Estimating the rate constant of cyclic GMP hydrolysis by activated phosphodiesterase in photoreceptors
aa r X i v : . [ q - b i o . S C ] N ov Estimating the rate constant of cyclic GMP hydrolysis by activatedphosphodiesterase in photoreceptors
J¨urgen Reingruber
Department of Computational Biology, Ecole Normale Sup´erieure, 46 rue d’Ulm 75005 Paris, France.
David Holcman
Department of Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel andDepartment of Computational Biology, Ecole Normale Sup´erieure, 46 rue d’Ulm 75005 Paris, France
The early steps of light response occur in the outer segment of rod and cone photoreceptor. Theyinvolve the hydrolysis of cGMP, a soluble cyclic nucleotide, that gates ionic channels located in theouter segment membrane. We shall study here the rate by which cGMP is hydrolyzed by activatedphosphodiesterase (PDE). This process has been characterized experimentally by two different rateconstants β d and β sub : β d accounts for the effect of all spontaneously active PDE in the outersegment, and β sub characterizes cGMP hydrolysis induced by a single light-activated PDE. So far,no attempt has been made to derive the experimental values of β d and β sub from a theoretical model,which is the goal of this work. Using a model of diffusion in the confined rod geometry, we deriveanalytical expressions for β d and β sub by calculating the flux of cGMP molecules to an activatedPDE site. We obtain the dependency of these rate constants as a function of the outer segmentgeometry, the PDE activation and deactivation rates and the aqueous cGMP diffusion constant. Ourformulas show good agreement with experimental measurements. Finally, we use our derivation tomodel the time course of the cGMP concentration in a transversally well stirred outer segment. I. INTRODUCTION
The modern theory of chemical reactions originates back to Arrhenius [1], who showed in 1889 that thebackward rate constant k b of two reactants depends exponentially on the temperature and the activationenergy barrier. However, the molecular description of the backward rate started with the seminal paperof Kramers in 1940 [2] (see also [3, 4]). The constant k b is used to describe the chemical reaction ofabundant species in solution and the concentration of the product resulting of the interaction of twomolecules is calculated by using the mass action law. But, The computation also involves the forward k f rate and the concentration of the two species. At a molecular level, k f reflects the mean time for oneof molecule to meet the other by diffusion, and the probability to react upon encounter. For diffusionlimited chemical reactions, based on the mean time for a uniform concentration of particles inside aninfinite 3-dimensional space to hit a sphere of radius a , von Smoluchowski obtained in 1914 the firstestimate k f = 4 πaD [5]. The Smoluchowski formula was later on extended to the case of a partiallyabsorbing sphere [6, 7] .Diffusion plays in many cases a prominent role in the determination of the forward binding rate [8, 9,10, 11, 12], and numerous fundamental processes in cellular biology rely on the rate at which diffusingmolecules hit a small target site: examples are trapping in patchy surfaces [13], receptor dwell time insidea synapse [14] and many more. When the number of molecules is not large, the mass action law is notsufficient to account for the random nature of the chemical reactions and other approaches are required[15]. In addition, in a confined geometry, the Smoluchowski formula does not describe the refine structureof the bounded space. For that purpose, the small hole approximation was developed, which is the meantime for a Brownian particle to escape a confined domain through a small window [15, 16, 17]. However,all these computations rely on the assumption that the reaction volume is quite homogenous and has ashape close to a convex domain (no bottle neck).In photoreceptor outer-segment, a diffusing molecule needs to find a specific target site in a degenerateddomain, where one dimensional length is much smaller than the others, and thus previous related to thesmall hole formula do not apply. This problem contains two difficulties: first, the target site occupiesonly a tiny portion of the boundary, and second, the diffusion occurs in a narrow domain.We shall now be specific and explain what is our goal in the context of phototransduction: Rodphotoreceptors are highly specialized biological devices that can detect a single photon absorption [18,19, 20, 21]. The photon absorption activates a cascade of chemical reactions in the outer segment, whichultimately hyperpolarizes the cell [21, 22, 23, 24, 25, 26]. The inner structure of a rod outer segmentis very specific and can be considered as a cylinder that contains a densely packed stack of paralleland uniformly distributed discs (see Fig. 1). The discs divide the outer segment into almost separatecompartments that are loosely connected through a narrow gap between the disc perimeters and the outersegment membrane, which we refer to as the outer shell [27]. Compartments are also linked through discincisures, however, since their impact is small [28] we will neglect them in first approximation . Thechemical reactions involved in the early steps of phototransduction occur on the surface of the internaldiscs, and result in the activation of the phosphodiesterase molecule (PDE) via a G-protein coupledactivation cascade [21, 22, 23, 26]. A photon-excited rhodopsin activates many transducin molecules,which bind to and thereby activate PDE. The number of activated PDE molecules following a singlephoton absorption was studied both experimentally and theoretically [29, 30, 31, 32, 33, 34]. We referto PDE molecules that become activated via the phototransduction cascade as light-activated PDE. Inaddition to the transduction pathway, PDE can also spontaneously activate, leading to a non-vanishingbackground activity even in darkness [30, 35].Cytoplasmic diffusible cGMP molecules controlling the opening of cationic channels in the plasmamembrane are hydrolyzed by activated PDE, and the reduction in the cGMP concentration leads tochannel closure and photoreceptor hyperpolarization. From another chemical pathway catalyzed byguanyl cyclase (GC), a molecule attached to the disc surfaces and the outer segment membrane, cGMPmolecules are synthesized from cytoplasmic GTP, a reaction which is calcium dependent. The magnitudeof the photoresponse signal depends significantly on the number of closed ionic channels, and thereforeon the drop in the cGMP concentration, which is controlled in part by the number of activated PDE andthe rate of GMP hydrolysis of a single activated PDE.cGMP hydrolysis is characterized by two rate constants β d and β sub , which are both derived fromexperimental measurements [22, 23, 27, 30, 33, 36, 37]. Our goal here is to derive these constants frommolecular considerations and biophysical theory, and thus obtain explicit analytical expressions. Tounderstand at an intuitive level how these rates are defined, we recall that in most photoresponse modelsthe cGMP concentration in the outer segment is well-stirred, a simplification that neglects diffusion andthe complex geometry of the outer segment. The effective differential equation for the well-stirred cGMPconcentration C ( t ) is [22, 23, 33] ddt C ( t ) = α ( t ) − β d C ( t ) − β sub P ∗ l ( t ) C ( t ) , (1)where P ∗ l ( t ) is the number of light-activated PDE molecules and α ( t ) the rate of cGMP synthesis. Theterm β d C ( t ) accounts for cGMP hydrolysis due to spontaneous PDE activation, and β sub P ∗ l ( t ) C ( t ) due