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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 94, Issue 9, 3363-3383, 1 May 2008

doi:10.1529/biophysj.107.114058

Biophysical Theory and Modeling

Diffusion of the Second Messengers in the Cytoplasm Acts as a Variability Suppressor of the Single Photon Response in Vertebrate Phototransduction

Paolo Bisegna^*, Giovanni Caruso^†, Daniele Andreucci^‡, Lixin Shen^§, ¶, Vsevolod V. Gurevich^§, Heidi E. Hamm^§, ¶ and Emmanuele DiBenedetto^¶, ‖, ,  

^* Department of Civil Engineering, University of Rome, Tor Vergata, Italy
^† Construction Technologies Institute, National Research Council, Rome, Italy
^‡ Department of Mathematical Methods and Models, University of Rome, La Sapienza, Italy
^§ Department of Pharmacology, Vanderbilt University Medical Center, Vanderbilt University, Nashville, Tennessee
^¶ Biomathematics Study Group, Vanderbilt University Medical Center, Vanderbilt University, Nashville, Tennessee
^‖ Department of Mathematics, Vanderbilt University, Nashville, Tennessee

Address reprint requests to Emmanuele DiBenedetto.

Vertebrate rod photoreceptors are capable of detecting the absorption of a single photon with a wavelength of ∼500nm 1,2. Moreover, the resulting responses are highly reproducible, in the sense that the peak amplitudes and the shapes of the photocurrent as a function of time are very similar. Quantitatively, repeated single photon activations yield peak photocurrents with coefficient of variation (CV), defined as the ratio of the standard deviation to the mean, of ∼20% 3,4. It is generally believed that a key contributor to this high fidelity of the single photon response (SPR) is the amplification part of the cascade 5,6,7,8,9,10,11,12.

One of the main points of this article is to challenge this assumption. We demonstrate that the diffusion of the second messengers cyclic guanosine-monophosphate (cGMP) and Ca²⁺ in the rod cytoplasm with characteristic complex geometry 13,14,15, after the activation of the photocascade, is the key determinant of the high reproducibility of the response.

The experimental results of the literature 6,7,8 have a photon as input and the photocurrent as output, and variability of the resulting photocurrent is statistically estimated. We show that the variation of the photocurrent is determined by two distinct modules, the activation cascade and the diffusion of cGMP and Ca²⁺, each contributing differently to the reproducibility of the response. Our modeling shows that the activation cascade and the random shutoff mechanism of a photoisomerized rhodopsin (Rh*) yields a CV of the total number of activated effectors, of ∼60%, whereas the diffusion of the second messengers reduces it to the observed 40% for the integration time and to ∼20% for the photocurrent at peak time. These two components, which cannot be distinguished using existing experimental techniques, can be mathematically separated into two modules and analyzed separately.

We show that the random walk of the Rh* after photoactivation, and the randomness of the activation site, contribute negligibly to the CV of the response. The main contributor to the variability seems to be the random shutoff mechanism of Rh*.

Finally, an experimentally observed CV of ∼20% for the current amplitude at peak time is obtained with Rh* shutting off through 2–3, at most, phosphorylated states. This is determined by numerical modeling and simulations of the CV as a function of the underlying biochemistry.

Molecules of receptor rhodopsin (Rh), transducin G-protein (T), and effector phosphodiesterase (E), are regarded as freely diffusing particles on the two-dimensional disk surface, each with their own specific diffusivity. The original signal of a single photon-activating molecule of Rh is amplified in the sense that an Rh* activates along its random walk, during the time it remains active, dozens of transducins (T→T*), by catalyzing GDP/GTP exchange on their α-subunit. Each molecule of T* associates, one-to-one, with a catalytic subunit of the effector forming a T*·E complex, denoted by E*, and called the activated-effector. The full-activation hypothesis postulates that a molecule of PDE is active only if both its subunits are bound to a molecule of T*. Thus, assuming full activation, [PDE*]=1/2[E*].

