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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 5, 2139-2149, 1 September 2008

doi:10.1529/biophysj.107.118505

Biophysical Theory and Modeling

A Simulation Study on the Activation of Cardiac CaMKII δ-Isoform and Its Regulation by Phosphatases

Hiroaki Chiba^*, †, Natalie S. Schneider^*, Satoshi Matsuoka^*, ‡, 1 and Akinori Noma^*, ‡, ,  

^* Cell/Biodynamics Simulation Project, Kyoto University, Kyoto, Japan
^† Pharmacology Laboratory, Mitsubishi Tanabe Pharma Corporation, Osaka, Japan
^‡ Department of Physiology and Biophysics, Graduate School of Medicine, Kyoto University, Kyoto, Japan

Address reprint requests to Akinori Noma, Dept. of Physiology and Biophysics, Graduate School of Medicine, Kyoto University, Yoshida-konoe, Sakyo-ku, Kyoto, 606-8501, Japan.

¹ Satoshi Matsuoka's present address is Innovation Center for Immunoregulation Technologies and Drugs, Graduate School of Medicine, Kyoto University, Kyoto, 606-8501, Japan.

With increasing heart rate, myocardial relaxation is accelerated to allow appropriate refilling of the ventricular cavity with the venous return 1,2. The Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) has been implicated in this autoregulation of frequency-dependent acceleration of relaxation (FDAR) 3,4,5. CaMKII is activated through the binding of Ca²⁺-bound calmodulin (CaM) during the transient increase in the intracellular Ca²⁺ concentration ([Ca²⁺]_i). In cardiac myocytes, activated CaMKII molecules phosphorylate many intracellular target proteins, including major components involved in excitation-contraction coupling 6,7, such as the sarcolemmal L-type Ca²⁺ channel, the ryanodine receptor, and the Ca²⁺ pump on the sarcoplasmic reticulum. With a rise in the frequency of the Ca²⁺ transient, the lifetime of activated CaMKII molecules is increased by intersubunit autophosphorylation, leading to an accumulation of the active CaMKII. Phosphorylated CaMKII maintains its catalytic activity even after the Ca²⁺ transient until it is inactivated by constitutive phosphatase activity. This was shown first for the brain-specific α and β isoforms of CaMKII and implicated in long-term potentiation, a mechanism playing a role in memory and learning 8,9. In heart, the predominant CaMKII isoform δ is found in two splice variants, δ_B localized to the nucleus and δ_C to the cytoplasm 10. Both variants were shown to undergo autophosphorylation 11. Autophosphorylation of CaMKII could potentiate the action of CaMKII during cyclic Ca²⁺ transients and thereby help to decode the stimulation frequency 12. However, the role of CaMKII and its autophosphorylation in the FDAR is still not well understood and is a subject of controversy 3,5,13,14. Furthermore, it has not yet been examined quantitatively how CaMKII activity is regulated by changes in heart rate.

Several computer models simulating CaMKII function have been developed based on in vitro experimental data, with the majority focusing on the neuronal CaMKII α and β isoforms 15,16,17,18,19. Although simple models for studying CaMKII activity in cardiac myocytes have been proposed, they are not directly based on experimental data on the δ isoform 20,21, nor do they consider deactivation of CaMKII by protein phosphatases (PPs) 22. It was experimentally demonstrated that the δ isoform has a higher affinity for CaM (K_d=33.5 nM) compared to α (K_d=62.4 nM) and a higher autophosphorylation rate, suggesting functional differences among the isoforms 22. Here, we introduce a novel CaMKII model that reflects the molecular properties of the δ isoform. This model achieves a good accordance with experimental data in vitro. The roles of CaMKII autophosphorylation and dephosphorylation by PPs in the frequency-dependent activation of CaMKII were demonstrated employing a cardiac Ca²⁺ transient model.

The CaMKII holoenzyme is a macromolecular complex consisting of two stacked ring-shaped hexamers. Binding of fully Ca²⁺-bound CaM (CaMCa₄) to the autoinhibitory domain of CaMKII exposes the catalytic site, which is capable of phosphorylating a wide range of target proteins. In addition, an activated CaMKII subunit is able to autophosphorylate neighboring subunits of the holoenzyme at Thr²⁸⁷. A phosphorylated CaMKII subunit has a 1000-fold higher affinity for CaMCa₄ than a nonphosphorylated one. Furthermore, in the phosphorylated state, CaMKII shows partial activity even after dissociation of CaMCa₄. CaMKII is completely deactivated only after dephosphorylation by PPs 9.

