INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Long-term potentiation (LTP) is the most studied experimental model of memory (Bliss and Collingridge, 1993; Huang et al., 1996; Malenka and Nicoll, 1999). Experimental data suggest that LTP relates to intermediate memory, which can be rewritten, as shown by phenomena of depotentiation and long-term depression (Linden, 1994; Bear and Malenka, 1994; Bear, 1996). Because memories of such type often consist of bistable elements, it is natural to ask whether some of the biophysical and biochemical systems, which participate in LTP, can operate as bistable switches. Lisman (1985) suggested that elements of the synaptic memory can use bistability of protein phosphorylation, and analyzed a bistable system that consisted of a protein kinase capable of intermolecular autocatalytic autophosphorylation and a phosphatase that dephosphorylates the kinase.

Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) may be a key component of such a bistable molecular switch. It is known that autophosphorylation of CaMKII results in Ca²⁺/CaM-independent (autonomous) kinase activity (Braun and Schulman, 1995; Hanson and Schulman, 1992; Soderling, 1995). During induction of LTP, short-term high elevations of Ca²⁺ concentration produce autophosphorylation of CaMKII that lasts 30 min or longer (Fukunaga et al., 1993, 1995; Miller and Kennedy, 1986; Ouyang et al., 1997). As long as CaMKII remains autophosphorylated it is able to phosphorylate the LTP related targets, such as AMPA receptors (Barria et al., 1997), even after the return of [Ca²⁺] to its resting level (Lisman et al., 1997).

Although a crucial role of CaMKII autophosphorylation in induction of LTP is widely accepted, its participation in maintenance of LTP remains controversial (Giese et al., 1998; Kennedy, 1998; Lisman, 1994; Lisman et al., 1997). To prove the latter, it is necessary to show that a permanent level of autonomous activity of CaMKII lasts in the involved synapses as long as LTP does. Some indirect supporting evidence exists. Zhao et al. (1999) have found that a lasting increase in the in vivo phosphorylation level of CaMKII accompanies long-term memory in chicks. Also, it seems almost evident that long-term autophosphorylation of CaMKII should be connected with some specific structures in neurons. The basal autonomous CaMKII activity constitutes ~5% of the maximum activity in neurons, while it is only ~0.03% in the purified enzyme (Hanson and Schulman, 1992). This relatively high level of autonomous activity cannot be maintained in cytosol of neurons, where protein phosphatase activity is relatively high and essentially Ca²⁺-independent (Strack et al., 1997). Thus, it takes only several minutes to dephosphorylate CaMKII added to cell extracts from the hippocampal CA1 region (Fukunaga et al., 2000). The postsynaptic densities (PSD) (Harris and Kater, 1994; Ziff, 1997) are the structures that could maintain high levels of CaMKII autophosphorylation because protein phosphatase activity in PSD is Ca²⁺-dependent in vivo and should be rather low at the resting level of Ca²⁺ (Mulkey et al., 1994; Strack et al., 1997). Indeed, ~10% of CaMKII remained phosphorylated in the total PSD fraction when isolation was done in the presence of inhibitors of protein phosphatases (Strack et al., 1997).

In view of these data, it is of interest to determine whether CaMKII autophosphorylation can be bistable in a wide range of [Ca²⁺] and whether such bistability can play a role in LTP.

Several mathematical models have been developed to study the dynamics of CaMKII autophosphorylation in response to variations of intracellular calcium concentration (Lisman and Goldring, 1988; Michelson and Schulman, 1994; Matsushita et al., 1995; Dosemeci and Albers, 1996; Coomber, 1998; Kubota and Bower, 1999). They showed a strong dependence of the level of autophosphorylation on the amplitude and duration of Ca²⁺ pulses. Recently, Okamoto and Ichikawa (2000) have demonstrated that CaMKII autophosphorylation is bistable in their model when [Ca²⁺] varies from 0.28 to 0.33 µM. This range is quite narrow and does not include the resting Ca²⁺ concentration.

