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Conceived and designed the experiments: YK MNW. Performed the experiments: YK. Analyzed the data: YK. Wrote the paper: YK MNW.
Calmodulin (CaM) is a ubiquitous Ca2+ buffer and second messenger that affects cellular function as diverse as cardiac excitability, synaptic plasticity, and gene transcription. In CA1 pyramidal neurons, CaM regulates two opposing Ca2+-dependent processes that underlie memory formation: long-term potentiation (LTP) and long-term depression (LTD). Induction of LTP and LTD require activation of Ca2+-CaM-dependent enzymes: Ca2+/CaM-dependent kinase II (CaMKII) and calcineurin, respectively. Yet, it remains unclear as to how Ca2+ and CaM produce these two opposing effects, LTP and LTD. CaM binds 4 Ca2+ ions: two in its N-terminal lobe and two in its C-terminal lobe. Experimental studies have shown that the N- and C-terminal lobes of CaM have different binding kinetics toward Ca2+ and its downstream targets. This may suggest that each lobe of CaM differentially responds to Ca2+ signal patterns. Here, we use a novel event-driven particle-based Monte Carlo simulation and statistical point pattern analysis to explore the spatial and temporal dynamics of lobe-specific Ca2+-CaM interaction at the single molecule level. We show that the N-lobe of CaM, but not the C-lobe, exhibits a nano-scale domain of activation that is highly sensitive to the location of Ca2+ channels, and to the microscopic injection rate of Ca2+ ions. We also demonstrate that Ca2+ saturation takes place via two different pathways depending on the Ca2+ injection rate, one dominated by the N-terminal lobe, and the other one by the C-terminal lobe. Taken together, these results suggest that the two lobes of CaM function as distinct Ca2+ sensors that can differentially transduce Ca2+ influx to downstream targets. We discuss a possible role of the N-terminal lobe-specific Ca2+-CaM nano-domain in CaMKII activation required for the induction of synaptic plasticity.
Calmodulin is a versatile Ca2+ signal mediator and a buffer in a wide variety of body organs including the heart and brain. In the brain, calmodulin regulates intracellular molecular processes that change the strength of connectivity between neurons, thus contributing to various brain functions including memory formation. The exact molecular mechanism as to how calmodulin regulates these processes is not yet known. Interestingly, in other excitable tissues, including the heart, each of two lobes of calmodulin responds differentially toward Ca2+ influx and toward its target molecules (e.g., ion channels). This way, calmodulin precisely controls the Ca2+ dynamics of the cell. We wish to test if a similar mechanism may be operational in neurons so that two lobes of calmodulin interact differentially with Ca2+ ions to activate different downstream molecules that control the strength of connections between neurons. We constructed a detailed simulation of calmodulin that allows us to keep track of its interactions with Ca2+ ions and target proteins at the single molecule level. The simulation predicts that two lobes of calmodulin respond differentially to Ca2+ influx both in space and in time. This work opens a door to future experimental testing of the lobe-specific control of neural function by calmodulin.
Calmodulin (CaM) is a ubiquitous Ca2+ buffer and signaling molecule in cells. In the excitatory synapse of hippocampal CA1 pyramidal neurons, the activation of CaM dependent enzymes results in the induction of synaptic plasticity (e.g., long-term potentiation (LTP) and long-term depression (LTD)) . The induction of NMDA receptor dependent LTP and LTD require increased Ca2+ and subsequent activation of CaM-dependent downstream enzymes: CaM-dependent protein kinase II (CaMKII) and calcineurin. Injection of CA1 pyramidal cells with peptides that block CaMKII activity inhibited the induction , , but not maintenance  of LTP, while injection of the activated form of the enzyme also produced LTP-like plasticity , . LTD is also critically dependent on Ca2+ and it appears that the CaM-dependent phosphatase, protein phosphatase 2B (calcineurin) is involved in LTD induction . The simplest correlative explanation for these results is that LTD is induced by intermediate levels of Ca2+ that activate CaM and subsequently calcineurin but not CaMKII. Conversely, higher levels of Ca2+ initiate CaM-dependent CaMKII activation and autophosphorylation, leading to LTP induction. However, it is still unknown how Ca2+ and CaM regulate two opposing processes as distinct as LTP or LTD in such a precise and controlled manner.
Besides being a major signaling molecule, CaM also functions as a primary Ca2+ buffer in CA1 pyramidal neurons . In fact, most CA1 pyramidal neurons contain CaM but not other EF-hand Ca2+ binding proteins (e.g., parvalbumin and calretinin) (reviewed in ). An exception is calbindin-D28K, which is expressed in a subpopulation of CA1 pyramidal neurons but only in rat (, ). CaM binds four Ca2+ ions, two in its N-terminal lobe and two in its C-terminal lobe . The binding sites in the N-terminal lobe are lower affinity  but exhibit faster kinetics as opposed to the higher affinity, slower kinetics of the C-terminal lobe sites , . Surprisingly little is known as to how such a protein with multiple Ca2+ binding sites influences the diffusion of Ca2+ in the cell. Most pre-existing theories of Ca2+ binding and diffusion assume a fast binding of Ca2+ and single Ca2+ binding site for the buffer (see reviews by ). In addition, recent experimental data suggest that each lobe of CaM has different affinity toward its downstream target (CaMKII and calcineurin) , , . As each lobe differentially responds to Ca2+ signals and downstream targets, it is possible that these lobe specific properties play distinct biological roles in synaptic spines (see Discussion for more details). This motivated us to dissect the spatial-temporal dynamics of lobe specific Ca2+-CaM interaction in detail at the single molecule level.
Many elegant experimental measurements have been made of dendritic spine Ca2+ , , , , . These measurements largely rely on a spatially averaged Ca2+ signal generated from fluorescence imaging of dyes whose quantum efficiency changes upon Ca2+ binding. As such, they contain no direct information relative to the issue of possible micro- or nano-domains of intracellular Ca2+. The problem is exacerbated by the high diffusion coefficients of free and dye bound Ca2+ which additionally smears the spatial signal in time frames relevant for Ca2+-imaging experiments. These and other caveats related to dye-based Ca2+-imaging experiments were recently reviewed . In addition, we do not have an effective fluorescence reporter to detect and monitor Ca2+ binding to each lobe of CaM at the single molecule level. As such, mathematical models and computer simulations are presently the only tractable means of investigating this critical aspect of synaptic physiology. Furthermore, in a medium size dendritic spine (i.e., sphere-shaped spine head of 500 nm diameter), the concentration of 1 µM of any chemical species corresponds to ~40 molecules. The basal (resting) level of spine Ca2+ is 50~100 nM which corresponds to 2~4 molecules of Ca2+ ions. Under such a circumstance, the behavior of single molecules within synaptic spines is not well described by the concentration-based mathematical approach such as reaction diffusion equation.
Here we report the single molecule level analysis of Ca2+-CaM interaction within a dendritic spine using a novel particle-based event-driven Monte Carlo algorithm, which we call Cellular Dynamics Simulator (CDS, ). Unlike other commonly used Monte Carlo simulation (e.g., MCell, ), it explicitly takes account of volume exclusion and collision between diffusing molecules in order to accurately simulate chemical reactions in the cellular interior. Using this simulator and first passage time theory, we dissect the mechanisms that influence the dynamics of Ca2+-CaM interaction at the single molecule level. We use a model of CaM built upon detailed kinetic data and ask if the lobe specific spatial-temporal micro-domain of Ca2+-CaM activation can exist and if so how it is biophysically regulated in a small sub-cellular compartment like dendritic spines. We employ a statistical spatial point pattern analysis  to understand the spatial profile of Ca2+-CaM interactions. The combination of spatial point pattern analysis and particle based Monte Carlo simulation is a unique computational strategy used in this study. Our analysis shows a higher sensitivity of the N-terminal lobe to the location and influx rate of Ca2+ from typical receptor/channel sources. Each lobe of CaM functions as distinct Ca2+ sensors and responds differentially to Ca2+ influx both in space and in time. Coupled with the experimental knowledge that different enzymes bind preferentially to either the N- or C-lobes of Ca2+ saturated CaM, we propose a possible explanation for how two opposing Ca2+/CaM-dependent enzymes can be differentially activated.
