A kinetic analysis of mouse rod and cone photoreceptor responses

Abstract

Most vertebrates have rod and cone photoreceptors, which differ in their sensitivity and response kinetics. We know that rods evolved from cone-like precursors through the expression of different transduction genes or the same genes at different levels, but we do not know which molecular differences were most important. We have approached this problem in mouse retina by analyzing the kinetic differences between rod flash responses and recent voltage-clamp recordings of cone flash responses, using a model incorporating the principal features of photoreceptor transduction. We apply a novel method of analysis using the log-transform of the current, and we ask which of the model’s dynamical parameters need be changed to transform the flash response of a rod into that of a cone. The most important changes are a decrease in the gain of the response, reflecting a reduction in amplification of the transduction cascade; an increase in the rate of turnover of cGMP in darkness; and an increase in the rate of decay of activated phosphodiesterase (PDE), with perhaps also an increase in the rate of decay of light-activated visual pigment. Although we cannot exclude other differences, and in particular alterations in the Ca²⁺ economy of the photoreceptors, we believe that we have identified the kinetic parameters principally responsible for the differences in the flash responses of the two kinds of photoreceptors, which were likely during evolution to have resulted in the duplex retina.

INTRODUCTION

In most vertebrate retinas there are rod photoreceptors, which are more sensitive and mediate vision in dim light; and cone photoreceptors, which are less sensitive but respond at higher temporal frequencies and are used for most of bright-light vision. As Schultze first proposed over 150 years ago (Schultze, 1866), these two photoreceptor types form a duplex retina, which we now know must have appeared very early during the evolution of vertebrates (Asteriti et al., 2015; Morshedian & Fain, 2015). Rods appear to have evolved from more primitive cone-like photoreceptors through gene duplication, modification of transduction enzymes, and changes in protein expression levels to accommodate vision in dimmer light (see Fain et al., 2010; Morshedian & Fain, 2017; Lamb, 2019).

Although rods and cones both use a similar transduction cascade, there are many molecular differences in the proteins they utilize, which are thought to be responsible for their different response properties. For several of the components of transduction, the two kinds of photoreceptors use different protein isoforms (see Ingram et al., 2016). That is true of their photopigments, transducin G proteins, phosphodiesterase (PDE6) effector enzymes, and cyclic nucleotide-gated channels. In other cases, the rods and cones are thought to use the same protein isoform but with different levels of expression. The GTPase activating proteins (GAPs) and guanylyl cyclase are both more highly expressed in cones than in rods (Cowan et al., 1998; Zhang et al., 2003; Takemoto et al., 2009). These molecular differences must in some way have been responsible for the evolution of the differences in the physiology of the two photoreceptor types. Rods and cones also differ in the morphology of their outer segments, with rods having cytoplasmic disks and cones membrane lamellae. This difference is however now known not to contribute to the different sensitivities and kinetics of the rod and cone photocurrent responses (Asteriti et al., 2015; Morshedian & Fain, 2015).

Many attempts have been made to determine which of the different isoforms or different expression levels are responsible for the differences in sensitivity and response kinetics of rods and cones. The cone isoforms of visual pigment (Sakurai et al., 2007; Shi et al., 2007; Fu et al., 2008), transducin (Deng et al., 2009; Chen et al., 2010b; Mao et al., 2013), and the phosphodiesterase-6 effector enzyme (PDE, Deng et al., 2013; Majumder et al., 2015) have been expressed in rods with variable results, sometimes producing alterations in response properties and sometimes producing no change at all. In other experiments, the expression of the GAP proteins in rods was increased (Krispel et al., 2006; Chen et al., 2010b), which may mimic the greater expression of these proteins in cones. In none of these experiments did a single molecular change in a rod produce responses similar to those of a cone.

These studies indicate that many different alterations in isoforms and expression levels must have been required during evolution to produce the duplex retina. It is however still unclear which alterations were most important. Because it may be difficult to make further progress with purely experimental methods, we approached this problem by combining experiment and theory. We began with a model of the mouse rod photoreceptor response that incorporates the principal features of the transduction cascade (Pugh & Lamb, 1993; Rieke & Baylor, 1996; Andreucci et al., 2003; Hamer et al., 2003; Hamer et al., 2005; Reingruber & Holcman, 2008; Chen et al., 2010c; Gross et al., 2012a; Reingruber et al., 2013). Although similar models have been proposed for cones (see for example Klaus et al., 2019), these attempts have suffered from a paucity of cone recordings. Suction-electrode recordings from mouse cones have been obtained by several investigators (Nikonov et al., 2005; Nikonov et al., 2006; Sakurai et al., 2011; Cao et al., 2014; Sakurai et al., 2015), including our own laboratory (Ingram et al., 2016; Kaylor et al., 2017), but these recordings are unlikely to provide an accurate reflection of the kinetics of the light-dependent change in outer-segment conductance because of the rapid kinetics of the cone response and the high capacitance of the cone membrane (Perry & McNaughton, 1991). Whole-retina measurements of massed cone responses (as in Sakurai et al., 2011; Sakurai et al., 2015; Morshedian et al., 2019) reflect the change in the cone membrane potential rather than the change in outer-segment conductance.

A more accurate assessment of changes in cone conductance could be obtained from voltage clamp (Perry & McNaughton, 1991). We have recently described voltage-clamp recordings from dark-adapted mouse cones in retinal slices (Ingram et al., 2019), not only from wild-type (WT) cones but also from cones with mutations in key transduction proteins. We now use these responses together with responses from mouse rods to define the minimal changes required to transform the dynamics of a rod response into those of a cone. We focus our study on dark-adapted responses to dim and non-saturating flash intensities, from which dynamical parameters can already be inferred.

In agreement with the molecular biology, we show that no one change is sufficient but that several changes need be made, particularly in the amplification of the G-protein cascade, the dark rate of cGMP turnover, the rate of decay of activated PDE, and perhaps also the rate of decay of activated visual pigment. Our analysis also predicts that a much lower number of PDEs are activated per photon in cones than in rods. These studies help clarify the physiological differences between the two photoreceptor types and may indicate which molecular changes were most important in forming the duplex retina over 500 million years ago.

MATERIALS AND METHODS

Animals and Ethical Approval

The data we used to determine model parameters came from experiments performed in accordance with the rules and regulations of the NIH guidelines for research animals, as approved by the Institutional Animal Care and Use Committee of the University of California, Los Angeles, USA, and were in conformity with regulations as described by Grundy (2015). Mice were kept under cyclic light (12-on/12-off) with adlib food and water in approved cages. Retinas lacking the genes for the two GCAPs (GCAPs^−/−) were obtained from Jeannie Chen of the University of Southern California, Los Angeles, CA (Mendez et al., 2001). Mice in which the rod-specific alpha subunit of the G-protein transducin had been deleted (Gnat1^−/−) were originally made in the laboratory of Janice Lem at Tufts University, Boston, MA (Calvert et al., 2000), and were obtained locally from the laboratory of Dr. Gabriel Travis at UCLA in Los Angeles, CA. Mice with deletions of both Gnat1 and the genes for the GCAPs (Gnat1^−/−;GCAPs^−/−) were made by mating animals from the two lines. At least three generations of double-deletion animals were produced before data collection commenced. All animals were on a C57BL/6J background; approximately equal numbers of male and female mice were used to obtain the recordings. Mice were euthanized by cervical dislocation.

Recordings

Wild-type (WT, C57BL/6J) rod responses and all cone responses were recorded with patch electrodes and voltage clamp from retinal slices (Ingram et al., 2019). Rod GCAPs^−/− responses were recorded with suction electrodes and have been previously published (Chen et al., 2010c). These earlier studies should be consulted for details of the experimental procedure. In both sets of experiments, temperature was measured with a thermocouple inside the chamber and was adjusted to 37°C for the suction recordings and 35°C for the voltage-clamp recordings; the value of the temperature was regulated to within 1°C with feedback controllers (TC-324B; Warner Instruments). Cone recordings were made from mice that lacked the rod transducin alpha subunit (Gnat1^−/−), in order to prevent rod signals contributing to cone responses (Ingram et al., 2019). Cell membrane potential was clamped near the dark resting membrane potential (−40 mV for rods and −50 mV for cones).

