A Simplified Model of Activation and Deactivation of Human Rod Phototransduction—An Electroretinographic Study

Abstract

Purpose: To test the hypothesis that a simple model having properties consistent with activation and deactivation in the rod approximates the whole time course of the photoresponse.

Methods: Routinely, an exponential of the form f = α·(1 – exp(–(τ·(t – t_eff)^s–1))), with amplitude α, rate constant τ (often scaled by intensity), irreducible delay t_eff, and time exponent s–1, is fit to the early period of the flash electroretinogram. Notably, s (an integer) represents the three integrating stages in the rod amplification cascade (rhodopsin isomerization, transducin activation, and cGMP hydrolysis). The time course of the photoresponse to a 0.17 cd·s·m⁻² conditioning flash (CF) was determined in 21 healthy eyes by presenting the CF plus a bright probe flash (PF) in tandem, separated by interstimulus intervals (ISIs) of 0.01 to 1.4 seconds, and calculating the proportion of the PF a-wave suppressed by the CF at each ISI. To test if similar kinetics describe deactivation, difference of exponential (DoE) functions with common α and t_eff parameters, respective rate constants for the initiation (I) and quenching (Q) phases of the response, and specified values of s (s_I, s_Q), were compared to the photoresponse time course.

Results: As hypothesized, the optimal values of s_I and s_Q were 3 and 2, respectively. Mean ± SD α was 0.80 ± 0.066, I was 7700 ± 2400 m²·cd^–1·s^–3, and Q was 1.4 ± 0.47 s^–1. Overall, r² was 0.93.

Conclusions: A method, including a DoE model with just three free parameters (α, I, Q), that robustly captures the magnitude and time-constants of the complete rod response, was produced. Only two steps integrate to quench the rod photoresponse.

Vision is initiated in the photoreceptors—the rods and cones that transform light into neural signals. In human retina, rods are much more numerous.¹ The photoresponse of the rod involves a sequence of biochemical events that begins with the absorption of a photon by rhodopsin, the visual pigment, and leads to the closing of cyclic nucleotide-gated (CNG) cation channels in the plasma membrane of the rod outer segment (ROS). This closure results in the collapse of the “dark current” and generates an electrical signal that can be measured as the electroretinographic (ERG) a-wave.^2–4 Thus, the a-wave is extensively used to study rod photoresponses in healthy and diseased human eyes.^5–8 

The dark current refers to the steady-state flow of cations (mainly sodium and calcium) into the dark-adapted ROS, which depolarizes the cell and causes a continuous release of neurotransmitter (glutamate) at the synapse.^9–11 Because the dark current enters the extracellular space through metabolic pumps in the rod inner segments (RISs) and returns into the cell at the ROSs, it must circulate through the cilia, which have high resistance, and thus a current dipole is created. The collapse of this dipole consequent to the closure of ROS cation channels (i.e., reduced influx of cations and consequent hyperpolarization of the rod) presents as the a-wave.^3,4,12 Therefore, the a-wave reflects the initial phase—activation—of the photoresponse.¹³ 

Rod activation is one of the best-documented physiologic processes.^12,14–16 In brief, the absorption of a photon causes rhodopsin to change conformation, converting it to an active isomer (R*) called metarhodopsin II. R* activates a G-protein called transducin, which is bound to the cytoplasmic surface of the disc membrane. Activated transducin (G_T*) then binds to and activates an enzyme called phosphodiesterase (PDE*), which catalyzes the hydrolysis of cyclic guanosine monophosphate (cGMP*), a second messenger that maintains the CNG cation channels in an open state, intromitting the dark current. The hydrolysis of cGMP consequently causes the closure of these channels. Three steps—R*, G_T*/PDE*, and cGMP*—combine to amplify the rod photoresponse. Each metarhodopsin II molecule can activate multiple transducin molecules, and each transducin-bound PDE can hydrolyze multiple cGMP molecules, leading to amplification due to an exponential increase in the number of activated molecules at each of these steps. The contribution of each of these integrating steps to the amplification of the rod photoresponse is hotly pursued,^17–20 and the kinetics of the a-wave have been elegantly shown to reflect the efficiency of the phototransduction activation cascade in isolated rods.^3,21 

