Coupled Molecular Feedback Loops Maintain Synaptic Long-Term Potentiation: A Computational Model

The brain stores memories by adjusting the strengths of connections between neurons. Such synaptic plasticity comes in different forms that strengthen or weaken synapses and range from very short-lived to long-lasting. One of the most well-studied forms of plasticity is long-term potentiation, LTP, a phenomenon whereby synaptic strength is persistently increased in response to stimulation. Different forms of LTP are known to play important roles in both short-term and long-term memory. Many different proteins have been identified in the sub-cellular molecular processes that are involved in LTP. An important question is how these proteins, with lifetimes measured in hours or days, can maintain memories for months or years. We present a computational model that shows how this problem can be solved by two interconnected feedback loops of molecular reactions, one involving the atypical protein kinase PKM{\zeta} and its messenger RNA, the other involving PKM{\zeta} and GluR2-containing AMPA receptors. The model demonstrates that robust bistability - stable equilibria in the synapse's potentiated and unpotentiated states - can arise from simple molecular reactions. Our simulations are able to account for a wide range of empirical results, including induction and maintenance of late-phase LTP, cellular memory reconsolidation and the effect of different pharmaceutical interventions.


Glossary
AMPA α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid, an artificial glutamate analog AMPAR AMPA receptor, a glutamatergic receptor that is activated by AMPA, in addition to glutamate. anisomycin an antibiotic that inhibits protein synthesis by blocking the formation of peptide bonds BRAG2 a protein that plays a key role in AMPAR endocytosis CA1, CA3 cornu ammonis area 1 and 3, regions of the hippocampus consolidation conversion of synaptic potentiation from a short-lived to a lasting form; formation of L-LTP dendrite protrusion on a neuron where most of the incoming synapses are located E-LTP early-phase LTP endocytosis of receptors: removal from the cell membrane by internalization EPSP excitatory postsynaptic potential GluR23Y a synthetic peptide that blocks endocytosis of GluR2 AMPARs glutamate the most common excitatory neurotransmitter in the central nervous system hippocampus a structure in the mammalian brain GluR2 one of four AMPA receptor subunit types kinase an enzyme that catalyzes phosphorylation L-LTP late-phase LTP LTM long-term memory LTP long-term potentiation mRNA messenger RNA, a molecule that specifies the sequence of amino acids in a protein NMDA N-Methyl-D-aspartic acid, a synthetic glutamate analog NMDAR NMDA receptor, a glutamatergic receptor that is activated by NMDA, in addition to glutamate peptide short chain of amino acids. Chains longer than about 50 amino acids are called proteins. phosphorylation addition of a phosphate group to a molecule, e.g. a protein, which may thereby be activated or deactivated PKMζ "PKM zeta", a constitutively active protein kinase found in brain tissue protein kinase a kinase that phosphorylates proteins PSD postsynaptic density, a region of cell membrane that forms the receiving side of a synapse PSI protein synthesis inhibition (or inhibitor) reconsolidation restabilization of LTP after reactivation-induced destabilization RNA ribonucleic acid, a molecule produced by transcription of DNA ZIP zeta-inhibitory peptide, a molecule that inhibits PKMζ activity

