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 (τ_D) 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.

Effects of cell size and bicarbonate on single photon response variability in retinal rods

Accurate photon counting requires that rods generate highly amplified, reproducible single photon responses (SPRs). The SPR is generated within the rod outer segment (ROS), a multilayered structure built from membranous disks that house rhodopsin. Photoisomerization of rhodopsin at the disk rim causes a local depletion of cGMP that closes ion channels in the plasmalemma located nearby with relative rapidity. In contrast, a photoisomerization at the disk center, distant from the plasmalemma, has a delayed impact on the ion channels due to the time required for cGMP redistribution. Radial differences should be greatest in large diameter rods. By affecting membrane guanylate cyclase activity, bicarbonate could impact spatial inhomogeneity in cGMP content. It was previously known that in the absence of bicarbonate, SPRs are larger and faster at the base of a toad ROS (where the ROS attaches to the rest of the cell) than at the distal tip. Given that bicarbonate enters the ROS at the base and diffuses to the tip and that it expedites flash response recovery, there should be an axial concentration gradient for bicarbonate that would accentuate the base-to-tip SPR differences. Seeking to understand how ROS geometry and bicarbonate affect SPR variability, we used mathematical modeling and made electrophysiological recordings of single rods. Modeling predicted and our experiments confirmed minor radial SPR variability in large diameter, salamander rods that was essentially unchanged by bicarbonate. SPRs elicited at the base and tip of salamander rods were similar in the absence of bicarbonate, but when treated with 30 mM bicarbonate, SPRs at the base became slightly faster than those at the tip, verifying the existence of an axial gradient for bicarbonate. The differences were small and unlikely to undermine visual signaling. However, in toad rods with longer ROSs, bicarbonate somehow suppressed the substantial, axial SPR variability that is naturally present in the absence of bicarbonate. Modeling suggested that the axial gradient of bicarbonate might dampen the primary phototransduction cascade at the base of the ROS. This novel effect of bicarbonate solves a mystery as to how toad vision is able to function effectively in extremely dim light.

Phototransduction in retinal cones: Analysis of parameter importance

In daylight, cone photoreceptors in the retina are responsible for the bulk of visual perception, yet compared to rods, far less is known quantitatively about their biochemistry. This is partly because it is hard to isolate and purify cone proteins. The issue is also complicated by the synergistic interaction of these parameters in producing systems biology outputs, such as photoresponse. Using a 3-D resolved, finite element model of cone outer segments, here we conducted a study of parameter significance using global sensitivity analysis, by Sobol indices, which was contextualized within the uncertainty surrounding these parameters in the available literature. The analysis showed that a subset of the parameters influencing the circulating dark current, such as the turnover rate of cGMP in the dark, may be most influential for variance with experimental flash response, while the shut-off rates of photoexcited rhodopsin and phosphodiesterase also exerted sizable effect. The activation rate of transducin by rhodopsin and the light-induced hydrolysis rate of cGMP exerted measurable effects as well but were estimated as relatively less significant. The results of this study depend on experimental ranges currently described in the literature and should be revised as these become better established. To that end, these findings may be used to prioritize parameters for measurement in future investigations.

Multi-scale, numerical modeling of spatio-temporal signaling in cone phototransduction

Mammals have two types of photoreceptors, rods and cones. While rods are exceptionally sensitive and mediate vision at very low illumination levels, cones operate in daylight and are responsible for the bulk of visual perception in most diurnal animals, including humans. Yet the mechanisms of phototransduction in cones is understudied, largely due to unavailability of pure cone outer segment (COS) preparations. Here we present a novel mathematical model of cone phototransduction that explicitly takes into account complex cone geometry and its multiple physical scales, faithfully reproduces features of the cone response, and is orders of magnitude more efficient than the standard 3D diffusion model. This is accomplished through the mathematical techniques of homogenization and concentrated capacity. The homogenized model is then computationally implemented by finite element method. This homogenized model permits one to analyze the effects of COS geometry on visual transduction and lends itself to performing large numbers of numerical trials, as required for parameter analysis and the stochasticity of rod and cone signal transduction. Agreement between the nonhomogenized, (i.e., standard 3D), and homogenized diffusion models is reported along with their simulation times and memory costs. Virtual expression of rod biochemistry on cone morphology is also presented for understanding some of the characteristic differences between rods and cones. These simulations evidence that 3D cone morphology and ion channel localization contribute to biphasic flash response, i.e undershoot. The 3D nonhomogenized and homogenized models are contrasted with more traditional and coarser well-stirred and 1D longitudinal diffusion models. The latter are single-scale and do not explicitly account for the multi-scale geometry of the COS, unlike the 3D homogenized model. We show that simpler models exaggerate the magnitude of the current suppression, yield accelerated time to peak, and do not predict the local concentration of cGMP at the ionic channels.

