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Department of Neurobiology and Anatomy, W.M. Keck Center for the Neurobiology of Learning and Memory, The University of Texas Medical School at Houston, Houston, Texas
Correspondence: Address reprint requests to John H. Byrne, Dept. of Neurobiology and Anatomy, W.M. Keck Center for the Neurobiology of Learning and Memory, The University of Texas-Houston Medical School, PO Box 20708, Houston, TX 77225. Tel.: 713-500-5602; Fax: 713-500-0623; E-mail: john.h.byrne{at}uth.tmc.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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-burst, pairing-induced, and chemical L-LTP, as well as L-LTP due to synaptic tagging. The model also simulates inhibition of L-LTP by inhibition of MAPK, CAMKII, PKA, or CAMKIV. The model predicts results of experiments to delineate mechanisms underlying L-LTP induction and expression. For example, the cAMP antagonist RpcAMPs, which inhibits L-LTP induction, is predicted to inhibit ERK activation. The model also appears useful to clarify similarities and differences between hippocampal L-LTP and long-term synaptic strengthening in other systems. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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1–2 h after electrical stimulation or after application of forskolin or BDNF. L-LTP is hypothesized to be essential for storing long-term memories (1
) and has been reported to last for months (2). Because of this apparently fundamental role of L-LTP in learning, it is desirable to develop a conceptual representation of L-LTP induction and maintenance. An important component of such a representation is a mathematical model describing the role of key biochemical processes in L-LTP induction and expression. Such a model should be able to predict the outcomes of proposed experiments, and also suggest experiments to clarify aspects of L-LTP induction and expression.
Although models have been developed to describe aspects of the induction of early LTP (E-LTP) (3–5), no model of L-LTP induction and expression appears to have been developed. In contrast to E-LTP, L-LTP requires transcription and protein synthesis (6,7), and is associated with induction of numerous genes (8). L-LTP is a complex process involving the activation of numerous kinases, phosphatases, and genes. Although a complete understanding of the molecular processes underlying L-LTP is not available, we believe it is valuable to develop a model representing key processes that have been characterized experimentally. Such a model may guide further hypotheses and experimental tests, and may provide a framework for understanding core mechanisms underlying long-term synaptic change and memory.
The development of the model was based on data concerning induction of L-LTP at Schaffer collateral synapses in the hippocampal CA1 region. The Schaffer collateral pathway has been the focus of numerous studies because damage limited to CA1 inhibits the formation of declarative memory (9,10). Also, selective deletion of the NR1 subunit of NMDA receptors in the CA1 region impairs spatial memory and LTP (11). Experiments have suggested that a number of kinases are essential for the induction and expression of L-LTP in CA1. Therefore, the model focuses on representing the postsynaptic roles of protein kinase A (PKA), MAP kinase (MAPK), and other necessary kinases. The model provides insight into dynamic features, such as biochemical nonlinearities, which are essential for generating thresholds for L-LTP induction and for translating brief electrical stimuli into long-lasting synaptic changes. The model also predicts outcomes for experiments that would further delineate the mechanisms underlying L-LTP induction and expression.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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,5). It keeps the number of equations manageable and promotes intuitive understanding of model dynamics.
The model does not consider stochastic fluctuations in molecule copy numbers. This simplification appears reasonable because average copy numbers are not well constrained for the species in our model. However, we note that fluctuations in molecule copy numbers would affect the rate and extent of biochemical reactions, and hence, introduce a random component into the L-LTP produced by a stimulus protocol. Fluctuations affecting the amount of L-LTP would arise not only from varying copy numbers of enzymes and substrates, but also from fluctuations in the transcription and translation of gene products essential for L-LTP. The origins and consequences of such fluctuations in gene expression have recently been reviewed (12). As more data are obtained to define the biochemical and genetic pathways responsible for L-LTP, modeling of stochasticity in these pathways will become feasible.
The model consists of 23 ordinary differential equations, and is schematized in Fig. 1. The model represents L-LTP as an increase in a synaptic weight W. Increases in W represent experimentally observed increases in excitatory postsynaptic potential amplitude or slope. The model does not consider L-LTP as dependent on prior E-LTP. Experimental evidence suggests these processes are independent, because application of forskolin or BDNF appears to induce a slowly developing L-LTP without E-LTP (13,14). However, essential upstream events, such as activation of specific kinases, may be common to the induction of both E-LTP and L-LTP.
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) and MAPK (16) phosphorylate proteins that enhance translation in the vicinity of synapses subjected to electrical stimuli. If this translation is inhibited, L-LTP is significantly impaired (17,18). Inhibition of CAMKII blocks induction of L-LTP (19). PKA phosphorylation of an unidentified substrate also appears necessary to set a tag at activated synapses (20). L-LTP occurs only at tagged synapses. The tag appears to allow capture of plasticity factors (proteins or mRNAs) produced after stimulation (21,22). In the model, activated CAMKII, PKA, and MAPK each phosphorylate a synaptic substrate, and all three phosphorylations are necessary for L-LTP.
Nuclear CAMKIV is activated by Ca2+ influx subsequent to electrical stimuli, and can phosphorylate transcription factors such as cAMP response element binding protein (CREB) (23) and CREB binding protein (24). L-LTP induction by tetanic or -burst stimuli is strongly attenuated by inhibition of CAMKIV (25). In the model, elevation of nuclear Ca2+ activates CaM kinase kinase (CAMKK). CAMKK and nuclear Ca2+ cooperate to activate CAMKIV (Eqs. 2 and 3 below). CAMKIV is assumed to phosphorylate a transcription factor denoted TF-1, and this phosphorylation is necessary for L-LTP (Fig. 1).