Fig. 1A illustrates the Ca2+ binding and unbinding pathway for each lobe of CaM. As shown, Ca2+ binding to the N-terminal lobe and the first Ca2+ binding event to the C-terminal lobe are diffusion limited while the second Ca2+ binding to the C-terminal lobe is the rate-limiting step in achieving the fully Ca2+-saturated state. If this Ca2+ binding step at the C-terminal lobe is much slower than the diffusion of Ca2+, the majority of Ca2+ ions that entered the spine head will have moved away from the channel without saturating local CaM molecules. The spatial profile of the C-terminal lobe or full Ca2+ saturation of CaM may then be less sensitive to the location of Ca2+ channels. On the other hand, if the N-terminal lobe Ca2+ saturation is fast as compared to the Ca2+ diffusion, its Ca2+ saturation may be more closely localized to the Ca2+ channels. Thus, three biophysical factors become important in understanding the spatial domain of Ca2+-CaM interactions. The first is how fast each lobe of CaM becomes Ca2+ saturated with a given concentration of Ca2+. The second is how fast Ca2+ ions escape from the spine. The third is how steep or flat the gradient of Ca2+ ion distribution will be in the spine head with a given Ca2+ injection rate through Ca2+ channels.
In this section, we analyze the first biophysical factor, which we call the (mean) first passage time: the (average) length of transition time required for each lobe of CaM molecule to reach the Ca2+ saturated state from a basal (apo-) state. In fact, a mathematical formula is already available to calculate this mean first passage time (Equations 5, 29 in ). In their single molecule biophysical analysis, Shaevitz et al.  used an algebraic recursive method to derive the Laplace transform of the first passage time distribution. Fig. 1B and Eq. 1~2 explain their formalism applied to Ca2+-CaM interactions. Here we define State “0” as a Ca2+ free (apo) form, State “1” as one Ca2+ ion bound form, and State “2” as a two Ca2+ ion bound form of a given lobe. The symbols kX ij in Fig. 1B denotes the rate constant between State i and State j (i, j=0, 1, 2) of lobe X (=N or C). Thus, each lobe has three states and the whole CaM molecule has nine states (Fig. 1C).
The resultant Laplace transform of the distribution of first passage time is:
where [Ca] is the given concentration of Ca2+ (Note, in order to apply Eq. 29 in , we needed to multiply the association rate constant by the concentration of Ca2+). Here we assume the system is well-stirred and the concentration of Ca2+ is constant (time-invariant). Then, the mean first passage time (<t>) can easily be found through differentiation (see Eq. 5 in ):
Note that the dissociation rate () of the second Ca2+ is not included in the formula. The latter rate determines the lifetime of fully Ca2+ saturated state of each lobe but it does not influence the first passage time. Therefore, three kinetic rates (, , ) and Ca2+ concentration determine the lobe specific first passage time. Note that both lobes have similar association rates for the first Ca2+ ions () (Fig. 1A). The difference in the second Ca2+ binding rates () is large as compared to the dissociation of the first Ca2+ ion () (Fig. 1A). Thus, in Eq. 2, the second Ca2+ binding rates (, ) determine the difference of the first passage time between the N- and C-lobes.
Fig. 2A is a numerical display of this formula showing that the first passage time sharply increases as we decrease the Ca2+ concentration (the unit of time, y-axis, is in seconds). As predicted, the mean first passage time for the C-terminal lobe (magenta) is much longer than the N-lobe (blue). For comparison, we show the first passage time for full Ca2+ saturation of CaM; the mean first passage time to reach the state N2C2 in Fig. 1C. As one can see from the diagram in Fig. 1C, this first passage time depends on all Ca2+ association and dissociation pathways for both lobes and is influenced by the lifetime of the Ca2+ saturated states of each lobe. The corresponding mathematical formula will be much more complicated than Eq. 1 and 2 and therefore, we calculated this quantity numerically using an extended version of the Gillespie type stochastic algorithm (see ,  for more details). The results presented in Fig. 2A suggest that the N-terminal lobe may respond to a short Ca2+ transient but the C-terminal lobe may not if the transient is shorter than the first passage time of C-lobe Ca2+ saturation. For example, NMDA receptor type Ca2+ transients (~1 µM peak with duration of ~80–200 ms) may not result in significant CaM saturation in the spine. In fact, at a ~1 µM Ca2+ concentration, the mean first passage time for the C-terminal lobe (or full Ca2+ saturation of CaM) is much longer than the duration of the Ca2+ transient (Fig. 2A upper right inset).
Such straightforward interpretation of the first passage time analysis, however, could be misleading. Note that we have only discussed the mean but not the entire distribution (or standard deviation) of the first passage time. In addition, we ignored the fact that the number of Ca2+ ions may be limited in the dendritic spines and that their concentration is not constant as postulated in Eq. 1~2: the N-terminal lobe and the C-terminal lobes on the same or different CaM molecules will compete for the limited number of Ca2+ ions. As for the stochastic fluctuation, we can derive the standard deviation of the first passage time using the same analytic method described above:
The resultant standard deviation is very close to the mean first passage time for all Ca2+ concentrations (i.e., the coefficient of variation is >0.9 for all [Ca2+]<10 µM). The second term in the right-hand side of Eq. 3 () is small because the first Ca2+ binding rate () for both lobes are high and therefore the ratio of the right-hand sides of Eq. 3 and Eq. 2 approaches 1.
Fig. 2B and C show the histograms of the first passage time distribution for the N-terminal lobe (blue) and the C-terminal lobe (magenta) Ca2+ saturation, respectively, taken from a single stochastic simulation (the same bin size, 5ms, for both lobes and the total number of CaM molecules is 400). Fig. 2B clearly shows that the Ca2+ saturation of the C-terminal lobe is possible even if the mean first passage time is shorter than that of Ca2+ transient. However, the inset of Fig. 2B and 2C, i.e., the histogram up to 80 ms, predict that the N-terminal lobe Ca2+ saturation predominates and precedes that of the C-terminal lobe during the short Ca2+ transient. Knowing that two lobes of CaM compete for the limited amount of available Ca2+ ions in the dendritic spines, we predict that the N-terminal dominance for the short Ca2+ transient is more prominent in neurons.
This type of analysis, however, is further complicated when taking into account the non-homogeneous spatial distribution of molecules. When Ca2+ ions enter the spine head through a Ca2+ channel, a steep spatial gradient of Ca2+may be formed around the channel mouth (depending on the Ca2+ injection rate). At a single molecule level, it is the transient local (microscopic) “concentration” of Ca2+ (i.e., the number of Ca2+ collision events) felt by a CaM molecule that determines the probability of Ca2+ saturation of a given lobe of each CaM molecule. A CaM molecule can experience much higher (local) Ca2+ “concentration” than indicated by the bulk Ca2+ transient depending on its location with respect to the Ca2+ source.
The present work aims to describe a detailed analysis of this spatial stochastic phenomenon. However, before going into the detailed simulations, it is necessary to dissect each of the biophysical factors that we discussed at the beginning of this section. The last two of these factors determine the space- and time- dependent Ca2+ profile in the spines. Without such a systematic dissection, the interpretation of simulation results when trying to determine the spatial/temporal profile of CaM activation would not be possible. We next explored how fast Ca2+ ions escape from the spine.
The second factor that will determine the spatial profile of CaM activation is the escape rate of Ca2+ from the spine. Ca2+ ions that enter the spine through ion channels will eventually diffuse into the dendrites or be extruded by the Ca2+ pumps . Here we focus on the impact of spine geometry and Ca2+ pumps on the escape rate of Ca2+ from the spines. We carry out this analysis in a stepwise manner. We first analyze the escape of Ca2+ via pure diffusion without Ca2+ pumps (or buffers) and establish the impact of spine morphology on the Ca2+ escape rate (Fig. 3A and B). Then we add Ca2+ pumps to examine their impact (Fig. 3C). This way we can isolate and understand the contribution of each of these factors in the regulation of the Ca2+ escape rate. In neurons, Ca2+ buffers such as CaM also influence this escape rate but in a highly complicated manner. We will study the effect of Ca2+ binding proteins (CaM) in the later sections when we combine all known biophysical factors in the detailed simulations.
