Pattern formation of stationary transcellular ionic currents in Fucus

Model Description

To describe the dynamics of both ions and membrane electric potential difference, we consider electrodiffusive models coupled to the membrane conductances to account for the membrane transport processes. For the sake of simplicity, we only consider two ions 1 and 2 of charge number z ₁ and z ₂, which diffuse at different rates in solution. D_j is the coefficient of diffusion of the ion j. In the initial state, the membrane potential V and each ionic concentration, respectively, are equal to V ₀ and C_{j 0} (j = 1, 2).

Membrane Currents. The current of each ion occurs through membrane proteins and is characterized by a current-voltage I–V curve: I ₁(V) and I ₂(V). The differential conductances G ₁ and G ₂ around the initial state V = V ₀ control the linear response of the cell to all perturbations. In all of the following, we assume that G ₁ is negative and that G ₂ and the sum G ₁ + G ₂ are positive (Fig. 1). This assumption is supported by experiments showing the presence of both voltage-dependent calcium channels and a potassium leak in Fucus (26). We model such a current by a cubic variation to have a N shape (Fig. 1): where ν is the dimensionless membrane potential ν = (V – V ₀)/|V ₀|, F is the Faraday constant, ν₁ and ν₂ are two characteristic dimensionless membrane potentials, and P is a permeability coefficient that evaluates the dependence of the current with the intracellular concentrations. For the sake of simplicity, we take the same value for I_j and ignore the variations with the extracellular concentrations. We also define a dimensionless parameter , which quantifies the variation of the currents with the concentrations versus the variation with the membrane potential. The results reported here do not depend on the exact shapes of the currents, which affect only the magnitude of the results.

2D and 3D Models. We consider a circular cell of radius R. In the extracellular and intracellular media, each ionic concentration C_j and the electric potential φ satisfy a continuity equation based on Nernst–Planck electrodiffusive flux and the Poisson equation: The boundary conditions at cell membrane are: where is the molar electrodiffusive flux, F is the Faraday constant, is the outer normal unit vector, ε is the bulk permittivity, and C_m is the membrane capacitance: C_m = ε_m/d, where d is the membrane thickness. The subscripts i and e refer to the inside and the outside of the cell. The membrane potential V is equal to φ_i – φ_e. Eq. 5 is valid at the two sides of the membrane, which means that the current of each ion through the membrane, I_j, is equal to the outer and inner electrodiffusive ones. Eq. 6 provides the continuity of the electric field. Eq. 7 is the dielectric one where the right member is approximately a charge per unit area because of mobile ions integrated across the Debye layer. Eqs. 6 and 7 have been calculated from Maxwell boundary conditions at each interface (extracellular medium-membrane and membrane-intracellular medium), assuming the absence of fixed charges (due to lipids or proteins). We have assumed that the normal component of the electric field inside the membrane is a constant. This assumption is valid in biological cells where the characteristic length of membrane potential variations is large compared with the membrane thickness. Then, in the framework of this model, we determine explicitly the ionic charge density in the extracellular and intracellular media; the ionic charge density is not zero, notably, in the Debye layers. Far from the cell, the electric potential and the concentrations are equal to the initial values.

The 1D Model. To understand and characterize the results obtained in two dimensions, we have studied in details a 1D case that has the great advantage to be less time-consuming. It does not change the qualitative results. The 1D model (a cell cylinder of radius r) is based on the 1D Nernst–Planck equation with an additional capacitive relation: Several authors have applied this model to relatively small intracellular structures such as dentritic spines (27). We have checked that the 3D model reduces to the 1D one for a cylindrical cell if the correlation space constant is large compared with the cell radius.

Numerical Methods. All the simulations are performed with dimensionless parameters, which vouches for the scope of the results. The dimensionless coordinates X and Y in two dimensions are equal to x/λ₂ and y/λ₂, where λ₂ is the characteristic correlation space constant of the current , where σ is the solution conductivity. The dimensionless time T is equal to , where D̃ is a mean ionic diffusion coefficient: D̃ = δ₁ D ₁ + δ₂ D ₂, with . We recall that . For all the figures, z ₁ = 2, z ₂ = 1, δ_j = 0.5, ν₁ = –0.2, ν₂ = 0.15, , and e|V ₀|/k _B T = 4. These values are not critical for the results. p = 0.1 for all of the figures except the stability diagram (see Fig. 5). Other parameters are provided in the following section. We assume an initial white noise of the membrane potential and analyze the response of our system by computing the electrodynamic equations.