MAPK activation leads to phosphorylation of transcription factors such as CREB and Elk-1. Elk-1 participates in induction of zif-268 (26), a gene necessary for L-LTP (27). Induction of Arg3.1/Arc, necessary for L-LTP, is blocked by MAPK inhibition (28). L-LTP is blocked by MAPK inhibition (29,30). The MAPK isoforms that appear necessary for L-LTP induction are extracellular-regulated kinase (ERK) I/II (13,31). In the model, MAPK denotes these ERK isoforms. Active MAPK is assumed to translocate to the nucleus before phosphorylating a transcription factor denoted TF-2 (Fig. 1). Empirically, MAPK complexed with the CREB kinase RSK-2 translocates to the nucleus after depolarization by KCl (31). Dominant negative PKA, or inactive cAMP analogs, inhibit this translocation. The model therefore assumes PKA activity is necessary for MAPK nuclear translocation. Phosphorylation of TF-2 by MAPK and of TF-1 by CAMKII is assumed to induce expression of a representative gene essential for L-LTP. The concentration of the gene product protein is denoted [GPROD].
After tetani, cAMP is elevated in hippocampal slice ((32,33); see, however, (34)). PKA is activated (33). PKA inhibition strongly attenuates tetanic L-LTP (35) and L-LTP can be induced by applying an active cAMP analog (36). In the model, L-LTP-inducing stimuli elevate [cAMP], activating PKA. In electrically stimulated neurons, elevation of [cAMP] appears to be downstream of [Ca2+] elevation, with [Ca2+] elevation activating adenylyl cyclase isoforms 1 and 8 (37,38). Because data are insufficient for detailed modeling of adenylyl cyclase activation and cAMP production, the model does not describe Ca2+ activation of cAMP production. Instead, we have simulated [cAMP] elevations with prescribed amplitudes and durations that appear consistent with the data available (discussed further below).
Each synaptic stimulus is modeled with simultaneous elevations of the concentrations of four independent variables: synaptic Ca2+ 
nuclear Ca2+ concentration 
[cAMP], and an activation rate kf,Raf for Raf kinase (Eq. 5). Further details of stimulus parameters are provided in the following subsection. The concentrations, in µM, of active forms of enzymes and substrates are used as dependent variables.
In the model, 12 of the 23 dependent variables represent molecular species in the vicinity of the synapse. Stimuli activate synaptic CAMKII. Stimuli also activate synaptic Raf, which activates MAPKK, which activates MAPK. Five synaptic variables (Eqs. 5–12 below) describe the dynamics of this MAPK cascade. Activated CAMKII, MAPK, and PKA each phosphorylate a synaptic substrate, thereby generating a synaptic tag (Eqs. 15 and 16). These three synaptic tag substrates are dependent variables (Eq. 16). [GPROD], the concentration of a gene product necessary for L-LTP, is also a synaptic variable. The remaining two synaptic-dependent variables are the synaptic weight W and the concentration of a protein P, which limits increase of W (Eqs. 18 and 19). Stimuli also activate PKA via cAMP. Concentrations of active PKA and of cAMP are each represented by an averaged (lumped) variable that does not distinguish between the synapse and the soma. To allow for coupling of stimuli to activation of nuclear MAPK, the model also represents activation of a somatic Raf-MAPK cascade. Five somatic dependent variables describe this cascade. The model assumes that identical equations and parameters describe the somatic and synaptic MAPK cascades, except for a modified somatic Eq. 12, and two terms describing nuclear import and export of somatic MAPK (Eq. 13). The remaining five dependent variables are nuclear. These are the concentrations of active nuclear MAPK, CAMKK, and CAMKIV, and the degrees of phosphorylation of the transcription factors TF-1 and TF-2.
For simplicity, a minimal representation of the coupling between synaptic, somatic, and nuclear processes is adopted. Phosphorylation of TF-1 and TF-2 is assumed to directly increase the rate of synthesis of the synaptic gene product GPROD (Eq. 17). Therefore, the transport of GPROD from nucleus to synapse is not modeled. Activated somatic MAPK is transported into the nucleus (Eqs. 13 and 14), and the active nuclear MAPK can then phosphorylate TF-2. No other coupling between cellular compartments is represented.
Activation of CAMKII by synaptic Ca2+ is described by the following differential equation, which uses a Hill function of 
:
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4 (39). Use of a Hill coefficient >4 for CAMKII did not significantly affect the simulations discussed below. For CAMKIV activity, a steep [Ca2+] dependence is likely, given CaM-Ca4's obligatory binding to both CAMKIV and CAMKK.
Electrical or chemical stimuli are also assumed to elevate [cAMP]. For cAMP to activate PKA, two cAMP molecules must bind cooperatively to the regulatory (R) subunit of the PKA holoenzyme (40). Therefore, one qualitative representation of PKA activation assumes the activation rate is a Hill function of the second power of [cAMP]. The level of active PKA, [PKAact], is also assumed to undergo first-order decay due to deactivation. The resulting differential equation is
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Stimuli that induce L-LTP are assumed to phosphorylate and activate the first kinase in a synaptic MAPK cascade, commonly Raf-1 or B-Raf in neurons (41,42). Active Raf phosphorylates MAP kinase kinase (MAPKK) twice, activating MAPKK. MAPKK then phosphorylates MAPK twice, activating MAPK. These phosphorylations can be described by the following differential equations (43):
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). L-LTP-inducing stimuli briefly elevate kf,Raf.
