Minimal models of electric potential oscillations in non-excitable membranes

Abstract

Sustained oscillations in the membrane potential have been observed in a variety of cellular and subcellular systems, including several types of non-excitable cells and mitochondria. For the plasma membrane, these electrical oscillations have frequently been related to oscillations in intracellular calcium. For the inner mitochondrial membrane, in several cases the electrical oscillations have been attributed to modifications in calcium dynamics. As an alternative, some authors have suggested that the sustained oscillations in the mitochondrial membrane potential induced by some metabolic intermediates depends on the direct effect of internal protons on proton conductance. Most theoretical models developed to interpret oscillations in the membrane potential integrate several transport and biochemical processes. Here we evaluate whether three simple dynamic models may constitute plausible representations of electric oscillations in non-excitable membranes. The basic mechanism considered in the derivation of the models is based upon evidence obtained by Hattori et al. for mitochondria and assumes that an ionic species (i.e., the proton) is transported via passive and active transport systems between an external and an internal compartment and that the ion affects the kinetic properties of transport by feedback regulation. The membrane potential is incorporated via its effects on kinetic properties. The dynamic properties of two of the models enable us to conclude that they may represent alternatives enabling description of the generation of electrical oscillations in membranes that depend on the transport of a single ionic species.

Appendices

Appendix 1: A simple kinetic model of a cation pump

Figure 2a depicts a simple kinetic diagram of the active transport mediated by system 1 (Fig. 1a), in which the transport of y is coupled to the reaction S ↔ P. From analysis of this kinetic diagram, the net flux of transport J _p in the y → x _o direction is given by:

$$ J_{\text{p}} = (N_{\text{p}} /\Upsigma_{\text{p}} )\,(u_{01} v_{10} Sy-u_{10} v_{01} Px_{o} ), $$

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$$ {\text{with}}\,\Upsigma_{\text{p}} = u_{10} + v_{10} + u_{01} Sy + v_{01} Px_{\text{o}} . $$

In these expressions, N _p is the total membrane density of the enzyme; u ₀₁, v ₁₀, u ₁₀, and v ₀₁ are rate constants (cf. Fig. 2a), and S, P, x _o, and y are the concentrations of the corresponding species. If the process operates in an irreversible fashion, J _p will be given by:

$$ J_{\text{p}} = (N_{\text{p}} /\Upsigma_{\text{p}} )\,(u_{01} v_{10} Sy). $$

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Equation 20 can be identified with the term [δ y/(1 + γ y)] of Models II and III (Eqs. 6 and 10, respectively), with δ and γ given by:

$$ \begin{gathered} \delta = (N_{\text{p}} u_{01} v_{10} S)/(u_{10} + v_{10} + v_{01} Px_{\text{o}} ) \quad {\text{and}} \\ \gamma = (u_{01} S)/(u_{10} + v_{10} + v_{01} Px_{\text{o}} ). \hfill \\ \end{gathered} $$

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Note that under some of the conditions in this study substrate S can be metabolite M.

Appendix 2: Derivation of expressions for the feedback effect of y from analysis of explicit kinetic models of channel activation and inactivation

Explicit kinetic schemes for the portion of ion transport mediated by system 3 (Fig. 1a) are shown in Fig. 2b, c. These kinetic models represent simple cases of channel activation (Fig. 2b) and inactivation (Fig. 2c) by the transported ligand, where the channel pre-exists in two conformational states, N ₀ and N ₀*, only one of which (N ₀) is capable of performing net ionic transport. For channel activation (Fig. 2b), the transition from the inactive to the active state is promoted by the state “y” of the transported species. From analysis of the kinetic diagram shown in Fig. 2b, the net flux J of transport of the cation in the x → y direction is given by:

$$ J = (Np/\Upsigma )\,(k_{01} l_{10} xy-k_{10} l_{01} y^{2} ) .$$

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In Eq. 22, N is the total density of the channel; p, q, k ₀₁, l ₁₀, k ₁₀, and l ₀₁ are rate constants (cf. Fig. 2b); x and y are the concentrations of the corresponding species, and Σ is the sum of all the directional diagrams of the model, given by:

$$ \Upsigma = (p + q)\,(l_{10} + k_{10} ) + py(l_{01} y + k_{01} x). $$

If the system operates irreversibly under conditions far from saturation, Eq. 22 can be approximated by:

$$ J \cong [ (Npk_{01} l_{10} /q )/(l_{10} + k_{10} )]xy. $$

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Equation 23 is formally analogous to the term β xy included in Model III (Eq. 10), with β given by (Npk ₀₁ l ₁₀/q)/(l ₁₀ + k ₁₀).

For channel inactivation by the transported species (Fig. 2c), y promotes the transition from the active (N ₀) to the inactive state (N ₀*). The net flux of transport J in the x → y direction is now given by:

$$ J = (Ns/\Upsigma )\,(k_{01} l_{10} x-k_{10} l_{01} y). $$

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where the symbols have analogous meanings to the previous case (cf. Eq. 22). The term Σ is now given by:

$$ \Upsigma = s(l_{10} + k_{10} + k_{01} x) + y[sl_{01} + r(k_{10} + l_{10} )]. $$

If the system operates under irreversible conditions and if the term s k ₀₁ x can be neglected, Eq. 24 can be approximated by:

$$ J = (Ns/\Upsigma )\,(k_{01} l_{10} x) $$

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$$ {\text{with}}\,\Upsigma = s(l_{10} + k_{10} ) + y[sl_{01} + r(k_{10} + l_{10} )]. $$

Equation 25 is formally analogous with the term β x/(1 + φ y) included in Model II (Eq. 6), with β and φ given by:

$$ \begin{gathered} \beta = (Nsk_{01} l_{10} )/s(l_{10} + k_{10} ) \quad {\text{and}} \\ \varphi = [sl_{01} + r(k_{10} + l_{10} )]/s(l_{10} + k_{10} ). \hfill \\ \end{gathered} $$

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Appendix 3: Stability analysis of model II

The eigenvalues λ _{1, 2} of the characteristic equation of the linear approximation to the model given by Eq. 8 are

$$ \lambda_{1,\,2} = - \theta /2 \pm \Updelta^{1/2} /2, $$

with:

$$ \begin{gathered} \theta = (k + r)\,(r - 1)/r \quad {\text{and}} \\ \Updelta = [ (k - r )(r - 1)/r]^{2} + 4(k/r)(r - 1). \hfill \\ \end{gathered} $$

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Because Δ is necessarily positive, Model II is not capable of exhibiting oscillatory behavior.

Appendix 4: Stability analysis of model III

The eigenvalues λ _1,2 of the characteristic equation of the linear approximation to the model given by Eq. 14 are

$$ \lambda_{1,\,2} = - \theta /2 \pm \Updelta^{1/2} /2, $$

with:

$$ \begin{gathered} \theta = - k/(r - 1) + (r-1)/r \quad {\text{and}} \\ \Updelta = \theta^{2} + 4(k/r)(1-r). \hfill \\ \end{gathered} $$

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If r ≫ 1, θ = 1 − (k/r) and Δ = [1 − (k/r)]² − 4k. If k = r, θ = 0, and Δ = −4k. Therefore, under this condition, because k is necessarily positive, sustained oscillations occur with the period T given by:

$$ T = 2\pi /k^{1/2} $$

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