The ChR2 photocurrent
Because the ChR2 photocurrent has already been well characterized (e.g. [6,15]), our purpose here is to simply highlight the major components of the photoelectrophysiological response. These components then instruct our quantitative modeling of ChR2 kinetics in the following sections. Under whole-cell voltage clamp recording, the response of a ChR2-expressing neuron to a stationary blue light stimulus has three main features: an initial peak with a fast decay, a steady-state plateau, and a postillumination decay back to baseline. Subsequent stimulus, applied shortly after the previous one was terminated (<10 s), induces a degraded transient response and the same steady-state one.
Under continuous illumination, the value of the peak current rises in a nonlinear fashion with light intensity. However, under short pulsed stimulation (1–10 ms) and lower light intensities the current can show a different type of peak, which appears because the number of active ChR2 channels have not reached the maximum value before the light was switched off (see Fig. 2a). In this case both the value of the peak current and the time to peak vary with pulse duration (Fig. 2b,c). Experiments with very short light pulses (1–5 ms) suggest that some ChR2 channels open after the light stimulus has terminated (Fig. 2b). For example, in the case of a 1 ms light pulse, the current peak occurred approximately 1.5 ms after the end of the stimulus. This is in agreement with spectroscopic measurements (12,13), which show that the activation of ChR2 goes through several intermediates and takes about 0.2–1 ms.
Useful information comes from the final part of the ChR2 photocurrent curve, when the light is turned off (see Fig. 3a top left inset). At the termination of the light stimulus, after an arbitrary long light stimulation, the ChR2 photocurrent undergoes a rapid decay to baseline. Close inspection of this portion of the response shows this decay to be biexponential (Fig. 3a) with two distinct time constants (Fig. 3a, bottom insets). This biexponential function rules out any explanation of the ChR2 ‘‘off’’ response being a simple capacitative resistive decay. Rather, there are two separate decay processes with independent time constants.
Figure 3. (a) The current decays biexponentially after the light is turned off (ChR2-expressing hippocampal cells). Inset: sets of values for two time constants τfast and τslow (bottom right), and the ratio of the amplitudes Islow/Ifast (bottom left), for the exponential fit vs light intensity, for n = 4 cells and three repeated measurements. Bars indicate the standard deviation. Light pulse duration is 500 ms. (b) ChR2 kinetics is intrinsic to the channel: (A) Kinetics of photocurrents recorded in voltage clamp mode (−70mV) in response to a 500 ms light stimulus are identical in neurons and HEK-293 cells. (B) ChR2 kinetics are not altered by the intracellular presence of the fast calcium chelator BAPTA (10 mm). (C) ChR2 kinetics are identical irrespective of whether the channel is tagged by either YFP or mCherry fluorophores, or by a nonfluorescent myc sequence.
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The dynamics of the ChR2 dark decay for the single light pulses of various durations is analyzed in Fig. 4. When the temporal width of light stimulation pulses is changed (between 1 and 500 ms), there is no discernible change in either decay time constant (Fig. 4a). However, the normalized photocurrent amplitude associated with each decay process does vary with pulse duration (Fig. 4b), indicative of population changes in different ChR2 states as the light stimulus persists. When the turning-off phase of the photocurrent was fitted with a single exponential function, the fit was not as good as in the biexponential case and the turning-off time constant (τoff) was reported in Ishizuka et al. (15) to slow down with increasing light duration (e.g.τoff ≈ 5 ms for 20 ms light pulse, but τoff ≈ 15 ms for 1 s pulse). This can now be explained by using results from Fig. 4, where for short light pulses only the fast decay component is present and it has a time constant of about 5 ms. However for longer pulses the slower component amplitude with a time constant 35 ms represents about 20% of the current, so the ‘‘composite’’ time constant which would represent a monoexponential process increases.
