). J. them. Phys. 60,3458. HAHN, H. S., ORTOLEVA, P. & Ross, J. (1973). J. theor. Biol. 41, 503. HERLITZKA, A. (1910). Wilhelm Roux Arch. Entwicklungsmech. Org. 10, 126. JAFFE, L. F. (1966). Proc. natn. Acad. Sci. U.S.A. 56, 1102. JAFFE, L. F. (1969). Develop. Biol. Supp. 3, 83. JAFFE, L. F. (1977). Nature 265, 600. JAFFE, L. F. & NUCCITELLI, R. (1977). Ann. Reu. Biophys. Bioeng. 6,445. JAFFE, L. F., ROBINSON, K. R. & NUCCITELLI, R. (1974). Ann. New York Acad. Sci. 9,372. KATCHALSKY, A. & SPANGLER, R. (1968). Quart. Rev. of Biophys. 1, 128. LARTER, R. & ORTOLEVA, P. (1980a, b) J. Theor. Biol. (submitted for publication). LARTER, R., STRICKHOLM, A. & ORTOLEVA, P. (1980) J. Ceil Sci. (submitted for publication).

NAYFEH, A. (1973). Perturbation Methods. New York: Wiley. NICOLIS, G. & PRIGOGINE, I. (1977). Self-Organization in Non-equilibrium Systems. New York: Wiley. NOYES, R. M. & FIELD, R. J. (1974). Ann. Rev. of Phys. Chem. 25,95. ORTOLEVA, P. & Ross, J. (1973). Dev. Biol. 34, F19. ORTOLEVA, P. & Ross, J. (1973). J. biophys. Chem. 1,87. ORTOLEVA, P. & Ross, J. (1974). J. them. Phys. 60, 5090. ORTOLEVA, P. (1980) “Developmental Bioelectricity”, PLONSEY, R. (1969). Bioelectric Phenomena. New York: McGraw-Hill. Poo, M. & ROBINSON, K. R. (1977). Nature 26S, 602. PRIGOGINE, I. & LEFEVER, R. (1968). J. them. Phys. 48, 1695. PRIGOGINE, I. & RICE, S. A. (1978). Adv. in Chem. Phys., Vol. 38. New York: Wiley. QUATRANO, R. S. (1973). Deo. Biol. 30,209. RASHEVSKY, N. (1960). Mathematical Biophysics. New York: Dover. SA~INGER, D. (1973). Topics in Stability and Bifurcation Theory. New York: SpringerVerlag. SCHMIDT, S. & ORTOLEVA, P. (1977). J. them. Phys. 67,377l. SCHMIDT, S. & ORTOLEVA, P. (1979). J. them. Phys. 71,lOlO.

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I’.

ORTOL.E\‘A

SRIVASTAVA, K. P. clc SAXENA, A. K. (1977). Int. S~rg. 62, 35. STATGOLD, I. (197 1). SIAM Rer. 13, 289. TURING, A. M. (1952). Phil. Trans. Roy. Sot.. London, Ser. B. 237, 37. VAN DYKE, M. (1975). Perturbation Methods in Fluid Mechanics. Amherst, Mass.: Paratwfll Press. WHEELER, P. C., WOLCOTT, L. E., MORRIS, J. L. & SPANGLER. M. R. (19681. In Neuroelectric Research. (D. U. Reynolds & A. Sjoberg. eds), pp. 83-99. Springfield. Illinok C. C. Thomas. WHITAHZR. D. M. 119401. J. Cell. camp. fhysiol. 15, 173.

APPENDIX Basic Equations

and Definitions

In this appendix we use a muhiple scale technique to derive a set of electrophysiological equations. Poisson’s equation (1) and the continuity equation (4) are valid under the assumptions discussed in sections 1 and 2. We begin by scaling (1) and (4) with typical values of the concentration, distance, etc. Let c, d and 7 be characteristic quantities. Then, we may use Einstein’s relation to find a characteristic mobility,

where R is the gas content and T is the absolute temperature. A characteristic voltage is, using dimensional analysis, RTI.97 In association with 7 and fi we may introduce a characteristic diffusion length, L,

L'=&. Also a characteristic

(A2)

rate of reaction can be written

From these quantities length, x:

we may construct

a characteristic

2 4lr9’~ x =-zE-. Introducing dimensionless space variable 4,

variables % = cc, 9 = r’=x

-16

Debye

inverse

t.44) DD,

T = 7t, etc. and scaled (A5 )

we obtain (6) and (7) where 9 refers to gradient with respect to the scaled variable 8 The parameter (Y, defined by (5) is the ratio of the Debye length to the characteristic diffusion length L. We are interested in cases where L is on the order of cellular dimensions and, hence, (Y presents itself as a natural smallness parameter.