Analysis of diffusion in curved surfaces and its application to tubular membranes

Diffusion of particles in curved surfaces is inherently complex compared with diffusion in a flat membrane, owing to the nonplanarity of the surface. The consequence of such nonplanar geometry on diffusion is poorly understood but is highly relevant in the case of cell membranes, which often adopt complex geometries. To address this question, we developed a new finite element approach to model diffusion on curved membrane surfaces based on solutions to Fick's law of diffusion and used this to study the effects of geometry on the entry of surface-bound particles into tubules by diffusion. We show that variations in tubule radius and length can distinctly alter diffusion gradients in tubules over biologically relevant timescales. In addition, we show that tubular structures tend to retain concentration gradients for a longer time compared with a comparable flat surface. These findings indicate that sorting of particles along the surfaces of tubules can arise simply as a geometric consequence of the curvature without any specific contribution from the membrane environment. Our studies provide a framework for modeling diffusion in curved surfaces and suggest that biological regulation can emerge purely from membrane geometry.

Simplified equation to extract diffusion coefficients from confocal FRAP data

Quantitative measurements of diffusion can provide important information about how proteins and lipids interact with their environment within the cell and the effective size of the diffusing species. Confocal fluorescence recovery after photobleaching (FRAP) is one of the most widely accessible approaches to measure protein and lipid diffusion in living cells. However, straightforward approaches to quantify confocal FRAP measurements in terms of absolute diffusion coefficients are currently lacking. Here, we report a simplified equation that can be used to extract diffusion coefficients from confocal FRAP data using the half time of recovery and effective bleach radius for a circular bleach region, and validate this equation for a series of fluorescently labeled soluble and membrane-bound proteins and lipids. We show that using this approach, diffusion coefficients ranging over three orders of magnitude can be obtained from confocal FRAP measurements performed under standard imaging conditions, highlighting its broad applicability.

Rhodopsin expression level affects rod outer segment morphology and photoresponse kinetics

BACKGROUND

The retinal rod outer segment is a sensory cilium that is specialized for the conversion of light into an electrical signal. Within the cilium, up to several thousand membranous disks contain as many as a billion copies of rhodopsin for efficient photon capture. Disks are continually turned over, requiring the daily synthesis of a prodigious amount of rhodopsin. To promote axial diffusion in the aqueous cytoplasm, the disks have one or more incisures. Across vertebrates, the range of disk diameters spans an order of magnitude, and the number and length of the incisures vary considerably, but the mechanisms controlling disk architecture are not well understood. The finding that transgenic mice overexpressing rhodopsin have enlarged disks lacking an incisure prompted us to test whether lowered rhodopsin levels constrain disk assembly.

METHODOLOGY/PRINCIPAL FINDINGS

The structure and function of rods from hemizygous rhodopsin knockout (R+/-) mice with decreased rhodopsin expression were analyzed by transmission electron microscopy and single cell recording. R+/- rods were structurally altered in three ways: disk shape changed from circular to elliptical, disk surface area decreased, and the single incisure lengthened to divide the disk into two sections. Photocurrent responses to flashes recovered more rapidly than normal. A spatially resolved model of phototransduction indicated that changes in the packing densities of rhodopsin and other transduction proteins were responsible. The decrease in aqueous outer segment volume and the lengthened incisure had only minor effects on photon response amplitude and kinetics.

CONCLUSIONS/SIGNIFICANCE

Rhodopsin availability limits disk assembly and outer segment girth in normal rods. The incisure may buffer the supply of structural proteins needed to form larger disks. Decreased rhodopsin level accelerated photoresponse kinetics by increasing the rates of molecular collisions on the membrane. Faster responses, together with fewer rhodopsins, combine to lower overall sensitivity of R+/- rods to light.