Rod WT responses and all cone responses were recorded in Ames’ medium bubbled with 95% O₂/5% CO₂ and buffered with 1.9 g per liter sodium bicarbonate, pH 7.3 to 7.4. Rod GCAPs^−/− responses were recorded in Dulbecco’s Modified Eagle’s medium supplemented with 15 mM NaHCO₃, 2 mM Na succinate, 0.5 mM Na glutamate, 2 mM Na gluconate, and 5 mM NaCl, bubbled with 5% CO₂, pH 7.4. Although rods in these two media have been reported to have somewhat different kinetic properties (Azevedo & Rieke, 2011), these differences are small in comparison to the differences in kinetics between rods and cones and are likely to result from a difference in Ca²⁺-dependent cyclase feedback, which would not affect responses from GCAPs^−/− rods. We attempted to record from mouse GCAPs^−/− rods with patch clamp in retinal slices under the same conditions we used for WT rods and cones. Despite considerable effort (4 animals, over 30 successful seals), we were unable to obtain stable voltage-clamp recordings from these cells, which seemed fragile and were unusually noisy by comparison to WT rods. The GCAPs^−/− rod responses even to dim flashes decayed with an extremely slow time course, and responses to bright flashes often “hung up” and failed to return to baseline during the duration of the recording. From that point onward, the rod no longer responded even to the brightest flash. We will show however that WT rods with patch-clamp in Ames medium (Fig. 1A) can be fitted with the same parameters as GCAPs^−/− rods recorded with suction-electrodes, with the addition to the WT model of GCAP-dependent Ca²⁺ feedback (see Fig. 4). Though the fits are not as good for WT as for GCAPs^−/− rods, we think this difference is more likely the result of oversimplification in our model for Ca²⁺ and feedback than methods of recording and perfusion solutions. Data were analyzed either with pCLAMP and Origin (for for suction recordings) or in MATLAB with custom scripts (for patch-clamp recordings).

Figure 1.

Experimental current recordings from mouse rods and cones. Currents from individual rods and cones have been normalized to their corresponding steady-state current in darkness I_d, with $\widehat{I}=\frac{I}{I_d}$ (such that $\widehat{I}=1$ in darkness) and then averaged point by point among cells to compute mean responses. Ordinates give the mean current $\widehat{\iota }=1-\widehat{I}$ (such that $\widehat{\iota }=0$ in darkness), which is equivalent to the change in current normalized to the change produced by a bright light closing all of the outer-segment channels. The shaded regions indicate standard error of the mean computed point by point. Saturating responses to the brightest stimuli were not used in our model calculations (see Methods) and are not shown. (A) Mean responses of 7 WT rods. (B) Rod GCAPs^−/−. Mean responses of 5 rods from Chen et al. (2010c). (C) Cone Gnat1^−/− (equivalent to WT or control, see text). Mean responses from 3 cones. (D) Cone Gnat1^−/−;GCAPs^−/− (equivalent to cone GCAPs^−/−, see text). Mean responses from 3 cones. Flash durations are 10ms in (A), 20ms in (B) and 5ms in (C, D). The figure inserts give the expected number of pigment isomerizations R*, computed with collecting areas κ = 0.2 (A) or 0.5 (B) for rod responses, and κ = 0.013 for cone responses.

Figure 4.

Adjusting the rod model. (A-B) To estimate the rod parameters ξ, μ_tr and ${\widehat{K}}_{\alpha }$ we concurrently fit the WT and GCAPs^−/− traces from Fig. 1A and 1B to the non-linear Eqs. (11) in the Methods using the kinetic parameters from Table 2 (in black) as input and ξ , μ_tr and ${\widehat{K}}_{\alpha }$ as fitting parameters. The best fit was obtained with ξ = 0.45, μ_tr = 23.8 s⁻¹ and ${\widehat{K}}_{\alpha }=0.87$. Black traces show the data, red traces show simulations. (C) Comparison of the log-transforms from (B). (D) Comparison of the normalized log-transforms from (C).

Derivation of the phototransduction model

We have used a simplified, spatially homogeneous model for photoreceptors (see for example Hamer et al., 2005; Reingruber & Holcman, 2008; Chen et al., 2010c; Korenbrot, 2012; Reingruber et al., 2013; Wang et al., 2018). The photoreceptor current, I, is taken to be a function of the cyclic guanosine nucleotide (cGMP) concentration c_cg according to:

$$I=I_d{\left(\frac{c_{\mathit{cg}}}{c_{\mathit{cg},d}}\right)}^{n_{\mathit{ch}}}$$ (1)

where I_d and c_cg,d are the current and cGMP concentration in darkness, and n_ch is the cooperativity constant of the channel, which we will take to be 2.5 (see for example Haynes & Yau, 1985; Zimmerman & Baylor, 1986; Yau & Baylor, 1989). We will make the simplifying assumption that changes in Ca²⁺ during the flash response are sufficiently rapid for mouse cones that they can be assumed to occur in proportion to the change in current, that is

$$\frac{c_{\mathit{ca}}}{c_{\mathit{ca},d}}=\frac{I}{I_d}$$ (2)

where c_ca is the calcium concentration, and c_ca,d is the calcium concentration in darkness. We make this assumption because light-dependent changes in Ca²⁺ are much faster in cones than in rods in salamander (Sampath et al., 1999), and because this simplifying assumption gave a reasonable fit of the model to our cone responses. We further assume that α, the Ca²⁺-dependent rate of synthesis of cGMP, quickly adapts to the changing Ca²⁺ concentration. We then compute the rate from the Ca²⁺ concentration according to

$$\alpha (c_{\mathit{ca}})={\alpha }_{\mathit{min}}\frac{c_{\mathit{ca}}^2+r_{\alpha }{K}_{\alpha }^2}{c_{\mathit{ca}}^2+K_{\alpha }^2}$$ (3)

where K_α determines the sensitivity of the rate to Ca²⁺, α_min is the minimum value of the cyclase rate at high Ca²⁺, and r_α is the ratio of maximum to minimum rates, i.e. α_max/α_min.

To simplify the solution of the equations, we introduce dimensionless quantities by normalizing with steady-state values in darkness. That is, we set

$$\widehat{I}=\frac{I}{I_d},{\widehat{c}}_{\mathit{cg}}=\frac{c_{\mathit{cg}}}{c_{\mathit{cg},d}},{\widehat{c}}_{\mathit{ca}}=\frac{c_{\mathit{ca}}}{c_{\mathit{ca},d}},\widehat{\alpha }=\frac{\alpha }{{\alpha }_d},\text{and}{\widehat{K}}_{\alpha }=\frac{K_{\alpha }}{c_{\mathit{ca},d}}$$ (4)

The rate of synthesis of cGMP in darkness is α_d = β_dc_cg,d where β_d is the rate constant of cGMP hydrolysis in darkness (turnover rate). With our assumption that the Ca²⁺ concentration is directly proportional to the current, ${\widehat{c}}_{\mathit{ca}}=\widehat{I}$, we can express the normalized the Ca²⁺-dependent rate of synthesis of cGMP as a function of the current as

$$\widehat{\alpha }(\widehat{I})=\frac{{\widehat{I}}^2+r_{\alpha }{\widehat{K}}_{\alpha }^2}{{\widehat{I}}^2+{\widehat{K}}_{\alpha }^2}\frac{1+{\widehat{K}}_{\alpha }^2}{1+r_{\alpha }{\widehat{K}}_{\alpha }^2}\approx \frac{1+{\widehat{K}}_{\alpha }^2}{{\widehat{I}}^2+{\widehat{K}}_{\alpha }^2}.$$ (5)

The approximation to the right in Eq. (5) is valid for $r_{\alpha }{\widehat{K}}_{\alpha }^2\gg 1$ and was used in most of our calculations. Using the full expression, we verified r_α largely satisfied this condition (see for example Woodruff et al., 2007).

