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It has long been speculated that cellular microdomains are important for many cellular processes, especially those involving Ca2+ signalling. Measurements of cytosolic Ca2+ report maximum concentrations of less than few micromolar, yet several cytosolic enzymes require concentrations of more than 20 μM Ca2+ to be activated. In this paper, we have resolved this apparent paradox by showing that the surface topology of cells represents an important and hitherto unrecognized feature for generating microdomains of high Ca2+ in cells. We show that whereas the standard modeling assumption of a smooth cell surface predicts only moderate localized effects, the more realistic “wrinkled” surface topology predicts that Ca2+ concentrations up to 80 μM can persist within the folds of membranes for significant times. This intra-wrinkle location may account for 5% of the total cell volume. Using different geometries of wrinkles, our simulations show that high Ca2+ microdomains will be generated most effectively by long narrow membrane wrinkles of similar dimensions to those found experimentally. This is a new concept which has not previously been considered, but which has ramifications as the intra-wrinkle location is also a strategic location at which Ca2+ acts as a regulator of the cortical cytoskeleton and plasma membrane expansion.
The ability to restrict enzyme activation to sub-domains within the cell is crucial for cell behaviour, such as migration, directed pseudopodia formation and cell polarization. Although calpains are known to be important in these activities [1–3] since they are relatively non-specific proteases, their unrestricted activation would wreak havoc within the cell. As these enzymes are only activated by very high Ca2+ concentrations, i.e. concentrations that are much higher than reached within the bulk cytosol, the activation signals must be restricted to strategic locations within the cell. While the existence of high Ca2+ microdomains within cells has long been discussed [1–3], theoretical considerations have suggested that this level of Ca2+ can only exist transiently in limited cytosolic space within 100 nm off the open mouth of Ca2+ influx channels [2,3]. However, these models have been based on topological smooth cell surfaces, rather than more realistic micro-topologies which often include irregular surface wrinkles.
The surface of cells is rarely smooth and often appears wrinkled when viewed with sufficient resolution, such as scanning electron microscopy or atomic force microscopy [4,5]. Typical non-tissue cells, such as neutrophils, macrophages, lymphocytes and mast cells [5,6], have multiple cells wrinkles, which when viewed by transmission microscopy appear as microvilli. These wrinkles are permanent or semi-permanent structures which have a specific spectrum of surface molecules, such as integrins and selectins on neutrophil  and lymphocytes  and are sub-light microscopic, being about 100 nm wide and projecting 800 nm from the cell surface . In macrophages and neutrophils, these wrinkles act as a membrane reservoir for the “expansion” of the cell surface area during phagocytosis [9,10] and spreading . The cytosolic free Ca2+ signal which accompanies these events [12,13] permits the unwrinkling of the membrane and involves activation of the Ca2+ dependent protease calpain-1 [10,11,13] which probably cleaves proteins such as talin and ezrin  that hold the wrinkles in place. As the concentration of Ca2+ required for calpain-1 activation is at least 2 orders of magnitude higher than the resting level of cytosolic free Ca2+, i.e. 10–50 μM [15–17], this activation signal must clearly be restricted to strategic locations within the cell. Experimentally, transient Ca2+ puffs can be observed within the bulk cytosol as Ca2+ is released from storage sites within a number of non-excitable cell types [18–20]. However, the cytosolic free Ca2+ concentration reached is within the physiological range of 0.1–1 μM similar to that in the bulk cytosol during Ca2+ influx. Nevertheless, the existence of high Ca2+ microdomains near the plasma membrane has long been suspected [1,2] and apparently physiological secretion of secretory granules seem to require high (50–100 μM) Ca2+ concentrations , suggesting that high cytosolic free Ca2+ is generated physiologically. Recently, it has been shown that TRPM2 channel opening is also activated by high micromolar cytosolic Ca2+ concentrations [22,23].