Activated MAPK can undergo PKA-driven nuclear translocation (31). To model nuclear MAPK activity, it is necessary to represent stimulus-induced activation of a somatic MAPK cascade and nuclear translocation of somatic MAPK. To represent somatic Raf and MAPKK activation, equations identical to Eqs. 5–9 are used. Kinetic parameters (Table 1) and stimulus-induced Raf activation are assumed identical for the somatic and synaptic cascades. Current data do not allow differences between somatic and synaptic parameters to be well specified, thus our assumption of identity appears reasonable for a qualitative representation. Parameter alterations during simulations (e.g., inhibition of MAPKK activation) are applied identically to the synaptic and somatic MAPK cascades. To represent somatic MAPK dynamics, equations identical to Eqs. 10–12 were used with two modifications. Equation 11 was altered to incorporate nuclear import and export (Eq. 13 below) and imported nuclear MAPK, [MAPknuc], was subtracted from the right-hand side of the somatic conservation relation, Eq. 12. The synaptic and somatic MAPK cascades are assumed not to interact due to their spatial separation.
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undergoes nuclear import at a rate proportional to PKA activity ([PKAact]). The concentration of active nuclear MAPK is denoted [MAPKnuc]. Nuclear export of MAPK is modeled as a first-order process. The above assumptions are expressed by the following differential equations for [MAPKnuc] and 
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). However, other kinases are needed to induce the labile state. Postsynaptic CAMKII activity is required for L-LTP. Synaptic MAPK is also likely to contribute by phosphorylating proteins that enhance local translation (16,45). Therefore, setting a synaptic tag appears to require CAMKII, MAPK, and PKA. In the model (Fig. 1), tagging is assumed to require phosphorylation of three substrates: Tag-1, Tag-2, and Tag-3. These species are respectively substrates of CAMKII, PKA, and synaptic MAPK. A molecular candidate for Tag-1 is the mRNA translation factor termed cytoplasmic polyadenylation element binding protein (CPEB), because CAMKII stimulates protein synthesis through phosphorylation of CPEB (46). The fractions of the kinase substrates that are phosphorylated are represented as deterministic variables denoted Tag-1P–Tag-3P. Their values range from 0 to 1. For simplicity, the model assumes that these three phosphorylations by different kinases are independent. With this assumption, the amount of synaptic tag, denoted as TAG, can be represented as the product of the phosphorylated fractions:
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The rate of synthesis of the gene product GPROD that is incorporated into tagged synapses is a saturable function of the degrees of phosphorylation of TF-1 and TF-2. [GPROD] also undergoes first-order decay, yielding the following differential equation for [GPROD]:
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A synaptic weight W represents changes in synaptic strength due to L-LTP induction, which requires both synaptic tagging and increased gene product level. The rate of increase of W is assumed proportional to the overlap, or product, of the tag with the gene product level. As discussed further below, the increase in W is assumed to be limited by the availability, for synaptic incorporation, of another precursor molecular species denoted P. These considerations yield the following differential equation:
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Phosphatidylinositol-3-kinase (PI3K) inhibition has been reported to block the expression of E-LTP (47), but these experiments were of insufficient duration to establish the role of PI3K in L-LTP. Therefore, the model does not currently represent dynamics of PI3K activity (but see Discussion). PI3K can activate the atypical protein kinase C isoform termed PKM/PKC
(48,49). However, this pathway has not been well studied in neurons.
Data do not generally exist to accurately determine concentrations of active enzymes in neurons. Therefore, we were not able to quantitatively fit time courses of concentrations or enzyme activities to data. However, we did obtain semiquantitative constraints from estimates of Bhalla and Iyengar (5) for concentrations of MAPK, PKA, PKC, CAMKII, MAPKK, and Raf ((5), see http://www.mssm.edu/labs/iyengar/ssupplementary_materials.shtml, henceforth denoted B&I). We set [MAPK]tot, [MAPKK]tot, and [Raf]tot to 0.25 µM, close to the B&I estimates of 0.36 µM, 0.18 µM, and 0.2 µM, respectively. Active PKA, [PKAact], peaks at 0.6 µM during simulated forskolin application, whereas B&I estimate 0.5 µM for the R2C2 tetramer. This tetramer is 80% of total PKA in unstimulated cells (50). The simulated peak concentration of active CAMKII is 7.9 µM (simulation of Fig. 3 A before scaling output). The B&I estimate of total CAMKII is 70 µM. Thus, the simulated peak concentration of active CAMKII is 11% of the estimated total. Simulated peak concentrations of active CAMKIV and CAMKK due to tetanus are 0.05 and 0.1 µM, respectively. These values are 5–10% of the total CAMKIV and CAMKK concentrations, which B&I estimate at 0.5–1 µM. The qualitative simulation results discussed below (Figs. 3–7) are not sensitive to these parameter values. The concentration time course of any variable can be rescaled with preservation of the model dynamics, if kinetic rate constants relating that variable to others are rescaled. For example, [CAMKIIact] can be doubled by doubling kact in Eq. 1, but if kphos in Eq. 16 is also halved, the rate of the phosphorylation catalyzed by CAMKII stays the same and the dynamics are unchanged.