The light-dependent rate constant of cGMP hydrolysis by PDE depends on the number of activated PDE molecules, P*, according to β_subP*, where β_sub is the rate constant of cGMP hydrolysis by a single light-activated PDE. The equation for the change in normalized cGMP concentration is

$$\frac{d}{\mathit{dt}}{\widehat{c}}_{\mathit{cg}}={\beta }_d\widehat{\alpha }-({\beta }_d+{\beta }_{\mathit{sub}}{P}^{\ast }){\widehat{c}}_{\mathit{cg}}$$ (6)

Log transform of current

To facilitate the analysis of the rod and cone flash responses, we introduce the log transform of the normalized current $y=-\mathit{ln}\widehat{I}$ so that $\widehat{I}=e^{-y}$. We reasoned as follows. The initial phase of the light response was shown by Pugh and Lamb (1993) to follow to a first approximation an equation of the form,

$$\widehat{I}=e^{-\frac{1}{2}\kappa \varphi \Delta \mathit{tA}{(t-t_{\mathit{eff}})}^2}$$ (7)

with κ the collecting area, ϕ the light intensity, Δt the flash duration, t_eff an effective delay at the beginning of the response, and A the amplification constant. If we let $\widehat{I}(t)=e^{-y(t)}$, then the initial waveform of y(t) becomes linear with the flash intensity ϕ. We shall show that for responses to dim flashes, and over much of the intensity range when the guanylyl-cyclase-activating proteins (GCAPs) have been knocked out and there is no Ca²⁺-dependent cyclase feedback (Mendez et al., 2001), not only the initial phase but the entire waveform of y(t) is, to a good approximation, linear with the flash intensity ϕ. If therefore the waveforms y(t) of responses at different flash intensities are normalized by their corresponding peak amplitudes, the waveform of the normalized log transform of a flash response is to a good approximation invariant with light intensity (see Fig. 2). In this way, the kinetics of flash responses can be pooled and analyzed together.

Figure 2.

Normalized log transforms of response waveforms. The GCAPs^−/− responses from Figs. 1B and 1D were transformed, first by taking the negative log of the current, $y(t)=-\mathit{ln}\widehat{I}(t)$, and then by normalizing the value of y(t) to give $\widehat{y}(t)=\frac{y(t)}{y_{\mathit{peak}}}$. (A) Plots of $\widehat{y}(t)$ for rod GCAPs^−/− responses of Fig. 1B. (B) Plots of $\widehat{y}(t)$ for cone Gnat1^−/−;GCAPs^−/− responses of Fig. 1D. (C) Comparison of mean values of $\widehat{y}(t)$ from (A) and (B).

From Eq. (1) after normalization from Eqs. (4), we get $\widehat{I}={{\widehat{c}}_{\mathit{cg}}}^{n_{\mathit{ch}}}=e^{n_{\mathit{ch}}\mathit{ln}{\widehat{c}}_{\mathit{cg}}}$. From this expression we get $y=-n_{\mathit{ch}}\mathit{ln}{\widehat{c}}_{\mathit{cg}}$, and with Eq. (6) we obtain

$$\frac{d}{\mathit{dt}}y=n_{\mathit{ch}}{\beta }_{\mathit{sub}}{P}^{\ast }-{\beta }_{d}H(y)$$ (8)

with

$$H(y)=n_{\mathit{ch}}\left[e^{\frac{y}{n_{\mathit{ch}}}}\widehat{\alpha }(e^{-y})-1\right]=n_{\mathit{ch}}\left[e^{\frac{y}{n_{\mathit{ch}}}}\frac{1+{\widehat{K}}_{\alpha }^2}{e^{-2y}+{\widehat{K}}_{\alpha }^2}-1\right].$$ (9)

Transduction equations

To obtain a closed system of equations for the rod or cone light response, we additionally need equations for the production and decay of the components of the transduction cascade. We assume that light-activated visual pigment activates the G-protein transducin, which activates PDE6. We let R* be the number of pigment molecules activated in the rod or cone by the flash, T* be the number of activated transducins, and P* be the number of activated PDEs. We have

$$\begin{gathered} \frac{d}{\mathit{dt}}{R}^{\ast }=\varphi (t)\kappa -{\mu }_{\mathit{rh}}{R}^{\ast } \\ \frac{d}{\mathit{dt}}{T}^{\ast }=k_{\mathit{act}}{R}^{\ast }-{\mu }_{\mathit{tr}}{T}^{\ast } \\ \frac{d}{\mathit{dt}}{P}^{\ast }={\mu }_{\mathit{tr}}{T}^{\ast }-{\mu }_{\mathit{pde}}{P}^{\ast } \\ \frac{d}{\mathit{dt}}y=n_{\mathit{ch}}{\beta }_{\mathit{sub}}{P}^{\ast }-{\beta }_{d}H(y) \end{gathered}$$ (10)

For flashes with duration Δt and flash-intensity ϕ we have ϕ(t) = ϕ (θ(t) − θ (t − Δt)), where θ is the unit step or Heaviside function (i.e., θ(t) = 0 for t < 0 and θ(t) = 1 for t ≥ 0). All of the parameters in Eqs. (10) are defined in Table 1. For the collecting areas κ we used 0.5 μm² for GCAPs^−/− rods recorded with suction electrodes (Field & Rieke, 2002), and 0.2 μm² for WT rods and 0.013 μm² for Gnat1^−/− and Gnat1^−/−;GCAPs^−/− cones recorded in retinal slices with voltage clamp (Ingram et al., 2019). The parameter k_act is the rate of activation of transducin by a single activated visual pigment, μ_rh is the deactivation rate of a light-activated visual pigment, μ_pde is the deactivation rate of a light-activated PDE, and μ_tr is the rate by which a T* is converted into P* but can be more generally viewed as an effective parameter that accounts for intermediate processes that contribute to PDE activation and delay the response without amplifying it.

Table 1.

Parameter descriptions for rod and cone model.

Parameter	Description
κ (μm2)	Photoreceptor collecting area
nch	CNG channel cooperativity (Hill coefficient)
βd (s−1)	Rate constant of cGMP hydrolysis in darkness (cGMP turnover rate in darkness)
μrh (s−1)	Rate of deactivation of an activated visual pigment
μpde (s−1)	Rate of deactivation of a light-activated PDE
μtr (s−1)	Rate for the transformation of light-activated transducin into light-activated PDE
${\widehat{K}}_{\alpha }$	Sensitivity of the cyclase activity on the Ca2+ concentration scaled by the dark Ca2+ concentration
ξ	$\text{Gain}\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}$
kact (s−1)	Rate of transducin activation by an activated visual pigment
βsub (s−1)	Rate constant of cGMP hydrolysis by a light-activated PDE

To simplify the analysis of Eqs. (10), we transform the variables according to ${\tilde{P}}^{\ast }=n_{\mathit{ch}}{\beta }_{\mathit{sub}}{P}^{\ast }$, ${\tilde{T}}^{\ast }=n_{\mathit{ch}}{\beta }_{\mathit{sub}}{\mu }_{\mathit{tr}}∕{\mu }_{\mathit{pde}}{T}^{\ast }$ and ${\tilde{R}}^{\ast }=n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}∕{\mu }_{\mathit{pde}}{R}^{\ast }$. With the transformed variables Eqs. (10) become

$$\begin{gathered} \frac{d}{\mathit{dt}}{\tilde{R}}^{\ast }={\mu }_{\mathit{rh}}(\varphi (t)\kappa \xi -{\tilde{R}}^{\ast }) \\ \frac{d}{\mathit{dt}}{\tilde{T}}^{\ast }={\mu }_{\mathit{tr}}({\tilde{R}}^{\ast }-{\tilde{T}}^{\ast }) \\ \frac{d}{\mathit{dt}}{\tilde{P}}^{\ast }={\mu }_{\mathit{pde}}({\tilde{T}}^{\ast }-{\tilde{P}}^{\ast }) \\ \frac{d}{\mathit{dt}}y={\tilde{P}}^{\ast }-{\beta }_{d}H(y) \end{gathered}$$ (11)

where we introduced the transduction gain

$$\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}.$$ (12)

Eqs. (11) reveal that y(t) and the current do not depend upon the individual values of κ, n_ch, β_sub and k_act but only on their product. We call ξ the gain because it determines the amplitude of ${\tilde{P}}^{\ast }(t)$ and that of the dim flash responses (see next paragraph).

