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Rate equation model of phototransduction into the membranous disks of mouse rod cells

A theoretical model was developed to investigate the rod phototransduction process in the mouse. In particular, we explored the biochemical reactions of several chemical components that contribute to the signaling process into/around the membranous disks in the outer segments of the rod cells. We constructed a rate equation model incorporating the molecular crowding effects of rhodopsin according to experimental results, which may hinder the diffusion of molecules on the disk mem- brane. The present model could effectively reproduce and explain the mechanisms of the following phenomena observed in experiments. First, the activations and relaxation of the wild-type mouse rod cell progressed more slowly than those of mutant cells containing half the amount of rhodopsin on the disk membrane. Second, the strong photoactivated state of the cell was sustained for a longer period when the light stimuli were strong. Finally, the lifetime of photoactivation exhibited a logarithmic increase with increasing light strength given exposure to strong light stimuli.


INTRODUCTION
Most living systems have evolved the ability to sense and adapt to environmental fluctuations through several internal biochemical processes. Photoreception is one of the most important signaling processes for several higher organisms. This signaling process starts from activation of the photoreceptor rhodopsin on the membranous disk, which is regulated by negative feedback through activation and repression cascades mediated by several membrane-associated proteins, low-molecular weight signaling molecules, and ions in each rod cell [1][2][3][4].
A high volume fraction of the photoreceptor rhodopsin tends to be maintained in the outer segments of the membranous disks of the rod cells of vertebrates [1][2][3][4][31][32][33][34]. Indeed, this high packing density of photoreceptors has been proposed to ensure efficient photon capture.
On the other hand, recent experimental studies of mouse rod cells showed that activation and relaxation of this signaling pathway in wild type (WT) mouse rod cells was slower than those of mutant cell containing half the amount of rhodopsin on the disk membrane. This fact suggests that the crowding of rhodopsin might actually limit the diffusion of signaling proteins on the disk membrane to slow down the phototransduction processes after rhodopsin activation. [4,33].
Recently, theoretical and experimental studies revealed the influence of the crowding of rhodopsin or other molecules on the disk membrane on the activation of the phototransduction process [32,34,35]. Moreover, the influence of the secondary messengers in the cyto-plasm, such as cGMP , disk membrane shape, and multi-level phosphorylations of rhodopsin on the phototransductions have also been explored with reaction diffusion models [33,[36][37][38][39][40].
These studies have successfully depicted the photoresponse behaviors in response to weak light stimuli. However, some of the mechanisms to explain characteristic phenomena observed in experiments have not yet been sufficiently explained yet, including: (i) activation and relaxation of the WT mouse rod cell progresses more slowly than those of mutant cells containing half the amount of rhodopsin on the disk membrane; (ii) the time interval of the photoactive state is drastically prolonged in response to strong light stimuli compared to weak stimuli; (iii) the lifetime of photoactivation exhibits a logarithmic increase with light strength under strong light stimuli.
To address these questions, in this study, we constructed a theoretical model of the whole process of phototransduction occurring in the outer segment of WT and mutant mouse rod cells in order to unveil the characteristics and the mechanism underlying the observed photoresponse behaviors for a wide range of light intensities. Recent experiments have shown that ∼ 20 types of signaling components such as proteins, low-molecular weight molecules, ions, and their complexes contribute to these processes. Based on these experimental findings, we constructed a rate equation model incorporating the molecular crowding effects of rhodopsin that could hinder the diffusion and reactions of molecules on the disk membrane. We confirmed that the present model can qualitatively reproduce and explain the light intensity-dependent temporal phototransduction behaviors, lifespan of activation, and rhodopsin volume fraction-dependent restrictions of the phototransduction speed in mouse rod cells, as observed in experiments.

Signaling components and reaction network of phototransduction
The molecular networks of vertebrate phototransduction have been extensively investigated experimentally [1][2][3][4]. After activation of rhodopsin (R) as R → R * ( * indicates the active form) by a light stimulus, the signal goes through a series of activation and relaxation states via the following reactions processes (Fig. 1).