Accordingly, the a-wave is routinely fit to an exponential function of the form f = α·(1 – exp((τ·ι·(t – t_eff)^s–1))), where α is the amplitude, τ is the rate constant, ι is the intensity of the stimulus, t_eff is an irreducible delay introduced by the amplifier, and s is an integer that represents the number of integrating stages.⁴ Conventionally, when modeling the response of the rod to a flash of light, the value 3 is used for s, representing R*, G_T*/PDE*, and cGMP*.^5,6,22 However, this approach only models the early phase of the response (i.e., the initiation of phototransduction) and does not provide a complete picture of the photoresponse; notably missing from the equation is an accounting of deactivation (i.e., the quenching of phototransduction). Quenching the rod response involves a series of processes that rejuvenate the dark current and “reset” the rods for the next stimulus.²³ 

Closure of the CNG channels reduces calcium influx, which leads to several effects. One effect is the acceleration of the phosphorylation of R* by rhodopsin kinase.²⁴ Rhodopsin kinase phosphorylates metarhodopsin II at certain serine and threonine residues, which decrements its efficiency in activating transducin and promotes its binding with arrestin. Recoverin is a calcium-sensing protein²⁵ that interacts with rhodopsin kinase to greatly stimulate rhodopsin phosphorylation. Once R* is bound by arrestin, its ability to activate downstream effector proteins is blocked.²⁶ Another effect is the increase in activity of regulator of G-protein signaling 9 (RGS9). RGS9 enhances the intrinsic GTPase activity of transducin, thereby turning off the signaling cascade by accelerating the hydrolysis of guanosine triphosphate (GTP) to guanosine diphosphate (GDP).²⁷ Additionally, calcium regulates the production of cGMP by guanylate cyclase (GC) via GC-activating proteins (GCAPs) and affects the affinity of the CNG channels for cGMP.²⁸ 

Mathematical models have been developed to describe the quenching process and have provided valuable insights into the interactions of these molecules.^17–19 These models are necessarily complex and suggest that two integrating stages in quenching are the phosphorylation of rhodopsin by recoverin (and its subsequent binding by arrestin) and the GTPase activity of transducin.²⁹ The time constants of these steps are dramatically impacted by calcium and RGS9 levels, among other factors.^30,31 

The intact full-field ERG is not well suited to the study of quenching because quenching is obscured by other retinal activity. However, the cleverly designed double-flash electroretinogram (dfERG) can reveal the time course of recovery of the dark current.³² The dfERG involves delivering a conditioning flash (CF) followed by a second, bright probe flash (PF) at variable interstimulus intervals (ISIs). The suppression of the dark current by the CF leads to a smaller a-wave in the response to the PF. The waning and waxing amplitude of the PF a-wave reflects the initial suppression and subsequent recovery of the dark current. We used the dfERG to monitor both the initiation and quenching phases of the rod photoresponse and created a simplified numeric model that captures its entire time course. 

Our hypothesis was that, if the initiation and quenching processes have similar kinetics, then a difference of exponentials (DoE) would provide a suitable, readily interpretable summary of these concurrent processes. In our formulation, the time constant of initiation (I), the time constant for quenching (Q), and the amplitude, α, are determined by empirical fit to the data. The appropriate number of steps, s (s_I, s_Q), must be specified and must be consistent with known biochemical properties of rod activation and quenching. The value of t_eff is mediated by the recording equipment. Specifically, we tested the hypothesis that a DoE with three integrating stages in initiation (s_I = 3) and two integrating stages in quenching (s_Q = 2) captures the magnitude and time constants of the complete rod photoresponse. Such a model would be readily applicable to clinical disorders of the rods. Herein we have evaluated our hypothesized model. 

