A Novel Explanation for Observed CaMKII Dynamics in Dendritic Spines with Added EGTA or BAPTA

Abstract

We present a simplified reaction network in a single well-mixed volume that captures the general features of CaMKII dynamics observed during both synaptic input and spine depolarization. Our model can also account for the greater-than-control CaMKII activation observed with added EGTA during depolarization. Calcium input currents are modeled after experimental observations, and existing models of calmodulin and CaMKII autophosphorylation are used. After calibration against CaMKII activation data in the absence of chelators, CaMKII activation dynamics due to synaptic input via n-methyl-d-aspartate receptors are qualitatively accounted for in the presence of the chelators EGTA and BAPTA without additional adjustments to the model. To account for CaMKII activation dynamics during spine depolarization with added EGTA or BAPTA, the model invokes the modulation of Ca_V2.3 (R-type) voltage-dependent calcium channel (VDCC) currents observed in the presence of EGTA or BAPTA. To our knowledge, this is a novel explanation for the increased CaMKII activation seen in dendritic spines with added EGTA, and suggests that differential modulation of VDCCs by EGTA and BAPTA offers an alternative or complementary explanation for other experimental results in which addition of EGTA or BAPTA produces different effects. Our results also show that a simplified reaction network in a single, well-mixed compartment is sufficient to account for the general features of observed CaMKII dynamics.

Introduction

CaMKII is a ubiquitous serine/threonine phosphatase that is activated during the induction phases of long-term potentiation (1,2) and long-term depression (3). It is activated through a series of events that begin with an elevation in calcium concentration. In dendritic spines, two prominent calcium sources are n-methyl-d-aspartate receptors (NMDARs) and voltage-dependent calcium channels (VDCCs). NMDARs open after binding glutamate, but their pores are normally blocked by a Mg²⁺ ion. To relieve the block, the spine must either be depolarized or bathed in a solution that does not contain any Mg²⁺ ions. VDCCs can be activated via depolarization alone. Calcium currents through these two sources have different temporal dynamics (4) and may enter the spine in different locations. Once in the spine, the calcium binds to a variety of buffers, including calmodulin, and is eventually removed by calcium pumps. Calmodulin has two lobes (the N- and C-lobes), both of which can bind up to two calcium ions but with different kinetics and affinities (5,6).

When a lobe binds a calcium ion, it undergoes a conformational change. This activated calmodulin may then bind to CaMKII subunits, which can be in either an open or a closed state. CaMKII subunits switch back and forth between these two states, and only subunits in the open state can become catalytically active (6). Upon binding calcium-loaded calmodulin, open CaMKII subunits become catalytically active and can phosphorylate neighboring subunits at Thr²⁸⁶, a process referred to as autophosphorylation. Phosphorylation at Thr²⁸⁶ greatly increases the binding affinity of calmodulin to CaMKII (by approximately three orders of magnitude), extending the time that calmodulin is bound to CaMKII and the time in which CaMKII is catalytically active. Effectively, this enables CaMKII to integrate calcium input over timescales on the order of a minute.

Note, also, that CaMKII’s affinity for calmodulin increases with the number of calcium ions bound to calmodulin (7), and that calmodulin has an increased affinity for calcium when CaMKII-bound (8). While CaMKII-bound, calmodulin can continue to bind and dissociate from calcium ions. Phosphatases that dephosphorylate CaMKII are also present in dendritic spines. Calmodulin can dissociate from CaMKII before a subunit is dephosphorylated, and such a CaMKII subunit remains partially active. A CaMKII subunit phosphorylated at Thr²⁸⁶ but without calmodulin bound can phosphorylate itself at Thr³⁰⁵ and Thr³⁰⁶, rendering itself inactive. These sites can also be dephosphorylated but it is not known if these three sites are dephosphorylated in any particular order.

Several models of CaMKII activation have been developed in the past. (For a summary of some of the work conducted between 1998 and 2004, see Sundstrom (9).) Theoretical work on CaMKII models has, among other things, explored the possibility of bistable (10) and tristable (11) biochemical switches formed from the pairing of CaMKII to a suitable phosphatase, as well as the effect of manipulating the frequency of calcium stimulation (12,13). Other work (6–8) has focused on measuring the kinetic constants of some of the reactions involved in activating CaMKII in vitro, or finding a simplified way to model autophosphorylation (14). However, to our knowledge, no work thus far has attempted to model the high-resolution experimental data on CaMKII activation in dendritic spines provided by Lee et al. (15). These data were gathered in exquisite detail, and have proven to be interesting.

The model presented in this article builds on much of the previous work on CaMKII activation. We attempt to develop a simplified model of CaMKII activation that can also account for the experimental results of CaMKII activation in dendritic spines presented by Lee et al. (15). To measure CaMKII activation in dendritic spines, Lee et al. (15) tagged CaMKII subunits with a fluorescence resonance energy transfer pair, making it possible to observe CaMKII activation in vivo. Calcium came from two sources: NMDARs and VDCCs, which could be controlled independently of one another. NMDARs were activated by the combination of photolysis (uncaging) of caged glutamate and depolarization (depolarization relieves the Mg²⁺ block of NMDARs) or by photolysis of caged glutamate in 0 mM extracellular Mg²⁺. In both cases, glutamate was uncaged using laser pulses aimed directly in front of spines at a rate of 0.5 Hz. VDCCs were activated by depolarization of the spine for 16 s (depolarization alone does not activate NMDARs). Both uncaging and depolarization experiments were repeated with EGTA or BAPTA perfused into spines, resulting in different and surprising effects on CaMKII activation.

