Monotonicity and Bistability of Calcium/Calmodulin-Dependent Protein Kinase-Phosphatase Activation

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1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 FrB13.2 Monotonicity and Bistability of Calcium/Calmodulin-Dependent Protein Kinase-Phosphatase Activation Ming Wu and Douglas A. Lawrence Abstract Calcium/calmodulin-dependent protein kinase II (CaMKII) is thought to be a key contributor to the induction of long-term potentiation (LTP). Researchers have developed a variety of mathematical models of CaMKII activation intended to produce simulation output that agrees with empirical observations. This paper focuses on one such model to which recent theoretical results for input-output monotone systems are applied. Several key findings in the literature are reproduced using simple algebraic computations as opposed to exhaustive, simulation-based analysis. In particular, it is confirmed that bistability exists over a range of calcium concentrations that nominally includes the intracellular resting level and switching between high activation and low activation states is described by a hysteresis curve. I. INTRODUCTION Long-term potentiation (LTP) is a prolonged enhancement of synaptic strength that has been experimentally linked to intermediate memory. Calcium/calmodulin-dependent protein kinase II (CaMKII) is thought to be a key contributor to the induction of LTP in post-synaptic densities (PSDs). Recently, researchers have developed a variety of mathematical models that capture with varying degrees of fidelity the calciumdependent autophosphorylation of CaMKII combined with calcium-dependent dephosphorylation by a protein phosphatase [4], [6], [7], [10]. In this paper, we adopt the mathematical model developed by Zhabotinsky ([10]) because it is of moderate complexity and yet capable of generating important experimentally-observed phenomena. The systems biology area has experienced rapid growth in recent years. In particular, the work of Sontag et al. (e.g., [1], [2], [3], [5], [8], [9]) has focused on the exploration of new systems and control problems inspired by biological phenomena. It has been observed that dynamic models of biological processes inherently possess properties such as input-output monotonicity and multi-stability that are less prevalent in more traditional application areas for systems and control. This paper investigates the application of some of the recent results for input-output monotone systems to the study of bistability and hysteresis in Zhabotinsky s model of CaMKII activation. The remainder of this paper is organized as follows. Zhabotinsky s model is described in the next section and a nonlinear state equation suitable for the ensuing analysis is formulated. Monotonicity properties are analyzed in Section M. Wu is with the School of Electrical Engineering and Computer Science, Ohio University, Athens, OH 45701, USA mw103302@ohio.edu D. Lawrence is with the School of Electrical Engineering and Computer Science, Ohio University, Athens, OH 45701, USA dal@ohio.edu III and bistability and hysteresis is characterized. Some conclusions are drawn and topics for further investigation are discussed in Section IV. II. MATHEMATICAL MODEL In this section we review in detail the mathematical model presented in [10] and ultimately formulate a nonlinear state equation amenable to monotonicity and bistability analysis. Values for all model parameters are listed in Table I which is reproduced from [10, Table 1]. CaMKII is a holoenzyme that consists of 8 to 12 subunits arranged in a ring configuration. Zhabotinsky s model uses 10 subunits with each subunit occurring in either a phosphorylated or unphosphorylated state. Subunit phosphorylation is governed by two competing processes: autophosphorylation and dephosphoryation by a protein phosphatase. These are discussed in the following subsections. A. Autophosphorylation Autophosphorylation must first be initiated by having two neighboring subunits each bind the calcium-calmodulin complex (Ca 2+ ) 4 CaM. This process is described by the reactions 4Ca 2+ + CaM C P 0 + C P 0 C P 0 C + C P 0 C 2 P 0 C 2 P 1 C 2 (1) in which C, P 0, P 1 denote the calcium-calmodulin complex, unphosphorylated CaMKII, and 1 fold phosphorylated CaMKII, respectively. Cooperative binding of (Ca 2+ ) 4 CaM to CaMKII subunits is modeled by the empirical Hill equation with positive Hill number n = 4 and Hill constant K H1. This yields the fraction of CaMKII subunits bound to (Ca 2+ ) 4 CaM F = ([Ca2+ ]/K H1 ) ([Ca 2+ ]/K H1 ) 4 The velocity of the initiation step is given by V 0 = 10k 1([Ca 2+ ]/K H1 ) 8 (1 + ([Ca 2+ ]/K H1 ) 4 ) 2 p 0 (2) in which k 1 is the rate constant of the last reaction in (1) and p 0 is the concentration of P 0. Once initiated and otherwise unimpeded, autophosphorylation is assumed to propagate in one direction. Further, under the assumption that the ability of a phosphorylated subunit /10/$ AACC 5923

2 to act as a catalyst in the phosphorylation of its neighbor is (Ca 2+ ) 4 CaM-independent, propagation of autophosphorylation is characterized by the reactions P i + C P i C P i C P i+1 (3) for i = 1,...,9 in which P i represents i fold phosphorylated CaMKII. However, dephosphorylation occurs at random subunit locations which results in a random distribution of phosphorylated and unphosphorylated subunits in the CaMKII enzyme. To address this under the assumption that all distinguishable configurations having the same number of phosphorylated subunits occur with equal probability, an effective number of phosphorylating pairs for i phosphorylated subunits is determined from w i = w 10 i = i j=1 jm j i j=1 m j i = 1,...,9 where m j is the number of distinguishable configurations with j autophosphorylating pairs. This yields the values w 1 = w 9 = 1.0, w 2 = w 8 = 1.8, w 3 = w 7 = 2.3, w 4 = w 6 = 2.7, and w 5 = 2.8 [10]. These weights scale the rate of the second reaction in (3) according to V i = w i k 1 ([Ca 2+ ]/K H1 ]) ([Ca 2+ ]/K H1 ]) 4 p i i = 1,...,10 (4) where p i is the concentration of P i. B. Dephosphorylation Of the four protein phosphatases that dephosphorylate phosphorylated CaMKII, PP1 is the only one known to dephosphorylate CaMKII in postsynaptic densities (PSDs) and is the one incorporated into this model. Dephosphorylation proceeds according to the reactions P i + PP1 P i PP1 PP1 + P i 1 for i = 1,...,10. Upon invoking the standard Michaelis- Menten approximation for this class of reactions, the dephosporylation rate for i fold phosphorylated CaMKII is V i = k 2 e p K M + N k=1 kp ip i (5) k in which e p is concentration of free PP1, k 2 is the associated catalytic rate constant, and K M is the associated Michaelis constant. PP1 activity is indirectly influenced by calcium via an inhibitor I1 as follows. The inhibitor I1 is phosphorylated by the c-amp protein kinase PKA to produce I1P which deactivates PP1. Calcineurin (CaN) is activated by cooperative binding with the calcium-calmodulin complex denoted C 3 (with empirical Hill constant 3). This, in turn, dephosphorylates I1P. This process is represented by the reactions: I1 + PKA I1 PKA I1P 3Ca 2+ + CaM C 3 CaN + C 3 CaN C 3 I1P + CaN C 3 I1P CaN C 3 I1 P i + PP1 P i PP1 PP1 + P i 1 PP1 + I1P PP1 I1P P i PP1 + I1P P i PP1 I1P As asserted in [10], when the the concentration of free I1 is constant and much less than Michaelis constant of PKA and the concentration of free I1P is much less than Michaelis constant of CaN then the phosphatase-inhibitor kinetics can be represented by ė p = k 3 I e p + k 4 (e p0 e p ) I = k 3 I e p + k 4 (e p0 e p ) + v PKA I 0 (6) ([Ca 2+ ]/K H2 ) 3 v CaN I 1 + ([Ca 2+ ]/K H2 ) 3 in which e p is the concentration of free PP1, e p0 is the total concentration of PP1, I is the concentration of free I1P, and I 0 is the concentration of free I1. Remaining parameters are listed in Table I. TABLE I MODEL PARAMETERS [10, TABLE 1 & FIG. 2 CAPTION] Parameters Symbol Value Units Concentration of CaMKII e k 20 µm Concentration of PP1 e p µm Concentration of free I1 I µm V CaN / K M2 v CaN 1.0 s 1 V PKA / K M3 v PKA 1.0 s 1 Michaelis constant K M 0.4 µm Ca 2+ Hill cons. of CaMKII K H1 4.0 µm Ca 2+ Hill cons. of CaN K H2 0.7 µm Catalytic const. of autophos. k s 1 