Subunit exchange enhances information retention by CaMKII in dendritic spines

Molecular bistables are strong candidates for long-term information storage, for example, in synaptic plasticity. Calcium/calmodulin-dependent protein Kinase II (CaMKII) is a highly expressed synaptic protein which has been proposed to form a molecular bistable switch capable of maintaining its state for years despite protein turnover and stochastic noise. It has recently been shown that CaMKII holoenzymes exchange subunits among themselves. Here, we used computational methods to analyze the effect of subunit exchange on the CaMKII pathway in the presence of diffusion in two different micro-environments, the post synaptic density (PSD) and spine cytosol. We show that CaMKII exhibits multiple timescales of activity due to subunit exchange. Further, subunit exchange enhances information retention by CaMKII both by improving the stability of its switching in the PSD, and by slowing the decay of its activity in the spine cytosol. The existence of diverse timescales in the synapse has important theoretical implications for memory storage in networks.


Introduction 8
Memories are believed to be stored in synapses, encoded as changes in synaptic strength (1)(2)(3). Long Term Potentia-9 tion (LTP), an activity dependent change in synaptic strength, is considered to be the primary post-synaptic memory 10 mechanism (4, 5). Various behavioural experiments strongly suggest a critical role for CaMKII in induction of LTP 11 (6,7). In the CA1 region of Hippocampus, blocking CaMKII activity blocks the induction of LTP (8). After LTP 12 induction, several other pathways including protein synthesis (9), clustering of receptors (10), receptor translocation 13 (11) and PKM-ζ activation (12), have been suggested as mechanisms for long-term maintenance of synaptic state. 14 state despite the potent resetting mechanisms of chemical noise and protein turnover. In the small volume of the 18 synapse (∼ 0.02 µm 3 (14)), the number of molecules involved in biochemical processes range from single digits to a 19 few hundred, thereby increasing the effect of chemical noise. Lisman proposed that a kinase and its phosphatase 20 could form a bistable molecular switch able to maintain its state for a very long time despite turnover (15). It has 21 been shown by various mathematical models that CaMKII and its phosphatase PP1 may form a bistable switch 22 (16,17) which can retain its state for years despite stochastic chemical noise and protein turnover (18). Although 23 there is experimental evidence that CaMKII/PP1 is bistable in in vitro settings (19,20), experimental evidence for in 24 vivo bistability is lacking. In spine cytosol, CaMKII has been shown not to act like a bistable switch but rather a leaky 25 integrator of calcium activity (8). However, CaMKII may be bistable in special micro-environments such as the "core" 26 PSD where it attaches to NMDA receptor (21,22). 27 From computational perspective, the CaMKII/PP1 bistable system is an attractive candidate for memory storage 28 (23). Bistability provides a plausible solution to the problem of state maintenance. Previous modeling work has 29 shown that CaMKII/PP1 system may form a very stable switch despite protein turnover and stochastic noise in 30 the small volume of the synapse (11). The stability increases exponentially with the number of holoenzymes (18). 31 It is important to note that this model exhibits bistable behaviour only in a narrow range of PP1 concentrations in 32 the PSD. This strict restriction may be met because phosphorylated CaMKII is protected from phosphatases in PSD 33 except PP1 (24) which is tightly regulated in the PSD (25). 34 CaMKII has another remarkable property which was hypothesized by Lisman (26) but discovered only recently, 35 namely, subunit exchange. In this process, two CaMKII holoenzymes can exchange active subunits leading to spread 36 of CaMKII activation (27). 37 In this paper, we adapt the model of Miller and Zhabotinksy (MZ) (18) to include subunit exchange and diffusion, 38 and quantify the effects of subunit exchange on the properties of the CaMKII-PP1 system in two adjacent neuronal 39 micro-environments: PSD and spine cytosol. 40 In the PSD, PP1 is tightly regulated and CaMKII is protected from other phosphatases. In the spine cytosol, 41 CaMKII is accessible to other phosphatases along with PP1. We examine how state switching lifetimes in the PSD are 42 affected by subunit exchange in different contexts of PP1 levels, turnover, and clustering of CaMKII. In the spine 43 cytosol we show how the integration of calcium stimuli generates two time-courses of CaMKII activity as a result of 44 subunit exchange (8).

