Mathematical Analysis of Electrical Responses in Photoreceptor Cells Reveals Distinct Chemical Control of Visual Signal Transduction in Rods and Cones.


 Retinal photoreceptor cells, rods and cones, convert photons of light into chemical and electrical signals as the first step of the visual transduction cascade. Although the chemical processes in the phototransduction system are very similar to each other in these photoreceptors, the light sensitivity and time resolution of the photoresponse in rods are functionally different than those in the photoresponses of cones. To systematically investigate how photoresponses are divergently regulated in rods and cones, we have developed a detailed mathematical model on the basis of the Hamer model. The current model successfully reconstructed light intensity-, ATP- and GTP-dependent changes in concentrations of phosphorylated visual pigments (VPs), activated transducins (Tr*s) and phosphodiesterases (PDEs), as well as cyclic nucleotide-gated currents (ICNG) in rods and cones. In comparison to rods, the lower light sensitivity of cones was attributed not only to the lower affinity of activated VPs for Trs but also to the faster desensitization of the VPs. The assumption of an intermediate inactive state, MIIi, in the thermal decay of activated VPs was pivotal for inducing faster inactivation of VPs. In addition to the faster inactivation of VPs, calculating a faster rate of RGS9 intervention for PDE-induced Tr* inactivation in cones was indispensable for simulating the electrical waveforms of the light intensity-dependent ICNG at higher temporal resolution in experimental systems in vivo.


Introduction
concentrations and binding constants (Kd) of signaling factors and the rate constant (k), 151 maximum activity (Vmax) and half-maximal effective concentration (K1/2) of substances for 152 activation of enzymes, among other parameters, are listed in Table 3. The initial values 153 of some variables are also listed in Table 4. Model development and simulation-based 154 analyses were both performed with simBio [16]. The time integration of the differential 155 equations was conducted using the Euler method with a time step of 1 µs. (1) 172 173 Activated VP immediately after light stimulation (MII * 0 at t = 0) is thus determined by 174 (2). 175 In the current simulation study, the stimulated VP, MII * (MII * 0 -MII * 3, see Figure 1), is 179 assumed to be capable of activating Tr, while being simultaneously inactivated through 180 two distinct mechanisms: phosphorylation and thermal decay.  Figure 5). The notation "pre" and "post" in above reaction 208 distinguish RK-bound states of R* before and after phosphorylation, respectively. 209 Tachibanaki et al. [17] showed that the steady-state phosphorylation level in cones was 210 ~3, which is in good agreement with the number of phosphates necessary for the 211 complete suppression of the activated VPs in rods [18]. Although a maximum of 9 212 phosphorylation sites in the C terminus of MII have been suggested [19], only 3 sites 213 were considered in the current model (for Figures 2-4, 0 ≤ n ≤ 2 for R.1.1 -1.8) based 214 on biochemical experiment observations in vitro [18] [17]. For the calculation of the 215 visual pigment phosphorylation shown in Figure 2, the total number of phosphate groups 216 incorporated into activated visual pigments was determined by Eq. (  model were designed to vary as the phosphorylation reactions proceed, whereas those for 225 the current model were set as constant parameters since the rate of phosphorylation up 226 to 3Pi/VP * total does not seem to vary significantly (see Fig. 1 in Tachibanaki et al., [17]). 227 The phosphorylation rate constants for MII i , k i RK1 -k i RK6, were assumed to be one-half 228 those of MII * (kRK1 -kRK6) to reproduce continuing phosphorylation processes even after 229 VP activity was completely terminated (based on the comparison of the time course of 230 VP phosphorylation (see Figure 2A) with that of Tr activation in the presence of ATP 231 ( Figure 3A)) [17] [20]. 232 For the simulation of the electrical waveforms of the light intensity-dependent 233 photoresponses shown in Figure 5B, the phosphorylation reaction was calculated only 234 for a single site (n = 0 for R.1.1 -1.8), assuming a complete loss of VP activity quickly 235 after arrestin (Arr) binds to MII * 1 in the "intact" system. In this case, MII * 0 was assumed 236 to be only capable of activating Tr (see the Discussion section). 237 238 Thermal decay of visual pigments 239 240 Thermal decay of VPs is another pathway for the inactivation mechanism. The process 241 is mediated by conformational changes in which MII * 0 transitions to MIII [21] directly 242 or after undergoing a transition to an intermediate state, MII i [22] (see the Discussion 243 section), as well as by bleaching (dissociation of retinal from opsin) and the subsequent recycling of VPs [23]. MIII may also undergo bleaching processes (see the reaction 245 scheme in Figure 3, reaction formulae R.2.1 -2.6 below, and corresponding equations, 246 Eqs. S1 -4, S14 -17, S27 - 28 where n indicates the number of phosphorylated sites (n ≥ 0).