Figure 4. The turning-off phase was fitted with biexponential function with time constants τfast and τslow and amplitudes Ifast and Islow. The graphs show the dependence of these parameters on the duration of the illumination pulse (1 mW mm−2 intensity, n = 3 cells). The average values and standard deviations for the time constants are: 〈τ1〉 = 3.8 ms, στ1 = 0.65 ms and 〈τ2〉 = 34 ms, στ2 = 5 ms. The standard deviations calculated for each individual cell are on average only one half of the value calculated for all cells. The lines are calculated using Eqs. (33) and (34) based on the theory given in Appendix 2 (parameters: Imax = 1 nA, ɛ1 = 0.5, ɛ2 = 0.15, γ = 0.1, Gd1 = 0.11 ms−1, Gd2 = 0.025 ms−1, e12 = 0.01 ms−1, e21 = 0.015 ms−1, wloss = 1.1, τChR = 0.2 ms−1).
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Neither time constant of the ChR2 ‘‘off’’ response shows any dependence on light intensity (Fig. 3a, bottom right), and the ratio of the amplitudes of the two exponential functions does not appear to be light intensity dependent (see Fig. 3a, bottom left). This observation is commented on in the Discussion and Appendix 2.
To be certain that the kinetic properties of the ChR2 photocurrent are intrinsic to the channel itself, and not specific to the neuronal context in which it is normally expressed, we also performed patch-clamp recordings on HEK-293 cells transfected with ChR2-YFP. Photocurrents recorded in response to long 500 ms light pulses were qualitatively indistinguishable in HEK-293 cells vs hippocampal neurons, and were also quantitatively indistinguishable (see Fig. 3b: A). In both cell types, it is still possible that channel kinetics might be modulated by the internal cellular environment. In particular, changes in internal free calcium levels can directly or indirectly alter the properties of various ion channels (e.g.19,20). To inhibit any such calcium changes, in particular those arising from entry through the open ChR2 channel itself, we loaded neurons with the fast Ca chelator BAPTA (10 mm) via the patch pipette. Nevertheless, blocking internal calcium transients in this manner had no effect, either qualitatively or quantitatively, on ChR2 photocurrent kinetics (Fig. 3b: B). Finally, it remains a possibility that the kinetics of the ChR2 channel are affected by the YFP fluorophore attached to its C-terminus. However, replacing this YFP with the mCherry fluorophore had no effect on ChR2 photocurrent kinetics, and we also saw no change in kinetics after replacing the fluorophore sequence with a short nonfluorescing myc tag sequence (these neurons were co-transfected with EYFP) (Fig. 3b: C). The ChR2 photocurrent kinetics described here, and the models we use to fit them, are therefore reflective of intrinsic channel properties of the ChR2 protein.
The three-state model
The process of recovery from desensitization (D C) is not yet fully understood. Its duration varies from a few seconds to a minute in the dark (depending on the pHo), but in the light the process appears to take only 60 ms (6,16). For the three-state model to be correct, the recovery rate must be modulated by light flux. Alternatively, the recovery process could occur only partially via the D state, with some molecules going directly from the O to the C state. Having tested both hypotheses, the light flux-modulated recovery seems more plausible.
We have adopted a three-state model as shown in Fig. 1a, which can be presented by a set of two rate equations:
where O, D and C are the numbers of ChR2 molecules in a cell which are in the corresponding O, D and C states, Gd = 1/τd and Gr = 1/τr are the rates of the channel closure and recovery of photosensitivity, and τd(r) are the corresponding time constants. F is the number of photons absorbed by a ChR2 molecule per unit time and ɛ is the quantum efficiency. We define the quantum efficiency as the probability that a photon absorbed by the retinal molecule in the ChR2 activates the channel (the exact value depends on the specific distribution of amino acids around the retinal binding pocket and the light wavelength, but a typical value would be ɛ ≈ 0.5 for rhodopsin ). If φ is the photon flux per unit area, then F = σretφ/wloss, where wloss is the measure of the loss of the photons from the original flux φ resulting from light absorption or scattering by the solution, and σret is the retinal cross-section (σret ≈ 1.2×10−20m2 ). If the total number of ChR2 molecules in a cell is N then N = O + D + C and we assume that N stays constant for the observed cell, i.e. that there is no ChR2 degeneration. The current measured by whole cell voltage-clamp recording in ChR2-expressing neurons is directly proportional to the number of open channels: I = Vg1O, where g1 is the conductance of an individual ChR2 ion channel and V is the holding voltage across the cell membrane (here we can assume that the reversal potential for ChR2 is approximately 0 mV ).