During glutamate uncaging in 0 mM extracellular Mg²⁺, Lee et al. (15) observed that CaMKII activation increases for ∼6–8 s, reaches a peak value, and then decreases approximately linearly for the remainder of the stimulation protocol. Similarly, during depolarization, CaMKII activation increases for 2–6 s, reaches a peak, and decays approximately linearly until the spine repolarizes and VDCCs close (15). After cessation of input protocols, CaMKII activation exhibits a biexponential decay. Perfusion of the chelators EGTA or BAPTA into spines, however, results in divergent behavior between uncaging and depolarization experiments. During glutamate uncaging, perfusion of either EGTA or BAPTA in equimolar concentrations results in similar reductions of CaMKII activation. Likewise, during depolarization, perfusion of BAPTA also results in decreased CaMKII activation. However, perfusion of EGTA results in increased CaMKII activation. This last result is surprising, because a reduction of calcium (which triggers CaMKII activation) results in greater CaMKII activation.

To explain their experimental results, Lee et al. (15) invoked calcium nanodomain effects, a mechanism that is frequently used to account for divergent effects between EGTA and BAPTA. Calcium nanodomain effects rest on the idea that EGTA and BAPTA affect local calcium (in the vicinity of calcium channels) and global calcium (away from calcium channels) differently owing to their different kinetics. BAPTA, with its nearly 100-fold faster binding rates (16), has been theoretically shown to be able to bind calcium nearer to calcium pores than EGTA (17), thereby reducing local calcium more effectively than EGTA. Global calcium, however, is thought to be affected about equally by both EGTA and BAPTA, because they have very similar dissociation constants (K_D).

Lee et al. (15) proposed that the similar decreases in CaMKII activation seen with EGTA and BAPTA during glutamate uncaging implied that nanodomain local calcium from NMDARs was insufficient to activate CaMKII and that global calcium was therefore responsible (15). In other words, had local calcium been responsible for activating CaMKII, there would have been a difference in the effects of EGTA and BAPTA. However, because perfusion of either chelator resulted in the same reduction of CaMKII activation during glutamate uncaging, Lee et al. (15) reasoned that local calcium in the nanodomain of NMDARs did not activate CaMKII, and that global calcium was instead responsible.

The divergent results seen with EGTA and BAPTA during depolarization, on the other hand, could be explained by positing that EGTA and BAPTA had different effects on CaMKII activation and inhibition processes. The activation process was posited to be near the pore of VDCCs (activated by local calcium), while the inhibiting process was posited to be in the interior of the spine (activated by global calcium). If EGTA were less effective at binding calcium in the vicinity of VDCCs (local calcium) but approximately as effective as BAPTA at binding calcium in the interior of the spine (global calcium), then perfusion of EGTA into the spine would inhibit the CaMKII inhibiting process in the interior of the spine, resulting in greater activation of CaMKII than would be seen without added chelator. The lower activation of CaMKII seen with perfusion of BAPTA into the spine could be explained by positing that CaMKII-inhibiting calcium in the interior of the spine was reduced alongside CaMKII-activating nanodomain calcium near VDCCs, effectively reducing overall CaMKII activation (15).

To explore these hypotheses, we built (18) spatially resolved models of calmodulin activation in dendritic spines using the reaction-diffusion packages MesoRD (19) and STEPS (20). Our hypothesis was that if we could find spatial inhomogeneities in calmodulin activation during calcium influx, then this would offer a path to explain the experimental results of Lee et al. (15) in line with their explanations, because calmodulin activation underlies both CaMKII activation and calcineurin activation (calcineurin is a phosphatase that may inhibit CaMKII activation). However, although we could not rule out the explanations proposed by Lee et al. (15), we were unable to find significant spatial inhomogeneities in calmodulin activation on timescales larger than milliseconds, and then, only for calmodulin species with calcium bound on the N-lobe (21). Calmodulin species with calcium bound to the C-lobe were very nearly uniformly distributed in our model during simulated calcium influx through VDCCs, with or without added EGTA or BAPTA (18).

This motivated us to investigate the model described in this article, which consists of a simplified model of CaMKII activation in a single, well-mixed compartment. We will show that this model can account for the general features of the experimental results presented by Lee et al. (15). Our model is built on recently-developed calmodulin and calbindin models (5) and the infinite subunit holoenzyme approximation (ISHA) autophosphorylation model (14). We also include a single phosphatase that we model after protein phosphatase 2 (PP2A). After calibration against experimental data without chelators, the model reproduces CaMKII activation time courses seen during glutamate uncaging and depolarization. Glutamate uncaging results with added EGTA or BAPTA are qualitatively reproduced without changes to the model. To account for experimental results during depolarization with added EGTA or BAPTA, our model employs the modulation of R-type VDCC currents seen with added EGTA or BAPTA (22) and shows good qualitative agreement. We believe that our results support the idea that a simplified model of CaMKII activation may be sufficient to capture the key details of CaMKII activation in the cytosol of dendritic spines. We also hope to offer a new perspective on the divergent experimental results sometimes seen with added EGTA or BAPTA.

Materials and Methods

Model simplifications and assumptions

We assume that dendritic spines can be modeled as well-mixed volumes, that is, reactants are distributed with such small spatial inhomogeneities that including them makes no significant difference in model results. Thus, we do not differentiate between nanodomain (local) calcium and cytosolic (global) calcium and a single variable is sufficient to keep track of the free calcium concentration. We do not dispute the existence of calcium nanodomains, but nanodomain effects may not be important in modeling CaMKII activation in the cytosol of dendritic spines (although they may be important in understanding calcium-dependent inhibition of VDCCs; see the Supporting Material for a discussion).

We also assume that spines contain a homogeneous population of VDCCs. This may be a significant simplification, because there is evidence that a small number of L-type VDCCs are responsible for CaMKII activation (15), while the majority of VDCCs that open during depolarization are R-type VDCCs (23) and possibly N-type VDCCs (22). In this work, we treat all VDCCs as being R-type (Ca_V2.3) VDCCs.