281
For the current model, the constraint of mass conservation was newly introduced to the 282 concentration of Tr to prevent the continuous increase in Gα-GTPγS in response to a 283 stronger light stimulus (see Figure 3B). Since GTPγS is not hydrolyzable, the initial rate 284 of the change in Gα-GTPγS ( Figure 3A) purely reflects the rate of Tr activation because 285 the VP* desensitization (inactivation and phosphorylation) reactions progress more 286 slowly than those of Tr activation [25]. The rate constants for the reactions of Tr 287 activation, kG1,n -kG7, were thus estimated by model fitting to the initial rate of Tr 288 activation in response to light flash stimulation (0.0085% for rods and 0.25% for cones) 289 as well as the light intensity-dependent activation of Tr in vitro [25] [20] (see Figure 3). 290 Note that the affinity of Gα-GTP for MII * 0 (kG1,0/kG2) was approximately 6-fold higher in 291 rods than in cones. The estimation was comparable to that of Chen et al. [26]. 292 The Gα-GTP-binding rate to MII * 1 (kG1,1) was estimated to be ~60% of that to MII * 0 (kG1,0), 293 while the rate (kG1,2 and kG1,3, see R.3.6) was assumed to further decrease through 294 successive phosphorylation of the VPs (to MII * 2 and MII * 3, respectively), based on Gibson 295 et al. [27] (see the Discussion section for more details). 296 The reaction rates for VP inactivation were verified by reproducing the experimental 300 data obtained in the absence of ATP ( Figure 3) under the condition that VP is not 301 phosphorylated. The values of kG1,n -kG7 determined in the present study were 302 comparable to these of the Hamer model [10]. 303 Physiologically, activated Tr * (Gα-GTP) undergoes inactivation by the hydrolysis of its 304 bound GTP to GDP on a minute time scale [20] (see the reaction scheme in Figure 1 The GTPase activity of PDE-associated Gα dramatically increases as the PDE・GαGTP 316 complex binds to Regulator of RGS9, a GTPase-accelerating protein (see the reaction 317 formulae R.7.1 -7.2 below and corresponding equations, Eqs. S69 and 71, in the 318 Supplementary materials under Equations). The RGS9 effects on free Tr * (Gα-GTP) were 319 assumed to be negligible since the binding affinity of RGS9 for free Tr * was considerably 320 lower than it was for PDE-bound Tr * [28] The rate constants for RGS9-dependent hydrolysis, kRGS1 -kRGS3, were estimated by 325 model fitting to the time courses of the electrical waveforms of the light intensity-326 dependent photoresponses, which are shown in Figure 5 recorded in a physiological 327 condition [8] (see the Discussion section). Based on Tachibanaki et al. [20], an 328 approximate 20-fold higher expression of RGS9 in cones than in rods was assumed for 329 the simulation (see Table 3). The effect of RGS9 was excluded from the current model for The catalytic activity of PDE in hydrolyzing cGMP at rest is elevated when the 337 inhibitory γ subunit is removed from the enzyme upon binding activated Tr (PDE * ・Gα-338 GTP, see R.5.1 -6.2). The activity of PDE thus decreases as Gα-GTP is hydrolyzed by its 339 GTPase activity. For the Hamer model, cGMP hydrolysis by PDE was simply described 340 by a rate constant [10]. In contrast, in the current study, the chemical reaction of cGMP 341 binding to PDE was calculated to indicate that cGMP hydrolysis (R.5.4) is also a [cGMP]i-342 dependent process (see the reaction scheme in Figure 1  Based on Kawamura et al. [8], Gα-GTP was also assumed to be eluted from the localized 355 outer membrane complex where phototransduction takes place in the in vitro system 356 (see R.6.1 and the Discussion section). The eluted Gα-GTP was also presumed to be 357 inactivated, as described in the previous section (see R.6.2). 358 The elution of Gα-GTP was estimated to be negligible in the "intact" system when 359 simulating the electrical waveforms of the light intensity-dependent photoresponses in 360  were comparable to those estimated by Hamer et al. [40]. simulation study clarified that the difference in the apparent rates of phosphorylation in 433 these two types of photoreceptors was due to distinct amounts of receptor kinases (12 434 µM in rods and 120 µM cones) and reaction rates for each chemical process during the 435 phosphorylation of the VPs in the rods and cones (kRK1 -kRK6 and k i RK1 -k i RK6, see Table   436 3), as predicted by Tachibanaki et al. [17]. with GTP (open symbols) and GTPγS (filled symbols) was not drastically different in rods 517 and cones, indicating that the rate of GTP hydrolysis is slower than that of GTP-518 dependent or GTPγS-dependent activation of PDE. However, the peak PDE activity was 519 significantly more sensitive to light when the phosphorylation of VPs was prohibited, 520 that is, when ATP was absent (the comparison is shown in Figure 4Bb to a). clamp experiments seems to be faster than these recorded under unclamped condition, 551 most likely due to unchanging driving force for ICNG [49]. MIII in both types of photoreceptors [22]. Since the transition from MII * 0 to MIII was 582 suggested to be an event on the order of a few minutes in vitro [21] [50], considering that 583 the state MII i is indispensable to the termination of visual pigment activity: within ~ 10 584 sec in rods and ~ 1 sec in cones (see Figure 3Aa and b, respectively).