tolight-activated PDE. Eq. 1 shows an important difference in modeling cGMP hydrolysis by spontaneously-and light-activated PDE: whereas β d is the rate constant for the change in the cGMP concentration dueto all spontaneously activated PDE in the outer segment, β sub denotes the change in the well stirredcGMP concentration due to a single light-activated PDE.In the literature, β d and β sub are considered as independent parameters, a hypothesis that is strength-ened by the finding that the experimental values for β d and β sub are around 1 s − and 10 − s − respectively,and therefore are extremely different in appearance [22, 23, 25, 32].Based on the diffusional encounter process between a cGMP and an activated PDE molecule in thecomplex rod outer segment geometry, we obtain explicit estimates for β d and β sub . Our analysis ismotivated by several known results: First, in darkness, in average around one spontaneously activatedPDE molecule is present in a single compartment [23, 30], which suggests that diffusion is rate limitingfor hydrolysis. Second, experimental data [25, 32] indicate that activated PDE is a nearly perfect effectorenzyme and hydrolyzes cGMP with a very high efficiency, which also hints that cGMP-hydrolysis islimited by diffusion. Third, a diffusion limited hydrolysis reaction couples the cytosolic cGMP level moststrongly to the activation status of PDE, which is at the basis of photoreceptor adaptation [23, 38].One of the main results of this paper is formula 32, β d = D cG πρµ + µ −
84 ln( Ra ) − , (2)which relates β d to the spontaneous PDE activation and deactivation rates µ + and µ − , the PDE surfacedensity ρ , the effective reaction radius a , the radius R of the outer segment, and the cytoplasmic cGMPdiffusion constant D cG . Furthermore, by comparing this purely diffusional cGMP hydrolysis rate toexperimental results, we can estimate the impact of the details of the chemical hydrolysis reaction.The paper is organized as follows: we first determine the rate constant of cGMP hydrolysis due to asingle activated PDE as a function of the cGMP concentration, the cGMP diffusion constant and thegeometrical structure of the outer segment. Using this result, we then compute the analytical expressionsfor β d and β sub . We find that β d is proportional to the mean number of spontaneously active PDEmolecules in a compartment and not in the whole outer segment. We compare our analytical estimationswith experimental measurements, and find good agreement. Our analysis suggests that the main reasonfor the discrepancy between β d and β sub is their incompatible definitions. By deriving β d and β sub frommolecular events, we show that they are no longer two independent parameters. Finally, we use ouranalysis to model the spatio-temporal time course of a photoresponse in a transversally well-stirred outersegment. II. RATE OF CGMP HYDROLYSIS BY ACTIVATED PDE
In this section, we estimate cGMP hydrolysis rate constant by a driven by single activated PDEmolecule P ∗ when diffusion is the limiting step. Later on, we use this result to derive expressions for β d and β sub . To illustrate our approach, we start with the molecular model for cGMP hydrolysis: cGM P + P ∗ k f ⇄ k b cGM P · P ∗ k −→ P ∗ + GM P . (3)A cGMP molecule binds to a P ∗ molecule with a forward rate k f and forms an intermediate complex cGM P · P ∗ . This complex can either dissociate with a backward rate k b , or cGMP becomes hydrolyzed toGMP with a rate k . We are interested in the rate k h by which cGMP molecules are hydrolyzed, which,at steady state, balances the cGM P production rate. In the restricted rod outer segment, the overallforward binding rate is k f G c , where G c is the number of cGMP molecules in a single compartment. As anexample, in darkness, G c is roughly in the range 100-1000, depending on the radius of the outer segment[23]. From Eq. 3, using the overall forward binding rate and Michaelis-Menton approximation, we obtain k h = k k f G c P ∗ k + k b + k f G c . (4)In the physiological range of cGMP concentrations, we assume that k ≫ k f G c , which implies that thehydrolysis of cGM P · P ∗ proceeds much faster compared to the formation of a new complex. Furthermore,since P ∗ hydrolyzes cGMP with very high efficiency [25, 32]), we suppose that k ≫ k b , and thereforeneglect the backward rate. Under these circumstances, Eq. 4 reduces to k h = k f G c P ∗ , (5)which has exactly the form of the hydrolysis term in Eq. 1. Eq. 5 can be formally obtained by setting k = ∞ , which means that cGMP hydrolysis occurs instantaneously after the formation of the the complex cGM P · P ∗ . In contrast, if we assume that k is small ( k ≪ k f G c and k b ≪ k f G c ), using Eq. 4, thisimplies that hydrolysis proceeds independently of the cGMP concentration with a rate k P ∗ , a scenariothat is not experimentally supported [23].For large values k , the cGMP hydrolysis rate in Eq. 5 is determined by the forward binding rate k f ,whose value depends on two parameters: the encounter rate k e of cGMP molecules with the P ∗ site,and the probability p that cGM P · P ∗ is formed upon encounter. The probability p depends on (largelyunknown) molecular properties of cGMP and activated PDE. In order to extract the impact of diffusionon cGMP hydrolysis, we set p = 1 and presume that the complex cGM P · P ∗ is formed each time acGMP molecule encounters P ∗ . Finally, by neglecting the molecular details of activated PDE, we donot distinguish between spontaneously- and light-activated PDE, and we consider only activated versusnon-activated PDE. If mainly diffusional issues are relevant for cGMP hydrolysis, then the catalyticactivities of spontaneously- and light-activated PDE should be very similar, as was already suggested byexperimental findings [30]. Symbol Description L Length of a rod outer segment R Radius of a disc d Gap between disc and outer segment membrane l Distance between two adjacent disc l d Width of a disc a P Radius of a PDE molecule a cG Radius of a cGMP molecule a = a P + a cG Sum of the radii of a cGMP and PDE molecule ρ PDE surface density µ + Spontaneous PDE activation rate µ − Spontaneous PDE deactivation rateTABLE I: Description of the parameters used in the model.
Before starting the analysis, we give range values for the main parameters: cGMP diffuses in thecytosolic volume V cyto of the outer segment with a diffusion coefficient D cG ≈ µm /s − [35, 37, 39, 40].In contrast, PDE molecules are attached to the disc surfaces, where they diffuse with a diffusion coefficient D P DE ≈ . µm /s − [29]. The exact geometrical dimensions of a rod outer segment varies betweenspecies [23, 41]: for example, the length L and radius R + d of the outer segment in a toad rod are 60 µm and 3 µm , whereas in a mouse rod they are 20 µm resp. 3 µm [23]. The longitudinal distance l betweentwo adjacent discs (the height of a compartment) and the width of a disc l d (see Fig. 1) vary around15 nm [41]. The width d of the outer shell is comparable to l [41]. The total number of compartments N c = Ll + l d in the outer segment is of the order N c ∼ . Finally, we assume that the radii a P of a PDEmolecule and a cG of a cGMP molecule are both comparable to the radius of a rhodopsin molecule, whichis around 1-2 nm [29]. The parameters are summarized in Table I. 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L Rzl
Outer Shell d (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Compartment l d FIG. 1: Section through a cylindrical rod outer segment, containing a densely packed stack of parallel anduniformly distributed discs. The volume delimited by two adjacent discs is called a compartment.