We have chosen to not include plasma-membrane calcium ATPase pumps or smooth endoplasmic reticulum (SER) calcium uptake. Plasma-membrane calcium ATPase pumps appear to not be important to our work because they contribute significantly less to calcium removal than NCX pumps (24) and also activate more slowly, possibly after the experimental protocols used by Lee et al. (15) had already ended (25). On the other hand, while the SER is responsible for ∼30% of total calcium removal (4), not every spine has an SER body (26). Similarly, although there are other known calcium buffers in the spine besides calbindin and calmodulin (27), we have chosen to omit them for simplicity. With these assumptions about calcium sources, calcium buffers, and calcium pumps, straightforward models of NMDAR and VDCC currents are able to reproduce the calcium time courses experimentally measured by Lee et al. (15). Note that these simplifications appear to be of second-order importance because the inputs to our model are the free calcium time courses experimentally measured by Lee et al. (15), which we reproduce by tuning calcium current parameters for NMDARs and VDCCs.

We make several assumptions and simplifications in our treatment of CaMKII activation. First, we assume that only calmodulin subspecies with calcium bound to the C-lobe (CaM_C and CaM_CC) are responsible for activating CaMKII. This assumption is supported by experimental evidence that calmodulin with a permanently active N-lobe but inactivated C-lobe binds to CaMKII with a roughly 100-fold lower affinity than calmodulin with a permanently active C-lobe and inactivated N-lobe (6 μM vs. 70 nM) and also activates CaMKII more slowly (8). We note, however, that binding to targets increases both calmodulin lobes’ affinity for calcium, so it is possible that a significant fraction of calmodulin binds calcium on the N-lobe after binding to CaMKII. Our model does not include the possibility that the N- and C-lobes bind to CaMKII at different times: We assume that once calmodulin with calcium bound to its C-lobe binds to CaMKII, that the N-lobe also binds to CaMKII.

Second, we assume that both of calmodulin’s C-lobe binding sites experience the same affinity increase for calcium when calmodulin binds to CaMKII. We assume that this affinity increase is the same, regardless of whether calmodulin is bound to a closed CaMKII subunit, an open CaMKII subunit, or an open and Thr²⁸⁶-phosphorylated CaMKII subunit. The increased affinity of CaMKII-bound calmodulin is modeled as a decreased off-rate of calcium dissociating from calmodulin, while the decreased affinity of CaMKII for CaM_C compared to CaM_CC is modeled as a decreased on-rate of CaM_C binding to CaMKII. We also assume that calmodulin dissociates from CaMKII when it loses its last calcium, effectively barring apocalmodulin from binding to CaMKII subunits.

Third, we treat CaMKII subunits phosphorylated at Thr²⁸⁶ but without calmodulin bound as having the same amount of activation as all other activated CaMKII subunits. However, we treat CaMKII subunits phosphorylated at Thr²⁸⁶, Thr³⁰⁵, and Thr³⁰⁶ as totally inactive. Also, consistent with previous work (6), we assume that phosphorylation at Thr²⁸⁶ stabilizes CaMKII subunits in the open state, so that, Thr²⁸⁶-phosphorylated CaMKII subunits cannot close and become inactive.

Finally, we assume that CaMKII phosphorylated at Thr²⁸⁶, Thr³⁰⁵, and Thr³⁰⁶ must be dephosphorylated at Thr³⁰⁵ and Thr³⁰⁶ before it can be dephosphorylated at Thr²⁸⁶. For the dephosphorylating agent of CaMKII, we include only a single PP2A-like phosphatase. PP2A is activated by protein kinase A (28) and is responsible for ∼70% of dephosphorylation activity toward CaMKII in cytosol, while the remaining 30% is due to PP1 and PP2C (29). Hence, PP2A appears to be the dominant agent dephosphorylating CaMKII in cytosol. We assume that our PP2A-like phosphatase dephosphorylates CaMKII phosphorylation sites (Thr²⁸⁶, Thr³⁰⁵, and Thr³⁰⁶ in our model) at equal rates. As PP2A activity does not appear to be modulated by CaMKII activity, there is no mechanism in our model for CaMKII to affect the activity of our PP2A-like phosphatase.

We have taken the concentrations of the key reactants to be 30 μM calbindin (5), 100 μM calmodulin (5), and 100 μM CaMKII. This concentration of CaMKII is between the range of values reported in Lee et al. (15) and Otmakhov and Lisman (30).

Finally, to compare experimental results with simulation results, we developed a simple interpretation of the CaMKII activation data reported by Lee et al. (15). These data present the average CaMKII fluorescence lifetime change that results from activation of CaMKII through binding of calcium-loaded calmodulin and/or phosphorylation at Thr²⁸⁶. To convert these data into the concentration of activated CaMKII, we assumed that a CaMKII enzyme could be in one of only two states, active and inactive, and that these states had different fluorescence lifetimes. We then sought an estimate of the change in fluorescence lifetime of a single CaMKII enzyme between its inactive and active states, which gives an upper bound on average (population) CaMKII fluorescence lifetime change. In fact, Supporting Material Fig. 11 of Lee et al. (15) provides data on what appears to be a maximal change of 0.2 ns in CaMKII fluorescence lifetime under conditions of glutamate uncaging and added phosphatase inhibitors, conditions that are optimal for maximum CaMKII activation. From this datum, we extrapolated that an average lifetime change of 0.2 ns indicated that all CaMKII in the spine was active and that average lifetimes less than this meant that a fraction of CaMKII enzymes were inactive. To determine the concentration of active CaMKII enzymes, we used the following expression:

$$\left[{\text{CaMKII}}_{\text{active}}\right]=\left[{\text{CaMKII}}_{\text{total}}\right]\times \frac{\text{Δ}\text{average}\text{fluorescence}\text{lifetime}}{0.2\text{ns}}.$$

Calcium- and voltage-dependent inhibition of VDCCs

As a group, VDCCs exhibit a wide range of phenomena that affect the currents through them, although not each subtype exhibits each phenomenon. VDCCs can undergo voltage-dependent facilitation (VDF), voltage-dependent-inhibition (VDI), calcium-dependent facilitation (CDF), and calcium-dependent inhibition (CDI) (23,31,32). VDF and CDF are processes in which the presence of a voltage prepulse or calcium prepulse increases current through VDCCs (CDF has only been observed in P/Q-type channels (22)). In VDI, VDCC current decreases with time, merely because they have been opened, regardless of the charge carrier flowing through them (typically Ca²⁺ or Ba²⁺).

