All mouse experiments were approved and performed in accordance with the guidelines and regulations by the local authority, the Lower Saxony State Office for Consumer Protection and Food Safety (Niedersächsisches Landesamt für Verbraucherschutz und Lebensmittelsicherheit), and executed at the Georg August University, Göttingen. Briefly, all animals had free access to the shelter and to water and food, and every effort was made to minimize animal suffering and the number of animals used. For removal of tissues, the animals were deeply anesthetized by CO₂ inhalation at a fixed concentration and rapidly killed by cervical dislocation.

Cultured hippocampal neurons were obtained from neonatal mice (NMRI, P3–P6), as described previously (Mironov, 2015). During the experiments, the coverslips with cells were mounted in a recording chamber that was continuously superfused at 34°C.

The electrophysiology and imaging in cultured hippocampal neurons are performed as described previously (Mironov, 2015; Mironov, 2019). Bath and pipette solutions contain 30 mM Tris buffer (pH 7.4) and 154 mM NaCl or 88 mM CaCl₂. The solutions have an osmolality of 305–315 mosmol/l. Fluo-4 and Fluo-4 dextran (M.W. 30 kDa) were obtained from Thermo Fisher (Osterode, Germany). Both dyes were used at a concentration of 0.3 mM. Fluo-4 dextran was dissolved at 1% concentration. Dextran (10 kDa) has a diffusion coefficient of D = 15 μm²/s (Shuai and Parker, 2005), which is close to the measured value for calmodulin, D = 10 μm²/s (Sanabria et al., 2008). Given D _Ca = 200 μm²/s, both values are close to the relative diffusion coefficients d = 0.03 (Mironova and Mironov, 2008) used in this study (Table 1).

Patch electrodes were pulled from borosilicate glass (1.4 mm o. d., WPI, Berlin, Germany). The pipettes have a small bore (<0.1 µm) and a long shank (≈5 mm), with a resistance of ≈40 MOhm. Coverslips with the hippocampal neurons are placed on the microscope stage. The cells are patched, inside-out patches are excised, and synuclein (α-Syn) channels are forced to incorporate into the membrane (Mironov, 2015). The horizontally oriented pipette is then lowered to the bottom to enter the TIRF illumination layer (≈200 nm above the coverslip).

For imaging, an upright microscope (Axioscope 2, Zeiss) is used with a ×63 objective lens (N. A. 1.4). The fluorescence is excited by 488 nm light from an SLM Diodenlaser (Soliton, Gilching, Germany). The laser beam is delivered from below through a prism (TIRF Labs, Inc. Cary NC) at an angle appropriate to evoke TIRF. The emission (525 nm) passes through a dichroic mirror filter at 535 nm. The images are captured by using a cooled CCD camera, iXON Ultra 888 EMCCD (ANDOR, Oxford Instruments), operated under ANDOR software. The line-scan mode (512 × 2 pixels at 12-bit resolution) has an acquisition time of 0.1 ms. Because the fluorescence of fluo-4 increases >10-fold after Ca²⁺ binding, this minimizes out-of-focus effects that appear in imaging of living cells. The cell-free experimental configuration also provides good control of the solutions bathing the patch.

Single-channel events were recorded in 12 patches obtained from different neurons with the pipettes filled with Fluo-4-dextran solution. Eight patches with synuclein channels were recorded with Fluo-4 in the pipette. The episodes of channel opening were collected at different holding potentials. They were analyzed offline to examine the dependence of bound Ca²⁺ distributions upon single-channel Ca²⁺ flux and its duration.

To start with, consider Ca²⁺ diffusion from a single channel in the presence of a single buffer as a system of two partial differential equations (PDE) ∂ Ca 2 + / ∂ t   = D C a Δ Ca 2 + –   k o n   Ca 2 + F + k o f f B ,   ∂ F / ∂ t = D b Δ F   –   k o n   Ca 2 + F + k o f f B . (1) Here, Δ stands for the Laplacian; [Ca²⁺], [F], and [B] are the concentrations of calcium and buffer in free and Ca²⁺-bound forms, respectively; k _on and k _off are the rate constants of calcium-binding to and dissociation from the buffer. The ratio k _off/ k _on = K _d is the dissociation constant of the buffer and measures its affinity to calcium. D _Ca and D _b are the diffusion coefficients for calcium and buffer, respectively. For simplicity, the equations are set for a single buffer, but generalization for multiple buffers is straightforward, as discussed below. The definitions and typical parameters are summarized in Table 1.

