Theoretical Model of Neurotransmitter Release during In Vivo Vesicular Exocytosis Based on a Grainy Biphasic Nano-Structuration of Chromogranins within Dense Core Matrixes

At first glance, thermodynamics might appear to require swelling of condensed polyelectrolyte matrixes when they release. Indeed, to account for electroneutrality the outward flux of neurotransmitter cations is necessarily compensated by an equal influx of monocations associated to an inflow of water molecules (probably hydrated Na⁺ and H₃O⁺ ions owing to their prevalence in the extracellular fluid) and possibly by inward ATPases-driven proton fluxes across the vesicle membrane. These hydrated monocations cannot create a dense hydrogen-bonds network with the polyanionic matrix peptidic groups as catecholamine cations do. This enforces dipole-dipole repulsions between chromogranins carboxylate moieties that must then mutually fend off to allow sufficient screening of the local dipole-dipole electrostatic repulsions. Hence, an isolated vesicle matrix should not retain its initial compacted stage during release and should swell concomitantly. This has been validated in vitro by experiments performed on isolated giant mast cells granules from beige mice whose membranes had been stripped off.^37,50–52 The same was observed during exocytosis of "naked" cortical granules from Lytechinus pictus eggs,⁵³ and may even be observed in electron microscopy (EM) micrographs in which a considerable fraction of the former matrix is literally extruded through large fusion pores.⁵⁴ However, there are several indications that under physiological conditions, i.e., when the vesicles membranes are left intact, swelling is kinetically delayed. For example, the same authors as in Reference 53 reported that under in vivo conditions giant mast cells granules from beige mice exhibited fusion pore expansions before the granules could swell to any observable extent.⁵⁵ This indicates that observations performed in vitro on "naked" vesicular granules cannot be readily generalized to physiological in vivo conditions.

Tanaka and Fillmore established theoretically that the swelling rates of spherical gels submitted to a positive osmotic pressure π₀ is given by Equation 1:³⁵

where R^{t = 0}_gel is the initial gel radius, R^{t → ∞}_gel = R_gel^{t = 0}/[1 − π₀/(3K)] the final one achieved when expansion stops because the initially applied osmotic pressure π₀ has been fully relaxed, and K the bulk modulus of the polymeric network. This prediction was fully validated by the same authors through measurements performed on a series of millimetric-sized spherical polyacrylamide gels. Interestingly, the swelling time constant τ_swell scales with the square of the final particle radius:

where μ is the shear modulus of the gel and f the friction coefficient between the gel network and the fluid medium which penetrates into it during swelling. Note that D_swell = (K + 4μ/3)/f has the dimension of a diffusion coefficient and is called the "diffusion coefficient of the gel" for this reason³⁵ though this may be misleading outside of this context. Hence, τ_swell = (R^{t → ∞}_gel)²/D_swell is formally identical to the time constant of diffusional leakage from a spherical body of identical radius when its entire outer surface is fully permeable to the diffusing particles. In fact, this formal mathematical analogy is total as evidenced by the long time limit of Equation 1 which is achieved as soon as t > τ_swell:

Indeed, introducing q_gel(t) = ω[R^{t → ∞}_gel − R_gel(t)], where ω is a suitable constant quantity with the units of moles/length (yet, see later an important caveat about such formal definition), Equation 3 results strictly identical to that giving the quantity, q(t):

of molecules or ions remaining at time t inside a fully exposed spherical body of radius R_ves initially filled with a quantity q^{t = 0} = ω(R^{t → ∞}_gel − R_gel^{t = 0}) of ions or molecules that diffuse out of it with a constant diffusion coefficient D_ves, as soon as t > τ_diff = (R_ves)²/D_ves. Note that this formal identity is true at any time even if we insist here only on the long time limits for simplicity in comparing Equations 3 and 4. However, this mathematical analogy between swelling and releasing rates is wrong and misleading because it rests on the introduction of an ad-hoc quantity, ω, that has no physical sense. Hence, any such formal identity between a swelling radius and a diffusing quantity has no physical meaning at all as is established immediately hereafter, though this has misled previous authors.³⁶