However, theoretical models [1–3] based on smooth spherical (or other shaped) surfaces have suggested that this level of Ca2+ can only exist transiently very near the open mouth of Ca2+ influx channels (within 100 nm). Therefore only molecules very close to open channels would sense such signals. Since it is speculated that high Ca2+ would be strategically important within wrinkles, we therefore sought to construct a model which included a wrinkled cell surface in order to establish whether the wrinkled topology had a significant influence on the near membrane Ca2+ concentration during Ca2+ influx. We show here that the wrinkled surface of cells provides a mechanism for generating high Ca2+ domains where the concentration of Ca2+ reaches tens of micromolar while the bulk cytosol remains sub-micromolar and that any cell with a wrinkled surface topology can have high Ca2+ microdomains due to this effect. These anatomical structures provide a hitherto unrecognised mechanism for restricting the activation of Ca2+ activated enzyme activity to near membrane microdomains within the cell.
In order to generate a wrinkled cell surface to investigate the effect of Ca2+ influx, we created a 2D wrinkled surface segment (Fig. 1b) with intracellular node points from which Ca2+ concentration was calculated using finite element method. All simulations were made in 2D using axial symmetry and cylindrical coordinates. By rotation of this segment about its z-axis, the corresponding 3D surface was created which included parallel wrinkles (Fig. 1c). Although the wrinkles on the surfaces of actual cells are at random orientations (Fig. 1a), the model wrinkles have the same appropriate cross-section in 2D and are extended membrane folds as in the real-life situation (Fig. 1c). Furthermore, the surface area of the wrinkles matches that in real cells. The same algorithms were used to calculate cytosolic free Ca2+ changes in both the wrinkled surface and the smooth surfaced model (Fig. 1d). The Ca2+ concentration was calculated using standard equations for diffusion of free and buffered Ca2+, influx of Ca2+ and ATP driven Ca2+ extrusion. The model includes the three variables: cytosolic free Ca2+, Ca2+ bound to intracellular buffer and free Ca2+ buffer. The partial differential equations used to calculate changes in the concentration of free Ca2+, protein and protein bound Ca2+ are;
where Di is diffusion constants and R is the reaction:
where kf and kr are the rate constants (see Table 1). At the central axis axial symmetry was used for all species. For Buffer and Ca2+ bound buffer (Ca2+:Buffer) the boundary at the membrane was modelled with symmetry/insulation. For cytosolic free Ca2+ the boundary condition at the membrane surface is given by the flux:
The first term in this expression represents the resting flux of Ca2+ across the membrane. The second term represents the pumping of Ca2+ out of the cell, while the third term represents the Ca2+-influx following stimulation of the cell. Although it is possible that some Ca2+ influx channels may be localized to surface projections such as sensory microvilli , we have taken the conservative assumption that Ca2+ channels were distributed equally over the cell membrane. In this way, the model had not an in-built bias towards higher cytosolic free Ca2+ within the wrinkled areas of membrane.
The parameters used in the equations and the initial conditions are listed in Tables 1 and 2, respectively. Using the same numerical values for Ca2+ influx, efflux, buffering and diffusion, we have modelled two cases; one of a smooth spherical cell surface, and the other for a more realistic topology with a “wrinkled” surface. The terms describing the flux of Ca2+ across the membrane is taken from neutrophils, because in these cells, surface wrinkles are of a particular interest and partly because many of the parameters required have been quantified in these cells. The model, however, employs the essential features of Ca2+ modelling from other models [2,25–29] and diffusion terms which are assumed to be general for all cells. The model can therefore be generalised to any cell type and the effect of surface wrinkling on the generation of high Ca2+ microdomains established.