Fig. 3A shows the time courses of Ca2+ decay for three different spine neck geometries. Here, we randomly placed a fixed number of Ca2+ ions (=400 that corresponds to ~10 µM) in the head of a spherical spine and let them diffuse out of the spine to the dendrite. The diffusion coefficient () of Ca2+ was set to 200~225 µm2/s (nm2/µs) . Each curve in Fig. 3A represents the average of 100 simulation runs. Clearly, the longer and the narrower the neck, the slower the Ca2+ decay process. This is a so-called narrow escape problem and has been extensively investigated , . As predicted by these theoretical studies, the simulated Ca2+ decay transient is well approximated by a single exponential decay term. These decay time constants fit well (the relative error <5%) with one of the pre-existing mathematical formula (the left-hand side of Eq. 4 below):
where , , and are the volume of the spine head, the length and the radius of spine neck, and the volume of neck, respectively . Fig. 3B summaries our simulation results for different spine geometries. We plot the narrow escape time () against the ratio of spine head and neck volume () (x-axis). As shown all data points are aligned on straight lines, indicating that the narrow escape time is a linear function of the volume ratio () (see the right-hand side of Eq. 4). Note that Eq. 4 was previously tested against experimental data of molecular diffusion (using photo-bleaching recovery of fluorescein-dextran and enhanced green fluorescent protein) across spine-dendrite junctions in CA1 neurons , . In other words, Eq. 4 is consistent with escape of diffusing molecules from real spines on CA1 neurons. Additional simulations confirm that Eq. 4 fits well with real spines when morphologies from 3D EM reconstructions are used (http://synapses.clm.utexas.edu/) (data not shown).
Another biophysical factor that regulates the Ca2+ decay from spines is Ca2+ pumps , . The main Ca2+ extrusion mechanisms in CA1 spines are Na+/Ca2+ exchangers (NCX, NCKX) and plasma membrane Ca2+ ATPase (PMCA) , . We have modeled both of them using the kinetic scheme used in  (see Methods for more details). Fig. 3C shows a Ca2+ clearance process with standard spine morphology (500 nm spine head diameter, 500 nm spine neck length and 150 nm spine neck diameter) with (dashed black line) and without pumps (solid black line). The fast decay time constant of Ca2+ in the presence of pump is ~45% of the narrow escape time without pumps (~5–6 ms). In this analysis, we have included NCX/NCKX and PMCA at the concentration close to the highest level known in the literature to examine the maximal impact that Ca2+ pumps would have on Ca2+ clearance. The Ca2+ transients with reduced number of pumps lie between the dashed and solid lines (data not shown).
Overall, the analyses in Fig. 3 show that the narrow escape time of Ca2+ without buffers in a standard spine in the presence of pumps is ~5 ms or shorter. In the subsequent section, we will show that a major Ca2+ buffer in CA1 pyramidal neurons (i.e., CaM) slows down the Ca2+ decay to ~10~20 ms (the latter is close to that observed in the Ca2+ imaging analyses . It is this brief time window that each lobe of CaM becomes Ca2+ saturated or not during each Ca2+ spike. The first passage time becomes a critical factor to understand the spatial profile of Ca2+-CaM interactions.
Having established the impact of spine geometry on the Ca2+ extrusion process, we now analyze the third biophysical factor that influences the spatial gradient of spine Ca2+: the Ca2+ injection rate of channels. Since the kinetics of the voltage-gated Ca2+ channels and NMDA receptors are highly complicated, we used a “model stochastic Ca2+ channel” in this section. A single stochastic Ca2+ channel was placed on the top of the head of a standard spine (black circle in Fig. 4A; see Fig. 3C for the standard morphology of CA1 dendritic spine). This channel injects Ca2+ at a given (average) rate and we examine the relation between the Ca2+ injection rate and the spatio-temporal profile of Ca2+ transients in the spine. To realize the impact of Ca2+ injection rate in isolation on the spatiotemporal Ca2+ profile, there are no pumps or Ca2+ binding buffers in this model spine. Once injected, Ca2+ ions travel via simple diffusion until they are absorbed from the compartment at the spine-dendrite boundary (see the vertical arrow in Fig. 4A). We varied the rate, but the total number of injected Ca2+ ions was set to 700 so that the peak Ca2+ concentration would be in a physiological range (~6–16 µM, i.e., ~250–650 Ca2+ ions; see panel C of Fig. 4). Note these numbers are taken from the lowest estimated Ca2+ injection rate of NMDA receptors and the higher Ca2+ injection rates of voltage gated Ca2+ channels (, , ).
Fig. 4 Panel A and B show the location of Ca2+ ions (not to scale) at designated time points after the start of Ca2+ injection. The mean Ca2+ injection rates in Panel A and B are 1.4 and 0.07 Ca2+ ions per microsecond, respectively. At the higher injection rate (1.4 ions/µs), there is a build-up of Ca2+ ions near the channel (Panel A) while such a build up is not evident in Panel B. Note that the time points chosen for Panel A and B are 20-fold different so that the total number of Ca2+ injected by the indicated time points in Panel A (10, 20, 100, 200 µs) and B (200, 400, 2000, 4000 µs) are identical. Ca2+ ions can travel ~140 nm from the channel via diffusion before the next Ca2+ ion exits the channel at injection rate of 0.07 ions/µs. At a higher Ca2+ injection rate, Ca2+ ions will accumulate near the channel pore before they diffuse away (red in Fig. 4C). As anticipated, the lower Ca2+ injection rate (black) leads to a much lower peak Ca2+ number (concentration) than the higher Ca2+ injection rates. Ca2+ ion can travel more than 1 µm away from the channel during 1 ms. During a 10 ms Ca2+ injection period, a significant fraction of Ca2+ ions has already left the spine. Thus, we have lower Ca2+ peak than at the higher Ca2+ injection rate. After the peak, the Ca2+ level decreases with a time constant of ~7–8 ms for all Ca2+ injection rates. This decay process is controlled by the diffusion and is consistent with the narrow escape rate we calculated in Fig. 3.
Fig. 4 clearly shows the impact of Ca2+ injection rates on the spatial and temporal dynamics of Ca2+ transients in dendritic spines. The relative lack of a Ca2+ gradient in Fig. 4B and the long first passage time of the C-terminal lobe of CaM in Fig. 2 suggest that a spatial gradient of the Ca2+-saturated C-terminal lobe may not form. However, as mentioned at the beginning of this section, we need to include CaM and examine the combined effect of all of these biophysical factors on the spatial profile of Ca2+-CaM interactions. The second half of Results provides this analysis.
In the previous sections, we studied the impact of three biophysical factors: the first passage time (Fig. 2), the narrow escape time (Fig. 3), and the impact of Ca2+ injection rate on the Ca2+ micro-domain (Fig. 4). In this section, we wish to study the combined effects of these factors on the spatial-temporal pattern of Ca2+-CaM interaction. As a first step, we placed a single “model Ca2+ channel” as in Fig. 4 but add CaM to assess the impact of Ca2+ injection rates on the Ca2+-CaM interaction. Besides the “artificial” model channel, we included CaM and Ca2+ pumps. We distributed 1600 molecules of CaM (i.e., 40 µM) uniformly within the spine volume (the estimated concentration of CaM in CA1 dendritic spines is 10~100 µM, ). Before injecting Ca2+ ions the entire system is equilibrated at basal Ca2+ conditions, i.e., ~40–46 Ca2+ bound CaM molecules with ~2 free Ca2+ ions (the latter correspond to 50 nM of basal free Ca2+ concentration). At this basal condition, majority of CaM molecules are Ca2+ free or in a single Ca2+ bound form and none of their lobes are Ca2+ saturated. The diffusion coefficient of CaM varies between 2~20 µm2/s (nm2/µs) , . In this section, we set it to 20 µm2/s (nm2/µs) (but see our comments below).