In the following derivations, all concentrations are normalized by dividing by the total buffer concentration B _o . Equation 1 is then written as c t = Δ c   –   c f + γ 1   −   f , (2a) f t = d Δ f   −   c f + γ 1   –   f , (2b) where the subscript denotes the time derivative, c = [Ca²⁺]/B _o and f = [F]/B _o are the normalized concentrations of free calcium and free buffer, respectively; d = D _b /D _Ca is the relative diffusion coefficient, and γ = k _off /k _on B _o = K _d /B _o = 0.002 is the normalized dissociation constant. The last term on the right-hand side in Eq. 2a describes Ca²⁺ dissociation, and b = 1–f is the normalized concentration of the Ca²⁺-bound buffer. The dimensionless times and distances are defined as τ = 1/k _on B _o = 50 μs and r _o = √D _Ca /B _o k _on = 0.1 μm, respectively. These intrinsic constants of Ca²⁺-buffering (Table 1) are calculated for diffusion-limited k _on = 10⁸ M^-1s^-1 and B _o = 0.2 mM, typical values for cytoplasmic calcium-binding proteins, and D _Ca = 200 μm²/s.

The boundary condition is specified by assuming the constant flux through the open channel c r = − i / 2 π z R 2 D C a F , (3) where i is a single-channel current, F is the Faraday constant, and R is the exit radius (R = 0.5 nm, (Malasics et al., 2009)). For radial diffusion, it is useful to transform c = u/r, which replaces the Laplacian Δ = 1/r ² d (r ² dc/dr) with the second derivative Δ = d ² u/dr ² . The boundary condition transforms into R u r   –   u   ≈   − u = − A R , (4) where A is the Ca²⁺ concentration at the channel lumen. Because R is very small, the first term can be neglected. For i = 0.1 pA, the constant is A = 0.7 mM.

The linearized version (EBA) assumes a fast irreversible binding of Ca²⁺ to buffer. This sets f = 1 and γ = 0 in Eq. 2a and leads to a simple ordinary differential equation (ODE) of the second order Δ c = c , (5) which has an exponential–hyperbolic solution c = A / r exp − r / r o , (6)

The solution of the transcendental Equation 6 is given by the Lambert function defined as W = exp (-W) (Olver et al., 2011). The reference point may be r = 0.57 r _o for c = A. In the 1D case, the 1/r factor is absent, and r = 0.37r _o .

The PDEs derived in the text below are solved with Du Fort–Frankel numerical integration (Du Fort and Frankel, 1953). Its performance is better than that of the Crank–Nicolson algorithm and unconditionally stable. For discretization of diffusion equation c _t = Dc _xx , the concentrations at times t _n = nΔt and space points x _m = mΔx are needed. The following parameters are calculated d 1 = 1   −   α / 1 + α , d 2 = α / 1 + α , with  α = 2 D Δ t / Δ x 2 , (7) and the reaction–diffusion (RD) equation is then solved explicitly at each mesh point as c m n + 1 = d 1   c m n − 1 + d 2   c m + 1 n + c m − 1 n −   Δ t   c m n   f m n . (8)

The last term is the reaction term (-cf) specified by respective terms in the PDEs derived below.

Ca²⁺ diffusion in the presence of a single immobile buffer is described by the PDE system as c t = Δ c   –   c f , (9a) f t = − c f   . (9b)

For the initial concentration of free buffer f _o = 1, integration of the second equation gives f = f o e − y , (10) where a new variable, y = ∫ c d t , (11) the integral of Ca²⁺ concentration, is introduced. Integration of Eq. 9a then gives a non-linear PDE y t = Δ y   −   1 + e − y , (12) with the boundary condition y _R = tc _R . Equation 12 has no general solution even in the 1D case except a self-similar particular solution in moving coordinates z = ax–bt (Polyanin and Zaitsev, 2004).