The passive Ca2+ leak across the plasma membrane (Prest([Ca2+]ext − [Ca2+])) (Eq. (5)) is balanced by a pumped efflux across the membrane (Jefflux[Ca2+]/(Km + [Ca2+])) mediated by a Ca2+-ATPase (Eq. (5)) [30,31]. The maximal efflux of Ca2+ (Jefflux) was therefore calculated to balance influx resultant from a resting permeability coefficient for Ca2+ of 8 cm/s [32–34]. Other pumping rates were calculated assuming Henri–Michaelis–Menten kinetics, which were qualitatively similar to results obtained if we instead used Hill kinetics . The affinity of Ca2+ for the Ca2+-ATPase was set as constant Km = 1.5 μM. Although the affinity may increase following calmodulin activation , the model was not qualitatively sensitive to changes in this parameter over the time scale of our simulations (0.1–5 s) (data not shown).
A number of different stimuli generate a large increase in cytosolic Ca2+ in neutrophils. There is evidence to suggest that part of the influx is mediated by the non-selective cationic TRP channels, especially TRPM2 [35,36]. We estimate that the number of open channels must be at least 100–150 cell−1 during the first second of stimulation for a Ca2+ influx sufficiently large to generate the observed change in bulk cytoplasmic Ca2+ concentration . It is assumed that the opening of Ca2+ channels is randomly distributed on the cell surface in the same way as Ca2+-ATPase. The effect of channel opening on cytosolic free Ca2+ (kopen) depends on the time at which there is an increased open probability, which is shown in Fig. 2 for 1 s (Fig. 2A) and for 0.25 s (Fig. 2B) when we simulate the influx in the wrinkled model.
When we simulate the Ca2+ influx in the model without wrinkles we use the functions shown in Fig. 2C and D which essentially are the same as Fig. 2A and B, but has a slightly lower value such that the change in the bulk concentration of Ca2+ is the same. The additional influx is modelled using a built-in continuous function to simulate a step function. The extra influx is modelled as if it is independent on the extracellular Ca2+ concentration and we have normalized the influx such that the total Ca2+ influx is the same in the smooth surface and the wrinkled surface model. This implies that the influx pr membrane area is higher in the model without wrinkles.
Cytosolic Ca2+ buffering results from binding of Ca2+ to both proteins and small molecules, which in neutrophils is equivalent to a total buffer concentration of 0.76 mM with an average Kd of 0.5 μM . We model the buffering with an equilibrium reaction and model all three species (Eqs. (1)–(3)). Although the buffer is a mixture of a diverse group of molecules, we have used previously published diffusion constants for Ca2+ and the molecules that buffer Ca2+ .
The radius of the spherical surface of the neutrophil was taken as 5 μm (Fig. 1b), on which were superimposed wrinkles perpendicular to the membrane pointing away from the centre of the cell (Fig. 1e). The wrinkles are based on an ellipse which is 1400 nm long and 100 nm wide. The ellipse is connected to the “cell” 100 nm away from the cell using two circles with a radius of 100 nm. Thus, the wrinkles are 100 nm wide 100 nm from the cell surface and more than 200 nm wide at the surface (Fig. 1e). In a related myeloid cell type, similar wrinkles have been shown to have a width at the base of 100 nm and a height above the spherical surface of 800 nm . In neutrophils scanning electron microscopy suggests that their surface wrinkles are essentially similar . The model of the cell therefore has wrinkles perpendicular to the cell surface and pointing directly at the centre of the cell (Fig. 1b). By revolving the segment about its z-axis, a 3D surface is generated which has similarity to the wrinkled surface of a neutrophil. The same approach was used to generate the smooth cell model. The surface area of the wrinkled cell is 871 μm2, where 73% of the membrane is in the wrinkles. The smooth cell has a surface area of 314 μm2. The total length of the wrinkles is 446 μm as measured from where the wrinkle begins to protrude out of the cell membrane (see Fig. 1e). Approximately 4% of total cell volume is in wrinkles.