The results in Fig. 5 show the dynamics of Ca2+/CaM with a channel of high Ca2+ injection rate (1.4 Ca2+ ions/µs and a total of 700 Ca2+ ions are injected as in Fig. 4). Fig. 5A shows the number of Ca2+ saturated N- and C-lobes (blue and magenta, respectively) and fully Ca2+-saturated CaM. The number of free Ca2+ ions in the spine is shown in Fig. 5B. Both Fig. 5A and 5B are taken from the same single simulation run. The result of stochastic simulation varies from one simulation run to the other; however, the overall qualitative dynamics in Fig. 5A and 5B are similar among different simulation runs. The N-terminal lobe of CaM binds Ca2+ much faster than the C-terminal lobe (Fig. 2). As a consequence, the number of Ca2+ saturated N-terminal lobes increases rapidly as Ca2+ is injected (blue line in Fig. 5A). After the termination of Ca2+ injection (at 500 µs), the N-terminal lobes quickly release Ca2+ and the C-terminal lobes slowly bind the available Ca2+ (Fig. 5A). Once bound, Ca2+ remains associated with the C-lobe for a relatively long time (the decay time constant is ~120 ms) and the C-lobes therefore trap Ca2+ in the spine (Fig. 5A). The free Ca2+ level eventually returns to the basal level after a few hundred ms (data not shown). Another important point to note is that even at this high Ca2+ injection rate, the total number of fully Ca2+-saturated CaM molecule is less than ~7. This number varies from simulation to simulation, but with a single Ca2+ channel, the number remains below 10 (over 100 simulation runs), a remarkably low number.
Panels C, E, and G of Fig. 5 show the spatial dynamics of each lobe of CaM taken from 15 simulation runs. During the early rising phase of their Ca2+ saturation, each lobe of CaM exhibits a nano-domain near the channel pore. For example, in Fig. 5E, we record the location (red circle) of each CaM molecule when its N-lobe becomes first Ca2+ saturated. We plot these accumulated locations of “first Ca2+ saturation event” up to the different designated time point in the figure (note each lobe may undergo multiple cycles of Ca2+ saturation, but only the first one is recorded in Panel C, E, and G in Fig. 5 and in subsequent figures). The formation of a Ca2+/CaM nano-domain is clear. A similar but less obvious nano-domain is observed for the C-terminal lobe (Panel C) and for the fully Ca2+-saturated CaM.
To further confirm these observations, we performed spatial point pattern analysis (see Methods and , , ). In this statistical analysis, we counted the number of the Ca2+ saturation events (e.g., as shown in Fig. 5E for the N-terminal lobe) and then randomly distributed the same number of points within the spine volume. We calculated a so-called (Besag's) L-function (see Methods for details) for this random point pattern. We repeated this process 1000 times and calculated the mean and the maximum and minimum envelope of the L-function (the black dotted lines in Fig. 5F) for the set of 1000 randomly generated spatial patterns. We then calculated the L-function for the original data point pattern of Ca2+ saturation and compared this (the red line in Fig. 5F) with that of complete spatial randomness (the black lines in Fig. 5F). The L-function of data (red) is outside of the maximum and minimum envelopes (black dotted lines) indicating that the given point pattern is not random. In this case, L-function is larger than the maximum envelope and it is typical of spatial clustering. We performed a similar analysis for the C-terminal lobe (Fig. 5D) and fully Ca2+ saturated CaM (Fig. 5H) and obtained the same conclusion (non-randomness). For all cases, we also performed (two-sample) Kolmogorov-Smirnov (goodness-of-fit hypothesis) test (significance level=0.05)  to verify the conclusion of envelope test.
In summary, the high Ca2+ injection rate results in a transient Ca2+-CaM nano-domain (for both lobes of CaM). The N-terminal lobe responds to and senses the Ca2+ gradient much faster than the C-lobe (blue Fig. 5A). The C-lobe's response is resistant to the Ca2+ gradient because of its longer first passage time (i.e., slow binding kinetics of Ca2+). Note we recorded and analyzed only the first Ca2+ saturation events for each lobe of each CaM molecules. The relatively widespread C-terminal lobe Ca2+ saturation in Panel C, therefore, is not because the high affinity C-terminal lobe carries Ca2+ ions while diffusing away from the channel.
What if we reduce the Ca2+ injection rate? Fig. 4 indicates that the spatial gradient of Ca2+ is less prominent with a reduced Ca2+ injection rate. One possible scenario is that, under such a condition, only N-terminal lobe with higher Ca2+ binding kinetics (Fig. 2) can detect and sense the spatial gradient. The Ca2+ saturation of C-terminal lobe and/or full Ca2+ saturation of CaM may show relatively homogeneous spatial patterns under this condition. Fig. 6 shows results to test this prediction. The simulation conditions are the same as in Fig. 5 except the Ca2+ injection rate is reduced to 0.07 Ca2+ per microsecond. This is close to the lowest Ca2+ injection rate observed for a single NMDA receptor Ca2+ current , , . Panel A and B in Fig. 6 show the population dynamics of Ca2+ saturated N- and C-terminal lobe, fully Ca2+ saturated CaM (A), and free Ca2+ ions (B). The difference in the rising phase of Ca2+ saturated N- and C- terminal lobes observed in Fig. 5A becomes less obvious at these lower rates of Ca2+ influx. The Ca2+ saturated N- and C-terminal lobes increase at a similar rate but the N-terminal lobe exhibits a larger fluctuation due to its fast Ca2+ dissociation rate. Again, the number of fully Ca2+ saturated molecules is small (less than 5~10) over the course of a 25 ms simulation experiment.
In addition, the location of Ca2+ saturation for each lobe becomes less localized around the channel (Fig. 6C and 6E). It still looks like the N-lobe exhibits a nano-domain but it is unclear by a simple inspection of the data as to whether a nano-domain exists for the C-terminal lobe. Up to the time points 2 ms and 4 ms, the Ca2+ saturation of the C-terminal lobe takes place throughout the entire spine head. The distribution of these points appears to be random. To confirm whether this pattern is random or not, we carried out the same statistical analysis as that used in Fig. 5 (panel D, F, H). Clearly, the data point patterns in Panel D and F (red line) are closer to the maximum envelope (black dotted line) of complete spatial randomness but the N-terminal lobe data pattern shows a deviation from the complete spatial randomness. This result was again confirmed by Kolmogorov-Smirnov test. The spatial pattern of the C-terminal lobe and full Ca2+-CaM saturation lie within the maximum/minimum envelope and did not suggest significant deviations from the spatial randomness.
In conclusion, the N-terminal lobe exhibits a transient Ca2+-activated nano-domain at both lower and higher Ca2+ injection rates. This indicates that the kinetic property of the N-terminal lobe (Fig. 1 and and2)2) is the major determinant of the spatial pattern formation by the N-terminal lobe. In fact, we repeated simulations used to produce Figs. 5 and and66 with different spine morphologies (with shorter and longer spine neck as shown in Fig. 3A) and obtained similar results as to the N-terminal lobe specific nano-domain (Fig. S1 and Fig. S2). We also set the diffusion coefficient of CaM to 2 µm2/s (nm2/µs) and repeated simulations in Fig. 5 and and66 (Fig. S1 and Fig. S2). As long as CaM molecules are randomly distributed within the spine volume (at time 0), neither the diffusion coefficient nor the concentration of CaM (even when reduced to 10 µM) affected the high sensitivity of the N-terminal lobe to the Ca2+ influx. It appears that the Ca2+ binding kinetics of CaM (first passage time) is the major determinant of the lobe specific spatial pattern formation during Ca2+ influx.
