Introduction

It is well known that the retina is one of the most energy-consuming tissues with the highest oxygen demand in the human body. Most of the energy is mainly spent on photoreceptor metabolic requirements. However, the detailed mechanisms are less well understood. It is generally agreed that photoreceptor energy demand in darkness is greater than that in light conditions. ATP expenditure by rods is reported to be approximately 9 \(\times\) 107 molecules of ATP s−1 in darkness and 2 \(\times\) 107 molecules of ATP s−1 in bright illumination1. Energy is predominantly consumed via the Na+/K+ ATPase pump (INaK) and plasma membrane Ca2+ ATPase pump (IPMCA) for removing excess Na+ and Ca2+, respectively. One ATP molecule is hydrolyzed to transport three Na+ ions out and two K+ ions into the cell through INaK, whereas IPMCA has a stoichiometry of 1:1, i.e., one Ca2+ per hydrolyzed ATP.

The initial steps of vision represented by photoreceptors via light-induced isomerization of 11-cis-retinal to all-trans-retinal trigger the phototransduction cascade and the subsequent decline in cGMP concentration, leading to the closure of light-sensitive current and neuronal hyperpolarization. It has been suggested that the energy expenditure of the complex transduction cascades at saturating illumination is relatively small compared to the total energy consumption in darkness1. This evidence may support the view that photoreceptors use most of their energy to maintain ion homeostasis via INaK and IPMCA in darkness and in light. Understanding the fundamentals of ion homeostatic regulatory mechanisms may allow for the development of effective treatment strategies for retinal diseases related to photoreceptor energy consumption.

Photoreceptors absorb nutrients and oxygen to generate energy from the retinal pigment epithelium. Mitochondrial oxidative phosphorylation is the primary fuel source for photoreceptors. A continuous O2 supply is essential for photoreceptors because it cannot be stored. A disruption in the balance between oxygen supply and energy demand impairs oxidative phosphorylation, leading to many retinal diseases, such as retinitis pigmentosa, age-related macular degeneration, and retinal detachment. Rods are more sensitive than cones. Morphological abnormalities in human rods can be observed after age 302. In age-related macular degeneration, rods are affected earlier and more seriously than cones3. A decrease in mitochondria reduces the cellular membrane potential and energy supply, leading to increased reactive oxygen species and photoreceptor loss4. To date, no effective treatment has been established for photoreceptor loss, but numerous therapeutic approaches have reported some success in slowing photoreceptor degeneration by downregulating energy demand.

In attempts to improve photoreceptor survivability and prevent vision loss, several studies have reported some neuroprotective therapies, such as an adenosine monophosphate-activated protein kinase activator for preserving ATP levels under photostress5, a ketogenic and low-protein diet for slowing retinal degeneration6, and nicotinamide mononucleotide for restoring retinal detachment7. Furthermore, some studies have shown improvement in mitochondrial function and reduction in cell loss by long wavelengths absorbed in mitochondrial respiration4,8. A better understanding of the crucial energy demand in photoreceptors, i.e., ionic current activities, may provide an effective tool for investigating retinal pathophysiological changes and analyzing novel therapeutic treatments related to the energy consumption of photoreceptors.

Mathematical models have provided extensive insights into complex biophysiological phenomena and are widely applied to describe clinical and experimental studies. Several conductance-based models of rod photoreceptors have well described the changes in electrical properties, ionic currents, and light-sensitive currents in response to light in lower vertebrate photoreceptors9,10,11. However, the alterations in intracellular ion concentrations essential for evaluating each ionic current’s driving force and estimating the energy expenditure via INaK and IPMCA have not been detailed. Furthermore, electrophysiological and morphological differences exist between lower vertebrate and mammalian rod photoreceptors.

In this preliminary study, we aimed to clarify the composition of the ATP consumption factors of mouse rod photoreceptors in darkness and in light. The mathematical formulas for ionic current activities combined with a phototransduction model were introduced to form a new mathematical model for estimating the energy expenditure of each ionic current. The proposed model should provide a better understanding of the critical role of physiological phenomena with detailed mechanisms that could explain how photoreceptors maintain ion homeostasis in response to light and darkness.