CaM, a highly conserved protein, possesses at its C-terminal lobe two high-affinity Ca²⁺-binding sites with a K_d of ∼1–2μM and at its N-terminal lobe two low-affinity sites with a K_d of ∼2.6–13μM, depending on experimental conditions 12,23,24. Since dissociation of Ca²⁺ from the C-terminal lobe is slow, the fraction of C- and N-terminal lobes occupied with Ca²⁺ might increase with increasing frequency of the Ca²⁺ transient. This mechanism may play an essential role in the activation of CaMKII. Therefore, we used the sequential four-step Ca²⁺ binding model described by Holmes 16 (CaM, CaMCa, CaMCa₂, CaMCa₃, CaMCa₄) (Figure 1A). This model includes cooperative Ca²⁺ binding within each lobe and assumes that the C-terminal Ca²⁺ binding sites are occupied before the N-terminal sites. The time-dependent changes of individual CaM states are determined as shown below:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

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Figure 1

(A) Scheme of a CaM model (see Eqs. (1) for kinetic dynamics). (B) Scheme of a CaMKII model (see Eqs. (10) for kinetic dynamics). This model consists of one inactive state (CaMKII) and three active states (CaMKII_CaMCa₄ (CaMCa₄ bound to CaMKII); CaMKIIP_CaMCa₄ (phosphorylated CaMKII_CaMCa₄); and CaMKIIP (CaMCa₄ dissociated from phosphorylated CaMKII dissociated from CaMCa₄)). For dissociation of CaMCa₄ from CaMKII_CaMCa₄ (A2) or CaMKIIP_CaMCa₄ (C1), two pathways are assumed, as indicated by numerals 1 and 2, respectively.

A four-state CaMKII model was developed, as shown in Figure 1B, with a nonactive state (CaMKII) and three active states, a CaMCa₄-bound state (CaMKII_CaMCa₄), an autophosphorylated CaMCa₄-bound state (CaMKIIP_CaMCa₄), and an autophosphorylated CaMCa₄-dissociated state (CaMKIIP). All active states are assumed to have the same activity. The rate A1 for the binding of CaMCa₄ to CaMKII is calculated with an association rate constant k_asso as follows:

(10)

(11)

(12)

(13)

Autophosphorylated CaMKII is dephosphorylated by several types of PPs 30,31, and 90% of PP activity in mammalian heart is mediated by PP1 and PP2A 32,33. Recently, PP1 was reported to dephosphorylate CaMKII in intact rat cardiac myocytes 14. In this study, only PP1 is considered. The dephosphorylation from CaMKIIP_CaMCa₄ to CaMKII_CaMCa₄ (Figure 1B, rate B2) is defined based on Michaelis-Menten kinetics, as shown below:

(14)

(15)

The dissociation of CaMCa₄ from CaMKIIP_CaMCa₄ (rate C1) is calculated in the same way as for A2. However, the dissociation rate constants k_disso2 and k_dissoCa2 have a 1000-fold lower value compared to k_disso and k_dissoCa, respectively, because autophosphorylated CaMKII shows a 1000-fold higher affinity for CaMCa₄19.

(16)

(17)

The time-dependent changes of individual CaMKII states are calculated as:

(18)

(19)

(20)

(21)

Since experimental data for the heart-predominant δ isoform are limited, a new kinetic model was first constructed for the brain-specific α isoform of CaMKII, for which experimental data obtained with different protocols are available. Then the δ isoform model was developed by modifying the parameter set of the α model according to the comparative experimental study carried out on the four isoforms of CaMKII by Gaertner et al. 22.

All experimental data referred to in this article were obtained from rat or mouse tissue. Since CaMKII shows a high sequence homology among species 35, the assumption might be justified that the same kinetic scheme and CaMKII activity are applicable to other species.

The hypothetical Ca²⁺ transient described by Negroni and Lascano 36 was used to test the frequency-dependent activation of CaMKII. Ca²⁺ release from sarcoplasmic reticulum is given by Q_rel and Ca²⁺ uptake by Q_pump:

(22)

(23)

(24)

The models were implemented in Java using the simBio package 37, software for cell simulation. Differential equations are solved using a Euler method with dynamically adjusted time steps.

