Describing the model architecture

We have previously described PKC Apl II translocation and its desensitization in response to 5HT application in the presence of PKA and protein synthesis inhibitors [13], [18], [19]. We showed that PKC translocation differentially desensitizes to spaced and massed applications of 5HT, and that this differential desensitization was dependent on protein translation and PKA activity. In order to understand the molecular mechanisms underlying desensitization of PKC Apl II translocation we designed a signaling network based on our previous experimental findings and biochemical mechanisms known to underlie G protein-coupled receptor (GPCR) desensitization. Our network consists of the translocation of PKC, the cycling of a GPCR, the translation of two hypothetical proteins, and activity of PKA. We have tried to simplify the network whenever possible, including bundling multiple biochemical reactions into one single rate in order to simplify its architecture. The reasoning behind the network's architecture is given in this section and the model equations are given in the Materials and Methods section.

The basic unit of the model is the 5HT GPCR (S) that once activated leads to the production of diacylglycerol (DAG), which is capable of activating and translocating PKC Apl II to the membrane [20], [21]. While this pathway consists of multiple steps, such as G-protein activation of phospholipase C and phospholipase D [18], these are not likely to be important for modeling of desensitization, since in most systems the amount of the activatable GPCR is the rate-limiting quantity that is decreased during desensitization [22], [23], [24].

GPCRs can enter a number of different pathways, such that S can exist in several different states, where the change in concentration of each state with respect to time is modeled. The base component of our model includes the activation and inactivation of S without any desensitization dynamics. This component corresponds to how quickly PKC Apl II translocates to the membrane after 5HT application and how quickly it dissociates from the membrane after 5HT is washed away. It is known that application of 5HT results in a maximal translocation of PKC Apl II within one minute, after which it remains at this maximal level for at least five minutes [18], [19]. Washing off 5HT prompts the complete dissociation of PKC Apl II within one minute [13], [18], [25]. To replicate these findings, we used a simple network architecture, whereby in the presence of 5HT, S_OFF becomes S_ON, which then transforms to S_IN1. S_OFF represents the inactivated receptor that can become activated by 5HT, turning S_OFF into S_ON, which then produces DAG allowing for the translocation of PKC Apl II. S_IN1 is an inactivated receptor that needs to be recycled before it can become activated by 5HT again. At a biochemical level, the transitions from S_ON to S_IN1 to S_OFF involve multiple molecular steps including GPCR phosphorylation by G protein receptor kinases, binding of beta arrestin, possible internalization of the receptor, unbinding of the ligand, and then recycling of the receptor back to its initial state [22], [23], [24]. For simplicity, we have reduced these multiple steps into the two steps (S_ON to S_IN1 to S_OFF) since (i) this is sufficient to capture the behavior required to understand the questions we are addressing (see below) and (ii) we have no specific knowledge concerning regulation of these pathways in Aplysia. The major constraint from the data is that PKC comes off the membrane in less than one minute after 5HT is washed off. Thus S_ON to S_IN1 must be fast enough to account for this inactivation. However, in the first 5 min of 5HT activation, there is little desensitization of PKC Apl II translocation. Thus, S_IN1 to S_OFF must be rapid enough to prevent appreciable desensitization in the first five minutes. The transitions between states of S were modeled using mass action kinetics. These model parameters were fit to the previously described PKC dynamics [13], [18], [25] (equations, parameter values, and parameter estimation methods can be found in the Materials and Methods section). Once an appropriate fit was found these parameters were set and we were able to begin expanding the model and modeling data related to PKC Apl II desensitization.