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Simulation of L-LTP-inducing stimuli
Stimulation protocols (Fig. 2) lead to elevation of [Ca2+] and [cAMP] and activation of the MAPK signaling cascade. Details of Ca2+ dynamics were not modeled, given that the model of Fig. 1 is a qualitative representation of the roles of kinases essential for L-LTP induction. Instead, the Ca2+ response to stimuli was modeled in the simplest plausible manner. Two independent variables were used, synaptic [Ca2+] and nuclear [Ca2+]. Basal and 
values were 40 nM. Tetanic and -burst stimuli were modeled as square-wave increases in and 
For tetanic stimuli, three tetani were usually simulated, with an interstimulus interval of 5 min (Fig. 2 A). Each 1-s, 100-Hz tetanus was simulated as a 3-s increase of synaptic Ca2+ to 1 µM and nuclear Ca2+ to 500 nM. A similar duration of Ca2+ elevation is suggested by data. One study (51) used a photolabile Ca2+ buffer to terminate postsynaptic Ca2+ elevation after tetani. Delaying buffer photolysis for 2.5 s did not attenuate LTP, whereas photolysis within two seconds inhibited LTP. More recent imaging data also suggest a time constant of 1–3 s for decay of Ca2+ transients after tetanus (52), although another study (53) found a more rapid decay and a higher peak [Ca2+] (4–6 µM). Changes in [cAMP] and MAPK activity produced by the simulated tetani and other protocols are discussed below.
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-burst stimulus protocols, 10–12 bursts of four 100-Hz pulses are typically delivered 200-ms apart (total duration 2.2 s) (14). This protocol was simulated with a 5-s square-wave increase in (to 1 µM) and (to 500 nM) (Fig. 2 B). We also simulated L-LTP induction with the pairing protocol used in Huang et al. (54), which uses multiple pairings of a single action potential (AP) in the potentiated synaptic pathway with a burst stimulus in a second pathway. Sixty bursts of three 100-Hz APs are given 5-s apart, for a total duration of 5 min. We modeled each AP-burst pairing with a brief (1.2 s) elevation of (to 400 nM) and (to 180 nM) (Fig. 2 C).
As noted above, the kinetics of cAMP production and its activation by Ca2+ have not been well characterized. Therefore, we assumed each tetanus or -burst induced a prescribed, square-wave elevation of [cAMP]. Observations suggest that the time for [cAMP] to return to basal levels after stimulation is 1–2 min (55,56). Therefore, we assumed [cAMP] remained elevated for 1 min during and after stimulation. The pairing protocol (54) was assumed to elevate [cAMP] for 6 min (protocol duration + 1 min). Specific values for [cAMP] were 0.05 µM (basal), 0.15 µM (tetanic), 0.35 µM (-burst), and 0.15 µM (pairing).
Neuronal MAPK can be activated by Ca2+ elevation acting via CaM kinase I (57) or by cAMP elevation (58–60) or by a Ca2+-independent pathway involving mGluR5 (61). Raf activation is the convergence point for these mechanisms of MAPK cascade activation. Rather than modeling these complexities in detail, we assumed each tetanus or -burst increased the rate constant kf,Raf for synaptic and somatic Raf phosphorylation and activation (Eq. 5). In the absence of detailed data, we assumed a square-wave increase lasting for 1 min for tetanic and -burst stimuli and 6 min for the pairing protocol. Values for kf,Raf were 0.0075 min–1 (basal), 0.16 min–1 (tetanus), 0.41 min–1 (-burst), and 0.16 min–1 (pairing). As discussed above, identical equations and kf,Raf values describe synaptic and somatic Raf activation. kf,Raf and [cAMP] elevations needed to be higher for -bursts than for tetani, so that similar peak MAPK activation, gene induction, and L-LTP resulted after one -burst versus after three tetani.
We also simulated chem-LTP, in which application of forskolin or BDNF activates PKA and MAPK (14,62). Typical experimental applications last 30 min. For 30 min, kf,Raf was elevated to 0.3 min–1 and [cAMP] was elevated to 0.4 µM. Synaptic and nuclear [Ca2+] were slightly elevated, by 60 nM, for 30 min. Data suggests neuronal [Ca2+] is elevated by exposure to forskolin or BDNF (63,64).
Modeling synaptic tagging and heterosynaptic L-LTP
The model was extended to simulate sequential tetanic stimulation of two synapses, A and B, with GPROD synthesis blocked during tetanization of synapse B (Fig. 7 below). Experimentally, if protein synthesis is blocked during tetanization of synapse B, L-LTP of synapse B still results (22). The synaptic tag hypothesis (21,22) suggests that the tetanus to synapse B activates synaptic kinases and phosphorylates tag substrates. L-LTP results because gene expression and protein synthesis was induced by the prior tetani at synapse A. The necessary proteins are then captured by the tagged synapse B.
The model extension was carried out as follows. The differential equations for the 12 dependent synaptic variables were duplicated (Eqs. 1, 4–12, 16, 18–19) and the synaptic tag was duplicated (Eq. 15). The independent stimulus variables [cAMP], kf,Raf, and were duplicated for synapse B. Tetanus of either synapse was simulated by brief elevations of these stimulus variables at only the tetanized synapse. Tetanus of either synapse elevates 
activating CAMKIV, and also elevates somatic kf,Raf, activating the somatic MAPK cascade. PKA is also activated, enhancing MAPK nuclear translocation. For all stimulus variables, the basal and elevated levels are identical for stimulus of synapses A and B. These values are as given above (see preceding subsection).
The only coupling between synapses A and B is via the nucleus. Stimulation of either synapse induces activation of the nuclear kinases (CAMKK, CAMKIV, and MAPK) and elevation of the level of GPROD at both synapses. In Fig. 7, to simulate the experimental block of protein synthesis by anisomycin, GPROD synthesis is blocked (ksyn and ksynbas in Eq. 17 are set to zero) during and after tetanus of synapse B.
This extension of the model simulates tagging and L-LTP of synapse B when GPROD synthesis is blocked during and after tetanus of synapse B (Fig. 7). However, to simulate more general stimulus protocols with multiple synapses, it would be necessary to represent cumulative activation of somatic PKA, which drives nuclear import of active MAPK. Separate variables would be required to represent PKA activity at the soma and at each synapse.