Background
In his address to the Royal Society in 1894, Santiago Ramon y Cajal hypothesized that the brain stores information by adjusting the strengths of associations between neurons, as well as by growing new connections (Cajal, 1894). In the years since, the existence of both of these mechanisms, now known as synaptic plasticity and synaptogenesis, respectively, has been well established, and there is ample evidence that synaptic plasticity plays an important role in learning and memory (Frankland & Bontempi, 2005;Kandel, Dudai, & Mayford, 2014;Sossin, 2008). It may thus be regulated on either the pre-or postsynaptic side, and mechanisms of synaptic plasticity have been shown to operate in both compartments (Kandel et al., 2014). Plasticity may either strengthen or weaken a synapse, and the effect may be short-lived or long-lasting. Shortterm synaptic plasticity, lasting from milliseconds to minutes, is primarily due to presynaptic mechanisms that adjust the amount of transmitter release, whereas postsynaptic modifications that adjust the number and sensitivity of receptors are important for long-term plasticity (Sossin, 2008). In particular, this is true of long-term potentiation (LTP), a type of persistent strengthening of synapses in response to stimulation (Bliss & Collingridge, 1993), which has been studied extensively in the CA3-CA1 synapses of the rodent hippocampus (Sossin, 2008), and is known to depend on an increase in the number of receptors inserted in the postsynaptic membrane (Malenka & Bear, 2004). There are at least two forms of LTP: Moderately strong stimulation induces early-phase LTP (E-LTP), which persists for at most a few hours. When the stimulation is stronger, E-LTP may be followed by late-phase LTP (L-LTP), which can last for days, months or longer (Malenka & Bear, 2004), and is believed to be an important mechanism for the storage of long-term memories (Frey, Krug, Reymann, & Matthies, 1988;Kandel, 2000).
The establishment of L-LTP, known as synaptic or cellular memory consolidation, is a process that takes less than an hour (Bourtchouladze et al., 1998;Fonseca, Nägerl, & Bonhoeffer, 2006) and requires synthesis of new protein. This has been demonstrated by showing that infusion of protein-synthesis-inhibiting drugs such as anisomycin can prevent establishment of L-LTP (Frey, Huang, & Kandel, 1993;Huang, Li, & Kandel, 1994;Nguyen & Kandel, 1996). On the behavioral level, protein synthesis inhibition (PSI) has been shown to impair the formation of long-term memory, consistent with the notion of L-LTP as a memory mechanism (Davis & Squire, 1984). Once long-term memory is established, it is in general no longer vulnerable to infusion of a protein synthesis inhibitor (Davis & Squire, 1984). However, memory retrieval can induce a state of transient instability, during which the memory is again susceptible to protein synthesis inhibition (Misanin, Miller, & Lewis, 1968;Nader, Schafe, & Le Doux, 2000;Przybyslawski & Sara, 1997). This susceptibility of memory to post-retrieval PSI infusion has been shown to correlate with instability of L-LTP at the neural level (Hong et al., 2013;Rao-Ruiz et al., 2011), providing further evidence of the importance of LTP as a mechanism of longterm memory. The synaptic destabilization that is triggered by memory retrieval is followed by a period of restabilization which has similarities with the initial synaptic consolidation that follows memory acquisition. It has therefore become known as memory reconsolidation (Przybyslawski & Sara, 1997), more specifically synaptic (or cellular) reconsolidation, to avoid confusion with the related but distinct phenomenon systems reconsolidation, a temporary dependence on the hippocampus for restabilization of a memory after reactivation (retrieval). For reviews of reconsolidation research, see Baldi and Bucherelli (2015), Besnard, Caboche and Laroche (2012), Nader and Einarsson (2010). For a computational model of systems reconsolidation, see Helfer and Shultz (2017).

Glutamatergic synapses
In this report, we focus on L-LTP induction and maintenance at glutamatergic synapses, the most abundant type of synapse in the vertebrate nervous system (Mayer & Armstrong, 2004;Meldrum, 2000). Glutamatergic synapses contain several kinds of receptors that are activated by the neurotransmitter glutamate. Of particular interest for LTP are the α-amino-3-hydroxy-5methyl-4-isoxazolepropionic acid receptor (AMPA receptor or AMPAR), which mediates synaptic transmission (Maren, Tocco, Standley, Baudry, & Thompson, 1993), and the N-methyl-D-aspartate receptor (NMDA receptor or NMDAR), which is involved with regulatory functions including the regulation of synaptic strength (Ben Mamou, Gamache, & Nader, 2006;Walker & Davis, 2002).
AMPARs are ion channels that open when activated by the neurotransmitter glutamate. The opening of the channel allows positively charged ions, mainly sodium and potassium, to flow through the cell membrane (Henley & Wilkinson, 2013). This causes a partial depolarization of the membrane, which at rest is polarized by a net negative charge inside the cell. The partial depolarization is known as an excitatory postsynaptic potential, or EPSP, and the amplitude of the EPSP produced by a single action potential arriving at a synapse is a measure of synaptic strength. Among other factors, the EPSP amplitude depends on the number of AMPARs inserted in the postsynaptic density (PSD), the area of cell membrane that constitutes the receiving side of the synapse (Henley & Wilkinson, 2013). Thus mechanisms that control the trafficking of AMPARs into and out of the PSD play an important part in the regulation of synaptic strength.
AMPARs are heterotetramers, i.e. they consist of four non-identical subunits. The subunits are of four different kinds, named GluR1, GluR2, GluR3 and GluR4, and AMPARs can be made up of different combinations of these (Isaac, Ashby, & McBain, 2007). GluR2 is of particular interest here, because L-LTP is associated with an increase in the number of GluR2-containing AMPARs inserted in the PSD (Hong et al., 2013;Migues et al., 2010;Yao et al., 2008).
AMPA receptors are not permanently inserted in the PSD, but are constantly being recycled. Certain proteins transport AMPARs into the PSD from pools maintained in adjacent areas, while others remove them (a process known as internalization or endocytosis) and either recycle them to stand-by pools or mark them for degradation (Huganir & Nicoll, 2013;Malinow & Malenka, 2002).