The complete model architecture is presented in Figure 2. The model components (color coded) were developed sequentially, with maroon and black first then blue, red, and finally green. The maroon component represents only the translocation of PKC to the plasma membrane and its subsequent dissociation from the membrane. The black component represents the desensitization pathway in the presence of a protein translation inhibitor and a PKA inhibitor. In the presence of these inhibitors, PKC Apl II translocation desensitizes during exposure to 5HT [13]. Thus, there must be a protein translation-independent and PKA-independent desensitization pathway, or a homologous desensitization pathway, which we model as an alternate recycling pathway from S_IN1 to S_OFF, passing through S_IN2 (Figure 2; black network only, equations can be found in the Materials and Methods section). Here S_IN2 acts as a secondary inactivated state that requires a longer processing time than S_IN1 before recycling back to S_OFF. At the biochemical level, this represents the sorting of the GPCR in the endocytic compartment from a rapid recycling pathway into a slow recycling pathway or degradative pathway. This architecture was chosen because of the abundant literature supporting this mechanism for desensitization of GPCRs [22], [23], [24].

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Figure 2. Complete model network.

The maroon network describes the translocation of PKC to the plasma membrane and its subsequent dissociation from the membrane. The module denoted in black represents the homologous desensitization pathway. The blue network defines the PKA mediated desensitization of PKC Apl II. The red network illustrates the AD pathway responsible for the rescuing PKC from desensitization. Finally the D pathway, which is antagonistic to the AD pathway and causes the increase in desensitization, is specified by the green network. In green are the additional roles of AD needed to counteract D during massed training.

https://doi.org/10.1371/journal.pcbi.1002324.g002

PKA, which is activated by 5HT, has been shown to increase desensitization of PKC Apl II translocation in the absence of protein translation [13]. The condition where PKA is active and protein translation is inhibited is modeled by the combination of the black, maroon, and blue components. In order to model PKA-mediated protein synthesis-independent desensitization, we included a reduced and modified version of a previous model of PKA activity [26]. Our modifications to this PKA model are described in the next sections. Activity of PKA is capable of converting S_OFF directly into S_PKA, where S_ON is not immediately attainable and PKC Apl II cannot be activated (Figure 2; black and blue networks, equations can be found in the Materials and Methods section). At the biochemical level, this network would represent phosphorylation of the receptor, or receptor-associated protein, by PKA causing the endocytosis of the GPCR from the plasma membrane to an endocytic compartment distinct from S_IN1 and S_IN2, probably representing a regulated recycling endosome [27]. It is important to note that since PKA can convert S_OFF to S_PKA, conversion to S_PKA does not require S to go through the active state, S_ON, such as the desensitization mediated by S_IN2. This network architecture is required to account for the observation that PKA activity between pulses of 5HT, when S would not be activated, is capable of desensitizing PKC Apl II translocation [13] and is consistent with data on heterologous desensitization of GPCRs in other systems [28]. This consideration also removed the alternate topology where S_PKA would represent alternate sorting from S_IN1, since the receptor is only in the S_IN1 state when the receptor goes through the active state.

The recycling of S_PKA back into S_OFF is inhibited by PKA. This inhibition was not initially part of the architecture, but it was not possible to replicate both the massed training and spaced training data sets without including the PKA inhibition of S_PKA recycling (see results below). At a biochemical level, this suggests that PKA activity is not only required to induce sorting of the receptor to the regulated recycling endosome but its retention in this compartment as well.

The reverse situation, with PKA activity inhibited but protein translation allowed to function is modeled by the combination of the black, maroon, and red components. Protein translation in the absence of PKA activity leads to a reduction in the desensitization of PKC Apl II translocation only during massed 5HT application and not spaced [13]. This observation requires that a protein, which protects PKC Apl II translocation from the constitutive desensitization pathway be translated during massed training. We name this hypothetical protein Anti-Desensitizer (AD) and its effects on the network are represented by the black, maroon and red components combined. We modeled the mechanism of AD mediating this protection by having AD convert S_OFF into S_AD, a form of S preserved from the desensitization pathways leading to S_IN2 or S_PKA, but similar to S_OFF in its ability to become activated by 5HT and cause the translocation of PKC Apl II (Figure 2, black, maroon and red pathway; equations can be found in the Materials and Methods section). At the biochemical level, this would represent the AD protein binding to the receptor, or receptor associated protein, preventing its inactivation and internalization [29], [30], [31], [32]. Since a protein-synthesis dependent protection from desensitization is seen in massed, but not spaced, training protocols, we would expect AD to be synthesized only after massed training. In order for this differential synthesis to occur, we made production of AD proportional to the mathematical integration of the level of active PKC Apl II. PKC Apl II is constantly active during massed training, but not during spaced training; thus, integrating PKC activity allows for selective activation of AD during massed training. PKC is known to regulate the translational machinery in many systems [33], [34] including Aplysia [35], [36], but the exact mechanism by which PKC regulates translation in this case is not known and is not explicitly modeled here.

