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Cell membranes contain numerous nanoscale conductors in the form of ion channels and ion pumps1–4 that work together to form ion concentration gradients across the membrane, which can be triggered to release an action potential (AP) 1, 5. We ask if artificial cells can be built to utilize ion transport as effectively as natural cells. Here we used the electrogenic cell (electrocyte) of an electric eel to model the formation of AP by tracking the conversion of ion concentration gradients into APs across the different nanoscale conductors. Using the parameters extracted from the model, we designed an artificial cell based on an optimized selection of conductors. The resulting cell is similar to the electrocyte but has higher power output density and greater energy conversion efficiency. We suggest methods for producing these artificial cells that have applications in powering medical implants and other tiny devices.
The electrocyte in an electric eel (Electrophorus electricus) can generate potentials of about 600V2, 3, 6 to stun prey and ward off predators (Fig. 1a). The transmembrane proteins in the electrocytes are asymmetrically distributed across two primary membranes, one innervated and the other non-innervated (Fig. 1b), and are separated by an insulating septa (wall). The non-innervated membrane has numerous sodium potassium ATPase pumps (Na+/K+) and both K+ and chloride (Cl−) channels. The innervated membrane contains high densities of acetylcholine receptors (AChRs), voltage-gated Na+ channels (which are responsible for activating APs), voltage-gated K+ channels (Kvs)7, inward rectifier K+ channels (Kirs, which are ion channels that stop ion flow when the membrane is depolarized)6 and leak channels.
When the chemical agonist, acetylcholine (ACh), is released into the junction between the AChR and another nearby excitable cell (synapse), AChRs bind with ACh and become permeable to the cations, Na+ and K+. This opens the AChRs and depolarizes the innervated membrane, raising the probability that voltage-gated Na+ channels will open (Fig. 1c)3, 6, 8. Depolarization causes the normally negative innervated cell membrane potential to become positive with respect to the potential on the non-innervated membrane. With Na+ flowing into the cell, the innervated membrane potential further increases, causing the opening of additional voltage-gated Na+ channels. This cascade of AChRs opening large number of Na+ channels results in AP formation on the innervated membrane. The Kir channels are closed during this stage, which speeds the increase of the membrane potential. The maximum innervated membrane potential is +65 mV5, 6. The non-innervated membrane potential remains at approximately −85mV due to ATPase, K+ channel and Cl− activity2, 5, 6. After the peak of the AP, the innervated membrane is repolarized with the inactivation of Na+ channels8 and the opening of Kir and Kvs channels. Ion flux through leak channels further expedites the restoration of membrane potential to the resting state (−85mV). The ion concentration gradient is sustained by Na+/K+-ATPase with energy from ATP hydrolysis9. Ion flux, through the open ion channels, results in a net electric current from the innervated membrane to the non-innervated membrane.
Many mathematical models of axons have been produced, including the Hodgkin-Huxley model, which depicts the AP formation on the squid giant axon (SGA) by ion transport through Na+ channels, Kvs and leak channels1; the natural electrocyte is more complicated as it is a polarized cell. Cell polarity is the necessary feature for directional transport and formation of transcellular potential10. Our model is based on cells arranged in series (Fig. 2a) powering an external circuit. In it, the innervated membrane is described by the Hodgkin-Huxley model, with additional current through Kirs and AChRs. The ion channels on the non-innervated membrane are described by the Goldman-Hodgkin-Katz current equation8, 11, which depicts the ion flux as a function of membrane potential and the transmembrane ion gradient. To compare the performance of different cell types, we constructed a pseudo squid giant axon (PSGA) model from the known characteristics of the SGA and a generic model for the non-innervated membrane (this model is referred to as "pseudo" because the SGA is not a polarized cell). Detailed equations and parameters are included in the supplementary information.