Catalytic const. of phosphatase k s 1 Association rate of PP1 I1P k µm 1 s 1 Dissociation rate of PP1 I1P k s 1 C. State Equation In this section, we cast the autophosphorylation and dephosphorylation processes described in the preceding subsections, specifically the rate equations associated with the reaction rates (2), (3), and (5), in the form of a nonlinear state equation amenable to monotonicity and bistability analysis. For CaMKII activation which initializes autophosphorylation, we introduce the calcium-dependent exogenous input w a = ([Ca2+ ]/K H1 ) ([Ca 2+ ]/K H1 ) 4 (7) in terms of which we define the function α 0 (w a ) = 10k 1 w 2 a 5924

3 along with constants α i = w i k 1, i = 1,...,9. To include the effects of dephosporylation in the state equation formulation, we set w d = e p for notational consistency (which is generated by the calcium-dependent phosphatase-inhibitor kinetics (6)) and also define the function k 2 δ(p) = K M + 10 k=1 kp k in which p = (p 1,p 2,...,p 10 ). Under the assumption that the total CaMKII concentration is a constant denoted p tot (= e k in Table I), this constraint can be used to eliminate p 0 from the analysis via p 0 = p tot (p p 10 ) and the interaction between autophosphorylation and dephosphorylation is governed by a system of coupled first-order ordinary differential equations in the concentration variables p = (p 1,p 2,...,p 10 ) which evolve on {p = (p 1,...,p 10 ) p i 0 and p p 10 p tot } While p 1,p 2,...,p 10 seem to be a natural choice for the state variables, it will prove to be more convenient to adopt x i := p i + + p 10, i = 1,...,10 which yields ẋ 1 = α 0 (w a )x 1 δ(x)w d (x 1 x 2 ) + α 0 (w a )p tot ẋ 2 = α 1 w a (x 1 x 2 ) 2δ(x)w d (x 2 x 3 ) ẋ 3 = α 2 w a (x 2 x 3 ) 3δ(x)w d (x 3 x 4 ) ẋ 4 = α 3 w a (x 3 x 4 ) 4δ(x)w d (x 4 x 5 ) ẋ 5 = α 4 w a (x 4 x 5 ) 5δ(x)w d (x 5 x 6 ) ẋ 6 = α 5 w a (x 5 x 6 ) 6δ(x)w d (x 6 x 7 ) ẋ 7 = α 6 w a (x 6 x 7 ) 7δ(x)w d (x 7 x 8 ) ẋ 8 = α 7 w a (x 7 x 8 ) 8δ(x)w d (x 8 x 8 ) ẋ 9 = α 8 w a (x 8 x 9 ) 9δ(x)w d (x 9 x 10 ) ẋ 10 = α 9 w a (x 9 x 10 ) 10δ(x)w d x 10 (8) in which, with a slight abuse of notation, we now write k 2 δ(x) = K M + 10 k=1 x k By definition of the state variables x i, i = 1,...,10 it follows that δ(x) = δ(p). Upon defining, with e 1,...,e 10 denoting the standard basis vectors on R 10, 10 A a (w a ) = α 0 (w a )e 1 e T 1 w a α i 1 e i (e i e i 1 ) T along with A d (w d ) = w d ( 9 i=1 and finally i=2 ie i (e i e i+1 ) T + 10e 10 e T 10 B a (w a ) = α 0 (w a )e 1 (8) can be written compactly as ẋ = (A a (w a ) δ(x)a d (w d ))x + B a (w a )p tot (9) which evolves on the convex, compact subset of R 10 X = {x = (x 1,...,x 10 ) 0 x 10 x 1 p tot } ) In the analysis to follow, it will be useful to view (9) as resulting from the open-loop system ẋ = (A a (w a ) ua d (w d )) x + B a (w a )p tot y = δ(x) (10) together with unity feedback u = y. III. ANALYSIS In this section, we apply the results established in [1], [2] to the the dynamic model developed in the preceding section in order to confirm the bistability properties presented in [10] without resorting to exhaustive, simulation-based steady-state analysis. A. Monotonicity Analysis Input-Output Monotonicity of a dynamic system represented by a nonlinear state equation is defined in terms of closed convex cones in the state, input, and output spaces that define a partial ordering on the respective space. Having obtained (10) via the previously described change of variables allows us to work with the nonnegative orthant K = R 10 0 in the state space. Henceforth, the notation x 1 x 2 means that x 1 x 2 K so that each component of x 1 is greater than or equal to the corresponding component of x 2. For the inputs (u,w a,w d ), we will adopt the partial ordering defined by K in = K (1,0,1) = {(ξ 1,ξ 2,ξ 3 ) R 3 ξ 1 0, ξ 2 0, ξ 3 0}. Thus (u 1,wa,w 1 d 1) (u2,wa,w 2 d 2) corresponds to (u 1,wa,w 1 d 1) (u2,wa,w 2 d 2) K in from which u 1 u 2, wa 1 wa, 2 and wd 1 w2 d. Finally, we will use the negative ordering on the real line R for the scalar output y and directly write y 1 y 2. Following [1], [2], the system (10) is monotone if the following implication holds for all t 0: and x 1 (0) x 2 (0) ( u 1 (t),w 1 a(t),w 1 d(t) ) ( u 2 (t),w 2 a(t),w 2 d(t) ) imply that the respective solutions (assumed to exist for all t 0) satisfy x 1 (t) x 2 (t) The output relationship in (10) is monotone if x 1 x 2 implies y 1 y 2. We seek to verify input-output monotonicity of (10) by checking the following equivalent infinitesimal conditions ([1, Corollary III.3]): (x,u,w a,w d ) x j 0 i j u (x,u,w a,w d ) 0 (x,u,w a,w d ) w a 0 (11) (x,u,w a,w d ) w d