A quantitative approach to analyze binding diffusion kinetics by confocal FRAP

Most of the important types of interactions that occur in cells can be characterized as binding-diffusion type processes, and can be quantified by kinetic rate constants such as diffusion coefficients (D) and binding rate constants (k(on) and k(off)). Confocal FRAP is a potentially important tool for the quantitative analysis of intracellular binding-diffusion kinetics, but how to dependably extract accurate kinetic constants from such analyses is still an open question. To this end, in this study, we developed what we believe is a new analytical model for confocal FRAP-based measurements of intracellular binding-diffusion processes, based on a closed-form equation of the FRAP formula for a spot photobleach geometry. This approach incorporates a binding diffusion model that allows for diffusion of both the unbound and bound species, and also compensates for binding diffusion that occurs during photobleaching, a critical consideration in confocal FRAP analysis. In addition, to address the problem of parametric multiplicity, we propose a scheme to reduce the number of fitting parameters in the effective diffusion subregime when D's for the bound and unbound species are known. We validate this method by measuring kinetic rate constants for the CAAX-mediated binding of Ras to membranes of the endoplasmic reticulum, obtaining binding constants of k(on) ∼ 255/s and k(off) ∼ 31/s.

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.

Dynamics of mouse rod phototransduction and its sensitivity to variation of key parameters

The deep understanding of the biochemical and biophysical basis of visual transduction, makes it ideal for systems-level analysis. A sensitivity analysis is presented for a self-consistent set of parameters involved in mouse phototransduction. The organising framework is a spatio-temporal mathematical model, which includes the geometry of the rod outer segment (ROS), the layered array of the discs, the incisures, the biochemistry of the activation/deactivation cascade and the biophysics of the diffusion of the second messengers in the cytoplasm and the closing of the cyclic guanosine monophosphate (cGMP) gated cationic channels. These modules include essentially all the relevant geometrical, biochemical and biophysical parameters. The parameters are selected from within experimental ranges, to obey basic first principles such as conservation of mass and energy fluxes. By means of the model they are compared to a large set of experimental data, providing a strikingly close match. Following isomerisation of a single rhodopsin R * (single photon response), the sensitivity analysis was carried out on the photo-response, measured both in terms of number of effector molecules produced, and photocurrent suppression, at peak time and the activation and recovery phases of the cascade. The current suppression is found to be very sensitive to variations of the catalytic activities, Hill's coefficients and hydrolysis rates and the geometry of the ROS, including size and shape of the incisures. The activated effector phosphodiesterase (PDE *) is very sensitive to variations of catalytic activity of G-protein activation and the average lifetimes of activated rhodopsin R * and PDE *; however, they are insensitive to geometry and variations of the transduction parameters. Thus the system is separated into two functional modules, activation/deactivation and transduction, each confined in different geometrical domains, communicating through the hydrolysis of cGMP by PDE *, and each sensitive to variations of parameters only in its own module.

A generalization of theory for two-dimensional fluorescence recovery after photobleaching applicable to confocal laser scanning microscopes

Fluorescence recovery after photobleaching (FRAP) using confocal laser scanning microscopes (confocal FRAP) has become a valuable technique for studying the diffusion of biomolecules in cells. However, two-dimensional confocal FRAP sometimes yields results that vary with experimental setups, such as different bleaching protocols and bleaching spot sizes. In addition, when confocal FRAP is used to measure diffusion coefficients (D) for fast diffusing molecules, it often yields D-values that are one or two orders-of-magnitude smaller than that predicted theoretically or measured by alternative methods such as fluorescence correlation spectroscopy. Recently, it was demonstrated that this underestimation of D can be corrected by taking diffusion during photobleaching into consideration. However, there is currently no consensus on confocal FRAP theory, and no efforts have been made to unify theories on conventional and confocal FRAP. To this end, we generalized conventional FRAP theory to incorporate diffusion during photobleaching so that analysis by conventional FRAP theory for a circular region of interest is easily applicable to confocal FRAP. Finally, we demonstrate the accuracy of these new (to our knowledge) formulae by measuring D for soluble enhanced green fluorescent protein in aqueous glycerol solution and in the cytoplasm and nucleus of COS7 cells.