Numerical methods
The forward Euler method was used for integration, with a time step of 15 ms. Simulations verified that further reductions in the time step did not significantly improve the accuracy of the results illustrated in Figs. 3–7
. To further verify accuracy, the simulations of Figs. 3 and 6 were repeated using the second-order Runge-Kutta integration method (65). No significant differences in the time courses of the model variables resulted.
Initial values for the model variables were as follows. Somatic and synaptic [Raf], [MAPKK], and [MAPK] were respectively set to 0.5 x [Raf]tot, 0.5 x [MAPKK]tot, and 0.5 x [MAPK]tot. [MAPKnuc] was set to 0.2 x [MAPK]tot. The remaining 16 dependent variables were set to 0.001. and were set to 40 nM. To allow the model to reach equilibrium, simulations were run for at least four simulated days before L-LTP induction. During the equilibration simulation only, to ensure complete equilibration, the variables with the slowest time constants (W and [P]) were set equal to their steady-state values as determined by the other model variables. We verified that integration for even longer times did not alter the equilibrium state. The model was programmed in Java and simulated on Pentium 3 microcomputers. Programs are available upon request.
To allow concurrent visualization of variables of different magnitudes, amplitude scaling factors were applied when plotting simulation results (Figs. 3–7), as follows. The time courses of [Rafp] and [CAMKIVact] were vertically scaled (multiplied) by 10. [CAMKIIact] was vertically scaled by 0.1; MAPK species concentrations were scaled by 5.0; and TAG was scaled by 110. [GPROD] was scaled by 0.4. In Figs. 3–7, the variables representing enzyme concentrations and the variables [P] and [GPROD] have units of µM. The other variables, such as W and TAG, are nondimensional.
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RESULTS |
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5 min and CAMKIV for 45 min (Fig. 3 A). The time required for decay of CAMKIV activity is similar to data (23). PKA activity increases by 100% during L-LTP induction, which is consistent with data (33). Simulated synaptic and somatic MAPK activity ([MAPKact] and 
both last 2 h (Fig. 3 B). Data concerning the duration of MAPK activity are contradictory. One recent study suggests MAPK remains phosphorylated, and presumably active, for at least 8 h after tetanus (66). However, earlier studies (67,30) suggest a much briefer activation of 30 min. Because long-lasting MAPK activity could regulate transcription and other processes involved in L-LTP, we suggest further experimental study of MAPK kinetics is warranted. Simulated basal [MAPKact] is 15% of peak [MAPKact]. L-LTP induction nears completion in 2 h (Fig. 3 D, time course of W). Similarly, induction of L-LTP with BDNF (bypassing E-LTP) requires 2 h (13). In Fig. 3 D, W increases by 142%. This amplitude is similar to the excitatory postsynaptic potential increase observed after three or four 1-s, 100-Hz tetani (68,30).
In Fig. 3 C, the synaptic tag variable and gene product level are both plotted to illustrate their overlap. Equation 18, describing the increase in W, represents the amount of L-LTP as proportional to this overlap. The time course of [P] is illustrated in Fig. 3 D. In the model, P is assumed to limit the amount of L-LTP generated by prolonged stimuli, with synaptic incorporation of P both increasing W and diminishing [P] (Eqs. 18 and 19). With the parameters of Fig. 3, simulation of four tetani does generate a significantly greater elevation of W (174%). However, simulation of 10 tetani causes only a slightly greater W elevation (186%), because [P] declines to 0.
Effects of supralinear stimulus-response relationships
The model incorporates three supralinear stimulus-response relationships. First, the rates of activation of CAMKII, CAMKK, and CAMKIV are determined by nonlinear Hill functions of [Ca2+]. Second, active Raf phosphorylates MAP kinase kinase (MAPKK) twice. MAPKK-PP then phosphorylates MAPK twice. Only MAPK-PP phosphorylates MAPK substrates at a significant rate. These multiple phosphorylations of MAPKK and MAPK generate supralinearity in the output of the MAPK cascade (MAPK activity) as a function of the input (the rate of Raf activation) (69). Third, multiple kinase activities converge to increase W. The rate of increase of W is proportional to gene product concentration ([GPROD]) and to the synaptic tag (TAG). The rate of GPROD formation is proportional to phosphorylation of two transcription factors and therefore to the activities of CAMKIV and nuclear MAPK (with saturation at high activities). TAG is proportional to the phosphorylation of three sites and therefore to the activities of synaptic CAMKII, MAPK, and PKA. Thus, if the activities of CAMKII, CAMKIV, PKA, and MAPK are doubled, the rate of increase of W can increase by up to 16-fold.
Empirically, a 2–3 s, 10–20-fold elevation of Ca2+ (from 40 nM basal levels to 1 µM in the vicinity of tetanized synapses, or 300 nM at the nucleus) suffices for long-lasting gene induction (induction of Arg3.1/Arc and other LTP-associated genes lasts >30 min) (8,28). Such amplification of a brief input into a long-lasting output requires steep, supralinear relationships of input (Ca2+ elevation) to output (gene induction or synaptic weight changes). Without supralinearity, a 20-fold elevation of [Ca2+] lasting for 3 s would drive only a negligible increase in a variable such as gene product concentration. The much longer time-constant of the latter variable would almost completely damp its response to the brief stimulus.
To quantify the effect of the three supralinearities discussed above, we repeated the simulation of Fig. 3 in three different ways, with supralinearity reduced as follows.