The smooth and wrinkled models were implemented and solved in COMSOL Multiphysics, Chemical Engineering Module (COMSOL AB.). They were simulated using Direct (Pardiso) solver, with relative and absolute tolerances of 1E−7 and 1E−8, respectively, and with the time step restricted to maximum 0.1 s. The models are available as a COMSOL model report (supplementary material 1).
As a high sub-plasma membrane Ca2+ concentration is proposed to be functionally important in both exocytsosis  and membrane unwrinkling during phagocytosis  and cell “spreading”  of neutrophils, we have in the simulations used parameters which are applicable to micro-anatomy and Ca2+ in these cells. The values and published sources of these parameters are given in Tables 1 and 2. The exact nature, number or distribution of Ca2+ influx channels on the surface of neutrophils is not known, but it can be estimated that there are at least 100–150 channels/cell (see above). Assuming a random distribution of the channels 70–100 of these are situated in the wrinkles. When simulated Ca2+ influx was run for a Ca2+ influx phase of 1 s period, the bulk cytosolic free Ca2+ rose after a delay of about 100 ms to a value of about 800 nM (Fig. 3c) in both the smooth and the wrinkled cell. The timing and extent of this rise agree with the timing and magnitude experimentally determined in neutrophil populations, suggesting that simple diffusion of influxing Ca2+ is the dominant mechanism for the bulk Ca2+ signal as in chick sensory neurones . Similar relationships were found for Ca2+ influx occurring uniformly across either the smooth or the wrinkled spherical surface. However, Ca2+ concentrations 5 nm from the end of the horizontal wrinkle (i.e. parallel to the z-axis, as indicated in Fig. 3b), were significantly higher than 5 nm beneath the plasma membrane of the smooth sphere (Fig. 3a and b). In all subsequent analyses, we have similarly taken these two points as measures of “the near membrane Ca2+ concentration”. The initial rise in cytosolic Ca2+ occurred at the onset of Ca2+ channel opening, and was followed by a further rise during open channel period. In the simulation, this was followed by an abrupt decline to the bulk cytosolic level when the channels were closed (Fig. 3a and b).
The additional Ca2+ within the wrinkles arose in part because Ca2+ influx occurred across a larger surface area than for the equivalent sub-membrane region in the smooth model. However, as the larger surface area also included additional Ca2+ extrusion pumps, the extent of the Ca2+ rise will depend on the rate of diffusion of free and bound Ca2+ out of the narrow mouth of the wrinkle. As the values for Ca2+ diffusion have not been accurately determined in neutrophils, we used the published values for free and bound Ca2+ diffusion for oocyte cytosol as ; Dbuffer = 13 μm2/s; . While the diffusion of free Ca2+ is unlikely to differ significantly in different cells, the diffusion of “bound Ca2+” would depend on the nature of the cellular Ca2+ buffer. As neutrophils have a high cytosolic Ca2+ buffering capacity [37,41] we therefore investigated the effect of diffusion of the buffer on the model. In the smooth surface model, reduced diffusion of buffered Ca2+ would have little effect on the peak Ca2+ concentration (Fig. 4a). However, the peak Ca2+ concentration in the wrinkles is sensitive to this parameter and rises steeply as the diffusion constant of buffered Ca2+ is reduced (Fig. 4b). However, as this parameter cannot be measured locally in neutrophils, in subsequent simulations the “standard” diffusion parameters have been used.