Phototransduction starts with the activation of the G-protein transducin (G), which  diffuses on the disk membrane via collision with photoactivated rhodopsin (R * ), as where R * catalyzes the exchange of a GTP for a GDP on G. G * then diffuses along the membrane surface, and two G * molecules eventually bind to a single cyclic GMP phosphodiesterase (PDE) to construct a complex, (2G * · P DE), as A membrane-associated regulator of G-protein signaling, RGS9, binds to (2G * · P DE) as Then, the G-bound GTP molecules are hydrolyzed and this complex is dissociated as cGMP is synthesized by guanylate-cyclase-activating proteins in the cytoplasm which opens the Ca 2+ channel (Ch) as Here, Ch c and Ch o indicate the closed and open channel, respectively, and n refers to the number of cGMP molecules required to open a channel [1][2][3]. In this study, we assume n = 2. The hydrolysis of cGMP is accelerated by (2G * · P DE) as which results in the closure of Ca 2+ channels in the membrane.
Ca 2+ flows into the cytoplasm of the outer segment through the Ca 2+ channel and is extruded by an Na + /Ca 2+ , K + exchanger as If the concentrations of Ca 2+ become reduced in the outer segment, the rod cell moves into a hyperpolarization state, which can activate the downstream neurons.
Two Ca 2+ s bind to the cytoplasmic protein recoverin (Rec c ) as and bind to the membrane as (Rec m indicates Rec on membrane.).
Rhodopsin kinase (RK) accelerates the inactivation of R through inducing the phospholyration of activated rhodopsin R * as which triggers the relaxation of this phototransduction. RK is then inactivated by coupling and eventually returns to the active form through dissociation of the complex as / [Area of the disk membrane] for signaling components in the cytoplasm ( Fig. 1(b) where X t i indicates the total concentration of signaling components i (rhodopsin, G-protein, P DE, RGS9, RK, and Rec) on the disk membrane.
Based on the reaction processes described in the previous subsection, the rate equations of X i after instantaneous exposure to a light stimulus are obtained aṡ Here, since the time scales of gene expression and protein degradation are much slower than those of the phenomena of focus, we assume conservation of the total concentrations of the following molecules, X t i , in the outer segment of the rod cell: and In the present arguments, the instantaneous light stimulus is implemented as follows. It is noted that the time course of the presented rate equation model always induces relaxation to a unique stationary state for a given parameter set such as reaction coefficients and X t i , independent of the initial concentrations of the signaling components. Here, the concentrations of signaling components of this stationary state correspond to those of the rod cells kept in darkness for a sufficiently long period of time. Thus, to consider the behaviors of this model in response to an instantaneous light stimulus, we set the state of the system at t = 0 as follows. First, we make the system relax to the stationary state with long-term simulation. Here, X R * relaxes to = 0. Next, after inducing the relaxation, we change the , and set the time as t = 0. Here, L = X 0 R * /X t R is considered to be proportional to the intensity of the light stimulus. Then, we define L as the light intensity.
In experiments, the photoactivation of cells was estimated by monitoring the Ca 2+ flux through Ca 2+ channels, φ, as r = 1 − φ φ dark , where φ dark indicates the Ca 2+ flux in the darkness [1][2][3][4]33]. In the present model, the Ca 2+ flux is proportional to X Cho (t). Thus, we measure to estimate the photoactivation of the system.

Assumptions of the parameters for WT and mutant models
The present rate equation model involves a large number of parameters such as the reaction coefficients and the total numbers of chemical components. Now, we assume the following parameter sets for the model to most accurately reproduce and explain several of the experimental results observed in mouse rod cells.