Figure 5. The peak (open squares) and the plateau currents (full squares) in the experiment of Ishizuka et al. (15) and our corresponding theoretical values (full lines), according to the three-state model (τd = 11 ms and τr = 165 ms). The broken line shows the simulation results when Gr depends logarithmically on the light flux, see Eq. (13). Inset: typical response curves (three-state model, Eqs.  and ) for long light pulses (∼1 s) for different light intensities.
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When the light is turned off, the current decreases exponentially:
The major success of the three-state model in the quantitative modeling of the cell's response was in the relatively accurate prediction of the peak and plateau values of the current (shown in Fig. 5). A Michaelis–Menten-type equation for the plateau current given in Eq. (4) was empirically found by Ishizuka et al. (15).
The analytical form for the current also gives an opportunity to compare the values of the photocycle parameters obtained from the simulations with those determined from experimentally measured ChR2 currents. For example, in Bi et al. (3) the values for λ1 and λ2 were determined from curve fitting using the exponential function: I(t) = a0 + a1 (1 − exp (−t/τact) + a2 exp (−t/τdeact), where τs are related to Eq. (6) as τact = 1/λ2 and τdeact = 1/λ1, and represent the activation and inactivation time constants, respectively. This form we readily recognize in Eq. (3). When λ1 and λ2 are determined as fitting parameters from the experimental results, we can estimate the physical parameters τd = 1/Gd and τr = 1/Gr as using equations Eqs. (6)–(8) we get:
Precisely determining the values for the ChR2 excitation rate P(=ɛF) might be problematic in a specific experiment. Usually we know the photon flux φ, but not F, the actual number of photons absorbed by a molecule per unit time. However, typical experimental light intensities, as in Table 1, give λ2 ≫ λ1, and λ2 ≫ Pmax (where Pmax is the maximum activation rate when no photons are lost due to scattering or absorption by the medium), hence the value for P can be estimated and the error will not have a big impact on the final result. Equations (11) and (12) then give: τd ≈ τact (see Table 1). Results in Table 1 suggest a strong dependence of the two time constants on light intensity. τd decreases when φ increases, in proportion with ln(φ), whereas τr first decreases for smaller values for φ, but then stabilizes at ∼50 ms for higher values (Fig. 6). However, τr has much stronger dependence on the intensity of light for lower light intensities: its duration in the dark (τr ≈ 10 s) is far longer than under 1 mW mm−2 illumination (τr ≈ 0.1 s). Here we have to assume that the recovery process (D C) is light assisted.
Table 1. The open-state (τd) and recovery (τr) lifetimes according to the three-state model, for the experimental results for the ‘‘activation’’ (τact) and ‘‘deactivation’’ (τdeac) times from Bi et al. (, fig. 2G,H). Light pulse duration is 1 s. Pmax = ɛφσret, where ɛ is estimated to be 0.5 and σret is approximately 10−8 μm2.