4 for all i,j {1,2,...,10} for all x in the interior of X and for all u > 0, w a > 0, w d > 0 in which f(x,u,w a,w d ) represents the vector-valued map on the righthand side of the first equation in (10). For the first set of identities (x,u,w a,w d ) = iuw d x i+1 0 i = 1,...,9 (x,u,w a,w d ) = α i 1 w a x i 1 0 i = 2,...,10 with all other partial derivatives zero for i j. For the second set of identities, u (x,u,w a,w d ) = iw d (x i x i+1 ) 0 i = 1,...,9 f 10 u (x,u,w a,w d ) = 10w d x 10 0 Next, f 1 (x,u,w a,w d ) = 2α 0 (w a )(p tot x 1 ) w a 0 (x,u,w a,w d ) = α i (x i 1 x i ) w a 0 and finally i = 2,...,10 w d (x,u,w a,w d ) = iu(x i x i+1 ) 0 i = 1,...,9 f 10 w d (x,u,w a,w d ) = 10ux 10 0 where these inequalities follow from the definition of the set X. Thus we conclude that the system (10) is monotone. Also, we observe the output map δ(x) is monotone since x 1 x 2 gives δ(x 1 ) δ(x 2 ). It is of further interest to investigate monotonicity of the overall system consisting of the cascade interconnection of the combined CaMKII and CaN activation subsystems (with input v = [Ca 2+ ] and output (w a,w d )) and the autophosphorylation-dephosphorylation dynamics (8). The former is represented by the nonlinear state equation derived from (6) and (7) ė p = k 3 I e p + k 4 (e p0 e p ) I = k 3 I e p + k 4 (e p0 e p ) + v PKA I 0 (v/k H2 ) 3 v CaN I 1 + (v/k H2 ) 3 w a = (v/k H1 ) (v/k H1 ) 4 (12) w d = e p Another application of the infinitesimal monotonicity characterization reveals that (12) is monotone with respect to the orthant K 1 = K (1,0) = {(ξ 1,ξ 2 ) R 2 ξ 1 0,ξ 2 0} for the state (e p,i) and the negative ordering on R for the scalar input v. Further, the CaMKII/CaN activation subsystems are I/O monotone with respect to the orthant K out = K (1,1) = {(ξ 1,ξ 2 ) R 2 ξ 1 0,ξ 2 0} for the output (w a,w d ). The fact that K out and the (w a,w d ) portion of K in are incompatible seemingly precludes the possibility that the cascade interconnection of (12) and (9) is inputoutput monotone from the calcium input (with respect to the negative ordering on R) to the composite state with positivity cone K K 1. Nevertheless, several interesting conclusions can be reached for the unity feedback system (10) with (w a,w d ) treated as external stimuli [2]. For instance, strong monotonicity of the closed-loop system (9) for constant (w a,w d ) can be concluded essentially because the discussion surrounding the second example in [2, Section 8] applies to (9). Moreover, it follows that int(x) is positively invariant and so trajectories of (9) are bounded. B. Input-State and Input-Output Characteristics := For constant calcium concentration [Ca 2+ ], constant w a is determined by (7) and constant w d corresponds to the equilibrium phosphatase concentration given by e p0 γ k 3 v PKA k 4 v CaN I 0 + γ (γ ([Ca2+ ]/K H2 ]) ([Ca 2+ ]/K H2 ]) 3 ) Open-loop equilibria of (10) for constant u and constant w := (w a,w d ) are characterized by from which 0 = ( A a (w a ) ua d (w d ) ) x + B a (w a )p tot x = ( A a (w a ) ua d (w d ) ) 1 Ba (w a )p tot := k x (u,w) y = ( δ k x ) (u,w) := k y (u,w) We observe that for any nonnegative, constant values for u, w a, and w d, the equilibrium state x = k x (u,w) is globally exponentially stable because the dynamics (9) reduce to the parameterized linear time-invariant system d dt [x k x(u,w)] = ( A a (w a ) ua d (w d ) ) [x k x (u,w)] and it can be verified that the constant matrix A a (w a ) ua d (w d ) has strictly negative real-part eigenvalues for each constant (u,w a,w d ). As in [1], [2], we refer to the map k x (, ) as the input-state (I/S) characteristic and the map k y (, ) as the input-output (I/O) characteristic from u to y, in this case parameterized by constant calcium concentration through the implicit dependence of w a and the equilibrium value of w d given above. Notice that in the absence of dephosphorylation (u = 0), k x (0,w) = p tot [ ] T which leads to the equilibrium concentrations of i fold phosphorylated holoenzyme molecules given by p i = 0, i = 1,...,9 and p 10 = p tot. This implies that all holoenzyme molecules become fully phosphorylated at equilibrium in the absence of dephosphorylation. 5926

