Co-localization and confinement of diphosphohydrolases and ecto-nucleotidases modulate extracellular adenosine nucleotide pools

Nucleotides comprise small molecules that perform critical signaling and energetic roles in biological systems. Of these, the concentrations of adenosine and its derivatives, including adenosine tri-, di-, and mono-phosphate are dynamically controlled in the extracellular-space by diphosphohydrolases and ecto-nucleotidases that rapidly degrade such nucleotides. In many instances, the close coupling between cells such as those in synaptic junctions yields tiny extracellular ‘nanodomains’, within which the charged nucleotides interact with densely-packed membranes and biomolecules. While the contributions of electrostatic and steric interactions within such nanodomains are known to shape diffusion-limited reaction rates, less is understood about how these factors control the kinetics of sequentially-coupled diphosphohydrolase/nucleotidase-catalyzed reactions. To rank the relative importance of these factors, we utilize reaction-diffusion numerical simulations to systematically probe coupled enzyme activity in narrow junctions. We perform these simulations in nanoscale geometries representative of narrow extracellular compartments, within which we localize sequentially- and spatially-coupled enzymes. These enzymes catalyze the conversion of a representative charged substrate such as adenosine triphosphate (ATP) into substrates with different net charges, such as adenosine monophosphate (AMP) and adenosine (Ado). Our modeling approach considers electrostatic interactions of diffusing, charged substrates with extracellular membranes, and coupled enzymes. With this model, we find that 1) Reaction rates exhibited confinement effects, namely reduced reaction rates relative to bulk, that were most pronounced when the enzyme was close to the pore size and 2) The presence of charge on the pore boundary further tunes reaction rates by controlling the pooling of substrate near the reactive protein akin to ions near trans-membrane proteins. These findings suggest how remarkable reaction efficiencies of coupled enzymatic processes can be supported in charged and spatially-confined volumes of extracellular spaces.


Figure 1:
Left) Schematic of the synapse-like junctional space formed between the membranes of adjacent cells. Nucleotidases confined within the junctional space hydrolyze ATP into AMP and Ado. Right) A model geometry based on the schematic, for which the reservoirs correspond to the non-junctional space. The spatial and electrostatic configuration of the mock synapse influence the reactivity of confined nucleotidases CD39 and CD78. .
Within the pore we consider two sequential, CD39-and CD73-catalyzed ATP and AMP hydrolysis reactions that are in steady-state. With this model, we 152 examine how enzyme co-localization, 'tethering' the enzymes to the pore wall, 153 and charges on the enzyme and pore surfaces shape enzyme kinetics within the 154 idealized pore volume. Although we assume that the enzymes are spherical 155 with uniform reactivity and charge, we have found that such representations 156 are reasonable approximations of structurally-detailed, non-uniformly charged We first validate our model against an analytical solution for the diffusion limited reaction rate coefficient on a uniformly reactive sphere. [54] Here, the association rate, k on , for the reactive sphere embedded in an infinite domain is given by where R is the radius of the enzyme and D is the substrate diffusion coeffi-185 cient. For the purpose of validation, we evaluate this rate at the sphere (0.5 186 nm radius) by assuming a uniform concentration for ATP (1.0 mM) at both the 187 reservoir and pore (11.5 nm diameter). We will later assume no-flux (reflective) 188 boundary conditions for the concentration along the pore to emulate a typical 189 non-reactive pore boundary. Under the aforementioned conditions, we numer-sults using reflective pore boundaries (see Table 1), in Fig. 2 we demonstrate a 197 numerical prediction of k on,AT P =1.151 × 10 −3 nm 3 ns −1 , which is within 1.8% 198 of the analytical result, k smol,local =1.13 × 10 −3 nm 3 ns −1 , for which we used the    Fig. 2) we present normalized association rates 207 for ATP with enzyme CD39,k on ≡ k on,A /k on,Bulk , subject to a constant en-208 zyme radius (r E = 1.0 nm) and varied pore diameters (r E ≈ r p with r p =1.3 nm 209 , r E < r p with r p =3.0 nm, and r E r p with r p =5.5 nm). Confinement of 210 the enzyme to the pore reduced the reaction rate coefficient by roughly 70% 211 relative to the corresponding rate in bulk (k on = 1). This can be qualitatively 212 rationalized by the concentration profiles manifest in the channel (see Fig. S1).