It is important to remark that under the conditions where Equation 3 applies, the local swelling "wave" performs under quasi-steady state across the whole gel,³⁵ i.e., the swelling intensity along one gel diameter becomes independent of time when normalized to that of R_gel(t) (note that the same is true for concentration profiles when Equation 4 applies). Therefore, when a swelling gel may release material only through swelling (i.e., upon assuming no diffusion limitation but that ions and molecules are released infinitely fast to only cope with the local amount of swelling), the quantity of material, q_swell(t), still contained inside the spherical gel at any time t > τ_swell is given by (see Appendix):

where R_gel(t) is given by Equation 1 or Equation 3 according to the time scale relative to τ_swell. This readily follows through remarking that the total number of unit cells, viz., filled and empty, in the gel matrix remains constant (as would chromogranin anionic moieties during exocytosis) while their local volumes expand with time due to the partial local exchange between catecholamine cations by hydrated monocations. Since Equation 3 shows that R_gel(t) approaches its limiting value R^{t → ∞}_gel in a strict single-exponential mode, its involvement to the cubic power in Equation 5 implies that q_swell(t) time-variations cannot follow a single-exponential mode. More precisely, q_swell(t) time-variations are given by a linear combination of three exponentials with perfectly related time constants: τ_swell, τ_swell/2 and τ_swell/3. This is then also to be the case for, φ_swell, the released flux of material since φ_swell = −dq_swell/dt. This cannot be reconciled with the single-exponential decay predicted for release by diffusion from a homogeneous matrix (Equation 4). Furthermore, this is in clear contradiction with most amperometric measurements that establish that descending branches of amperometric spikes (see Figure 2) follow a strict single-exponential mode decay or, as recently reported by Ewing and coll., may display a two-exponential one involving two unrelated time constants (see below). This clearly shows that swelling and diffusion cannot be simultaneous active during vesicular release. Either swelling occurs rapidly compared to diffusion or, if swelling actually occurs, this happens only after release fluxes have reached too low amplitudes to be observed by amperometry.

This conclusion, is made even stronger when remarking that the above analysis considered the most favorable situation for swelling. Indeed, Equations 1 and 3 are only valid for a "naked" matrix although in the case of vesicular exocytosis matrixes are still, at least in part, wrapped by their membranes. Tanaka and Fillmore rate law thus overestimate vesicular matrixes swelling rates through neglecting the Laplace tension-induced pressure, π_surf, exerted by the membrane onto the matrix over most part of its surface and that increases with time as detailed below. This ought to compensate the osmotic pressure π₀ driving force thus reducing the swelling trend. For small fusion pores, π₀ needs to be replaced by (π₀ − π_surf) since the membrane almost covers the whole matrix. Conversely, for large fusion pores, (π₀ − π_surf) applies only where the matrix is still covered by its membrane, so that whenever a large pore may be formed, π₀ may induce swelling over the membrane-free pore area. This may possibly lead to an extrusion of the swelling matrix section through large fusion pores as observable in some EM micrographs.⁵⁴

It is thus understood that the presence of the membrane affects swelling vs. the case of a gel as investigated by Tanaka et al. because of the role of π_surf. So, let us precisely examine the role of π_surf in limiting swelling. For this purpose we assume that at a given time swelling provokes an increase ΔR_ves of the vesicle radius R_ves(t). To account for this increased radius the surface area of the membrane vesicle needs to increase by 8πR_vesΔR_ves if the radius of the fusion pore R_pore is constant (if not, e.g., at the very beginning of release, the membrane surface area increase is 2π(4R_vesΔR_ves − R_poreΔR_pore) where ΔR_pore is the simultaneous increment of the pore radius). The duration of exocytotic release, being in a few milliseconds range, is too short to allow any biological mechanism to incorporate intracellular bilipids inside the vesicle membrane bilayer to compensate for such rapid surface increase. Since there is not any observation establishing that some of the cell membrane could slip by the pore area to compensate for this increase (in fact the rare TIRFF observations evidencing membrane flows show exactly the opposite²⁰), any ΔR_ves would increase the membrane surface tension energy by 8πγR_vesΔR_ves, where γ is the membrane surface tension coefficient. Thus, unless the vesicle membrane structure is totally ruptured or forms a high number of pores due to the surface tension building up⁵⁶ the osmotic driving pressure is rapidly counteracted by the Laplace tension pressure exerted by the vesicle membrane. Therefore, the impeded swelling accumulates a positive energy inside the matrix core while release occurs. According to Laplace law, the surface tension coefficient, γ, of the vesicle membrane increases with time proportionally to the pressure difference, ΔP(t), building between the vesicle inside due to the impeded swelling and that, ca., 1 atm, prevailing in the cytoplasm: γ(t) = R_vesΔP(t)/2. This evidences that even if a small amount of initial swelling occurs initially and provokes the rapid fusion pore expansion, it is rapidly contained by the membrane surface tension.