Protein kinase M zeta (PKMζ)
Many proteins have been implicated in the induction and maintenance of LTP, including CaMKII, PKA, MAPK and several isoforms of PKC (for a review, see Malenka & Bear, 2004).
An atypical isoform of PKC, protein kinase Mζ (PKMζ), is believed to play an important role for L-LTP. The level of PKMζ has been shown to increase as the result of NMDA receptor stimulation (Osten, Valsamis, Harris, & Sacktor, 1996;Sacktor et al., 1993), consistent with its proposed role in L-LTP induction. Inhibition of PKMζ activity results in disruption of established L-LTP (Ling et al., 2002;Sajikumar, Navakkode, Sacktor, & Frey, 2005;Serrano, Yao, & Sacktor, 2005), and perfusion of PKMζ into a neuron can induce L-LTP (Ling et al., 2002) . PKMζ activity is believed to increase the number of inserted GluR2-containing AMPARs at the synapse both by facilitating the trafficking of these receptors into the PSD and by inhibiting their removal (Sacktor, 2011). GluR2-containing AMPARs are held at extrasynaptic pools by the protein PICK1 which binds to the GluR2 subunit (Yao et al., 2008). PKMζ facilitates interaction between the trafficking protein NSF and the GluR2 subunit, which results in its release from PICK1, freeing the AMPARs to diffuse laterally into the PSD (Yao et al., 2008). Furthermore, once GluR2 AMPARs are inserted in the PSD membrane, PKMζ prevents their removal by inhibiting the interaction between the protein BRAG2 and the GluR2 subunit (Henley, Barker, & Glebov, 2011), an interaction that plays a key part in endocytosis of GluR2containing AMPA receptors (Hardt, Nader, & Wang, 2014;Sacktor, 2011).
While GluR2-containing AMPARs are important for the stabilization of L-LTP, there is evidence that GluR2-lacking AMPARs play an important role in the induction of early-phase LTP (E-LTP), and also in reconsolidation. Several studies have shown that GluR2-lacking AMPARs are initially inserted at the time of memory acquisition or LTP induction, and then gradually replaced by GluR2-containing AMPARs during consolidation (Clem & Huganir, 2010;Kessels & Malinow, 2009;Plant et al., 2006). Hong et al. (2013) showed that memory reactivation triggers an abrupt replacement of GluR2 AMPARs by GluR2-lacking AMPARs. This is followed by a gradual reversion, i.e. the GluR2 AMPARs are restored and the number of GluR2-lacking AMPARs declines, as the potentiated state of the synapse is restabilized (Hong et al., 2013). Because the temporary removal of GluR2 AMPARS is compensated for by an increase in GluR2-lacking AMPARs, the synaptic strength remains more or less constant during the period of instability (Hong et al., 2013). Rao-Ruiz et al. (2011) reported similar results, although they observed a brief period of reduced synaptic strength between the GluR2 AMPAR removal and non-GluR2 AMPAR insertion. Taken together, these results suggest that the stabilization of LTP, both initially during consolidation, and after reactivation-induced destabilization, requires insertion of GluR2-containing AMPARs, and that PKMζ plays an important role in maintaining the GluR2 AMPARS at the synapse.
An important question is how L-LTP, which can last for months or longer, can be maintained by a protein like PKMζ, with a half-life that probably does not exceed several hours or at most a few days (Hernandez, Oxberry, Crary, Mirra, & Sacktor, 2014;Sacktor, 2010;Westmark et al., 2010). A proposed answer to this question involves local translation of messenger RNA (mRNA) in dendritic spine heads. Most synapses are formed at spine heads, with one synapse per spine (Schwartz & Westbrook, 2000). PKMζ mRNA is transported from the cell body to dendritic spine heads (Doyle & Kiebler, 2011;Muslimov et al., 2004), but the mRNA in its basal state is translationally repressed by molecules that bind to it, or to the complex of proteins required to initiate translation (Bramham & Wells, 2007;Doyle & Kiebler, 2011;Westmark et al., 2010). There is evidence that PKMζ catalyzes reactions that lift this translational block (Sacktor, 2010), possibly through inhibition of the PIN1 protein (Sacktor, 2011), resulting in a positive feedback loop (Sacktor, 2010). By promoting its own synthesis in this manner, PKMζ may be able to remain at an increased level, and thus maintain L-LTP, for a long time, perhaps indefinitely.
It has also been suggested that the increased amount of inserted GluR2-containing AMPARs at a potentiated synapse captures the PKMζ molecules and keeps them from dissipating away from the synaptic compartment (Sacktor, 2011). This hypothesis is supported by several studies that show that blocking of GluR2 AMPAR endocytosis can prevent depotentiation under protocols that otherwise cause disruption of L-LTP (Migues et al., 2010;Rao-Ruiz et al., 2011;Yu, Chang, & Gean, 2013). Together with PKMζ's inhibiting effect on AMPAR endocytosis this constitutes a second feedback loop, a reciprocal relationship in which PKMζ and GluR2-containing AMPARs prevent each other's removal from the synapse. As we shall see, the interaction between these two feedback loops plays a central role in our explanation of synaptic bistability, that is that synapses have two stable equilibrium states, unpotentiated and potentiated. Transient stimuli can cause a synapse to transition between these two states, but in the absence of such signals it tends to remain in one state or the other.