5 C. Bistability and Hysteresis Equilibria of (9) correspond to fixed points of the relationship u = k y (u,w). To investigate further, we let C = [ ] and H(u,w) = K M C ( A a (w a ) ua d (w d ) ) 1 Ba (w a )p tot in terms of which, exploiting the form of the nonlinearity δ(x), fixed points of u = k y (u,w) satisfy H(u,w)u = k 2 H(u,w) is a proper rational function in u of degree 10 for all constant calcium-dependent w = (w a,w d ). Letting n(u,w) and d(u,w) denote the degree 10 numerator and denominator polynomials in u, respectively, fixed points of u = k y (u,w) correspond to real roots of the degree 11 polynomial n(u,w)u k 2 d(u,w) = 0 (13) The I/S characteristic k x (u,w) is non-degenerate by virtue of the global stability property mentioned above. The I/0 characteristic k y (u,w) is non-degenerate provided that ( k y / u)(u,w) 1 at each fixed point. For Ca 2+ concentrations ranging from approximately 0.09µM to 0.7µM and parameter values in Table I, the polynomial equation (13) has three real roots for each [Ca 2+ ] value. This is depicted in Fig. 1 for [Ca 2+ ] = 0.2µM in which the I/O characteristic is plotted on a log-log scale. The 3 fixed points correspond to intersections of the I/O characteristic with the line y = u. The I/O characteristic essentially shifts from left to right with increasing [Ca 2+ ] value with a corresponding effect on the fixed points. Since the logarithmic scaling on both axes preserves the slope of the tangent to the characteristic at each point, Fig. 1 indicates that the I/O characteristic is non-degenerate whenever the [Ca 2+ ] value yields three fixed points. The total concentration of phosphorylated subunits is in general determined from the i fold phosphorylated CaMKII concentrations via p 1 + 2p 2 + 3p p 10 = x 1 + x 2 + x x 10 The equilibrium concentration of phosphorylated subunits as a function of u and w is therefore determined from the I/S characteristic according to Ck x (u,w) The value of w determined by the constant calcium concentration and the u values determined by the fixed points of the I/O characteristic yield closed-loop equilibrium concentrations of phosphorylated subunits. These values are plotted versus constant calcium concentration in Fig. 2. This graph is identical to [10, Fig. 2] and was generated by polynomial root finding as opposed to exhaustive simulation-based asymptotic (steady-state) analysis. It is also possible, as an application of the main result [2, Theorem 3], to validate the stability claims made in [10]. For constant calcium concentration values in the 0.09µM to 0.7µM range yielding a nondegenerate I/O characteristic with three fixed points, these fixed points are in a one-to-one correspondence with closedloop equilibria (as determined from the I/S characteristic). Then, having already asserted that the closed-loop system is strongly monotone with bounded trajectories, almost all initial conditions yield trajectories that converge to the two stable equilibria corresponding to the two fixed points at which ( k y / u)(u,w) < 1 which is easily determined graphically as indicated in Fig [Ca 2+ ] = 0.2 µm Unstable Fig. 1. Input-output characteristic with fixed points Unstable Fig. 2. Bistability shown in a range of [Ca 2+ ] Critical calcium concentration values of 0.09µM and 0.7µM yield the two I/O characteristics plotted in Fig. 3 that each display a degenerate fixed point along with a nondegenerate fixed point with ( k y / u)(u,w) < 1. These critical [Ca 2+ ] values correspond to equilibrium bifurcations. For [Ca 2+ ] < 0.09µM, there is a single stable equilibrium corresponding to a low level of CaMKII phosphorylation (inactive state). For [Ca 2+ ] > 0.7µM, there is a single stable equilibrium corresponding to a high level of CaMKII phosphorylation (active state). These critical [Ca 2+ ] values 5927