Methods

Volunteers

We studied 21 healthy young adults with normal vision, all of whom provided written, informed consent before participating. This project was approved by the Boston Children's Hospital Institutional Review Board. 

Electroretinography

For each volunteer, we dilated one pupil using tropicamide and then, following 30 minutes of dark adaptation, placed the electrodes under dim red illumination. During electrode placement, dilation of the pupil to approximately 8 mm was confirmed in every volunteer. Three volunteers were tested using a Burian-Allen (BA) bipolar electrode (Hansen Labs, Coralville, IA, USA), and 18 were tested using a DTL electrode (Diagnosys LLC, Lowell, MA, USA). When using the BA, we placed the ground on the skin over the ipsilateral mastoid; when using the DTL, we placed the reference at the temple and the ground at the forehead. Our ERG system was an Espion E2 with ColorDome ganzfeld (Diagnosys LLC). All responses were amplified and filtered using a bandpass of 5 to 1000 Hz and digitized at a sampling rate of 2 kHz. 

To estimate the rate of rhodopsin photoisomerization produced by our stimuli, we followed the approach detailed in Hansen et al.³³ Briefly, we calculated the Troland value (Td·s·m^–2) for retinal illumination by multiplying the stimuli, measured in candelas, by 50 (i.e., π·\({( {\frac{8}{2}} )^2}\)) and then multiplying the result by 8.5, a factor that accounts for various properties of the rods and the quantum efficiency of rhodopsin.³⁴ As shown in Figure 1, we derived the time course of the rod response to a white LED (0.17 cd·s·m^–2, ∼75 R* rod^–1) CF that elicited a small a-wave, as follows: first, we recorded the response to the CF alone. Then, we recorded the response to an intense white LED (20 cd·s·m^–2, ∼8500 R* rod^–1), rod-saturating PF and measured its amplitude 10 ms after presentation (just before the trough of the a-wave). We took this measurement as proportionate to the amplitude of the maximal rod dark current, a_max. Next, we presented 10 pairs of stimuli, CF then PF, distinguished by ISI. The ISIs were 10 ms, 20 ms, 50 ms, 0.1 s, 0.15 s, 0.2 s, 0.4 s, 0.7 s, 1 s, and 1.4 s. We used the response to the CF recorded alone as the baseline for measuring the amplitude of the response to the PF at each interstimulus time t, a_sat,t. Therefore, the fraction of the dark current suppressed by the CF at elapsed time t, SF_t, was given by SF_t = 1 – a_sat,t/a_max. Twelve volunteers completed two runs in the same session. 

Figure 1.

 

Representative example of the “double-flash” protocol for estimating suppression of the dark current, SF_t. Top: In the dark-adapted eye, the responses to an a-wave–saturating (20 cd·s·m^–2) PF (maroon trace) and a 0.17 cd·s·m^–2 CF (green trace) were each recorded alone. Then the CF and PF were presented in tandem, separated by 10 ISIs; in these double-flash cases, the responses were aligned 10 ms prior to presentation of the PF. The amplitudes of the responses to the PF (orange lines) were measured 10 ms after PF presentation, taking the response to the CF, recorded alone, as the baseline for the measurement of the double-flash responses (black traces) at each ISI. Bottom: This shows the proportion of the rod response suppressed by the PF (maroon point) and by the CF at each ISI (black points). The orange lines help to visualize the waning and waxing of the dark current “underneath” the intact ERG (green trace).