First, we set the total concentration of each signaling component X t i according to the following assumptions for constructing both the WT model and mutant model of cells. As shown in recent experiments, the area fractions of rhodopsin are expected to be ∼ 0.27 in WT mouse rod cells, in contrast to ∼ 0.135 for mutant cells [4,33]. It is noted that the area fration is restricted to be ≤ 1. In the previous subsection, we set the same restriction for the total concentration of signaling components on the disk membrane. Thus, we assume X t R = 0.27 for the WT model and X t R = 0.135 for the mutant model. It has also been shown that the expression levels of genes encoding signaling proteins are almost identical between WT and mutant cells, except for rhodopsin [4]. Thus, we assume that X t P DE , X t G , X t RGS9 , and X t RK are the same between the WT and mutant models, and their sum obeys ≤ 0.4 − X t R , since the volume fractions of molecules in normal cells are estimated to be 0.2 ∼ 0.4. In the following arguments, we mainly focus on the cases of X t G = 0.01, X t P DE = 0.1, X t RGS9 = 0.0005, and X t RK = 0.0005, and the total concentration of Rec in the cytoplasm and that of the channel on the cell membrane are assumed to be X t Rec = 0.05 and X t Ch = 0.05, respectively. Next, we consider the reaction coefficients. The reaction coefficients are given as the combinations of the association and/or dissociation rates of molecules and their diffusion constants. However, not all of these values are precisely determined or reported in experiments. Thus, each reaction coefficient is set according to the following naturally assumed facts aiming for simplicity. A1) In general, the diffusion of molecules on the lipid membrane is slower than that in the cytoplasm, suggesting that two-body reactions occur more frequently in the cytoplasm than on the membrane. Thus, the reaction coefficients for two-body reactions on the disk membrane are considered to be smaller than those in the cytoplasm. In this argument, for simplicity, we assume that the reaction coefficients on both the disk membrane and cytoplasm are respective constant values k mem and k cyt , independent of signaling components, when the concentrations of the molecules are diluted.
Thus, each reaction in a molecularly crowded environment is slower than that in an enviroment with diluted molecules. In this argument, we assume that the effective reaction rate of two-body reactions on the disk membrane and cytoplasm, k ′ mem and k ′ cyt , obey k mem : k ′ mem = 1 : (1 − ρ) and k cyt : k ′ cyt = 1 : (1 − ρ c ), where ρ c refers to the total volume of molecules in the cytoplasm, and is assumed to be constant since the fluctuation of the molecular density in the cytoplasm seems to be small and relax immediately.
Then, the coefficients of the two-body reaction processes on the disk membrane are A, B, C + , D, E, S, T = k ′ mem = k mem (1 − ρ), and H, J, P , Q = k ′ cyt . Here, we regard the degradation of cGMP by (2G · P DE) and the opening of the Ca 2+ channel by cGMP (reaction (6)) as the representative cytoplasmic reactions, since the diffusion of cGMP in the cytoplasm is an essential determinant of the frequency of these reactions. Moreover, the binding rate of 2Ca 2+ · Rec c to the disk membrane is assumed to be similar to that of the reactions on the membrane, since this process is also influenced by the molecular concentration on the disk membrane.
A3) The diffusion of Ca 2+ in the cytoplasm is faster than any other reaction process. Here, we assume that the inflow and outflow rates of Ca 2+ are the same with M = N = k ion , which is set as a large value. A4) If any cGMP molecules do not bind to the Ca 2+ channel, this channel is considered to close quickly. Thus, K = k c is set to be a large value.
A5) The spontaneous reactions are considered to be slower than catalytic reactions. For simplicity, we assume that the coefficients of spontaneous degradation of signaling components are small constant values of C − , F , I, R = k d .
A6) The synthesis of molecules is also considered to be a relatively slower reaction. Thus, the production rate of cGMP , G = k cG , is set to be a small value.
Based on these assumptions, we classify the parameters as follows: L) k ′ cyt , k ion , and k c have large values; S) k d and k cG have small values; and M) k mem has an intermediate value.

RESULTS AND DISCUSSION
First, we focus on the time course of r(t) for both the WT and mutant models of mouse rod cells for an appropriately large L ( Fig. 2(a,b)). The results showed that the relaxation of the mutant model was faster than that of the WT model for each L, which is consistent to experimental observations [4,33]. Notably, the photoactivated state with large r(t) was prolonged with the appearance of the plateau of r(t) for large L values, a result that has not been obtained in recent theoretical models of the mouse rod cell [32][33][34][35][36][37][38][39][40] but has been observed in experiments [1][2][3][4]33].