Finally, allowing both protein translation and PKA activity to proceed normally results in an increase in the desensitization of PKC Apl II translocation during spaced training [13]. This increase in desensitization was observable only when both PKA activity and protein translation are allowed to proceed, meaning a translated protein is mediating this increase in desensitization, and its rate of translation is dependent on PKA activity. We name this hypothetical protein Desensitizer (D), and we model its mechanism of action similarly to that of PKA by transforming S_OFF into S_PKA and inhibiting its recycling back to S_OFF (Figure 2, complete network; equations can be found in the Materials and Methods section). Another possible architecture would have been to generate another state of S (S_D), but there was not a good biochemical rationale for this and the model worked well (see below) without this additional state. At the biochemical level, D would be a protein that promotes endocytosis [29], particularly to the PKA-dependent pathway. The rate of translation of D is dependent on the amount of PKA activity, similar to the dependence of AD translation on PKC Apl II activity. One difference between the translation of D and AD is that D's production is delayed by 10 min after its induction. The use of a delay was necessary to account for the observation that desensitization of PKC Apl II translocation after a 5 min pulse of 5HT did not begin until after a 10 min wash [13]. At a biochemical level, there may be many reasons for a delay, ranging from requirements for post-translational modification, cellular trafficking, or delay in the activation of proteins synthesis. Finally, while trying to model the data we found that for D to cause enough desensitization during spaced training resulted in too powerful an inhibition during massed training. This over-inhibition resulted from the fact that unlike AD, D is synthesized during both spaced and massed training since PKA is active in both scenarios [10]. To diminish the role of D during massed training, we introduced two additional effects of the AD protein. First, AD inhibited the transition from S_OFF to S_PKA, and second, it could transform not only S_OFF to S_AD but also S_PKA to S_AD (Figure 2; complete network). At a biochemical level, this corresponds to the ability of the AD protein to prevent endocytosis to the PKA-dependent pathway, and moreover, to bind to the GPCR in the regulated recycling endosome and enhance its recycling, similar to the mechanism by which decreased PKA activity enhanced recycling from this compartment. We also attempted to model the system with AD preventing the translation of D as opposed to opposing its actions, but were unable to achieve a good fit to the data with this architecture.

For simplicity, we made the assumption that during the time course of our experiments an insignificant amount of new S is created. This assumption was also made partially because for S to enter the S_OFF state, the GPCR would not only have to be synthesized, but processed through the endoplasmic reticulum, Golgi apparatus, and transported back to the membrane, so new S could only contribute to the later parts of the experimental paradigm. We do not have a term for destruction of S, however, as described below, the S_IN2 pathway may be equivalent to a degradation pathway, where the GPCR enters late endosomes and lysosomes.

Modeling the homologous desensitization pathway finds slow rate of recovery from desensitization

PKC Apl II translocation still desensitizes during exposure to 5HT even when both protein translation and PKA have been inhibited [13]. Thus, there must be a homologous desensitization pathway (Figure 3A; black network only, equations can be found in the Materials and Methods section). Parameter values were estimated by fitting the model to PKC Apl II translocation measurements taken during a continuous 90 min application of 5HT in the presence of the protein translation inhibitor anisomycin and the PKA inhibitor KT5720 [13]. Several parameter estimation methods were used, and surprisingly, all of them yielded recycling rates of S_IN2 back to S_OFF (k_A5) that were near zero (parameter values can be found in Table 1), resulting in an excellent fit to the data as can be seen in Figure 3C (R²>0.99). Note that throughout the paper, data presented in blue represents data obtained from Farah et al. (2009) used to train the model, while data presented in red represents experiments performed to confirm predictions of the model. The model predicted very little recycling of the signaling complex from S_IN2 during massed training in the absence of protein translation and PKA activity (Figure 3C, D). This was unexpected, since our earlier experiments showed that the desensitization seen after a 5 min pulse of 5HT recovered completely within 45 min, suggesting efficient recycling of the signaling complex [13]. However, these experiments were not done in the presence of a PKA inhibitor.