A single molecule of E*, during its lifetime, hydrolyzes in excess of 50 molecules of the second messenger, cGMP 16,17. Diffusion of cGMP away from the cationic channels that it keeps open causes channel closure, and thereby suppresses the inward current. Low variability (high fidelity) of the SPR is experimentally assessed in terms of this photocurrent 3,4,6,7,8.

The strength of the output signal depends on the number of activated effectors E*, which in turn depends on the active lifetime of Rh*, and their own active lifetime. Inactivation of Rh* occurs essentially by two molecular events. First Rh* is phosphorylated by rhodopsin kinase (RK), by the sequential attachment of one or more phosphates at its C-terminal serine and threonine residues 18. Then phosphorylated Rh* is capped by arrestin (Arr), which shuts it off by making it inaccessible for T 19,20,21,22. The catalytic activity of E* terminates when activated transducin dissociates from the complex T*·E after its intrinsic GTPase hydrolyzes GTP to GDP. This latter process is greatly accelerated by RGS9 23.

Absolute sensitivity of the visual system is limited by dark noise due to isomerization of Rh by thermal fluctuations and spontaneous activation of E 1,6,7,24,25,26. The system is so sensitive that it can act, at least for dim flashes, as a photon counter 27, permitting one absorbed photon to be distinguished from two 4.

This high reproducibility of SPR is intriguing, as the process contains several elements of randomness. For example, the disk activated by a quantum of light is a random one among the 1000 disks forming the rod outer segment (ROS); and the activation site is random within the activated disk; Rh* randomly diffuses within the activated disk, and remains active at random time t_Rh*. Finally, the number of Rh*-phosphorylations before quenching by arrestin is random.

Several hypotheses have been proposed to explain low SPR variability. Among these is that t_Rh* has little effect on the process, that is, either t_Rh* is itself little-variable, or the photocurrent is relatively insensitive to variations of t_Rh*8. Another is that a multistep Rh* shutoff stabilizes the output photocurrent 6,7,28.

It was pointed out in Pugh 10 that a full account of the single photon response has to include an analysis of the spatiotemporal diffusion of the second messengers cGMP and Ca²⁺ in the layered geometry of the ROS. This is the key point of this study. In a series of articles, we have created a mathematical and computational model of the spatiotemporal dynamics of cGMP and Ca²⁺ in the ROS by resolving the layered geometry. This was done by means of the mathematical theories of homogenization and concentrated capacity 13,14,15,29,30. The model is a tool that permits one, in almost real-time, to separate and check the effects of all the biochemical and physical parameters involved, even the random ones, including diffusivities, reaction rates, catalytic coefficients, and shutoff times. The geometrical parameters of the ROS including disk incisures, their shape, and their geometrical arrangements, can also be varied and their effects on the response can be calculated.

By means of this model, we separate and test the effects of the various random events contributing to the variability of the response. These include the random activation site, the random walk of Rh*, and the hypotheses of a multistep or abrupt random shutoff of Rh*.

We find that neither a random activation site nor a random walk of Rh* contribute significantly to the CV of E*; it is the multistep Rh* inactivation mechanism that is the main contributor to the CV. However, the number of steps in deactivation is not a main contributor to variability suppression. The surprising result of the simulations is that the diffusion of cGMP and Ca²⁺ damp out the variability of the SPR.

The dynamics of the second messengers cGMP and Ca²⁺ is modeled by taking the homogenized-concentrated limit of their physical, pointwise dynamics within the interdiscal spaces and in the outer shell. The limiting homogenized geometry is simpler in that the outer shell and the disks disappear and are replaced by dynamic equations on their limiting geometries, linked by equations expressing their mutual balance of fluxes. In particular, incisures in the homogenized-concentrated limit tend to segments (Fig. 1). We denote by D_R the disk of radius R and by D_eff the effective domain of the activation cascade—that is, D_R from which all the segments have been removed:

Display large version of this figure

Figure 1

(Left) Transversal cross section of the ROS bearing an incisure. (Right) Limit of such a cross section, bearing the limiting incisure ν.

Indeed the function (x, y, t)→E*(x, y, t) for (x, y) ranging over an activated disk, serves as an input only, and its dynamic can be modeled independently. In this respect, the dynamics of the second messengers cGMP and Ca²⁺ is a module of the visual transduction cascade and the dynamics of the activated effector E* is a separate module.