Eq. 12 (Smith et al., 2001) was solved numerically using the Du Fort–Frankel algorithm described in above. Figure 1 shows the calculated radial (3D) and 1D profiles for free Ca²⁺ and Ca²⁺-bound buffers for different Ca²⁺ fluxes. They are coded as relative Ca²⁺ levels at channel lumen A = [Ca²⁺]/B _o . It is seen that [Ca²⁺] decays rapidly from the entry site. In the 3D case, [Ca²⁺] is very big due to the hyperbolic factor 1/r, intrinsic in radial symmetry, Eq. 6 (Mironov, 2019). Large free Ca²⁺ values strongly violate the EBA assumption that [Ca²⁺]<<B _o . The spatial profiles of the Ca²⁺-bound buffer expand with time as traveling waves. They appear already for a very small relative Ca²⁺ flux A = 0.01 and widen with increasing A (Figure 1). The increment between subsequent curves decreases with time, indicating a decrease in wave speed.

A simple analytical argument helps understand the appearance of traveling waves. Free Ca²⁺ approach fast exponential–hyperbolic steady-state solution given by Eq. 4 in EBA. Under such conditions, the bound Ca²⁺ is obtained from Eq. 9b as Ca 2 + − b o u n d = b = 1 −   f = 1   − exp − A t / r o . exp − r / r o . (13)

The expression describes the traveling wave spreading out according to the time constant proportional to the Ca²⁺ level at channel lumen, A.

Another interesting feature emerges when we identify, for a moment, the bound buffer with an immobile Ca²⁺-sensor (McMahon and Jackson, 2018). The temporal and spatial patterns of sensor activation should then be given by the profiles for the Ca²⁺-bound buffer, as shown in Figure 1. From this, one may deduce that a crucial factor in the Ca²⁺-mediated functional response may be the time when a sensor at a given location is saturated with Ca²⁺. A suggestion with such a role of bound Ca²⁺ underlines a statement that “messages diffuse faster than messengers” (Pando et al., 2006) coined by simulating the diffusion of particles in the presence of traps as applied for GFP-tagged glucocorticoid receptors in the nuclei of mouse adenocarcinoma cells. The mathematical formulation of the problem resembles that considered in this study.

Both the 3D and 1D cases of Ca²⁺ diffusion are physiologically relevant because they present Ca²⁺ diffusion defined by specific geometric constraints. For radial diffusion, Ca²⁺ spreads out in a semi-infinite cytoplasm free from intracellular organelles of macromolecular aggregates that may hinder diffusion. 1D diffusion may imitate Ca²⁺ diffusion through a tortuous pathway that extends from the point of Ca²⁺ entry to the sensor. A well-known 1D example is the Ca²⁺ waves in neuronal dendrites (Augustine et al., 2003; Mironov, 2008). Of note, in linear models, the 3D and 1D solutions are related: the former is obtained from the latter by simple division of the solution by the distance from the channel lumen. The application of such transformation is shown in Supplementary Figure SB.

A general conclusion from this section is that bound Ca²⁺ distribution in the presence of an immobile buffer is dynamic rather than static. A simple time-dependent model shows inherent contradictions with the linear EBA model: (i) free buffer is not in excess even for small Ca²⁺ fluxes (i < 0.1 pA), (ii) not constant, and (iii) shows the time- and space-dependent changes in bound Ca²⁺ that mirror buffer depletion. The conclusions are also valid for low-mobility buffers, as examined in the following section.

The results in the previous section are extended below to the case of a slowly moving buffer. Eq. (2) is now given as c t = Δ c   –   c f , f t = d .   Δ f   −   c f . (14)

The system was solved in Supplementary Appendix SA1. The calculations presented in Figure 2 were made for calmodulin, the relative diffusion coefficient of which is set to d = 0.03. This is compatible with a relative diffusion coefficient d = 0.07 for an endogenous “immobile” buffer in chromaffin cells (Neher and Augustine, 1992). The radial and linear Ca²⁺ profiles resemble the data obtained for the immobile buffer (the dots in Figure 2). A close relationship validates expansion on a small parameter d applied in deriving the solution of Eq. B1 in Supplementary Appendix SA1. It also indicates that immobile buffer approximation serves as a good approximation for slowly moving buffer.