Results

Let us consider a circular cell of radius R corresponding to the Fucus zygote (Fig. 2). The relevant ions are the calcium (ion 1) and potassium (ion 2) ones. The relevant value of the intracellular calcium diffusion coefficient must take into account the binding to buffers, reaction with organelle, and so on. We have collected the intracellular apparent values from different species. They vary from 2.010^–6 to 10^–8 cm²/s (4, 28, 29). We have used an intermediate value, D _Ca²⁺ ≈ 2.010^–7 cm²/s. This value provides a large ratio of coefficients of diffusion D ₂/D ₁ = 100. The differential membrane conductances are assumed to be of the same order: G ₁/G ₂ = –0.8. For a dimensionless radius 2G ₂ R/σ = 0.01, a stable stationary dipolar circulation of ions appears, breaking the initial symmetry as observed in Fucus (Fig. 2). Calcium ions and the electric field enter at the depolarized pole in agreement with experiments (9–12). An asymmetry exists between the magnitudes of the electric field at the two poles, reflected by the relative length of the arrows. It arises from the asymmetry of the N shape of I ₁(V).

To understand this mechanism fully, we needed to return to the 1D model that allows both reasonable time computations and analytical calculations. In one dimension, for D ₂/D ₁ = 100 and G ₁/G ₂ =–0.8, the system is still unstable; any perturbation is amplified. After a characteristic time, a stable spatial modulation of finite wavelength λ of the membrane potential has developed along the cellular axis. It arises from ionic currents (Fig. 3A) that break the symmetry along the cell. On Fig. 3B, we have also computed the patterns of the intracellular ionic concentrations C ₁ and C ₂. The variations of the extracellular concentrations are opposite. The width and the amplitude of both depolarized and hyperpolarized bands are different. Such a property is uncommon in pattern formation (30). The characteristic time is typically an ionic diffusive time: T ≈ R ²/D ₁ in the limit of large ratio D ₂/D ₁ (Fig. 4). In agreement with experiments, the characteristic time is ≈1 min for the Fucus zygote (9–12). This time is shorter than that required for any cytological modification [F-actin polymerization, notably (17)], which suggests that ionic current pattern could be the initial event leading to the symmetry breakage. The difference between the total ionic current and the ohmic current is described in Discussion.

To investigate the domain of occurrence of this cellular diffusive instability, we vary the two parameters D ₂/D ₁ and G ₁/G ₂. For instance, for a typical ratio of diffusion coefficients, D ₂/D ₁ = 0.7, which corresponds to K⁺ (ion 1) and Na⁺ (ion 2) and the same ratio of differential membrane conductances, G ₁/G ₂ = –0.8, the membrane potential difference V relaxes to zero (Fig. 4 Inset) on a characteristic electric time t ≈ C_m/(G ₁ + G ₂) as predicted in literature (1, 3). This response corresponds to domain 1 of the stability diagram (Fig. 5). Note the tremendous difference between the diffusive and electric time relaxation (four orders of magnitude). When G ₁/G ₂ < –1, we recover the expected domain of wave propagation for a total negative differential conductance (Fig. 5, domain 3). The stability diagram depends also on the variation of the currents with the concentrations (Fig. 5, domain 2). We have computed this diagram for three values of p. The large ratio of coefficients of diffusion D ₂/D ₁ used here arise from the low calcium mobility caused by the intracellular buffering activity (4, 28, 29). As shown in Fig. 5, for small p, the diffusive cellular instability can appear as soon as D ₂/D ₁ > 1, indicating that a huge value is not essential.