Here, I develop a simple model of autophosphorylation of CaMKII in the presence of a phosphatase. I show that if the total concentration of CaMKII subunits is significantly higher than the phosphatase Michaelis constant, two stable steady states of CaMKII autophosphorylation can be found over a wide range of concentrations of the intracellular calcium. This range includes the resting Ca²⁺ concentration, and is wide enough to ensure that the background firing activity cannot induce transition from the low- to the high-phosphorylated states of CaMKII. I demonstrate that Ca²⁺ transients that induce such transitions are in the same range of amplitudes and frequencies as the Ca²⁺ transients that induce LTP. Moreover, concentrations of CaMKII, which are necessary to produce such a bistability, are shown to be of the same order of magnitude as the values found in the postsynaptic density. These results show that the CaMKII-phosphatase bistability can play an important role in the long-term synaptic modifications. They also suggest a plausible explanation for the very high concentrations of CaMKII found in postsynaptic densities of cerebral neurons.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Model

It is known that autophosphorylation of Thr²⁸⁶ in -subunits of CaMKII results in Ca²⁺/CaM-independent (autonomous) kinase activity (Hanson and Schulman, 1992). The autophosphorylation is an intraholoenzyme, intersubunit process (Hanson et al., 1994; Mukherji and Soderling, 1994). CaMKII consists of 8-12 subunits that form a petal structure (Hanson and Schulman, 1992). During autophosphorylation, one subunit acts as a catalyst and another as a substrate. Either binding of Ca²⁺/calmodulin (Ca²⁺/CaM) or phosphorylation of Thr²⁸⁶ is necessary for activation of the catalytic subunit. The substrate subunit must also bind Ca²⁺/CaM to be phosphorylated (Hanson et al., 1994).

Here I use a model with many simplifications. I ignore any difference between - and -subunits. Binding of ATP to a subunit does not appear explicitly in the model. Binding of Ca²⁺/CaM and phosphatase to a subunit is independent of the state of other subunits. Activation of subunits by Ca²⁺ obeys the Hill equation. Binding of phosphatase to a phosphorylated subunit and the rate of dephosphorylation is independent of binding of Ca²⁺/CaM. I consider catalytic activities of the unphosphorylated subunit bound to Ca²⁺/CaM and the phosphorylated subunit as being equal.

I use a model with asymmetric interaction of the neighbor subunits as proposed by Hanson and Schulman (1992). Essentially, their model is a ring along which autophosphorylation propagates in one direction. I use a model with 10 subunits in simulations and, in what follows, all the coefficients related to the number of subunits in a holoenzyme correspond to the 10-subunit model. Fig. 1 shows a schematic of the first two steps of autophosphorylation of a holoenzyme with six subunits. The left column presents initiation of autophosphorylation. After two neighbor subunits bind (Ca²⁺)₄CaM, designated C, the first subunit phosphorylates the second one in the clockwise direction. The following scheme corresponds to the initiation step:

(1)

(2)

(3)

(4)

Here P₀ is the unphosphorylated holoenzyme and P₁ is the 1-fold phosphorylated holoenzyme. I describe the (Ca²⁺)₄CaM activation of subunits by the empirical Hill equation:

(5)

Here F is the fraction of subunits bound to (Ca²⁺)₄CaM and K_H1 = [Ca²⁺]₅₀ is the calcium Hill constant of CaMKII. The probability of (Ca²⁺)₄CaM binding to two neighbor subunits is low when [Ca²⁺] is significantly less than K_H1. In this case, the rate of the initiation step is:

(6)

Here k₁ is the rate constant of Reaction 4, 10 is the statistical factor.

View larger version (62K): [in this window] [in a new window]   FIGURE 1   Schematic of the first two steps of autophosphorylation of CaMKII holoenzyme. C designates the Ca2+/calmodulin complex, P is orthophosphate.

The middle and right columns in Fig. 1 present two routes of propagation of autophosphorylation. Autonomous activity of the CaMKII is usually 60-80% of the (Ca²⁺)/CaM-dependent activity (Hanson and Schulman, 1992). I assume here that catalytic activity of the autophosphorylated subunit does not depend on binding of (Ca²⁺)₄CaM. This means that I neglect differences between the middle and right columns, and in this case the rate of propagation of autophosphorylation depends only on the following reactions:

(7)

(8)

Correspondingly, the rate of the second autophosphorylation step is:

(9)

Here k₁ is the rate constant of Reaction 8, which is the same as of Reaction 4.