In CDI, as originally understood, the presence of calcium hastened the inactivation of VDCCs, although some evidence pointed to calmodulin. Determining the central mediator of these effects in Ca_V1.1–Ca_V1.4 (L-type) VDCCs was controversial for many years (33), but consensus eventually settled on calmodulin (34–36). However, there remains a scarcity of quantitative data (37). Calmodulin also appears to be the mediator for VDI and CDI in Ca_V2.3 (R-type) VDCCs (38), which appear to be the dominant VDCC subtype in dendritic spines (23). Given calmodulin’s apparently central role in mediating VDI and CDI and that calmodulin is activated by calcium, it appears sensible that the presence of calcium chelators, such as EGTA and BAPTA, would modify VDCC currents by modulating VDI and CDI in L-type (39) and R-type (22,40) VDCCs.

As of this writing, quantitative understanding of these phenomena in VDCCs remains elusive. The development of state models that capture the full range of VDCC behavior appears, at the outset, to be very complex compared to other types of ion channels (33). In the absence of detailed models, our model of R-type VDCC currents includes only a few key features of their modulation in the presence of chelators. These are, in short, a reduction in CDI (a slowed decay of VDCC inactivation and increased steady-state current) and an increase in the peak current in the presence of EGTA and a reduction in CDI concomitant with increased steady-state current in the presence of BAPTA (22).

Model form

In dendritic spines, CaMKII activation can take on a range of values in the steady-state phase of depolarization and subsequently shows a biexponential decay trend after cessation of input (15). The ability to take on a range of values during sustained depolarization and the decay of activation after decay of Ca²⁺ transients suggests that the bistable behavior found in some models of CaMKII activation does not occur in dendritic spines (in-line with previous work (41), which found that bistability was possible but precarious). We therefore sought to construct a model in which CaMKII activation always decayed after cessation of input. The biexponential decay trend, on the other hand, suggested the use of Michaelis-Menten kinetics to describe phosphatase activity toward CaMKII.

Calcium influx

VDCC currents are modeled as the sum of two components: A steady-state component consisting of a constant current and a transient component that starts with a rapid rise in calcium current followed by an exponential decay. Collectively, these two components can describe the experimentally measured calcium time courses observed by Lee et al. (15) during their depolarization protocol. Specifically,

$$I_{\text{VDCC}}\left(t\right)=I_{\text{VDCC},\text{peak}}\times \left(1-\text{exp}\left(-t/{\tau }_{\text{VDCC},\text{rise}}\right)\right)\times \text{exp}\left(-t/{\tau }_{\text{VDCC},\text{decay}}\right)+I_{\text{VDCC},\text{steady}\text{state}}.$$ (1)

Here, τ_{VDCC, rise} controls the time required for the transient current to reach its peak, while τ_{VDCC, decay} controls the time required for it to decay to zero and I_{VDCC, peak} sets its maximum value. I_{VDCC, steady state} accounts for VDCC currents in the steady-state phase of depolarization, after the initial transient current has decayed. For simulations of CaMKII activation during depolarization without added chelators, we calibrate these values so that the model reproduces the calcium time course measured by Lee et al. (15) during depolarization. For simulations of CaMKII activation during depolarization with added EGTA, we increase the values of I_VDCC, peak, τ_{VDCC, decay}, and I_{VDCC, steady state} consistent with the experimental results of Liang et al. (22). For simulations of CaMKII activation during depolarization with added BAPTA, we increase only I_{VDCC, decay} and I_{VDCC, steady state}, also consistent with the experimental results of Liang et al. (22).

NMDAR currents are modeled as having two components with different decay rates (42):

$$I_{\text{NMDAR}}=I_{\text{NMDAR},\text{fast}}\times \left(1-\text{exp}\left(-\varphi /{\tau }_{\text{NMDAR},\text{rise}}\right)\right)\times \text{exp}\left(-\varphi /{\tau }_{\text{NMDAR},\text{fast}\text{decay}}\right)+I_{\text{NMDAR},\text{slow}}\times \text{exp}\left(-\varphi /{\tau }_{\text{NMDAR},\text{slow}\text{decay}}\right).$$ (2)

The ϕ-value is the phase of the glutamate uncaging stimulus, meaning that it starts at 0 s, counts to 2 s, and then resets to 0 s every time a new synaptic stimulus is started in our simulation. This mimics the experimental protocol used by Lee et al. (15), in which glutamate was uncaged at a rate of 0.5 Hz. Here, also, τ_{NMDAR, fast decay} and τ_{NMDAR, slow decay} control the rate at which the fast and slow components, respectively, decay. Parameters for NMDAR currents were calibrated to reproduce the calcium time courses experimentally measured by Lee et al. (15) during glutamate uncaging. See Table S1 in the Supporting Material for parameter values of both NMDAR and VDCC currents and Figs. S1 and S3 in the Supporting Material for a comparison of simulated calcium time courses to experimental data during depolarization and glutamate uncaging.