The basal synthesis rate of P was elevated 10-fold in Cases 1–3 to ensure decrease of L-LTP was not due to depletion of P. L-LTP (the increase in W) was reduced to 5.8% (Case 1), 87% (Case 2), and 5.5% (Case 3), compared to 142% in Fig. 3 D. Therefore, high [Ca2+] Hill coefficients and convergence of multiple kinases (Cases 1 and 3) contribute substantially to simulated L-LTP. The double phosphorylations of MAPKK and MAPK (Case 2) contribute considerably less.
Supralinear stimulus-response relationships also cause simulated L-LTP to exhibit threshold behavior. In Fig. 3 D, W increases by 142% after three tetani. If only two tetani are simulated, the amount of L-LTP decreases by more than half, and if only one tetanus is simulated, L-LTP decreases by a further 80%. Such threshold dynamics may help explain the experimental requirement of 3–4 tetani for the reliable induction of L-LTP.
Sensitivity of L-LTP induction to parameters and stimulus pattern
Biochemical and genetic systems are commonly observed to be robust to significant changes in the values of parameters, such as mutations that alter enzyme activities. Therefore, a plausible model of L-LTP induction should be robust, such that simulated stimulus responses should not exhibit very high sensitivity to small changes in parameter values. However, it is also desirable to use modeling to predict parameters to which L-LTP induction may be most sensitive. Some of these high-sensitivity parameters could function as physiological control parameters to regulate LTP induction, and might serve as targets for pharmacological intervention to augment L-LTP and memory.
A standard method defines a set of relative sensitivities Si, with the index i ranging over all parameters pi (70,71). Let R denote the amplitude of a simulated stimulus response. For each pi, a small change is made, and the resulting change in R is determined. The relative sensitivity Si is then defined as the relative, or fractional, change in R divided by the relative change in pi,
 | (20) |
and All of the Si-values were found to have an absolute value <3. Thus, the model is not unduly sensitive to variations in any one parameter. The range of Si values was (–2.39, 2.65). Of the 46 Si-values, 10 had an absolute value above 1. Eight Si-values had absolute value >1.4, corresponding to the parameters [Raf]tot (Si = 2.65), kf,MAPKK (Si = 2.65), kb,MAPKK (Si = –2.39), kb,Raf (Si = –2.30), kf,Raf(basal) (Si = 1.64), kf,MAPK (Si = 1.54), Kcamp (Si = –1.77), and [cAMP]basal (Si = –1.53). All of these parameters except Kcamp and [cAMP]basal characterize the kinetics of the MAP kinase cascade. As discussed above, multiple phosphorylations within this cascade generate a supralinear relationship between Raf activation and MAPK activation. Thus, the magnitude of L-LTP induction exhibits a rather sensitive dependence on kinetic parameters of the MAPK cascade.
The relative sensitivities calculated with small parameter changes may not always predict the response of the model to larger parameter changes. Therefore, the calculation of the Si-values was repeated, using substantial (40%) increases in each parameter pi. Interestingly, an overall damping of the Si-values was observed. Of the 46 Si-values, 41 decreased in absolute value. The Si range decreased to (–1.7, 0.60). Only four Si-values had absolute value >1.0, corresponding to the parameters kb,MAPKK (Si = –1.69), kb,Raf (Si = –1.68), kb,MAPK (Si = –1.08), and Kcamp (Si = –1.35). The magnitude of L-LTP remains rather sensitive to MAPK cascade kinetics. The damping of the Si-values with larger parameter changes suggests the model is reasonably robust to parameter variability, as is necessary for a plausible model of intracellular signaling and responses to stimuli.
Can the model predict a pattern of tetanic interstimulus intervals (ISIs) that is optimal for induction of L-LTP? To examine this question, we first determined the dependence of L-LTP on the ISI for a group of three tetani, simulated as for Fig. 3, with the ISI varying from 0 to 300 min in steps of 1 min. For each simulation, the amount of L-LTP (the increase of W) was determined 24 h post-tetanus. Only a small enhancement of L-LTP by stimulus spacing was found. L-LTP was 135% for an ISI of 1 min, increasing slightly to a peak of 145% for ISIs of 9–15 min. Above 15 min L-LTP declined smoothly, to 98% for an ISI of 60 min and 36% for an ISI of 300 min. The model therefore predicts relatively little enhancement of hippocampal tetanic L-LTP when the ISI is increased from 1 min to 5 min or longer.
However, the observed decline of L-LTP for long ISIs (60 min) suggests that for long ISIs, a strong enhancement of L-LTP can be produced by grouping stimuli into bursts. To explore this enhancement, we simulated six tetani, delivered in two protocols: 1), equal separation by ISIs of three hours versus 2), two bursts of three tetani, with ISIs of 10 min within bursts and 860 min between bursts. Both protocols have a duration of 15 h. Twenty-four h after stimuli, the L-LTP induced by Protocol 1 was 95%, whereas Protocol 2 induced a much greater L-LTP, 250%. Similar enhancements of L-LTP (not shown) were observed for grouping of stimuli into four-tetanus bursts, and for replacement of tetanic stimuli by 10-min chemical stimuli. Two-tetanus bursts induce much less L-LTP as discussed previously, and bursts of more than four tetani induce little additional L-LTP due to depletion of the precursor protein P (Eq. 19). Therefore, the model predicts that a stimulus pattern maximizing induction of L-LTP can be obtained by grouping stimuli into bursts of 3–4 tetani each. Within each burst, the ISI should be 10–15 min.