Many electrophysiological parameters for the natural electrocyte, such as the channel rate constants and densities, are unknown or span a broad range12, 13. We found accurate values for these parameters numerically by using a nonlinear least square difference method to match initial model results with the nuances of the rise and fall of AP in published electrocyte data5 (Fig. 2b). The ion channel configuration of the innervated membrane extracted from the AP curve using this method appear in Table 1; other physiological parameters, including the channel rate constants and the ion transporter configuration of the non-innervated membrane, were analyzed and are reported in the supplementary information. In the model, the conductance of Na+ channels on the innervated membrane was found to be 1.57×103 ps·μm−2, equivalent to a channel density of≈143 channels·μm−2, which falls between reported values of ≈100 channels·μm−2,12, 13 to≈500 channels·μm−2 13. The innervated membrane capacitance was found to be 12.1 μF·cm−2, which has been experimentally reported as 15.6μF·cm−2 5; this high capacitance can be due to high local tortuosity or modification to the lipid5, 14. The derived resting resistance, 16.3 Ω·cm2, is also consistent with experimental data2, 5. The model results for channel opening and closing sequences (Fig. 2c) are also in agreement with observations from the natural electrocyte 6, 8.
As is well known for normal voltaic batteries, the voltage output of our model cell decreases when the external resistance (R) decreases (Fig. 3), influencing the single pulse energy output (Ws). The dependence of energy output (Ws) on external resistance has been reported phenomenologically15, 16. The cell has maximum Ws when the voltage output is around half of the open-circuit voltage.
The energy output of the artificial cell was optimized numerically (Table 1). The design started as a list of nanoconductors (channels and pumps), basic physical parameters (two membranes, known ion permeability, etc.) and a mathematical equation, known as an objective function, that quantifies the desired criteria (i.e. energy output or energy density). Numerical methods are used to impartially maximize the value of the objective function given the starting parameters, equations and constraints. The output is the configuration for an artificial cell that maximizes the value of the objective function. Constrained nonlinear numerical optimization was used to guide the parametric design for the artificial cell. One constraint was to maintain the overall channel number as a constant on each membrane; the numbers of individual types of ion channels were varied in the optimization algorithm. The resulting artificial cell, with the same total channel/pump density as the natural electrocyte, produces 41 % more energy per pulse (Table 1), with longer AP duration (Fig. 3). These results are based on the comparison between the models of the artificial cell and the electrocyte there may a small difference between the model and true physical behaviour, but comparing the model results eliminates this error.
The power output of a cell is affected by both R and the stimulus interval. At any given R, the cell produces maximum power output (P) when it is excited at an interval slightly longer than the refractory period (the time taken to ready for another AP) plus the AP duration (Fig. 4a). The natural electrocyte has a peak power output of 0.427 W·m−2, while one PSGA produces only 0.0233 W·m−2. One artificial cell can produce 0.545 W·m−2 (Table 1); without increasing channel or pump density, the peak power output for the artificial cell surpasses the natural electrocyte by 28%.
The transmembrane ion gradient of the cell is sustained by Na+/K+-ATPases, which are fuelled by ATP hydrolysis9 using the energy from the oxidization of glucose or fatty acids17. In our model the natural electrocyte has an energy conversion efficiency of 14.7 %; it has been reported experimentally to be 15.4 %15. The PSGA converts energy more efficiently (19.7 %) than the electrocyte, but the SGA serves a different function and the natural SGA and PSGA model have different electrophysiological parameters (such as unitary conductance’s) than the electrocyte. The peak efficiency of the artificial cell was found to be 19.2 % (Table 1).
Our numerical optimization of the cell design has produced the design for an artificial cell with both higher power output density and greater energy conversion efficiency than the natural electrocyte. The lack of voltage-gated K+ channels in the artificial cell (Table 1) is consistent with the understanding of their function to shorten AP duration. The AP from the PSGA has smaller amplitude, shorter duration and shorter refractory period. These differences are well suited to its role to conduct nerve pulses at high speed with little energy use. The electrocyte must generate large pulses less frequently to shock prey and defend against predators; its AP has a broader duration and higher amplitude; the AP of the artificial cell has even longer duration.
The reported configurations all fall into a stable excitable region, where at most one action potential is elicited by one stimulus. Intrinsic in the model is a regime in which the cell oscillates (repetitive firings of action potentials without recurring stimuli). In this regime, even at the resting potential, the inward sodium current through the Na+ channels exceeds the outward current through the Kir and leak channels, which depolarizes the membrane, makes the Na current regenerative and forms repetitive APs. As both power output and energy conversion efficiency are lower in this regime, it was not included in the optimization.