Result for PDE activation

The solution of Eqs. (11) for PDE is ${\tilde{P}}^{\ast }(t)=\kappa \xi {\int }_0^t\varphi (s)g_p(t-s)\mathit{ds}$ with the Green’s function

$$g_p(t)={\mu }_{\mathit{rh}}{\mu }_{\mathit{tr}}{\mu }_{\mathit{pde}}\left(\frac{e^{-{\mu }_{\mathit{rh}}t}}{({\mu }_{\mathit{rh}}-{\mu }_{\mathit{tr}})({\mu }_{\mathit{rh}}-{\mu }_{\mathit{pde}})}+\frac{e^{-{\mu }_{\mathit{tr}}t}}{({\mu }_{\mathit{tr}}-{\mu }_{\mathit{rh}})({\mu }_{\mathit{tr}}-{\mu }_{\mathit{pde}})}+\frac{e^{-{\mu }_{\mathit{pde}}t}}{({\mu }_{\mathit{pde}}-{\mu }_{\mathit{rh}})({\mu }_{\mathit{pde}}-{\mu }_{\mathit{tr}})}\right)$$ (13)

Note that g_p(t) is symmetric in μ_rh, μ_tr and μ_pde. For a short flash of duration Δt we have ${\tilde{P}}^{\ast }(t)=R_0^{\ast }\xi g_p(t)$, where $R_0^{\ast }=\varphi \kappa \Delta \mathrm{t}$ is the number of isomerizations generated by the flash. The number of activated PDEs is ${\mathrm{P}}^{\ast }(t)=\frac{{\tilde{P}}^{\ast }(t)}{n_{\mathit{ch}}{\beta }_{\mathit{sub}}}=R_0^{\ast }\frac{\xi }{n_{\mathit{ch}}{\beta }_{\mathit{sub}}}{g}_p(t)$. The number of PDEs activated during a single-photon response is obtained with $R_0^{\ast }=1$.

Asymptotic result for dim flashes

The equation for y(t) is non-linear due to H(y) (see Eq. (9)) and cannot be solved analytically. However, for GCAPs^−/− photoreceptors where non-linear Ca²⁺ feedback is absent, and for dim flashes where y(t) remains small, we can use the first order approximation β_dH(y) ≈ βy with

$$\beta ={\beta }_d(1-n_{\mathit{ch}}{\widehat{\alpha }}^{\prime })$$ (14)

and ${\widehat{\alpha }}^{\prime }=\frac{d}{\mathit{dx}}{\phantom{\mid }\widehat{\alpha }(x)\mid }_{x=1}=-\frac{2}{1+{\widehat{K}}_{\alpha }^2}$ to derive an asymptotic expression for the current. The solution of Eqs. (11) with the linear approximation β_dH(y) = βy is $y(t)=\kappa \xi {\int }_0^t\varphi (s)g_y(t-s)\mathit{ds}$, where

$$g_y(t)={\mu }_{\mathit{rh}}{\mu }_{\mathit{tr}}{\mu }_{\mathit{pde}}\left(\frac{e^{-{\mu }_{\mathit{rh}}t}}{({\mu }_{\mathit{tr}}-{\mu }_{\mathit{rh}})({\mu }_{\mathit{rh}}-{\mu }_{\mathit{pde}})({\mu }_{\mathit{rh}}-\beta )}+\frac{e^{-{\mu }_{\mathit{tr}}t}}{({\mu }_{\mathit{rh}}-{\mu }_{\mathit{tr}})({\mu }_{\mathit{tr}}-{\mu }_{\mathit{pde}})({\mu }_{\mathit{tr}}-\beta )}+\frac{e^{-{\mu }_{\mathit{pde}}t}}{({\mu }_{\mathit{rh}}-{\mu }_{\mathit{pde}})({\mu }_{\mathit{pde}}-{\mu }_{\mathit{tr}})({\mu }_{\mathit{pde}}-\beta )}+\frac{e^{-\beta t}}{({\mu }_{\mathit{rh}}-\beta )(\beta -{\mu }_{\mathit{tr}})(\beta -{\mu }_{\mathit{pde}})}\right).$$ (15)

For short flashes we get $y(t)=\varphi \kappa \Delta \mathrm{t}\xi g_y(t)=R_0^{\ast }\xi g_y(t)$. For photoreceptors in which the genes for the GCAPs have been deleted and which therefore lack Ca²⁺-dependent feedback of cyclase activity, ${\widehat{\alpha }}^{\prime }=0$ and β = β_d. Because the dim flash analysis is valid for WT and GCAPs^−/− photoreceptors, we conclude that, to first approximation, a WT cell behaves like a GCAPs^−/− cell with an increased cGMP turnover rate of ${\beta }_d(1-n_{\mathit{ch}}{\widehat{\alpha }}^{\prime })$.

By dividing y(t) by its peak value y_peak we define the normalized log-transform

$$\widehat{y}(t)=\frac{y(t)}{y_{\mathit{peak}}}\approx \frac{g_y(t)}{g_{y,\mathit{peak}}}.$$ (16)

For dim flashes the normalized log-transform is independent of the light intensity ϕ and the gain ξ and depends only on the dynamical parameters, and the waveform of $\widehat{y}(t)$ characterizes the shape of a flash response.

Fitting Procedure

Fitting was done with the Data to Dynamics framework (Raue et al., 2013; Raue et al., 2015). The optimization algorithm that we used for fitting and parameter estimation is the nonlinear least-squares solver LSQNONLIN from MATLAB. Because the model is non-linear in parameters, we give goodness of fit values as root-mean-squared error (RMSEs, see for example Dekking et al., 2005). Lower values of RMSE indicate a better fit of the model to the data.

RESULTS

Fig. 1 illustrates dark-adapted photoresponses of rods and cones, normalized to the maximum response to light bright enough to close all of the outer-segment channels. Saturating responses to the brightest stimuli were not used in our model calculations (see Methods) and are not shown. For our analysis we consider two different expressions for the normalized currents: first $\widehat{I}$, which is unity in darkness and decreases with light; and the complementary current $\widehat{\iota }=1-\widehat{I}$, which is zero in darkness and increases to unity in bright light. The latter expression is more common in the literature and is depicted in Fig. 1.

The responses of WT mouse rods in Fig. 1A are averaged responses from 7 rods recorded with voltage-clamp from retinal slices (see Methods). As we explain in the Methods, we were unable to obtain similar voltage-clamp recordings from GCAPs^−/− mouse rods which were sufficiently stable to be used for model calculations. The responses of GCAPs^−/− mouse rods in Fig. 1B are therefore averaged suction-electrode recordings from 5 rods taken from previously published experiments (Chen et al., 2010c). The GCAPs^−/− rods lack the guanylyl-cyclase activating proteins (GCAPs), which are closely associated with guanylyl cyclase in both rods and cones and mediate Ca²⁺-dependent modulation of the rate of the cyclase. The use of this mutant to disable Ca-dependent feedback simplifies the fitting of the model.

Cone responses are recent voltage-clamp recordings from mouse retinal slices recorded under the same conditions as the WT rod responses in Fig. 1A (Ingram et al., 2019). Recordings from cones were made from Gnat1^−/− retinas lacking Gnat1, the alpha-subunit of the rod G protein transducin. Cones in the mouse retina receive input from rods through connexin-36 gap junctions (Asteriti et al., 2017; see Fain & Sampath, 2018), and this input is deleted in Gnat1^−/− cones (Ingram et al., 2019). In Fig. 1C, we show averaged responses from three Gnat1^−/− cones, which are effectively WT cone responses lacking rod input. Responses of a very similar waveform were recorded from Cx36^−/− retinas lacking connexin 36 (Ingram et al., 2019). In Fig. 1D we show averaged responses from three Gnat1^−/−;GCAPs^−/− cones lacking both Gnat1 and the GCAP proteins, which are effectively GCAPs^−/− responses lacking any rod input.