Methods

Model characteristics

The genes encoding ion channels in mouse rods have been revealed using single-cell RNA-Seq analysis12. The proposed model consisted of various ion channels, ion pumps, exchangers, and transporters identified using a gene expression database incorporating an available electrophysiological recording of each specific gene (Table 1).

Table 1 Various ion channels, ion pumps, exchangers, and transporters analyzed in the current study.
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Rod photoreceptors consist of four parts: outer segment, inner segment, cell body, and synaptic terminal. For simplification, as in earlier studies9,10, the rod photoreceptor in the current research is divided into two compartments: (1) outer segment and (2) inner segment with cell body and synaptic terminal (Fig. 1).

Figure 1
figure 1

Schematic of the mouse rod photoreceptor model in the current study. (Upper) various ion channels, ion pumps, exchangers, transports, intracellular calcium system, and essential ion species (K+, Na+, and Cl−), (Lower) two main compartments: (1) outer segment and (2) inner segment with cell body and synaptic terminal.

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The cyclic nucleotide-gated ion channel (ICNG) and potassium-dependent sodium/calcium exchanger (INCKX) are only found in the plasma membrane of the outer segment. ICNG, a nonselective cation channel with no significant anion permeability, is controlled by the light-sensitive conductance of two messenger molecules, cGMP and Ca2+. ICNG forms heterotetrameric complexes composed of three Cnga1 subunits and one Cngb1 subunit28, mediating Na+ and Ca2+ influxes and K+ efflux. The permeability ratios of Na+ relative to K+ and Na+ relative to Ca2+ are approximately 1 and 2–6, respectively29,30,31,32. INCKX has a stoichiometry of exchange of four Na+ for one Ca2+ and one K+ and plays a vital role in maintaining Ca2+ efflux function in the outer segment. The homomeric channel of INCKX is formed by Slc24a1 genes or NCKX1 proteins strongly expressed in mouse rods33. The amplitude of INCKX in salamander photoreceptors is ~ 10% of ICNG, and it has been suggested that it is similar to that of mice17.

Earlier studies have revealed at least five types of ion channels identified in the inner segment and synaptic terminal of amphibian photoreceptors, including hyperpolarization-activated channel (Ih), delayed rectifying K+ channel (IKx), voltage-gated Ca2+ channel (ICa), Ca2+-dependent K+ channel (IKCa), and Ca2+-dependent Cl- channel (IClCa)34,35,36. We assumed that the electrophysiological behaviors of each animal species might have a similar pattern if the genes encoding ion channels have no fundamentally different gene expression profiles. Ih is encoded by the Hcn1 gene, which is strongly expressed in mouse rods12,23. Hcn1 forms a functional homomeric channel for Na+ influx and K+ efflux with a relative permeability ratio for Na+ and K+ (PNa/PK) of 0.3–0.3637,38,39. This channel has a vital role in shaping the initial transient in the membrane hyperpolarization of rod photoreceptors in response to light. IKx or noninactivating K+ channels have recently been classified as voltage-gated K+ channels (IKv) or delayed rectifier potassium currents21. They are comprised of heteromeric tetramers of four pore-forming α subunits, including three Kcnb1 subunits and one Kcnv2 subunit22. Similar to ICNG, ICa is mainly responsible for continuous Ca2+ influx into the synaptic terminal. This channel is composed of the pore-forming α1 subunit and the auxiliary α2δ and β subunits. The Cacna1f or Cav1.4 gene encodes the α1F-subunit of the voltage-gated L-type Ca2+ channel (ICaL), predominantly expressed in mouse rods40. The α1F-subunit consists of four homologous domains with six α-helical transmembrane-spanning segments in each domain. IKCa and IClCa, Ca2+-dependent channels, have been identified in photoreceptors. IKCa has been reported in the tiger salamander retina36, but it remains unclear whether IKCa can be found in mouse rods12,41. IClCa has a critical physiological function and is widely expressed in many cells. In mouse rods, IClCa regulates signal transmission at synaptic terminals26. The electrophysiological characteristics of IClCa are controlled by the membrane voltage and free intracellular Ca2+ concentration9,10.