L-LTP, LTM and pharmacological interventions
The notion that L-LTP is an important neural correlate of long-term memory (LTM) has been supported experimentally by demonstrating that pharmacological interventions that block L-LTP induction also interfere with the establishment of LTM (Dudai, 2004), and that interventions that disrupt established L-LTP also impair consolidated memories (Pastalkova et al., 2006). Here we consider three types of pharmaceuticals that have been shown to produce significant results with respect to both L-LTP induction and maintenance, and to related behavior-level memory phenomena.

Protein synthesis inhibitors. Infusion of protein synthesis inhibitors (PSIs) such as
anisomycin into brain tissue can prevent the induction of L-LTP (Dudai, 2004), and also interferes with memory consolidation, the establishment of LTM (Judge & Quartermain, 1982;Schafe & LeDoux, 2000). Once L-LTP is established, it becomes resistant to infusion of anisomycin (Bourtchouladze et al., 1998;Fonseca et al., 2006). This does not mean that L-LTP can be maintained indefinitely without ongoing protein synthesis, but rather that it can tolerate an interruption of protein synthesis for the amount of time that anisomycin remains active after infusion.
Reactivation of a consolidated memory, e.g. by a reminder, can temporarily return it to a labile state in which it is again vulnerable to PSI infusion (Judge & Quartermain, 1982;Nader et al., 2000). The putative molecular process underlying this phenomenon has been termed cellular or synaptic memory reconsolidation (Nader et al., 2000;Sara, 2000). Concordant with the hypothesis that L-LTP is the neural correlate of LTM, the temporary post-reactivation vulnerability of LTM to PSI infusion can be explained as destabilization of L-LTP, followed by a restabilization phase that requires protein synthesis, hence the susceptibility to PSI. The destabilization has been shown to require the activity of NMDA receptors (Ben Mamou et al., 2006), and to depend critically on endocytosis of GluR2 AMPARs (Yu et al., 2013;Yu, Huang, Chang, & Gean, 2016).
Thus protein synthesis inhibition is known to both prevent establishment of L-LTP and to block reconsolidation.

ZIP.
Much of the work demonstrating the role of PKMζ in L-LTP is based on administration of the synthetic peptide ZIP (zeta-inhibitory peptide), which binds to the catalytic region of the PKMζ molecule, thus blocking its enzymatic activity (Serrano et al., 2005). On the behavioral level, infusion of ZIP into brain tissue has been shown to impair consolidated LTM (Pastalkova et al., 2006). On the neural level, ZIP is known to disrupt established L-LTP when applied during the maintenance phase (Ling et al., 2002;Sajikumar et al., 2005;Serrano et al., 2005;Shema, Sacktor, & Dudai, 2007). These results are consistent with the notion of a positive feedback loop: Inhibiting PKMζ's enzymatic activity prevents it from catalyzing its own synthesis; the PKMζ concentration then drops, the AMPAR endocytosis rate increases, and the synapse returns to its basal state. On the other hand, ZIP does not prevent L-LTP induction when applied only during or immediately after stimulation. This was demonstrated by Ren et al. (2013) in an in-vitro experiment where they were able to precisely control the onset and duration of ZIP application.
3. GluR2 3Y . GluR23Y is a synthetic peptide that blocks regulated endocytosis of GluR2 AMPARs (Ahmadian et al., 2004;Brebner et al., 2005). Infusion of GluR23Y has been shown to block both the destabilizing effect of PSI infusion after memory reactivation (Hong et al., 2013;Yu et al., 2013) and the depotentiating effect of ZIP during L-LTP maintenance (Migues et al., 2010). The GluR23Y peptide is modeled on a sequence of the GluR2 subunit's carboxyl tail and its endocytosis-inhibiting effect is believed to be due to competitive disruption of the binding of endocytosis-related proteins to this sequence on GluR2 subunits (Brebner, Phillips, Wang, & Wong, 2006)

Computational model
The findings described above suggest a model of L-LTP maintenance with two connected feedback loops: (1) PKMζ maintains its own mRNA in a translatable state and translation of the mRNA in turn replenishes PKMζ; (2) PKMζ maintains GluR2-containing AMPA receptors at the synapse, and these in turn keep PKMζ molecules from dissipating away from the synaptic compartment. Below we describe a computational model that incorporates these relationships and investigate its ability to account for results reported in the empirical literature.