In addition, the spatial pattern of fully Ca2+ saturated CaM was also influenced by the Ca2+ injection rate (Fig. 5A, ,6A,6A, ,5H,5H, and and6H).6H). Recall that Ca2+ dissociation from the C-terminal lobe is slower than from the N-terminal lobe (Fig. 1A). The C-terminal lobe remains fully Ca2+ saturated for extended time (>100 ms) during which CaM (or any Brownian particle of the same diffusion coefficient) can travel a distance equal to or larger than the entire spine head volume. CaM can reach its fully Ca2+ saturated state when additional Ca2+ binds to the N-terminal lobe (note again, the first Ca2+ saturation event of the C-terminal lobe is less sensitive to the location of the Ca2+ source as compared to the N-terminal lobe). Alternatively, if Ca2+ injection rate is high and the transient Ca2+ concentration is adequate, CaM can reach the fully Ca2+ saturated state via N-terminal lobe Ca2+ saturation before Ca2+ saturates the C-terminal lobe because the first passage time for the N-terminal lobe is shorter than the C-terminal lobe (Fig. 2). The latter pathway may be responsible for the nano-domain of fully Ca2+ saturated CaM observed in Fig. 5G and Fig. 5H. If these two modes of Ca2+ saturation exist, they would have different physiological impacts of CaM signaling system as the two lobes of CaM have distinctive binding affinities for different targets. A detailed inspection of Fig. 5 and Fig. 6 simulation results in the next section reveals and confirms these two Ca2+ saturation pathways of CaM and their dependence on the Ca2+ injection rates.
Fig. 7 presents results from studies on the Ca2+ saturation pathway of CaM at the single molecule level. In Fig. 7A and 7B, we randomly selected a CaM molecule from the simulation presented in Fig. 5, and analyzed its spatial location and Ca2+ binding state. We plot the trajectory of this molecule in the spine with different colors representing the different Ca2+ occupied states. The red is for the fully Ca2+ saturated state (State N2C2 in Fig. 7A or Fig. 7E), magenta for State N1C2 and N2C1 (three Ca2+ bound state), yellow for State N1C1, N0C2 and N2C0 (two Ca2+ bound state), green for State N0C1 and N1C0 (one Ca2+ bound state), and blue for State N0C0 (apo CaM) (see Fig. 7A and Fig. 7E for the notation). Note the direct state change between the states of the same color will never occur (see Fig. 7E). The choice of color for different states seems complicated but by using this strategy, we can explicitly show the state changes of a CaM molecule with a minimum number of colors.
The CaM molecule we selected for Fig. 7A and B was located relatively close to the channel at time 0 (in blue, but not clearly visible behind other colors in Fig. 7B). It went through N0C1 (green) and N1C1 (yellow) states, reached the N-terminal Ca2+ saturated state (N2C1, magenta), and then fully Ca2+ saturated (N2C2, red) near the channel (use Fig. 7A and 7E to follow these state changes). In other words, this CaM molecule follows the sequence of N-terminal lobe Ca2+ saturation before becoming fully Ca2+ saturated (indicated by the arrow in Fig. 7A). There is no C-terminal lobe Ca2+ saturation before the N-terminal lobe. After becoming fully Ca2+ saturated, the molecule started to move away from the channel but its C-terminal lobe remained Ca2+ saturated and stays in the N2C2 (red), N1C2 (magenta), and N0C2 (yellow) states as it explores the space close to the channel (Fig. 7B).
Fig. 7C and 7D show the single molecule analysis for the low injection rate (0.07 Ca2+ ions/µs). We randomly selected a CaM molecule from the simulation presented in Fig. 6 and kept track of its state change (Fig. 7C) and spatial location (Fig. 7D). This CaM molecule was located in the middle of the spine head at the beginning of the simulation and explored a large area in the spine head in N0C0 (blue) state before reaching the N0C1 (green) state. It briefly went into the N1C1 (yellow) state and returned to the N0C1 (green) state and then it reached the N0C2 (yellow) state, the Ca2+ saturated state of the C-terminal lobe (indicated by the arrow in Fig. 7C; also follow these state changes in Fig. 7E). After the C-terminal lobe saturation, it undergoes a rapid Ca2+ binding to the N-terminal lobe (at time ~6.5 ms) via states N1C2 (magenta) to reach the fully Ca2+ saturated state (N2C2, red) (Fig. 7C). After Ca2+ is released from the fully Ca2+ saturated C-terminal lobe, this CaM molecule undergoes multiple state changes between N0C0 (blue), N0C1 (green), and N1C1 (yellow) states (see Fig. 7C and E).
These analyses (Fig. 7A and 7C) revealed two distinctive Ca2+ saturation pathways: N-terminal first pathway and C-terminal first pathway (see Fig. 7E). Fig. 7F and 7G present results that address the generality of the single examples shown in 7A and 7C. In these figures, we use the data from Fig. 5//66 and plot the number of CaM molecules that have reached the Ca2+ saturated state (for the first time) up to each time point (cumulative sum). We plot the number of CaM molecules who have reached saturation via N-terminal lobe saturation first (blue) and via C-terminal lobe first (magenta). At the lower Ca2+ injection rate, the C-terminal lobe first is the dominant pathway (Fig. 7F). At the higher Ca2+ injection rate, the probability of CaM reaching the fully Ca2+ saturated state via the N-terminal lobe first pathway is significantly increased, especially during the first 5 ms (Fig. 7G). Note it is this first ~5 ms time period that the number of Ca2+ saturated N-terminal lobes exceed that of the Ca2+ saturated C-terminal lobe (Fig. 5A). Overall, the C-terminal lobe first pathway exists for both low and high Ca2+ injection rates. The Ca2+ saturation of CaM via the N-terminal lobe dominant pathway only becomes prominent at higher Ca2+ injection rates.
So far we have analyzed the lobe-specific Ca2+-CaM spatial domains using a “model” channel. The purpose of this arrangement was to systematically analyze the impact of Ca2+ injection rates that may underlie possible lobe-specific Ca2+-CaM nano-domains. We now explore the same issue under a more realistic situation. Instead of a single “model” Ca2+ channel, we place multiple NMDA receptors on the spine head and analyze the impact of their spatial distribution on the lobe-specific Ca2+-CaM nano-domain.
As stated earlier, NMDA receptors are the major Ca2+ source in CA1 spines . The estimated number of NMDA receptors lie between 5~20 , . The number and distribution of NMDA receptor may vary from one spine to the other. To gauge the impact of the spatial localization of NMDA receptors, we decided to create two extreme cases. In Fig. 8, we placed 20 NMDA receptors in a 200 nm diameter area of the spine membrane to mimic NMDA receptors embedded in the post-synaptic density. In Fig. 9, we uniformly distributed the same number of NMDA receptors over the entire spine head. In both cases, we populated the spine volume with the same number of CaM molecules and Ca2+ pumps as in Fig. 5 and and66 (see Methods for more details of simulation).
In panel A and B of Fig. 8 and and9,9, we show the Ca2+ binding kinetics and free Ca2+ transients of single simulation runs of each case. The stochastic fluctuation (opening and closing) of NMDA receptors dictates the Ca2+ transient as predicted by previous work . Interestingly, we could not find any significant differences between the two different distribution patterns of NMDA receptors (Fig. 8 and Fig. 9) in terms of overall Ca2+ (or Ca2+ binding to CaM) transients. To show the spatial patterns of Ca2+ saturation, we compiled the results of 20 simulation runs (of 20~25 ms, for each NMDA receptor distribution pattern) and plot the locations of the Ca2+ saturated N- and C-lobe and fully Ca2+ saturated CaM as before (Fig. 8 C~H and Fig. 9 C~H).