A. Analysis of cGMP hydrolysis due to a single activated PDE
To describe the time course of cGMP concentration in the outer segment, we consider different players:cGMP molecules are independent and diffuse freely inside the outer segment domain Ω. Whenever acGMP molecule hits the boundary area ∂ Ω h occupied by the P ∗ molecules, it becomes instantaneouslyhydrolyzed. The synthesis of cGMP occurs on the surface ∂ Ω − ∂ Ω h with a rate α σ ( x , t ). We account forthese interactions by using the density C ( x , t ) of cGMP molecules at position x and time t , it satisfiesthe diffusion equation with the appropriate boundary condition [4], ∂∂t C ( x , t ) = D cG △ C ( x , t ) , for x ∈ Ω (6) D cG ∂∂n C ( x , t ) = − α σ ( x , t ) , for x ∈ ∂ Ω − ∂ Ω h (7) C ( x , t ) = 0 , for x ∈ ∂ Ω h . (8)We shall now study Eq. 6 for a single compartment Ω c . Approximation of cGMP hydrolysis in a single compartment (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Ω c ∂ Ω P ∗ (a) ra R l z ∂ Ω P ∗ Ω a (b) FIG. 2: (a) Elementary cylindrical compartment with a P ∗ molecule located centrally on the upper surface. (b)Cross-section view of the compartment. A cGMP molecule is hydrolyzed when reaching the boundary of thevolume Ω a at r = a . We now consider a compartment Ω c in which a single PDE molecule is activated on either one of thetwo disc surfaces (in Fig 4a P ∗ is attached to the upper surface). In our approximation, cGMP hydrolysisrate is given by the flux J h of cGMP to the surface area ∂ Ω P ∗ occupied by a P ∗ molecule. To compute J h , we shall make some approximations:We consider a uniform and time independent cGMP synthesis rate α σ . Because cGMP synthesis iscalcium dependent, this corresponds to a calcium clamped outer segment or at equilibrium ( this is thecase in darkness). Furthermore, because the height l of a compartment is around a few nm , and muchsmaller compared to the radius R ∼ µm , the time scale for longitudinal equilibration l /D cG is muchshorter than the one for radial equilibration ∼ R /D cG . Hence, newly synthesized cGMP moleculesat the surface quickly equilibrate in longitudinal direction before encountering a P ∗ molecule, which isusually located far away compared to the compartment height l (except for the negligible amount cGMPsynthesized in direct neighborhood of P ∗ ). This scenario is equivalent to having cGMP synthesized insidethe compartment, and we therefore replace cGMP synthesis on the surface by synthesis inside the volume.The volume synthesis rate α v is linked to α σ by α v = 2 α σ l , (9)where the factor 2 accounts for the two disc surfaces enclosing a compartment. With a volume synthesisrate α v , the diffusion equation for cGMP is ∂∂t C ( x , t ) = D cG △ C ( x , t ) + α v , for x ∈ Ω c . (10)The boundary conditions are given by Eq. 8 and Eq. 7 with α σ ( x , t ) = 0.Because cGMP diffuses much faster than PDE ( D P DE ≪ D cG ) we neglect PDE motion [29, 42, 43].Since at leading order approximation the exact position of the activated PDE is not relevant [44, 45, 46],we position P ∗ at the center of the disc. We will discuss this issue in more detail in section III. Inaddition, we approximate cGMP molecules by infinitesimal points, and use the effective reaction radius a = a P + a cG for a P ∗ molecule [29, 42, 43].Because the effective diameter 2 a ∼ nm of the boundary area ∂ Ω P ∗ occupied by a P ∗ molecule iscomparable to the compartment height l ∼ nm , and the radius R ∼ µm is much larger than a and l , the main limiting factor for cGMP hydrolysis rate is the speed by which cGMP molecules find P ∗ .We note that once a cGMP molecule enters into a neighborhood of ∂ Ω P ∗ , since 2 a is comparable to l ,it has a high probability to hit ∂ Ω P ∗ and become hydrolyzed. In a first approximation, we model thehydrolysis reaction by assuming that a cGMP molecule entering the small cylindrical volume Ω a (givenin cylindrical coordinates by r ≤ a ) above or below ∂ Ω P ∗ is instantaneously hydrolyzed (see Fig. 2). Thecorresponding boundary condition is C ( x , t ) = 0 , for r = a . (11)This condition leads to an overestimation of the true hydrolysis rate because cGMP molecules enteringthe domain Ω a can as well leave this region without touching the surface ∂ Ω P ∗ . However, in appendixA, we show that the overestimation is in the range of a factor 2 (see also the discussion in section III).Having discussed the approximations, we shall now proceed to estimate the cGMP flux J h into ∂ Ω a .We are particularly interested in J h as a function of the cGMP concentration inside the compartment.From there, we will extract cGMP hydrolysis rate due to a single activated PDE. With this result, wewill then derive the expression for β d (see Eq. 1). Using the cylindrical symmetry, Eq. 10 reduces to ∂∂t C ( r, t ) = D cG r ∂∂r r ∂∂r C ( r, t ) + α v , (12) C ( r, t ) = 0 for r = a . (13)Integrating Eq. 12 over the compartment volume yields an equation for the time dependent number ofcGMP molecules G c ( t ) in Ω c , ddt G c ( t ) = − J R ( t ) − J h ( t ) + α v πR l (cid:18) − a R (cid:19) , (14)where G c ( t ) = 2 πl Z Ra C ( r, t ) rdr , (15) J h ( t ) = 2 πalD cG ∂∂r C ( r, t ) (cid:12)(cid:12)(cid:12) r = a , (16) J R ( t ) = − πRlD cG ∂∂r C ( r, t ) (cid:12)(cid:12)(cid:12) r = R . (17)The flux J R ( t ) is maintained by cGMP molecules that diffuse between compartments. To derive anexpression for J h , we consider the steady state regime where the flux J R is given. The steady stateconcentration C ( r ) obtained from Eq. 12 is given by C ( r ) = α v πR πD cG (cid:18) ln (cid:16) ra (cid:17) − r − a R (cid:19) − J R πD cG l ln (cid:16) ra (cid:17) . (18)To obtain the number of G c molecules inside a compartment, we insert Eq. 18 into Eq. 15 and for aR ≪ G c = α v πR l R D cG (cid:18)
12 ln (cid:18) Ra (cid:19) − (cid:19) − J R R D G (cid:18)
12 ln (cid:18) Ra (cid:19) − (cid:19) . (19)We define the times τ and τ and the corresponding rates k and k as τ = 1 k = R D cG (cid:18)
12 ln (cid:18) Ra (cid:19) − (cid:19) , (20) τ = 1 k = R D cG (cid:18)
