Kinetic model of the effects of electrogenic enzymes on the membrane potential

Predictions from a mathematical approach to model ionic signaling for rapid responses of Sertoli cells exhibit similarities to pharmacological approaches

Sertoli cells are essential for the male reproduction system as they provide morphological support and nutrients for germ cells to guarantee ongoing spermatogenesis. The aim of this work was to predict the electrical properties at the plasma membrane that trigger Sertoli cell rapid responses by involving ionic channels. The rapid responses of Sertoli cells in culture were monitored using patch clamp electrical measurement and compared to data obtained using pharmacological tools (from intact seminiferous tubules). A mathematical model was used to define the roles of potassium channels and the ATP-dependent Na+/K+pump in these responses. Mathematical data verification was also performed to determine the resting and hormonal stimulated membrane potentials of Sertoli cells in the intact seminiferous tubules and of Sertoli cells in culture (patch clamp measurements). The prediction of these data based on mathematical modeling demonstrated, for the first time, the involvement of potassium channels and the activation of Na+/K+pump in the hyperpolarization of Sertoli cells and their consequent rapid responses. Moreover, the mathematical analysis showing the involvement of ionic balance in the rapid responses of these cells to hormones, such as follicle-stimulating hormone, is consistent with previous reports obtained using pharmacological techniques in Sertoli cells. Thus, the validation of such data is reliable and represents a first step in the proposition for a mathematical model to predict rapid responses of Sertoli cells to hormonal stimuli.

Quantitative model analysis of the resting membrane potential in insect skeletal muscle: Implications for low temperature tolerance

Abiotic stressors, such as cold exposure, can depolarize insect cells substantially causing cold coma and cell death. During cold exposure, insect skeletal muscle depolarization occurs through a 2-stage process. Firstly, short-term cold exposure reduces the activity of electrogenic ion pumps, which depolarize insect muscle markedly. Secondly, during long-term cold exposure, extracellular ion homeostasis is disrupted causing further depolarization. Consequently, many cold hardy insects improve membrane potential stability during cold exposure through adaptations that secure maintenance of ion homeostasis during cold exposure. Less is known about the adaptations permitting cold hardy insects to maintain membrane potential stability during the initial phase of cold exposure, before ion balance is disrupted. To address this problem it is critical to understand the membrane components (channels and transporters) that determine the membrane potential and to examine this question the present study constructed a mathematical "charge difference" model of the insect muscle membrane potential. This model was parameterized with known literature values for ion permeabilities, ion concentrations and membrane capacitance and the model was then further developed by comparing model predictions against empirical measurements following pharmacological inhibitors of the Na⁺/K⁺ ATPase, Cl^- channels and symporters. Subsequently, we compared simulated and recorded membrane potentials at 0 and 31 °C and at 10-50 mM extracellular [K⁺] to examine if the model could describe membrane potentials during the perturbations occurring during cold exposure. Our results confirm the importance of both Na⁺/K⁺ ATPase activity and ion-selective Na⁺, K⁺ and Cl^- channels, but the model also highlights that additional electroneutral flux of Na⁺ and K⁺ is needed to describe how membrane potentials respond to temperature and [K⁺] in insect muscle. While considerable further work is still needed, we argue that this "charge difference" model can be used to generate testable hypotheses of how insects can preserve membrane polarization in the face of stressful cold exposure.

Quantitative Model for Ion Transport and Cytoplasm Conductivity of Chinese Hamster Ovary Cells