To explore the degree of cooperativity in CaMKII activation, simulation results shown in Figure 2 and Figure 5 were fitted to the Hill equation (Eq. (25)) by nonlinear regression analysis using SigmaPlot (Version 10; SPSS, Chicago, IL)

(25)

Since fitting the CaMKIIα model to different experimental conditions was done manually, a meaningful parameter identifiability analysis could not be applied. However, the simulation results were superimposed onto experimental data to assess the appropriate parameter fitting.

The dependency of CaMKIIα activation on [CaMCa₄] (steps A1 and A2 of the model) was analyzed in the absence of ATP, i.e., no autophosphorylation. Figure 2A shows the experimental results described by De Koninck et al. 38 (solid circles) and Bradshaw et al. 34 (open squares), which were obtained by fixing the [Ca²⁺] and [CaMKII] and measuring the activation level of CaMKII in a quasi-steady state after addition of different [CaM] in vitro. With the assumption of a 1:1 binding of CaMCa₄ to CaMKII, i.e., n_H=1.0, our CaMKIIα model reconstructed well these experimental results (Figure 2A, solid line). The K_0.5 (k_disso/k_asso) of 66.7 nM is within the range of various experimental K_0.5 values, such as 48±6 nM 34 and 79±8 nM 38 or the K_d value for CaMCa₄ binding (62.4±25.1 nM) 22. The assumption of 1:1 binding, however, failed to simulate experimental results obtained by Gaertner et al. 22 (Figure 2B), who applied various [CaMKII]s to a solution containing fixed [Ca²⁺] and [CaM], and plotted the CaM-bound fraction against free [CaMKII]. Gaertner et al. 22 obtained a steeper slope with n_H=1.9 in their experiments, which they suggested was due to a positive cooperativity in the binding of CaMCa₄ to individual CaMKII molecules within the enzyme complex. However, the noncooperative binding of CaMCa₄ to CaMKII is further supported by experimental data from Bradshaw et al. 34, where the reaction of 0.2μM CaMKII with 5μM CaM was measured after adding 1–100μM [Ca²⁺] for 1min (Figure 2C, solid circles). Simulation data obtained by applying the same experimental procedure yielded n_H=2.98±0.03, which is in line with the experimental data. This high cooperativity results from cooperative binding of Ca²⁺ to CaM (Figure 2C, solid line).

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Figure 2

Quasi-steady-state activation of CaMKIIα by CaM. Simulations with the CaMKIIα model were carried out using in vitro experimental protocols. Solid circles and open squares represent experimental data, and solid lines the simulation results. (A) Experimental data are cited from Bradshaw et al. 34 (squares) and De Koninck and Shulman 38 (circles): In the presence of 500μM Ca²⁺, but the absence of ATP, 5 nM CaMKIIα was incubated with different [CaM] (1–10,000 nM) for 1min. The ratio of [CaMKII_CaMCa₄] to [CaMKII]_total is plotted against [CaM]_total (n_H=1.02; K_0.5=69.85±0.02 nM). (B) Experimental data are cited from Gaertner et al. 22: In the presence of 500μM Ca²⁺, but the absence of ATP, 100 nM CaM was incubated with different [CaMKIIα] (1–10,000 nM) for 1min. The ratio of [CaMKII_CaMCa₄] to [CaM]_total is plotted against the free [CaMKII] (n_H=1.00; K_0.5=67.46±0.00 nM). (C) Experimental data are cited from Bradshaw et al. 34: In the absence of ATP and in the presence of different [Ca²⁺] (0.1–100μM), 0.2μM CaMKII was incubated with 5μM CaM for 1min. The ratio of [CaMKII_CaMCa₄] to [CaMKII]_total is plotted against [Ca²⁺]; (n_H=2.98±0.03; K_0.5=3.46±0.01μM).