46
Model validation. The basic computational units in our model are individual CaMKII subunits and a CaMKII ring 47 consisting of 6 or 7 CaMKII subunits. We treat the CaMKII ring as a proxy for the CaMKII holoenzyme, which consists 48 of two such rings stacked over each other (28,29). In our model, CaMKII exists in 15 possible states compared to 2 in 49 (18) (see Materials and Methods). This leads to many more reactions than the MZ model. Since analytical comparison 50 of the two models was not possible, we first compared numerical results from our model without diffusion and 51 without subunit exchange with the MZ model ( Fig. 1). 52 Our model exhibited all the key properties of the MZ model: 1. CaMKII/PP1 under basal calcium stimulus 53 conditions formed a bistable switch in the PSD (Fig. 1C, D), 2. The stability of the switch increased exponentially 54 with system size (Fig. 1E), 3) Increased number of PP1 molecules (N PP1 ) shut off the switch (Fig. 2), and 4. bistability 55 was robust to slow turnover of CaMKII (Fig. 3). 56 Thus, our baseline model exhibited all the key properties that have previously been predicted for the bistable 57 CaMKII switch. However, the subunit exchange and diffusion introduce several interesting additional properties, 58 which we examine now. red sigmoidal fit in Fig. 2B).

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Subunit exchange also had a strong effect on time spent by the switch in transition from one stable state to another 73 (relaxation time). When subunit exchange was enabled, the relaxation time was reduced (red v/s blue dotted line in 74  phosphorylate all inactive holoenzymes produced by turnover, and the switch started to show exponential decay of 81 stability. As expected, turnover increased the number of switching events in the regime of bistability in both cases.

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Thus subunit exchange increases the range of N PP1 and turnover rate over which the switch remains bistable.

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Subunit exchange facilitates the spread of CaMKII activity. As suggested in (27), we found that subunit exchange 84 facilitates spread of CaMKII activation (Fig. 4). When subunits were allowed to diffuse, they could be picked by 85 neighbouring inactive CaMKII holoenzymes. This effectively overcame the first slow step of CaMKII phosphorylation 86 (Eq. (1)) thereby facilitating the spread of activation. 87 We put N CaMKII =18 inactive holoenzymes in a cylinder with the volume of 0.0275 µm 3 and the length of 540 nm 88 representing the PSD. The cylinder was divided into 18 voxels (1 holoenzyme in each voxel). Each voxel was 89 separated by 30 nm, which is the average nearest neighbour distance for CaMKII holoenzymes (31). Each voxel 90 was considered to be a well-mixed environment i.e. diffusion was instantaneous within the voxel. Diffusion was 91 implemented as cross-voxel "jump" reactions (See Materials and Methods). We did not try 2D/3D diffusion because 92 of its simulation complexity and because it would be expected to be qualitatively similar (32). 93 We fixed the diffusion coefficient of PP1 (D PP1 ) and quantified the effect of varying the diffusion coefficient of 94 subunits (D sub ) and basal calcium levels. We used D PP1 =0.5 µm 2 s −1 which is the observed value of the diffusion 95 coefficient of Ras, which is a similar sized protein (33). We ran simulations for 4 hours at basal calcium concentration 96 [Ca 2+ ]=80 nM and without subunit exchange (i.e. D sub =0). Here the system showed no significant CaMKII activity. 97 When we enabled subunit exchange by setting D sub =0.1 µm 2 s −1 , Fig. 4B), CaMKII activity rose to maximum within 98 4 h. As expected, the effect of subunit exchange (rise time quantified as the time taken by CaMKII to rise from 10% 99 to 90%) was stronger when the basal Ca 2+ was higher (Fig. 4C). Increasing D sub decreased the rise time of CaMKII 100 activity.

101
Subunit exchange did not have any impact on the average CaMKII activity at longer time scales (Fig. 4E) though 102 we found that long-time average CaMKII activity increased when subunit exchange was enabled. This change was 103 independent of D sub (not due to subunit exchange) but was strongly dependent on D PP1 . This is due to the fact that 104 potency of PP1 reduced with increased D PP1 (Fig. S2) which led to decreased PP1 activity and hence CaMKII activity. 105 Thus subunit exchange facilitates the spread of kinase activity at short time scale but does not influence its long 106 time activity.   where receptors and CaMKII holoenzymes are clustered at the synapse.

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When there is no subunit exchange across voxels i.e. D sub =0, these switches are expected to switch independently 112 like multiple coins flipped together, resulting in a binomial distribution of activity. The clustered system had 3 113 relatively stable bistable systems (long residence time, Fig. 1E). As expected, without subunit exchange, activity in 114 this system had a binomial distribution (Fig. 5B, red plot).

115
Then we allowed PP1 and CaMKII subunits to undergo linear diffusion. We fixed D PP1 =0.5 µm 2 s −1 as before and 116 varied D sub to quantify effect of subunit exchange. Subunit exchange led to synchronization of switching activity. 117 The population of clustered CaMKII acted as a single bistable switch (Fig. 5B, blue plot). This effect was strong and 118 robust to variation in D sub . Even for a very small value of D sub =0.01 µm 2 s −1 , we observed strong synchronization 119 (Fig. 5D). The synchronization disappeared completely for diffusion coefficient less than D sub =10 −4 µm 2 s −1 for 120 distance d > 30 nm (Fig. 5D).