6 can be interpreted as switching thresholds between inactive and active states, with the intermediate range functioning as a hysteresis band as depicted in Fig. 4. It is noted in [10] that the resting [Ca 2+ ] nominally lies within the hysteresis band. Thus, if the intracellular calcium concentration exceeds the upper threshold long enough, the state trajectory will be attracted to an equilibrium corresponding to the active state. Even as [Ca 2+ ] retreats to the resting value, a high level of activation will persist. LTP has been attributed to this type of phenomena. It is also observed in [10] that cellular mutations leading to off-nominal parameter values can have the effect of shifting or narrowing the hysteresis band in such a way that a high level of activation is no longer maintained by the resting [Ca 2+ ] level Ca 0.09µM Ca 0.70µM Fig. 3. Threshold values of [Ca 2+ ] study of bistability and hysteresis in a mathematical model of CaMKII activation, a biochemical process associated with long-term potentiation (LTP). Several of the key findings reported in [10] were reproduced here using simple algebraic computations afforded by the theoretical machinery expounded in [2], [1], [8] as opposed to exhaustive, simulationbased steady-state analysis. In particular, bistability exists for a range of calcium concentrations that nominally includes the intracellular resting level and switching between high activation and low activation states is described by a hysteresis curve. Other issues such as parameter sensitivity of the bistability/hysteresis phenomena exhibited by Zhabotinsky s model will be explored using a variety of systems-theoretic tools. Also of interest is the study of more complex, higher fidelity mathematical models such as the one developed by Kubota and Bower [7]. Finally, many biologists have observed that the response of biochemical processes such as CaMKII activation is sensitive to the frequency of the underlying stimulus. Examining this phenomena from a dynamic systems perspective will also be pursued. V. ACKNOWLEDGEMENTS The authors would like to thank Dr. William R. Holmes and Dr. Jose Ambrose-Ingerson in the Department of Biological Sciences at Ohio University for many stimulating and fruitful discussions. REFERENCES [1] D. Angeli and E. D. Sontag. Monotone control systems. IEEE Transactions on Automatic Control, 48(10): , [2] D. Angeli and E. D. Sontag. Multi-stability in monotone input/output systems. Systems & Control Letters, 51: , [3] D. Angeli and E. D. Sontag. Oscillations in i/o monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55: , [4] C. J. Coomber. Site-selective autophosphorylation of Ca 2+ / calmodulin-dependent protein kinase II as a synaptic encoding mechanism. Neural Computing, 10: , [5] G. A. Enciso and E. D. Sontag. Monotone bifurcation graphs. Journal of Biological Dynamics, 2(2): , [6] W. R. Holmes. Models of calmodulin trapping and CaM kinase II activation in a dendritic spine. Journal of Computational Neuroscience, 8:65 85, [7] Y. Kubota and J. M. Bower. Transient versus asymptotic dynamics of CaM kinase II: Possible roles of phosphatase. Journal of Computational Neuroscience, 11: (17), November [8] E. D. Sontag. Some new directions in control theory inspired by systems biology. Systems Biology, 1(1):9 18, [9] E. D. Sontag. Monotone and near-monotone systems. In G. Garcia I. Queinnec, S. Tarbouriech and S-I. Niculescu, editors, Biology and Control Theory: Current Challenges (Lecture Notes in Control and Information Sciences, volume 357, pages Springer-Verlag, Berlin, [10] A. M. Zhabotinsky. Bistability in the Ca 2+ /calmodulin-dependent protein kinase-phosphatase system. Biophys. J., 79(5): , Fig. 4. Hysteresis curve IV. CONCLUDING REMARKS This paper has investigated the application of recent theoretical results for input-output monotone systems to the 5928