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Figure 3. Modeling and experimental validation of homologous desensitization pathway.

A, Model network pathways of homologous desensitization pathway (black) and PKA-mediated desensitization pathway (black and blue). B, Representative confocal fluorescence images of sensory neurons expressing eGFP-PKC Apl II during a 90 min exposure to 5HT followed by a 45 min wash and then a 5 min 5HT application, all in the presence of anisomycin and KT5720. C, Quantification of PKC Apl II translocation (bars) and modeling output (line). Blue bars are data used from Farah et al. (2009) to fit the model parameters. Red bars are data from the present study (n = 8 cells). Error bars are SEM. D, Modeling of S dynamics in response to experimental protocol from B. Black line represents the ratio of S_OFF and S_ON to total S and the red line the ratio of S_IN2 to total S. The times of addition of 5HT and pharmacological agents are indicated below the figure.

https://doi.org/10.1371/journal.pcbi.1002324.g003

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Table 1. Model parameters.

https://doi.org/10.1371/journal.pcbi.1002324.t001

To test the prediction of the model that desensitization seen in the absence of PKA activity was not reversible, we conducted a new experiment. The rate of S_IN2 recycling was predicted to be slow enough that a wash period after massed training with anisomycin and KT5720 would result in little recovery of translocation to initial values. Thus, in a simulation of a 90 min exposure to 5HT followed by a 45 min wash and then a 5 min pulse of 5HT, all in the presence of anisomycin and KT5720, the 5 min pulse of 5HT should only cause a small amount of PKC Apl II translocation, since a majority of S is held in the inactivated state S_IN2 (Figure 3C, D). To test this prediction of the model, we used this protocol in a new imaging experiment using Aplysia sensory neurons expressing eGFP-PKC Apl II. The initial massed training caused a similar amount of translocation to that previously observed by Farah et al. (2009) (Figure 3B, C). Furthermore, the amount of desensitization after the 5 min pulse of 5HT matched the modeling prediction extremely well, demonstrating that recovery from desensitization under these conditions was indeed very slow (Figure 3B, C).

This protocol required that the neurons be imaged for a total of 140 min. To ensure that the lengthy exposure to room temperature (20–23°C) and the drugs anisomycin and KT5720 had no effect on the health of the neurons, or their ability to translocate PKC Apl II, two 5 min pulses of 5HT were applied with a 130 min wash in between, all in the presence of both drugs. Recovery from a 5 min pulse of 5HT occurs after 45 min [13], so we expect that a 130 min wash should result in complete recovery and that any depression in PKC Apl II translocation would be caused by injury to the neurons due to prolonged exposure to room temperature and drugs. There was no significant difference in the amount of PKC Apl II translocation between the first and second pulse of 5HT (mean+/−sem; 1.08+/−0.18, n = 5). Thus the persistent desensitization observed in the previous experiment is due only to accumulation of S in S_IN2, as predicted by the model and not due to injury to the neurons.

Modeling desensitization induced by PKA confirms rapid rate of recovery

PKA, which is activated by 5HT, has been shown to increase desensitization of PKC Apl II translocation during both massed and spaced training [13]. In order to model PKA mediated desensitization, we included a reduced and modified version of a previous model of PKA activity [26]. We reduced the complexity of this model to only include only the dynamics of cAMP production and the association and dissociation of the subunits of PKA. This simplification was done since our experiments and simulations do not occur over long enough time periods for us to expect a contribution from the persistent activity of PKA, which was a major feature of their model. We modified the Pettigrew et al. model by altering the basal level of cAMP and the association rate of the PKA subunits to refine PKA dynamics to better match published data demonstrating PKA activity persisting for a small period after washout of 5HT [10], [37], [38]. This revision was necessary since PKA activity during the wash period is required for desensitization [13]. The new PKA dynamics to massed and spaced training can be seen in Figure 4A–C. Furthermore, we removed any synthesis or degradation of PKA subunits since, similar to PKC Apl II, we do not expect a significant change in the amount of protein during the time course of our experiments [10].