In the absence of dephosphorylation, the subsequent steps of propagation of autophosphorylation are identical to the first one because there is always only one autophosphorylating pair in which the unphosphorylated subunit is adjacent to the phosphorylated neighbor in the clockwise direction. However, protein phosphatases dephosphorylate subunits at random. As a result, there is a random distribution of phosphorylated and unphosphorylated subunits in the holoenzyme. I assume that all distinguishable configurations of subunits in the holoenzyme with a given number of phosphorylated subunits exist with equal probabilities. In this case, the effective number of autophosphorylating pairs is:

(10)

where m_j is number of distinguishable configurations with j autophosphorylating pairs.

Values of w_i are: w₁ = w₉ = 1.0, w₂ = w₈ = 1.8, w₃ = w₇ = 2.3, w₄ = w₆ = 2.7, w₅ = 2.8.

Then, the rate of autophosphorylation of holoenzymes with i phosphorylated subunits is:

(11)

where

(12)

is the per-site rate of propagation of autophosphorylation.

Dephosphorylation of subunits proceeds according to the Michaelis-Menten scheme:

(13)

Here S is an unphosphorylated subunit and SP is a phosphorylated one. Accordingly, the rate of dephosphorylation of holoenzymes with i phosphorylated subunits is:

(14)

where

(15)

is the per-subunit rate of dephosphorylation, e_p is the concentration of the active protein phosphatase, and k₂ and K_M are the catalytic and the Michaelis constants, respectively.

Four protein phosphatases dephosphorylate CaMKII-P: PP1, PP2A, PP2C (Strack et al., 1997), and a specific CaMKII phosphatase (Ishida et al., 1998; Kitani et al., 1999). According to Strack et al. (1997), activity of PP1 is 20% and PP2A is 60% of the total phosphatase activity in cytosol and, correspondingly, 50% and 8% in PSD. PP1 is the only protein phosphatase that dephosphorylates CaMKII in PSD according to Strack et al. (1997) and Yoshimura et al. (1999).

Activity of PP1 can be controlled by Ca²⁺/CaM via inhibitor I, calcineurin (CaN), and cAMP-dependent protein kinase (PKA) (Shenolikar and Nairn, 1991; Mulkey et al., 1994). PKA phosphorylates inhibitor 1 (I1) and calcineurin dephosphorylates it (Shenolikar and Nairn, 1991). The Hill number for Ca²⁺ activation of CaN is ~3 (Stemmer and Klee, 1994). Phosphorylated inhibitor-1 (I1P) deactivates PP1 with K_I = 1 nM (Endo et al., 1996). The corresponding scheme is:

(16)

I assume that the concentration of free I1 is constant and much less than K_M of PKA, and the concentration of free I1P is much less than K_M of CaN.

The complete model of autophosphorylation of CaMKII in the presence of Ca²⁺-dependent PP1 is:

(17)

Here P_i is the concentration of the i-fold phosphorylated CaMKII holoenzyme, e_p is the concentration of PP1 not bound to I1P, e_p0 is the total concentration of PP1, I is the concentration of free I1P, and I₀ is the concentration of free I1; v₁, v₂, and v₃ are given by the rate Eqs. 6, 12, and 15; k₃ and k₄ are, respectively, the association and dissociation rate constants of the PP1 · ·I1P complex; V_CaN = V_CaN/K_M2, V_CaN is the activity, K_M2 is the Michaelis constant, and K_H2 is the Hill constant of CaN, v_PKA = V_PKA/K_M3; V_PKA and K_M3 are the activity and the Michaelis constant of PKA.

During propagation of autophosphorylation I neglect the initiation terms because the probability of (Ca²⁺)₄CaM binding to two neighbor subunits is very low when [Ca²⁺] is significantly less than [Ca²⁺]₅₀.

I use a model with a Ca²⁺-independent protein phosphatase in some simulations. Inhibitor 1 is washed out during isolation of PSD. To simulate suggested in vitro experiments with isolated PSD, I set I₀ = 0 in Eq. 17. This results in e_p = e_p0, and is equivalent to elimination of the two last equations in Eq. 17 and treating e_p as a parameter in the remaining equations. I also use this truncated model to simulate kinetics of autophosphorylation and dephosphorylation of CaMKII in cytosol, where 80% of the phosphatase activity is Ca²⁺-independent, and I can neglect the Ca²⁺ dependence of the remaining 20%. Table 1 gives the values of parameters used in the simulations.