Modeling

We defined the DoE model from discrete exponentials representing the initiation and quenching phases of phototransduction as follows:  

\begin{eqnarray}{{{f}}_{\rm{I}}}\left( {{t}} \right) = {\rm{ \alpha }}\left( { 1 - {{\rm{e}}^{\left( {{{-I}}\,{\rm{\iota }}\,{{\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)}^{{{\rm{s}}_{\rm{I}}}{\rm{ - 1}}}}} \right)}}} \right) \cdot \left( {{\rm{1 + sgn}}\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)} \right)/{\rm{2}}\quad \end{eqnarray}

(1a)

 

\begin{eqnarray}{{{f}}_{\rm{Q}}}\left( {{t}} \right){\rm{ = \alpha }}\left( { 1 - {{\rm{e}}^{\left( {{{-Q}}\,{{\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)}^{{{{s}}_{\rm{Q}}}{\rm{ - 1}}}}} \right)}}} \right) \cdot \left( {{\rm{1 + sgn}}\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)} \right)/{\rm{2}}\quad \end{eqnarray}

(1b)

 

\begin{eqnarray} &&{\rm{SF}}\left( {{t}} \right) = {{{f}}_{\rm{I}}}\left( {{t}} \right){\rm{ - }}{{{f}}_{\rm{Q}}}\left( {{t}} \right) \nonumber \\ && ={{ -\alpha }}\left( {{{\rm{e}}^{\left( {{{-I}}\,{{\iota }}\,{{\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)}^{{{{s}}_{\rm{I}}}{\rm{ - 1}}}}} \right)}} - {{\rm{e}}^{\left( {{{-Q}}\,{{\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)}^{{{{s}}_{{\rm Q}}}{\rm{ - 1}}}}} \right)}}} \right) \nonumber \\ && \cdot \left( {{\rm{1\ + \ sgn}}\left( {{{t - }}{{{t}}_{{\rm{eff}}}}} \right)} \right)/{\rm{2}}\end{eqnarray}

(1c)

with α representing the magnitude of the suppression of the dark current, rate constants of initiation (I) and quenching (Q), the former scaled by flash intensity (ι), an irreducible delay introduced by the amplifier and other sources (t_eff), and time exponents related to the number of integrating steps in the rod amplification (s_I) and quenching (s_Q) cascades. We note that the units of I depend on the units of ι and the value of s_I, and the units of Q depend on the value of s_Q. We constrained s_I and s_Q to be integers. The inclusion of “(1 + sgn(t – t_eff))/2” (where sgn is the signum function) constrains fitting to t ≥ t_eff. Additionally, Equation 1c is only valid for 0 ≤ SF ≤ 1. Accordingly, we added this additional constraint to Equation 1c, prior to fitting, so that it became  

\begin{eqnarray} && \begin{array}{@{}l@{}} {\rm{SF}}\left( t \right) = \max \left[ {\rm{0, \min}}\left[ {\rm{1, -\alpha }} \left( {\rm{e}}^{\left(\vphantom{s_Q^A} {{-I}}\,{{\iota }}\,{{( {{{t - }}{{{t}}_{{\rm{eff}}}}} )}^{{{{s}}_{\rm{I}}} {\rm{-1}}}}\right)}\right.\right.\right.\\ \\ \qquad\left.\left.-\left. {\rm{e}}^{\left( {{-Q}}\,{( {{{t - }}{{{t}}_{{\rm{eff}}}}} )}^{{{{s}}_{\rm{Q}}} {\rm{-1}}} \right)} \right) \right] \right] \cdot ( {{\rm{1 + sgn}}( {{{t - }}{{{t}}_{{\rm{eff}}}}} )} )/{\rm{2}}\end{array} \end{eqnarray}

(2)

 

We performed all fitting in MATLAB (lsqcurvefit; The MathWorks, Natick, MA, USA). Sample MATLAB code is available in the Supplementary Materials. First, we determined I and t_eff for the volunteers’ responses to the PF by fitting, ensemble, Equation 1a to the first 10 ms of each PF response, as recorded alone, with s_I = 3, t_eff shared among volunteers, and I free to vary by volunteer. If an individual was tested twice, we used the mean PF response for the fit. Then, we fit Equation 2 to the values of SF_t in all volunteers, ensemble, where s_I and s_Q were specified, t_eff was set to the previously obtained value, and α, I, and Q were free to vary between volunteers. Again, if a volunteer had multiple data sets, we used the mean values of SF_t at each ISI in the fitting. Initially, we fit using s_I = 3 and s_Q = 2, following our hypothesis that this would be optimal. However, to test this assumption, we repeated the fits, to both the PF response and to the SF_t data, using all (two or eight, respectively) other combinations of values of 2 through 4 for each parameter (s_I, s_Q) and compared the results by inspecting the residual root mean square error (RMSE) and using the Akaike information criterion (AIC). 