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Figure 4. PKA dynamics.

A, Model network pathway of PKA dynamics. R and C represent the regulatory subunit and catalytic subunit of PKA, respectively, where the amount of PKA activity is considered identical to C activity. B, C, PKA activity in response to a 90 min 5HT application (B) or 5×5 min 5HT application (C). Black line represents the amount of cAMP activity; blue line represents C, and red line RC. R is not shown, as it is identical to C. (Model adapted from [70]). D, PKC Apl II translocation in response to 5×5 min application of 5HT with 15 min washes in between and anisomycin present throughout from Farah et al. (2009) (bars) and modeling output (line). E, Modeling of S dynamics in response to experimental protocol from D. Black line represents the ratio of S_OFF and S_ON to total S, the red line the ratio of S_IN2 to total S, and the blue line represents ratio of S_PKA to total S.

https://doi.org/10.1371/journal.pcbi.1002324.g004

The black and blue networks (Figure 3A) make use of the previously described PKA activity model to affect the desensitization of PKC translocation. Two data sets were used to estimate the parameters of the blue component of the model: one continuous 90 min application of 5HT in the presence of anisomycin and five pulses of 5HT each lasting 5 min with 15 min washes in between, all in the presence of anisomycin [13]. The parameters were estimated to fit both data sets. The conversion of S_OFF into S_PKA is modeled using mass action kinetics. The recycling of S_PKA back into S_OFF is inhibited by PKA and is modeled using a combination of mass action kinetics and an inhibitory Hill function (see Materials and Methods section). This network architecture resulted in an excellent fit to both data sets (R² = 0.99 for massed training and 0.88 for spaced training) (Figures 4D, 5B). It was not possible to replicate both the massed training and spaced training data sets without including the PKA inhibition of S_PKA recycling. Without this inhibition, fitting the massed training data set caused too much desensitization during spaced training and fitting the spaced training data set caused insufficient desensitization during massed training.

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Figure 5. Modeling and experimental validation of desensitization mediated by PKA pathway.

A, Representative confocal fluorescence images of sensory neurons expressing eGFP-PKC Apl II during a 90 min exposure to 5HT followed by a 45 min wash and then a 5 min 5HT application, all in the presence of anisomycin. B, Quantification of PKC Apl II translocation (bars) and modeling output (line). Blue bars are data used from Farah et al. (2009) to fit the model parameters. Red bars are from the present study (n = 10 cells). C, Modeling of S dynamics in response to experimental protocol. Black line represents the ratio of S_OFF and S_ON to total S, the red line the ratio of S_IN2 to total S, and the blue line represents ratio of S_PKA to total S. D, Comparing PKC translocation at 135 min after the second 5HT pulse with PKA inactive (anisomycin and KT5720) and PKA active (anisomycin). Student's unpaired two-tailed T test conducted and statistical significance of p<0.01 illustrated by *.

https://doi.org/10.1371/journal.pcbi.1002324.g005

Massed training in the absence of protein synthesis leads to more desensitization of PKC Apl II translocation when PKA is active [13]. However, the model predicts that soon after 5HT is washed away, PKA becomes inactive and S_PKA can recycle back to S_OFF. This recycling suggests that unlike S_IN2 mediated desensitization, PKA induced desensitization recovers quickly. Thus, when we simulate a 90 min exposure to 5HT followed by a 45 min wash and then a 5 min pulse of 5HT (as above, but in the absence of a PKA inhibitor), the model predicts a considerable recovery of PKC translocation (Figure 5B, C). This recovery happens because during the 90 min stimulation, the majority of S is held in S_PKA, and during the wash most of S_PKA recycles back to S_OFF. This recycling allows for a greater amount of PKC translocation compared to when PKA was inhibited and the majority of S is found in S_IN1 (Figure 3C). To test this prediction of the model, we conducted a new imaging experiment, measuring the translocation of eGFP-PKC Apl II during the application of the above protocol (Figure 5A). The translocation of PKC Apl II caused by the 5 min pulse of 5HT after the 45 min wash is in agreement with the modeling prediction, thus validating this component of the model (Figure 5B). The amount of desensitization of PKC Apl II translocation during the massed training is equivalent to that observed by Farah et al. (2009) and, as in that study, PKA increases the amount of desensitization during massed training. However, despite this increased desensitization in the presence of PKA, active PKA increases the recovery from desensitization, as predicted by the model. The large difference between the recovery in the presence or absence of the PKA inhibitor, KT5720, is illustrated in Figure 5D.