A molecule of rhodopsin, activated at time t=0, becomes inactivated abruptly, after a random time t_Rh*, of average τ_Rh*. During the random interval [0, t_Rh*), however, Rh* goes through n molecular states Rh*_j, j=1,…, n, each with transducin-activation rate ν_j, which remains constant as long as Rh* remains in the state Rh*_j. For example, Rh*_j might be in different phosphorylated states of Rh*, identified by the number (j–1) of phosphates attached to Rh* by RK. The state (n+1) is identified with Rh* being quenched by Arr binding. The transitions from the state j to (j+1) occur at random transition times 0<t_j≤t_n=t_Rh*, with remaining constant during the time interval (t_j–1, t_j], for j=1,…, n, and where t_o=0. Denote by δ_x(t) the Dirac mass in R², concentrated at x(t)=(x(t), y(t)), and dimensionμm⁻², and by the characteristic function of the interval (t_j–1, t_j]. Then the rate equations for T* and E* are

(1)

(2)

(3)

In Eq. (1), the constant k_T*E is the rate of formation of the T*·E complex or equivalently the rate of formation of E*. The constant k_E* is the rate of deactivation of E* by the hydrolysis of GTP by T*, within the T*·E complex. The constant k_j, in s−1, is the constant of activation of T* by Rh* through a successful encounter at time t at the position x(t) on the random path of Rh*. The activation rate is proportional to the relative number ([T] – [T*])/[T] of transducin molecules available for activation. It is assumed that T*(x,y,t)<[T](x,y,t) at all (x, y)∈D_eff, and at all times, so that local depletion of T is negligible, which is true for dim light responses including SPR. Alternatively, at bright light the activation process obeys Michaelis-Menten kinetics with Michaelis constant K, and activation occurs in the saturation limit, i.e.,

Models like Eq. (1) involve deterministic parts, such as the diffusion processes appearing on the left-hand side, deterministic first-order reactions with given rates, and stochastic terms. Indeed, the transitions times t_j for j=1,…, n are random variables, and the path t→x(t) is random.

A newly created Rh* is in the state 1 (denoted by ). It undergoes a transition to the state 2 after some time s₁, which is an exponentially distributed random variable with mean τ₁. More generally, Rh* reaches the state j (denoted by ) after (j–1) transitions from . Then it undergoes a transition to the state (j+1) after time s_j has elapsed from the birth of . The quantity s_j is an exponentially distributed random variable with mean τ_j. After n transitions, Rh* is turned off, reaching the state n+1. The random variables s₁,…, s_n are mutually independent; their sum, denoted by t_Rh*, is the lifespan of Rh*, which itself is a random variable with mean τ_Rh*. The s_j are connected to the transition times and their mean values must satisfy . The theoretical calculation of the probabilities P_j(t) of Rh* being in the j^th state at time t, hinges upon the structure of the sequences of the mean times {τ₁,…, τ_n}, and the catalytic .constants {ν₁,…, ν_n}. The structure of such sequences in turn depends on the underlying biochemistry. A theoretical choice could be that τ_j and ν_j are the same in each state. More biochemically motivated choices would allow for a {τ_j} and {ν_j} to be variable from state to state. The CV of some quantities can be computed theoretically, a priori, in terms of the sequences {ν_j} and {τ_j} irrespective of their structure (Appendix B ). We will return to these explicit formulae in Eq. (14), to discuss their biochemical significance.

Experimental evidence suggests that the phosphorylated states are functionally different 31. In particular, their ability to activate T is different 32,33.