To examine the autophosphorylation rate, the dependency of the autophosphorylation level on either [CaM] or [Ca²⁺] was analyzed in the presence of ATP. Fig. 3 shows a comparison of experimental findings with simulation results. Since the dephosphorylation step, B2 in the CaMKII model, is suppressed in the absence of phosphatase activity, the phosphorylation level reached after a given activation time is dependent on the overall autophosphorylation rate, which is determined by steps A1, A2, and B1 in the model. The solid line in Figure 3A indicates the sum of [CaMKIIP_CaMCa₄] and [CaMKIIP] obtained after a 15-s activation with different [CaM] at 0°C, according to the protocol by Gaertner et al. 22. Figure 3B shows the reconstruction of an experiment as performed by DeKoninck et al. 38, who measured the autophosphorylated fraction after 6s at 30°C. Figure 3C shows a comparison of the simulation results with data from an experiment performed by Bradshaw et al. 34, who measured the autophosphorylation level 5min after each applied [Ca²⁺] at 0°C. The model parameters for the CaM-Ca²⁺ binding, and k_asso, k_disso, and kcat match well all experimental data tested above. The steeper slope in the relationship in Figure 3C is caused by the cooperative binding of Ca²⁺ to CaM. The saturation of the relationships in Fig. 3 is due to the saturation of the reaction rates of steps A and B at higher [CaM] or [Ca²⁺], and not to completion of autophosphorylation. Thus, the saturation level is determined by the duration of the activation time unique to each experimental protocol. In the case of saturating high [Ca²⁺], such as the 500μM (KmCaM=30 nM) used in the experiments shown in Figure 3AB (Eq. (11)), pathway 2 of step A2 is almost completely blocked. It should be noted that the fraction of CaMKIIP is a minor population in the sum of [CaMKIIP_CaMCa₄] and [CaMKIIP], because of the 1000-fold smaller k_disso2 compared with k_disso.

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Figure 3

Relationship between the autophosphorylation level of CaMKIIα and [CaM] or [Ca²⁺]. Solid circles represent the experimental data and solid lines the simulation results. (A) Experimental data are cited from Gaertner et al. 22: In the presence of 0.1mM ATP and 500μM Ca²⁺, 62 nM CaMKIIα was incubated with different [CaM] (1–10,000 nM) for 15s at 0°C. The ratio of ([CaMKIIP]+[CaMKIIP_CaMCa₄])/[CaMKII]_total) normalized to the maximum fraction is plotted against [CaM]_total. (B) Experimental data are cited from De Koninck and Shulman 38. In the presence of 0.25mM ATP and 500μM Ca²⁺, 5 nM CaMKIIα was incubated with different [CaM] (1–10,000 nM) for 6s at 30°C. The ratio of ([CaMKIIP]+[CaMKIIP_CaMCa₄])/[CaMKII]_total) is plotted against [CaM]_total. (C) Experimental data are cited from Bradshaw et al. 34: In the presence of 2mM ATP and different [Ca²⁺] (0.1–100μM), 0.2μM CaMKII was incubated with 50μM CaM for 5min at 0°C. The percentage of the autophosphorylated CaMKII fraction is plotted against [Ca²⁺]. Different values of kcat were used for the simulations to adjust the temperature (see Table 1).

The frequency-dependent activation of CaMKIIα was demonstrated by De Koninck et al. 38, who exposed CaMKII molecules immobilized on a membrane to a phosphorylation mixture containing Ca²⁺, CaM, and ATP for a 200-ms duration at 1.0, 2.5, and 4.0Hz. The CaMKII autophosphorylation level increased with time, as indicated in Fig. 4 (symbols), and the slope of this time-dependent increase was markedly accelerated with increasing phosphorylation frequency. As indicated by the solid lines in Fig. 4, the CaMKIIα model satisfactorily reconstructed these experimental data. During the time interval between the applications of the phosphorylation mixture, dissociation of CaMCa₄ from CaMKII_CaMCa₄ proceeded mainly along pathway 2 in the kinetic scheme in Figure 1B, in which Ca²⁺ dissociates from CaMKII_CaMCa₄ before the dissociation of CaM 23. When the application interval was shortened, at higher frequencies, the mean level of CaMKII_CaMCa₄ increased, which resulted in an accelerated accumulation of phosphorylated species. If hypothetical pathway 2 was eliminated from reaction step A2, the dissociation of CaMCa₄ from CaMKII_CaMCa₄ through pathway 1 was too slow during the stimulation interval, and the frequency dependence of the CaMKII autophosphorylation observed over the range of ∼1–4Hz was highly reduced (data not shown). It should be noted that the involvement of pathway 2 is negligibly small under the experimental conditions used in Fig. 3, but becomes a major route when [Ca²⁺] is low compared to the KmCaM of 30 nM 23 (see Eq. (11)). The fact that Ca²⁺ was chelated using EGTA during the intervals in experiments by De Koninck et al. 38 validates the addition of pathway 2 in our model.