A potential complication in reproducing calcium time courses during glutamate uncaging experiments is that Lee et al. (15) used two different protocols. CaMKII activation during glutamate uncaging was recorded in conditions of 0 mM extracellular Mg²⁺ (with the exception of one experiment), while calcium time courses (the input to our model) were recorded by first depolarizing the spine (to relieve Mg²⁺ block of NMDARs) and then uncaging glutamate. This means that measurements of calcium time course using the same protocol as used to measure CaMKII activation with added chelators are unavailable. To develop our model, we have assumed that calcium time courses during these two protocols were the same.

Calcium buffering and removal

We use the calmodulin and calbindin models developed by Faas et al. (5). EGTA and BAPTA are both modeled as having one binding site. NCX pumps are modeled with Michaelis-Menten kinetics using values for k_cat (reaction velocity) and K_M (Michaelis-Menten constant) determined in Markram et al. (43). Calcium leak channels are modeled as constant currents. For kinetic constants, see Table S1.

CaMKII

We will describe the CaMKII portion of our reaction network in three steps. First, we will discuss the reactions between a single CaMKII subunit and calcium-bound calmodulin, and between CaMKII-bound calmodulin and calcium. Next, we will discuss the various states in which a single CaMKII subunit can be found. Finally, we will combine these two sets of reactions into a single diagram and delineate which CaMKII species are considered catalytically active.

Recall from Model Simplifications and Assumptions that CaMKII subunits can be in open and closed states and that we have simplified our model by allowing only calmodulin with one or two calcium bound on the C-lobe to bind to CaMKII subunits. Such calcium-bound calmodulin can bind to CaMKII subunits in the open, closed, and Thr²⁸⁶-phosphorylated states (with increasing affinity from closed, to open, and to open and Thr²⁸⁶-phosphorylated states (6)), but only reactions involving closed CaMKII subunits will be considered first (see Fig. 1 A).

Figure 1.

Reactions between (A) Ca²⁺-bound CaM and CaMKII, and (B) CaMKII-bound CaM and Ca²⁺ in our model.

Calmodulin with one or two calcium bound (CaM_C and CaM_CC) can bind to and dissociate from closed CaMKII subunits. CaMKII-bound CaM_CC can lose one calcium to become CaMKII-bound CaM_C. CaMKII-bound CaM_C can gain a calcium, or it can lose its sole calcium. Recall from Model Simplifications and Assumptions that if it loses its sole calcium, we assume that this calmodulin also dissociates from CaMKII at the same time. The distinction between CaM_C dissociating from CaMKII and CaM_C losing its calcium and automatically dissociating from CaMKII at the same time is not trivial from a modeling perspective, because these two reactions happen at quite different rates. We will expand on this point after the entire CaMKII reaction network has been presented.

Having considered reactions between calmodulin and CaMKII subunits, we now consider transformations that can occur to a CaMKII subunit. A CaMKII subunit can be in four possible states: Closed; open; open and Thr²⁸⁶-phosphorylated; and open and phosphorylated at Thr²⁸⁶, Thr³⁰⁵, and Thr³⁰⁶ (Fig. 1 B). CaMKII subunits can convert from closed to open and vice versa (modeled as first-order reactions (6)). Open CaMKII subunits can be phosphorylated at Thr²⁸⁶ via an intersubunit reaction and dephosphorylated via our PP2A-like enzyme. As mentioned in the Introduction, intersubunit phosphorylation at Thr²⁸⁶ is modeled with ISHA, which greatly simplifies calculating the rate of CaMKII autophosphorylation at Thr²⁸⁶. Previous work (7,10,13) modeled this process by enumerating all of the possible Thr²⁸⁶ phosphorylation states of a CaMKII holoenzyme (e.g., subunits 1 and 3 are Thr²⁸⁶-phosphorylated, all others not) and treating each unique permutation as a unique species, resulting in a considerable (though not intractable) increase in model complexity. Here, unique permutation is in the sense of rotational symmetry, that is, a CaMKII holoenzyme Thr²⁸⁶-phosphorylated at subunits 1 and 3 would be treated as equivalent to a CaMKII holoenzyme Thr²⁸⁶-phosphorylated at subunits 4 and 6. In contrast, the ISHA models CaMKII subunits as a single, long chain in which each subunit can only phosphorylate its neighbor. It has been shown to produce very nearly the same behavior as a model in which CaMKII subunits are arranged in a ring of five or six subunits (14) and reduces the calculation of the total autophosphorylation rate at Thr²⁸⁶ to the sum of a pair of fractions (because we allow open CaMKII subunits and open and Thr²⁸⁶-phosphorylated CaMKII subunits to phosphorylate other subunits at different rates). Phosphorylation at Thr³⁰⁵ and Thr³⁰⁶ is an intrasubunit process and is modeled as a first-order reaction that phosphorylates both sites simultaneously. Dephosphorylation of the sites Thr²⁸⁶, Thr³⁰⁵, and Thr³⁰⁶ by our PP2A-like phosphatase is modeled with Michaelis-Menten kinetics. We assume that all sites are dephosphorylated at the same rate and that Thr³⁰⁵ and Thr³⁰⁶ are dephosphorylated before Thr²⁸⁶.

Fig. 2 shows the complete reaction network for CaMKII subunits, with each CaMKII subspecies marked as being either catalytically active or inactive. Note that the reactions between calcium-loaded calmodulin and CaMKII, and CaMKII-bound calmodulin and calcium are the same for closed, open, and open and T286-phosphorylated CaMKII subunits.

Figure 2.

CaMKII reaction network in our model. (Arrows) Addition of or removal of reactants (phosphate groups have been omitted for clarity). Reactions in which calcium dissociates from CaM_C followed by apoCaM dissociating from CaMKII have been highlighted.