Simulations of L-LTP inhibition
Empirically, inhibition of CAMKII during and after stimuli blocks LTP induced by tetani (19) or by a pairing protocol (72). However, if the CAMKII inhibitor was perfused postsynaptically immediately after either stimulus protocol, no inhibition of LTP was observed. The model can simulate these observations. Fig. 4 illustrates that a block of L-LTP results when CAMKII activity is inhibited for 1 h during and after three tetani. In contrast, if the 1-h CAMKII inhibition is assumed to begin 5 min after the tetani, L-LTP is not significantly attenuated. The window during which CAMKII activation is required is narrow, comprising the tetani and only a few minutes afterwards. Therefore, in the model, the rapid decay of CAMKII activity in 5 min after tetanus (Fig. 3 A) represents the disappearance of the requirement of CAMKII activity for L-LTP. Recent data suggest CAMKII activity may decay rapidly. Although hippocampal CAMKII phosphorylation persists for at least 30 min after tetani (73), the activity of CAMKII appears to decay within 5 min after tetanic or chemical stimuli (74).
Fig. 4 also illustrates the effect on tetanic L-LTP of simulated inhibition of MAPK signaling by the commonly used compounds U0126 or PD98059, which block MAPKK activation (75). Strong attenuation is simulated (Fig. 4) if inhibition of MAPKK activation is modeled as a 90% reduction in the rate constant kf, MAPKK (Eqs. 7 and 8) during tetani and for 10 min after (as noted in Methods, such parameter alterations are applied identically to the somatic and synaptic MAPK cascades). Experimentally, inhibiting MAPKK activation during and after tetanic stimulation blocks L-LTP induction (30); -burst L-LTP is also strongly attenuated by U0126 if this inhibitor is present during and for 10 min after stimulus (14). In the model, the dual action of MAPK to phosphorylate a transcription factor (TF-2) and a synaptic substrate (Tag-3) is necessary for strong L-LTP attenuation. A model variant in which MAPK phosphorylates only one substrate retains considerable residual L-LTP (not shown).
Fig. 4 also illustrates inhibition of L-LTP due to CAMKIV inhibition during and after tetanus. Empirically, transgenic mice expressing dominant-negative CAMKIV exhibit impaired L-LTP (24). In the simulation, CAMKIV was not inhibited before tetanus, although in the mice CAMKIV activity should be reduced at all times. In the model, inhibition of CAMKIV before tetanus reduces gene expression (the concentration of GPROD), thereby decreasing the basal value of the synaptic weight W, whereas experimentally, dominant negative CAMKIV does not reduce basal synaptic strength (25). This contradiction suggests that in vivo, a compensatory homeostatic mechanism preserves basal synaptic weights. For simplicity, the current model does not hypothesize a homeostatic mechanism. In the model, the lack of a homeostatic mechanism similarly leads to diminished basal synaptic strength with CAMKII, MAPK, or PKA inhibition. A planned extension will incorporate homeostatic regulation of basal synaptic strength, which may maintain neuronal activity and synaptic drive near set points (76).
Antisense Arg3.1/Arc mRNA oligonucleotides inhibited tetanic L-LTP by 40–60% (77). No effect was seen on baseline synaptic strength. To simulate this experiment, the rate of GPROD synthesis (Eq. 17) was decreased by 60% during and after three tetani. This alteration reduced the peak of [GPROD] by 59%, similar in magnitude to the empirical reduction in Arg3.1/Arc protein (77). Simulated L-LTP was reduced by 53%.
Tetanic L-LTP is blocked by a PKA inhibitor peptide, PKI (78). In the model, tetanic L-LTP was blocked when [PKAact] was reduced by 90% during and after stimulation. Empirically, tetanic L-LTP was also blocked by a brief application of RpcAMP, which competitively inhibits cAMP's activation of PKA (68). RpcAMP was washed out after the tetanus. We attempted to simulate this experiment by terminating PKA inhibition 5 min after three simulated tetani. However, this did not block L-LTP. Five minutes after the tetani, phosphorylation of the CAMKII and MAPK synaptic tag substrates remained high. When PKA inhibition was terminated, the PKA substrate was significantly phosphorylated by basal PKA activity. The synaptic tag variable therefore increased, and overlapped with increased synthesis of GPROD, inducing L-LTP.
One possible explanation for the experimental block of L-LTP by brief RpcAMP applications is that RpcAMP inhibits PKA-independent activation of the MAPK signaling cascade. We therefore examined whether simulated L-LTP was inhibited if both PKA activity and MAPKK activation (kf, MAPKK) were reduced by 90% during three tetani and for 5 min after. These reductions sufficed to inhibit L-LTP by 81%. There is experimental support for the suggestion that RpcAMP inhibits PKA-independent activation of MAPK. Activation by cAMP of the GTP-binding protein Rap1 can contribute to Raf activation (42) and this pathway appears independent of PKA (58,60).
Simulation of -burst, pairing-induced, and chemical L-LTP
Fig. 5 A illustrates that the model simulates similar amounts of L-LTP for four stimulus protocols. L-LTP is taken to be the increase in W above baseline 24 h after each protocol. The largest potentiation (142%) is for tetanic L-LTP induction. A -burst stimulus (TBS) protocol was also simulated, yielding L-LTP of 86%, which is similar to experimental values (14). Inhibition of MAPKK activation (reduction of kf, MAPKK by 90%) during and for 10 min after TBS attenuated L-LTP by 69%. A similar attenuation was observed experimentally (14). We also simulated (Fig. 5 B) the L-LTP induction protocol used in Huang et al. (54), which pairs stimulation of two synapses. Substantial L-LTP (103%) resulted. The relatively weak electrical stimuli of the pairing protocol yield lower nuclear Ca2+ and less CAMKIV activation. Therefore, to obtain substantial gene induction ([GPROD] elevation) and consequent L-LTP, the pairing protocol was assumed to strongly activate Raf and consequently MAPK (kf,Raf was elevated to 0.16 min–1 for 6 min as described in Methods). The strong MAPK activation compensated for the weak CAMKIV activation, yielding substantial induction of GPROD and L-LTP. An experimental prediction follows. Pairing-induced L-LTP should be less inhibited than tetanic L-LTP after dominant negative CAMKIV is introduced as in Kang et al. (25).