213
The concentration profile decreases from A=6.0 × 10 −4 nm −3 at the right-hand 214 side reservoir (Γ R ) and approaches zero at the left-hand side reservoir (Γ L ). As 215 the pore radius decreases from r p r E to r p > r E , the concentration of ATP 216 within the pore decreased relative to the reservoir. Hence, pore confinement 217 in essence reduces the substrate concentration at the enzyme surface, which 218 culminates in a reduced k on,AT P . 219 Figure 5: ATP, AMP and Ado concentrations at midpoint between enzymes for different pore wall boundary conditions. CD39 and CD73 are separated by a distance of 6.0 nm. . We additionally varied the proximity of the enzyme to the pore surface. 220 This variation serves as a proxy for probing the reactivity of enzymes that are 221 essentially free floating within the pore interior versus immobilized to the pore 222 surface. The reactivity of CD39 was additionally reduced, albeit negligibly, as 223 CD39 was localized to the pore surface. This can be rationalized by noting 224 the similarity between the time-independent diffusion equation and the Laplace 225 equation commonly used in electrostatics (see Eq. 15 with κ = 0.). 226 The total electric flux is dependent on the capacitance; as the sphere ap-227 proaches an insulator wall (J · n = 0), the capacitance decreases [55], which 228 decreases the total electric flux. The total electric flux is the electrostatic equiv-229 alent of the concentration flux in substrate diffusion, hence, the numerically-230 estimated k on,AT P values were smaller for immobilized enzymes relative to those 231 far from the surface. Altogether, these results demonstrate that restricting the 232 diffusion of ATP within the pore and to a slightly greater extent, near the pore 233 wall, suppress k on,AT P relative the the bulk.

234
Reduction of k on,AT P through enzyme confinement of enzymes is expected 235 to subsequently suppress production rates for AMP and Ado . However, co-236 localization of enzymes within 'nano reactors' is a common approach to tune 237 production rates of desired chemical products [44,46,56]. We therefore intro-238 duced a second enzyme, CD73, into the pore and simulated the steady state 239 reactions AT P → AM P at CD39 and AM P → Ado at CD73. In Fig. 3 To delineate the effects of pore confinement and enzyme colocalization on 260 k on,AM P and k prod,Ado independent of k on,AT P , we report in Fig. 4)   [60], we used charges of -2, -1 and 0 for ATP, AMP and Ado, respectively, 307 to exemplify effects on reactivity. Under these conditions, k on,AT P decreases 308 as the pore radius to enzyme radius decreases as was observed for the neutral 309 system (see Fig. 6). Importantly, k on,AT P for the charged system does assume 310 a higher rate coefficient than the neutral system. We next imposed a negative electric potential on the pore surface and 318 present the resulting reaction rate coefficients (red in Fig. 7) . We chose sur- . We first examine effects of the pore 323 electric potential, Φ pore , on reaction kinetics, assuming CD73 is uncharged.

324
In Fig. 7 we demonstrate that in general k on,AT P monotonically decreases re-325 gardless of the membrane charge. In the event that the pore interactions with 326 substrate ATP are repulsive (Φ p < 0), the reaction rate coefficient decreases at 327 a faster rate. However, in certain regimes the charge complementarity of the 328 pore surface was found to greatly accelerate k on,AT P relative to the neutral pore, 329 whereas a repulsive pore (Φ pore < 0, blue) attenuated k on,AT P by roughly 17% 330 or 3.77 × 10 −1 nm 3 ns −1 (see Fig. 6) for r p ∼ 6r E ). We attribute the enhanced  Interestingly, attractive pore/ATP interactions initially accelerate k on,AT P 339 as the pore diameter is reduced, whereafter the rate declines. We find the 340 maximal acceleration is achieved when the pore size is roughly six-fold higher 341 than the enzyme radius (see Fig. S3). This maximum is dependent on the wall 342 potential amplitude, namely as the attractive wall potential amplitude increases,

354
We initially anticipated that positioning the protein directly adjacent to the 355 membrane would improve the reactivity relative to the pore center. Further, we 356 observe that k on,AT P can be amplified when the enzymes are tethered to the 357 pore surface under specific conditions, namely wide pores and strong attraction, 358 but this advantage is generally minor and thus of limited consequence to NDAs 359 (Fig. S6). We found little difference for k on,AT P at modest (< |25| mV) pore 360 potentials for far in Fig. S7). This appears to be consequence of reduced access 361 to the enzyme as it approaches the wall, which counterbalances the increased the 362 concentration of ATP near the surface due to attractive electrostatic infractions.