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Figure 4

Frequency-dependent activation of CaMKIIα. Experimental data are cited from reference 38. The phosphorylation mixture (500μM Ca²⁺, 100 nM CaM, and 0.25mM ATP) was applied to CaMKIIα for 200ms at different frequencies; solid squares, open triangles, and solid circles represent 1, 2.5, and 4Hz, respectively. During the interval of applying the phosphorylation mixture, [Ca²⁺] was set at zero. Solid lines represent the simulation results.

Gaertner et al. 22 found that the CaMKIIδ isoform exhibited a higher CaM affinity, specifically, K_d=33.5 nM versus K_d=62.4 nM, and a faster autophosphorylation compared to the CaMKIIα isoform. To convert the CaMKIIα model to a CaMKIIδ model with a minimum of modifications, the rate constants k_disso, k_dissoCa, k_disso2, and k_dissoCa2 were decreased twofold and kcat was increased sixfold. Figure 5A, which corresponds to Figure 2B for the CaMKIIα isoform, compares the simulation results (solid line) with experimental data (filled circles) obtained with the CaMKIIδ isoform for the quasi-steady-state relationship between the free [CaMKII] and the CaMCa₄ bound fraction, [CaMKII_CaMCa₄]. Figure 5B represents the autophosphorylation level of CaMKIIδ accumulated after a 15-s application of different [CaM] (1–10,000 nM) at 0°C in the presence of 0.1mM ATP and 500μM Ca²⁺, corresponding to Figure 3A, which shows the same relationship for CaMKIIα. The CaMKIIδ model obtained after the above-described changes of only five rate constants well matches experimental findings.

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Figure 5

(A) Steady-state binding of CaM to CaMKIIδ and (B) CaMKIIδ autophosphorylation after a given activation time with [CaM]. Solid circles represent experimental data and solid lines the simulation results. Experimental data are cited from Gaertner et al. 22. (A) Experimental conditions used for the simulation are the same as described in Figure 2B; (n_H=1.0; K_0.5=33.77±0.03 nM). (B) Experimental conditions used for the simulation are the same as those described in Figure 3A.

For the reconstructions of the experimental findings described above (Figure 2 and Figure 3 and Figure 4 and Figure 5), Ca²⁺ was either applied continuously or as repetitive 200-ms pulses. However, to investigate the potential role of CaMKII phosphorylation in the FDAR 3,5,13,14, it is important to use the “physiological” myocyte Ca²⁺ transient. The upper panels in Figure 6A show the cardiac Ca²⁺ transient 36, which was applied in the examination of the cumulative CaMKII activation as shown in Figure 6B. First, the time courses of [CaMCa₂] and [CaMCa₄] were calculated at stimulus frequencies of 3 and 5Hz to examine whether active CaM accumulates with increasing stimulus frequency due to the slow dissociation of Ca²⁺ from the C-terminal lobe. As illustrated in the lower panels of Figure 6A, accumulation of active CaM is absent at 3Hz, but a slight accumulation was detected at 5Hz. This accumulation is due to a slight overlap of the Ca²⁺ tail with the next Ca²⁺ transient. On the contrary, the cumulative activation of CaMKII occurred progressively even at a stimulus frequency of 0.5Hz (Figure 6B, left panel). The rapid rise of the [CaMKII_CaMCa₄] with a rise in the [Ca²⁺] was followed by a relatively slow decay during each stimulus cycle. The phosphorylated fraction ([CaMKIIP_CaMCa₄]) increased steadily with time, eventually to 100%, since no phosphatase activity was provided in this simulation. Due to the slow relaxation during “diastole”, the rate of [CaMKII_CaMCa₄] increase was markedly enhanced at the higher stimulus frequency of 3Hz (Figure 6B, right panel). With time the [CaMKIIP_CaMCa₄] markedly increased at the expense of the [CaMKII_CaMCa₄]. It is suggested that the cumulative activation of CaMKIIδ is not attributable to the Ca²⁺ binding to CaM, but to the CaMKIIδ kinetics over the physiological range of the heart rate.