A second consequence of the membrane-refrained swelling is coherent with the fact that under normal physiological conditions exocytotic events appear to release significantly less than their total initial content in catecholamine.²¹ Considering that release occurs from a matrix not allowed to swell to any significant degree implies that the content of catecholamine cations inside the matrix cannot drop to too low values. Indeed, the presence of catecholamine cations stabilizes the condensed structure of the matrix (see below). Conversely, when the vesicle membrane is stripped off, swelling may occur and its whole initial content may be released.^21,36

The above established that if swelling seems a priori favored based on simple thermodynamic considerations it must be kinetically and thermodynamically hampered by the vesicle membrane while it contains the whole granule. This is certainly an important conclusion since it is predictable that changes of the membrane bilipid composition in different vesicles or over longer time scales may result in a modulation of release rates and quantities through controlling swelling upon increasing or decreasing π_surf.

The situation results entirely different for diffusion. Indeed, we established previously that fast diffusional release may easily occur even when the pore surface area is much smaller than that of the vesicle.^24,25 The fact that diffusion out of the matrix then occurs through a small fusion pore of radius R_pore = ρR_ves only results in a moderate apparent decrease of the outward-diffusion rate, i.e., from k^diff = (D_ves/R²_ves) for a fully exposed matrix to k^diff_ρ = Λ_ρ(D_ves/R²_ves), where D_ves is the cation diffusion coefficient inside the polyelectrolyte matrix and Λ_ρ is a factor that varies linearly with ρ = R_pore/R_ves when this is small.²⁴ Indeed, Λ_ρ ≈ Arcsin(ρ)/π² up to ρ ≈ 0.85, i.e., Λ_ρ ≈ ρ/π² for ρ < 0.4, a range that amply covers that of the maximal fusion pore extensions.²⁵ The descending branch of the amperometric current is then readily given by:

in full agreement with quantitative analyses of most experimental spikes^24,25,57 (sed vide infra). In other words, in drastic contrast with its huge effect on swelling, the presence of a membrane still covering most of the matrix during release just slightly slows down the rate of diffusional release.

The fact that diffusion dominates over swelling creates a delicate problem. Indeed, amperometric data for chromaffin or PC12 cells evidence that the diffusion rate D_ves/R²_ves of catecholamine cations inside the matrix must be in the order of 420 s⁻¹,^24,25 which corresponds to D_ves values in the range of 10⁻⁷ cm²s⁻¹ for vesicles with radii R_ves ranging between 100 and 150 nm. This is essential to account for the large measured release fluxes. However, diffusion coefficients of bulky ions such as catecholamine cations prone to create a dense network of hydrogen-bonds in a condensed polyelectrolyte such as exocytotic granules are expected to be rather small. More precisely, a high concentration (0.3 to 0.6 M) of catecholamine cations confined inside a spherical domain of radius, R_ves ≈ 100 − 150 nm, corresponds to a few millions of molecules.^2,21 Hence, assuming a homogeneous, viz., an isotropic distribution of catecholamine cations within the matrix points out that each cation-anion pair would occupy an average volume of 2–3 nm³. This is similar to what is expected for the global size of one peptidic anionic moiety and one catecholamine cation, i.e., should leave almost no significant free space to allow sufficiently fast diffusion. However, this difficulty does not hold if the matrix structure is not homogeneous, viz., if it contains domains with different compaction levels. This seems to contradict the general experimental knowledge based on the fact that EM investigations of prepared cell slices suggest that matrixes are rather homogeneous.⁵⁴