Results

In our initial inspection of the data, we found that, although the amplitudes of BA responses were larger than DTL responses by a factor of almost 2, there were no appreciable differences in the BA and DTL values of SF_t. Thus, we have combined the data obtained using these electrodes in the results presented below. 

In Figure 2, we plot the individual responses to the PF and their mean, as well as the mean fit to the first 10 ms of the PF to Equation 1a, with s_I = 3. The data comport excellently to the model. For comparison, we also show the best-fitting models with s_I = 2 and s_I = 4. With t_eff determined, we fit each individual volunteer's SF_t data to Equation 2. In Figure 3, we plot the fits of our hypothesized model that assumes three integrating steps in the initiation (i.e., s_I = 3) and two integrating steps in the quenching (i.e., s_Q = 2) of the rod photoresponse. Across all volunteers, α was 0.80 ± 0.066, I was 7700 ± 2400 m²·cd^–1·s^–3 (∼18 ± 5.6 rod·R*^–1·s^–2), and Q was 1.4 ± 0.47 s^–1 (mean ± SD). Thus, the coefficients of variation for these estimates were 0.082, 0.31, and 0.32, respectively. 

Figure 2.

 

Modeling the response to the PF. The first 20 ms of each volunteer’s response to a bright (20 cd·s·m^–2) stimulus are shown (gray traces), as is their mean (black trace). Fits of the first 10 ms (circles) of these data to Equation 1 with s_I values of 2 (stippled cyan line), 3 (red line), and 4 (stippled purple line) are shown. For the best-fitting model (s_I = 3), the value of t_eff was ∼3.2 ms (arrow).

Figure 3.

 

Plots of the suppression of the dark current, SF_t, derived using the double-flash ERG protocol detailed in Figure 1, in every volunteer (black lines and circles) and fits of the hypothesized DoE model (red lines) with three integrating steps in the initiation and two integrating steps in the quenching of the rod photoresponse. The overall r is 0.97 (Table).

Metrics for the goodness of the fits to the PF (Equation 1a), assessing initiation of the rod photoresponse, and to DF data (Equation 2), assessing both initiation and quenching of rod photoresponse, are given in the Table. The columns labeled PBTB (probability better than below) show the likelihood, rounded to the nearest whole percentage, that the model with the indicated value of s_I (PF) or combination of s_I and s_Q (DF) is better than the next-best model (i.e., the fit presented immediately below), as determined using AIC. When we inspect only the PF side of the table, we are left to consider values of both 3 and 4 for s_I. However, when we add the DF side, it is clear that we must discard the value of 4 for s_I. Furthermore, it is plain that the value of s_Q must be 2. That is, our hypothesized model, consistent with three integrating steps in initiation and two integrating steps in quenching, matches the data significantly better than any other combination. 

Table.

 

Goodness of Fits

 

In Figure 4, we plot the individual DF data and their mean, as well as the three best-fitting models: our hypothesized model, with s_I = 3 and s_Q = 2, and the similar models but assuming three or four integrating steps in quenching (i.e., s_Q = 3 and 4). These alternate models fit poorly. 

Figure 4.

 

Replot of the individual SF_t values from Figure 3 (gray) and superimposed mean SF_ts (black) on a linear (top) and—to provide a better view of the rapid initiation phase of the photoresponse—a logarithmic (bottom) time scale. The fits of these data to Equation 2, with s_I = 3 and s_Q values of 2 (red line), 3 (stippled cyan line), and 4 (stippled purple line), are shown.