12 ln (cid:18) Ra (cid:19) − (cid:19) , (21)we can rewrite expression 19 as G c = α v πR lk − J R k . (22)At steady state, the value of J h is fixed by the balance of fluxes, and Eq. 14 gives for aR ≪ J h = − J R + α v πR l . (23)Using Eq. 22 we can express α v as a function G c and J R , α v πR l = k G c + k k J R . (24)Finally, inserting Eq. 24 into Eq. 23 yields J h = k G c + k − k k J R . (25)Formula 25 gives the steady state hydrolysis rate J h as a function of G c and J R . This result dependsstrongly on the diffusional and geometrical properties of the microdomain. Whereas Eq. 23 gives a directexpression for J h as a function of the synthesis rate α v , and does not involve diffusion, Eq. 25 is relatedto α v indirectly via the value of G c , and therefore involves diffusion.In appendix A we obtain an interpretation for the two times τ and τ , and, thus, for the rates k and k : τ (see Eq. A21) is the mean time for uniformly distributed cGMP molecules to reach the absorbingboundary at r = a , given reflecting boundary conditions at r = R ; τ (see Eq. A17 for r = R ) is the meantime to reach r = a , when the initial position is uniformly distributed at r = R . In reality, there is noreflecting boundary at r = R , however, a vanishing flux J R is mathematically equivalent to a reflectingboundary condition at r = R . B. Derivation of the rate constant β d for spontaneous PDE activation In darkness, spontaneous PDE activation leads to a uniform cGMP hydrolysis in the outer segment[22, 23, 30] with an overall hydrolysis rate (see Eq. 1 integrated over the cytoplasmic volume) J d,os = β d G os , (26)where G os is the total number of cGMP molecules in the outer segment. To derive an analytical expressionfor the rate constant β d , we start from Eq. 25. Because spontaneous PDE activation occurs uniformlythroughout the outer segment, apart from fluctuations, the flux J R between compartments vanishes indarkness. Thus, the steady state hydrolysis rate J h of a single P ∗ molecule given in Eq. 25 can be writtenas J h = k G c . (27)To obtain the dark hydrolysis rate J d,c per compartment, we have to further consider the mean number ofspontaneously activated PDE molecules P ∗ s,c in a compartment. As long as the number P ∗ s,c is small andthe P ∗ molecules are geometrically well separated [47], the rate J d,c increases linearly with P ∗ s,c . Hence,we obtain J d,c = k P ∗ s,c G c . (28)The hydrolysis rate in the whole outer segment J d,os is obtained by summing J d,c over all N c compart-ments. Since G os = N c G c well approximates the total number of cGMP molecules in the outer segment(the volume 2 πRdL of the outer shell is negligible compared to the volume πR lN c of all compartments),we obtain J d,os = k P ∗ s,c G os . (29)Finally, by comparing Eq. 29 with Eq. 26 and by using Eq. 20, we obtain β d = k P ∗ s,c = D cG R
84 ln (cid:0) Ra (cid:1) − P ∗ s,c . (30)We conclude that β d is determined by the mean number of spontaneously activated PDE in a compart-ment, and not in the outer segment [30]. Furthermore, we will now relate β d to the spontaneous PDEactivation rate µ + , the deactivation rate µ − , and the PDE surface density ρ . The number of PDE on thedisc surfaces attached to a single compartment is P c = 2 πR ρ , and P ∗ s,c is given by P ∗ s,c = P c µ + µ − = 2 πR ρ µ + µ − . (31)Together with Eq. 20 and Eq. 30 we obtain the final expression β d = D cG πρµ + µ −
84 ln( Ra ) − . (32) C. Effective set of equations to model cGMP dynamics
By generalizing our previous results, we shall now derive an effective set of equations to model cGMPdynamics following a photon absorption. Since a photon absorbtion transiently generates an elevatedamount of P ∗ molecules inside the affected compartment, it induces an increased cGMP hydrolysis anda cGMP gradient in the outer segment. In this case, the fluxes J R between compartments are no longerzero after a photon absorption.We start the derivation by extending the equilibrium expression for J h given in Eq. 25 to time dependentsituations. Because free cGMP diffusion is fast, cGMP equilibrates quickly inside a compartment. Incontrast, G c ( t ) and J R ( t ) fluctuations are determined by the effective longitudinal diffusion between thecompartments, which is strongly hindered by the compartmentalization of the outer segment [27, 40, 48].Thus, we consider that G c ( t ) and J R ( t ) fluctuate on a slower time scale compared to the equilibrationtime scale inside a compartment. Under this condition, a first approximation of the time dependenthydrolysis rate J h ( t ) of a single P ∗ molecule is given by the equilibrium expression in Eq. 25 with timedependent J R ( t ) and G c ( t ): J h ( t ) = k G c ( t ) + k − k k J R ( t ) . (33)We note that the expression for J h given in Eq. 25 can be extended to time dependent cases, whereasthis is not possible starting from Eq. 23.To obtain the set of equations for the time dependent number of cGMP molecules inside a compartment,we start when a photon is absorbed in compartment n , while the other compartments n = 1 . . . N c , n = n remain unperturbed. In the regime considered here, cGMP hydrolysis depends linearly on P ∗ l ( t ).Using Eq. 14, the equation for the number G ( n ) c ( t ) of cGMP molecules in a compartment n is given by( δ n,n is the Kronecker-Delta) ddt G ( n ) c ( t ) = − J ( n ) R ( t ) − ( P ∗ s,c + P ∗ l ( t ) δ n,n ) J ( n ) h ( t ) + α v πR l . (34)Inserting the expression for J ( n ) h ( t ) given in Eq. 33, and using the definition of β d in Eq. 30, we obtain ddt G ( n ) c ( t ) = − (cid:18) P ∗ s,c + P ∗ l ( t ) δ n,n ) k − k k (cid:19) J ( n ) R ( t ) − β d G ( n ) c ( t ) − k P ∗ l ( t ) δ n,n G ( n ) c ( t ) + α v πR l . (35)By approximating the transversally well stirred cGMP concentration in a compartment by C ( n ) ( t ) ≈ G ( n ) c ( t ) πR l , and by using Fick’s law, the fluxes J ( n ) R ( t ) are approximated by J ( n ) R ( t ) = − D cG πRd (cid:18) C ( n +1) ( t ) − C ( n ) ( t ) l + l d + C ( n − ( t ) − C ( n ) ( t ) l + l d (cid:19) = − D cG dRl G ( n +1) c ( t ) − G ( n ) c ( t ) l + l d + G ( n − c ( t ) − G ( n ) c ( t ) l + l d ! . (36)Eqs. 35 and 36 constitute a close system of equations for the G ( n ) c ( t ) (which can be transformed intoequations for the concentrations C ( n ) ( t )). Furthermore, Eq. 35 models the impact of spontaneously- andlight-activated PDE in an equivalent way. The simulation in Fig. 3 shows the time course of the numberof cGMP molecules, scaled with respect to the dark equilibrium value, after the absorption of a