Free Ca²⁺ approached the steady state quickly, but the bound Ca²⁺ was never stationary. [CaB] waves became denser with time. The speed of changes in free and bound Ca²⁺ changes with time and can be characterized by the instantaneous velocity. This was estimated as the x-distance between the half-amplitude points divided by the time increment between the respective curves. For linear diffusion with small Ca²⁺ flux (A = 0.1, the right panels in Figure 2), it subsided from 56 to 3 μm/s, as estimated at 0.1 and 5 ms after the pulse. The velocity changed at the same time points from 78 to 18 μm/s for bigger Ca²⁺ fluxes (A = 1). In contrast, the bound Ca²⁺ “moved” much faster. For A = 0.1, the velocities at 0.1 and 5 ms after beginning the pulse were 84 and 46 μm/s, and for A = 1, they were 107 and 85 μm/s, respectively. For the radial diffusion (Figure 2, the left panels), the values are around 40% smaller. This can be explained by the role of the damping hyperbolic factor (1/r). The 1D values are in the range for Ca²⁺ waves measured in neuronal dendrites (Augustine et al., 2003; Mironov, 2008). The data thus demonstrate that the bound Ca²⁺ apparently “moves” much faster than free Ca²⁺ and overtakes it within a few milliseconds after switching on Ca²⁺ influx.

In order to test theoretical predictions, I sought for experimental proof-of-principle and aimed to obtain the 1D profiles of the Ca²⁺-bound buffer. They were imaged after opening of single channels, and bound Ca²⁺ was monitored within a thin pipette filled with fluo-4(X) indicator dyes (inset in Figure 3A). α-Synuclein forms in the plasma membrane of hippocampal neurons’ Ca²⁺-permeable channels (Mironov, 2015). A single channel in the inside-out patch influx was used to generate controllable “outward” Ca²⁺ influx into the patch pipette. The data shown are representative of recordings made in 12 patches for fluo-4-dextran and in eight patches for fluo-4.

This 1D experimental system for examination of CaB waves has several advantages: (i) the number of pixels to collect is equal to N, not to N ³ as in the 3D case, which allows faster measurements; (ii) 1D profiles are more extended because they do not include a “hyperbolic factor”, 1/r, a distance from the source (Figures 1, 2); (iii) the openings are longer than generated by voltage-dependent Ca²⁺ channels, which open normally less than for a couple of milliseconds, which allows more room for the waves to develop.

The channels in the inside-out patch show stable activity for about 20 min. The patch was held at 0 mV, where the current was 0, and the potential was periodically shifted for 1 s. Between 10 and 20 openings are observed during the pulse (Figure 3A). Changes in fluorescence in response to channel opening were stereotypic and depended only on opening duration and single-channel current. The distribution of bound Ca²⁺ expanded with time and formed a traveling wave (Figure 3B). After returning to the closed state, the fluorescence decreased fast due to Ca²⁺ diffusion and reaction with free buffer molecules. In the 1D case, the diffusional time constant can be estimated as x ²/2D _Ca = 1 μm²/(2.200 μm²/s) = 2.5 ms. When only 10% free buffer is available, the reaction time constant is small, 1/k _on B _o = 1/(2 × 10^−5 M) (10⁸ M^-1s^-1) = 0.5 ms, which promotes fast decay of bound [Ca²⁺]. The vertical cross-sections of the kymograph in the rightmost panel in Figure 3B show the relative fluorescence in time and space and indicate a spread out of bound Ca²⁺. Experimental traces are overlaid upon thick theoretical curves calculated using the model described above and plotted in Figure 2. Both sets of data show a good agreement.

The experiments were also carried out with Fluo-4-filled pipettes. The distribution of bound Ca²⁺ was narrower and quickly attained the steady state. This is expected as Fluo-4 has a diffusion coefficient of approximately 100 μm²/s (Mironov, 2019), which is close to that of Ca²⁺. These experiments may be treated within the EBA framework by setting the relative diffusion coefficient in Eq. (2) to d = 1. The solution is obtained in Supplementary Appendix SB1 and used to plot the thick traces in Figure 3C. The theoretical predictions show good correspondence with experimental traces.