363
Electrostatic enhancement of k on,AT P is generally expected to promote k on,AM P and 364 k prod,Ado , thus we examined the extents to which the intermediate species' k prod,Ado increases as the enzymes are brought into close proximity (d E1E2 → 0).

370
As observed in the preceding section, the absorbing pore boundaries show the 371 greatest sensitivity to enzyme distance, with favorable AMP /CD73 electrostatic 372 interactions for Φ CD73 < 0 yielding faster k prod,Ado reaction rate coefficients relative to neutral CD73, and conversely slower rates for positively-charged CD73.

374
The enhancement in the former case reflects both the electrostatic attraction 375 of substrate AMP toward CD73, while ATP is electrostatically repelled toward 376 CD39, which culminate in increased k on,AM P and k prod,Ado , respectively. In the 377 latter case, the positively charged CD73 repels AMP and attracts the negatively- Effects of charge Effect of charge sign composition on reactivity, given z AT P =-1, z AM P =+1, z Ado =0 and Φ CD39 > 0 Effect of pore electrical potential on reactivity of enzyme for different sizes of the pore. We define an effective pore radius for charged pores, to be obtained from the green dashed line. For relatively small pores, the effective pore radius is larger for attractive pores, and smaller for repulsive pores. For relatively larger pores, competition between the pore wall and the enzyme becomes more significant, leading to a decline in the reactivity of attractive pores (see Fig. S3 in which with decreasing the size of enzyme we allow larger relative sizes of pore to enzyme radius). .

Figure 7:
Effects of charge Effect of charge sign composition on reactivity, given z AT P =-2, z AM P =-1, z Ado =0 and Φ CD39 > 0 Normalized reaction rate coefficient for production of Ado as a function of distance between enzymes. .

Figure 8:
Effects of charge Efficiency of sequential enzymes for different pore wall electric potentials, given z AT P =-2, z AM P =-1, z Ado =0 and Φ CD39 > 0. Corresponding values for k on,AT P are provided in Fig. S7 .
In the previous section, we highlighted reaction rate coefficients and efficiencies without accounting for electrostatic screening by common electrolytes. To model physiological conditions characterized by roughly 100 millimolar monovalent ion concentrations, we solved the linearized Poisson-Boltzmann equation, assuming Debye lengths on the order of 1 nm. This Debye length signifies that electrostatic interactions are significantly screened, which will in turn modulate reaction kinetics. To assess effects on k on,AT P , it is helpful to compare rates as a function of (1 + aκ), where a is the enzyme radius and κ is the inverse Debye length (see Fig. 9). This functional form is motivated by the relationship where I is the ionic strength, while A and B are generally fitting parameters Effect of Debye Length on reactivity of sequential enzymes: a) Efficiency and first enzyme reactivity for large and small Debye lengths for co-localized and separated enzymes for Φ CD39 > 0, Φ p > 0 and Φ CD73 < 0 z AT P =-1 and z AM P =1 b)Relationship between ln(kon) and (1 + aκ) −1 for three different pore electrostatic potentials. Subpanel shows fitting of curve to simulation data points. .  To systematically probe the potential for these factors to impact NDA activ-   in turn increased k prod,Ado . In contrast, efficiency was strongly reduced when 583 nucleotides were depleted at the surface (absorbing), as might be expected for 584 significant nucleotide uptake by plasma membrane adenine nucleotide translo-585 cases [66]. As discussed in the next section, this reduced efficiency for absorbing 586 membranes could be countered by co-localizing the two-enzymes to favor AMP 587 's reaction on CD73 relative to diffusing toward the membrane Ultimately, these 588 findings suggest that nucleotide pools capable of activating targets such as ADP 589 sensitive P2Y channels will be strongly regulated by the relative activity of pro-

597
We had thus expected that co-localizing NDAs within junctions would improve 598 reaction efficiency. However, we found that close spatial coupling was advanta-599 geous only when the junction membrane significantly interacted with the inter-