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Thus for most physiologically plausible values of diffusion coefficient D sub , subunit exchange causes synchroniza-122 tion of switching activity of clustered CaMKII. encoded photoactivable inhibitory peptide after activating it by glutamate uncaging (34). In the spine, CaMKII is 127 more accessible to phosphatases than in the PSD, where our previous calculations had been located. To model the 128 increased availability of phosphatases, we increased the number of PP1 by an order of magnitude, and increased 129 the volume of the compartment to match the volume of a typical spine head i.e. 0.02 µm 3 (14). We found that 130 CaMKII acted as a integrator of calcium activity with typical exponential decay dynamics (Fig. 6A). We then enabled 131 the diffusion of CaMKII subunits and PP1 with same diffusion coefficient D sub =D PP1 =1 µm 2 s −1 . These conditions 132 decreased the rate of dephosphorylation of CaMKII holoenzymes significantly. The decay dynamics could not be 133 fitted using a simple exponential (Fig. 6B). With our values of parameters, decay rate decreased by an order of 134   magnitude when subunit exchange was enabled (Fig. 6B). 135 We expected subunit exchange to have a strong effect on the decay activity of clustered CaMKII in spine cytosol 136 (e.g. CaMKII bound to actin) because of the close proximity of holoenzymes, leading to rapid exchange. Our 137 simulations supported this prediction. If there are populations of clustered as well as non-clustered CaMKII in the 138 spine, we expect that they will exhibit long and short time-courses of activity decay.

139
Thus we suggest that subunit exchange may be a mechanism that leads to CaMKIIα activity decaying with two 140 time-courses in spine cytosol (8).

142
Here we have shown that subunit exchange strongly affects the properties of CaMKII/PP1 pathway, both in its role 143 as a bistable switch in PSD and as an integrator of calcium activity in spine cytosol. In the PSD, where the model was 144 tuned to elicit bistable dynamics from clustered CaMKII, subunit exchange improved the stability of CaMKII/PP1 145 switch by synchronizing the kinase activity across PSD (Fig. 6). It also improved CaMKII tolerance of PP1 and 146 turnover rate ( Fig. 2 and Fig. 3). In the case where CaMKII was uniformly distributed in PSD, subunit exchange 147 facilitated more rapid activation of CaMKII (Fig. 4BCD) (27). These simulation results predict that a CaMKII mutant 148  lacking subunit exchange would be deficient in the switch stability and slower to activate.

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In the spine head, subunit exchange facilitated integration by prolonging the decay time-course of kinase activity 150 (Fig. 6). The fact that CaMKII dynamics changed from an integrator to bistable switch as we moved from spine 151 cytosol (a phosphatase rich environment) to the PSD (where PP1 is tightly controlled) suggests an interesting 152 sub-compartmentalization of functions in these microdomains. Furthermore, we observed that the clustering of 153 CaMKII had important implications for its sustained activity.

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CaMKII is non uniformly distributed in PSD. Most of it is concentrated in a small region of 16 nm to 36 nm 155 below synaptic cleft (22) where it may exist in large clusters given that CaMKII has multiple binding partners in 156 the PSD. Our study predicts that subunit exchange may lead to synchronization when CaMKII is clustered, or 157 more rapid activation by calcium when it is uniformly distributed. Given that CaMKII can form clusters with 158 N-methyl-D-asparate (NMDA) receptors, it would be interesting to study the mixed case where some CaMKII is 159 clustered and rest is uniformly distributed. This would require detailed 3D simulation and is beyond the scope of 160 this study.

161
Subunit exchange is unlikely to have any impact on neighbouring spines. The mean escape time of a single CaMKII 162 subunit from a typical spine is between 8 s to 33 s (35). In a real synapse, this time would be even larger given that 163 CaMKII interacts with many other molecules. Any phosphorylated subunit is almost certain to be de-phosphorylated 164 by PP1 during this time. We therefore predict that the effects of synchronization are local to each PSD, where PP1 165 is known to be tightly controlled. Subunit exchange loses its potency in the phosphatase rich region of the bulk 166 spine head or dendrite. We therefore consider it unlikely that CaMKII subunit exchange plays any role in intra-spine 167 information exchange such as synaptic tagging.