Discussion

Critical Size Evaluation. A linear analysis and numerical computations permit also the determination of the dispersion relation (Fig. 6). Contrary to previous works on electrodynamic instabilities (25), for D ₂/D ₁ = 100, when the conductance G ₁ is decreased from positive values to negative ones, the membrane potential becomes unstable at a finite characteristic wavelength and, beyond, a finite range of wavelength exists for which there is instability (Fig. 6). If the cellular radius is less than a critical one, R _c, the membrane potential is stable. Above R _c, the first mode that is unstable is the dipolar mode as shown in Fig. 2. If the radius is further increased (or D ₁ is decreased), the quadrupolar mode is the most probable and so on [mechanism of wavelength selection (15, 16)]. After ≈10 h, the quadrupolar mode will generate two opposite growing poles similar to the mode observed in experiments in response to a change of calcium diffusion (31) or in the presence of a plane-polarized light (32). The induction of double rhizoids indicates a mechanism of selection of the developmental axis, supporting such a theoretical approach. In the limit of high D ₂/D ₁, R _c ≈ (D ₁/D ₂)σ/|G ₁|. In Fucus, after the fertilization, the total membrane resistance becomes small, a few kiloohms (26), which provides an order of magnitude of |G ₁|. Therefore, for σ ≈ 0.5 Ωm, we get R _c ≈ 31 μm, which is in agreement with the Fucus radius. The presence of a finite wavelength domain of instability occurrence is a fundamental property to establish some analogy with reaction-diffusion system (17–20, 33).

Ohmic Versus Total Extracellular Currents. The vibrating probe consists of a microelectrode that oscillates on a small amplitude farther from the membrane than the Debye layer but sufficiently close to measure a signal (12). The vibrating probe measures the local electric field which, according to Ohm's law, is assumed to be proportional to the total extracellular current. However, this assumption fails in the case of transcellular ionic currents. Indeed, outside the membrane at a distance longer than the Debye length, it is valid to assume electroneutrality: ρ = z ₁ F(C ₁ – C ₁₀) + z ₂ F(C ₂ – C ₂₀) ≈ 0, where ρ is the charge density. In this approximation, the total electrodiffusive current is equal to: where and are the local electric field and the local concentration gradient, respectively. The total current reduces to its ohmic part when the ions flowing through the membrane have a similar diffusion coefficient (e.g., Na⁺, K⁺, and Cl^–) or when concentration gradients are neglected (34). Unfortunately, in Fucus, calcium diffuses slower than potassium or sodium ions. So, a significant test of the validity of our mechanism is to compute the ohmic part, I _ohmic, of the total current. Outside the membrane at a distance longer than the Debye length, using electroneutrality, we calculate: On Fig. 3A, the computation of D̃(I ₁/D ₁ + I ₂/D ₂) shows that a hyperpolarized (depolarized) band matches an output (input) of the ohmic current in agreement with experiments. In agreement with Eq. 10 the total extracellular current is opposite to the ohmic current (Fig. 3A). Note that in the limit where dominates, the vibrating probe can evaluate the current of ion 1: .