For both distribution patterns of NMDA receptors, the N-terminal lobe Ca2+ saturation exhibits deviations from spatial randomness (Fig. 8F and Fig. 9F). In the case of NMDA receptor clusters (Fig. 8), a transient nano-domain of Ca2+ saturated N-terminal lobe is formed close to the receptor cluster and visible in the 2D projection of the data. In contrast, there is no detectable focus of clustering of Ca2+ saturated N-terminal lobe for homogenous NMDA receptor distributions (compare Fig. 8E and and9E9E at 4 ms). However, our methodology (Ripley's K-function/Besag's L-function) still detected a slight deviation from complete spatial randomness (Fig. 9F). This may suggest that the N-terminal lobe is still sensitive to the location of NMDA receptors but their spatial pattern of Ca2+ saturation was not clearly visible in the 2D projection of the data. The C-terminal lobe exhibits a minor and weak deviation from the spatial randomness for both cases. Overall, the N-terminal lobe shows a nano-domain regardless of the spatial distribution pattern of NMDA receptors.
We have analyzed the lobe specific spatial and temporal pattern of Ca2+-CaM interactions at the single molecule level in synaptic spine compartments. Ca2+ metabolism in neuronal spines is a dauntingly complicated process that involves nonlinear interactions between channels, pumps, CaM, and other potential Ca2+ binding proteins. We focused on three primary biophysical factors, Ca2+ binding kinetics of CaM, Ca2+ clearance from the spine compartment, and Ca2+ injection rate, and dissected the spatial pattern of Ca2+-CaM interactions in a stepwise manner. Our results indicate that the N-terminal lobe and the C-terminal lobe of CaM have different functions in decoding Ca2+ signals in space and time. The N-terminal lobe is more sensitive to the Ca2+ transients while the C-terminal lobe is relatively resistant to the spatial gradient of Ca2+. Our systematic dissection (Fig. 2~9) strongly indicated that the Ca2+ binding kinetics to each lobe of CaM is the key regulatory mechanism of the spatial pattern of the Ca2+-CaM system. Our simulation study also identified two Ca2+ saturation pathways and their Ca2+ injection-rate dependencies: the C-terminal lobe first vs. the N-terminal lobe first pathways. The simulation results showed that the former is especially prominent with the low Ca2+ injection rate.
What are the implications of the lobe specific functionalities of CaM, especially for the CaM-and NMDA receptor-dependent synaptic plasticity that involves CaMKII and calcineurin? In order to understand this issue, one must pay close attention to the details of Ca2+-CaM-target interactions. Each lobe of CaM (as well as the entire CaM molecule) undergoes a series of conformational changes upon Ca2+ and/or target binding. In fact, the Ca2+ binding and target association are thermodynamically coupled (see ). Target binding increases or decreases the affinity of Ca2+ of CaM while Ca2+ binding in turn changes the binding kinetics of CaM towards its targets (see below for more discussion). The changes in the Ca2+ binding kinetics upon target binding (i.e., due to the different conformational states of CaM) is a critical factor that may affect the spatial profile of Ca2+-CaM-target activation.
Another important issue to consider is that a fraction of CaM molecules may already exist in a complex with its target even at basal Ca2+ concentrations. Interestingly, recent experimental and modeling work suggested that the N-terminal lobe of CaM preferentially interacts with CaMKII before the C-terminal lobe , . In fact, these kinetic studies suggest that CaM remained bound to CaMKII for extended periods at basal Ca2+ concentrations via the N-terminal lobe. This mode of CaM-CaMKII interaction is different from the so-called CaM-trapping by auto-phosphorylated CaMKII (see  for full discussion of this issue). Once bound to CaMKII via the N-terminal lobe, the C-terminal lobe of the same CaM molecule interacts with CaMKII. When bound to CaMKII, the Ca2+ binding kinetics of the C-terminal lobe are accelerated by the law of detailed balance . As shown in Fig. S3, CaMKII bound C- and N-terminal lobes both have faster Ca2+ binding kinetics (Panel A) and shorter first passage time for Ca2+ saturation (Panel B). The present work (Fig. 2, ,55~9) predicts that CaMKII-bound CaM may exhibit a nano-domain as observed in the target-free N-terminal lobe as long as the distribution of CaMKII is homogeneous within the spines. The latter assumption (homogenous distribution of CaMKII) may not be the case. However, recent experimental results indicated the presence of a nano-domain of CaMKII activation in CA1 spines . Since CaMKII plays a key role in LTP (long-term potentiation) induction, further investigation of this CaMKII nano-domain is critical.
What if the C-terminal lobe preferentially interacts with calcineurin which underlies LTD (long-term depression) induction? Then, each of the two lobes of CaM differentially regulates these two opposing processes of synaptic plasticity. This may seem like an attractive hypothesis and in fact, our preliminary modeling study indicated that the C-terminal lobe of CaM has a higher affinity toward calcineurin than the N-terminal lobe. However, the affinity of calcineurin for CaM is extremely high  and as a consequence, most of the calcineurin molecules may already be bound to Ca2+-CaM even at the basal free Ca2+ concentrations in CA1 spines. On the other hand, for full activation, additional Ca2+ must bind the regulatory subunit (subunit B, CnB) of calcineurin . If the Ca2+ binding kinetics of CnB is similar to that of the C-terminal lobe of CaM, one would expect a spatial and temporal pattern of calcineurin activation to be similar to the C-lobe specific Ca2+-CaM activation domain. Detailed experimental characterization of the Ca2+ binding kinetics of the “CaM-like” subunit of calcineurin (CnB) is necessary.
In CA1 pyramidal neurons, another critical factor, RC3 (neurogranin, Ng), regulates the induction of NMDA-receptor and CaM-dependent synaptic plasticity. RC3 is highly enriched in CA1 spines and is known to regulate the transition between the induction of LTP vs. LTD , . The biochemical analysis of RC3-CaM interactions suggested that it may have an additional impact on the spatial nano-domain of Ca2+-CaM. RC3 binds CaM (even in the absence of Ca2+) and accelerates the Ca2+ dissociation from the C-terminal lobe thereby decreasing its affinity toward Ca2+ , . The thermodynamic reciprocal interaction between Ca2+ binding and target interaction that we mentioned earlier may play an important role in determining the spatial dynamics of Ca2+-CaM-RC3 interactions. The released Ca2+ ion can bind the N-terminal lobe of the same or another CaM molecule. We predict that RC3 has a positive impact on the N-terminal specific Ca2+-CaM nano-domain and on the nano-domain of CaMKII bound CaM. In addition, RC3 is known to interact with membrane phosphatidic acid . The spatial distribution of RC3 and the mobility of CaM-RC3 may have an additional significant impact of the spatial dynamics of Ca2+-CaM activation. Overall, genetic studies clearly suggest a critical role of RC3 in the regulation of Ca2+ dynamics in spines , . Together with CaMKII, RC3 is another molecular target for future study using the particle-based Monte Carlo simulation.
The persistent existence of N-terminal lobe specific Ca2+-CaM nano-domain (Fig. 5~Fig. 9) may at first seem reminiscent of the traditional view on Ca2+ micro-domains. However, we must point out that “Ca2+ domains” and “Ca2+-CaM domains” are, strictly speaking, different concepts. A “Ca2+ nano-domain” is defined by the mean distance traveled by Ca2+ ions before being captured by buffer (Ca2+ binding protein) or being extruded. Only under certain conditions, for example, when the Ca2+ binding rate is faster than the diffusion of Ca2+, are “Ca2+ domain” and “Ca2+-buffer” domain closely related in space. Clearly, the C- and N- terminal lobe specific Ca2+-CaM domains respond differently for the same Ca2+ input (Fig. 5 and and6)6) and the spatial profile (and size) of the C-terminal lobe domain is different from the “(free) Ca2+-domain”. Fig. S4 illustrates this point and shows the distributions of Ca2+ ions, Ca2+ saturated N-terminal and fully Ca2+ saturated CaM from a single simulation run in Fig. 5 and and6.6. Clearly, the size and spatial profile of these domains are not identical.
The spatial profile of the “Ca2+” signal ([Ca]i below), in the presence of excess unsaturable mobile buffers, is given by the following equation :
where, is the single channel Ca2+ current, is the diffusion coefficient of Ca2+ (defined earlier), the distance from the channel, [Ca]0 is the bulk Ca2+ concentration, and denotes the mean path length of a Ca2+ ion travels before being captured by buffer, B is the buffer concentration is the Ca2+ binding rate, and F is the Faraday constant. This and many other mathematical formulas have been developed (see reviews in ) but they are not very useful to study the spatial profile of Ca2+-CaM or for any other protein or buffer with multiple Ca2+ binding sites of different binding kinetics.
