A model of the roles of essential kinases in the induction and expression of late long-term potentiation

Abstract

The induction of late long-term potentiation (L-LTP) involves complex interactions among second-messenger cascades. To gain insights into these interactions, a mathematical model was developed for L-LTP induction in the CA1 region of the hippocampus. The differential equation-based model represents actions of protein kinase A (PKA), MAP kinase (MAPK), and CaM kinase II (CAMKII) in the vicinity of the synapse, and activation of transcription by CaM kinase IV (CAMKIV) and MAPK. L-LTP is represented by increases in a synaptic weight. Simulations suggest that steep, supralinear stimulus-response relationships between stimuli (e.g., elevations in [Ca(2+)]) and kinase activation are essential for translating brief stimuli into long-lasting gene activation and synaptic weight increases. Convergence of multiple kinase activities to induce L-LTP helps to generate a threshold whereby the amount of L-LTP varies steeply with the number of brief (tetanic) electrical stimuli. The model simulates tetanic, -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.

FIGURE 1

Schematic of the model. Synaptic stimulation elevates Ca²⁺ and cAMP and activates the MAPK cascade. Ca²⁺ activates CAMKII and CAM kinase kinase (CAMKK). CAMKK and Ca²⁺ activate CAMKIV. cAMP activates PKA. Activated MAPK, PKA, and CAMKII phosphorylate synaptic substrates (Tag-1–Tag-3). CAMKIV and MAPK phosphorylate transcription factors (TF-1, TF-2). A variable TAG, denoting the synaptic tag needed for L-LTP, equals the product of the fractional phosphorylations of Tag-1–Tag-3. For L-LTP induction, a gene product must be induced. Induction requires phosphorylation of TF-1 and TF-2. L-LTP induction is modeled as an increase in a synaptic weight W. The rate of increase is proportional to the value of the synaptic tag and to the amount of gene product.

FIGURE 2

Simulations of L-LTP inducing protocols. (A) Tetanic protocol. Each of three tetani briefly elevates [Ca²⁺], [cAMP], and the rate constant k_f,Raf for Raf activation. The red bar represents concurrent elevations in both cytosolic and nuclear [Ca²⁺]. (B) θ-burst protocol, simulated with a single brief increase in and [cAMP], and k_f,Raf. The elevations in [cAMP] and k_f,Raf are larger than with the tetanic protocol. For this and the other protocols, the relative heights of the red, green, and blue bars qualitatively reflect the differing amplitudes of the [Ca²⁺], [cAMP], and k_f,Raf elevations, respectively. (C) Pairing protocol. Sixty short bursts of action potentials are each simulated with a relatively small, brief increase in and [cAMP] and k_f,Raf are elevated during the 5-min protocol and for 1 min afterwards. (D) Chem-LTP. During a 30-min interval, and are slightly elevated, whereas [cAMP] and k_f,Raf are elevated more than in any other protocol.

FIGURE 3

(A) Changes in active CAMKII, active CAMKIV, and active Raf during and after three simulated tetanic stimuli. A–D use the same stimulus protocol. (B) Changes in synaptic and somatic active MAPK, nuclear active MAPK, and the synaptic tag. (C) Changes in the synaptic tag and the gene product assumed necessary for L-LTP. (D) Changes in the synaptic weight variable W and the concentration of the precursor protein P. For plotting, time courses were vertically scaled (but not horizontally scaled) as described in Numerical Methods. 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.

FIGURE 4

Time courses of the synaptic weight W after the tetanic protocol of Fig. 3. Four cases are simulated: 1), no kinase inhibition (control); 2), the concentration of active CAMKII in Eq. 16 is scaled down by 90% during the tetanic stimulation and for 50 min after (− CAMKII); 3), the concentration of active CAMKIV is scaled down by 90% during and at all times after stimulation (− CAMKIV); and 4), MAPKK activation (the rate constant k_f,MAPKK) is inhibited by 90% during stimulation and for 10 min after (− MAPKK). For the cases of CAMKII and CAMKIV inhibition, the W time courses are virtually identical.

FIGURE 5

(A) Changes of the synaptic weight W after four stimulus protocols: 1), the tetanic stimuli used in Fig. 3; 2), a θ-burst stimulus protocol (TBS); 3), application of chemical for 30 min (Chem); and 4), a paired stimulus protocol (53). (B) Changes of [MAPK_act], [GPROD], the synaptic tag, and W during and after the paired stimulus protocol.

FIGURE 6

(A) Changes of [MAPK_act], the synaptic tag, and active CAMKII during and after a simulated 30-min chemical application. (B) Changes of [GPROD] and W. Also shown is the attenuated W time course (− MAPKK) observed when k_{f, MAPKK} is reduced by 90% during and for 5 min after the chemical application.

FIGURE 7

(A) Schematic of the simulation of synaptic tagging. Three tetani, identical to Fig. 2 A except with an interstimulus interval of 10 min, are applied to synapse A. One hour after the first tetanus to synapse A, synapse B is likewise given three tetani. Only the tetani to synapse A activate gene expression (GPROD synthesis). (B) Time courses of the tag at synapse A, the tag at synapse B, [GPROD], and W for synapses A and B. The W (tetanic) time course represents L-LTP of synapse A, the W (tagged) time course represents L-LTP of synapse B. Synthesis of GPROD is blocked 35 min after the first set of tetani by setting the rate constants k_syn and k_synbas (Eq. 17) to zero.