Limitations and Future directions
In order to work with the system that was numerically solvable, we made several 704 assumptions. Firstly, we assumed all enzymatic reactions were fast compared to 705 the diffusion of nucleotides between reactive centers. NDAs are known to rapidly 706 manage nucleotide pools with reaction rates on the order of 1 µM s −1 [31]. Since 707 the intrinsic reaction rates of these enzymes vary quite considerably depending 708 on the isoform and cell type, we assumed reaction-limited conditions for simplic-709 ity and generality. It may also be appropriate to consider feedback inhibition, 710 given evidence that productions can hinder NDA-catalyzed AMP hydrolysis. 711 We additionally assumed spherical shapes for the proteins; while this may seem   The enzymes are therefore approximated as spheres.

820
Three species of substrate are included: A, AMP , and Ado . ATP is converted to the AMP product when it encounters the surface of CD39, followed by AMP's conversion to adenosine on enzyme CD78: We define the concentration of a given species S as c S , which is an unknown Under steady state conditions the concentration of species S does not vary in time, and so the governing differential equation is: To reduce the computational burden, an alternate form of the Smoluchowski 834 equation is used. The substrate flux is expressed as: The equivalence of these two expressions for the flux can be readily verified 836 using the product rule for gradients.   We also define the integrated flux over any surface Γ as wheren is the unit normal to the surface Γ.

843
The reaction kinetics at an enzyme are assumed to follow a simple rate law. For the reactions in Equation 5, the rate laws are given by which defines reaction rate coefficients k CD39 = k on,AT P = k prod,AM P and 844 k CD73 = k on,AM P = k prod,Ado .  Table 1.

853
Boundary Surface Γ CD39 Γ CD73 Γ R Γ L Substrate ATP c AT P = 0 j AT P = 0 c AT P = C 0 c AT P = 0 Substrate ADP j AM P = −j AT P c AM P = 0 c AM P = 0 c AM P = 0 Substrate Ado j Ado = 0 j Ado = −j AM P c Ado = 0 c Ado = 0

861
The system of partial differential equations and boundary conditions de-      Effective pore radius for larger relative size of the pore to enzyme radius. . Figure S4: Effects of charge Effect of charge sign composition on reactivity, given z AT P =-1, z AM P =+1, z Ado =0 and Φ CD39 > 0 Normalized reaction rate coefficient for production of Ado as a function of distance between enzymes. .

Figure S5:
Effects of charge Efficiency of sequential enzymes for different pore wall electric potentials, given z AT P =-1, z AM P =+1, z Ado =0 and Φ CD39 > 0 . Figure S6: For large potential of the pore in comparison with enzyme potential, as the enzyme get close to the pore membrane, due to the attraction between pore membrane and substrate ATP the concentration of ATP is more and so it leads to an increase to the reactivity of enzyme. However, when the enzyme is getting too closer to the pore membrane, the competition between pore membrane and enzyme , especially for very large potential on membrane, is increasing, leading to a sudden decrease in reactivity.This obtained for kappa=1, and Dp=5DE. .

Figure S7:
Coloc/tethering effects on keff. These data obtained for the case which maximize the efficiency with V E1 > 0 , V E2 < 0, Dp=8 and Vp=25mV. The konA does not change when enzymes get close to the wall for all different charges of the pore wall. However, I have tried a case when Dp=11, kappa=1 and Vp=100 and saw that the konA increase as the enzymes get close tho the wall for attractive case between species ATP and pore wall (here positive), although the difference is not considerable(around 1.5 percent change). There is two factors affecting konA when it get close to the wall: The capacitance change when the enzyme get close to the wall. However, the concentration of ATP species is more near the wall of the pore. . Figure S8: Effect of Debye Length on reactivity of sequential enzyme: a) Reactivity of the first enzyme and second enzyme as a function of pore to enzyme distance for electrostatical compositions with maximum (red) and minimum (blue) reactivity along with the efficiency (black) c) b)effective pore radius based on potential of the pore and Debye length. The green curve shows the data for uncharged reactivity of the first enzyme as a function of real physical size. based on the data we obtained from charged simulation, we can match an effective pore size which for attractive is bigger and for repulsive is less than its real size. .