Furthermore, in a small sub-cellular compartment, like CA1 spines, the number but not the concentration of molecules is important. As an illustration, when the equation for the steady-state Ca2+ concentration profile is applied to an L-type Ca2+ channel, it predicts a sharp Ca2+ gradient which results in 100 µM Ca2+ concentration at a distance of ~4 nm from the channel (see Fig. 1C in ). 100 µM of Ca2+ within 4 nm distance of a channel is more than sufficient to saturate the C-terminal lobe of CaM, but it corresponds to less than 1 molecule of Ca2+ ion, leading to a contradiction. In order to understand the spatial information flow of the Ca2+ signaling system in dendritic spines, one must explicitly calculate the first passage time distribution of Ca2+ saturation of CaM and their spatial profile using an accurate particle-based Monte Carlo algorithm and appropriate data analysis method (e.g., statistical point pattern analysis) as we did in this study.
In addition, it is important to note that the two lobes of CaM, with almost opposite impacts on Ca2+-CaM nano-domains, reside in the same molecule and are competing for a limited amount of Ca2+ as we discussed in the Results (Fig. 2). This again implies that the N- and C- terminal lobes decode Ca2+ signals in a different manner, and potentially serve distinct cellular functions. The current work is the first step to understand this unique functionality of CaM at the single molecule level.
The Ca2+ transient in dendritic spines is regulated by highly nonlinear interactions between voltage-gated Ca2+ channels, K+ channels, and glutamate receptors. This important issue was recently reviewed in . Clearly, Ca2+-activated K+ channels (SK channels) in hippocampal neurons shape the Ca2+ transients in spines and a direct coupling between voltage-gated Ca2+ channels and SK channels via “Ca2+ nano-domains” is a critical regulatory factor of spine Ca2+ metabolism. In addition, CaM itself regulates the activities of Ca2+ channels and Ca2+ pumps (PMCA) . Without the detailed knowledge of these issues, we are not able to quantitatively address their impacts on spine Ca2+ dynamics. It is also difficult to make correct interpretations of pre-existing Ca2+ imaging experimental data (e.g., roles of pump in the diffusional coupling between dendrites and spines). For these reasons, in this study we focused on the initial rising phase of Ca2+ transients and therefore only studied the impacts of Ca2+ injection rate that are relevant for any Ca2+ channels.
With these limitations in mind, we repeated all simulations in Fig. 5~9 without Ca2+ pumps and discovered that the resultant spatial profile of lobe specific Ca2+-CaM domains were similar to the results with Ca2+ pumps (data not shown). As long as Ca2+ pumps are uniformly distributed, the Ca2+ binding kinetics of CaM dictates the spatial and temporal pattern of the Ca2+-CaM interaction. We have not, however, tested spatially non-uniform distribution of Ca2+ pumps such as clusters of PMCA/NCX/NCXK tightly coupled to Ca2+ channels. This is an open area of future research.
Finally, the smooth endoplasmic reticulum (SER) is another source of Ca2+ that potentially influences Ca2+ transients in the spine. Although our simulator is fully capable of implementing SER structures and Ca2+ release from this source, only a small subset of dendritic spines on CA1 pyramidal neurons contain SER . Furthermore, a recent study suggested a strong link between the SER containing spines and metabotropic glutamate receptor dependent synaptic depression  which is an interesting but different topic than the focus of the present work.
CaM is a bi-lobed molecule that has two Ca2+-binding sites within each lobe. Fig. 1A shows how this kinetic mechanism is modeled. Each lobe of CaM has three different states dependent on the number of bound Ca2+ ions: (apo)-CaM, (Ca2+)-CaM and (Ca2+)2-CaM (the horizontal arrows in Fig 1A). The resultant CaM model has nine Ca2+ binding states (Fig. 1C). We assume that Ca2+ binding to the C-lobe and N-lobe are independent and that inter-lobular cooperativity is not considered. The rate constants of Ca2+ binding to each lobe are taken from our previous work , . This model is a simplification of our more elaborate model of CaM . In the latter modeling scheme, Ca2+ association and dissociation at each Ca2+ binding site of CaM were explicitly modeled. Further refinement of the latter detailed model is also possible by taking into account of open (relaxed) and inactive closed (tense) states of each EF-hand of CaM as proposed by Stefan et al. . We repeated the first passage time analysis in Fig. 2 using the former detailed model and confirmed that there is no qualitative difference between the detailed and simplified models. Future efforts will be made to incorporate the idea of relaxed and tense states in our simulations to specifically examine their consequences on Ca2+/CaM/target interactions.
The Ca2+ transient in the spine (head) is regulated by a highly complicated set of nested feedback loops . This includes ionotropic glutamate receptors (AMPA receptors and NMDA receptors), CaV2.3 voltage-sensitive Ca2+ channels, small conductance Ca2+-activated K+ channel (SK channels), and voltage-gated Na+ channels. The role of voltage-gated CaV2.3 channels and Na channels have been largely unknown until recently , . On the other hand, the nature of ionotropic glutamate receptors such as NMDA receptors, the major source of Ca2+ influx into the spine compartment, has been extensively studied in the past and we used a recently published model for our simulation (Fig. 8 and and9)9) .
The functional roles , , ,  and molecular expression , ,  of Ca2+ pumps have been studied; however, very limited quantitative information is available regarding the number, (intra-spine) distribution, and detailed kinetics properties of these Ca2+ pumps. The membrane densities of the plasma membrane Ca2+-ATPase (PMCA) and the Na+-Ca2+ exchanger (NCX) are 150~300/µm2 of membrane and 32~60/µm2 membrane, respectively , . Since we do not have reliable data for the intra-spine distribution of these pumps, we decided to use the maximum estimated membrane densities for each pump to evaluate their impacts on Ca2+ dynamics (Fig. 3C). The PMCA kinetic constants are 0.2 µM Km for Ca2+ and a turnover rate of ~100 s−1 and NCX has a Km of 3 µM and a turnover rate of ~1000 s−1 . For initial investigations we fixed the resting extrusion at 25 ions per second and 48 ions per second for PMCA and NCX, respectively .
The reaction scheme for the Ca2+ pump is similar to the one in :
where , , and are Ca2+ inside the spine, extruded Ca2+, pump, and Ca2+-pump complex. PMCA hydrolyzes one ATP molecule per Ca2+ ion transported, i.e., exchanges one Ca2+ for one H+ (see recent reviews by Di Leva et al. ). NCX exchanges three Na+ for one Ca2+ and NCKX imports four Na+ while exporting one Ca2+ and one K+ (reviewed in ). Provided that we do not model the diffusions of Na+ or K+ or ATP hydrolysis, Eq. 6 captures the essential characteristics of these Ca2+ pumps (see Discussion for Ca2+-CaM dependent regulation of PMCA). Finally, we randomly incorporated Ca2+ leak channels so that the net flux of Ca2+ is 0 at rest (50 nM Ca2+).
The NMDA receptor kinetics was taken from previous modeling work . Although our CDS simulator is fully capable of simulating glutamate release and diffusion in the synaptic cleft, this issue was not a focus of the present study. Instead, we assumed that each NMDA receptor was exposed to a constant level of glutamate as in previous modeling work , i.e., we stimulated the NMDA receptors for 0.1 ms with 1 mM of glutamate application and observed the subsequent Ca2+/CaM activation in the spine. The stochastic fluctuation of Ca2+ influx is then due to the stochastic kinetics of NMDA receptors.
All other numerical analyses including spatial point pattern analysis and first passage time calculation were carried out under the Matlab environment (The MathWorks, Inc., Natick, MA, USA).