585
In cones, the current simulation experiments estimated that the rate of MII * 586 inactivation was ~ 120-fold faster than that was in rods. The molecular nature of VP in 587 cones (cone opsin), however, was indistinguishable from that in rods (rhodopsin), at least 588 in terms of their light sensitivity in expression systems [26] [9]. The mechanisms 589 underlying the faster inactivation of cone opsin may be due to other environmental 590 factors and need to be further explored in future experimental studies. 591 On the other hand, to reproduce continuing phosphorylation of VP even after complete 592 termination of its activity, the phosphorylation rate constants for MII i (k i RK1 -k i RK6) were 593 assumed to be slower (1/2) than those for MII * . 594 When complete abolition of VP activities, within ~10 sec after stimulus, PDE-dependent 595 cGMP hydrolysis still progressed for ~ 30 sec in rods and cones (shown as + ATP/+GTP 596 in Figure 4Aa and b, respectively) due to a longer lifetime of Tr * in the in vitro experimental systems. The assumption of faster inactivation of Tr * by PDE was therefore 598 indispensable to obtain a faster recovery of ICNG, which was recorded in vivo for both 599 types of photoreceptors ( Figure 5). In the current study, the mechanism underlying the 600 fast inactivation of Tr * was attributed to the RGS9-mediated reaction, which was 601 presumed to be intact under physiological conditions. Compared to that in rods, nearly 602 20 hold higher expression of RGS9 was assumed in cones ( [20], see Table 3); therefore, 603 the higher temporal resolution of the electrical waveforms of the light intensity-604 dependent ICNG was achieved in these cells (see Figure 5). Note that RGS9-independent 605 degradation of Tr * was also assumed to be faster in cones due to the expression level of 606 RGS9 [20]. Although RGS9 was simply assumed to be translocated from the 607 membranous disk at the outer segments to the inner segments of photoreceptor cells 608 during sample preparation in vitro experiments [29], different mechanisms may be 609  Table 3), were estimated by model fitting to the 630 initial rate of Tr activation in response to light flash stimulation as well as light 631 intensity-dependent activation of Tr in vitro [25] [20] (see Figure 3). However, the 632 simulation experiments revealed that the time course and the light intensity-dependent activation of Tr, as shown in Figure 3, may be reproduced with various sets of kG1,0 -kG7 634 values. Specifically, the higher light sensitivity of rods can be reproduced even when the 635 affinity of the activated VP to Tr ratio is lower than it is in cones when the subsequent 636 molecular reactions (kG1,1 -kG7) are faster. Since these parameters have not been 637 conclusively determined by experimental studies, kG1,1 -kG7 were set by referring to 638 former simulation studies [11] [10] [40]. Furthermore, the Gα-GTP-binding rate in MII * 0 639 -MII * 3 (kG1,0 -kG1,3) was assumed to decrease with successive phosphorylation of the VPs 640 [27]. The reduction rate was determined by the parameter denoted by ω in R. 3 al. [27], the value for cones was not given in the literature. In this study, ω was estimated 646 by fitting the experimental data, as shown in Figures 3 and 4, and found to be 0.9 for 647 cones. 648