The parameter which gives the largest effect in our model is the Ca2+ channel opening time (Fig. 4c and d). If this period is reduced, while the number of open channels is adjusted to give the same Ca2+ influx (see Fig. 2), there is little effect on the peak sub-membrane Ca2+ generated in the smooth surface model (Fig. 4c). In contrast, reducing the Ca2+ channel opening time to 0.25 s or 0.1 s, increases the peak Ca2+ in wrinkles to 20 μM and 80 μM, respectively (Fig. 4d). In our initial simulations, we took the Ca2+ rise time in neutrophil populations as an estimate of the timing of this increased Ca2+ channel open probability to about 1 s. However, the responses of neutrophils are asynchronous in the subsecond time scale, with individual cells having variable delays . The population response is thus a time-averaged signal. When the rise of cytosolic free Ca2+ is monitored in individual neutrophils, the Ca2+ rise is actually faster, occurring over 100–250 ms (Fig. 4e). Similar kinetics to those observed experimentally are predicted by our simulations for the period of Ca2+ channel opening of around 250 ms (Fig. 4e and f). With an influx time of 0.25 s, the model predicts that cytosolic Ca2+ concentration at the centre of the cell will start to increase about 100 ms after the Ca2+ influx is initiated and that the plateau is reached 0.8 s after initiation of Ca2+ influx (Fig. 4f). The apparent time delay between the onset of Ca2+ influx and until the maximum concentration is reached in the cell is also within the same order as observed experimentally  (see Figs. 3 and 4 ).
Using experimentally determined timing and magnitudes for the bulk Ca2+ signal, the model predicts significantly raised intra-wrinkle Ca2+ concentrations of near 20 μM (Fig. 5a and b, and movie 3 (supplementary material)) over a significant volume of cytosol (about 4% of the total cell volume as mentioned earlier). Under these conditions, the distribution of Ca2+ concentrations with respect to the distance to the cell surface within the cell at the time when Ca2+ peaked (0.25 s after the additional influx is activated), shows a clear boundary approximately 1 μm within the cell (Fig. 5c and d), where the cytosolic Ca2+ is essentially the same as predicted for the smooth surface. However, in the outer most micron, Ca2+ rises with a different gradient as the wrinkle is entered (Fig. 5c and d). The sudden rise in free Ca2+ (Fig. 5b) exists because the amount of free buffer is exhausted locally. The ratio between free Ca2+ and bound Ca2+ is plotted in Fig. 6. Before the influx is started the ratio is 7.9 × 10−4 close to membrane, as a result of the steady state between influx and efflux of Ca2+. The ratio changes less than a factor of 10 following Ca2+ influx. However, in the wrinkled surface model, the ratio changes by a factor of 30 during 0.25 s Ca2+ influx and the buffer is almost depleted locally (Fig. 6d). It should be noted that the total cellular buffer capacity is not exhausted (Fig. 6d), but that the effect is localised to the wrinkles. This is because free buffer from the bulk cytosol cannot diffuse into the wrinkles sufficiently fast to replace the Ca2+ bound buffer. This is in contrast to the smooth surface model, where diffusion of the free buffer is unrestricted (Fig. 6c). The general conditions required for establishing a local high Ca2+ domain can therefore be defined as a region of cytosol having rapid access to Ca2+ whose buffering capacity is limited or not easily refilled. In such cases, a microdomain of high Ca2+ may be generated. This outermost micron is thus a microdomain whose Ca2+ concentration rises to significantly higher levels than the bulk cytosolic concentration. This concentration of Ca2+ may well represent a lower limit since the diffusion constants of buffered Ca2+ in small cells may be lower than in the larger oocytes which we used here. The presence of organelles close to the membrane could add further restrictions to the diffusion and increase the magnitude of the microdomains.