Experimentally, chemical L-LTP (chem-LTP) is induced by forskolin or BDNF, without electrical stimulation. We first attempted to model chem-LTP by activation of Raf and PKA, without elevation of Ca2+. However, significant L-LTP could not be simulated, because without some CAMKII activation, the level of synaptic tag remains very low, and without CAMKIV activation, the gene product level [GPROD] remains very low. We therefore assumed that synaptic and nuclear Ca2+ were slightly elevated during the 30-min chemical application. Elevations of 60 nM for and were assumed. Substantial chem-LTP (135%) was then simulated. Similar L-LTP magnitudes are observed experimentally (14,79). Fig. 6, A and B, illustrates the simulation of chem-LTP. A large increase in the synaptic tag variable TAG is seen, partly due to very strong PKA activation and almost complete phosphorylation of the PKA tag substrate Tag-P2. The CAMKII activation that phosphorylates Tag-P1 and contributes to TAG elevation is small compared to that in electrical stimulus protocols (Fig. 6 A, rise in CAMKII activity slightly above baseline).
Empirically, it is plausible that forskolin or BDNF application elevates [Ca2+]. In GnRH neurons, increased cAMP augments [Ca2+] (63). BDNF application to cultured hippocampal neurons increased [Ca2+], apparently due to IP3-gated Ca2+ release from intracellular stores (64).
Inhibition of MAPKK activation by U0126 or PD98059 suffices to block chem-LTP even when the inhibitor is washed out immediately after BDNF or forskolin application (13,14). The model simulates this behavior. If MAPKK activation is inhibited by 90% during and for 5 min after the chemical stimulus, L-LTP is strongly attenuated (the increase in W is reduced by 78%, Fig. 6 B).
We examined whether simulated -burst, pairing-induced, and chemical L-LTP exhibited threshold behavior, i.e., a supralinear increase in the amount of L-LTP versus the stimulus duration. The threshold for tetanic L-LTP was discussed above. We reduced the duration of the -burst, pairing, and chemical protocols by 40%. L-LTP was reduced by greater percentages; 80% (-burst), 68% (pairing), and 67% (chemical). These greater percentage reductions illustrate that a supralinear increase of L-LTP with stimulus duration exists for all protocols, and this supralinearity is steepest for the -burst protocol and the tetanic protocol.
Simulation of synaptic tagging
We examined whether the model could simulate the primary synaptic tagging experiment presented in Frey and Morris (22) (their Fig. 1). In that experiment, one synapse, synapse A, was first given three tetani (100 Hz for 1 s, interstimulus interval of 10 min), inducing L-LTP. Thirty-five min later, protein synthesis was halted by anisomycin. A second synapse, synapse B, was then given three tetani. One hour separated the first tetanus to synapse A and that to synapse B. Despite the presence of anisomycin, synapse B underwent L-LTP. This experiment has been interpreted (21,22) as supporting the hypothesis of synaptic tagging, with synapse B tagged by the second set of tetani. Synapse B can then capture the gene products that were previously synthesized as a consequence of the tetani to synapse A.
To model this experiment, the model of Fig. 1 was extended to represent two synapses, as described in Model Development, above. For synapse A, the first set of three tetani activated synaptic kinases, somatic and nuclear MAPK, and GPROD synthesis, yielding substantial L-LTP (traces for TAG-A, [GPROD], and W(tetanic), Fig. 7 B). No L-LTP of synapse B resulted, because kinases at synapse B were not activated. To model the effect of anisomycin, synthesis of GPROD was halted 35 min after the tetani to synapse A. The second set of tetani, to synapse B only and with anisomycin, had no effect on synapse A. However, these tetani activated kinases at synapse B, setting the synaptic tag (trace for TAG-B, Fig. 7 B). Substantial L-LTP of synapse B resulted (112% increase in W (tagged), Fig. 7 B) because the TAG-B time course for synapse B overlapped the GPROD time course resulting from prior stimulation of synapse A. The TAG-B time course subsequently decays within 3 h, similarly to data (21,22).
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In the model, L-LTP inducing stimuli are represented by separate increases in [Ca2+], [cAMP], and synaptic and somatic Raf activation. However, cAMP elevation in electrically stimulated neurons appears to follow [Ca2+] elevation and activation of adenylyl cyclase 1 and 8 (37,38), and Raf activation appears at least partly driven by [Ca2+] elevation (57). Therefore, the increase in synaptic weight seen in L-LTP is predominantly driven by very brief (1–5 s) increases in intracellular [Ca2+]. As discussed in Results, the model represents a supralinear relationship between the stimulus of Ca2+ elevation and the response of synaptic weight increase, and this supralinearity is essential for amplifying a brief [Ca2+] increase into a long-lasting increase in the synaptic weight W. The supralinearity also results in threshold dynamics, in that the amount of L-LTP increases steeply with the number of stimuli (see Results).
Empirically, a similar supralinear relationship between [Ca2+] elevation and synaptic weight increase has been found. Moderate stimuli, such as low-frequency electrical pulses, produce LTD, whereas with stronger stimuli, there is a crossover to LTP. The kinetic profiles of Ca2+ signals generated by stimuli in cortical slices have recently been compared with the plasticity outcome (80). An abrupt crossover from LTD to LTP occurred when peak [Ca2+] incre