The homogeneous condensation of chromogranins is unlikely to prevail under in vivo conditions based on the physics of polyelectrolytes. de Gennes and coll., established the physical laws that govern the condensation of long polyanionic polymer chains such as chromogranins depends drastically on the counter-cations charges.^38,39 Briefly, the too small electrostatic energies provided by monocations is rapidly overran by the increasing negative entropy due to the dense folding of the polyanionic chain when the size of the compacted nodules increases. To relax this entropic load the polyanion-monocations ensemble adopts a pearl necklace type structure as occurs for long DNA single strands, in which small highly compacted nanodomains are separated by polyanionic strands around which monocations are less tightly bound (Figure 3). Dications lead to a similar structure except that the dense nodules size result larger due to the increase in electrostatic interactions and to the possibility of locking electrostatically topologically distant negative moieties into hairpin configurations. Conversely, trications provide sufficiently high electrostatic energies to allow a high compaction over macroscopic distances. Note that this may be the reason for the apparent homogeneity of dense core matrixes as revealed by EM since most cell slices preparations involve the use of lanthanides trications for providing a high density contrast between the matrix (hence the common name of "dense core matrixes") and the surrounding cytoplasmic components.⁵⁴ In fact, molecular dynamic simulations of chromogranins folded structures^58,59 agree with the above prediction based on the nowadays classical de Gennes' "blob" theory since they evidence a coexistence of two types of nanoscale domains with different compaction as predicted here.

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Catecholamine cations (Cat⁺) carry a single charge but their phenolic OH groups and ammonium N-H ones are prone to form a dense network of hydrogen-bonds with peptidic groups along the chain. This provides additional energy vs. simple 1:1 local electrostatic interactions. It is then presumable that they enforce a stronger compaction than that expected from hydrated monocations (e.g., Na⁺,n H₂O; H₃O⁺,n H₂O) though weaker than that enforced by dications such as Ca²⁺ and far less than trivalent counter-cations do.⁶⁰ In this respect, it is worth recalling that in addition to a high concentration of catecholamine cations, vesicle matrixes contain also non-negligible concentrations of strongly chelated calcium ions (ca. 40 mM)⁶¹ whose long term permanence inside vesicles is maintained by specific calcium channel pumps.^61–64 Though the relative amount of Ca²⁺ vs Cat⁺ is small, it is noted that calcium ions have a high affinity for hydroxyl groups. Besides the effect of Ca²⁺ dications per se (see above) they may chelate up to 3–6 Cat⁺ per Ca²⁺. If so within the matrix, this will form highly charged aggregates, Ca_nCA_nx^(2+x)n where x ≤ 6, contributing both to a higher level of compaction than Ca²⁺ alone would do and to firmly sequestrate up to a few tens percents of catecholamine cations within these highly compacted nodules. The remaining Cat⁺ should then be contained in the less tight environments formed by the less folded chromogranin chain sections.^58,59 Not only these latter are less bound to the matrix backbone than those in the dense nodules, but they should also be able to diffuse much faster across the matrix through the interconnected less compact domains (Figure 4A) than they would if the matrix structure was homogeneously compacted.

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Evidently, this model has to remain theoretical in essence up to when more physiologically compatible means of investigation at the subnanometric scale become available. However, it is firmly rooted on established fundamental physical laws of polyelectrolytes condensation. So, presently, its validity can be established only by reference to the de Gennes' theory^38,39 and through the validity of the predictions that it allows in view of the body of amperometric data present in the literature. So, let us examine what predictions can be derived from this model.

A first evident conclusion is that the existence of a highly compact polyelectrolyte nano-reservoirs allows increasing the free space around the less-bound Cat⁺ cations stored in the less compacted polyelectrolyte. Though this is not readily quantifiable presently, this is expected to allow significant diffusion rates compared to what would occur in a homogeneously compacted matrix as already perfectly observable in the outcome of molecular dynamic simulations.^58,59

A second direct consequence of this two-reservoir model is that the strongly bound Cat⁺ ions contained in the tight nodules should not be easily available to release unless there is some non-negligible kinetic exchange between the two types of reservoirs during release or if the whole matrix may partially swell before the membrane reaches a critical tension and blocks swelling. The consequences of possible kinetic exchange between the two types of reservoirs during release will be examined in a next section, so let us focus here on the second possibility. As established above, whenever the matrix membrane is removed completely or at least over a large section of the matrix surface area, swelling is not anymore impeded so that the swollen dense nodules may release their content so the whole content of the matrix may be discharged. Conversely, under physiological conditions, i.e., when release occurs through a small fusion pore connecting the vesicle inside to the extracellular medium, swelling of the dense nodule should not occur to any significant extent as established above. Hence, amperometric measurements of the vesicles load must result less by a few tens percents than the overall content present in vesicles (sed vide infra if a kinetic exchange between the two types of reservoirs happens at a sufficient rate during release). Interestingly, this is exactly what Ewing's group has observed.²¹ In a series of seminal experiments they convincingly demonstrated that under in vivo physiological conditions PC12 cells vesicles release stops (i.e., gives rise to amperometric currents, if any, that become indistinguishable from the background current) while the vesicles still contain in average between ca. 40 to 60% of their initial catecholamine content. Oppositely, release from the same PC12 vesicles was total after their membranes were stripped off.