Each artificial cell can produce 0.545 W·m−2. The power output density of cells in series would scale linearly with the number of layers. 300 μW could be produced continuously with a device of 4.3 mm × 4.3 mm × 3.9 mm, with a channel density on the innervated membrane of≈350 channels·μm−2. It may be possible to increase this power density further given more than five-fold higher channel density measurements in other cells8, 18; however, the maximum energy conversion efficiency is most dependent on the performance of individual channels and the ratio of different channel types rather than the overall channel density. The artificial cell would be able to supply electrical energy for powering medical implants, including retinal prostheses19. The supply of ATP can be sustained either by coupling a proton-gradient-driven FoF1- ATPase with a light-driven proton pump such as bacteriorhodopsin 20, or by inserting isolated mitochondria21 or bacteria22 into the artificial cell. The model was also used to find the simplest cell capable of generating an AP from an electrical stimulus; the simplest cell requires voltage-gated Na channels and K+ leak channels on the innervated membrane and both Kv channels and Na+/K+-ATPases on the non-innervated membrane. The performance of the simplest cell is substantially lower than the electrocyte or the artificial cell (Table 1).
The artificial cell would consist of two artificial cell membranes (mimicking innervated and non-innervated membranes) separating aqueous solutions. Supported bilayers can maintain the function of inserted ion transporters23; each membrane could be a supported lipid bilayer on a porous structure, such as mesoporous silica24. Many natural ion channels can be purified by affinity chromatography and inserted into lipid bilayers by vesicle fusion25, 26. The numbers of inserted ion channels/pumps can be monitored by measuring the membrane current during insertion. Artificial ion channels might be made by precisely tailoring the pore size of SiO2 nanopores using approaches such as TiO2 atomic layer deposition27 and organic group functionalization28. Synthetic channels can also be engineered using α-hemolysin (α-HL); their properties (e.g. unitary conductance, gating, and ion selectivity) can be tailored by protein or chemical engineering29. Artificial channels may also be synthesized from peptides or other molecules30. These artificial channels can mimic natural channels and may offer enhanced stability and tailored electrophysiology. The artificial cells would be connected in series, to increase the total voltage output, or in parallel, to raise the current output. A number of layered cells could be fired by injecting current at the top or bottom of the stack.
It is interesting to note that this work has mapped out changes in the system level design of the electrocyte that could produce higher energy density and convert energy more efficiently. These changes may show the next step in the evolution of these cells; it is also possible that there are advantages to the species in maintaining this lower energy conversion efficiency.
Models were written in MATLAB 7.5 (The Mathworks, Natick, MA) running under a 64-bit operating system (Windows Server 2003, Microsoft Corporation, Redmond, WA) on a multiprocessor server with 8 GB of RAM. Unknown channel parameters were extracted by matching published experimental data to the model using regression methods. Once the natural channel parameters were found, the parametric design inputs for the model were varied using a non-linear optimization algorithm to search for the optimal configuration of the artificial cell to maximize the objective functions. Objective functions are the mathematical relationships that quantify the “goodness” of the design, such as the energy conversion efficiency or the energy density of the devices. The optimization algorithm would vary the design parameters and calculate the formation of many action potentials until steady state conditions were reached; at that point the objective function would be evaluated and the optimization routine would compare the current value of the objective function to previous values and decide to either stop the optimization (if a maximum was reached) or vary additional parameters in an effort to further maximize the objective function.
We thank Fred Sigworth, Eric Jakobsson, Shreedhar Natarajan, Janet Novotny, T.P. Ma and Sandy Yulke for their discussions and comments. The full description of the procedures used in this paper requires the identification of certain software and operating systems and their suppliers. The inclusion of such information should in no way be construed as indicating that such software or operating systems are endorsed by NIST or are recommended by NIST or that it is necessarily the best software or operating system for the purposes described. This work is supported by the National Center for Design of Biomimetic Nanoconductors, funded by Grant Number PHS 2 PN2 EY016570B from the National Institutes of Health through the NIH Roadmap for Medical Research.
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Supplementary Information accompanies the paper on www.nature.com/naturenanotechnology
J.X. and D.A.L. conceived and designed the experiments: J. X. performed the experiments: J.X. and D.A.L. analyzed the data and co-wrote the paper.