To establish and maintain ion homeostasis, additional ionic currents are needed. In the current study, we have added five ionic currents and five unidentified leak currents, including INaK, IPMCA, Na+/Ca2+ exchanger (INCX), the flux of Na+–K+–2Cl− cotransporter (JNKCC1), the flux of K+–Cl− cotransporter (JKCC2), outer segment Ca2+ leak channel (IL,Caos), inner segment Ca2+ leak channel (IL,Cais), K+ leak channel (IL,K), Na+ leak channel (IL,Na), and Cl− leak channel (IL,K). Two types of ATP-dependent pumps are found in mouse rod photoreceptors, INaK and IPMCA. INaK has been shown to have high expression levels in the inner segment of mouse rods42,43. INaK consists of two subunits, α and β subunits. The α3β2 isoform (Atp1a3/Atp1b2) is mainly expressed in mouse rods12, and Na+, K+, and ATP binding sites are located at the α3-subunit. INaK plays a pivotal role in maintaining the concentration gradients for Na+ and K+. One ATP molecule is hydrolyzed to pump three Na+ ions out and two K+ ions into the cell. IPMCA has a crucial role in regulating intracellular Ca2+ homeostasis. One Ca2+ is pumped out for each hydrolyzed ATP via IPMCA pumps. The Atp2b1 isoform of IPMCA is highly expressed in the IPMCA of mouse rods12. INCX extrudes one Ca2+ from the cell in exchange for three Na+ ions entering the cell. There are three isoforms of INCX, including the Slc8a1, Slc8a2, and Slc8a3 isoforms. It remains unclear which isoform is expressed in mouse rods. Johnson et al.44 showed high expression of the Slc8a1 isoform. However, Clark et al.12 only found low expression of the Slc8a3 isoform in mouse rods. Two electroneutral types of cotransporters have been identified in photoreceptors consisting of Na+–K+–2Cl− and K+–Cl− cotransporters12,45. The Na+–K+–2Cl− cotransporter is a major Cl− uptake transporter predominantly expressed in pre- and postsynaptic regions of the outer plexiform layer in the mouse retina45. In contrast, the K+–Cl− cotransporter is a primary Cl− extruder. Both Na+–K+–2Cl− and K+–Cl− cotransporters are essential for maintaining cell volume and Cl− homeostasis. To date, the role of Cl− in mouse rods remains unclear. Additionally, five unidentified leak channels were necessary for model fitting, consisting of three unidentified leak channels, including K+, Na+, and Cl−, and two unidentified Ca2+ leak channels divided by location (outer and inner segments).

Mathematical model

Earlier mathematical models have been proposed with significant ionic currents9,10,11; however, they have some limitations for maintaining ion homeostasis, which is necessary for estimating energy consumption. In the current study, a novel mathematical model of the mouse rod was developed from our previous works46. We introduced mathematical formulas for ionic current activities combined with a phototransduction model. The changes in ionic currents with the alteration in ion concentration were newly defined by the classical Goldman–Hodgkin–Katz (GHK) constant field and Nernst equations, which are widely used for explaining cell membrane electrophysiological phenomena. The membrane voltage of the mouse rod photoreceptor was determined by the ordinary differential equation (Eq. 1).

$${C}_{\mathrm{m}}\frac{d{V}_{\mathrm{m}}}{dt}=-{I}_{\mathrm{All}},$$
(1)

where ‘Cm’ and ‘Vm’ are the membrane capacitance and potential, respectively. ‘IAll’ is the sum of all ionic currents (Fig. 1, Table 1). The ionic currents ‘I’ were calculated by using a combination of available electrophysiological recordings and mathematical formulas similar to those in earlier studies10,23,47,48,49,50,51,52,53,54, which can be classified into three groups: (1) single ion flux channel (Eq. 2) for IKv, IKCa, and IClCa, (2) multiple ion flux channels (Eq. 3) for ICNG, Ih, and ICaL, and (3) simplified formulas from published data for INCKX, IPMCA, INCX, INaK, JNKCC1, and JKCC2 (Supplementary Materials).