Method Deterministic vs. stochastic simulation
Systems of chemical reactions can be modeled either by deterministic methods based on ordinary differential equations (ODEs) or by stochastic simulation. When the numbers of molecules are small, stochastic simulation is the better choice, because random fluctuations then have significant effects that are not captured by deterministic methods (Rao & Arkin, 2003). In particular, random fluctuations can cause a small system to spontaneously transition from one steady state to another; the resulting impact on system stability can be studied in a stochastic simulation, but not in a deterministic model (Cao & Samuels, 2009), because the latter only accounts for average reaction rates over a large number of molecules.
The molecules of interest for our simulation are present in small numbers in a dendritic spine head, e.g. fewer than a hundred PKMζ molecules (see Appendix A) and at most ca 150 AMPA receptors (Matsuzaki et al., 2001;Takumi, Matsubara, Rinvik, & Ottersen, 1999). This is well below the size of system that can be realistically simulated by deterministic methods (Cao & Samuels, 2009; Gonze, Abou-Jaoude, Ouattara, & Halloy, 2011). We therefore base our simulation on the Gillespie algorithm (Gillespie, 1977), a well-established and widely used approach to discrete and stochastic simulation of reaction systems (Cao & Samuels, 2009;Gonze et al., 2011;Rao & Arkin, 2003). The following description presents the model at a relatively high level of abstraction; for a detailed account, please refer to Appendix B. The model consists of four inter-dependent pairs of processes (see Figure 1):

GluR2 AMPAR trafficking into and out of the PSD
The model includes a fixed-size population of GluR2-containing AMPARs. At any time, a subset of the AMPARs are inserted in the PSD while the remainder are maintained in extrasynaptic pools. The molecular mechanisms that transport AMPARs into the PSD are not explicitly modeled. Instead, AMPAR insertion is simulated as a probabilistic process where the rate of insertion is proportional to the number of uninserted GluR2 AMPARs in stand-by pools. Removal (endocytosis) is enabled by the protein BRAG2. In addition to the BRAG2facilitated endocytosis, a constitutive process removes AMPARs from the synapse at a low rate.

Inhibition and disinhibition of BRAG2-GluR2 interaction
The mechanism by which PKMζ inhibits the interaction between BRAG2 and the GluR2 subunit to block AMPAR removal from the PSD is not known, but presumably involves phosphorylation of some substrate. We model the inhibition as phosphorylation of the BRAG2 molecule itself; other possibilities include phosphorylation of a site on the GluR2 subunit or of another participating protein. The BRAG2-GluR2 interaction is restored through dephosphorylation of the same substrate by a phosphatase, which is assumed to be present in fixed concentration.

Synthesis and degradation/dissipation of PKMζ
Synthesis consists in local translation of PKMζ mRNA. Inserted GluR2 AMPARs inhibit degradation and/or dissipation of PKMζ away from the synaptic compartment by sequestering the PKMζ molecules (Sacktor, 2011). This is modeled as an affinity of PKMζ for inserted GluR2 AMPARs, with a reduced dissipation/degradation rate while so attached.

Activation/deactivation of PKMζ mRNA
PKMζ Although the increase in PKMζ level that is associated with L-LTP induction is known to depend on NMDAR activation (Sacktor et al., 1993), the underlying biochemical pathways are unknown.
In the model this mechanism is represented by an unspecified enzyme that we call E1 which, when activated by a reaction cascade triggered by NMDAR activation, has the ability to lift the translational block on PKMζ mRNA, thereby enabling PKMζ synthesis.
Similarly, the destabilizing effect of memory reactivation has been shown to depend on NMDAR activity and on GluR2 AMPAR endocytosis (Hong et al., 2013;Migues et al., 2016;Yu et al., 2013), but the biochemical cascades that connect these event have not yet been identified. In our model, reactivation is simulated as an increase in the level of a second unspecified enzyme E2 with the ability to catalyze GluR2 AMPAR endocytosis.
In addition to these processes, the model includes simulation of the effects of the three pharmaceuticals described in the introduction. The time intervals that these drugs remain at a high enough concentration to inhibit their targets depend on the doses infused and also on their specific rates of decay or metabolism. The intervals used here are based on activity periods reported in the cited references: PSI: Infusion of a protein synthesis inhibitor is simulated by disabling PKMζ synthesis for nine hours, the amount of time that the protein synthesis inhibitor anisomycin remains active after infusion into brain tissue (Wanisch & Wotjak, 2008). GluR2 3Y : Perfusion of GluR23Y is simulated by disabling regulated GluR2 AMPAR endocytosis for twelve hours (Migues et al., 2016). (GluR23Y does not affect constitutive GluR2 AMPAR endocytosis (Brebner et al., 2005).)

Objectives
Our computational model simulates the regulation of PKMζ concentration at the postsynaptic density and its role in the induction and maintenance of L-LTP. The goal for the model is to simulate the empirical results described in the introduction and summarized below:

ZIP during stimulation does not prevent L-LTP induction
ZIP treatment during and immediately after stimulation does not prevent establishment of L-LTP (Ren et al., 2013)

Reactivation followed by PSI infusion does disrupt LTM
PSI administered within a time window after reactivation disrupts LTM (Debiec et al., 2002;Nader et al., 2000) 8

GluR2 3Y blocks the LTM-disrupting effect of PSI
GluR23Y administered together with PSI after reactivation blocks the LTM-disrupting effect of PSI (Hong et al., 2013;Yu et al., 2013Yu et al., , 2016

Results
In the following graphs, P denotes the total number of PKMζ molecules in the synaptic compartment, corresponding to the sum of the P and AI•P variables in the model (Appendix B), and AI denotes the number of AMPA receptors inserted in the PSD, with and without bound PKMζ molecules, corresponding to the sum of the AI and AI•P variables in the model. Reaction numbers refer to the reactions described in Appendix B.