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Figure 6

Cumulative activation of CaMKIIδ by Ca²⁺ transients in the absence of PPs. (A) Time courses for [CaMCa₂] (lower panels, black lines) and [CaMCa₄] (red lines) evoked by applying the Ca²⁺ transient of Negroni and Lascano 36 indicated in the upper panel at stimulus frequencies of 3Hz (left panel) and 5Hz (right panel). Stimulation was started at time zero after a long pause. (B) Simulation results obtained with the CaMKIIδ model activated with the Ca²⁺ transient at 0.5Hz (left panel) or 3Hz (right panel) at 37°C. Concentrations used are 5mM [ATP], 6μM [CaM], and 0.1μM [CaMKIIδ]. Upper panels depict the Ca²⁺ transients, and lower panels show the time-dependent changes of the CaMKII_CaMCa₄ fraction (black lines) and the CaMKIIP_CaMCa₄ fraction (red lines). The changes in the CaMKIIP fraction were too small to be detected in these graphs.

In vivo dephosphorylation by PPs affects the frequency-dependent activation of CaMKII. We first examined whether our CaMKIIα model could reproduce experimental data obtained for the neuronal CaMKIIα. The steady-state level of autophosphorylation was measured by Bradshaw et al. 34. The experimental findings are shown in Fig. 7 in comparison with simulation results obtained with essentially the same experimental protocol used in Figure 3C, but in the presence of 1.25μM of PP1. The experimentally determined values of kcat_PP1 and Km_PP1 inserted in this CaMKIIα model could well reproduce these experimental data 34. Due to the lack of experimental data for the δ isoform, the same pair—kcat_PP1 and Km_PP1—was used in the CaMKIIδ model to calculate this relation (Fig. 7, red line). The observed leftward shift of the [Ca²⁺] dependency is due to the higher rates in A2 and B1, which were validated in Fig. 5.

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Figure 7

Ca²⁺-mediated activation of CaMKIIα and CaMKIIδ in the presence of PP1. Solid circles represent experimental data and solid lines the simulation results for CaMKIIα (black) and CaMKIIδ (red). Experimental data are cited from Bradshaw et al. 34. In the presence of 2mM ATP and different [Ca²⁺] (0.1–100μM), 1μM CaMKII and 1.25μM PP1 were incubated with 5μM CaM at 0°C. The steady-state value of the autophosphorylated CaMKII fraction is plotted against [Ca²⁺].

In Figure 8A, the influence of PP1 on the frequency-dependent activation of CaMKIIδ was examined. To our knowledge, however, the cellular concentration of PPs has not been precisely measured, and region-specific variations in PP expression were found in the human heart 33. Therefore, we examined the opposing influence of PP1 on the autophosphorylation of CaMKII by systematically changing the [PP1] from 0.01 to 3μM in the simulations. The steady-state values of the activated fraction, given by ([CaMKII]_total – [CaMKII])/[CaMKII]_total, are plotted against the frequency (Figure 8A). At a low [PP1] of 0.01μM, the activated fraction of CaMKIIδ is highly frequency-dependent, ranging from 14% at 0.5Hz to 96% at 5Hz, displaying a hyperbolic relationship. With increasing [PP1], the activation-frequency curve changes to a sigmoidal shape over the medium range of [PP1] and to an exponential relation for high phosphatase levels. At a [PP1] of 3μM, CaMKIIδ autophosphorylation was almost completely suppressed.

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Figure 8

Effects of [PP1] variation on the frequency-dependent activation of CaMKII. The cardiac Ca²⁺ transient, shown in Figure 6A was applied at different frequencies (0.5, 1, 2, 3, 4, and 5Hz) to the CaMKIIδ (A) and the CaMKIIα model (B). The steady-state values of ([CaMKII]_total – [CaMKII])/[CaMKII]_total are plotted against the frequency of the Ca²⁺ transient. The simulations were performed in the presence of 5mM ATP, 6μM CaM, and a total [CaMKII] of 0.1μM for different [PP1] (0, 0.01, 0.03, 0.1, 0.3, 1, and 3μM) at 37°C.

To demonstrate functional differences between the CaMKII isoforms, the same simulation protocol was applied to the CaMKIIα model. As shown in Figure 8B, CaMKIIα activation was much more sensitive to [PP1]. A [PP1] of 0.1μM could almost suppress the activation. These results are due to the lower affinity of CaMKIIα for CaM and its slower autophosphorylation rate.