photonat time t = 0 in the middle of the outer segment. The parameters for the simulation are suitable for atoad rod [23]. The input function P ∗ l ( t ) is obtained using the set of equations published in [34]. D. Derivation of the rate constant β sub in a well stirred outer segment We now derive an analytic expression for the rate constant β sub using the approximation of a wellstirred outer segment [23]. Since the volume of the outer shell is negligible compared to the combinedvolume of all compartments, the total number of cGMP molecules in the outer segment is G os ( t ) = N c X n =1 G ( n ) c ( t ) . (37)By summing Eq. 35 over all compartments, and using that P N c n =1 J ( n ) R ( t ) ≈ ddt G os ( t ) = − k − k k P ∗ l ( t ) J ( n ) R ( t ) − β d G os ( t ) − k P ∗ l ( t ) G ( n ) c ( t ) + α v πR lN c . (38)In a well stirred outer segment we have G ( n ) c ( t ) = G os ( t ) /N c . By further neglecting the term − k − k k P ∗ l ( t ) J ( n ) R ( t ) (the flux J R vanishes in a well stirred outer segment), we get ddt G os ( t ) = − β d G os ( t ) − k N c P ∗ l ( t ) G os ( t ) + α v πR lN c . (39)Finally, dividing Eq. 39 with the cytosolic volume V cyto ≈ πR lN c yields the standard equation for thewell stirred cGMP concentration C ( t ) = G os ( t ) /V cyto , ddt C ( t ) = − β d C ( t ) − k N c P ∗ l ( t ) C ( t ) + α v . (40)0 Time (s) c G M P l e v e l (a)
600 800 1000 1200 14000.70.750.80.850.90.951
Compartment (b)
Time (s) a c t i v . P D E (c) FIG. 3: cGMP dynamics after a photon absorption at time t = 0 in compartment n = 1000 ( N c = 2000). Thesimulation is performed using Eqs. 35 and 36. The cGMP concentration is scaled with the equilibrium value. (a)Time dependent cGMP level averaged over the outer segment. (b) cGMP level per compartment for various timepoints. (c) Number of light-activated PDE molecules obtained using the equations published in [34]. By comparing Eq. 40 with Eq. 1 we obtain for β sub the expression β sub = k N c = β d N c ¯ P ∗ s,c . (41)Since N c is of the order 10 , it follows that β sub is much smaller than β d . Using Eq. 20 for k and V cyto ≈ πR lN c , Eq. 41 can be written as β sub = πD cG lV cyto
84 ln (cid:0) Ra (cid:1) − . (42)By comparing expression 42 with the standard definition of β sub given by [22, 23] (we neglect cytoplasmicbuffering for cGMP [32, 37]) β sub = k sub K m N Av V cyto , (43)1we obtain a new formula for k sub K m given by ( N Av is the Avogadro number) k sub K m = N Av V cyto k N c = 8 πD cG lN Av (cid:0) Ra (cid:1) − . (44) III. COMPARISON WITH EXPERIMENTAL RESULTS
To validate our computations, we now compare our analytical results for β d and β sub (Eq. 30 andEq. 41) with experimental measurements [23, 27, 30, 33]. We start with β d . Using data available fortoad rods, ρ = 100 µm − , R = 3 µm , µ + = 4 × − s − , µ − = 1 . s − , D cG = 100 µm /s − , a = 3 nm , l = 15 nm [23, 27, 30], and inserting these values into Eq. 20, Eq. 31 and Eq. 30, we obtain k = 3 . s − , P ∗ s,c = 1 .
26 and β d = k P ∗ s,c ≈ . s − . (45)This analytic result has to be compared to the experimentally found value β d ≈ s − [23], which isapproximately four times smaller than this prediction. Eq. 32 shows that β d depends only logarithmicallyon the compartment radius R , and thus it is very similar across species that differ mostly on the radius ofthe outer segment, in agreement with experimental findings [23]. The discrepancy between our theoreticalprediction and the experimental value for β d can be attributed to several factors:1. We made the assumption that a cGMP molecule already becomes hydrolyzed when reaching theinner cylinder Ω a at r = a . Thus, the time τ in Eq. 20 is shorter than the true time needed toarrive at the P ∗ site. Hence, Eq. 20 overestimates the hydrolysis rate. In appendix A, we derivean accurate estimate for the mean time τ a cGMP molecule reaches the P ∗ site located on thesurface of a compartment (see Eq. A23). Compared to τ (Eq. 20), the new estimate for τ includesspecifically the mean time τ a a cGMP molecule starting on the boundary of Ω a reaches the P ∗ molecule on the surface. By considering the additional time τ a , we replace τ and τ with the moreaccurate expressions ˜ τ = τ a + τ and ˜ τ = τ a + τ . Accordingly, the rates k and k have to bereplaced by ˜ k and ˜ k , given by˜ k = 1˜ τ = 1 τ a + τ , ˜ k = 1˜ τ = 1 τ a + τ . (46)For toad rod values with l/a ∼ R/a ∼ g (5) ≈ . g (5) is obtained from Fig. 5b), we find that ˜ k ≈ . k . By using ˜ k instead of k in Eq. 30 weobtain the new estimation β d = ˜ k P ∗ s,c = 2 . s − , (47)which is closer to the experimental observation.2. Our assumption that every encounter between cGMP and P ∗ results in cGMP hydrolysis will cer-tainly lead to an overestimation of the hydrolysis rate. Moreover, since we neglected the moleculardetails of the hydrolysis reaction, this will also induce an error. Nevertheless, since our analyticresult for β d is very close to the experimental finding, we conclude that cGMP hydrolysis by P ∗ has to be largely diffusion limited, and in addition has to be quite efficient, such that nearly everyencounter between cGMP and activated PDE leads to a hydrolysis reaction. This is supportedby the experimental observations that activated PDE hydrolyzes cGMP with very high efficiency[25, 32].3. Uncertainties in the experimental values for D cG , µ + and µ − , involved in the computation of β d , introduce ambiguities in our analytical prediction. For example, there is still considerabledisagreements about the exact value of the diffusion constant D cG [35, 37, 39, 40]. Furthermore, atfirst approximation, we used for the effective reaction radius a the sum of the molecular radii of aPDE and cGMP molecule. A more precise value for a will affect β d in Eq. 47 mainly via τ a , since τ depends only logarithmically on a (see Eq. 32.24. The value of β d was computed by fixing the position of P ∗ at the disk center and neglectingpossible fluxes between compartments. In general, spontaneous PDE activation and diffusion leadsto P ∗ positions that are uniformly distributed over the disk surface, and different P ∗ positions inneighboring compartments induce small fluxes. We left open here the computation of the varianceof the cGMP hydrolysis rate constant coming from random locations of P ∗ molecules. However, the P ∗ position should not much influence the rate constant for cGMP hydrolysis: The rate constantis determined by the MFPT of a cGMP molecule to find the P ∗ target. Outside a small boundarylayer around P ∗ (the radius of the boundary layer is of the order of the reaction radius a ), theleading order term of the MFPT in dimension 2 depends only logarithmically on the distancebetween cGMP and P ∗ , and in dimension 3 it is a constant [44, 45]. Hence, since almost all cGMPmolecules are outside the boundary layer, the exact position of P ∗ is not important for their meantime to hydrolysis. We conclude that our expression for β d should remain a valid approximation atfirst order, even when considering random P ∗ positions.We shall now compare expressions Eqs. 41,42 for β sub , and the ratio k sub K m (Eq. 44) with experimentalmeasurements. Eq. 41 reveals that β sub is a factor N c P ∗ s,c smaller than β d . Since N c is of order 10 and P ∗ s,c of order 1 −
10, this agrees with the experimental findings that β sub is around 10 − times smallerthan β d [23, 25]. From Eq. 44, we obtain the prediction k sub K m = N Av V cyto k N c ≈ . × M − s − . This estimation can be further improved by using the rate ˜ k = 0 . k instead of k , giving k sub K m ≈ . × M − s − , which has to be compared to k sub K m ≈ . × M − s − obtained from experiment [32]. It is importantto note that our analytic results for k sub K m and β d (see Eq. 47) are both around two times larger than theexperimental findings, which is an indirect confirmation of our assumption that β d and β sub (note that k sub /K m is proportional to β sub ) are not two independent rate constants, but can be derived from thesame underlying hydrolysis reaction. Despite of the encouraging results, we would also like to indicatesome difficulties related to the definition and derivation of the parameters β sub and k sub K m : First, weextracted the formula for k sub K m using the expression for β sub given in Eq. 42. This approach is problematicbecause the definition of β sub involves the assumption of a well stirred cGMP concentration during aphotoresponse, which is not very accurate (see Fig. 3 and [27, 37]). Second, if diffusion limits the rate ofcGMP hydrolysis in the physiological range, the experimentally observed value for k sub K m does not reflectan intrinsic property of the chemical reaction. Instead, it depends strongly on diffusional and geometricaldetails, and, therefore, on the experimental setup. For example, measurements of the Michaelis constant K m were performed using fragments of disrupted rod outer segments with a length only a fraction ofthe intact outer segment length [32, 49]. Eq. 40 shows that the rate of cGMP hydrolysis increases withdecreasing fragment length L (since N c ∼ L ). Thus, the apparent value of the Michaelis constant K m ( k sub is assumed to be a true constant) that is needed to fit the rate of cGMP hydrolysis will be higher ina suspension containing large fragments compared to a suspension with small fragments, as it has beenobserved [32, 49].In this work we have assumed that cGMP hydrolysis in the physiological range is diffusion limited,and is independent of whether PDE is spontaneously- or light-activated. The agreement between ourtheoretical results and experimental measurements indicates that the large disparity between β sub and β d is largely due to their definition, and not due to biochemical differences. For example, in [27] the effect ofspontaneously- and light-activated PDE was modeled using two very different rates k = 0 . µM − s − and k ∗ = 110 µM − s − . We will show now that the large discrepancy between k and k ∗ in [27] essen-tially originates from modeling needs. Indeed, cGMP hydrolysis by spontaneously activated PDE wasmodeled as k [ P DE ] σ [ cGM P ], where [ P DE ] σ is the surface concentration of PDE. In contrast, hydrolysisby light-activated PDE was modeled as k ∗ [ P DE ∗ ] σ [ cGM P ], with [ P DE ∗ ] σ as the surface concentrationof light-activated PDE. By introducing the mean surface concentration of spontaneously activated PDE,[ P DE ∗ s ] σ = [ P DE ] σ µ + µ − , we rewrite k [ P DE ] σ [ cGM P ] as k µ − µ + [ P DE ∗ s ] σ [ cGM P ], which now has the same3form as k ∗ [ P DE ∗ ] σ [ cGM P ]. Inserting the values µ + = 4 × − s − and µ − = 1 . s − found in [30], weobtain k µ − µ + = 189 µM − s − , which is now comparable to k ∗ = 110 µM − s − . We conclude that model-ing cGMP hydrolysis by spontaneously- and light-activated PDE in a similar way involves comparableparameters, indicating that hydrolysis may be indeed independent of whether PDE is spontaneously- orlight-activated. IV. CGMP HYDROLYSIS IN CONES
After having discussed in detail cGMP hydrolysis in rods, we now briefly explore hydrolysis in cones.Similar to [48], our analysis for rods can be adapted to cones. Unlike rods, cones do not contain disc inthe outer segment. However, the membrane invaginations in cones can be modeled similarly to discs inrods. Since the radius of the cone outer segment decreases from the bottom versus the top, we can adaptour formulas to cones by replacing the disc radius R with a compartment dependent radius R n . Thus, incones, the rates ˜ k and ˜ k depend on the compartment n . Therefore, the response to a photon absorptionin cones varies on the location where the photon has been absorbed. Since k P ∗ s,c depends logarithmicallyon the compartment radius R n (see Eq. 30), we suggest that the value for the dark hydrolysis rate β d incones should be of the same magnitude as found in rods, see also [35]. V. SUMMARY AND DISCUSSION