Cat⁺ cations stored in each of the two types of nanoreservoirs differ by the strength of their attachment to the chromogranins backbone (i.e., different partition coefficients vs. the extracellular fluid) and by their mobility within their local environments (i.e., different diffusion rates). Both factors affect the rates of release into the extracellular environment. To be released, the strongly bound Cat⁺ cations need to exchange with the less bound ones, while the latter ones diffuse at significant rates along tortuous paths around the highly compacted domains (Figure 4A). As a first approximation, this duality may be represented by the following mechanism:

where q_slow and q_fast represent the time dependent quantities (moles) stored in the highly and less compacted reservoirs respectively and q_out that which has been already released in the extracellular fluid, hence been detected electrochemically. With these definitions, q_slow + q_fast + q_out = q^{t = 0}_total where q^{t = 0}_total = q_slow^{t = 0} + q^{t = 0}_fast is the initial vesicle load at t = 0. k₁, k₂ and k^diff_ρ = Λ_ρ(D_ves/R²_ves) are kinetic parameters globally equivalent to classical rate constants though they combine chemical and physicochemical components. In fact, they should be more appropriately viewed as reciprocals of the time constants featuring each of the three phenomena represented in Equation 7. Note that k₁/k₂ = q^{t = 0}_fast/q^{t = 0}_slow if the matrixes were in thermodynamic equilibrium before release started. Note that considering Ewing's reports that between ca. 40 and 60% of the catecholamine cations remain trapped within the matrix at the end of release,²¹ k₁/k₂ values are then expected to range between 2/3 and 1.5. Within this framework, the amperometric current is then simply expressed as:

In the following, we assume for simplification that the fusion pore suddenly opens at its maximal aperture R_pore = ρR_ves × Θ(t) where Θ(t) is the Heaviside step function. In other words, the following analysis considers only the descending branches of amperometric spikes. Indeed, these are the spikes currents components that essentially report about the diffusion-controlled kinetics of release. To produce full simulated current spikes the following results need to be convoluted with specific pore opening time functions, R_pore(t), as previously reported in detail.^24,25 This only alters the short-time currents, i.e., before and shortly after the spike maximum, so for the sake of simplicity and generality we do not consider this issue here.

The system in Equations 7 and 8 can readily be solved analytically or numerically to afford different spikes shapes as a function of the relative values of k₁, k₂ and k^diff_ρ as illustrated in Figure 5 for a selected choice of χ = k₁/k^ρ_diff and κ = k₁/k₂ values (note that the current and time scales are shown in dimensionless format to account for any possible values of the initial vesicle load q^{t = 0}_total or of the apparent diffusion rate k^diff_ρ).

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In fact, three main limiting situations may be observed. A first one corresponds to k₁ and k₂ being larger than k^diff_ρ so that at each moment the content of the two types of nano-reservoirs rapidly equilibrate, i.e., q_fast(t) = (k₁/k₂)q_slow(t), so that their global content is released in phase through the action of k^diff_ρ. Solving the kinetic system in Equation 7 then shows that the current is given by:

so that the total electrical charge, , of the amperometric spike is:

Hence, the whole vesicle initial content is released through a spike displaying a classical single-exponential decay branch. However, this occurs with a slower rate than predicted for release from a homogeneously loaded matrix for the same k^diff_ρ value due to the process of equilibration between the two types of nano-reservoirs. Unless k^diff_ρ may be known independently, the current variations in Equation 9 cannot be distinguished experimentally from those predicted for a fully homogeneous matrix since (k^diff_ρ)^app = k_ρ^diff/(1 + k₂/k₁). Would such behavior be observed experimentally it would then be easily mistaken for a normal one and would lead to the determination of a smaller maximal opening of the fusion pore than the actual one: (R^max_pore)^app ≈ R_pore^max/(1 + k₂/k₁) = R^max_pore/(1 + q^{t = 0}_slow/q^{t = 0}_fast). Considering the above evaluation of q^{t = 0}_slow/q_fast^{t = 0} based on Ewing's experimental results,²¹ the real maximal opening of the fusion pore would be between ca. 5/3 and 2.5 larger than that determined through a straight application of Equation 6. This would evidently lead to some ambiguity onto the precise determination of R^max_pore value. Though, this uncertainty is not large enough to obliterate our former conclusions that at their maximal expansion fusion pores radii are much smaller than those of their vesicles.²⁵ Yet, as will be made clear in the following this situation is not likely to be observed in amperometric traces.