NMDAR stimulation induces L-LTP
We model the result of strong NMDAR stimulation as a rapid increase of the population of active E1 enzyme molecules. This causes the translational repression of PKMζ mRNA to be lifted (reactions 28-30) and synthesis of PKMζ to start (reaction 7). It takes the model between 30 and 60 minutes of simulated time to complete the switch to its potentiated steady state in which there is a high number of inserted GluR2 AMPARs, (Figure 2). This is consistent with the observed cellular consolidation window (Davis & Squire, 1984;Dudai, 2004).
Although the spike of activated E1 enzyme phosphorylates the mRNA-binding proteins, resulting in a high level of activated PKMζ mRNA (RA in the model), translation is prevented by the protein synthesis inhibitor, and PKMζ synthesis is not initiated (Frey et al., 1988;Osten et al., 1996). When the E1 enzyme returns to its inactive form the mRNA becomes repressed again, and the model remains in its unpotentiated state. Like the potentiated state, the unpotentiated state is very stable: No spontaneous potentiation events are observed even when running the model for a year of simulated time.
By introducing a delay between stimulation and PSI infusion, we can study the model's consolidation window, the time interval after induction during which PSI prevents establishment of L-LTP. As shown in Figure 4, the window is between 30 and 40 minutes, consistent with empirical results (Bourtchouladze et al., 1998;Fonseca et al., 2006).

ZIP during and immediately after stimulation does not prevent L-LTP induction
ZIP application during stimulation and the first 10 minutes thereafter after does not prevent L-LTP induction, (Figure 5).

Figure 5: ZIP immediately after stimulation does not prevent L-LTP induction. RA is active
PKMζ mRNA, P is PKMζ, AI is inserted GluR2 AMPARs, E1A is activated E1 enzyme.
Presence of ZIP during the first ten minutes after stimulation does not prevent L-LTP induction (Ren et al., 2013). The stimulation lifts the translational block and PKMζ production gets started. Even though PKMζ's enzymatic activity is inhibited, the mRNA stays activated long enough to ride out the ZIP activity. When the ZIP is washed out, PKMζ becomes active and drives the synapse into its potentiated state.

PKMζ infusion induces L-LTP
L-LTP can be induced by diffusion of PKMζ into a neuron (Ling et al., 2002;Serrano et al., 2005). We simulate infusion by rapidly increasing the number of PKMζ molecules in the synaptic compartment to 100. This causes the model to settle into its potentiated state, (Figure 6).

PSI blocks PKMζ-infusion-induced potentiation
The same level of PKMζ infusion that induces L-LTP in the previous experiment (100 molecules) fails to do so in the presence of PSI (Figure 7). Although the PKMζ infusion initially inhibits BRAG2, allowing a temporary increase in the number of inserted GluR2 AMPARs, the PSI prevents replenishment to compensate for PKMζ degradation and dissipation and the model returns to its unpotentiated state (Serrano et al., 2005).

PSI does not disrupt established L-LTP
Once L-LTP is established, PSI infusion does not depotentiate the synapse (Debiec et al., COMPUTATIONAL MODEL OF PKMζ REGULATION 28 2002;Nader et al., 2000). The number of inserted GluR2 AMPARs decreases as the result of PKMζ degradation/dissipation without replenishment, but not enough to switch to the depotentiated state within the time interval that the PSI remains active, nine hours.

Figure 8: PSI infusion during L-LTP maintenance does not cause depotentiation. RA is active
PKMζ mRNA, P is PKMζ, AI is inserted GluR2 AMPARs, E1A is activated E1 enzyme.
As seen in Figure 8, although the synapse does not depotentiate, the interruption of PKMζ synthesis causes a gradual decrease in the number of inserted GluR2 AMPARs. If the model is correct, then this decrease may be detectable as a reduced EPSP current several hours after PSI infusion. However, it is possible that the GluR2 AMPARs are replaced by GluR2-lacking AMPARs (Clem & Huganir, 2010), in which case the synaptic strength would be maintained. If this is the case, then it may instead be possible to detect a transient increase in rectification index, because GluR2-lacking AMPARs, but not GluR2-contaning ones, are characterized by a slight inward rectification (Clem & Huganir, 2010;Hong et al., 2013). The model thus predicts that one or the other of these two effects (EPSP reduction or rectification) should be detectable a few hours after PSI infusion.

Reactivation destabilizes, but does not disrupt, L-LTP
The effect of memory reactivation is simulated as a rapid increase in the amount of active E2 enzyme. This results in endocytosis of the inserted GluR2 AMPARs and release of the bound PKMζ molecules which then start to dissipate. However, due to continued synthesis, the PKMζ level is kept from dropping below threshold and the model settles back into the potentiated steady state (Debiec et al., 2002;Nader et al., 2000).