The algorithmic principle of the event-driven particle-based Monte Carlo simulator (CDS) is described in  and the software is downloadable from our website (http://nba.uth.tmc.edu/cds). The CDS algorithm uses the discretized Brownian motion and relies on the first passage theory and event-driven simulation scheme. The overview of pre-existing particle-based Monte Carlo simulations (Smoldyn , GFRD , the coarse-grained molecular simulator described by Ridgway et al. , and MCell ) and differences between these simulator and CDS are also discussed in .
Under the CDS algorithm scheme, we calculate the first passage time (and probability) of molecular collisions and chemical reactions for each molecule in the simulation and create a list of all possible future events and their timing. We execute all of these molecular collisions and chemical reactions exactly as they happen one-by-one while moving all molecules simultaneously in the space. Every time we execute an event, we update the event list based on the new location or chemical status of the molecules. The time interval between two consecutive events varies from one simulation step to the other. Therefore, unlike time-driven Monte Carlo algorithms (e.g., MCell and Smoldyn), there is no fixed time step in CDS. This event-driven scheme is the only accurate way to handle molecular collisions in a crowded cellular environment. In some cases, the interval between two successive events (collision or chemical reaction) becomes long and may result in the non-Brownian motion of molecules. To avoid this situation, we add “change of direction of move” to the event list so that the direction of molecular motion is constantly randomized at least once every10 ns (the jump length of Ca2+ ion during this time period is smaller than the size of CaM molecule).
In the CDS simulations, the radius of gyration of CaM (2.2 nm) was used to set the size of CaM molecules. The radius of Ca2+ ion was set to 0.2~0.25 nm (larger than its atomic radius) taking into account its hydration shell , i.e., we modeled Ca2+ as a solvated ion while simulating its diffusion and interactions with proteins. The diffusion coefficient () of Ca2+ in non-buffered cytoplasm is 200~225 µm2/s (nm2/µs) .
The idea behind the Ripley's K-function, or its derivative Besag's L-function, is that if the distribution of the points is random, the number of points within a distance is proportional to if there is no spatial boundary in the system. Suppose we have a 3D spatial distribution of points (~) and denotes the number of all points within a distance of the particular point . The Ripley's K-function is defined by
where is the density of particles, the average number of particles in a unit ball , . The expected value for a random Poisson distribution in infinite space is . The Besag's L-function is a derivative of K-function and is defined by
so that its expected value for a random Poisson process in infinite space is (linear). A deviation of L-function from the spatial randomness indicates a clustering or repulsion of the point distribution. We can calculate K-functions with respect to a specific point in space such as a Ca2+ channel (instead of 's), but in this work, we calculated (Besag's) L-function for all points in space. The latter type of L-function is important and very useful as the clustering of points (the location of Ca2+ saturation) can happen in the middle of the spine head when multiple channels exist or when multiple cycles of Ca2+ binding and unbinding to the same CaM molecule take place (Fig. 8 and and9).9). Our data represent an analysis of inter-point (inter-Ca2+-saturation point) distance distribution at all distance scales and over the entire spine compartment. The important point to note is that in a confined and complicated geometry such as dendritic spines, a simple mathematical formula of Besag's L-function for complete spatial randomness is unavailable. To overcome this constraint, we created 1000 sets of randomly distributed points in the spine of the same number of data points and then calculated the L-function for the data and for the simulated random point patterns. If the resultant L-function of the data deviates from the simulated point pattern, we can conclude that the data points are not randomly distributed.
Domain of Ca2+-CaM in a spine with a long neck. We randomly placed the same number of CaM molecules (1,600) as in Fig. 5~6 in a spine with a longer and narrower neck (spine head radius 250 nm, neck radius 50 nm, and neck length 750 nm; see also Fig. 3A). We carried out the analyses as in Fig. 5 and and66 to test the impact of spine morphology on the spatial domain of Ca2+-CaM interactions. Here we show the summary of these analyses. Panel A shows the locations of Ca2+ saturation of C- terminal or N-terminal lobes up to the designated time point with a higher (as in Fig. 5) or lower (as in Fig. 6) Ca2+ injection rate. We carried out the same statistical spatial point pattern analysis as in Fig. 5~6. Panels B and C are the Besag's L-functions for lower and higher Ca2+ injection rates, respectively. Each row of these panels show the L-function for the Ca2+-saturation of the C-terminal, N-terminal lobes and the full Ca2+ saturation of CaM up to the designated time point. The Besag's L-function in Panel D show the Ca2+ saturation of the C-terminal lobe at a low Ca2+ injection rate and the N-terminal lobe at a high Ca2+ injection rate, respectively, with the reduced diffusion coefficient of CaM (DCaM=2 µm2/s). No spatial gradient is formed for the C-terminal lobe with a lower Ca2+ injection rate. At a higher Ca2+ injection rate, Ca2+-saturation of the N-terminal lobe exhibits a gradient around the channel. The results did not depend on the diffusion coefficient of CaM as long as the CaM molecules were randomly placed in the spine.
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Domain of Ca2+-CaM in a spine with short neck. The same analyses (as in Fig. S1) were carried out in a spine with a short and wide neck (spine head radius 250 nm, neck radius 100 nm, and neck length 125 nm; see also Fig. 3A). Panel A shows the locations of Ca2+ saturation of N- and C- terminal lobes up to the designated time point with a lower or higher Ca2+ injection rate. Panels B and C are the Besag's L-functions for lower and higher Ca2+ injection rates, respectively. The Ca2+-saturation of the C-terminal, N-terminal lobes and the full Ca2+ saturation of CaM follow the same pattern as in Fig. 5 and and6.6. In summary, these results (Fig. S1~Fig. S2) confirm that, as long as CaM is uniformly distributed in the spines, the Ca2+ binding kinetics is the major factor that controls the spatial domain of Ca2+-CaM saturation, regardless of spine morphology and the diffusion coefficient of CaM.
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Mean first passage time of Ca2+ binding to CaMKII-bound CaM. (A) A kinetic diagrams showing the interactions between Ca2+ and each lobe of CaMKII-bound CaM. The kinetic rates of Ca2+ binding to each lobe of CaMKII-bound CaM are shown in Fig. 1A. Each arrow in the panel represents the corresponding rate constant. The rightward arrows indicate the Ca2+ association rate and the leftward arrows are the Ca2+ dissociation rates. These values were obtained via parameter optimization as in . Note the second Ca2+ association rate for the C-terminal lobe is no longer slow (compare with Fig. 1A). (B) Mean first passage time (mFPT) of Ca2+ binding to CaMKII-bound CaM lobes are compared to target free CaM and displayed as a function of the Ca2+ concentration (blue and red for the N- and C-terminal lobe, respectively). Target free lobes are indicated by solid lines and CaMKII bound lobes are shown by dashed lines of the same color. The unit of first passage time (y-axis) is seconds. The range of Ca2+ concentrations considered is from 0.05 µM (resting level)~12 µM (close to the peak Ca2+ concentration during the synaptic stimulation). The inset shows the blow up of mFPT near ~1 µM Ca2+ concentrations.
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The spatial domain of Ca2+ and lobe-specific Ca2+-CaM domain. These panels are snapshots from the simulations showing the spatial correlation between free Ca2+ ions with respect to Ca2+ saturated CaM. For illustration purpose, the sizes of molecules in the figure were artificially enlarged and are not proportional to their physical dimensions. In fact, the large light brown dots are free Ca2+ ions at the designated time points. Each of these panels are taken from a single simulation run used in Fig. 5 (Panel A and B, high Ca2+ injection rate) and Fig. 6 (Panel C and D, low Ca2+ injection rate). The small blue dots are the locations of (first) N-terminal lobe saturation (for each CaM molecule) taken from the same simulation. The red points are the locations of (first) full Ca2+ saturation of CaM up to the designated time point. Note Ca2+ ions are still being injected in Panel D at the designated time point but there is no significant spatial gradient of Ca2+ or the N-terminal lobe Ca2+ saturation. In Panel B (high Ca2+ injection rate), the (first) full Ca2+ saturations of CaM took place away from the channel.
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The authors have declared that no competing interests exist.
This work was supported by United States National Institutes of Health grants GM069611 and NS038310. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.