In this paper, we have studied the rate constant of cGMP hydrolysis by activated PDE in rod and conephotoreceptors. Our analysis is based on the assumption that cGMP hydrolysis is diffusion limited anddetermined by the encounter rate between cGMP and activated PDE. We derived an explicit formulafor the rate constant of cGMP hydrolysis by a single activated PDE molecule as a function of theconfined outer segment geometry and the cGMP diffusion constant (Eq. 27). Our calculation takes intoaccount the complex structure of the rod outer segment, uniformly divided by a stack of parallel discsinto homogenous microdomains, called compartments, and coupled to each other via cGMP diffusion.We obtained analytical expressions for the rate constants β d and β sub . In addition, we give a set ofeffective equations that allow to model the transversally well stirred cGMP concentration after a photonabsorption.Interestingly, we found that only the amount of spontaneously activated PDE in a single compartmentis needed to calculate the dark hydrolysis rate β d (see Eq. 30). This result differs from [30], where thecompartmentalization was not considered, and all spontaneously activated PDE in the outer segmentadditively contribute to β d . Because the number of spontaneously activated PDE in the outer segmentis by a factor N c ∼ larger compared to a single compartment, the catalytic activity of an excitedPDE in [30] was estimated much lower compared to what we found here. We computed the PDE activity(given by the rate ˜ k in Eq. 46) to be around 1 s − , whereas in [30] it is around 10 − s − . Using therates for spontaneous PDE activation and deactivation [30], we estimate that the average number ofspontaneously activated PDE molecules in a single compartment is around one. Together with our resultfor the PDE activity, this naturally explains the experimental value β d ∼ s − . For the derivation of β d ,it was essential to assume that cGMP hydrolysis occurs locally at the activated PDE site. In contrast, ifcGMP hydrolysis occurred uniformly over the disc surface, then the experimental value for β d could notbe recovered without introducing additional adjusting parameters (see appendix B).We have derived a set of equations (Eqs. 35 and 36) that allow to calculate the time course of thetransversally well stirred cGMP concentration following a photon absorption. These equations modelcGMP hydrolysis by spontaneously and light-activated PDE in a similar way. Under the assumptionthat cGMP concentration in the outer segment is well stirred, we derived an expression for the rate β sub (Eq. 41,42), and the ratio k sub K m (Eq. 44).Eq. 41 connects β sub to β d and gives a direct explanation why β sub is found to be so much smaller than β d . Our result suggests that the large discrepancy between β d and β sub is largely due to their definitions: β d incorporates the effect of all spontaneously activated PDE in the outer segment, while β sub accountsfor only a single light-activated PDE.4 APPENDIXAPPENDIX A: MEAN TIME TO HYDROLYSIS IN A COMPARTMENT (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Ω c ∂ Ω P ∗ (a) x ra R l z T a Ω a Ω o ∂ Ω P ∗ T o (b) FIG. 4: (a) Cylindrical compartment Ω c with an activated PDE molecule located centrally on the upper surface.(b) The first time a cGMP molecule hits ∂ Ω P ∗ , when starting at position x in Ω o , is given by the sum of twotimes. First, the time T o for a molecule starting at x to arrive at the boundary ∂ Ω a , and second, the time T a forthe particle starting at ∂ Ω a to arrive at ∂ Ω P ∗ . In this part of the appendix, we shall obtain a precise estimate for the mean time τ a cGMP moleculestarting uniformly distributed inside the cylindrical compartment Ω c reaches the activated PDE molecule,defined as the small surface patch ∂ Ω P ∗ with radius r = a (see Fig 4). The motion of the cGMP moleculeis Brownian in Ω c . It is reflected all over the boundary except at ∂ Ω P ∗ , where is is absorbed. We denoteby T ( x ) the random initial time a cGMP molecule starting at x ∈ Ω c hits ∂ Ω P ∗ . Due to rotationalinvariance, the Mean First Passage Time (MFPT) τ ( x ) = E[ T ( x ) | x (0) = x ] depends only on r and z .Using a cylindrical coordinate system x = ( r, ϕ, z )), we decompose the domain Ω c into the inner cylinderΩ a = { x ∈ Ω c | r ≤ a } , (A1)and the hollow cylinder Ω o = Ω c − Ω a = { x | a ≤ r ≤ R } . (A2)We define the mean time τ as the average over a uniform initial distribution in Ω c , τ = 1 | Ω c | Z Ω c τ ( r, z ) dV . (A3)Using that Ω c = Ω o + Ω a , we can rewrite Eq. A3 as τ = | Ω c | − | Ω a || Ω c | | Ω o | Z Ω o τ ( r, z ) dV + | Ω c | − | Ω o || Ω c | | Ω a | Z Ω a τ ( r, z ) dV = 1 | Ω o | Z Ω o τ ( r, z ) dV − | Ω a || Ω c | (cid:18) | Ω o | Z Ω o τ ( r, z ) dV − | Ω a | Z Ω a τ ( r, z ) dV (cid:19) , (A4)where | Ω o | R Ω o τ ( r, z ) dV is the mean time for particles starting uniformly distributed in Ω o to hit ∂ Ω P ∗ ,and | Ω a | R Ω a τ ( r, z ) dV is the mean time to hit ∂ Ω P ∗ for particles starting uniformly distributed in Ω a .Because particles originating from Ω o have to reach Ω a before hitting ∂ Ω P ∗ , it is plausible (and can alsobe shown) that the mean time to ∂ Ω P ∗ for particles starting uniformly in Ω o is larger than the meantime for particles starting uniformly in Ω a . Furthermore, for a ≪ R , we have | Ω a | ≪ | Ω c | . From this, wefinally obtain τ = 1 | Ω c | Z Ω c τ ( r, z ) dV ≈ | Ω o | Z Ω o τ ( r, z ) dV . (A5)5Thus, for a ≪ R , the mean time τ is well approximated by the mean time for particles starting uniformlyin Ω o to hit ∂ Ω P ∗ .We shall now estimate τ ( r, z ), by considering the equation [4, 50] D cG (cid:18) ∂ ∂r + 1 r ∂∂r + ∂ ∂z (cid:19) τ ( r, z ) , = − < z < l , ≤ r < R (A6) τ ( r, z ) = 0 , z = l , r < a∂∂z τ ( r, z ) = 0 , z = l , r > a∂∂z τ ( r, z ) = 0 , z = 0 ∂∂r τ ( r, z ) = 0 , r = R .
We will first estimate the average time τ ( r ) = 1 l Z l τ ( r, z ) dz . (A7)Using the dimensionless variables x = ra , y = za , ˆ τ ( x, y ) = D cG R τ ( r, z ) , α = aR , β = la , x α = 1 α , (A8)equation A6 becomes (cid:18) x ∂∂x x ∂∂x + ∂ ∂y (cid:19) ˆ τ ( x, y ) = − α , < y < β , ≤ x < x α (A9)ˆ τ ( x, y ) = 0 , y = β , x < ∂∂y ˆ τ ( x, y ) = 0 , y = β , x > ∂∂y ˆ τ ( x, y ) = 0 , y = 0 ∂∂x ˆ τ ( x, y ) = 0 , x = x α , and Eq. A7 ˆ τ ( x ) = 1 β Z β ˆ τ ( x, y ) dy . (A10)We integrate Eq. A9 over the variable y to derive an equation for ˆ τ ( x ) for x ≥
1. Taking into accountthe boundary conditions at y = 0 and y = β , we obtain1 x ∂∂x x ∂∂x ˆ τ ( x ) = − α , x > ∂∂x ˆ τ ( x ) = 0 for x = x α The solution is given ˆ τ ( x ) = f ( α, β ) + 12 ln( x ) − α ( x − , (A12)with f ( α, β ) = ˆ τ (1) = 1 β Z β ˆ τ (1 , y ) dy . (A13)6Hence, for r ≥ a , we have τ ( r ) = R D cG f ( α, β ) + R D cG (cid:18)
12 ln( x ) − α ( x − (cid:19) (A14)= τ a + τ o ( r ) , (A15)where we defined τ a = R D cG f ( α, β ) , (A16) τ o ( r ) = R D cG (cid:18)
