Pepperberg plot: Modeling flash response saturation in retinal rods of mouse.

Retinal rods evolved to be able to detect single photons. Despite their exquisite sensitivity, rods operate over many log units of light intensity. Several processes inside photoreceptor cells make this incredible light adaptation possible. Here, we added to our previously developed, fully space resolved biophysical model of rod phototransduction, some of the mechanisms that play significant roles in shaping the rod response under high illumination levels: the function of RGS9 in shutting off G protein transducin, and calcium dependences of the phosphorylation rates of activated rhodopsin, of the binding of cGMP to the light-regulated ion channel, and of two membrane guanylate cyclase activities. A well stirred version of this model captured the responses to bright, saturating flashes in WT and mutant mouse rods and was used to explain "Pepperberg plots," that graph the time during which the response is saturated against the natural logarithm of flash strength for bright flashes. At the lower end of the range, saturation time increases linearly with the natural logarithm of flash strength. The slope of the relation (τD) is dictated by the time constant of the rate-limiting (slowest) step in the shutoff of the phototransduction cascade, which is the hydrolysis of GTP by transducin. We characterized mathematically the X-intercept ( Φ o ) which is the number of photoisomerizations that just saturates the rod response. It has been observed that for flash strengths exceeding a few thousand photoisomerizations, the curves depart from linearity. Modeling showed that the "upward bend" for very bright flash intensities could be explained by the dynamics of RGS9 complex and further predicted that there would be a plateau at flash strengths giving rise to more than ~107 photoisomerizations due to activation of all available PDE. The model accurately described alterations in saturation behavior of mutant murine rods resulting from transgenic perturbations of the cascade targeting membrane guanylate cyclase activity, and expression levels of GRK, RGS9, and PDE. Experimental results from rods expressing a mutant light-regulated channel purported to lack calmodulin regulation deviated from model predictions, suggesting that there were other factors at play.

A model for enhanced and selective transport through biological membranes with alternating pores.

We investigate the outflux of ions through the channels in a cell membrane. The channels undergo an open/close cycle according to a periodic schedule. Our study is based both on theoretical considerations relying on homogenization theory, and on Monte Carlo numerical simulations. We examine the onset of a limiting boundary behavior characterized by a constant ratio between the outflux and the local density, in the large volume limit. The focus here is on the issue of selectivity, that is on the different behavior of the ion currents through the channel in the cases of the selected and non-selected species.

Monte Carlo study of gating and selection in potassium channels.

The study of selection and gating in potassium channels is a very important issue in modern biology. Indeed, such structures are known in essentially all types of cells in all organisms where they play many important functional roles. The mechanism of gating and selection of ionic species is not clearly understood. In this paper we study a model in which gating is obtained via an affinity-switching selectivity filter. We discuss the dependence of selectivity and efficiency on the cytosolic ionic concentration and on the typical pore open state duration. We demonstrate that a simple modification in the way in which the selectivity filter is modeled yields larger channel efficiency.

Identification of key factors that reduce the variability of the single photon response.

Rod photoreceptors mediate vision in dim light. Their biological function is to discriminate between distinct, very low levels of illumination, i.e., they serve as reliable photon counters. This role requires high reproducibility of the response to a particular number of photons. Indeed, single photon responses demonstrate unexpected low variability, despite the stochastic nature of the individual steps in the transduction cascade. We analyzed individual system mechanisms to identify their contribution to variability suppression. These include: (i) cooperativity of the regulation of the second messengers; (ii) diffusion of cGMP and Ca(2+) in the cytoplasm; and (iii) the effect of highly localized cGMP hydrolysis by activated phosphodiesterase resulting in local saturation. We find that (i) the nonlinear relationships between second messengers and current at the plasma membrane, and the cGMP hydrolysis saturation effects, play a major role in stabilizing the system; (ii) the presence of a physical space where the second messengers move by Brownian motion contributes to stabilization of the photoresponse; and (iii) keeping Ca(2+) at its dark level has only a minor effect on the variability of the system. The effects of diffusion, nonlinearity, and saturation synergize in reducing variability, supporting the notion that the observed high fidelity of the photoresponse is the result of global system function of phototransduction.

Kinetics of rhodopsin deactivation and its role in regulating recovery and reproducibility of rod photoresponse.