Second, the lifespan of the photoactivation of cells, τ , was measured, which is defined as the time interval from the point at which r(t) reaches its maximum r max to the point at which r(t) = 0.8r max , based on a recent experiment [4] (Fig. 3(a)). For the cases in which L is larger (than ∼ 2.0 × 10 −4 ), τ for the WT model was always longer than that for the mutant model for the same L, as observed in the experiment [4]. Moreover, in both models, the slope of τ increased with an increase in L, and tau showed a logarithmic increase when L was so large that r(t) reached a plateau. Such phenomena were also observed in the experiment [4]. In addition, both the activation and relaxation of the mutant model were faster than those of the WT model in the case of the same X 0 R * (L = L w for the WT model and L = 2L w for the mutant model) when L is larger than ∼ 2.0 × 10 −4 but X 0 R * is small (Fig. 3(d)), as observed in experiments with weak light intensity [4,33].
The present model also demonstrated a decrease in τ for smaller L values (less than ∼ 2.0 × 10 −4 ) with an increase in L (Fig. 3(a)). Such behavior has not been reported in any experiments. We speculate that this was due to acceleration in the relaxation of the activation of each rod cell with increasing light intensity within the context of very weak light stimuli. However, this conjecture should be confirmed in future experiments to evaluate the validity of the present model. Now, we consider the mechanism of the phenomena obtained in the present model. In particular, we focus on the accelerations of the activation and relaxation due to the mutations and the appearances of the plateau of r(t) for large L.
The reason for the accelerations in the activation and relaxation in the mutant cell with a reduced rhodopsin density can be easily understood as follows. The limiting steps of the Next, we consider the mechanism of the slow relaxation of r(t) with the appearance of the plateau for large L values. The dominant contribution of the appearance of the plateau is considered to result from the deficiency of RGS9; in particular, its concentration is substantially reduced compared to G and P DE, which slows down the degradation of (2G * · P DE) to 2G + P DE. If L is small, the concentration of G * is not so large, so that the concentration of (2G * · P DE) is similar or less than that of RGS9. In this case, (2G * · P DE) is rapidly transformed to (2G * · P DE · RGS9) and the degradation of cGMP is immediately suppressed. On the other hand, if L becomes large, the concentration of G * becomes so large that the concentration of (2G * · P DE) becomes larger than that of RGS9. Consequently, most of the RGS9 becomes involved in (2G * · P DE · RGS9), and cGMP continues to be degraded by the remaining (2G * · P DE). Thus, the transformation from (2G * · P DE) to (2G * ·P DE ·RGS9) is hindered by the conflict of RGS9, which induces the observed plateau of r(t) for large L These arguments were confirmed by simulations of the model using larger values of X t RGS9 than 0.0005 ( Fig. 4(a, b)) where the length of the plateau, the range of L with which the plateau appears, was changed. Moreover, the maximum r(t) was reduced with an increase in RGS9. Thus, the deficiency of RGS9 has a dominant contribution to the amplification and reduction in the rate of relaxation of photoactivation of rod cells.
Finally, we show the typical behaviors obtained by the present model when the parameters are changed from the previously considered ideal cases. If X t G (X t P DE ) becomes so large (small) to approach X t P DE (X t G ) or X t RK becomes larger, a slow increase in τ with an increase in L appeared again for larger L values (Fig. 4(c,d)). Similar results were obtained when some reaction coefficients were randomly changed. However, these behaviors have not yet been reported experimentally.

CONCLUSION
In this study, we investigated the mouse rod phototransduction process through the development of rate equation models consisting of the chemical components that contribute to the signaling processes into and around the membranous disks in the outer segments of rod cells. By considering the effects of the molecular crowding of rhodopsin on the disk membrane, we could explain the mechanism observed both experimentally and theoretically by which photoactivation and relaxation in the WT mouse rod cell are slower than those of the mutant cell containing half the amount of rhodopsin. We also clarified that the deficiency of RGS9 compared to the other molecules on the disk membrane is primarily responsible for inducing prolongation of the photoactivated state of rod cells in response to strong light stimuli; these observations and explanations have not previously been obtained in recent theoretical models.
The present model includes a large number of parameters, and only a portion of these parameters are experimentally derived. Thus, in the presented arguments, we have adopted simple but reasonably appropriate assumptions for estimating the unknown parameters.
The accumulation of further experimental data for several of the relevant parameters in the future would allow us to address quantitative considerations and derive several predictions for the realistic behaviors of mouse rod cells. Moreover, the reduction and mathematical analysis of a few effective valuables and their rate equations from the presented model should be conducted in the future in order to unveil more detail properties and mechanisms of these signaling processes.