From the recordings in Fig. 1, we will proceed in the following way: we first use the normalized log-transform of GCAPs^−\− current responses from Figs. 1B and 1D, which lack Ca²⁺ feedback, to compare the striking difference in kinetics between mouse rods and mouse cones (Fig 2). We then analyze both WT and GCAPs^−/− rod responses together to define a set of rod parameters (Fig. 4), which we use as reference and starting values for our cone analysis. In the model calculations of Fig. 5 we identify those dynamical parameters that have to be considerably different between rods and cones. We then concurrently fit the cone WT and GCAPs^−/− responses to estimate values for the unknown cone parameters, including the gain and Ca²⁺ sensitivity of the cyclase rate; and we probe our cone model by fixing certain parameters and varying others (Fig. 6). Finally, we use our simulations to compare the single-photon response and the number of PDE molecules activated per photon between a rod and a cone (Fig. 7).

Figure 5.

Transition from rod to cone kinetics. The black traces are normalized log-transforms $\widehat{y}(t)$ from cone Gnat1^−/−;GCAPs^−/− responses from Fig. 2B. The red traces show a simulation of a normalized log-transform computed with rod parameter values for μ_rh, μ_tr, μ_pde, and β_d as given in Table 2, except for the parameter indicated in each of the panels, which has been estimated by fitting the normalized cone Gnat1^−/−;GCAPs^−/− responses with Eq. (11). (A-D) The best-fitting parameter values were β_d = 200 s⁻¹, μ_rh = 200 s⁻¹, μ_tr = 200 s⁻¹, and μ_pde = 200 s⁻¹, corresponding to the upper limit of 200 s⁻¹ that we imposed for the parameter range. RMSEs were as follows: 0.14 (A), 0.21 (B), 0.15 (C), and 0.20 (D). (E) The best-fitting parameter values were μ_rh = μ_tr = μ_pde = 130 s⁻¹. RMSE was 0.1. (F) The best-fitting parameter values were β_d = 31 s⁻¹ and μ_pde = 183 s⁻¹. RMSE was 0.02.

Figure 6.

Adjusting the cone model. To estimate cone parameter values, the currents $\widehat{\iota }=1-e^{-y(t)}$ of Gnat1^−/−;GCAPs^−/− cones and Gnat1^−/− cones (effectively wild type) from Fig. 1C and 1D have been fitted concurrently with the non-linear model in Eq. (11) from the Methods with fitting parameters ξ, μ_rh, μ_tr, μ_pde, β_d, and ${\widehat{K}}_{\alpha }$. We explored four different fitting scenarios as explained in the text. The best-fitting parameter values for each scenario are summarized in Table 2 (fixed parameter values are in black, fitted values are in blue). (A) and (B). Current data (black traces) and the corresponding simulation results obtained with the parameters for the cone scenario 1 (red traces) for Gnat1^−/− (A) and Gnat1^−/−;GCAPs^−/− cones (B).

Figure 7.

Single-photon response (SPR) for rod and cone model. (A) Comparison of the SPR current for WT and GCAPs^−/− rod with Gnat1^−/− (effectively wild type) and Gnat1^−/−;GCAPs^−/− cones. The currents are computed with the analytic formula ${\widehat{\iota }}_{\mathit{spr}}(t)=1-e^{-\xi g_y(t)}$ with g_y(t) from Eq. (15) in the Methods and parameter values from Table 2 (scenario 1 for cone). (B) Comparison of the time-dependent number of activated PDEs corresponding to the response in (A). The number of activated PDEs is computed as ${\mathrm{P}}^{\ast }(t)=\frac{\xi }{n_{\mathit{ch}}{\beta }_{\mathit{sub}}}{g}_p(t)$, with g_p(t) from Eq. (13) in the Methods with parameter values from Table 2 and β_sub= 0.07.

Normalized log-transform of the rod and cone light response

We begin by considering the log-transform of $\widehat{I}$ of rod and cone GCAPs^−/− responses, since we have shown (see Eq. (15) in the Methods) that the entire waveform of the log-transform of the photoreceptor response in the absence of cyclase feedback should be proportional to the strength of the stimulus over much of the range of stimulus intensities. To demonstrate this phenomenon, we have taken the logarithms of the GCAPs^−/− responses $\widehat{I}=1-\widehat{\iota }$ of Figs. 1B (for rods) and 1D (for cones) and have normalized each curve to its peak value. That is, we have calculated the negative log of the current, $y=-\mathit{ln}\widehat{I}$, and then normalized it to the peak value to give $\widehat{y}=\frac{y}{y_{\mathit{peak}}}$ (see also Eq. (16) in the Methods). For both rods (Fig. 2A) and cones (Fig. 2B), $\widehat{y}$ follows a very similar time course at each light intensity which is, however, quite different for the two kinds of photoreceptors. In Fig. 2C, we compare on the same time base the mean waveforms of $\widehat{y}$ for rods and cones from Figs. 2A and 2B. This figure illustrates the considerable difference in the kinetics of the rod and cone responses.

Model for the rod and cone response

As we describe in the Methods section, we used a parsimonious, spatially homogeneous model of vertebrate phototransduction. Our model was derived from earlier models (see for example Hamer et al., 2005; Reingruber & Holcman, 2008; Chen et al., 2010c; Korenbrot, 2012; Reingruber et al., 2013; Wang et al., 2018), which were based on the known transduction reactions (see Eq. (10) in the Methods). In brief (see Fig. 3), we assume that light-activated visual pigment (R*) activates the G-protein transducin (T*), which activates PDE6 (P*). The P* hydrolyses the second messenger cGMP, which controls the opening of the cyclic-nucleotide-gated (CNG) channels in the outer-segment plasma membrane. In a WT rod or cone, the synthesis of cGMP by guanylyl cyclase is modulated by Ca²⁺ feedback via the GCAPs (not shown in Fig. 3). This feedback is eliminated in GCAPs^−/− photoreceptors. We use the same equations to model both the WT and GCAPs^−/− responses of both kinds of photoreceptors, because the biological transduction pathway is very similar between rods and cones (see Ingram et al., 2016). This molecular similarity suggests that, to a first approximation, the same model structure can be used to describe both rods and cones with differences faithfully captured by adjustment of model parameters, which we have circled in Fig. 3. Eqs. (15–16) in the Methods show that the difference in the kinetics of the rod and cone responses in Fig. 2 are generated by differences in the deactivation rate of a light-activated pigment μ_rh, the rate μ_tr by which an activated transducin is converted into an activated PDE, the deactivation rate of a light-activated PDE μ_pde, and the cGMP turnover rate in darkness β_d. The question then becomes, how many of these parameters are different, and which ones are most important?

Figure 3.

Schema of transduction cascade showing rates of activation and decay used in the model. Model parameters are given in bold and are circled; they are defined in Table 1. Abbreviations: hν, light; R, visual pigment; R*, activated visual pigment; T, transducin; T*, activated transducin; P, phosphodiesterase; P*, activated phosphodiesterase; GAP, GTPase-Activating Proteins; GTP, guanosine triphosphate; GDP, guanosine diphosphate; GMP, guanosine monophosphate; cGMP, 3’,5’-cyclic guanosine monophosphate.

Adjusting the rod model

To answer this question, we begin with the model for rods, where we have used kinetic parameters for the most part derived from the literature (see Table 2). We assumed a value for μ_rh of 28 s⁻¹ (for a time constant of 36 ms, see Krispel et al., 2006; Chen et al., 2010a), a value for μ_pde of 5 s⁻¹ (for a time constant of 200 ms, see for example Krispel et al., 2006; Tsang et al., 2006), and a value for β_d of 4.1 s⁻¹ (Gross et al., 2012b; Reingruber et al., 2013). In contrast to μ_rh, μ_pde, and β_d, no clear estimate for μ_tr can be found in the literature. The value of μ_tr is often assumed to be very large, in which case the model in Eqs. (11) can be further simplified by removing the intermediate state T* such that R* directly activates PDE (Pugh & Lamb, 2000; Gross & Burns, 2010; Gross et al., 2012). More generally, μ_tr can be viewed as an effective parameter that delays the initial rising of the response but does not contribute to amplification, provided each activated transducin (or each pair of transducins, see Qureshi et al., 2018) activates only a single PDE. To obtain a better fit to the initial rising phase of the responses, we decided to keep μ_tr as a model parameter.

Table 2.

Parameter values for rod and cone models. Fixed input values are in black and fitted parameter values are in blue. For each cone scenario the same Gnat1^−/− and Gnat1^−/−;GCAPs^−/− data have been used for the fitting. The RMSE makes possible the comparison of the goodness of fit between the different cone scenario (uL = the fitted parameter value reached the upper limit of the parameter range that we imposed).