Calcium signaling

In the time-dependent simulations, I use Ca²⁺ dynamics generated by a simple protocol for induction of LTP, which is a single tetanus with frequencies of excitation varying from 5 to 100 Hz, and duration in the range from 1 to 1800 s (Bliss and Collingridge, 1993; Huang et al., 1996). Ca²⁺ response to a single depolarization pulse is modeled by an instant elevation and the following exponential decay. I assume a simple summation of Ca²⁺ pulses during periodic excitation (Helmchen et al., 1996):

(18)

Here [Ca²⁺]_rest is the resting concentration of Ca²⁺, A is the amplitude of a single Ca²⁺ pulse, f is the frequency of excitation,  is the relaxation time of Ca²⁺ decay, and n is the number of pulses in the tetanus. I use the following values of the parameters: [Ca²⁺]_rest is 0.1 µM,  is 0.2 s (Helmchen et al., 1996; Magee and Johnston, 1997; Majewska et al., 2000). I take A equal to 0.4 µM to simulate background firing. To induce LTP, EPSPs and action potentials must coincide in time. In this case, amplitudes of the single Ca²⁺ pulses become rather large (Yuste et al., 1999). I use A equal to 1.0 µM to simulate induction of LTP.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Fig. 2 shows how the steady-state values of the total concentration of phosphorylated subunits of CaMKII depend on the concentration of Ca²⁺. In the shaded area of the region of bistability, three steady states correspond to each [Ca²⁺]. The top and bottom steady states are stable and the middle one is unstable. Arrows show directions of evolution of the system in time at constant concentrations of Ca²⁺. A is the static [Ca²⁺]-threshold for down-switching, and B is the up-switching static threshold. These thresholds are boundaries of the bistability region.

View larger version (71K): [in this window] [in a new window]   FIGURE 2   Bistability of the steady-state concentrations of phosphorylated subunits of CaMKII. In the shadowed bistability region, the top and bottom branches of the steady-state curve consist of the stable steady states (), while the middle branch consists of the unstable steady states (). Arrows show directions of evolution of the system in time at constant concentrations of Ca2+. A is the static [Ca2+]-threshold for down-switching, and B is the up-switching static threshold. Parameters (in µM): KH2 = 0.7, KM = 0.4, ek = 20.0, ep0 = 0.05, I0 = 0.1.

Fig. 3 shows how boundaries of the region of bistability depend on various parameters of the system. Fig. 3 A displays the bistability domain in the plane concentration of CaMKII (e_k) versus [Ca²⁺] at two values of K_M. One can see that the bistability region shrinks considerably when e_k decreases. Also, when K_M increases, the right boundary of domain shifts significantly to the left, while the left boundary moves only slightly at low values of e_k. Fig. 3 B shows in more detail how the region of bistability diminishes when K_M increases. If e_k equals 1 µM, bistability vanishes when K_M is larger than 12 µM. Fig. 3 C shows positions of the bistability domain in the plane concentration of PP1 (e_p0) versus [Ca²⁺] at two values of e_k. The domain boundaries shift to the right when e_p0 increases or e_k decreases. Fig. 3 demonstrates that e_k must exceed 10 µM, and K_M must be significantly lower than 1 µM to obtain a bistability range that includes the resting value of the intracellular [Ca²⁺] and is wide enough to prevent induction of LTP by random fluctuations of Ca²⁺ concentration.

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Effect of parameters on size and position of the bistability domain. The lines on left correspond to the down-switching thresholds and the lines on right correspond to the up-switching threshold as shown in Fig. 2. Parameters (in µM): KH2 = 1.4, I0 = 0.1. (A) The (ek, [Ca2+])-plane(ep0 = 0.3 µM): solid lines, KM = 0.4 µM; broken lines, KM = 10.0 µM. (B) The (KM, [Ca2+])-plane (ep0 = 0.3 µM): solid lines, ek = 20.0 µM; broken lines, ek = 1.0 µM. (C) The (ep0, [Ca2+])-plane (KM = 0.4 µM): solid lines, ek = 20.0 µM; broken lines, ek = 1.0 µM.