Origin of the Instability. To understand the origin of the stationary self-organized pattern described here, let us consider the effect of a local fluctuation of the membrane potential (Fig. 7A) around its resting-state value (Fig. 7A). As shown in Fig. 7A, a local membrane depolarization will occur in response to an increase of a local positive (negative) charge density inside (outside) the cell. This fluctuation generates membrane currents, I ₁ for ion 1 (Ca²⁺) and I ₂ for ion 2 (K⁺) (Fig. 7A). Because G ₁/G ₂ is negative, these currents are opposite. The first consequence is a local membrane current I ₁ + I ₂ (Fig. 7B), which tends to dissipate the fluctuation as G ₁ + G ₂ > 0. The cable model provides the characteristic dynamic coefficient of this process: D_m ≈ rσ/4C_m, where C_m is the membrane capacitance. For a cell diameter r of 10 μm, σ = 0.5 Ωm and C_m = 0.01 Fm ^–2, and D ₂ = 2.0 10^–5 cm²/s, D_m/D ₂ ≈ 10⁶ » 1. Thus, a very fast lateral inhibition will occur. The second consequence of the local membrane potential fluctuation is a variation of the ionic concentrations on either side of the membrane that, in turn, induces a lateral ionic electrodiffusive flux inside and outside the cell (Fig. 7C). These fluxes are opposite and, because both ions diffuse at different speeds, ionic diffusion of 1 and 2 induces spatial ionic charge differences that, in turn, generate an electric field determined (and computed) by Eq. 11. The direction of the electric field depends on the sign of G ₁/D ₁ + G ₂/D ₂, which is negative for D ₂/D ₁ = 100 (Fig. 8). In this case, the diffusion-induced electric field amplifies the initial perturbation of the membrane potential. The characteristic dynamic coefficient of this process is an ionic diffusive one, D ₁ D ₂/D̃. This second effect is a slow local self-activation. Thus, two antagonist effects occur in response to a membrane potential fluctuation: a very fast lateral inhibition arising from membrane potential propagation along the cell membrane due to the cable properties of cell membranes and a slow local amplification due to ionic diffusion-induced currents that enhance the local membrane potential fluctuation. The result is a balance between these two antagonistic effects. The unstable case happens as soon as the perturbation lasts longer than the time required for lateral inhibition, which permits the establishment of a significant ion gradient around the membrane to initiate the instability. Note that the total current and the electric field are opposite, breaking Ohm's law. This finding underlines why the instability cannot be predicted by a classic electric cable. For D ₂/D ₁ = 0.7 (K⁺/Na⁺), the electric field has the other direction (G ₁/D ₁ + G ₂/D ₂ > 0) (Fig. 8). Therefore, no local positive feedback occurs for D ₂/D ₁ < 1 because the two effects depicted in Fig. 7 inhibit the initial perturbation.

It has been suggested that self-organization of the fluid mosaic of charged channel proteins in membranes could explain the formation of the ionic current pattern. This phenomenon occurs with a larger time constant (hours) than that of the diffusive process described here (minutes). This finding suggests that electrodiffusion would be the early event for pattern formation, lateral diffusion of membrane proteins being important to sustain the pattern, at least in the case of Fucus. Other phenomena (e.g., membrane, cell wall, and cytoplasmic reorganization and actin polymerization) with large relaxation times could also contribute to maintaining the pattern and, beyond, the breaking symmetry as time elapses.

After fertilization, the polar axis remains labile for about 10 h. During this period it can be reoriented in the presence of an external cue (9, 10, 13, 14). Two kinds of reorientation can be differentiated. The first appears soon after the fertilization and consists of spontaneous reorientation of the polar axis measured with the vibrating probe (35). The framework of electrodiffusive instabilities provides a simple explanation for this fast reorientation: the current pattern will reorganize under the electric field fluctuation. The amplitude required to induce this reorientation will be larger than thermal fluctuations and of the order of the electric field generated by the pattern of transcellular ionic currents. The second kind of reorientation occurs in response to an environmental cue when the spatial organization of the cytoplasm is clearly visible. For example, a unilateral light can reorient the axis after few hours before a characteristic time (axis fixation). To understand this last process, it is necessary to take into account other phenomena (e.g., signal transduction pathways and cytoplasmic reorganization) with large relaxation times.

Experimental evidence shows a correlation of the polarizations of neighboring Fucus (36). In our model and in the self-aggregation model, the electric field generated by the dipolar transcellular ionic currents decreases like 1/r ³ far from the cellular membrane (r is the radial coordinate). Thus, if two Fucus are sufficiently close, the two loops of currents interact strongly and the instability will select a peculiar mode; their polarizations will be correlated. By symmetry, for two Fucus, the simplest mode is such that the axis is the same as in experiments (36). If the two Fucus are far from each other, thermal noise will prevent any correlation.

Finally, the mechanism described here casts a glance on the role of ionic diffusion in bioelectric self-organization and defines simple conditions for a new kind of instability. It gives a reasonable accurate representation of the spatiotemporal pattern of transcellular ionic currents observed in numerous cells and, notably, in Fucus. The mechanism seems to have the salient features of a Turing-like pattern (17–20, 33). It can be included in the general class introduced by Gierer and Meinhardt (33, 37): local self-activation (diffusion-induced currents) and lateral inhibition (electric relaxation). However, the condition D ₁ < D ₂ is only necessary to have the positive feedback in this mechanism. In Turing, D ₁ < D ₂ contributes to differentiate the timescales of positive and negative effects what underlies the microscopic difference between the two mechanisms.