$$I=g\cdot {m}^{\mathrm{M}}\cdot h\cdot \left({V}_{\mathrm{m}}-{E}_{\mathrm{ion}}\right);\mathrm{ ion }= \{{\mathrm{K}}^{+}, {\mathrm{Cl}}^{-}\}$$
(2)
$$I=g\cdot {m}^{\mathrm{M}}\cdot h\cdot \sum_{ion}{C}_{F,ion};\mathrm{ ion }= \{{\mathrm{Ca}}^{2+},{\mathrm{ K}}^{+},{\mathrm{ Na}}^{+}\}$$
(3)

where the activation variable, inactivation variable, and gating exponent are represented by ‘m’, ‘h’, and ‘M’, respectively. The modifications and the detailed equations are described in the Supplementary Materials. The ‘g’ refers to conductance. The reversal potential (Eion) of Ca2+, K+, Na+, and Cl- is based on the Nernst equation (Eq. 4), and ‘CF,ion’ is the GHK constant field equation (Eq. 5).

$${E}_{\mathrm{ion}}=\frac{R\cdot T}{z\cdot F\cdot {10}^{-3}}\cdot \mathrm{ln}\left(\frac{{\left[\mathrm{ion}\right]}_{\mathrm{o}}}{{\left[\mathrm{ion}\right]}_{\mathrm{i}}}\right);\mathrm{ ion }= \{{\mathrm{Ca}}^{2+}, {\mathrm{K}}^{+}, {\mathrm{Na}}^{+}, {\mathrm{Cl}}^{-}\}$$
(4)
$${C}_{\mathrm{F},\mathrm{ion}}=\frac{z\cdot F\cdot {V}_{\mathrm{m}}\cdot {10}^{-3}}{R\cdot T}\cdot \frac{{\left[\mathrm{ion}\right]}_{\mathrm{i}}-{\left[\mathrm{ion}\right]}_{\mathrm{o}}\cdot {e}^{-\frac{z\cdot F\cdot {V}_{\mathrm{m}}\cdot {10}^{-3}}{R\cdot T}}}{1-{e}^{-\frac{z\cdot F\cdot {V}_{\mathrm{m}}\cdot {10}^{-3}}{R\cdot T}}};\mathrm{ ion }= \{{\mathrm{Ca}}^{2+}, {\mathrm{K}}^{+}, {\mathrm{Na}}^{+}\}$$
(5)

where ‘F’ is Faraday’s constant, ‘T’ is the temperature in kelvins, and ‘R’ is the ideal gas constant. [ion]o and [ion]i are the concentrations of extracellular and intracellular ion species, respectively. The valence of the ion is ‘z’. The calculations of the intracellular calcium systems were developed by using the actual currents and buffering systems from earlier mathematical studies10,52. The concentration of Ca2+ in outer segments is controlled by a dynamic balance between influx via ICNG channels and extrusion via the INCKX exchanger (Eq. 6) (see Supplementary Materials 3 for details). At the inner segment, the calcium influx is organized by the ICaL current, and the calcium is released via two ionic currents, INCX and IPMCA (Eq. 7).

$$\frac{d{\left[{\mathrm{Ca}}^{2+} \right]}_{\mathrm{os}}}{dt}=\frac{-({I}_{\mathrm{CNG},\mathrm{Ca}}-2\cdot {I}_{\mathrm{NCKX}}+{I}_{\mathrm{L},\mathrm{Caos}})}{2\cdot F\cdot {V}_{\mathrm{os}}}-\frac{d{\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{os},\mathrm{b}}}{dt}-\frac{{J}_{\mathrm{dif}}}{{V}_{\mathrm{os}}}$$
(6)
$$\frac{d{\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{is}}}{dt}=\frac{-({I}_{\mathrm{CaL},\mathrm{Ca}}+{I}_{\mathrm{PMCA}}-2\cdot {I}_{\mathrm{NCX}}+{I}_{\mathrm{L},\mathrm{Cais}})}{2\cdot F\cdot {V}_{\mathrm{is}}}-\frac{d{\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{is},\mathrm{b}}}{dt}+\frac{{J}_{\mathrm{dif}}}{{V}_{\mathrm{is}}}$$
(7)