Fig. 4 A shows positions of the bistability domain in the plane (e_p0, [Ca²⁺]) at three values of the calcium Hill constant of calcineurin (K_H2). One can see that activity of PP1 has to decrease with decreasing K_H2 in order to obtain a proper Ca²⁺ range of bistability; if K_H2 equals 1.4 µM, e_p0 must be ~0.3 µM, if K_H2 equals 0.7 µM, e_p0 must be ~0.05 µM, and if K_H2 equals 0.3 µM, e_p0 must be below 0.01 µM. Fig. 4 B shows the bistability domain in the plane: (e_p0, K_H2) at the resting Ca²⁺ concentration equal to 0.1 µM. It demonstrates that the domain is quite large at e_k = 20 µM and becomes very small when e_k = 1.0 µM.

View larger version (15K): [in this window] [in a new window]   FIGURE 4   Dependence of the bistability domain on the calcium Hill constant of calcineurin and concentration of PP1. Parameters (in µM): KM = 0.4, I0 = 0.1. (A) Effect of KH2 on position of the bistability domain in the (ep0, [Ca2+])-plane: ek = 20.0; dotted lines, KH2 = 1.4 µM; solid lines, KH2 = 0.7 µM; broken lines, KH2 = 0.3 µM. (B) The bistability domains in the (ep0, KH2)-plane at the resting Ca2+ concentration equal 0.1 µM: solid lines, ek = 20.0 µM; broken lines, ek = 1.0 µM.

Fig. 5 shows kinetics of autophosphorylation during tetanic excitation. During the 10 Hz tetanus (Fig. 5 A), the average concentration of Ca²⁺ is well below K_H1, and it takes almost 60 s to reach the level of total autophosphorylation of 70 µM, which eventually leads to transition to the top steady state. At this set of parameters, 15 min of excitation at 5 Hz cannot induce transition from the bottom to the top steady state. Fig. 5 B demonstrates that the system needs 1 s to reach the level of 92 µM during the 100 Hz tetanus, when the constant component of [Ca²⁺] is 19.5 µM.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   Kinetics of autophosphorylation during tetanic excitation. Parameters (in µM): KM = 0.4, ek = 20.0, I0 = 0.1. (A) The 10 Hz, 60 s tetanus; the top plot shows the Ca2+ transient, the bottom plot displays the total concentration of phosphorylated subunits versus time; parameters (in µM): KH2 = 1.4, ep0 = 0.3. (B) The 100 Hz, 1 s tetanus; parameters (in µM): KH2 = 0.7, ep0 = 0.1. Initial conditions correspond to the low steady state at [Ca2+] = 0.1 µM.

In both cases shown in Fig. 5, the concentration of autophosphorylated subunits continues to rise after the end of excitation, but very slowly. At the resting concentration of Ca²⁺, the system is almost "frozen" because activity of CaMKII is extremely low. Fig. 6 A demonstrates that it takes >500 days to reach the stationary level of 186 µM. However, brain in vivo is not in a steady state, a permanent background firing (BF) is always present, and affects almost every neuron. I model BF by a permanent periodic firing with a low amplitude and frequency of 5 Hz, which corresponds to the major Fourier component of BF found in vivo (Karnup, 1996). Fig. 6 B shows that BF greatly accelerates the approach to the top phosphorylated state, which now takes only ~12 h. At the same time, this BF is unable to switch the system from the low to high states of autophosphorylation, but only insignificantly elevates the concentration of phosphorylated subunits in comparison with the low steady-state level. Fig. 6 B also demonstrates that BF can maintain the system in the top phosphorylated state even if the Ca²⁺ threshold for down-switching (0.15 µM in this case) is above the resting Ca²⁺ concentration (0.1 µM).

View larger version (14K): [in this window] [in a new window]   FIGURE 6   Kinetics of transition to the stationary high-phosphorylated state after tetanic excitation. (A) Autophosphorylation in the autonomous model (ep0 = 0.04 µM). (B) Acceleration of transition in the model with background firing (ep0 = 0.1 µm). Parameters (in µM): KH2 = 0.7, KM = 0.4, ek = 20.0, I0 = 0.1. Initial conditions correspond to the low steady state at [Ca2+] = 0.1 µM.

