Abstract
Calmodulin (CaM) is a major Ca2+ binding protein involved in two opposing processes of synaptic plasticity of CA1 pyramidal neurons: long-term potentiation (LTP) and depression (LTD). The N- and C-terminal lobes of CaM bind to its target separately but cooperatively and introduce complex dynamics that cannot be well understood by experimental measurement. Using a detailed stochastic model constructed upon experimental data, we have studied the interaction between CaM and Ca2+-CaM-dependent protein kinase II (CaMKII), a key enzyme underlying LTP. The model suggests that the accelerated binding of one lobe of CaM to CaMKII, when the opposing lobe is already bound to CaMKII, is a critical determinant of the cooperative interaction between Ca2+, CaM, and CaMKII. The model indicates that the target-bound Ca2+ free N-lobe has an extended lifetime and may regulate the Ca2+ response of CaMKII during LTP induction. The model also reveals multiple kinetic pathways which have not been previously predicted for CaM-dissociation from CaMKII.
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Acknowledgements
This work was supported by US National Institutes of Health grants NS26086, NS038310, and GM069611 and Robert A. Welch Foundation Grant AU1144. Y. Kubota also gratefully acknowledges support from the Institutional Training Grant on Neuroplasticity (NS 041226) during the early phase of this work.
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Appendix (Stoichiometry of CaM Binding)
Appendix (Stoichiometry of CaM Binding)
As stated earlier, our recent experimental data (Forest et al. 2008) confirmed that the stoichiometry of CaM binding to CaMKII is close to 1. However, there is experimental data (Gaertner et al. 2004a; Rosenberg et al. 2005) suggesting the Hill coefficient of CaM binding to CaMKII is ~ 1.9. The data reported by the same group indicates the CaMKII activation requires only one CaM molecule bound to the subunit. The observation by Gaertner et al. (2004a) of Hill coefficient ~1.9 data seems to contradict the CaMKII activation data as well as our recent CaM-CaMKII binding stoichiometry data (Forest et al. 2008). In this Appendix, we propose a possible explanation for these seemingly conflicting observations.
In their experiment, Gaertner et al. (9) mixed 100 nM of CaM-C75-IAEDANS, 500 μM Ca2+, and varying concentration of CaMKII (Section 2.4). After reaching the steady state, the relative fraction of CaMKII-bound CaM was plotted against the free CaMKII concentration in the system. Note, in Fig. 3, the fraction of CaMKII-bound CaM was plotted against the free CaM not the free CaMKII. In order to calculate the free CaMKII concentration, one must accurately measure the actual concentration of CaMKII bound CaM from the fluorescence signals. However, as we pointed out in Section 2.4 the fluorescence signal used in this experiment may not necessarily report all CaMKII-CaM complexes with 100% efficiency.
If the fluorescence signal does not report the accurate concentration of CaMKII bound CaM molecules, then it would result in an over-estimate of free CaMKII concentration, which in turn may potentially lead to a higher Hill coefficient. Figure 7(a) strongly supports this hypothesis. In this figure, we faithfully reproduced the experiment of Gaertner et al. (2004a) and plotted the fraction of CaMKII bound CaM as a function of real concentration of free CaMKII in the simulation (the circles). The Hill function fitted to this data (the dotted line) results in the Hill coefficient of ~1 and the dissociation constant of ~40 nM. If, however, we assume that only 10 % of CaMKII-bound CaM is reported by the fluorescence signal and if the fraction of CaMKII-bound CaM is plotted against the erroneously estimated concentration of free CaMKII (the crosses), the Hill coefficient will become ~2.1 (the Hill function curve fit is represented by the solid line). The resultant dissociation constant is ~65–70 nM. This simulation experiment seems to explain the higher Hill coefficient and higher dissociation constant reported by Gaertner et al (2004a) and by Rosenberg et al. (2005).
Stoichiometry of CaM Binding to CaMKII. (a) The steady-state binding kinetics of CaM to CaMKII. In this simulation experiment, 100 nM of CaM-C75-IAEDANS, 500 μM Ca2+ are mixed with varying concentration of CaMKII. After reaching the steady state, the relative fraction of CaMKII bound CaM was plotted against the free CaMKII concentration in the system. Note in Fig. 3, the fraction of CaMKII-bound CaM was plotted against the concentration of free CaM but not free CaMKII. The average of 50 simulations (the circles) was fitted with the Hill function (the dotted line) revealing a Hill coefficient 1 and a dissociation constant of 40 nM. If, however, we assume that only 10 % of CaMKII-bound CaM is reported by the fluorescence signal, the fraction of CaMKII-bound CaM is plotted against the erroneously estimated concentration of free CaMKII (the crosses) was best fit by the Hill function with the coefficient of ~2.1 (the solid line). The resultant dissociation constant is ~65–70 nM. (b) A simple binding reaction \( A + X \to Y \) can result in a Hill coefficient 2. The analytical result of Eq. (8) (the open circles) for this simple binding reaction was plotted with a Hill function curve fit (the dotted line of Hill coefficient 1). In Appendix A, we derived an analytic formula for the fraction of X-molecule bound A as a function of erroneously estimated concentration of free X. This analytical result (the crosses) was plotted with a Hill function curve fit (of the coefficient 2, the solid line)
In fact, the efficiency of fluorescence signal does not have to be 10%. The efficiency less than 100 % always results in a Hill coefficient larger than 1 and a dissociation constant higher than the expected value. Figure 7(b) illustrates the underlying principle. In this figure, we analyze a simple binding reaction:
in which a fluorescence labeled ligand A binds to X with a dissociation constant of K D . The fraction of A-bound X molecule expressed by the free A concentration is
In this and following equations, the symbols for the chemical species, e.g., A, X, and Y also denote the concentration of corresponding molecules. The fraction of X-bound A molecule expressed by the free X concentration is
Both of these expressions result in the curves of a Hill coefficient = 1 and of the same dissociation constant K D.
Suppose we assume only a fraction k (0 < k < 1) of X-bound A molecule is detected by the fluorescence signal, the steady state concentration of free A, X, and Y obey:
where X T and A T are the total concentrations of molecular species X and A, respectively. Note X estimated in Eqs (10 and 11 below) is the (erroneously) estimated concentration of the molecular species X from the fluorescence signal. Then the fraction of X-bound A molecule expressed by the free X concentration is
Equation (11) may result in a steady-state curve that fits to a Hill function with a coefficient higher than 1. Without going through further analysis (e.g., asymptotic expansion), Fig. 7(b) already shows this is the case with a numerical example for KD = 0.01 μM and k = 0.1. Even for this simple binding reaction, the incorrect estimate of X-bound A by fluorescence signal leads to a Hill coefficient ~2 and an overestimate of dissociation constant (the crosses in Fig.7(b) is the plot of Eq. (11) and the solid line is the Hill function fit to the analytical results). The analytical result for Eq. (8) is also shown (the circles and dotted line) for comparison.
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Byrne, M.J., Putkey, J.A., Neal Waxham, M. et al. Dissecting cooperative calmodulin binding to CaM kinase II: a detailed stochastic model. J Comput Neurosci 27, 621–638 (2009). https://doi.org/10.1007/s10827-009-0173-3
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DOI: https://doi.org/10.1007/s10827-009-0173-3