The relationship shown in Figure 8A suggests a role of PPs in the dynamic adjustment of CaMKIIδ activity over the physiological range of the heart rate. For example, in guinea pig heart, with a physiological frequency range of 3–6Hz, a [PP1] of 0.1–0.3μM might play a role, whereas in the human heart, with a rate of 1–3Hz, a much lower [PP1] of 0.01–0.03μM might be appropriate.

To clarify the dynamic regulation of the CaMKII activation, especially through variation of both the heart rate and the concentration of PPs, a novel and simple four-state CaMKIIδ model was developed that includes autophosphorylation and dephosphorylation by PP1. Although several cardiac CaMKII models were reported, they were not directly fitted to experimental data of the δ isoform 20,21 or did not include the deactivation of CaMKII by dephosphorylation 22. Both models, CaMKIIα and CaMKIIδ, used in this study could well reproduce experimental findings regarding the steady-state dose-response relationships for activation by CaMCa₄ (Figure 2A and Figure 5A), and the time-dependent accumulation of activated CaMKII fraction (Figure 3B and Figure 5B). Simulations for the frequency-dependent activation of CaMKIIα induced by a repetitive pulse-like application of the phosphorylation mixture interrupted by a Ca²⁺-free medium strongly supports the existence of pathway 2 of step A2 in the kinetic scheme (Fig. 1), where Ca²⁺ dissociates before CaM from CaMCa₄-bound CaMKII. The PP1 regulation of the frequency-dependent accumulation of activated CaMKII was clarified. In particular, PP1 within a physiological concentration range (0.01–0.1μM) provides a dynamic way to regulate the frequency-dependent activation of CaMKIIδ, and the heart rate variations are well covered. Our simulation results suggest that different phosphatase activities might be involved in frequency-dependent regulation of CaMKII activity, adjusting the activation level of CaMKII to the various physiological heart rates found for different species.

Total concentrations of CaM, PP1, and CaMKII greatly affect simulation results, as exemplified by varying the concentration of PP1 in Fig. 8. In this study, protein concentrations were chosen that represent the level expected for cardiac myocytes. A [CaM]_total of 6μM is in accordance with an experimental measurement in cardiac myocytes 12. Although it is difficult to directly measure [PP1] and [CaMKII] in experiments, we estimated a [PP1] based on experimental values of PP1 activity/total cellular protein, A_p, of 1.07 units/mg 39, and protein concentration, [P]_cell, of 37.5 mg/ml 40 measured in cardiac myocytes. Using a PP1-specific activity, A_s, of 13,300 units/mg and the PP1 molecular weight (M_PP1=36,000), [PP1] was determined as 0.084μM using the equation

(26)

The combination of 0.1μM [CaMKII], 0.1μM [PP1], and 6μM [CaM]_total in Figure 8A well reconciled the apparent dissociation of FDAR and protein phosphorylation by CaMKII suggested by Huke and Bers 14. They found no change in the amount of phosphorylated CaMKII after 2-Hz pacing for 5min in rat cardiac myocytes. A rise of CaMKII autophosphorylation was observed only after a strong inhibition of PP1 at the 2-Hz pacing. They suggested that CaMKII has no role in FDAR. Our simulation results, presented in Fig. 8, are in roughly good agreement with their experimental results. In particular, in the presence of a physiological [PP1] of 0.1μM, the fraction of activated CaMKII is negligibly small at 2-Hz stimulation, but gains significance progressively with decreasing PP1 activity, which would correspond to an experimental blocking of PPs. However, it is interesting to note that the simulation predicts that CaMKII is indeed a feasible candidate for regulating FDAR over the physiological heart rate (∼240–400min⁻¹, or 4–7Hz) in rat heart. At these higher frequencies, dynamic changes in the autophosphorylation of CaMKII are apparent in Fig. 8 even at high [PP1] such as 1μM. We conclude that CaMKII is capable of mediating cardiac FDAR, with CaMKII autophosphorylation playing a role.