where ‘[Ca2+]’ is the calcium concentration, ‘V’ is the cell volume, and subscripts ‘os’, ‘is’, ‘b’, and ‘dif’ refer to the outer segment, inner segment, buffer, and diffusion, respectively. The changes in each [ion]i were estimated by the flow of individual ionic currents and given by \(d\left[\mathrm{ion}\right]/dt=I/(z\cdot F\cdot V)\). The energy expenditure from ionic current activities was estimated by the flows of Na+ uptake via ICNG, INCKX, Ih, ICaL, INCX, JNKCC1, and IL,Na. Every ATP molecule is hydrolyzed to transport three Na+ out of the cell via INaK (Eq. 8). Ca2+ uptake and release were described by Eq. (7); however, one ATP molecule is required for the extrusion of excess Ca2+ via IPMCA, and the required ATP was determined by Eq. (9) (see Supplementary Materials for d[Na+]influx/dt & d[Ca2+]influx/dt).

$${\text{ATP}}\,{\text{for}}\,{\text{pumping}}\,{\text{out}}\,{\text{excess}}\,{\text{Na}}^{ + } = \frac{{d\left[ {{\text{Na}}^{ + } } \right]_{{{\text{influx}}}} }}{dt} \cdot \frac{{N_{{\text{A}}} \cdot V_{{{\text{cell}}}} }}{3} \cdot 10^{ - 3}$$
(8)
$${\text{ATP}}\,{\text{for}}\,{\text{pumping}}\,{\text{out}}\,{\text{excess}}\,{\text{Ca}}^{2 + } = \frac{{d\left[ {{\text{Ca}}^{2 + } } \right]_{{{\text{influx}}}} }}{dt} \cdot N_{{\text{A}}} \cdot V_{{{\text{is}}}} \cdot 10^{ - 6}$$
(9)

where NA refers to the Avogadro constant. In the light environment, additional energy is required for the complex mechanism of the phototransduction cascade. The molecular mechanisms underlying the complex transduction cascade can be described by using the mathematical model from earlier studies50,51,52,53,54.

Time-integration and units

The model parameters of each ionic current were newly fitted with experimental findings and theoretical work. The model was based on the standard differential equation form, and numerical integration was performed by Euler’s method.

The following dimensions were applied in this model: millivolt (mV) for membrane potential, picoampere (pA), picoampere per picofarad (pA/pF) or femtoampere per picofarad (fA/pF) for ionic current, millimolar (mM) or micromolar (μM) for concentration, and second (s) and millisecond (ms) for time. All codes for the simulation program were prepared using Microsoft Visual Studio Community 2019 (Microsoft Corp.).

Results

Electrophysiological features at different light-flash intensities

When the photoreceptor is exposed to light, the transduction cascades are activated, leading to the closing of the light-sensitive channels and subsequently producing hyperpolarization. In the current study, we reproduced mouse rod photoreceptor responses to various light intensities described in earlier studies13,52. Figure 2 shows the simulation results of the mouse rod photoreceptor response to a variety of light-flash intensities. At high light-flash intensities, the membrane voltage was hyperpolarized to below − 55 mV, and ICNG and INCKX were close to zero. The amount of light-activated cGMP-phosphodiesterase required to produce a half-maximal response was approximately 43.5 photons μm−2 s−1, and the time to peak was 185 ms (Fig. 1S). Calcium homeostasis in the outer segment is regulated by a dynamic equilibrium between Ca2+ influx and Ca2+ extrusion via ICNG and INCKX, respectively. Hyperpolarization induces the reduction in ion transport events in most ionic currents, except Ih and IClCa. Both Ih and IClCa were activated in response to light exposure, and their maximum amplitudes were − 3.9 and − 0.4 pA/pF at a time to peak at 47 and 34 ms, respectively. The reduced ion transport activities led to increased [K+] and decreased [Ca2+], [Na+], [Cl−], and [cGMP]. However, K+, Na+, and Cl− concentrations required approximately 785, 723, and 680 s, respectively, to return to their initial value in darkness after high-intensity light-flash simulations. The peak of total energy expenditure via INaK, IPMCA, and phototransduction cascade mechanisms was approximately 7 \(\times\) 107 molecules of ATP s−1 (Fig. 3).