Discussion

We developed a method, including a DoE model (Equation 2) with just three free parameters (α, I, Q), that robustly captures the magnitude and time constants of the complete rod photoresponse; our optimal fixed parameters were s_I = 3, s_Q = 2, and t_eff = 3.2 ms (note that the value of t_eff is specific to our setup). The finding that three integrating steps are required to model the initiation of the rod photoresponse is consistent with the well-established phototransduction cascade of rhodopsin isomerization, transducin activation, and cGMP hydrolysis. Our results confirm that this “simple” model of activation kinetics applies, even for a relatively dim flash that elicits a subsaturating a-wave. In contrast, only two integrating steps were required to model the quenching of the rod photoresponse, suggesting that arrestin binding to phosphorylated rhodopsin and GTPase activity leading to transducin inactivation are the primary integrating steps in recovery. While quenching involves complex interactions between multiple pathways (touched on above), our model “collapses” these into a single time constant (Q). 

Certainly, this model oversimplifies the processes involved in quenching. For just one example, the time constant Q does not have any accounting for calcium feedback, unlike the way the time constant I scales with flash intensity (compare Equation 1a and Equation 1b). The model also ignores other potential contributors to the ERG a-wave, such as transient currents in the outer nuclear layer (ONL); in addition to the voltage change from suppressing the dark current, the a-wave may reflect capacitive currents that hyperpolarize the rod axon and synaptic terminal.³⁴ These currents are distinct from conventional photoresponse physiology but likely do not much affect the SF_t values in the double-flash paradigm.³¹ Despite these limitations, our model—with s_I = 3 and s_Q = 2—captures the complete time course of the rod photoresponse with high correlation (r = 0.97) and parameters with relatively low variance. This gives us courage to apply this model to study of diseases of the rods. 

Furthermore, it does so by using a CF that is below the intensity required to saturate the a-wave. The use of a dim stimulus makes our method considerably easier for the volunteer to tolerate than methods using a paired flash protocol with equally bright CF and PF, both because it requires presentation of half as many unpleasant flashes per trial and because the necessary delay between trials is shorter. That said, the model frequently fit to recovery data from bright flashes^32,35–38 is mathematically similar to Equation 1b and, notably, assumes s_Q = 2, so our model is certainly extensible to CFs of any intensity—however, for the reasons noted above, Q is unlikely to be stable over a wide range of CFs. 

Finally, while our model does not have a formal solution for maximal suppression (SF_max) and, consequently, no formal solution for the oft-reported time to 50% recovery of the dark current (t₅₀), these parameters can be solved numerically or estimated as follows: taking α as an approximation of SF_max, Equation 1b can be solved for t when f_Q = α/2, as  

\begin{eqnarray} {{{t}}_{{\rm{50}}}}{\rm{\ = \ }}{\left( {\frac{{{\rm{ln(2)}}}}{{{Q}}}} \right)^{\left( {\frac{{\rm{1}}}{{{{{s}}_{{\rm{Q - 1}}}}}}} \right)}}{\rm{ + \ }}{{{t}}_{{\rm{eff}}}}\end{eqnarray}

(3)

 

Supplying a value of 2 for s_Q, the formula becomes t₅₀ = ln(2)/Q + t_eff, which simplifies to t₅₀ ≈ ln(2)/Q since t_eff is brief. Likewise, for studies reporting t₅₀, Q can be estimated as Q ≈ ln(2)/t₅₀. These translations allow comparison of our model parameters to results from other reports found in the literature. 

Acknowledgments

Supported by National Institutes of Health R01EY028953 (JDA) and R01EY010597 (ABF). 

Disclosure: J.D. Akula, None; A.M. Lancos, None; B.K. AlWattar, None; H. De Bruyn, None; R.M. Hansen, None; A.B. Fulton, None