The structural and enzymatic differences among the α, β, γ, and δ CaMKII isoforms were extensively studied in vitro, and their differences were reported to be small 22. However, it was suggested that subtle changes among these isoforms might result in drastic differences in the activation dynamics under various cellular environments 22. For converting the α isoform model to the CaMKIIδ model based on experimental data, a twofold decrease in the CaM affinity and a sixfold increase in the autophosphorylation rate were required. To estimate the physiological impact of these differences, simulations were performed with both models using the same cardiac myocyte specific Ca²⁺ transient only for comparison, although neuronal Ca²⁺ transients are different 16 in both time course and peak magnitude from Ca²⁺ transients in cardiac myocytes. According to our simulation results (Figure 8B), CaMKIIα autophosphorylation was suppressed at all frequencies tested by a [PP1] of 0.1μM, which is close to our estimated cellular [PP1] in cardiac myocytes. This is in strong contrast to the results in Figure 8A, where CaMKIIδ was strongly activated at higher frequencies.

CaMKIIα is accumulated at the postsynaptic density in neurons and involved in long-term potentiation, which is induced by high stimulation frequencies (∼30Hz) 8,9,41. Therefore, lower frequencies, such as 5Hz, used in the preent simulations described here do not seem sufficient to activate CaMKIIα, especially in the presence of PPs. This clearly shows that CaMKII isoforms are adapted to meet the conditions in their respective cellular environment.

As our proposed model is a simple one, it exhibits limitations. For all three active states considered in the model (CaMKII_CaMCa₄, CaMKIIP_CaMCa₄, and CaMKIIP) the same autophosphorylation activity was assumed. This is different than the general assumption that some CaMKII states exhibit only partial activity, in particular, CaMKIIP 19. However, since the fraction of CaMKIIP is quite small compared to those of the other two active states, a variation of the autophosphorylation activity for different states might not remarkably influence our conclusions. Moreover, CaMKIIα autophosphorylated at Thr²⁸⁶ was shown to undergo further autophosphorylation at Thr³⁰⁵/Thr³⁰⁶ after dissociation of CaM, known as secondary or inhibitory autophosphorylation. In this state, sometimes called the capped state 19, CaMKII still exhibits some autonomous activity, but it cannot regain full activity through CaMCa₄ binding since these threonine residues reside in the CaM binding site. It has been reported that inhibitory autophosphorylation plays an important role in synaptic metaplasticity 42. Since, to our knowledge, it is not known whether secondary autophosphorylation occurs in cardiac myocytes, a capped state was not considered in our model.

Experimental data for the kinetic properties of CaMKIIδ are still very limited. Therefore, our model may have to be modified in the future according to newly obtained experimental results, especially concerning the localization of CaM, PPs, and CaMKII molecules within the cell, as well as their local concentrations. In this study, [PP1] and [CaMKII] were estimated based on average, probably cytoplasmic, concentrations within the cell (Eq. (26)). Any variation in the Ca²⁺ affinity of the N-terminal lobe of CaM largely affects reaction step A1 by varying the [CaMCa₄] (Fig. 1). K_d measurements for Ca²⁺ binding to the N-terminal lobe of CaM are variable, ranging from 2.6 to 13μM depending on experimental conditions 12,23,24. It should be noted that the simulation study presented here is useful in designing new experimental studies despite the above-stated limitations of the model.

Although CaMKII autophosphorylation might be insignificant in the normal heart, due to strong control by phosphatases, an almost threefold increase in CaMKII activity was found in patients with end-stage dilated cardiomyopathy 43. Furthermore, in mice in which pressure overload was induced by aortic constriction, Zhang et al. found that the expression of CaMKIIδ_C was upregulated and the kinase activity was increased due to autophosphorylation 44. The same group found, in addition, that transgenic mice overexpressing CaMKIIδ_C developed a dilated cardiomyopathy. In an immunoprecipitation compared to wild-type mice, a greater amount of the enzyme was associated with the ryanodine receptor, a CaMKII target, resulting in a high phosphorylation of the receptor and, subsequently, a strongly altered Ca²⁺ homeostasis 45, whereas PP1 and PP2A protein levels were unaltered 44. Moreover, in transgenic mice overexpressing the phosphatase calcineurin which resulted in a severe cardiomyopathy, CaMKII activity was also found to be increased 46. The above experimental findings clearly show that in cardiac myocytes, CaMKII and its PPs need to be highly regulated, and that a tiny disturbance of this balance could result in heart disease. Future work will incorporate the CaMKIIδ model presented here in a comprehensive cardiac myocyte model to analyze these complex mechanisms.