Figure 2
figure 2

Electrophysiological characteristics of membrane potential and a variety of ionic currents of the mouse rod photoreceptor in response to light-flash exposure. Simulation results of light responses were performed at a variety of light-flash intensities (Jhv); 1.7, 4.8, 15.2, 39.4, 125, 444, 1406, and 4630 photons μm−2 s−1 from gray to black (with color gradient), respectively. The responses started at time 0.5 s, and the stimuli consisted of 20-ms flashes. The collecting area was 0.43 μm252, and the membrane capacitance was 3.6 pF55.

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Figure 3
figure 3

ATP expenditure of mouse rod photoreceptors required for ion extrusion and phototransduction in darkness (D) and in light (L).

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Energy expenditure at different constant light intensities

The total ATP expenditure required for Na+ and Ca2+ extrusions was 7 \(\times\) 107 molecules of ATP s−1 in darkness for maintaining ion homeostasis, which significantly decreased after light exposure. Approximately 4.3 \(\times\) 107 molecules of ATP s−1 (62.4% of total energy expenditure in darkness) were required for ion extrusion and phototransduction under continuous high-intensity light simulation (Fig. 3). An extrusion of excess Na+ via INaK consumed 5 \(\times\) 107 molecules of ATP s−1 in darkness. ICNG (65%), INCKX (14%), and Ih (20%) had significant contributions to Na+ uptake in darkness (Fig. 4); the other ion channels (1%), including ICaL, INCX, JNKCC1, and IL,Na, however, had no significant impact on the total Na+ uptake (Fig. 3). In light, the proportion of total energy expenditure via Ih increased to 83%, whereas Na+ uptake via other currents was close to zero. On the other hand, approximately 2 \(\times\) 107 molecules of ATP s−1 were consumed for releasing excess Ca2+ ions via IPMCA in darkness, but only 6.3 \(\times\) 106 molecules of ATP s−1 were consumed in bright light.

Figure 4
figure 4

Changes in Na+ and K+ flow of major ionic currents across the plasma membrane in the current model of mouse rod photoreceptor in darkness and in light. OS outer segment, IS inner segment with cell body and synaptic terminal.

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The simulation of the mouse photoreceptor in response to 5 s of constant light showed a reduction in membrane potential amplitude. Nevertheless, the plateau phase remained in the range of − 45 to − 50 mV in the high-intensity light-flash simulations (Fig. 5A). The total ATP consumption significantly decreased after light exposure (Fig. 5B). In contrast, the energy consumption via the phototransduction cascade increased linearly with increasing intensity (Fig. 5C). Under high-intensity constant light, the ATP required for the extrusion of Na+ and Ca2+ decreased by approximately 27.6% and 65.9%, respectively, from the initial value in darkness (Fig. 5D). The levels of K+ and Cl− concentrations slightly changed, whereas Na+ and Ca2+ dramatically decreased under high-intensity constant light simulations (Fig. 5E,F).

Figure 5
figure 5

Membrane voltage, energy expenditure, and alterations in ion concentrations during constant light simulations. Simulations were performed at a variety of light intensities (Jhv): 1.7, 4.8, 15.2, 39.4, 125, 444, 1406, and 4630 photons μm−2 s−1 from gray to black (with color gradient). The responses started at time 0 s, and the stimuli consisted of 5 s of constant light. (A) membrane potential (mV), (B) total energy expenditure (molecules ATP), (C) energy consumption via phototransduction cascades (molecules s−1), (D) total energy expenditure and required ATP for Na+ and Ca2+ extrusions (molecules s−1), (E) alterations of [K+] and [Cl−] from initial values (% s−1), (F) alterations of [Na+], [Ca2+]os, and [Ca2+]is from initial values (% s−1), and (C–F) each parameter was recorded at 5 s.