The exactly opposite situation is met with when k₁ and k₂ are much smaller than k^diff_ρ. This prevents any kinetic exchange between the slow and fast nano-reservoirs over the duration of an amperometric spike. Then only the fast one contributes, i.e., q_slow = q^{t = 0}_slow at any time. The current again decays through a single-exponential mode, but its amplitude is smaller than expected for the total vesicle load since now release is restricted to the initial content of the fast nano-reservoir, the content of the slow one remaining trapped inside the matrix when the amperometric current returns to the baseline:

and:

Again, this situation (shown in Figure 5a) may be easily confused with that predicted by Equation 6. Indeed, unless q^{t = 0}_total is known independently, the identity between Equations 12 and 6 is total. Interestingly, this seems to exactly correspond to what Ewing and coll. reported in their recent works, i.e., that mean vesicles loads determined from the charge of amperometric spikes were statistically lower than the actual average values measured for the same vesicles when they had their membrane totally stripped off and could release their full neurotransmitter loads.²¹ Would these measurements correspond to this situation, it is inferred that q^{t = 0}_fast ≈ ηq_total^{t = 0} was determined by amperometry (with 0.4 ⩽ η ⩽ 0.6 according to Ewing's results) while q^{t = 0}_total was measured for membrane-free vesicles. Hence, Ewing's results suggest that the situation described by Equation 12 is extremely frequent.

Finally, a third limiting situation of extreme interest is observed when k₁, though small vs. k^diff_ρ is large enough for its effect not being anymore negligible during a spike duration. The fast nano-reservoirs then deplete rapidly as before but may now be slowly refilled by the slow ones. In other words, the fast nano-reservoirs empty quickly giving rise to the usual single-exponential behavior with a time constant 1/k^diff_ρ, though this adds onto a slower second single-exponential release with a time constant 1/k₁ featuring the slow but continuous kinetic refill from the slow nano-reservoirs. This is leading to a two-exponential formulation of the amperometric current:

where:

so that:

whatever is the value of Γ. This situation corresponds to the cases (b-e) shown in Figure 5. The two-exponential components are clearly visible in panels (d,e), however in (b,c) the second exponential is so slow relative to the first one that it seems to give rise to a near constant current. Its single-exponential nature would show up if the time scale had been extended up to t ≫ 1/k₁. However, in real amperometric traces, the time gap between two successive spikes is generally not sufficiently wide to allow such long monitoring of a single spike, so that spikes behaving as in (b,c) may appear as regular spikes superimposed to a constant current onto which the next spike would add. This is often observed in amperometric traces, but up to now such spikes were generally eliminated from quantitative analysis because they were thought to be normal spikes corrupted by a baseline drift. Furthermore, it is noted that the quantity released during the exponential fraction of the current is rather modest (Q_spike = 0.09nFq^{t = 0}_total for Figure 5c) so that, under experimental circumstances, such spikes would anyway be ascribed to small vesicles (compare to the small relative sizes of the corresponding current intensities compared to a normal spike shown in dashed line). Conversely, when χ = k₁/k^ρ_diff is closer to unity (Figures 5d,5e) spikes exhibit a clear two-exponential mode. Although this seems a rather frequent occurrence, i.e., concerns more than half of the spikes in an amperometric trace based on a recent systematic analysis from Ewing's group,²⁶ they were also generally eliminated from quantitative kinetic analyses. Indeed, before the present theory could justify such observations, these spikes were though to result from the accidental superimposition of two spikes, viz., a large sharp one overlaid over a small sluggish one.