Since Ca2+ storage organelles, endoplasmic reticulum and sarcoplasmic reticulum, can be located within 25 nm of the plasma membrane [44,45], and create Ca2+ microdomains [46,47], it was important to investigate whether such located organelles influence the topology generated Ca2+ microdomains. Assuming that these organelles take-up Ca2+ with standard kinetics (Jefflux [Ca2+]/(Km + [Ca2+]) and with unlimited capacity and with no leakage, Ca2+ microdomains within the wrinkles would be elevated further. To compensate for near extracellular concentrations of Ca2+ in this model we have scaled the influx parameter Jstim with (1 − [Ca2+]/[Ca2+]ext) in all the simulations. If only a part of the membrane was covered the near membrane Ca2+ concentration in the wrinkles increased to around 0.45 mM (Fig. 7a and c), or 1 mM when the entire membrane was sheltered (Fig. 7b and d). These effects are due to the organelle near the wrinkle obstructing diffusion of the buffer, bound Ca2+ and free Ca2+ out of the wrinkle. If the dimension of the uptake organelle covering a few wrinkles was wider than 0.8 μm near millimolar Ca2+ domains remained. Increasing the influx of Ca2+ into the organelle by a factor of 10 or 100 also had only little impact on the Ca2+ microdomains. To reduce the Ca2+ domains to 25 μM, the maximum uptake into the organelles must be increased by more than a factor of 1000 or the Ca2+ influx reduced to 13.5% of that used previously. When the influx of Ca2+ is just 20% of that in Fig. 5b, the model predicts the existence of Ca2+ microdomains with a concentration of 0.1 mM. In both cases the global Ca2+ concentration was only changed by less than 45 nM. These simulations show that organelles if very close to the wrinkles will increase the magnitude of the Ca2+ microdomains. It should be noted that in a number of cell types, including neutrophils, the endoplasmic reticulum does not extend to the plasma membrane, and can thus have only little influence on the intra-wrinkle Ca2+ concentration.
In this study, we have modelled the cell surface topology using the winkle dimensions reported by scanning electron microscopy . However, although these dimensions provide a good estimate of mean wrinkle structure, wrinkles may exist in a range of sizes. For example, under transmission electron microscopy, the wrinkles appear as “microvilli” with lengths of 50–1900 nm and base widths of 150–200 nm [48,49] and biophysical measurement suggest the functional lengths of the wrinkles to be only 300 nm . It is therefore important to establish what influence the wrinkle dimensions have on the intra-wrinkle Ca2+ concentration.
We have therefore repeated the modelling study using the extremes dimensions for the wrinkles but maintaining the overall geometry whereby approximately 70% of the membrane is localised in the wrinkles in accordance with reported estimates [6,48,49].
Within surface wrinkles which were 1500 nm long (Fig. 1e, L1 + L2 = 1500 nm) and 100 nm wide (base width), Ca2+ concentrations were even higher than in our previous model, reaching around 85 μM (Fig. 8a). When the width of these wrinkles was increased to 200 nm, the intra-wrinkle Ca2+ concentration was reduced but remained high at 40 μM (Fig. 8b). A similar lowering of intra-wrinkle Ca2+ was observed when the standard length (800 nm) wrinkles were widened to 200 nm, intra-wrinkle Ca2+ peaking at around 11 μM (Fig. 8c). Conversely, narrowing the wrinkle width to 50 nm elevated intra-wrinkle Ca2+ concentrations to 45 μM.
Not surprisingly, as the wrinkle length is reduced, the surface of the cell approximates more closely to the smooth sphere. However, even within wrinkles just 300 nm long (Fig. 1e, L1 + L2 = 300 nm) and 100 nm base width, the Ca2+ concentration is elevated at 5.2 μM (Fig. 8d). As before, increasing the base width of these wrinkles reduces the peak intra-wrinkle Ca2+. However, even with a short stubby wrinkle 300 nm long and 200 nm wide, the Ca2+ concentration within is higher than the bulk cytosol by an additional 1 μM.
It was concluded that high Ca2+ microdomains will be generated most effectively by the longer and more narrow membrane wrinkles, but that wrinkles of similar dimensions to those found experimentally can generate localised Ca2+ regions of nearly 0.1 mM.