|φ (× 109ph μm−2 s−1)||Pmax (ms−1)||λ1 (ms−1)||λ2 (ms−1)||τact = 1/λ2 (ms)||τdeact = 1/λ1 (ms)||τd (ms)||τr (ms)|
The ChR2 system shows the ability of spontaneous recovery in the dark, because retinal can re-isomerize to the all-trans ground state in dark, without the need for special enzymes (5). However the recovery from inactivation is relatively slow in the dark (τr,dark ≈ 5−10 s) (1,5,15). In addition to the re-isomerization, it appears that the process of protonation of one or more amino acids in or near the retinal binding pocket is part of the ChR2 recovery process (1) and hence strong pHo dependence of the recovery time (1). Our three-state model can qualitatively reproduce the response to a series of light pulses (see ref. , Fig. 5), which is affected by the dark recovery. However, for this the model requires two different values for τr: one for light (τr,light) and one for dark conditions (τr,dark) meaning that the recovery rate Gr = 1/τr may be dependent upon general light flux. This is not implausible, as in invertebrates chromophore recovery can be assisted via absorption of a second photon (22). For example in Drosophila, the metarhodopsin is thermally stable and can be re-isomerized back to rhodopsin by absorption of red light (23). After testing several functions we found that a logarithmic dependence for Gr(φ):
where c and φ0 are appropriate numerical constants, gives the plateau values in Fig. 5 which are a better match to the experimental values than the case when Gr(φ) = const.
While the three-state model is able to successfully reproduce several elements of the ChR2 photocurrent, the need to introduce a logarithmic dependence of deactivation on light flux shows that a simple three-state model cannot fully explain the response of ChR2-expressing neurons to a light stimulus. In addition, the three-state model, with its single decay phase, cannot account for the biexponential ChR2 current decay we observed following light offset (Fig. 3), nor can it explain the pH-induced shift in the action spectrum (12). Finally, the parameter space in our three-state model is not big enough to accurately fit the complete ChR2 photocurrent waveform (Fig. 7). To resolve these issues, we have developed a more detailed four-state model of ChR2 kinetics.
Figure 7. Experimental (full line) and the three-state model simulation results (broken line), with the parameters chosen to fit the peak and plateau of the experiment. The model gives too fast decrease of the currents.
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The four-state model
The different dark decay time constants seen in our experimental results in Figs. 3 and 4 indicate the potential existence of two open states, O1 and O2. In addition, Ernst et al. (12) have pointed to the possible existence of two ChR2 closed states (C1 and C2), and the photocycle experiments of Bamann et al. (13) indicate two closed (corresponding to P0 and P4) and two open (P2, P3) functional channel states. Therefore, if we neglect the short-lived (<1 ms) intermediate states, a four-state model, shown in Fig. 1b, becomes an attractive proposition.
Let us assume that a dark-adapted ChR2 molecule is completely settled in the dark state C1. Upon illumination C1 converts into the open state O1, within 1 ms, going through two fast intermediate states (resolved by the transient spectroscopy and described in Ernst et al. ). The model allows O1 to convert back to C1. Alternatively, there may be a transition to another open state O2, which could be less conductive than O1. The transition between the two open states is reversible. The O2 state decays to the second closed state C2. C2 thermally converts into C1, but this process is slow (of the order of seconds) in comparison with the other transitions. If in this model we assume that the light-assisted transitions are C1 O1 and C2 O2, then the model can be described by the following set of rate equations:
where O1, O2, C1 and C2 are the numbers of the ChR2 molecules in the open states 1 and 2, and closed states 1 and 2, respectively (N = O1 + O2 + C1 + C2), Gd1 and Gd2 are the rates of transition O1 C1 and O2 C2, Gr is the rate of thermal conversion of C2 to C1, and e12 and e21 are the rates of transition between O1 to O2 and vice versa. The photon absorption and isomerization of retinal in rhodopsin is a very fast process (∼200 fs ). However, the subsequent conformal change in the protein is a slower process, hence we use the activation rates Ga1 for the C1 O1, and Ga2 for the C2 O2 processes. They depend on the quantum efficiencies ɛ1 (ɛ2) as:
where i = 1,2 and τChR is the activation time of ChR2 ion channel and the function f(t,τChR) is given in Appendix 1. The experimentally measured photocurrent will then be:
where g1 (g2) is the conductance of a single ChR2 ion channel in the open state O1 (O2), γ = g2/g1 and V is the holding potential (voltage clamp measurements). Imax = Vg1N is the maximum possible current, when all ChR2 channels are in the O1 state, o1 = O1/N and o2 = O2/N are the fractional occupancies of the open states.