Earlier, we drew a distinction between CaM_C dissociating from CaMKII and calcium dissociating from CaM_C followed immediately by apoCaM dissociating from CaMKII. In our model, CaM_C binding to open and closed CaMKII subunits has nearly the same dissociation constant (K_D = 2.36 × 10⁻⁶) as calcium binding to the first binding site of the C-lobe of CaMKII-bound CaM (K_D = 1.29 × 10⁻⁶). However, CaM_C dissociates from closed and open CaMKII subunits with a kinetic constant of ∼1 s⁻¹, while calcium dissociates from CaMKII-bound CaM_C with a kinetic constant of ∼10² s⁻¹. Hence, loss of calcium followed by dissociation from CaMKII is the primary pathway by which calmodulin dissociates from CaMKII in our model. This pathway becomes even more dominant in the case of open and T286-phosphorylated CaMKII (CaMKII_open⋅P_T286) subunits, which have a 1000-fold greater affinity for calcium-loaded CaM (6) but no corresponding increase in affinity of CaMKII_open⋅P_T286-bound CaM for calcium (compared to closed CaMKII-bound CaM or open CaMKII-bound CaM). The lack of a corresponding increase in affinity of CaMKII_open⋅P_T286-bound CaM for calcium is a modeling decision we have made due to a lack of experimental evidence on this topic and may therefore be of interest to confirm experimentally. The reverse reaction (apoCaM binding to CaMKII subunits followed by CaMKII-bound apoCaM binding calcium) is not included because there appears to be no evidence for apoCaM binding to CaMKII.

Simulation scheme

Our model employs a deterministic, first-order ordinary differential equation (ODE) scheme. The ODEs are integrated in the software MATLAB (The MathWorks, Natick, MA) with the ode15s and ode23t solvers. We supply these solvers with an indicator matrix to specify what quantities can affect the time rate of change of each quantity (that is, the first derivative with respect to time), which greatly speeds up execution.

Model development

We built our model as laid out in Model Simplifications and Assumptions, adjusted two previously-measured parameters, and calibrated six other parameters in an effort to match the experimental results of Lee et al. (15) for CaMKII activation during depolarization and for CaMKII activation during glutamate uncaging in 0 mM extracellular Mg²⁺. The previously-measured parameters were the on-rate of calmodulin binding to CaMKII_open (6) (increased by 10%) and the rate of CaMKII_closed shifting to CaMKII_open (6) (increased by 80%, which is within measurement error (6)). The calibrated parameters are the rates of catalytically active CaMKII_open and CaMKII_open⋅P_T286 species phosphorylating CaM_C- or CaM_CC-bound CaMKII subunits in the ISHA, the affinity increases of CaMKII for CaM_CC over CaM_C and CaMKII-bound CaM for calcium over non-CaMKII-bound CaM, the Michaelis-Menten parameters k_cat and K_M for our PP2A-like phosphatase, and the rate of autophosphorylation at Thr³⁰⁵ and Thr³⁰⁶ (k_{cat, Thr306/Thr306}). Our value for K_M is in-line with previous estimates (10), but our value for k_{cat, Thr306/Thr306} is almost two orders-of-magnitude smaller than a previous estimate (14).

Results

We will present our model results in three steps. First, we will present model results of CaMKII activation in the absence of chelators during both depolarization and glutamate uncaging in 0 mM extracellular Mg²⁺. Second, we will present model results of CaMKII activation during glutamate uncaging in 0 mM extracellular Mg²⁺ with added EGTA or BAPTA. Finally, we will present model results of CaMKII activation during depolarization with added EGTA or BAPTA, including our adjustments to calcium currents based on experimental evidence of the effects of EGTA or BAPTA on Ca_V2.3 (R-type) VDCC currents (22).

For all simulation results, we will directly compare our model results to the experimental data of Lee et al. (15). Recall that we report CaMKII activation time course as the concentration of total activated CaMKII, rather than average fluorescence lifetime change. To facilitate comparison between our results and the experimental results of Lee et al. (15), we have converted their data on average fluorescence lifetime change to the concentration of active CaMKII using our conversion formula (presented in Model Simplifications and Assumptions). Total activated CaMKII in our results is the sum of the concentrations of CaMKII_open⋅CaM_C, CaMKII_open⋅CaM_CC, CaMKII_open⋅P_Thr286⋅CaM_C, CaMKII_open⋅P_Thr286⋅CaM_CC, and CaMKII_open⋅P_Thr286.

CaMKII activation in the absence of chelators

We will present model results of CaMKII activation during depolarization first, followed by model results of CaMKII activation during glutamate uncaging in 0 mM extracellular Mg²⁺ (hereon referred to simply as “glutamate uncaging”). During depolarization, free calcium concentration peaks rapidly (in under 32 ms) and decays to a steady-state value (above the quiescent concentration) so long as depolarization is maintained (15). (See Fig. S1 for a comparison of calcium time course in our model to the experimental data of Lee et al. (15).) Similarly, CaMKII activation reaches a peak value in under 2 s, but decays much less rapidly than the free calcium concentration, possibly reaching a steady-state value (15). Our model replicates these features well, with possibly slightly too much activation between 2 and 4 s (Fig. 3 A).

Figure 3.

CaMKII activation time course during (A) simulated depolarization and (B) glutamate uncaging. (A, blue line) Concentration of total activated CaMKII; (open circles) experimental data from Fig. 4 a of Lee et al. (15); (hatched marks) experimental data from Fig. 5 a of Lee et al. (15). (B, blue line) Concentration of total activated CaMKII; (open circles) experimental data from Fig. 5 g of Lee et al. (15). To see this figure in color, go online.

Lee et al. (15) report two sets of data for the same depolarization protocol (open circles and hatched marks in Fig. 3 A), which can be used to estimate the variability of CaMKII activation data. Under identical conditions, CaMKII activation varies by as much as 10 μM (using our conversion formula) at 12 and 14 s (while the spine was still depolarized) and as much as 15 μM after repolarization (20 and 22 s). If one assumes error bars of ∼±5 μM for all data points, our model results are in good agreement with the experimental data.