In mammalian cells cytoplasm ion concentrations and hence cytoplasm conductivity is an important indicator of their physiological state. Changes in the cytoplasm conductivity has been associated with physiological changes such as progression of cancer and apoptosis. In this work, a model that predicts the effects of physiological changes in ion transport on the cytoplasm conductivity of Chinese hamster ovary (CHO) cells is demonstrated. We determined CHO-specific model parameters, Na⁺/K⁺ ATPase pumps and ion channels densities, using a flux assay approach. The obtained sodium (P_Na), potassium (P_K) and chloride (P_Cl) permeability and Na⁺/K⁺ ATPase pump density were estimated to be 5.6 × 10^-8 cm/s, 5.6 × 10^-8 cm/s, 3.2 × 10^-7 cm/s and 2.56 × 10^-11 mol/cm², respectively. The model was tested by comparing the model predictions with the experimentally determined temporal changes in the cytoplasm conductivity of Na⁺/K⁺ ATPase pump inhibited CHO cells. Cells' Na⁺/K⁺ ATPase pumps were inhibited using 5 mM Ouabain and the temporal behavior of their cytoplasm conductivity was measured using dielectrophoresis cytometry. The measured results are in close agreement with the model-calculated values. This model will provide insight on the effects of processes such as apoptosis or external media ion concentration on the cytoplasm conductivity of mammalian cells.

An approach to expand description of the pump and co-transporter steady-state current

The membrane transporters (pumps and co-transporters) are the main players in maintaining the cell homeostasis. Models of various types, each with their own drawbacks, describe transporter behavior. The aim of this study is to find the link between the biophysically based and empirical models to face and solve their specific problems. Instead of decreasing the number of states and using few complex rate constants as is usually done, we use the number of states as great as possible. Then, each transition in the cycle can represent an elementary process and we can apply the mass action law, according to which if rate constants depend on concentrations the dependence is linear. Thus, the expression for the steady state transporter current can be transformed from a function of rate constants into a function of concentrations. When transporter states form a single cycle, it can be characterized by two modes of action - forward and backward ones. Specific mode is realized depending on the available free energy. Each mode of action is characterized by a set of transporter affinities together with a parameter that describes the maximal turning rate. Except standard affinities corresponding to the substances that are binding to the transporter, affinities for the substances that are released are also defined. Such scheme provides great possibilities to construct approximations as each individual affinity could be estimated from experiments as precisely as possible. The approximations may be used for not only description and study of the transporter current but also in cellular models that attempt to describe wide variety of processes in excitable cells.

The determinants of transverse tubular volume in resting skeletal muscle

The transverse tubular (t)-system of skeletal muscle couples sarcolemmal electrical excitation with contraction deep within the fibre. Exercise, pathology and the composition of the extracellular fluid (ECF) can alter t-system volume (t-volume). T-volume changes are thought to contribute to fatigue, rhabdomyolysis and disruption of excitation–contraction coupling. However, mechanisms that underlie t-volume changes are poorly understood. A multicompartment, history-independent computer model of rat skeletal muscle was developed to define the minimum conditions for t-volume stability. It was found that the t-system tends to swell due to net ionic fluxes from the ECF across the access resistance. However, a stable t-volume is possible when this is offset by a net efflux from the t-system to the cell and thence to the ECF, forming a net ion cycle ECF→t-system→sarcoplasm→ECF that ultimately depends on Na⁺/K⁺-ATPase activity. Membrane properties that maximize this circuit flux decrease t-volume, including P_Na(t) > P_Na(s), P_K(t) < P_K(s) and N_(t) < N_(s) [P, permeability; N, Na⁺/K⁺-ATPase density; (t), t-system membrane; (s), sarcolemma]. Hydrostatic pressures, fixed charges and/or osmoles in the t-system can influence the magnitude of t-volume changes that result from alterations in this circuit flux. Using a parameter set derived from literature values where possible, this novel theory of t-volume was tested against data from previous experiments where t-volume was measured during manipulations of ECF composition. Predicted t-volume changes correlated satisfactorily. The present work provides a robust, unifying theoretical framework for understanding the determinants of t-volume.