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Finally, we suggest that existence of diverse time-scales of CaMKII activity in the PSD and spine head has important 169 theoretical implications. A very plastic synapse is good at registering activity dependent changes (learning) but bad 170 at retaining old memories. On the other hand, a rigid synapse is good at retaining old memories but is not efficient for 171 learning. A theoretical meta-model which sought to strike a balance between these two competing demands requires 172 that diversity of timescales should exist at the synapse (36). In this model, complex synapses with state variables 173 with diverse time-scales are shown to form a memory network in which storage capacity scales linearly with number 174 of synapses, and memory decay follows 1/ √ t -a power-law supported by psychological studies (37). This model 175 requires memory trace to be first stored in a fast variable and then progressively and efficiently transferred to slower 176 variables. Our study suggests a concrete mechanism for such a process. Here, calcium concentration in PSD can 177 be mapped to the fastest variable. The CaMKII integrator in cytosol could represent the second slower variable to 178 which the trace is transferred from calcium. Further, the state information is transferred to the third slower CaMKII 179 bistable switch. The dynamics of CaMKII in the PSD forms an even slower bistable variable for longer retention of 180 the memory trace. It is possible that memory is transferred from here to even slower variables, such as sustained 181 receptor insertion (11), PKM-ζ activation (12), or local protein synthesis (9).

183
The kinase CaMKII has a rich history of modeling spanning over two decades with varying complexity (16). We based our study 184 on the work of Miller and Zhabotinksy (MZ model) (18). We extended their model to incorporate subunit exchange and diffusion. 185 We treat the CaMKII ring as proxy for the holoenzyme because vertical dimers (one subunit from the top ring and one from 186 the ring below) are inserted or released together (38) and we assumed that the top and the bottom subunits of a vertical dimer 187 phosphorylate and de-phosphorylate together.

188
In our model, a CaMKII ring with n subunits (n=6 or 7) can exist in n different states x a y n−a where a is the number of 189 un-phosphorylated subunits (represented by x), and n − a is the number of phosphorylated subunits (represented by y). Following Zhabotinksy, we also assumed that Ca 2+ /CaM binding to CaMKII ring is independent of the current state of the 201 ring. The activation of a subunit by Ca 2+ follows a Hill equation (Eq. (1)) (17).

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x a y n−a v 1 x a−1 y n−a+1 v 2 x a−2 y n−a+2 where n = 6 or 7, and 1 ≤ a ≤ n.

204
The dephosphorylation of the CaMKII ring and the single subunit follow a Michaelis-Menten like scheme. But rather than 205 using the Michaelis-Menten approximation, we implemented this as coupled mass-action chemical reactions as shown by Eq. (2). 206 PP1 + x a y n−a k + k − PP1.x a y n−a k 2 PP1 + x a+1 y n−a−1 where n = 6 or 7, and 1 ≤ a ≤ n.  x a y 7−a + x k + x k − x x a+1 y 6−a , for 0 ≤ a ≤ 6 x a y 6−a + y k + y k − y x a y 7−a , for 0 ≤ a ≤ 6 [3] 214 The values of k + x , k − x , k + y , and k − y are not available in the literature that we are aware of. We assumed that these reactions operate 215 at the timescale of second. Bhattacharya et. al. (38) speculate that upon activation, the hub of holoenzyme becomes less stable and 216 more likely to open up and lose a subunit. Therefore we assumed the rate of losing subunit (k − y , k − x ) to be larger than the rate of 217 gaining a subunit (k + y , k + x ). In all simulations, we maintained the following ratio k − x = 10k + x N CaMKII and k − y = 10k + y N CaMKII . 218 PP1 deactivation. In the PSD, PP1 is the primary -and perhaps only -phosphatase known to dephosphorylate CaMKII (39). In the 219 PSD, PP1 is inhibited by phosphorylated inhibitor-1 (I1P) and a dopamine-and cyclic-AMP regulated neuronal phosphoprotein 220 (DARPP-32) (an isomer of inhibitor-1 (I1)) (40, 41) (also see (42)).

221
I1 is phosphorylated by protein kinase A (PKA) and dephosphorylated by calcineurin (CaN). We followed Zhabotinksy's 222 assumption that I1 level are constant in the PSD because I1 exchanges rapidly with spine cytosol (17). We assumed the concentra-223 tion of I1 to be the same in both PSD and spine cytosol. We followed Miller (5) is also assumed to be fast (18).
Turnover. The turnover of CaMKII is a continuous process with rate v t s −1 . We modeled turnover by replacing a CaMKII ring 231 with a > 1 phosphorylated subunits by an inactive CaMKII ring of the same symmetry.

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x a y 6−a v t x 6 y 0 , for a ≥ 1 x a y 7−a v t x 7 y 0 , for a ≥ 1 of h namely h crit for which error is in acceptable bound i.e. < 1%. The value of h crit is determined by the fastest bimolecular 239 reaction (Eq. (2)) and the slowest diffusion coefficient. The lower bound on h i.e. h crit k + D PP1 +D sub where k is the reaction rate (44). 240 Based on our own numerical results (see SI) and other studies (44,45), we are confident that h ≥ 10h crit = k + D PP1 +D sub is a good 241 value (45). We have h crit ≤ 3.2 nm whenever D PP1 + D sub ≥ 0.5 µm 2 s −1 . For all simulations presented in main text, we maintain 242 h ≥ h crit . For a case where h is smaller than h crit in some trajectories see Fig. S2.