During glutamate uncaging, free calcium concentration also peaks rapidly (to ∼2 μM) and exhibits a biphasic decay during each glutamate uncaging stimulus. Glutamate was uncaged a total of 45 times and the average peak of calcium transients decreased with time (15) (see Fig. S3 for calcium time courses used during simulated glutamate uncaging). CaMKII activity increases markedly during each glutamate uncaging stimulus, reaching a peak of ∼50 μM at 12 s before decaying almost linearly until the end of the glutamate uncaging protocol (15). Our model results (Fig. 3 B) are in good agreement with these experimental data.

CaMKII activation during glutamate uncaging with EGTA or BAPTA perfused into the spine

With 1 mM EGTA or BAPTA perfused into spines, CaMKII activation is reduced by nearly equal amounts in spines (possibly less so with added BAPTA than with added EGTA). Adding 5 mM EGTA or BAPTA further reduces CaMKII activation by similar amounts for both chelators (15). This may be a surprising result, because BAPTA, with its faster kinetics, might be expected to reduce CaMKII activation more than EGTA; however, this is not the case. After adding chelators and no other changes (specifically, no modification of the calcium input currents), our model displays the characteristic effects of adding EGTA or BAPTA at 1- and 5-mM concentrations (Fig. 4, A and B).

Figure 4.

Model predictions and experimental data for CaMKII activation during glutamate uncaging after addition of EGTA or BAPTA. (A, Solid green and dot-dash red) Simulation results for 1 mM EGTA and 1 mM BAPTA, respectively. (Green circles and red matched marks) Experimental data from Fig. 5 g (1 mM EGTA) and Fig. 5 h (1 mM BAPTA) of Lee et al. (15). (B, Solid black and dot-dash magenta) Simulation results for 5 mM EGTA and 5 mM BAPTA. (Black circles and magenta hatched-marks) Experimental data from Fig. 5 g (5 mM EGTA) and Fig. 5 h (5 mM BAPTA) of Lee et al. (15). To see this figure in color, go online.

Model results agree well with experimental data at 1 mM added BAPTA, but predict ∼10 μM too much CaMKII activation between 10 and 50 s with 1 mM added EGTA. Model results also predict too much CaMKII activation at 5 mM added EGTA or BAPTA, again, by ∼10 μM. However, the model succeeds in replicating the very similar decreases in CaMKII activation seen with equivalent concentrations of EGTA and BAPTA.

CaMKII activation during depolarization with added EGTA or BAPTA

A way to reproduce experimental data on CaMKII activation with EGTA or BAPTA perfused into the spine is to modify VDCC currents in accord with experimental measurements of VDCC currents in the presence of EGTA or BAPTA. Recall that addition of EGTA appears to reduce CDI in R-type VDCCs and to increase the peak current through R-type VDCCs, while addition of BAPTA results in reduced CDI but does not appear to increase peak current for R-type VDCCs (22). During depolarization with EGTA perfused into the spine, CaMKII activation is greater than control (with slightly delayed activation kinetics), while during depolarization with BAPTA perfused into the spine, CaMKII activation is lower than control (and also with delayed activation kinetics) (15). To replicate experimental data on CaMKII activation during depolarization with added EGTA, we have increased the peak current, slowed the decay of VDCC current, and increased the steady-state current (Fig. 5 B). For experimental results with added BAPTA, we have slowed the decay of VDCC current and increased the steady-state current (Fig. 6 B). With these modifications, our model is able to qualitatively reproduce the experimental results of Lee et al. (15) of CaMKII activation during depolarization with perfused EGTA (Fig. 5 A) and BAPTA (Fig. 6 A).

Figure 5.

Model results of CaMKII activation during depolarization with added EGTA, including modulation of R-type VDCC CDI. (A) CaMKII activation during depolarization with added EGTA. (Solid green and dot-dash red) Simulation results with 5 mM and 20 mM added EGTA, respectively. (Green circles and red hatched marks) Corresponding experimental data from Fig. 5, b and c, respectively, of Lee et al. (15). (B) Comparison of calcium currents used in model with no added chelators (dashed blue), 5 mM EGTA (solid green), and 20 mM EGTA (dot-dash red). I_{VDCC, Peak}, τ_decay, and I_{VDCC, steady state} are increased 140, 150, and 150%, respectively, for 5 mM added EGTA, and increased 420, 110, and 110% for 20 mM added EGTA. To see this figure in color, go online.

Figure 6.

Model results of CaMKII activation during depolarization with added BAPTA, including modulation of R-type VDCC CDI. (A) CaMKII activation during depolarization with added BAPTA. (Solid black and dot-dash magenta) Simulation results with 5 and 20 mM added BAPTA, respectively. (Black circles and magenta hatched marks) Corresponding experimental data from Fig. 5, d and e, respectively, of Lee et al. (15). (B) Comparison of calcium currents used in model with no added chelators (dashed blue), 5 mM BAPTA (solid black), and 20 mM BAPTA (dot-dash magenta). The values τ_decay and I_{VDCC, steady state} are each increased 120% with 5 mM added EGTA, and increased 140% with 20 mM added BAPTA. To see this figure in color, go online.

With 5 mM and 20 mM added EGTA, our model is able to qualitatively replicate CaMKII activation dynamics after I_{VDCC, Peak}, τ_decay, and I_{VDCC, steady state} are increased. Qualitatively replicating CaMKII activation dynamics with 5 mM and 20 mM added BAPTA requires increasing only τ_decay and I_{VDCC, steady state}. These changes to VDCC currents are consistent with experimental findings on R-type VDCC current modulation in the presence of added EGTA or BAPTA (22). We believe this is an important result for two reasons. 1) It offers an explanation for the increased CaMKII activation seen during depolarization with added EGTA without relying on nanodomain effects (but still using existing experimental results). 2) Inclusion of VDCC current modulation by EGTA or BAPTA in models of CaMKII activation is, to our knowledge, a novel feature.