The dynamic up-switching threshold of autophosphorylation depends on amplitude and duration of Ca²⁺ transients and can be characterized by an amplitude-duration function analogous to the classical strength-duration curve for the firing threshold (Noble and Stein, 1966). However, there have been only a few attempts to directly control amplitude and duration of the intracellular Ca²⁺ transients in neurons (Yang et al., 1999). In most of the experiments, the frequency and duration of stimulation are the controlled parameters during induction of LTP (Bliss and Collingridge, 1993). Fig. 5 demonstrates that the shape of Ca²⁺ transients varies greatly with the frequency and duration of stimulation. Therefore, it is more convenient to plot the threshold curve directly in the frequency-duration plane (Fig. 7). One can see that with decreasing frequency duration of the threshold tetanus increases sharply; the duration threshold is ~0.12 s at 100 Hz, and ~25 min at 5 Hz.

View larger version (15K): [in this window] [in a new window]   FIGURE 7   The dynamic up-switching threshold plotted in the plane frequency versus duration of tetanic stimulation. Parameters (in µM): KH2 = 1.4, KM = 0.4, ek = 20.0, ep0 = 0.3, I0 = 0.1.

Fig. 8 demonstrates how mutations can affect bistability in our model. Substituting alanine in place of Thr³⁵ eliminates inhibitor 1 phosphorylation by PKA and its phosphatase inhibitor activity (Endo et al., 1996). In this case, the term v_PKAI0 is equal to zero in the model. Fig. 8 A shows that the bistability region shifts toward higher concentrations of Ca²⁺, and the down-switching static threshold now becomes 1.26 µM. Fig. 8 B demonstrates that after the end of 100 Hz, 1 s tetanus, the systems drops to the low-phosphorylated state within one hour, contrary to behavior of the "wild type" system shown in Fig. 6 B.

View larger version (16K): [in this window] [in a new window]   FIGURE 8   Dynamics of autophosphorylation in the system with a mutant inhibitor 1 (vPKA = 0.0). (A) The steady-state characteristic of the system. (B) Kinetics of autophosphorylation induced by 100 Hz, 1 s excitation in the model with background firing. Parameters (in µM): KM = 0.4, ek = 20.0, ep0 = 0.1. Initial conditions correspond to the low steady state at [Ca2+] = 0.1 µM.

Fig. 9 illustrates feasible in vitro experiments that could confirm the existence of such bistability. These can be done with isolated PSD, where I1 is washed out and PP1 becomes Ca²⁺-independent (Strack et al., 1997; Yoshimura et al., 1999). Fig. 9 A shows the steady-state curve of the system with I₀ equal to zero. The system moves along a loop of hysteresis, when [Ca²⁺] gradually changes. When [Ca²⁺] increases, the system moves along the ABCDE path; when [Ca²⁺] decreases, the path is EDFBA. Fig. 9 B demonstrates the dynamics of transitions between steady states, which follow sub and superthreshold step-wise changes in [Ca²⁺]. When [Ca²⁺] jumps from 1.3 µM to 1.8 µM, the concentration of autophosphorylated subunits increases only slightly (A), because the system remains on the bottom branch of the curve shown in Fig. 9 A; when [Ca²⁺] jumps from 1.3 µM to 2.2 µM, the system transits to the top steady state (B). When [Ca²⁺] is switched from 2.3 µM to 1.8 µM, the system moves to a new steady state on the top branch of the curve (C); when [Ca²⁺] jumps from 2.3 µM to 1.5 µM, the system transits to the bottom branch of the curve (D).

View larger version (25K): [in this window] [in a new window]   FIGURE 9   Hysteresis and switching in the system with the Ca2+-independent protein phosphatase in vitro. (A) The steady-state curve with the hysteretic loop (BCDF). Arrows show direction of movement when [Ca2+] slowly increases or decreases (B) Transients after sub- and superthreshold shifts of [Ca2+]: A, the systems remains on the bottom branch of the bistability curve when [Ca2+] jumps from 1.3 µM to 1.8 µM; B, when [Ca2+] jumps from 1.3 µM to 2.2 µM, the system transits to the top steady state; C, when [Ca2+] is switched from 2.3 µM to 1.8 µM, the system moves to a new steady state on the top branch of the curve; D, [Ca2+] jumps from 2.3 µM to 1.5 µM, and the system transits to the bottom branch of the curve. Parameters (in µM): KM = 0.4, ek = 20.0, ep0 = 0.3, I0 = 0.0.