Having developed the four-state model thus far, does it give a better fit to the experimental ChR2 photocurrents than the three-state model? In order to answer this question we have performed a numerical fitting procedure using Eqs. (14)–(18), and the results of this modeling are shown in Fig. 8a for two power densities. More details about the relationship between the experimentally derived parameters (e.g. the current decay factors Λ1 and Λ2) and physical parameters describing the kinetics of ChR2 are given in Appendix 2. The results in Fig. 8a show that the four-state model offers a very good fit to the experimental results, with a reasonable choice of parameters (see the table in Fig. 8).
Figure 8. (a) The experimental photocurrents (red lines) and the fitting results for the four-state model (black lines), for two light intensities: 0.9 and 11 mW mm−2 (500 ms light pulse, 470 nm wavelength, −70 mV holding potential). The current decay after the light is turned off with Λ1 ≈ 0.030 ms and Λ2 ≈ 0.15 ms (Λ1(2) are defined in Appendix 2). The optimized fitting parameters for the two light intensities are given in the table (Gr = 0.4s−1 and wloss = 1.1). (b) The time dependency of the occupation levels (o1 and o2) for the O1 and O2 states, for two light intensities.
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The results for the distribution of ChR2 molecules between the two open states, both as a function of the duration of illumination and light intensity (Fig. 8b) indicate a light adaptation process. Our fitting parameters show that the ratio of the conductances of the two states is γ ∼ 0.1, hence we could talk about high (O1) and low (O2) conducting states. After the start of the illumination the O1 state is populated first (Fig. 8b). As the illumination continues, the number of ChR2 molecules in that state reaches a peak and then decreases to a steady-state value. At the same time the fraction of the ChR2 molecules in the O2 state is initially low but increases with stimulus time, reaching an asymptotic value. As the light intensity increases, the fraction of molecules in the low conducting state increases faster than the fraction in the high conducting state.
Using the activation rate formulas derived in Appendix 1Eqs. (24) and (25) and the four-state model Eqs. (14)–(18) we also obtain a good fit for the ChR2 photocurrents recorded in response to short stimulus durations (1–10 ms, Fig. 9). The further rise in current after short light pulses (tlight < 3 ms) can be explained by the fact that ion channels are not simultaneously opened with photon absorption but the protein conformal change and subsequent channel opening is a slower process. The effect is visible for short pulses as the channels are not closing yet. The introduction of the activation rate function (see Eqs.  and  in Appendix 2) mathematically describes this process, and gives good results for modeling experimental measurements (Fig. 9). The fitting parameter here for the activation time constant of ChR2 is τChR = 1.3 ms, longer than in the case of the Volvox ChR (12), where the average time of the transition from a dark-adapted state to the conducting intermediate in the spectroscopic measurements is about 0.2 ms. The longer activation time in our experiments is probably due to the time constant of our light illumination system (which is about 1 ms), because the same activation rate function (Eqs.  and ) can describe the light source illumination increase.
Figure 9. The experimental photocurrents (red crosses) and the model results (full lines) for the four-state model, for short light pulses of 1, 2, 3, 5, 8, 10 and 20 ms. Light intensity is 0.65 mW mm−2, 470 nm wavelength, −70 mV holding potential. The light-off current decay exponents are Λ1 = 0.0282 ± 0.0013 ms−1 and Λ2 = 0.362 ± 0.0282 ms−1. The fitting parameters are: τChR = 1.3 ms, γ = 0.1, ɛ1 = 0.6, ɛ2 = 0.14, Gd1 = 0.35 ms−1, Gd2 = 0.02 ms−1, e12 = 0.01 ms−1, e21 = 0.02 ms−1, wloss = 1.3, Imax = 0.2 nA.
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