The last situation shown in Figure 5f represents a totally mixed behavior in which the spike descending branch cannot be properly resolved into its two single-exponential components but Q_spike = nFq^{t = 0}_total. This occurs when k₁ and k₂ have values only slightly higher than that of k^diff_ρ. It should be noted that even if the present model predicts the possible occurrence of such situations, they seem to be extremely rare based on our own analyses of amperometric traces (chromaffin cells).² In fact, the above cited systematic quantitative analysis by Ewing and coll. (PC12 cells) evidenced that the large majority of experimental spikes displayed descending current branches that could be described either by a clear single-exponential mode or by a clear two-exponential one exception taken of those eliminated because they were considered to be corrupted by a baseline drift (see above and Figures 5b,5c). This shows a fortiori that the probability of observing release events in which k₁ and k₂ have values higher than those of k^diff_ρ is extremely low.

As a consequence, the first limiting situation giving rise to Equations 9 and 10, though theoretically possible, seems very improbable. This suggests that in an amperometric trace, most of the spikes will belong to one of the three following categories: (i) single-exponential descending branch described by Equation 11 and Q_spike = nFq^{t = 0}_fast < nFq^{t = 0}_total, i.e., when k₁ and k₂ are negligible compared to k^diff_ρ (Figure 5a); (ii) single-exponential descending branch superimposed to a near constant current plateau (Figures 5b,5c) with Q^expal_spike = nFq_fast^{t = 0} < nFq^{t = 0}_total, viz., following the limit of Equation 13 when k₁ is small but not totally negligible compared to k^diff_ρ; (iii) two single-exponential descending branch according to Equation 13 and Q_spike = nFq^{t = 0}_total (Figures 5d,5e). To the best of our knowledge these three categories encompass integrally all the different spike shapes and properties that have been observable in our laboratory or have been reported in the literature. Furthermore, since in most quantitative analyses of release kinetics categories (ii) and (iii) have been generally excluded since they are thought to be corrupted by some accidental superimposition of different spikes (iii) or by some drift of the baseline (ii) it is probable that most quantitative analyses relied exclusively on spikes of category (i). This is important to note that they then correspond only to a partial release of the initial quantity of catecholamine cations stored in the vesicle before release. This is entirely coherent with Ewing's observations that at the end of release between ca. 40 and 60% of the neurotransmitter remains trapped within the vesicle.

A significant proportion of PC12 vesicles exhibit a "halo" around their dense core matrixes. These vesicles are morphologically distinct from dense core vesicles in mast and chromaffin cells, a fact that maybe related to the ability of PC12 to continuously reload their vesicles.^65–67 Beyond the very origin of this important morphological difference, it must be noted that the observation of these halos in EM micrographs provides a strong support for the co-existence of different compaction domains of chromogranins-catecholamine cations inside of vesicles.

The formation of such halos is fully consistent with the physical laws giving rise to the present description of dense core vesicles structure.^38,39 Indeed, when vesicles are detached from the Golgi apparatuses their initial content in chromogranins is not expected to change. Conversely, when cells are equipped with mechanisms allowing the continuous loading of their exocytotic vesicles with catecholamines, mechanisms which concomitantly tailor their vesicle membranes are also active as evidenced by the larger size of vesicles equipped with halos. This is expected to adjust to the larger surface areas demanded by larger granule sizes without increasing the membranes tension (see above). Hence, in contradiction with cells such as chromaffins whose vesicles cannot be loaded with catecholamines, the slow loading processes should occur while keeping the Laplace tensions more or less constant irrespective of the enlarged vesicle size.

Would these matrixes be homogeneous before loading one would expect that the Cat⁺ excess is rapidly distributed over the whole matrix. Indeed, such homogeneous-matrixes view imposes that Cat⁺ mobilities are large enough inside the whole matrix. Would that not be the case, amperometric current intensities of significant magnitude could not be detected when these vesicles release. However, fast mobilities during release is tantamount to saying that equilibration of the extra catecholamine cations all across the matrixes should a fortiori occur during the comparatively long incubation times. It is then deduced that the matrixes will remain homogeneous while loading, hence, halos should not be observable.