Finally, I show that the model correctly describes kinetics of autophosphorylation of CaMKII in cytosol of neurons. In cytosol, at least 80% of the protein phosphatase activity is Ca²⁺-independent, the principal phosphatase is PP2A, and the concentration of CaMKII is one order of magnitude lower than in PSD (Strack et al., 1997). Fig. 10 displays behavior of the model with the Ca²⁺-independent phosphatase and parameters that correspond to the cytosolic system. Fig. 10 A demonstrates that bistability is absent in this case. Fig. 10 B shows that during the 100 Hz, 1 s tetanus, the concentration of phosphorylated subunits reaches ~4.6 µM, and drops below 0.4 µm in <10 min.

View larger version (15K): [in this window] [in a new window]   FIGURE 10   Dynamics of autophosphorylation of CaMKII in cytoplasm (k3 = 0.0). (A) The steady state characteristic is single valued. (B) Kinetics of autophosphorylation induced by 100 Hz, 1 s tetanus. Parameters (in µM): KM = 15.0, ek = 1.0, ep0 = 0.05. Initial conditions correspond to the low steady state at [Ca2+] = 0.1 µM.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Recently, all-or-none potentiation was found at CA3-CA1 synapses in hippocampus (Petersen et al., 1998). This bistability of synaptic strength may depend on bistability of phosphorylation of participating receptors and anchoring proteins, which, in turn, may depend on bistability of activity of a major protein kinase at the resting [Ca²⁺]. Bistability of autonomous activity of CaMKII has been analyzed previously. However, Lisman (1985) did not consider the effect of Ca²⁺, and Lisman and Goldring (1988) assumed propagation of autophosphorylation to be Ca²⁺-independent. Okamoto and Ichikawa (2000) have found bistability at Ca²⁺ concentrations that are significantly higher than the resting [Ca²⁺] and did not study bistability in the vicinity of the latter.

My simulations show that two stable steady states of autophosphorylation of CaMKII can arise in the Ca²⁺ concentration range that includes the resting [Ca²⁺] if the concentration of CaMKII is very high and the phosphatase activity is Ca²⁺/calmodulin-dependent. These conditions are found in PSD. In cytoplasm, the concentration of CaMKII is relatively low and the protein phosphatase activity is essentially Ca²⁺-independent. In this case, according to the model, the bistability region is either shifted strongly to the right of the resting concentration of Ca²⁺, or absent, and the phosphorylated fraction of CaMKII should decay to a very low level after concentration of Ca²⁺ drops to the resting value. The last result corresponds to the experimental observations. Thus, according to the model, high levels of autophosphorylation can be maintained at the resting concentration of Ca²⁺ only in PSD. The model also shows that stimulation protocols, which induce LTP, switch the system from the low-phosphorylated to the high-phosphorylated state, while background firing is unable to switch. However, background firing can maintain the system in the top phosphorylated state even if the Ca²⁺ threshold for down-switching is above the resting Ca²⁺ concentration.

The existence of bistability of the CaMKII autophosphorylation in PSD can be tested in in vitro experiments. If one would maintain isolated PSDs in a medium with ATP and gradually change the Ca²⁺ level in both directions, hysteresis should be observed, approximately as shown in Fig. 9 A. The addition of inhibitor 1 should shift the bistability region to lower concentrations of Ca²⁺ and make it wider.

Experiments with transgenic animals can provide evidence that such bistability plays a role in the maintenance of LTP. It is known that substituting alanine in place of Thr³⁵ eliminates inhibitor 1 phosphorylation by PKA and its phosphatase inhibitor activity (Endo et al., 1996). According to the model, such a mutation should prevent autophosphorylation of CaMKII at resting [Ca²⁺] (Fig. 8); that could down-regulate or eliminate LTP.

Bistability is a robust phenomenon in this model. It persists even when any parameter in the model varies by an order of magnitude, or more. However, the parameters significantly affect the position and extent of the Ca²⁺ range of bistability (Figs. 3 and 4 A). Where possible, I use parameter values estimated from available experimental data.