Discussion

An alternative explanation for the divergent experimental results seen with added EGTA or BAPTA

Many experimental results with divergent outcomes that depend on whether EGTA or BAPTA is added to the cytosol are explained by invoking calcium nanodomain effects (44–47). BAPTA, with its nearly100-fold faster kinetics compared to EGTA (16), has been theoretically shown (17) to be able to bind calcium nearer to the source than EGTA. A typical explanation may leverage this mechanism by positing that the trigger for a calcium-dependent effect is colocated with the source of calcium. However, it is possible that when the length scales of calcium nanodomain effects are taken into consideration, some explanations appear less likely. Assuming that VDCC currents are unaffected by the presence of EGTA or BAPTA, significant differences in calcium concentration are predicted to fade away by ∼50 nm from the channel pore (45). Calculations of this sort are typically carried out in a deterministic model. Yet, stochasticity in chemical reactions and diffusion of reactants may add considerable variance to local calcium concentrations, especially when chelators are present (48), effectively reducing the distance from calcium pores that nanodomain effects can be expected to be prominent. Diffusion of reactants includes not just reactants in the cytosol, but may also include ion channels, which diffuse on the cell membrane and may not always maintain a close distance to the locations to which they are tethered.

Consider the experiments reported by Lee et al. (15). Assume that calcium nanodomain effects are prominent in a 50 nm radius away from VDCC pores. A typical spine has a diameter of ∼0.5 μm, with an estimated volume of ∼1 μm³ (49). Yet there are only between five and 15 VDCCs and ∼50 NMDARs found on spines (23). If each of these calcium channels has a nanodomain radius of 50 nm (17) in which nanodomain effects are expected to be important, then, for 50 NMDARs, the total volume in which nanodomain effects are expected to be important is still only ∼1% of the total spine volume. The total volume in which nanodomain effects would be expected to play a role for VDCCs is smaller still, due to the lower number of VDCCs on spines. Note also that calcium nanodomain effects should be greater during calcium influx from NMDARs (because of their greater number), but this is not what Lee et al. (15) observed.

Invoking the modulation of VDCC CDI and peak currents by the addition of EGTA or BAPTA offers an alternative explanation to the experimental results of Lee et al. (15) and others where addition of BAPTA produces an effect but addition of EGTA does not. For example, SK_Ca and BK_Ca channels are small- and large-conductance, respectively, Ca²⁺-activated potassium channels that appear to colocate with a range of VDCCs. Activation of SK_Ca and BK_Ca channels is inhibited by addition of BAPTA, but unaffected by addition of EGTA (45).

These results can be accounted for by nanodomain effects, but they can also be accounted for by increased calcium currents through VDCCs in the presence of EGTA. Although the addition of EGTA increases the buffering capacity of the cytosol, this may be compensated for by the increased currents through VDCCs, resulting in larger-than-expected calcium concentrations that are sufficient to activate SK_Ca and BK_Ca channels. It is also possible that both nanodomain effects and VDCC CDI modulation due to chelators are occurring at the same time.

In the Supporting Material, we offer some thoughts on a theory of VDCC CDI modulation that are consistent with the experimental evidence found by Liang et al. (22) and make use of calcium nanodomain effects to explain the different effects of EGTA and BAPTA on VDCC CDI.

Finally, we wish to make experimentalists aware that the common use of calcium-binding fluorescent dyes to measure the free calcium concentration may itself alter calcium currents through VDCCs, further complicating this type of measurement.

A simplified model of CaMKII activation in the cytosol of dendritic spines may be sufficient

The model presented in this work combines two disparate models (5,14) and several lines of evidence (6,8,22,29) that cover different aspects of CaMKII activation and VDCCs. Yet, despite this heterogeneity in sources, it appears to give a qualitatively satisfactory account of CaMKII activation in the cytosol of dendritic spines. If the model can be regarded as successful in this endeavor, we would encourage the further use and development of simplifying approximations like the ISHA, which effectively nullify the combinatorial explosion that can arise when modeling CaMKII activation. We would also like to point out the inclusion of the closed state of CaMKII subunits, to which we could find only one previous reference (6) but which plays a large role in this model.

Conclusions

We have assembled a simplified, well-mixed model of CaMKII activation in the cytosol of dendritic spines built on many existing ideas. This model is able to qualitatively reproduce experimentally measured (15) time courses of CaMKII activation in the cytosol of dendritic spines during depolarization and glutamate uncaging, with and without added chelators. The chief simplifications in our model are the treatment of dendritic spines as single homogeneous volumes, modeling CaMKII autophosphorylation using the ISHA (14), and allowing only CaM_C and CaM_CC to activate CaMKII. To fit the model, we adjusted two previously measured parameters within error bounds and calibrated six experimentally unknown parameters so that the model matched experimental data on CaMKII activation in the absence of chelators as closely as possible. Without further changes, the model can then qualitatively reproduce CaMKII activation time courses during glutamate uncaging with EGTA or BAPTA perfused into the spine.

Experimental results of CaMKII activation during depolarization with EGTA or BAPTA perfused into the spine can be accounted for by including the modulation of VDCC CDI observed in dendritic spines when EGTA or BAPTA are present (22). The explanatory capability of this mechanism may go further than this work and appears to offer an alternative explanation for the often divergent results seen in experiments with perfused EGTA or BAPTA. We have also presented some simple arguments based on the total volume of calcium nanodomains that may aid in deciding when calcium nanodomain effects are a likely explanation for an experimental result. We hope that our work will prove to be useful in further analysis of CaMKII dynamics and if it should prove to be successful, that more well-mixed models will be considered and developed for small compartments.