Reactivation followed by PSI disrupts L-LTP
Simulation of PSI infusion simultaneously with reactivation, or shortly thereafter, causes depotentiation ( Figure 10). Figure 10: Reactivation with simultaneous PSI infusion. RA is active PKMζ mRNA, P is PKMζ, AI is inserted GluR2 AMPARs, E1A is activated E1 enzyme, E2A is activated E2 enzyme.
In the absence of new protein synthesis, the PKMζ level drops below threshold and the model settles into its unpotentiated state (Debiec et al., 2002;Nader et al., 2000). By varying the delay between reactivation and PSI infusion, we can establish the model's reconsolidation window, the time interval after reactivation during which L-LTP is vulnerable to PSI. As shown in Figure 11, the window is between 20 and 30 minutes, consistent with empirical results (Nader et al., 2000;Xue et al., 2012).

GluR2 3Y blocks post-reactivation PSI-infusion from causing depotentiation
When the GluR23Y peptide is infused together with PSI after reactivation, it prevents the depotentiation that PSI otherwise causes (Yu et al., 2013(Yu et al., , 2016. Figure 12: Infusion of PSI and GluR23Y immediately after reactivation. RA is active PKMζ mRNA, P is PKMζ, AI is inserted GluR2 AMPARs, E1A is activated E1 enzyme, E2A is activated E2 enzyme. As before, reactivation triggers activation of the E2 enzyme, but here the GluR23Y peptide blocks its endocytotic effect. As a result, the GluR2 AMPARs remain inserted and although the PSI stops synthesis of new PKMζ, the existing population of PKMζ molecules, bound to the inserted GluR2 AMPARs, declines at a slow enough rate to maintain the synapse in its potentiated state while the PSI wears off (Figure 12).
ZIP inhibits all PKMζ enzymatic activity, including both the catalysis of its own synthesis and the blocking of GluR2 AMPAR endocytosis. The result is rapid removal of GluR2 AMPARs and depletion of PKMζ, and the synapse quickly settles into its unpotentiated state (Figure 13).

GluR2 3Y blocks depotentiation by ZIP infusion
When the Glur23Y peptide is infused together with ZIP during L-LTP maintenance, the disruptive effect of ZIP is blocked (Migues et al., 2010). Figure 14: Infusion of ZIP and GluR23Y during L-LTP maintenance. RA is active PKMζ mRNA, P is PKMζ, AI is inserted GluR2 AMPARs, E1A is activated E1 enzyme.
As before, ZIP inhibits PKMζ's catalysis of its own synthesis as well as its blocking effect on GluR2 AMPAR endocytosis, but in this case, even though BRAG2 remains active, the presence of GluR23Y prevents it from inducing endocytosis of the inserted GluR2 AMPA receptors. As a result, the GluR2 AMPARs remain inserted and continue to maintain the PKMζ molecules at the synapse. The number of PKMζ molecules declines only slowly and the potentiation is able to survive through the 12-hour period of ZIP activity (Figure 14).

Discussion
The model presented here is able to explain a range of results relating to the role of PKMζ in late-phase long-term synaptic potentiation, including L-LTP induction by NMDAR stimulation or by PKMζ infusion and the findings that whereas PSI, but not ZIP, can block induction of L-LTP, the reverse is true for disruption of established L-LTP. In addition, it accounts for cellular reconsolidation, reconsolidation blockade by PSI infusion and prevention of ZIP-or PSI-induced depotentiation by infusion of the GluR23Y peptide. While subsets of these results have been covered by earlier models (Jalil, Sacktor, & Shouval, 2015;Ogasawara & Kawato, 2010;Smolen, Baxter, & Byrne, 2012), ours is the first to account for all of them.
Our model demonstrates that a bistable mechanism for synaptic potentiation can arise from the interaction of two coupled feedback loops, neither of which needs itself be bistable. One of these, the mutual reinforcement between PKMζ and PKMζ mRNA, has been featured in previously published models of L-LTP maintenance (Jalil et al., 2015;Ogasawara & Kawato, 2010;Smolen et al., 2012). The second positive feedback relationship in our model is between PKMζ and inserted GluR2 AMPARs, which mutually maintain each other by inhibiting each other's removal from the synapse (Sacktor, 2011). The ability of inserted AMPA receptors to sequester PKMζ molecules at the synapse allows the model to account for findings involving the inhibition of regulated GluR2 AMPAR endocytosis (Migues et al., 2010;Yu et al., 2013