Parameter	Rod	Cone Scenario 1	Cone Scenario 2	Cone Scenario 3	Cone Scenario 4
κ (μm2)	0.2 (WT) 0.5 (GCAPs−/−)	0.013	0.013	0.013	0.013
nch	2.5	2.5	2.5	2.5	2.5
ξ	0.45	0.0018	0.0019	0.0020	0.0024
βd (s−1)	4.1	11.0	12.4	12.4	16.6
μpde (s−1)	5	37.8	85.9	40 (uL)	20 (uL)
μrh (s−1)	28	70.7	28	40 (uL)	101.5
μtr (s−1)	23.8	70.7	85.9	100 (uL)	101.5
${\widehat{K}}_{\alpha }$	0.87	0.84	0.82	0.78	0.80
RMSE		0.031	0.034	0.033	0.044

In addition to the kinetic parameters μ_rh, μ_tr, μ_pde, and β_d, the flash response further depends on the dimensionless gain parameter $\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}$ (see Eq. (12) in the Methods) and the cyclase Ca²⁺-sensitivity parameter ${\widehat{K}}_{\alpha }$. In the expression for ξ, n_ch is the cooperativity constant (Hill coefficient) of the channel, β_sub is the rate constant by which a single light-activated PDE molecule hydrolyses cGMP, and k_act is the rate constant of activation of T* by a single R*. To estimate the parameters ξ , μ_tr and ${\widehat{K}}_{\alpha }$ we concurrently fit the WT and GCAPs^−/− rod responses from Fig. 1A and 1B to the non-linear Eqs. (11) in the Methods using known kinetic parameters from Table 2 (indicated in black) as input and ξ , μ_tr and ${\widehat{K}}_{\alpha }$ as fitting parameters. In Figs. 4A and 4B we show the agreement between the best fitting simulation and the data for the current $\widehat{\iota }=1-e^{-y(t)}$. We emphasize that the fitting and simulations for WT and GCAPs^−/− rod were performed with exactly the same parameter values but with the cyclase feedback switched off for the GCAPs^−/− simulations. The best fitting parameter values were ξ = 0.45, μ_tr = 23.8 s⁻¹ and ${\widehat{K}}_{\alpha }=0.87$. The agreement between simulation and data is reasonably good for both WT and GCAPs^−/− rods, even though recordings were made with different techniques and in different perfusion solutions (see Methods). Discrepancies are greater for WT rods, probably because our assumptions about Ca²⁺ feedback were over-simplified.

Because rod fittings were performed with the averaged data from Fig. 1, we checked these values by additionally performing a single-cell analysis. We chose the five GCAPs^−/− cells whose mean currents are given in Fig. 1B together with five WT cells, randomly chosen from among the seven WT cells used for the averaging in Fig. 1A. We then performed concurrent fittings with 25 different combinations of single-cell data. We fitted the single-cell data in exactly the same way as we performed the fittings with the averaged data. The results for the fitted parameters were (mean ± SD) ξ = 0.49 ± 0.12, μ_tr = 27.9 ± 37 s⁻¹, ${\widehat{K}}_{\alpha }=0.85\pm 0.47$. Although there was considerable variability, the mean values were very close to the values we obtained from the fit to the mean responses. Confidence intervals calculated by Monte Carlo simulations of individual cell fit parameters showed that, for all rod parameters, values obtained by fitting to mean responses were within calculated 95% confidence intervals of values obtained by fitting to individual responses. Thus we conclude that, within an alpha rate of 5%, fitting to the mean response does not yield a value that is significantly different from the mean of fitting to individual responses.

In Fig. 4C, we plot data and simulations of the log-transforms of the GCAPs^−/− responses, and in Fig. 4D we show the agreement between simulation and data for the normalized log-transforms corresponding to Fig. 2B. Table 2 provides a summary of all the rod parameters, with black parameter values giving fixed input values, and blue parameter values giving results from fittings.

Transforming rod dynamics into cone dynamics

We next used the GCAPs^−/− responses to examine which of the parameters μ_rh, μ_pde, μ_tr, and β_d have to be modified in order to turn the dynamics of rod responses into those of cones, as shown in Fig. 2C. In Fig. 5A – 5D we show the effect of changing only a single one of μ_rh, μ_pde, μ_tr, and β_dark (indicated in the panels). That is, we kept three of the four parameters to be the same as in the rod model and determined the best fit to the cone responses for the fourth using Eq. (11). For all four possibilities, fits were poor. RMSE values were 0.14 for β_d alone (Fig. 5A), 0.21 for μ_rh alone (Fig. 5B), 0.15 for μ_pde alone (Fig. 5C), and 0.20 for μ_tr alone (Fig. 5D). For each of parameters, the fitting program always returned the value we set for the upper limit of the fitting range of 200 s⁻¹ (corresponding to a time constant of 5 ms). Higher values did not seem reasonable to us and, in addition, did not alter the results. Even with this very high value, a change in only a single parameter was insufficient to capture the kinetics of decay of the cone response.

We then tested whether we can turn a GCAPs^−/− rod into a GCAPs^−/− cone by altering only the dynamical parameters μ_rh, μ_pde and μ_tr. We therefore fitted the GCAPs^−/− responses using μ_rh, μ_pde, μ_tr as fitting parameters while keeping β_d equal to the rod value. Again, no reasonable fit could be obtained (Fig. 5E, RMSE of 0.1). Similarly, when we used β_d as a fitting parameter and kept rod values for μ_rh, μ_pde, μ_tr we could not obtain good fits (not shown, RMSE of 0.11). This shows that both β_d and PDE kinetics have to be altered. When we let both μ_pde and β_d vary, we could obtain a reasonable fit to the normalized log transform (RMSE of 0.02, Fig. 5F). In contrast, when we varied only μ_rh and β_d, or only μ_tr and β_d, we obtained poor fits; the fitting program returned rates for both parameters at the value of 200 s⁻¹ we had set for the upper limit of the fitting range, and RMSE was 0.11 and 0.12 (not shown). In summary, we find that, at a minimum, the rate of turnover of cGMP (β_d) and the PDE deactivation rate (μ_pde) need to be increased to fit the kinetics of the cone waveform.

Adjusting the cone model

To examine in more detail the alterations we need to make in the parameters of the model to fit cone responses, we first fitted concurrently both the Gnat1^−/− and Gnat1^−/−;GCAPs^−/− cone data from Figs. 1C and 1D with Eq. (11) from the Methods. We did not make any assumptions about the cone parameter values and used all the parameters ξ_, μ_rh, μ_tr, μ_pde, β_d and ${\widehat{K}}_{\alpha }$ as unconstrained fitting parameters. Although the gain $\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}$ depends on μ_rh and μ_pde, the product β_subk_act is unknown. We therefore used ξ as an independent fitting parameter. The best-fitting values for ξ_, μ_rh, μ_tr, μ_pde, β_d and ${\widehat{K}}_{\alpha }$ are given in Table 2 as scenario 1 (RMSE was 0.031). The reason why the RMSE is now higher than in Fig. 5F (RMSE was 0.02) is because in Fig. 5F we fitted only Gnat1^−/−;GCAPs^−/− responses, and in addition these responses were normalized, which reduced the variability due to the response amplitude. We verified that the model and the fitting parameters are identifiable (Raue et al., 2009). We show the results of these fits in Figs. 6A and 6B. We compare the current $\widehat{\iota }=1-e^{-y(t)}$ for the Gnat1^−/− and Gnat1^−/−;GCAPs^−/− data (black traces) to simulations (red traces) obtained with Eq. (11) from the Methods and the fitted parameter values. These simulations show that the model together with the fitted parameter values faithfully reproduces the Gnat1^−/−;GCAPs^−/− and the Gnat1^−/− cone responses.