Shutoff of Rh* occurs by phosphorylation by RK, followed by Arr binding. Like many G-protein coupled receptors, rhodopsin contains multiple sites for phosphorylation in its C-terminus. The contribution of various phosphorylation sites to Rh* interaction with T, RK, and Arr has been addressed by several investigators, in a series of in vitro experiments 32,34,35,36,37,38. Biochemical experiments using the competition of synthetic phosphorylated and unphosphorylated rhodopsin C-terminal peptides suggest that phosphorylation of rhodopsin by RK is a cooperative process, i.e., the incorporation of one or two phosphates increases the probability of further phosphorylation 35. This phenomenon was rationalized in terms of increased affinity of the substrate for RK with increased phosphorylation 35. This should tend to favor the formation of multiphosphorylated rhodopsin species. Another study using full-length proteins 39 came to the opposite conclusion: that phosphorylation of rhodopsin and/or autophosphorylation of RK progressively decreases its affinity for light-activated rhodopsin. This mechanism could favor the accumulation of Rh* species with low level of phosphorylation. These models predict very different outcomes at high levels of illumination, when the number of Rh* molecules is comparable to the number of RK molecules in the photoreceptor, creating conditions where Rh* molecules compete for RK. However, under conditions relevant for our analysis where single photon responses are recorded, i.e., in the dark-adapted rod with only one Rh* and ≈200,000 molecules of RK 16, the difference between the predictions of these two models is negligible.

There is no consensus in the literature regarding the quantitative effect of progressive rhodopsin phosphorylation on transducin activation and arrestin binding 18,32,33. We based our choice for the sequences {τ_j} and {ν_j} on the data obtained with chromatographically separated rhodopsin species with different levels of phosphorylation 18,32, rather than on results obtained with complex mixtures of different phosphorhodopsin species 33. The catalytic activity of Rh* decreases with increasing levels of phosphorylation, at a rate of ∼12% for each additional level of phosphorylation (Fig. 2 in 32). Thus,

(4)

(5)

Display large version of this figure

Figure 2

The red curve reports the average of an extensive set of experimental SPR responses for mouse, kindly provided to us by F. Rieke. The blue curve is our numerical simulation of the mouse SPR using the mathematical model of Appendix A for the set of parameters in Table 3. The agreement is excellent, thereby showing that this selection parameters accurately reflects the timecourse and amplitude of experimentally generated light responses.

This choice is often made 6,7,9,45,46, although it is purely theoretical and is not motivated by known biochemistry. Let E** denote the random total number of molecules of E* produced over the entire time duration of the process, after a single isomerization. In Appendix B , a theoretical formula has been derived for CV(E**), regardless of the structure of the sequences {τ_j} and {ν_j} (Eq. (15)). This equation implies that if ν_jτ_j=const for all j=1,…, n, then

(6)

(7)

A code for Eq. (1) presents two major difficulties. The first is the incised geometry of D_eff, which is dealt with by mathematical methods of numerical analysis. The second is the stochastic input on the right-hand side of the first equation in Eq. (1). This includes the random activation site, the random path of Rh*, and random shutoff mechanism. The model permits one to test independently the effects of these random components on the variability of the response. For example, one can separate the effects of the activation site and the subsequent random walk of Rh* on D_eff from the shutoff mechanism. Also, one can separate the effects of the shutoff mechanism from the Brownian motion of Rh*. To achieve this, we performed the following sets of simulations:

Fix the activation site x_o ∈ D_eff and let Rh* follow its random Brownian path t→x(t) starting at x(0)=x_o, with a prescribed second moment of its probability density. The shutoff time t_Rh* and the number n of states before ultimate quenching, are fixed. The mean half-lives of different states could be equal or different, and still deterministically fixed. It turns out that if one regards the activation site as random, its mean position is at distance 2/3 of the radius of the activated disk. Thus, in this case, The only random effect in this case is that of the Brownian motion of Rh*.

The activation site x_o is random and Rh* remains fixed at its initial location. The shutoff time t_Rh* and the number n of steps before quenching are fixed. The only randomness is the position of the activation site, which discriminates responses from each other.

The activation site is fixed and also Rh* remains fixed at x_o. Quenching of Rh* occurs at the random shutoff time t_Rh*, in n states. Random numbers τ_j are selected according to their exponential distribution and subject to either the statistic of Eq. (5), and the probabilities P_j(t), for j=1,…, n, are computed accordingly (Appendix B ). These simulations separate the random effects of the shutoff mechanism from those of the activation sites and the motion of Rh*.