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Discussion

At the outer segment, the total current composed of ICNG (94.4%) and INCKX (5.6%) was approximately – 15 pA under dark conditions, which has been shown through experimental observations56,57. It utilized approximately 4 × 107 molecules of ATP s−1 to extrude Na+ via INaK (3.3 × 107 molecules of ATP s−1 and 7 × 106 molecules of ATP s −1 via ICNG and INCKX, respectively). After bright light exposure, it fell to nearly zero by approximately 2.7 × 106 ATP s−1 per pA of ATP consumption rate. These results are consistent with earlier assumptions1, and the electrophysiological characteristics fit well with the experimental data13 and mathematical model52. The Na+ influx of ICNG (90% of total ICNG influxes) was larger than the Ca2+ influx in darkness. The rising phase and the intensity-response curve of ICNG were in good agreement with experimental data and theoretical work50,51,52,53,54.

Ih is responsible for shaping the membrane potential response to light, i.e., the characteristics of peak-plateau sag36. We modified the activation and inactivation parameters of Ih to fit with the peak-plateau sag of the membrane potential as previously reported9,10,11,17,18,19. The hyperpolarized peak membrane potential was below − 55 mV, and the plateau phase was in the range of − 45 to − 50 mV in high-intensity light simulations. Ih contributed to Na+ uptake of approximately 20% in darkness, and Na+ uptake increased 3.5 times more than in darkness under continuous high-intensity light simulations. Approximately 3.6 × 107 molecules of ATP s−1 were consumed for the extrusion of excess Na+ from Ih in bright light. These results differ from earlier reports1 because of a difference in the shape of the characteristic peak-plateau sag of the membrane potential. Energy consumption for the extrusion of excess Ca2+ via IPMCA in darkness was similar to that reported by Okawa et al.1, contributing approximately one-third of the total energy expenditure or half of the energy consumed via dark current in darkness. The model was fitted to maintain intracellular Ca2+ homeostasis via IPMCA and INCX for Ca2+ extrusion and ICaL and IL,Cais for Ca2+ uptake. However, the roles of Ca2+ extrusion via IPMCA and INCX in mouse rod photoreceptors are not fully understood. Immunohistochemically, mammalian photoreceptor synaptic terminals have revealed strong and very weak staining for IPMCA and INCX, respectively58. It was suggested that ICaL and IPMCA play significant roles in maintaining intracellular Ca2+ homeostasis in mammalian photoreceptors. The ICaL is responsible for the neurotransmission of visual signals. Mutations in the human Cav1.4 gene encoding ICaL have been associated with congenital stationary night blindness type 2. This disease has been functionally classified into three types: loss-of-function, gain-of-function, and C-terminal modulator function impairment59, which may lead to an increase or decrease in energy consumption for Ca2+ extrusion via IPMCA. However, whether there is an increase or decrease in IPMCA density has not yet been shown in any of the mutations in Cav1.4 genes.