Functional consequences of NKCC2 splice isoforms: insights from a Xenopus oocyte model

The Na(+)-K(+)-2Cl(-) cotransporter NKCC2 is exclusively expressed in the renal thick ascending limb (TAL), where it exists as three main splice isoforms, NKCC2B, NKCC2A, and NKCC2F, with the latter two predominating. NKCC2A is expressed in both medullary and cortical TAL, but NKCC2F localizes to the medullary TAL. The biochemical characteristics of the isoforms have been extensively studied by ion uptake studies in Xenopus oocytes, but the functional consequences of alternative splicing remain unclear. We developed a charge-difference model of an NKCC2-transfected oocyte. The model closely recapitulated existing data from ion-uptake experiments. This allowed the reconciliation of different apparent Km values reported by various groups, which have hitherto either been attributed to species differences or remained unexplained. Instead, simulations showed that apparent Na(+) and Cl(-) dependencies are influenced by the ambient K(+) or Rb(+) bath concentrations, which differed between experimental protocols. At steady state, under bath conditions similar to the outer medulla, NKCC2F mediated greater Na(+) reabsorption than NKCC2A. Furthermore, Na(+) reabsorption by the NKCC2F-transfected oocyte was more energy efficient, as quantified by J NKCC/J Pump. Both the increased Na(+) reabsorption and the increased efficiency were eroded as osmolarity decreased toward levels observed in the cortical TAL. This supports the hypothesis that the NKCC2F is a medullary specialization of NKCC2 and demonstrates the utility of modeling in analyzing the functional implications of ion uptake data at physiologically relevant steady states.

Regulatory volume decrease of rat kidney principal cells after successive hypo-osmotic shocks

Outer Medullary Collecting Duct (OMCD) principal cells are exposed to significant changes of the extracellular osmolarity and thus the analysis of their regulatory volume decrease (RVD) function is of great importance in order to avoid cell membrane rupture and subsequent death. In this paper we provide a sub-second temporal analysis of RVD events occurring after two successive hypo-osmotic challenges in rat kidney OMCD principal cells. We performed experimental cell volume measurements and created a mathematical model based on our experimental results. As a consequence of RVD the cell expels part of intracellular osmolytes and reduces the permeability of the plasma membrane to water. The next osmotic challenge does not cause significant RVD if it occurs within a minute after the primary shock. In such a case the cell reacts as an ideal osmometer. Through our model we provide the basis for further detailed studies on RVD dynamical modeling.

Relationships between resting conductances, excitability, and t-system ionic homeostasis in skeletal muscle

Activation of skeletal muscle fibers requires rapid sarcolemmal action potential (AP) conduction to ensure uniform excitation along the fiber length, as well as successful tubular excitation to initiate excitation–contraction coupling. In our companion paper in this issue, Pedersen et al. (2011. J. Gen. Physiol. doi:10.1085/jgp.201010510) quantify, for subthreshold stimuli, the influence upon both surface conduction velocity and tubular (t)-system excitation of the large changes in resting membrane conductance (G_M) that occur during repetitive AP firing. The present work extends the analysis by developing a multi-compartment modification of the charge–difference model of Fraser and Huang to provide a quantitative description of the conduction velocity of actively propagated APs; the influence of voltage-gated ion channels within the t-system; the influence of t-system APs on ionic homeostasis within the t-system; the influence of t-system ion concentration changes on membrane potentials; and the influence of Phase I and Phase II G_M changes on these relationships. Passive conduction properties of the novel model agreed with established linear circuit analysis and previous experimental results, while key simulations of AP firing were tested against focused experimental microelectrode measurements of membrane potential. This study thereby first quantified the effects of the t-system luminal resistance and voltage-gated Na⁺ channel density on surface AP propagation and the resultant electrical response of the t-system. Second, it demonstrated the influence of G_M changes during repetitive AP firing upon surface and t-system excitability. Third, it showed that significant K⁺ accumulation occurs within the t-system during repetitive AP firing and produces a baseline depolarization of the surface membrane potential. Finally, it indicated that G_M changes during repetitive AP firing significantly influence both t-system K⁺ accumulation and its influence on the resting membrane potential. Thus, the present study emerges with a quantitative description of the changes in membrane potential, excitability, and t-system ionic homeostasis that occur during repetitive AP firing in skeletal muscle.

Minimal models of electric potential oscillations in non-excitable membranes

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.