THEORETICAL

BASIS

FOR

SELF-ELECTROPHORESIS

625

Electrical effects are very geometry-dependent. Indeed, we expect the Debye layer to follow the shape of the membrane. Hence, any expansion procedure in the smallness parameter (Y must reflect this. To demonstrate our method we shall choose a spherical cell isolated in a three-dimensional culture medium. As we shall find, however, many of the final equations are, under biological conditions, of a completely general form. Multiple

Scale Development

The Debye layer is a region of local non-zero charge density that is localized, for biological ionic strengths, near a surface or membrane. Thus, we seek a development which reflects short scale variation in the direction perpendicular to the membrane, but only long length scale variation along it. For example, in the Fucus problem, the voltage varies on a scale of the cell diameter in the direction along the membrane. In Fig. 2 we show the geometry of interest for the spherical cell. The cell membrane is taken to be a sphere of diameter A. To emphasize the role of the membrane we introduce a new scaled radial variable l such that HA-6

CA61

a’= xA/2.

(A7)

where Z is the scaled cell radius, In the spherical geometry, variations on the spatial scale x-l are only expected with respect to 5 and not with respect to 8 or 4. Hence only the variations of c and V with respect to 5 are expected to have a multiply scaled structure. To reflect the variations on multiple length scales we introduce a sequence of scaled variables f;l ln = (Y“5. (A8) Furthermore, all descriptive parameter LY,

variables

(;)=.i(,(%,)ff”.

are expanded

in the smallness (A9)

The expansion coefficients c, (&,, [i, . . . , 8, c$, t) and V, (&,, 11, . . . , 8, q5, t) are taken to be functions of all the scaled lengths. Ensuring that the coefficients c, and V,, are well behaved at all l will provide important constraints that will help to fix these coefficients. The next step is to express the various operators in terms of &,, 8, d variables. For example we have (AlO)

626

R.

I-ARTFR

AND

t’.

ORTOi.FVA

where fi involves only derivatives with respect to the angular variables and < is a unit vector in the radial direction. Anticipating that the 0, d variation is on the long length scale we introduce a resealed cell radius a, Li!=CYa’ which

is a constant

(All)

as a + 0. With this we have

where R2 is the usual angular part of the Laplacian. We proceed with the perturbation scheme by putting the expansions (A9) and the operator relations (A12) and (A13) into the dimensionless equations (6) and (7) and collecting terms to various orders in LY. As is typical, the multiple scale theory must be accompanied by a degree of intuition. In particular, we expect the existence of a limit as a + 0, i.e. the existence of a well defined limit at infinite ionic strength. We expect that this is consistent with the existence of a zeroth order solution co, that depends only on the longer scales 11, lz, . . . , 8,$ and not on the short scale variable lo. This conjecture must, of course, bear up to the test of self-consistency-a situation we shall indeed find to be the case. Thus we assume that

V,,

ac, 0, -=

av” --=o.

x0

Collecting

4%

(A14)

terms of order CY” = 1 we get (A151 z*co = 0.

(A16)

The first equation is satisfied trivially via our assumption (A14) whereas (A16) is the central charge neutrality constraint forced on systems of high ionic strength, i.e., as a + 0. The development starts to get interesting to order (Y :

a'c,

D--Y+ al;,

a'v,

MCc, 2

z*c1=-2

= 0.

(Al7

a4k

a'v, ah '

(Al8

THEORETICAL

BASIS

FOR

627

SELF-ELECTROPHORESIS

These equations will be found to characterize the Debye layer. Note that the conditions in the double layer are expected to depend parametrically on the longer scales, which is seen to be the case due to the presence of co in (A17). Indeed, these equations are differential equations with respect to lo, whereas co is independent of lo and hence acts like a constant in these equations. To order (Y* we find, from Poisson’s equation: (A19) --B,= a2vo - 2 2+Jl+~ x1

s+ al,

~ fl’v, (cz+~,)'+~*~~.