The experimental evidence that Ca2+ is extremely high in the cytosol within wrinkles is difficult to obtain, as the wrinkles themselves are sub-light microscopical objects. In dendritic spines of neurones, which are small anatomical structures, high Ca2+ can be observed during Ca2+ signalling . These structures are more complex than simple cell wrinkles, having a “firewall” of ER at the base, which can release and take-up Ca2+. The underlying mechanisms for generating localised elevated intra-spine Ca2+ may therefore be different . However, in the simple wrinkled membrane of neutrophils, there is evidence for the functional existence of a high Ca2+ sub-plasma membrane domains [10,11,21] and near membrane Ca2+ reported by a membrane associated Ca2+ indicator, FFP-18, is over 30 μM . It should be noted that at non-wrinkled regions of tight neutrophil adherence to a solid substrate, at which total internal reflection fluorescence microscope measurement of Ca2+ can be made within 100 nm of the plasma membrane, Ca2+ peaks at only 1 μM . This experimentally determined difference between near membrane Ca2+ concentrations at wrinkled and non-wrinkled neutrophil surface is predicted by our model.
The high sub-membrane Ca2+ concentration within the surface wrinkles are sufficient to activate proteins with Kd's of tens of micromolar Ca2+, such as calpain-1 [15–17], TRPM2 [22,23] and some isoforms of protein kinase C [53–55]. As all these examples are proteins located at the plasma membrane or associated with the cortical cytoskeleton which holds the wrinkles in place, they are strategically placed for activation within the wrinkled membrane. Although we have used the topology of the neutrophil as an example of the wrinkled surface, Bergmann glial cells which are far more convoluted, having a surface-to-volume ratio 13 times higher than neutrophils  also have microdomains of high Ca2+ which are found in their membrane projections . Like neutrophils the endothelial cells also have microvilli , and Ca2+ microdomains have recently been detected in endothelial cells following influx of Ca2+ , and we conjecture that they could be generated due to the wrinkles. The microvilli of Drosophila photoreceptors also generate microdomains of high (20–200 μM) Ca2+ following light stimulation , which is crucial for the function of the receptor . It thus seems likely that surface topology is important in a number of cell types for directing Ca2+ signalling to specific proteins with the Ca2+ microdomain it generates. As the model we present is simple both in geometry and biological assumptions, this raises the possibility that any cell with a non-smooth surface topology may exploit localised Ca2+ signalling within the wrinkled surface.
Although the Gouy–Chapman–Stern theory also provides explanation for elevated Ca2+ near the plasma membrane, it is applicable only within 2 nm of the membrane. This effect has been suggested in part to explain why PKC is activated at a bulk concentration of 700 nM Ca2+ . We have not included the effect of the electrical double layer in the current model, but Gouy–Chapman theory  predicts the electrical surface potential (ψ0) as
where is the concentration of the divalent electrolyte, is the concentration of the monovalent electrolyte, σ is the electrical surface charge density, R is the gas constant, T is the absolute temperature and 0 and r are the absolute and relative permittivities, respectively.
Setting the concentration of monovalent cations to 100 mM and divalent cations to 5 mM and assuming that the surface charge density σ is between −0.02 C/m2 and −0.05 C/m2 the ψ0 was calculated. If the surface charge density is small, the potential will decay exponentially with distance from the membrane surface with the coefficient κ:
At the zeta potential where ψ0 equals κ (around 1 nm from the surface of the membrane) the Ca2+ concentration will be increased by a factor of 2–4 with respect to the bulk concentration. Thus, the effect of the electrical surface potential is far weaker than that due to surface topology and not sufficient to explain the presence of the high Ca2+ concentrations. However, the electrical surface potential will increase the concentration of cations near the membrane and hence may work as a local cation buffer. If we assume that the surface charge density is −0.02 C/m2, then the membrane can at most bind 1.8 × 10−16 mole of positive charges or 9 × 10−17 mole of Ca2+ if we neglect the presence of Mg2+ and assume that no other cations interfere with the membrane. The intracellular buffer can, on the other hand, bind 4 × 10−16 mole Ca2+. Therefore, the mobile buffer still remains the dominant buffer, which binds at least 4 times more Ca2+ than the membrane.