Conversely, the formation of kinetically stable halos is a natural consequence of the present model. Indeed, an increase of the [Cat⁺]/[Ca²⁺] ratio while keeping the quantity of calcium ions constant⁶¹ necessarily leads to an increase of the volume of the less-bound areas through swelling within the present view of dense core vesicle structure. Based on Tanaka and Fillmore's swelling mechanism in Equations 1 and 3,³⁵ the external areas of the initial matrixes must swell faster than the zones closer to their centers where swelling is necessarily delayed kinetically by the increasingly stronger shear forces. Hence, during long incubation times, some vesicles may achieve uniform swelling over their whole volume, while others may not have yet reached this equilibration stage. These latter ones should then display what will appear as a halo around a denser core in EM micrographs (Figure 4B). Conversely, those in which swelling had time to equilibrate all over the matrix volume should appear as large dense ones in EM microphotographs. Finally, those that were not appreciably loaded have to remain small and dense as in normal cells. Interestingly these three categories are observable in EM microphotographs of cell slices after L-DOPA loading, in full coherence with the present model. Note however that when a halo is present, the model has to include an additional rather disorganized reservoir (Figure 4B), in addition to the two previous ones considered above.

For those vesicles in which a halo is formed, the mechanism in Equation 7 should then be modified to account for this fact:

where q_halo represents the time dependent quantity (moles) of catecholamine cations stored in the halo. The halo is by definition supposed to be less condensed than the fast releasing reservoir (Figure 4B) and is directly exposed to the fusion pore entrance. It is thus legitimate to assume that release is now mostly occurring through the halo, this being constantly supplied by exchange from the fast reservoir. Furthermore, for a same ρ value, (k^diff_ρ)_halo ought to be larger than k^diff_ρ considered in the above section owing to the largely disorganized structure of the halo. Finally, for the same reason one may assume for simplification that the quantities stored in the halo and in the fast reservoir are in rapid equilibrium at any time: q_halo = (k₃/k₄)q_fast. Though exchange with the slow-releasing reservoirs can be accounted analytically, for simplification we consider here that this does not occur. Following the same lines as for the derivation of Equation 9, one obtains:

and:

where q^{t = 0}_{halo + fast} represents the quantity stored jointly in the halo and the fast reservoir, i.e., the total quantity stored in the vesicle diminished by what is poised in the slow-releasing nano-grains. These results need to be contrasted with those given above in Equations 11 and 12. It immediately follows that the overall released quantity (Equation 18 vs. Equation 12) is increased by the presence of the halo, in full agreement with the experimental observations. The relative magnitude of current intensities is more delicate to predict ex nihilo. Indeed, they depend on the global ratio:

in which the first two terms are larger than unity but their effect is modulated by the value of (1 + k₄/k₃) that is unknown a priori.

However, this problem may be solved experimentally. Indeed, another interesting observation from Ewing's group and ours is relative to the changes incurred by vesicles of PC12 cells supplemented with a L-DOPA enriched diet. Under such conditions, the natural vesicles "refilling" mechanisms and "membrane-tailoring" slowly allows incorporating a large excess of catecholamine cations inside doped vesicles. This has two experimentally observable effects. One is the formation of larger vesicles with much larger "halos" than in controls.^65–67 The second is that amperometric currents intensities increase by ca. 250% and that the total charge of amperometric spikes also increases by ca. 250%.^65–66 However, despite these extremely important changes, the time constants of the current spikes descending branches do not vary significantly for L-DOPA-loaded vesicles compared to controls. This suggests that even if (k^diff_ρ)_halo/k^diff_ρ is larger than unity, this gain is more or less compensated by the decrease associated to the presence of the factor 1/(1 + k₄/k₃) in the time constant in Equation 17. Hence, [(k^diff_ρ)_halo/k^diff_ρ]/(1 + k₄/k₃) has to be close from unity. This suggests that the increase in current magnitude (Equation 19) has to track approximately that of the released quantity (Equation 18). Although this conclusion is only based on a comparison of time-constants between L-DOPA-loaded vesicles compared to controls it is fully coherent with experimental amperometric observations on PC12 cells since both the current intensity and the released charged increased by ca 250% upon incubation of the cells with L-DOPA.

Finally, if the catecholamine cation kinetics of exchange between the slow and fast reservoirs is sufficient to contribute to slowly refill the fast reservoir while this one empties, Equation 17 should be modified to account for this slow release component. Hence, the current is predicted to obey a composite equation similar to that in Equation 13 and must display as well a two-exponential decay mode. Since no experimental data have been yet reported about such double-exponential behavior for L-DOPA loaded vesicles we did not wish to elaborate further on this possibility. However, it is most certainly highly probable since in vesicles with halos k₁ values in Equation 16 should be at least of the same order as in controls (Equation 7) or possibly faster owing to the partial disorganization induced by the partial swelling leading to halo formation.