Values of [Ca²⁺]₅₀ from 0.7 to 4 µM have been reported for Ca²⁺/calmodulin activation of CaMKII (Fährmann et al., 1998; Gupta et al., 1992; Kennedy et al., 1983; Kuret and Schulman, 1984). De Koninck and Schulman (1998) have shown that the dissociation constant of Ca²⁺/calmodulin from the unphosphorylated -subunit of CaMKII is ~0.08 µM. One can estimate that [Ca²⁺]₅₀ varies from 2 to 8 µM, when the concentration of free calmodulin varies from 40 to 0.2 µM (Huang et al., 1981; Cohen and Klee, 1988). I have chosen K_H1 = 4 µM.

The Hill number for Ca²⁺ activation of calcineurin is 3 ± 0.1, and [Ca²⁺]₅₀ can vary from 0.6 to 1.4 µM, when Ca²⁺/calmodulin activates purified calcineurin in the presence of 6 µM of Mg²⁺. When [Mg²⁺] decreases, [Ca²⁺]₅₀ also decreases (Stemmer and Klee, 1994). I ran simulations with K_H2 equal to 0.3, 0.7, and 1.4 µM.

The Michaelis constants of PP1 for phosphoproteins vary from 0.21 to 6.8 µM (Johansen and Ingebritsen, 1987), and from 3.8 to 31 µM for catalytic subunit of PP2A (Bialojan and Takai, 1988). I could not find direct data for dephosphorylation of CaMKII. I assume that the Michaelis constants of these phosphatases with CaMKII as a substrate are of the same order of magnitude as for the other phosphoproteins.

The average concentration of the -subunits of CaMKII is ~10 µM in forebrain according to Erondu and Kennedy (1985), McNeil and Colbran (1995), and Strack et al. (1997). Our estimation of this concentration in PSD is 80 µM on the basis of data published by Suzuki et al. (1994), and 200 µM according to data presented by Strack et al. (1997). The total phosphatase activity toward CaMKII in homogenates of hippocampus is ~0.1 µM · s¹ (Fukunaga et al., 2000).

I chose the value of k₁ according to Hanson et al. (1994). I made reasonable estimates of parameters involved in the Ca²⁺ regulation of the protein phosphatase activity. It is worth mention that the results of simulations are insensitive to absolute values of these parameters as long as they are not too small, and their ratios are kept constant.

Here I use a rather simplified model. The scheme of autophosphorylation is strictly valid only if [Ca²⁺] is significantly less than K_H1. However, [Ca²⁺] becomes much higher than K_H1 during high-frequency stimulation. For assurance, I have run several simulations with an extended model that included additional terms in the autophosphorylation scheme. The difference in results was <10%.

I also neglect any binding of the CaMKII holoenzymes to PSD. However, the accumulation of CaMKII in PSD is a result of such binding. It was shown that CaMKII can bind to the C-terminal tails of subunits of the NMDA receptor in PSD (Gardoni et al., 1999; Leonard et al., 1999; Strack and Colbran, 1998). To be kept in PSD, CaMKII must be either autophosphorylated (Gardoni et al., 1999; Strack and Colbran, 1998; Strack et al., 1997; Yoshimura et al., 1999), or bound to Ca²⁺/CaM (Shen and Meyer, 1999). One can see that these requirements are the same as for the CaMKII activation. This means that the catalytic and autoinhibitory domains of the CaMKII subunits must dissociate before binding to PSD, and suggests that one of these domains binds the NMDA receptor or another anchor protein. There are two variants, if only one subunit participates in binding. If the catalytic site remains active in the bound state and can interact with its neighbor in the clockwise direction, the bound subunit will become the initiation point of autophosphorylation. As a result, autophosphorylation can proceed faster in comparison with the unbound holoenzyme. However, if the catalytic site becomes incapacitated, the rate of autophosphorylation will decrease approximately twofold. In these two cases, there is a moderate quantitative change in the dynamics of the system due to the binding, and the results remain valid. However, autophosphorylation is severely hindered if the holoenzyme binds two or more anchor sites. In this case, the bistability region shifts to much higher concentrations of Ca²⁺, and the physiological importance of bistability becomes doubtful.

The very high concentration of CaMKII in PSD can be a result of translocation of CaMKII from cytosol during high elevations of [Ca²⁺] (Shen and Meyer, 1999; Strack et al., 1997). The preliminary simulations with a two-compartment model show that this translocation affects the characteristics of bistability only if the Ca²⁺-dependence of CaMKII activity is quite different in cytoplasm and PSD.