INaK plays critical roles in K+ uptake and Na+ extrusion to maintain a high level of K+ and a low level of Na+ concentrations in photoreceptors to establish the normal resting membrane potential. To date, INaK is the only pump that can release excess Na+ ions. The α3β2 isoform current-membrane potential relationship was characterized by the voltage-dependent current using the Boltzmann relation23,60. It can operate at approximately 80% and 65% of the saturated maximum of INaK at resting and Vm less than − 55 mV, respectively. The current model demonstrated a fast recovery of INaK activity after photoreceptor hyperpolarization, resulting in decreased Na+ and increased K+. However, the transitional state of INaK from darkness to light remains unresolved. Further experimental studies may need to consider the difference in ion concentration between darkness and light. An inability to sustain ion homeostasis, such as ATP depletion or ion pump failure, can lead to apoptosis and pathogenesis61. The excess K+ was mainly released from photoreceptors via IKv, ICNG, and Ih, with approximately 49.5%, 20.9%, and 19.3% of total K+ extrusion in darkness and 35.1%, 0%, and 61.8% of absolute K+ extrusion under continuous high-intensity light simulations, respectively (Fig. 4 and Table 2). The contributions of the ionic channels to the outward current in darkness were 35.5%, 35.5%, 26.2%, and 3% for IKv, INaK, IPMCA, and IKCa, respectively, which are similar to earlier assumptions23. However, IKCa channels are expressed at a low level, as shown in the gene expression database of mouse rods12.

Table 2 Percentage of total K+ extrusion in photoreceptors via principal ion currents.
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IClCa, INKCC1, IKCC2, and IL,Cl associated with chloride fluxes play important roles in stabilizing the membrane potential of photoreceptors during presynaptic activity and presynaptic Ca2+ channel modulation. Nevertheless, changes in intracellular Cl− concentration within the physiological range are not fully understood. Note that in the current study, chloride influx and efflux were adjusted to maintain the low energy requirement for Na+ extrusion. Using the mathematical formulas for IClCa9,10, the simulation results of the electrophysiological characteristics showed different features due to the effects of the electrochemical driving force manifested by the Nernst equation. On the other hand, two electroneutral types of cation-chloride cotransporters, INKCC1 and IKCC2, were crucial for cellular homeostasis (Fig. 6). INKCC1 is strongly expressed at rod terminals. This cotransporter is responsible for Cl− uptake, maintaining a Cl− equilibrium potential. Unlike INKCC1, IKCC2 is the main Cl− extruder that promotes fast hyperpolarizing postsynaptic inhibition in the brain62 and inner retina45, but the role of IKCC2 in mouse rods is still unclear. Several simple mathematical equations exist to evaluate chloride regulation in neurons49,63. These equations can elucidate the depolarizing effect of INKCC1 and the hyperpolarizing effects of IKCC2, a concept widely acknowledged in the field64. However, the mathematical formulations for mouse rods remain poorly understood.

Figure 6
figure 6

Chloride homeostasis in the mouse rod in the current study. IS inner segment with cell body and synaptic terminal.

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After high-intensity light-flash simulations, the amounts of intracellular K+, Na+, and Cl− rapidly changed; however, the times of return of K+, Na+, and Cl− to their initial values were approximately 700 s, which subsequently prolonged energy balancing times. The changes in the intracellular concentration showed no significant difference in reversal potential, and thus they did not affect ionic currents. Nevertheless, whether K+, Na+, and Cl− ion accumulation or depletion causes retinal disease in mouse rods is unclear.

This study focuses on the utilization of mathematical formulas to evaluate the activities of ionic currents and factors governing ATP consumption within mouse rod photoreceptors under both darkness and light exposure. The model effectively mirrors physiological ranges, thereby closely aligning itself with empirical observations. Based on the findings derived from controlled physiological experiments, the current proposed model narrows its focuses to the short-term regulatory mechanisms governing cell status. However, it does not encompass the broader context of long-term cellular remodeling processes. Future investigations will delve into exploring specific transitional states, particularly those involving shifts from physiological to pathophysiological conditions. This exploration will encompass the effects of mitochondrial dysfunction, hypoxia, and gene mutations on energy utilization. Conversely, a comprehensive scrutiny of the energy expenditure of cone photoreceptors, recognized for their elevated energy requirements compared to rods under both dark and light conditions17, holds the potential to yield profound insights into the fundamental distinctions between the energy regulatory mechanisms of rods and cones.

Furthermore, the current model can provide a practical framework for future clinical and experimental studies for a better understanding of the pathophysiology of the retina and improve potential future treatments. However, further research on coupling the photoreceptor model with experiments is urgently needed to fill the scientific gaps in the complex phenomenon and increase the model’s accuracy and specificity.