Control of cell volume in skeletal muscle

Regulation of cell volume is a fundamental property of all animal cells and is of particular importance in skeletal muscle where exercise is associated with a wide range of cellular changes that would be expected to influence cell volume. These complex electrical, metabolic and osmotic changes, however, make rigorous study of the consequences of individual factors on muscle volume difficult despite their likely importance during exercise. Recent charge-difference modelling of cell volume distinguishes three major aspects to processes underlying cell volume control: (i) determination by intracellular impermeant solute; (ii) maintenance by metabolically dependent processes directly balancing passive solute and water fluxes that would otherwise cause cell swelling under the influence of intracellular membrane-impermeant solutes; and (iii) volume regulation often involving reversible short-term transmembrane solute transport processes correcting cell volumes towards their normal baselines in response to imposed discrete perturbations. This review covers, in turn, the main predictions from such quantitative analysis and the experimental consequences of comparable alterations in extracellular pH, lactate concentration, membrane potential and extracellular tonicity. The effects of such alterations in the extracellular environment in resting amphibian muscles are then used to reproduce the intracellular changes that occur in each case in exercising muscle. The relative contributions of these various factors to the control of cell volume in resting and exercising skeletal muscle are thus described.

Extracellular charge adsorption influences intracellular electrochemical homeostasis in amphibian skeletal muscle

The membrane potential measured by intracellular electrodes, E_m, is the sum of the transmembrane potential difference (E₁) between inner and outer cell membrane surfaces and a smaller potential difference (E₂) between a volume containing fixed charges on or near the outer membrane surface and the bulk extracellular space. This study investigates the influence of E₂ upon transmembrane ion fluxes, and hence cellular electrochemical homeostasis, using an integrative approach that combines computational and experimental methods. First, analytic equations were developed to calculate the influence of charges constrained within a three-dimensional glycocalyceal matrix enveloping the cell membrane outer surface upon local electrical potentials and ion concentrations. Electron microscopy confirmed predictions of these equations that extracellular charge adsorption influences glycocalyceal volume. Second, the novel analytic glycocalyx formulation was incorporated into the charge-difference cellular model of Fraser and Huang to simulate the influence of extracellular fixed charges upon intracellular ionic homeostasis. Experimental measurements of E_m supported the resulting predictions that an increased magnitude of extracellular fixed charge increases net transmembrane ionic leak currents, resulting in either a compensatory increase in Na⁺/K⁺-ATPase activity, or, in cells with reduced Na⁺/K⁺-ATPase activity, a partial dissipation of transmembrane ionic gradients and depolarization of E_m.

Volumetric response of vertebrate hepatocytes challenged by osmotic gradients: a theoretical approach

In this study we use a theoretical approach to study the volumetric response of goldfish hepatocytes challenged by osmotic gradients and compared it with that of hepatocytes from another teleost (the trout) and a mammal (the rat). Particular focus was given to the multiple non-linear interactions of transport systems enabling hypotonically challenged cells to trigger a compensatory response known as volume regulatory decrease or RVD. For this purpose we employed a mathematical model which describes the rates of change of the intracellular concentrations of main diffusible ions, of the cell volume, and of the membrane potential. The model was fitted to experimental data on the kinetics of volume change of hepatocytes challenged by anisotonic media. In trout and rat hepatocytes, experimental results had shown that hypotonic cell swelling was followed by RVD, whereas goldfish cells swelled with no concomitant RVD (M.V. Espelt et al., 2003, J. Exp. Biol. 206, 513-522). A comparison between data predicted by the model and that obtained experimentally suggests that in trout and rat hepatocytes hypotonicity activates a sensor element and this, in turn, activates an otherwise silent efflux of KCl - whose kinetics could be successfully predicted - thereby leading to volume down-regulation. In contrast, with regard to the absence of RVD in goldfish hepatocytes the model proposed suggests that either a sensor element triggering RVD is absent or that the effector mechanism (the loss of KCl) remains inactive under the conditions employed. In line with this, we recently found that extracellular nucleotides may be required to induce RVD in these cells, indicating that our model could indeed lead to useful predictions.

Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E(m)) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E(m) due to its influence upon transmembrane ion fluxes and cell volume (V(c)). These interrelationships between E(m), V(c) and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman-Hodgkin-Katz equation and the Gibbs-Donnan equilibrium, to more recent "current-summing" models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the "charge-difference" modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E(m), V(c) and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E(m), V(c) and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E(m), V(c) and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.

The contribution of refractoriness to arrhythmic substrate in hypokalemic Langendorff-perfused murine hearts

The clinical effects of hypokalemia including action potential prolongation and arrhythmogenicity suppressible by lidocaine were reproduced in hypokalemic (3.0 mM K⁺) Langendorff-perfused murine hearts before and after exposure to lidocaine (10 μM). Novel limiting criteria for local and transmural, epicardial, and endocardial re-excitation involving action potential duration (at 90% repolarization, APD₉₀), ventricular effective refractory period (VERP), and transmural conduction time (Δlatency), where appropriate, were applied to normokalemic (5.2 mM K⁺) and hypokalemic hearts. Hypokalemia increased epicardial APD₉₀ from 46.6 ± 1.2 to 53.1 ± 0.7 ms yet decreased epicardial VERP from 41 ± 4 to 29 ± 1 ms, left endocardial APD₉₀ unchanged (58.2 ± 3.7 to 56.9 ± 4.0 ms) yet decreased endocardial VERP from 48 ± 4 to 29 ± 2 ms, and left Δlatency unchanged (1.6 ± 1.4 to 1.1 ± 1.1 ms; eight normokalemic and five hypokalemic hearts). These findings precisely matched computational predictions based on previous reports of altered ion channel gating and membrane hyperpolarization. Hypokalemia thus shifted all re-excitation criteria in the positive direction. In contrast, hypokalemia spared epicardial APD₉₀ (54.8 ± 2.7 to 60.6 ± 2.7 ms), epicardial VERP (84 ± 5 to 81 ± 7 ms), endocardial APD₉₀ (56.6 ± 4.2 to 63.7 ± 6.4 ms), endocardial VERP (80 ± 2 to 84 ± 4 ms), and Δlatency (12.5 ± 6.2 to 7.6 ± 3.4 ms; five hearts in each case) in lidocaine-treated hearts. Exposure to lidocaine thus consistently shifted all re-excitation criteria in the negative direction, again precisely agreeing with the arrhythmogenic findings. In contrast, established analyses invoking transmural dispersion of repolarization failed to account for any of these findings. We thus establish novel, more general, criteria predictive of arrhythmogenicity that may be particularly useful where APD₉₀ might diverge sharply from VERP.

A quantitative analysis of cell volume and resting potential determination and regulation in excitable cells

This paper quantifies recent experimental results through a general physical description of the mechanisms that might control two fundamental cellular parameters, resting potential (Em) and cell volume (Vc), thereby clarifying the complex relationships between them. Em was determined directly from a charge difference (CD) equation involving total intracellular ionic charge and membrane capacitance (Cm). This avoided the equilibrium condition dEm/dt = 0 required in determinations of Em by previous work based on the Goldman-Hodgkin-Katz equation and its derivatives and thus permitted precise calculation of Em even under non-equilibrium conditions. It could accurately model the influence upon Em of changes in Cm or Vc and of membrane transport processes such as the Na+-K+-ATPase and ion cotransport. Given a stable and adequate membrane Na+-K+-ATPase density (N), Vc and Em both converged to unique steady-state values even from sharply divergent initial intracellular ionic concentrations. For any constant set of transmembrane ion permeabilities, this set point of Vc was then determined by the intracellular membrane-impermeant solute content (X-i) and its mean charge valency (zX), while in contrast, the set point of Em was determined solely by zX. Independent changes in membrane Na+ (PNa) or K+ permeabilities (PK) or activation of cation-chloride cotransporters could perturb Vc and Em but subsequent reversal of such changes permitted full recovery of both Vc and Em to the original set points. Proportionate changes in PNa, PK and N, or changes in Cl- permeability (PCl) instead conserved steady-state Vc and Em but altered their rates of relaxation following any discrete perturbation. PCl additionally determined the relative effect of cotransporter activity on Vc and Em, in agreement with recent experimental results. In contrast, changes in Xi- produced by introduction of a finite permeability term to X- (PX) that did not alter zX caused sustained changes in Vc that were independent of Em and that persisted when PX returned to zero. Where such fluxes also altered the effective zX they additionally altered the steady state Em. This offers a basis for the suggested roles of amino acid fluxes in long-term volume regulatory processes in a variety of excitable tissues.