Limitations
Our model represents a subset of the processes believed to be involved in LTP induction and maintenance (Kandel et al., 2014;Sweatt, 1999). Some processes not included in our model are:  the induction and stabilization of early LTP, which likely involves GluR2-lacking AMPARs (Clem & Huganir, 2010), the MAPK/ERK signaling pathway and the proteins PKA, CaMKII (Sweatt, 1999) and PKCλ (Ren et al., 2013;Tsokas et al., 2016)  a later phase of L-LTP, sometimes called LTP3, which requires gene transcription as well as mRNA translation (Raymond, 2007) and may involve a "tagging and capture" mechanism for selectively targeting gene products to potentiated synapses (Frey & Morris, 1997;Sajikumar et al., 2005). For an interesting computational model of synaptic tagging and capture, see Smolen et al. (2012).
 polymerization/depolymerization of actin and restructuring of the cytoskeleton (Kelly, Yao, Sondhi, & Sacktor, 2007;Rudy, 2015) The processes that we have modeled thus form a subset of a more complex machinery.
Nevertheless, it is interesting to note that this relatively simple model is able to account for many of the empirical findings regarding the role of PKMζ in L-LTP induction and maintenance, and to exhibit the degree of stability required for a neural mechanism to support long-lasting memories.

Estimating the number of PKMζ molecules in a spine head
Zeta-inhibitory peptide (ZIP) disrupts the ability of PKMζ to potentiate synaptic transmission when applied extracellularly in concentrations around 1 μM (Yao et al., 2013).
Assuming that ZIP molecules enter neurons by diffusion only (i.e. they are not actively transported into cells), 1 μM would be an upper limit on the intracellular ZIP concentration.
The volume of a dendritic spine head is between 0.01 and 0.1 μm 3 (Bartol et al., 2015;Kasthuri et al., 2015). If we conservatively use the upper limit of this range, 10 -1 μm 3 = 10 -19 m 3 , then the maximum number of ZIP molecules that would be present in a spine head due to a 1 μM bath concentration would be ≈ 6 *10 20 * 10 -19 = 60.
Because ZIP's inhibitory action results from ZIP molecules binding to PKMζ molecules in a one-to-one ratio, we may assume that for ZIP to significantly disrupt PKMζ activity, the number of ZIP molecules must be at least of the same order of magnitude as the number of PKMζ molecules.
Consequently, the number of PKMζ molecules at a potentiated synapse may be estimated to be fewer than about one hundred.

Simulated Reactions
The simulated biochemical reactions are described below.

Activation of PKMζ mRNA
PKMζ mRNA is present in dendritic spines, but is translationally repressed in its basal state (Doyle & Kiebler, 2011;Sacktor, 2011) due to mRNA-binding proteins that prevent translation from being initiated (Bramham & Wells, 2007). PKMζ is able to lift the repression, possibly by phosphorylating these proteins, thus catalyzing its own synthesis in a positive feedback loop. We model mRNA with its associated proteins as a single molecule, represented by RI in its inactive repressed state, and by RA when activated. Activation is modeled using Michaelis-Menten kinetics (Gonze et al., 2011), i.e. a PKMζ molecule (P) and an inactive mRNA molecule (RI) form a complex P•RI. The complex may then either dissociate (reaction 2) or the catalytic reaction (3) may take place, producing active mRNA (RA): 1.
Each reaction has an associated Gillespie reaction constant, c1, c2, etc., such that ci dt is the average probability that a particular combination of the reactant molecules of reaction i will react during the next infinitesimal time interval dt (Gillespie, 1977). The values of these constants are given in Appendix C.

Deactivation of PKMζ mRNA
The PKMζ mRNA returns to its repressed state when the mRNA-binding proteins are dephosphorylated by a phosphatase which we denote by PP. This is also modeled with

Sequestering of PKMζ in the synaptic compartment
Our model implements the notion suggested by Sacktor (2011) and supported by empirical results (Hong et al., 2013;Migues et al., 2010;Yu et al., 2013)

NMDAR stimulation
The mechanism by which NMDAR activation causes an increase in PKMζ is unknown. We model the effect of strong NMDAR stimulation as a rapid increase in the number of active molecules of an unspecified enzyme E1 which, like PKMζ, activates PKMζ mRNA. E1I and E1A represent the E1 enzyme in its active and inactive states, respectively:

Reactivation
Reactivation of a consolidated memory causes it to become destabilized (Debiec et al., 2002;Nader, 2003;Nader et al., 2000). The molecular mechanism underlying this destabilization is not well understood, but has been showed to depend critically on GluR2 AMPAR endocytosis (Hong et al., 2013;Yu et al., 2013Yu et al., , 2016

Protein synthesis inhibition
The effect of PSI infusion is simulated by disabling synthesis of PKMζ (reaction 7).

Inhibition of PKMζ by ZIP
We simulate the effect of ZIP infusion by disabling all PKMζ enzymatic activity (reactions 1, 9, 22 and 25).

Inhibition of AMPAR endocytosis by GluR2 3Y
The effect of GluR23Y infusion is simulated by disabling regulated AMPAR endocytosis, whether catalyzed by BRAG2 (reactions 16 and 20) or by the E2 enzyme (reactions 32 and 33).

Reaction constants
The values of the reaction constants have been chosen to produce reaction rates consistent with those reported in the literature.