The single photon response (SPR) in vertebrate phototransduction is regulated by the dynamics of R* during its lifetime, including the random number of phosphorylations, the catalytic activity and the random sojourn time at each phosphorylation level. Because of this randomness the electrical responses are expected to be inherently variable. However the SPR is highly reproducible. The mechanisms that confer to the SPR such a low variability are not completely understood. The kinetics of rhodopsin deactivation is investigated by a Continuous Time Markov Chain (CTMC) based on the biochemistry of rhodopsin activation and deactivation, interfaced with a spatio-temporal model of phototransduction. The model parameters are extracted from the photoresponse data of both wild type and mutant mice, having variable numbers of phosphorylation sites and, with the same set of parameters, the model reproduces both WT and mutant responses. The sources of variability are dissected into its components, by asking whether a random number of turnoff steps, a random sojourn time between steps, or both, give rise to the known variability. The model shows that only the randomness of the sojourn times in each of the phosphorylated states contributes to the Coefficient of Variation (CV) of the response, whereas the randomness of the number of R* turnoff steps has a negligible effect. These results counter the view that the larger the number of decay steps of R*, the more stable the photoresponse is. Our results indicate that R* shutoff is responsible for the variability of the photoresponse, while the diffusion of the second messengers acts as a variability suppressor.

Diffusion of the second messengers in the cytoplasm acts as a variability suppressor of the single photon response in vertebrate phototransduction.

The single photon response in vertebrate phototransduction is highly reproducible despite a number of random components of the activation cascade, including the random activation site, the random walk of an activated receptor, and its quenching in a random number of steps. Here we use a previously generated and tested spatiotemporal mathematical and computational model to identify possible mechanisms of variability reduction. The model permits one to separate the process into modules, and to analyze their impact separately. We show that the activation cascade is responsible for generation of variability, whereas diffusion of the second messengers is responsible for its suppression. Randomness of the activation site contributes at early times to the coefficient of variation of the photoresponse, whereas the Brownian path of a photoisomerized rhodopsin (Rh*) has a negligible effect. The major driver of variability is the turnoff mechanism of Rh*, which occurs essentially within the first 2-4 phosphorylated states of Rh*. Theoretically increasing the number of steps to quenching does not significantly decrease the corresponding coefficient of variation of the effector, in agreement with the biochemical limitations on the phosphorylated states of the receptor. Diffusion of the second messengers in the cytosol acts as a suppressor of the variability generated by the activation cascade. Calcium feedback has a negligible regulatory effect on the photocurrent variability. A comparative variability analysis has been conducted for the phototransduction in mouse and salamander, including a study of the effects of their anatomical differences such as incisures and photoreceptors geometry on variability generation and suppression.

Modeling the role of incisures in vertebrate phototransduction.

Phototransduction is mediated by a G-protein-coupled receptor-mediated cascade, activated by light and localized to rod outer segment (ROS) disk membranes, which, in turn, drives a diffusion process of the second messengers cGMP and Ca2+ in the ROS cytosol. This process is hindered by disks-which, however, bear physical cracks, known as incisures, believed to favor the longitudinal diffusion of cGMP and Ca2+. This article is aimed at highlighting the biophysical functional role and significance of incisures, and their effect on the local and global response of the photocurrent. Previous work on this topic regarded the ROS as well stirred in the radial variables, lumped the diffusion mechanism on the longitudinal axis of the ROS, and replaced the cytosolic diffusion coefficients by effective ones, accounting for incisures through their total patent area only. The fully spatially resolved model recently published by our group is a natural tool to take into account other significant details of incisures, including their geometry and distribution. Using mathematical theories of homogenization and concentrated capacity, it is shown here that the complex diffusion process undergone by the second messengers cGMP and Ca2+ in the ROS bearing incisures can be modeled by a family of two-dimensional diffusion processes on the ROS cross sections, glued together by other two-dimensional diffusion processes, accounting for diffusion in the ROS outer shell and in the bladelike regions comprised by the stack of incisures. Based on this mathematical model, a code has been written, capable of incorporating an arbitrary number of incisures and activation sites, with any given arbitrary distribution within the ROS. The code is aimed at being an operational tool to perform numerical experiments of phototransduction, in rods with incisures of different geometry and structure, under a wide spectrum of operating conditions. The simulation results show that incisures have a dual biophysical function. On the one hand, since incisures line up from disk to disk, they create vertical cytoplasmic channels crossing the disks, thus facilitating diffusion of second messengers; on the other hand, at least in those species bearing multiple incisures, they divide the disks into lobes like the petals of a flower, thus confining the diffusion of activated phosphodiesterase and localizing the photon response. Accordingly, not only the total area of incisures, but their geometrical shape and distribution as well, significantly influence the global photoresponse.