As for the rod fittings, we performed a single cell analysis with the three Gnat^−/− and three Gnat^−/−;GCAPs^−/− cells that were used for the averaged data in Fig. 1. We concurrently fitted 9 different combinations of single-cell data for scenario 1. We found parameter values of (mean ± SD) ξ = 0.0019 ± 0.005, β_d = 11.8 ± 2.6 s⁻¹, μ_rh = 65.1 ± 20.2 s⁻¹, μ_tr = 69.9 ± 19.9 s⁻¹, μ_pde = 43.4 ± 8.1 s⁻¹ , ${\widehat{K}}_{\alpha }=0.81\pm 0.11$. These values were again very close to the values we obtained from fitting the mean responses (Table 2). Confidence intervals calculated by Monte Carlo simulations showed that, for all cone parameters, parameters fit to mean responses were within the calculated 95% confidence intervals of parameters fit to individual responses. Thus we again conclude, with an alpha rate of 5%, that fitting to the mean response does not yield a value that is significantly different from fitting to the mean of the individual responses.

We next varied possible values of the parameters μ_rh, μ_tr and μ_pde. As we showed in the Methods section (see Eq. 13), we cannot decide which of the fitted parameter values corresponds to which parameter because of the symmetry among μ_rh, μ_tr and μ_pde; that is, interchanging the values of μ_rh, μ_tr and μ_pde with fixed ξ results in identical responses. We also note that because ${\beta }_{\mathit{sub}}{k}_{\mathit{act}}=\xi \frac{{\mu }_{\mathit{rh}}{\mu }_{\mathit{pde}}}{n_{\mathit{ch}}}$, the predicted value of β_subk_act might depend upon the way the fitted parameter values are attributed to μ_rh, μ_tr and μ_pde. For the parameters in Table 2 for cone scenario 1 we chose μ_pde ≤ μ_rh ≤ μ_tr.

In order to better explore the parameter space, we adopted three additional fitting scenarios. In scenario 2 we performed exactly the same fitting procedure as in scenario 1, except that μ_rh was fixed to the rod value. The fitting waveforms were very similar to the ones from scenario 1 (not shown), and the RMSE was only slightly increased from 0.031 to 0.034. This result indicates that a satisfactory fit can be obtained by accelerating μ_pde alone without any change in μ_rh.

In scenario 1 and 2, either the rate of decay of activated visual pigment (μ_rh) or of activated phosphodiesterase (μ_pde) was rapid. In scenario 3 we tested the possibility that both rates are slow and below 40 s⁻¹ (for time constants of decay greater than 25ms). We also constrained the range for μ_tr to 100 s⁻¹. The best fitting parameter values for this scenario were the upper limits μ_rh = μ_pde = 40 s⁻¹. The fitting results were again very similar to scenarios 1 and 2, and the RMSE was 0.033. This result together with scenarios 1 and 2 shows that μ_rh can be the same as the rod value or faster, but it is unlikely to be slower.

In our final scenario 4, we tested the possibility that the PDE lifetime in cones is longer than 50ms, corresponding to the upper limit of 20 s⁻¹ for the fitting range of μ_pde. The best fitting values were the upper limit μ_pde=20 s⁻¹, and the fitted values for μ_rh and μ_tr were very large (101s⁻¹), indicating extremely rapid activation of transducin and pigment decay. Because in mouse the extinction of visual pigment is catalyzed by the same enzyme (GRK1) in both rods and cones, we think it unlikely that the rate of decay of visual pigment is several-fold larger in cones than in rods. Combined with the around 30% larger RMSE for scenario 4 (0.044) compared to the three other scenarios (0.031 – 0.034), we conclude that μ_pde in a cone is very likely to be larger than 20 s⁻¹ (with a time constant of PDE decay less than 50 ms) and several-fold larger than the value in rods. When we retained μ_pde at its rod value and attempted to fit the cone responses, the fit was again poor (RMSE 0.18).

The best-fitting parameter values corresponding to each scenario are summarized in Table 2 (fixed input values are in black, fitted parameter values are in blue). From these values we draw the following conclusions. First, the gain parameter ξ varies little among the various scenarios and is at least a factor of 200 smaller for cones than for rods. The value of β_d is consistently 3 – 4 times larger in cones than in rods, indicating a higher rate of cGMP turnover in darkness. The rate μ_pde is consistently higher in cones, indicating a much shorter time constant of decay of activated PDE probably reflecting the higher concentration of GAPs in cones. Because of the similarity of the RMSE for the first three scenarios nothing definite can be said about the relative rates of pigment decay (μ_rh), which may or may not be higher in cones than in rods.

Response to a single photon

The difference in sensitivity and dynamics between rods and cones can be well exemplified by comparing responses to a single-photon excitation. To compute the single-photon response current, we used the analytic formula ${\widehat{\iota }}_{\mathit{spr}}(t)=1-e^{-\xi g_y(t)}$ with g_y(t) from Eq. (15) and with parameter values from Table 2. Because y(t) is small during the single-photon response, this analytic result is in excellent agreement with a single-photon response simulation obtained with Eq. (11). In Fig. 7A we compare the single-photon response for WT and GCAPs^−/− rods with those for Gnat1^−/− and Gnat1^−/−;GCAPs^−/− cones, using for cones the parameters of scenario 1. Results with parameters from the other three scenarios were similar both in magnitude and waveform.

The calculated rod single-photon current in Fig. 7A is similar in amplitude and waveform to those of recorded mouse single-photon responses (for example Mendez et al., 2001; Sampath et al., 2005; Chen et al., 2010c; Azevedo & Rieke, 2011), with normalized peak amplitudes (r/r_max) and integration times (t_i) of 0.07 and 390 ms (WT), and 0.165 and 640 ms (GCAPs^−/−). The rod responses are about a factor of 150 (WT and 165 (GCAPs^−/−) larger than in cones (Fig. 6A), similar to previous estimates (Nikonov et al., 2006). The amplitude of the single-photon current is about a factor of 2.3 larger in a GCAPs^−/− rod than in a WT rod and a factor of 2.1 larger in a GCAPs^−/− cone than in WT cone, somewhat smaller than found by Sakurai et al. (2011).

Next, we investigated how the number of activated PDEs varies between rods and cones during a single-photon response. The number of activated PDEs does not depend on Ca²⁺ feedback and is therefore the same for WT and GCAPs^−/− photoreceptors. To compute the number of activated PDEs we used the analytic expression ${\mathrm{P}}^{\ast }(t)=\frac{\xi }{n_{\mathit{ch}}{\beta }_{\mathit{sub}}}{g}_p(t)=\frac{k_{\mathit{act}}}{{\mu }_{\mathit{rh}}{\mu }_{\mathit{pde}}}{g}_p(t)$, where g_p(t) is given by Eq. (13) in the Methods. For a rod we have assumed k_act = 350 s⁻¹ (Reingruber et al., 2013), which leads to a maximum of around 8 PDEs activated at any one time during the response. With k_act = 350 s⁻¹ and the rod parameters from Table 2, we compute ${\beta }_{\mathit{sub}}=\frac{\xi {\mu }_{\mathit{rh}}{\mu }_{\mathit{pde}}}{n_{\mathit{ch}}{k}_{\mathit{act}}}\approx 0.07s^{-1}$, which is similar to the value used in Reingruber et al. (2013). The parameter β_sub is the rate of hydrolysis of cGMP by a single P* and is proportional to the catalytic activity of an activated PDE (k_cat) and inversely proportional to the outer segment volume (V_os); that is, β_sub ~ k_cat/V_os (Pugh & Lamb, 2000; Reingruber & Holcman, 2008, 2009). Experiments expressing cone PDE in a rod indicate that k_cat is about a factor of 2 smaller for cone PDE than for rod PDE (Majumder et al., 2015). Since the ratio of the cone-to-rod volumes of the outer segments is about 0.4 (Nikonov et al., 2006), the value of β_sub for a cone may be similar to the rod value. We therefore assumed β_sub = 0.07 s⁻¹ for both rods and cones, and we computed the number of activated PDEs from ${\mathrm{P}}^{\ast }(t)=\frac{\xi }{n_{\mathit{ch}}{\beta }_{\mathit{sub}}}{g}_p(t)$. In Fig. 7B we compare the number of activated PDES corresponding to the single-photon simulations shown in Fig. 7A. The peak number of PDEs activated at any one time during the single-photon response is reduced by a factor of around 52 in a cone compared to a rod.