All the previous components are random (activation site, random path of Rh*, random shutoff time t_Rh* for a given number of steps). This is the biologically realistic case, although the previous cases extract the impact of each component of randomness.

A MATLAB-based, finite-element code has been written for the system in Eqs. (1), based on its weak formulation (Appendix A ). The output E*, as a function of the two variables (x, y) ∈ D_eff and time t, is then fed into the code for the homogenized system describing the dynamics of the second messengers (Appendix A ). Finally, local and global currents generated across the ROS lateral surface, are computed by the formulae

(8)

(9)

(10)

While J_loc(θ, z, t) is pointwise current defined on the outer shell, experimentally what is measured is the global current j_tot(t) as a function of time, and what is graphed is the relative local or global current drop,

(11)

(12)

(13)

Display large version of this figure

Figure 3

Mouse: comparing the CV of the total activated effectors at time t with the CV of the total relative charge up to time t. (NN) Shutoff of Rh* in n biochemical states of decreasing duration and catalytic activity (see Biochemical Sequences {τ_j} and {ν_j}). (EE) Shutoff of Rh* in n theoretical states of equal duration and equal catalytic activity (see Sequence for Which τ_jν_j=Const). All simulations assume all activation steps as random (Case 4 of Random Events Contributing to SPR Variability). In all cases, CV decreases with increasing n. (A and B) For the biochemical state NN, the CV of both E**(t) and I*(t) stabilizes asymptotically after 3–4 phosphorylated states. A CV of ∼60% for E**(t) at times past the peak time is reduced to a CV of ∼40% for the corresponding photocurrent I*(t). (C and D) For the theoretical state EE, increasing n gives in all cases a decreased CV although at a decreasing rate for increasing n. The CV comparison E**(t) (∼60%) to I*(t) (∼40%) is still present, thus pointing to an intrinsic variability reduction effect of the diffusion part of the process.

Display large version of this figure

Figure 4

Salamander: comparing the CV of the total activated effectors at time t with the CV of the total relative charge up to time t. (NN) Shutoff of Rh* in n biochemical states of decreasing duration and catalytic activity (see Biochemical Sequences {τ_j} and {ν_j}); (EE) Shutoff of Rh* in n theoretical states of equal duration and equal catalytic activity (see Sequence for Which τ_jν_j=Const). All simulations assume all activation steps as random (Case 4 of Random Events Contributing to SPR Variability). In all cases, CV decreases with increasing n. For NN, the CV stabilizes for n≥3 and it is essentially the same for n=3, 4, 5. For EE, the CV keeps decreasing with increasing n, although at a decreasing rate for increasing n. For the salamander at early times, the CV is initially large and then rapidly drops. No similar effect occurs in mouse. (A and B) For the biochemical state NN, the CV of both E**(t) and I*(t) stabilizes asymptotically after 3–4 phosphorylated states. A CV of ∼60% for E**(t) at times past the peak time is reduced to a CV of ∼50% for the corresponding photocurrent I*(t). (C and D) For the theoretical state EE, increasing n gives in all cases a decreased CV although at a decreasing rate for increasing n. The CV comparison E**(t) (∼60%) to I*(t) (∼50%) is still present, thus pointing to an intrinsic variability reduction effect of the diffusion part of the process. The suppression of CV for the photocurrent I*(t) with respect to CV of the activating E**(t), while present, is less dramatic than for mouse (Figure 3AB). In addition, we observe a sharp variability at early times, which is likely due to presence of the incisures and their distributed geometry. This is supported by the absence of such incipient CV, in lumped models insensitive to incisures geometry (see also captions of Fig. 5). This early-time high CV seems also to be due to the random position of the activation site. Indeed, photons absorbed close to the disk boundary, yield a faster response than those absorbed far away from the boundary, say near the center of the disk. After a short time, depending on the diffusivity coefficients on the disk and on the disk radius, this difference is reduced. In the mouse, where diffusivities are larger and the radius is smaller than similar parameters in the salamander, no significant increase of variability at early times is observed (see also Fig. 3).