An elementary kinetic model of energy coupling in biological membranes

The purpose of this work is to contribute to the understanding of the fundamental kinetic properties of the processes of energy coupling in biological membranes. For this, we consider a model of a microorganism that, in its plasma membrane, expresses two electrogenic enzymes (E(1) and E(2)) transporting the same monovalent cation C and electrodiffusive paths for C and for a monovalent anion A. E(1) (E(2)) couples transport C to the reaction S(1)<-->P(1) (S(2)<-->P(2)). We developed a mathematical model that describes the rate of change of the electrical potential difference across the membrane, of the internal concentrations of C and A, and of the concentrations of S(2) and P(2). The enzymes are incorporated via two-state kinetic models; the passive ionic fluxes are represented by classical formulations of electrodiffusion. The microorganism volume is maintained constant by accessory regulatory devices. The model is utilized for stationary and dynamic studies for the case of bacteria employing the electrochemical gradient of Na(+) as energetic intermediate. Among other conclusions, the results show that the membrane potential represents the relevant kinetic intermediate for the overall coupling between the energy donor reaction S(1)<-->P(1) and the synthesis of S(2).

Modeling cell volume regulation in nonexcitable cells: the roles of the Na+ pump and of cotransport systems

The purpose of this study is to contribute to understanding the role of Na+-K+-ATPase and of ionic cotransporters in the regulation of cell volume, by employing a model that describes the rates of change of the intracellular concentrations of Na+, K+, and Cl-, of the cell volume, and of the membrane potential. In most previous models of dynamic cellular phenomena, Na+-K+-ATPase is incorporated via phenomenological formulations; the enzyme is incorporated here via an explicit kinetic scheme. Another feature of the present model is the capability to perform short-term cell volume regulation mediated by cotransporters of KCl and NaCl. The model is employed to perform numerical simulations for a "typical" nonpolarized animal cell. Basically, the results are consistent with the view that the Na+ pump mainly plays a long-term role in the maintenance of the electrochemical gradients of Na+ and K+ and that short-term cell volume regulation is achieved via passive transport, exemplified in this case by the cotransport of KCl and NaCl.

Mutual interaction of ion uptake and membrane potential

The concentration dependence of cation uptake by the cell may be considerably complicated when this uptake is accompanied by a depolarization of the cell membrane. In case of carrier-mediated transport deviations from Michaelis-Menten kinetics may come to the fore comparable to those found in a dual mechanism of cation uptake or when substrate inhibition is involved. This remains true when only the maximum rate of uptake and not the Km is dependent upon the membrane potential. We have proven this by means of computer simulation of cation transport mediated by a non-mobile carrier. Under restricted conditions still apparent Michaelis-Menten kinetics may be found despite the fact that the membrane potential varies with increasing substrate cation concentration. But even then there are still differences with 'normal' transport kinetics. A non-competitive inhibitor does not only affect the maximum rate of uptake but also the apparent Km. Depolarization of the cells by a cation which passes the cell membrane by means of diffusion, affects the uptake of the substrate cation almost in the same way as a non-competitive inhibitor does and causes both a decrease in the maximum rate of uptake and an increase in Km. In the case of competitive inhibition the apparent affinity of the inhibitor for the carrier depends upon the rate of transfer of this inhibitor through the cell membrane. The mutual influence of cation uptake and membrane potential is dealt with for uniport of either monovalent or divalent cations and for cotransport of monovalent cation with protons, as well. Possible effects of the surface potential are accounted for.