12 ln (cid:16) ra (cid:17) − r − a R (cid:19) . (A17)Eq. A14 has an intuitive interpretation: the mean time τ ( r ) for a cGMP molecule, uniformly distributedat r > a , is the sum of the mean time τ o ( r ) to the boundary r = a plus the mean time τ a from the surface ∂ Ω a to go to ∂ Ω P ∗ (see Fig 4).By averaging over a uniform initial distribution ρ = π ( R − a ) in Ω o , the overall mean time τ in Eq. A5is given by τ = 1 | Ω o | Z Ω o τ ( r, z ) dV = 2 πρ Z τ ( r ) rdr = R D cG f ( α, β ) + R D cG − α ) − α − α − α (A18)= τ a + τ o , (A19)where we defined τ o as τ o = 2 πρ Z τ o ( r ) rdr = R D cG − α ) − α − α − α . (A20)The leading order expansion of τ o for α ≪ τ o = R D cG (cid:18)
12 ln (cid:18) Ra (cid:19) − (cid:19) . (A21)For α ≪ f ( α, β ) by f (0 , β ) = g ( β ) and obtain ( β = la ) τ a ≈ R D cG g (cid:18) la (cid:19) , α ≪ . (A22)Altogether, for α ≪
1, the mean time τ in Eq. A5 is given by τ = τ a + τ o ≈ R D cG (cid:20) g (cid:18) la (cid:19) + 12 ln (cid:18) Ra (cid:19) − (cid:21) . (A23)To derive an explicit expression for f ( α, β ) and g ( β ) is a difficult mathematical problem. Nonetheless,we shall obtain some asymptotic limits for f ( α, β ). For β → l → τ a → f ( α,
0) = 0. For β → ∞ , the time τ a diverges to infinity, and f ( α, β ) → ∞ . For a = R ,corresponding to α = 1, we have τ a = l D cG , from which it follows that f (1 , β ) = β . (A24)Finally, the small hole theory [44] predicts that when l ∼ R ≪ a (which implies that α ∼ β ), the meantime τ is asymptotically given by τ ≈ V D cG a = πR D cG β . (A25)7By comparing Eq. A25 with Eq. A23 (ln( α ) can be neglected compared to β for α ∼ β ), we obtain theasymptotic g ( β ) ∼ π β , β ≫ . (A26)So far, we have only an asymptotic expansion for β ≫
1. To explore a much larger parameter space, wedecided to run Brownian simulations (10000 cGMP molecules are initially uniformly distributed over thelateral surface of ∂ Ω a ) to estimate τ a and f ( α, β ). The numerical results for f ( α, β ) are summarized inFig. 5. Fig. 5b shows that f ( α, β ) can be well approximated by f (0 , β ) = g ( β ) for α . . α β =1 β =5 β =10 f ( α, β ) (a) α (b) β f ( , β ) (c) FIG. 5: Numerical evaluation of the function f ( α, β ) ( α = a/R and β = l/a ). Each data point is obtained usingthe Brownian simulation of 10000 cGMP molecules. (a) f ( α, β ) for different values α and β . For α = 1 we have f (1 , β ) = β (Eq. A24). (b) Same data as in (a), restricted to small α . (c) Plot of f ( , β ) ≈ g ( β ) (same dataas in (a) and (b)). The dashed curve represents the asymptotic π β (Eq. A26), achieved for α → β → ∞ .For values β ≤
10 used in the simulations, the behavior of g ( β ) is close, but not yet in full agreement with π β . APPENDIX B: MODEL WITH UNIFORM HYDROLYSIS ON THE DISC SURFACES
In this section, we consider a model that is based on cGMP hydrolysis occurring uniformly on the discsurface (see also [27, 37]). This is very different from the situation presented within the main body of thepaper, where we assumed that cGMP hydrolysis occurs locally at the P ∗ site. We show now that uniformcGMP hydrolysis leads to a dark rate constant proportional to D cG /l ∼ s − , which is very differentfrom Eq. 30 (see also section III). Thus, in order to account for the experimental value β d ∼ s − , oneis forced to introduce a small adapting parameter κ h .We will analyze two different scenarios: In one situation, synthesis and hydrolysis of cGMP are bothmodelled by boundary source terms. In another, only hydrolysis is modelled by a boundary source term,while synthesis occurs uniformly within the cytoplasmic volume.
1. Model with boundary source terms for hydrolysis and synthesis
The reaction-diffusion equation for cGMP concentration C ( z, r, t ) inside a compartment is given by ∂∂t C ( z, r, t ) = D cG ∆ C ( z, r, t ) , (B1) − D cG ∂C ( z, r, t ) ∂z (cid:12)(cid:12)(cid:12) z =0 z = l = − κ h C ( z, r, t ) (cid:12)(cid:12)(cid:12) z =0 z = l + α σ ( t ) (B2) D cG ∂C ( z, r, t ) ∂r (cid:12)(cid:12) r = R = 0 (B3)The steady state expressions for the concentration C of Eq. B1 and the hydrolysis rate J h are ( G c = πR C ) C = α σ κ h , (B4) J h = 2 πR κ h C = 2 κ h l G c . (B5)Because hydrolysis and synthesis are both modelled by fluxes originating from the same boundary, theequilibrium only reflects the balance of the fluxes, and does not involve cGMP diffusion.
2. Model with boundary source term for hydrolysis, and a volume synthesis rate
To include cGMP diffusion (see section II A), we now model cGMP synthesis by a uniform volumeproduction rate α v ( t ) = 2 α σ ( t ) /l . This model avoids the problems arising when cGMP synthesis andhydrolysis are both modelled by surface fluxes originating from the same boundary. The equation for thecGMP concentration reads ∂∂t C ( z, r, t ) = D cG ∆ C ( z, r, t ) + α v ( t ) , (B6) − D cG ∂C ( z, r, t ) ∂z (cid:12)(cid:12)(cid:12) z =0 z = l = − κ h C ( z, r, t ) (cid:12)(cid:12)(cid:12) z =0 z = l (B7) D cG ∂C ( z, r, t ) ∂r (cid:12)(cid:12)(cid:12) r = R = 0 (B8)The steady state solution of Eq. B6 is C ( z ) = ˆ C zl (1 − zl ) + ˆ C β , (B9)where we introduced the parameter β and the concentration ˆ C as β = κ h lD cG , ˆ C = α v l D cG . (B10)9At equilibrium, the number of cGMP molecules G c and the hydrolysis rate J h in a compartment are G c = πR Z l C ( z ) dz = ˆ C (cid:18)
112 + 12 β (cid:19) πR l , (B11) J h = πR D cG (cid:18) dC ( z ) dz (cid:12)(cid:12) z =0 − dC ( z ) dz (cid:12)(cid:12) z =1 (cid:19) = πR lα v = D cG l (cid:18)
112 + 12 β (cid:19) − G c . (B12)Contrary to Eq. B5, the flux J h in Eq. B12 depends on cGMP diffusion constant. In the limit β → ∞ ( κ h → ∞ ) (perfectly absorbing boundaries), we obtain J h = G c /τ , where τ = l D cG is the mean time fora molecule to reach the boundaries at z = l or z = 0. On the other hand, in the limit β →
0, we obtain J h = D cG l βG c = 2 κ h l G c . Thus, we recover the expression given in Eq. B5.Eq. B12 is formally equivalent to Eq. 27, however, the physical content is very different. J h in Eq. B12is proportional to the rate by which cGMP molecules collide with the disc surfaces, which is of the order D cG /l ∼ s − . In contrast, J h in Eq. 27 is determined by the rate by which cGMP molecules find P ∗ ,given by D cG /R ∼ s − . In order to obtain β d ∼ s − from Eq. B12, one needs a small value for theadapting parameter κ h . Acknowledgments