(A201

We shall assume that z * c2 is independent of f. and show this to be the case in hindsight. Because of this assumption, B1 is a constant with respect to f. and we may integrate (A19) to find

av2 2”1 -+2!3+ ~ = B~lo + Bz al1 &+u alo

(A21)

where BZ is independent of fo. But VI and V2 are finite for all lo, hence, B1 must vanish. This solubility condition, typical of multiple scale schemes, leads to

a2vo ~ 2 av, ~ n' --z*c2= as: +&+a zT+(il+ai'

"0.

L422)

Note that since the right-hand side of (A22) only depends on li, 4’*, . . . our assumption that Z*CZ is independent of lo is verified. A similar situation exists for the continuity equation. The solubility condition leads to ace

-

at

= Dv:Co

+ MC,&

“,

+ Ma&.

b, “,

+

G

(A231

where 0: and 9, are the indicated operators with respect to ti, i.e. (A241

v+““+al:

Similarly

2 ll+a

R2 d+-847 (rl+d2’

(A25)

we can rewrite (A22) in the form vTvo=--z*c*.

(A26)

At this point all seems hopeless since (A23), (A26) do not comprise a closed

628

R.

LARTER

AND

I’.

OR’I‘O1.El’A

system in co, V. but involve the as yet undetermined second order charge density z * c2! The key to this “paradox” is that we have not invoked charge neutrality for the zeroth order concentrations, i.e. z * co = 0. Since chemical reactions preserve charge we have z*G=O. Then dotting

where

(A23) with z, using (A26),

(A27) (A27) we get

cro is defined as: (A29)

f~o=z*MC~,

and is seen to be the conductivity of the medium. Combining (A28) with (A26) and (A23) we obtain the final equations, valid in the bulk of the solution: c+;‘MCoZ

V:V,=

*)(DV:C,+&Co.~,

v))+G(Co).

(A301 (A31)

--cr;‘Z*(DV:co+M~$,,.~,v,,).

These equations, in conjunction with the double layer equations (A17), (A18) provide a complete, self-consistent description of the system to order a, i.e. V-Vo+aV1, c=co+acI. Note that the factor (1 -(TO* MC~Z* 1 (where the z* implies “dot z into the quantity to the right”) acts like a projection operator, which guarantees that if the initial conditions co (t = 0) are charge neutral then co will be charge neutral thenceforth. Boundary

Conditions

The results of the previous part are not complete without a consideration of the boundary conditions. Indeed, it is not as yet clear that the first order corrections, cl, do not contribute to the boundary conditions. Although c, contributes to order (Y to c its gradients yield zeroth order terms, since it depends on lo, i.e. is rapidly varying. In the multiple scale scheme we envision the &‘,,as independent variables. It may be verified that the cl, VI equations have solutions which vanish far from the membrane. Hence we have to order cx (9), the boundary conditions on c and V as l[ol + 00. The qualitative picture is shown in Fig. 3. We see that the descriptive variables are essentially given by co and V,, everywhere but in

THEORETICAL

BASIS

FOR

SELF-ELECTROPHORESIS

629

the inner region, a regime of order 0
where L is a reaction-diffusion length, L* = 67. $a” also scales with the same factor. With this the scaled form of (11) becomes (12). In (12) the superscripts i and e indicate evaluations of c and V in the i and e regions just adjacent to the membrane. To determine the boundary conditions on the coefficients cn and V, of the CYexpansions (A9), one substitutes (A9) and the multiscale operator (A12) into (12) and collects terms. To order a” = 1 we obtain

ace

D-+MCo--alo

(A33)

630

R.

which is trivially

LARTER

AND

P.

ORTOLEL’A

satisfied in light of (A14).

To order N we obtain:

We now show that the cl and VI terms do not contribute to (A34). (A17) and (A18) may be combined to yield an equation for the first-order charge density .’ = I”‘Z * &, -$(z *qp (A351 a[0 where

I is the dimensionless

ionic strength, I“‘EZ*D

‘MC;‘.

t-436)

(A35) is easily solved, yielding (2 * qy where that

Q’,’ are constants.

= Q “‘exp( * JP’l,)

Substituting

ac1 D--= ah

Hence, we have the boundary

ace

D-+MCo--

al1

(A37)

-MCo---.

into (A17)

(A371 it is easy to show

av1

(A381

X0

condition,

(A39):

= rR’%,)

+ J‘%o,

Vu).

(A39)

Equation (A39) is an important result; it states that the zeroth order descriptive variables co and V. obey the full boundary conditions with respect to their slow scale variation (dependence on [r) and that the first order quantities cl and VI do not contribute to this lowest order boundary condition despite the rapid variation of the latter (i.e. despite the fact that cl and VI depend on lo).