Mathematical and computational modelling of spatio-temporal signalling in rod phototransduction

Rod photoreceptors are activated by light through activation of a cascade that includes the G protein-coupled receptor rhodopsin, the G protein transducin, its effector cyclic guanosine monophosphate (cGMP) phosphodiesterase and the second messengers cGMP and Ca2+. Signalling is localised to the particular rod outer segment disc, which is activated by absorption of a single photon. Modelling of this cascade has previously been performed mostly by assumption of a well-stirred cytoplasm. We recently published the first fully spatially resolved model that captures the local nature of light activation. The model reduces the complex geometry of the cell to a simpler one using the mathematical theories of homogenisation and concentrated capacity. The model shows that, upon activation of a single rhodopsin, changes of the second messengers cGMP and Ca2+ are local about the particular activated disc. In the current work, the homogenised model is computationally compared with the full, non-homogenised one, set in the original geometry of the rod outer segment. It is found to have an accuracy of 0.03% compared with the full model in computing the integral response and a 5200-fold reduction in computation time. The model can reconstruct the radial time-profiles of cGMP and Ca2+ in the interdiscal spaces adjacent to the activated discs. Cellular electrical responses are localised near the activation sites, and multiple photons sufficiently far apart produce essentially independent responses. This leads to a computational analysis of the notion and estimate of 'spread' and the optimum distribution of activated sites that maximises the response. Biological insights arising from the spatio-temporal model include a quantification of how variability in the response to dim light is affected by the distance between the outer segment discs capturing photons. The model is thus a simulation tool for biologists to predict the effect of various factors influencing the timing, spread and control mechanisms of this G protein-coupled, receptor-mediated cascade. It permits ease of simulation experiments across a range of conditions, for example, clamping the concentration of calcium, with results matching analogous experimental results. In addition, the model accommodates differing geometries of rod outer segments from different vertebrate species. Thus it represents a building block towards a predictive model of visual transduction.

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.

The DNA binding activity of p53 displays reaction-diffusion kinetics

The tumor suppressor protein p53 plays a key role in maintaining the genomic stability of mammalian cells and preventing malignant transformation. In this study, we investigated the intracellular diffusion of a p53-GFP fusion protein using confocal fluorescence recovery after photobleaching. We show that the diffusion of p53-GFP within the nucleus is well described by a mathematical model for diffusion of particles that bind temporarily to a spatially homogeneous immobile structure with binding and release rates k1 and k2, respectively. The diffusion constant of p53-GFP was estimated to be Dp53-GFP=15.4 microm2 s-1, significantly slower than that of GFP alone, DGFP=41.6 microm2 s-1. The reaction rates of the binding and unbinding of p53-GFP were estimated as k1=0.3 s-1 and k2=0.4 s-1, respectively, values suggestive of nonspecific binding. Consistent with this finding, the diffusional mobilities of tumor-derived sequence-specific DNA binding mutants of p53 were indistinguishable from that of the wild-type protein. These data are consistent with a model in which, under steady-state conditions, p53 is latent and continuously scans DNA, requiring activation for sequence-specific DNA binding.

Mathematical model of the spatio-temporal dynamics of second messengers in visual transduction

A model describing the role of transversal and longitudinal diffusion of cGMP and Ca(2+) in signaling in the rod outer segment of vertebrates is developed. Utilizing a novel notion of surface-volume reaction and the mathematical theories of homogenization and concentrated capacity, the diffusion of cGMP and Ca(2+) in the inter-disc spaces is shown to be reducible to a one-parameter family of diffusion processes taking place on a single rod cross section; whereas the diffusion in the outer shell is shown to be reducible to a diffusion on a cylindrical surface. Moreover, the exterior flux of the former serves as a source term for the latter, alleviating the assumption of a well-stirred cytosol. A previous model of visual transduction that assumes a well-stirred rod outer segment cytosol (and thus contains no spatial information) can be recovered from this model by imposing a "bulk" assumption. The model shows that upon activation of a single rhodopsin, cGMP changes are local, and exhibit both a longitudinal and a transversal component. Consequently, membrane current is also highly localized. The spatial spread of the single photon response along the longitudinal axis of the outer segment is predicted to be 3-5 microm, consistent with experimental data. This approach represents a tool to analyze point-wise signaling dynamics without requiring averaging over the entire cell by global Michaelis-Menten kinetics.