The wrinkles in this model are symmetrical structures and per se artificial, however, the real wrinkles also span the entire membrane and are connected in an almost similar geometric fashion (Fig. 1). Different approaches have been used to estimate the structure and dimension of the wrinkles [6,48,49]. All the estimates indicate that the width is between 100 nm and 200 nm, but caution must be taken with the estimates based on transmission electron microscopy as the wrinkles are 3D structures that are spanning the membrane and in case the section is not perpendicular to the direction of the wrinkle the width will be overestimated. In the model we present in Fig. 1 the wrinkles are 100 nm wide, 100 nm from the surface, and that is based on results obtained using scanning electron microscopy . Using transmission electron microscopy it has been reported that the wrinkles might be wider [48,49], and we found that doubling the width decreased the near membrane Ca2+ concentration slightly (Fig. 8b). If the length of the wrinkles is decreased the topology will approach that of a smooth cell and hence the micro-environment provided by the wrinkles will disappear. On the other hand physics also sets an upper limit because if the wrinkles are too long they will break due to shear stress. To become activated the μ-calpains must bind Ca2+ and their Kd is between 10 μM and 50 μM [63,64]. The concentrations of Ca2+ obtained with wrinkles that are 800 nm long (Fig. 5) can easily explain the activation of μ-calpains near the membrane and even in wrinkles only 300 nm long there would also be a transient μ-calpain activity. When we extended the length of the wrinkles we found that the concentration of Ca2+ became close to 0.1 mM (Fig. 8), which is far more than needed to activate, e.g. the μ-calpains. Though the numbers for the length of the wrinkles vary between 300 nm and 1900 nm, the width is reported to be in a narrower interval from 100 nm to 200 nm. The wrinkles that are 800 nm long and 100 nm wide as described in Fig. 1 sets the ideal conditions for creating Ca2+ domains in the micromolar range that can activate proteins which otherwise would show little if any activity in the cytosol. If the structures were much wider or a lot shorter the apparent wrinkle structure would be lost and Ca2+ domains would disappear.
We simulated the Ca2+ influx with a deterministic approach, which is based on the assumption that the individual behaviour of different molecules can be neglected due to the number of the species according to the law of large numbers. One result of the simulations is that it is very likely that there are at least 800 active channels in the plasma membrane, which corresponds to 0.9 channel per μm2. With the assumption of a homogeneous distribution of the channels this implies that there are at least 584 channels in the wrinkles or 1.3 channels per μm wrinkle. The wrinkles have a total volume of 0.219 fl, which corresponds to about 4% of the total cell volume. A concentration of Ca2+ ions of 100 nM corresponds to about 1300 free Ca2+ ions in the wrinkles and when the concentration is 25 μM there are 3.3 × 105 free Ca2+ ions in the wrinkles. Whether a system should be described as stochastic or deterministic depends on both the number of particles and the properties of the system as such. The number of free Ca2+ ions in the wrinkles following activation is around the deterministic limit as reported by Kummer et al. . The Ca2+ dynamics inside the wrinkles in single neutrophils have not yet been measured, but global cytosolic Ca2+ measurements of neutrophils suggests that there is a stochastic element .
It is important to stress that the microdomains of high Ca2+ predicted by our simulations are not generated by assuming non-uniform distributions of Ca2+ channels, pumps, or Ca2+ release sites or by proposing new molecular properties for Ca2+ channels. The zones of high Ca2+ arise simply by including the known micro-anatomy of cell surfaces in the simulation. The work we report here has therefore highlighted the importance of including membrane surface topology when considering a model of chemical behaviour in cells.
This research was funded by the Danish Natural Science Research Council (grant no 272-06-0345) and the Wellcome Trust (WT079962AIA). We thank Chris Von Ruhland, Cardiff University, for the SEM shown in Fig. 1a. JCB thanks European Science Foundation (FUNCDYN) for travel support.
Appendix ASupplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ceca.2010.01.005.