649
The reproduction of a single photon response [51] has long been considered one of the 650 most important characteristics to reproduce by a mathematical model. Prior theoretical 651 analysis has concluded that the VP phosphorylation process requires multiple steps, and 652 VP affinity for RK and Tr must exponentially decline as VP phosphorylation proceeds 653 [15] [40] [10]. However, these theoretical models failed to reproduce the time courses of 654 VP phosphorylation or Tr activation observed in the experiments, therefore, some 655 unknown mechanisms must be involved to control these chemical processes. Importantly  Figure 5B, based on the concentration of RGS9 and the rate 663 constants for GC-mediated cGMP production determined by Tachibanaki et al. [20] and 664 Kawamura and Murakami [14], respectively. Since GC-mediated cGMP production is 665 regulated by Ca 2+ , the total content of endogenous Ca 2+ buffer (eT) and the rate constants 666 for the reactions of Ca 2+ binding to the endogenous buffers (kb1 and kb2) also determine 667 the recovery time course of the light intensity-dependent ICNG. Although the parameters 668 for the endogenous Ca 2+ -buffering system estimated in the current study were comparable to those in the Hamer model [40], these values need to be re-evaluated if the 670 rate constants for RGS9-mediated inactivation of Tr * was experimentally determined.  Figure S1 A and B (in Supplement), respectively (results normalized). Figure   692 S1 A shows that the phosphorylation process was strongly related to the kRK1 -kRK6 693 parameters, however, especially for cone, they were also related with Tr activation and 694 inactivation, and PDE activation processes. Similarly, Figure S1 B shows that the ICNG 695 were strongly related with both phosphorylation, Tr activation and cGMP production 696 processes. To see the detail of the relation between these parameters and the rising phase 697 and the falling phase of ICNG, sensitivity analysis on the maximum and the minimum 698 slope of ICNG were performed (see Figure S2 in Supplement). The rising phase was closely 699 related with phosphorylation, Tr activation and PDE activation parameters ( Figure S2  Although the current models well demonstrated the phototransduction in response to light stimulation that lasted for only a short period, corresponding to experimental 706 conditions, the cellular responses during light/dark adaptation may not be accurately 707 predicted since the reactions for VP re-formation from opsin and 11-cis-retinal were not 708 included in the systems. For future studies, it will be worthwhile to include the molecular 709 reactions regarding the retinal cycle to predict the adaptation phenomena of the 710 nonlinear behaviors of photoreceptors in response to light stimuli, as they refer to 711 previous conditions. 712 It also should be emphasized that arrestin is another key molecule to determine VP 713 inactivation through binding to phosphorylated VP. Therefore, incorporating arrestin 714 reactions to the current model is fundamental to understand the entire biological process 715 of VP inactivation. However, experimental data available in literature, as far as we were 716 aware, were not sufficient enough to derive the binding rates especially when changes 717 depending on number of phosphorylation of VP. Accordingly, for the current model, 718 macroscopic data fitting was performed for arrestin-dependent VP inactivation, whereas 719 it needs to be updated when sufficient experimental evidences are available in the future. 720     Table 2 for abbreviations). B, Details of the phosphorylation 751 reactions. Phosphorylation of VP at 3 sites in vitro (Figures 2-4), where only 1 site in vivo 752 ( Figure 5) was assumed. Phosphorylation reactions and Tr* elution indicated in light 753 gray were not included, whereas RGS9-dependent inactivation of Tr* and GC-dependent 754 cGMP synthesis, depicted in dark gray, were added for simulating ICNG in Figure 5. Reaction scheme of visual phototransduction in rods and cones A, Visual phototransduction, including activation and inactivation of VP, Tr, and PDE, in rods and cones (see Table 2 for abbreviations). B, Details of the phosphorylation reactions. Phosphorylation of VP at 3 sites in vitro (Figures 2-4), where only 1 site in vivo ( Figure 5) was assumed. Phosphorylation reactions and Tr* elution indicated in light gray were not included, whereas RGS9-dependent inactivation of Tr* and GC-dependent cGMP synthesis, depicted in dark gray, were added for simulating ICNG in Figure 5.

Figure 2
Phosphorylation of visual pigments in rods and cones. A, The time courses for the phosphorylation of VPs (the number of phosphate groups incorporated into an activated visual pigment molecule) measured in the membrane preparations of puri ed frog rod (a, circle) and carp cone (b, triangle) in response to light ash at 1.3% and 2.5%, respectively, in the experiments [17]. The corresponding simulation results (dotted lines in a and b) are also shown in the gures. B, Maximum rates of phosphorylation reaction per activated visual pigment at different ash intensities in rods (a, circle) and cones (b, circle), determined 10 sec and 0.6 sec after light stimuli, respectively (data modi ed from Tachibanaki et al. [17]).

Supplementary Files
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