DISCUSSION

During the evolution of the vertebrate eye, more ancient cone-like precursors gave rise to photoreceptors with greater sensitivity, in part from a greater gain of the transduction cascade, and in part from a slower decay of the light response and longer time of integration of incoming photons (see Fain et al., 2010; Morshedian & Fain, 2017; Lamb, 2019). This process resulted in the duplex retina of the vertebrates with highly specialized rod and cone photoreceptors, which greatly differ in their sensitivity and response kinetics. Results from molecular biology (summarized in Ingram et al., 2016) indicate that evolutionary changes were produced gradually by small effects on several transduction parameters rather than a single effect on only one, but it is still unclear which adaptations were most important in determining the different response properties of the two kinds of photoreceptors.

We have approached this problem by comparing rod responses and recent voltage-clamp recordings of mouse cone responses. To quantify the differences between rods and cones, we used a spatially homogeneous model that focuses on the main features of the transduction cascade (Fig. 3). Because of similarities in the molecular transduction pathway (see Ingram et al., 2016), we used the same model for rods and cones but with different parameter values. In this way we could classify the differences between the two kinds of photoreceptors. Our analysis shows that the most important changes between rods and cones are likely to be the rate of turnover of cGMP in darkness (β_d), the rate of decay of the activated phosphodiesterase P* (μ_pde) and the gain $\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}$, where n_ch is the cooperativity constant of the CNG channels, β_sub is the rate constant of cGMP hydrolysis by an activated PDE and k_act is the rate of activation of transducin by activated visual pigment. The difference in ξ reflects the accelerated decay of P* and perhaps R*, but we will argue below that the product β_subk_act must also be significantly altered, indicating that the amplification of transduction at the level of the rates of activation of transducin by visual pigment and/or cGMP hydrolysis by PDE must be lower in a cone. These changes alone, provided they were large enough, would have been sufficient during evolution to alter the response of a cone to produce a rod capable of responding to single photons and mediating dim light vision. There may be other differences, which we made no attempt to include in this work. In particular, there may be differences in the Ca²⁺ economy of the two photoreceptors including buffering, mechanisms of Ca²⁺ extrusion, and differences in the Ca²⁺ permeability of the cyclic-nucleotide-gated channels (Perry & McNaughton, 1991; Sampath et al., 1999). These differences may be especially important during adaptation.

From our current model we cannot distinguish whether the changes in parameter values are generated by differences in the properties of transduction enzymes or differences in expression levels, but earlier studies have given some indication of the molecular basis for the changes we postulate. Our analysis indicates that the rate of cGMP turnover in darkness may be around 4 times faster in cones than in rods (see Table 2). A difference of 6-fold was inferred from the rate of decline of the salamander cone photocurrent in zero-Na⁺ solution (Perry & McNaughton, 1991; see also Cornwall et al., 1995), and an even higher rate of turnover was estimated for carp cones (Takemoto et al., 2009). One possible explanation for the increased turnover is an increase in the expression level of both the cyclase (Takemoto et al., 2009) and PDE (Reingruber et al., 2013). Direct biochemical measurement of the expression levels of a cone protein is difficult in mouse because there are so few cone photoreceptors.

Our calculations indicate that the rate of PDE decay (μ_pde) is also higher in cones such that the decay time constant may be reduced from about 200 ms in rods (Krispel et al., 2006; Tsang et al., 2006) to as little as 25 ms in cones (see Table 2). Such a short time constant may result from a higher level of expression of GAP proteins in cones (Cowan et al., 1998; Zhang et al., 2003; Takemoto et al., 2009) and is close to the value for Gnat1^−/− cones of the single-exponential time constant of decay (τ_rec), which is about 30 ms for small-amplitude responses (Ingram et al., 2019). It is possible that the rate of decay of light-activated visual pigment (μ_rh) is also different and more rapid in cones. On the one hand, mouse rods and cones use the same GRK1 kinase and similar arrestins (see Ingram et al., 2016), and we might expect that the rates of decay of R* would also be similar. On the other hand, it is possible that the expression levels of these proteins or the accessibility of pigment phosphorylation sites may be different for rods and cones. We hope to resolve this question by recording from cones with increased expression of cone visual-pigment kinase.

Our fittings in Table 2 indicate that the gain $\xi =\frac{n_{\mathit{ch}}{\beta }_{\mathit{sub}}{k}_{\mathit{act}}}{{\mu }_{\mathit{pde}}{\mu }_{\mathit{rh}}}$ is a factor of 200 to 250 smaller in cones. Of the five gain parameters, the Hill coefficient of the channels n_ch seems not to be significantly different between rods and cones (see for example Haynes & Yau, 1990). We can think of the other four parameters as given by the ratios k_act/μ_rh and β_sub/μ_pde. Since k_act is the rate of transducin activation by visual pigment, and since the inverse of μ_rh is the lifetime of R*, this first ratio gives the response amplification due to activation of transducin. For the second ratio, β_sub gives the rate of hydrolysis of cGMP by a single P*, and the inverse of μ_pde is the time constant of P* decay. This second ratio therefore gives the response amplification due to hydrolysis of the second messenger cGMP by PDE. Our simulations in Table 2 indicate that the product of these two ratios is of the order of 200 to 250 times smaller in cones. That is to say, amplification in cones is around 200 to 250 times less effective than in rods. This large difference in ξ is also the primary reason for the discrepancies in the SPR amplitudes between rods and cones in Fig. 7A. The ratios of the SPR amplitudes in Fig. 7A are not exactly equal to the ratios of ξ because the amplitude of the function g_y(t) from Eq. (15) is slightly different between rods and cones.

With the values for ξ, μ_pde and μ_rh from Table 2, we can compute that the product β_subk_act must be around 13 times smaller in cones. In the Results section (Response to a single photon), we argued that the value of β_sub may not greatly differ between rods and cones despite the difference in outer-segment volume and the molecular species of the PDE. These considerations would then imply that k_act, which is the rate of transducin activation by visual pigment, is likely to be much smaller in cones. A smaller rate of transducin activation may explain the reduced amplification constant of mouse cones (Nikonov et al., 2006; Ingram et al., 2019).

In this work we have focused on brief flashes to estimate the minimal changes that have to be made in a dark-adapted retina to transform rod dynamics into those of cones. We made a number of assumptions and constructed a parsimonious model which focused on the principal mechanisms of transduction to facilitate the analysis of the responses. For example, we assumed that the Ca²⁺ concentration changes instantaneously in proportion to the level of the current, we neglected effects of Ca²⁺ buffering, and we assumed that the cGMP synthesis rate by guanylyl cyclase changes instantaneously with the Ca²⁺ concentration. We also modeled cyclase modulation as the only form of transduction modulation occurring in the cones so that, in a GCAPs^−/− cone, there is no modulation of the response of any kind within the time course of the flash response. These assumptions are unlikely all to be true, and these over-simplifications may explain why our model is better able to fit responses of GCAPs^−/− photoreceptors in the absence of Ca²⁺ feedback than WT photoreceptors in its presence.

All of these assumptions will have to be modified for longer-duration stimuli where adaptation becomes important. A more extended model will be necessary to incorporate time-dependent changes in Ca²⁺ concentration and cyclase activity perhaps including additional mechanisms of transduction modulation, to explain why rods saturate in dim light but cones do not. As models become more sophisticated, our estimated parameter values are likely to alter, providing a more accurate reflection of the dynamics of the transduction cascade. We believe, however, that our calculations have already revealed some of the principal differences between the two kinds of photoreceptors, which we will use as input for more detailed future models required to explain rod and cone differences in their totality.

Key Points Summary.

• Most vertebrate eyes have rods for dim-light vision and cones for brighter light and higher temporal sensitivity.

• Rods evolved from cone-like precursors through expression of different transduction genes or the same genes at different expression levels, but we do not know which molecular differences were most important.

• We approached this problem by analyzing rod and cone responses with the same model but with different values for model parameters. We showed that, in addition to outer-segment volume, the most important differences between rods and cones are (1) decreased transduction gain, reflecting smaller amplification in the G-protein cascade; (2) faster rate of turnover of the second messenger cGMP in darkness; and (3) accelerated rate of decay of the effector enzyme phosphodiesterase and perhaps also of activated visual pigment.

• We believe our analysis has identified the principal alterations during evolution responsible for the duplex retina.