Rescue from desensitization by Anti-Desensitizer (AD) protein

The rescue from desensitization by the AD protein is modeled using the black and red network components in combination (Figure 6A, red pathway). Two data sets were used to estimate the parameters of this component of the model: 90 min application of 5HT in the presence of KT5720 and five 5 min pulses of 5HT with 15 min washes in between, all in the presence of KT5720 [13]. The model produced an excellent fit to both data sets (R² = 0.95 for spaced training and 0.99 for massed training) (Figure 6B, D). One of the unexpected predictions of the model was both a fast degradation of AD, (with a half-life of ∼5 min) and a slow rate of the S_AD→S_OFF recycling.

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Figure 6. Modeling the rescue from desensitization by Anti-Desensitizer (AD) protein.

A, Model network pathways of AD desensitization protection pathway (red network designates AD pathway). B, PKC Apl II translocation in response to 5×5 min application of 5HT with 15 min washes in between and KT5720 present throughout from Farah et al. (2009) (bars) and modeling output (line). C, Top panel represents S dynamics in response to experimental protocol from B. Black line represents the ratio of S_OFF and S_ON to total S and the red line the ratio of S_IN2 to total S, and green line represents the ratio of S_AD and S_ADON to total S. Bottom panel represents the amount of AD over time. D, PKC Apl II translocation in response to 90 min application of 5HT and KT5720 present throughout from Farah et al. (2009) (bars) and modeling output (line). E, Top panel represents S dynamics in response to experimental protocol from D. Line colours similar to in C.

https://doi.org/10.1371/journal.pcbi.1002324.g006

To validate this component of the model, we designed a protocol that would be sensitive to the fast degradation rate of AD. This protocol consisted of exposure to 25 min of 5HT in the presence of KT5720 then 65 min of 5HT in the presence of both KT5720 and anisomycin, with no wash in between. This protocol allows for the indirect observation of the degradation of AD and the recycling of S_AD back into S_OFF. The addition of anisomycin will terminate the translation of AD. During these last 65 min, the model predicts that AD will decay and thus be less effective at transforming S_OFF into S_AD (Figure 6C). The model further predicts that the absence of AD will cause the remaining S_AD to recycle back into S_OFF, where it will lose its protection from the homologous desensitization pathway, which will manifest in decreased PKC Apl II translocation. Thus, by observing the increased amount of desensitization of this protocol in comparison to when AD translation is present throughout, we can validate the model's predicted rate of AD degradation and rate of S_AD recycling back into S_OFF. To test these predictions of the model, a new imaging experiment was performed by applying this protocol to Aplysia sensory neurons expressing eGFP-PKC Apl II. As expected, the amount of PKC translocation observed in these neurons during the first 25 min of 5HT was equivalent to that observed during the 25 min of massed training in the presence of KT5720, carried out by Farah et al. (2009) (Figure 6A, B). However, the final 65 min of this protocol, where both KT5720 and anisomycin are present, caused a lower amount of PKC Apl II translocation compared to that caused by massed training in the presence of only KT5720, in agreement with the model prediction (R² = 0.99) confirming the fast degradation rate of AD and the slow rate of S_AD to S_OFF (Figure 7C).

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Figure 7. Experimental validation of AD dynamics.

A, Representative confocal fluorescence images of sensory neurons expressing eGFP-PKC Apl II during a 90 min exposure to 5HT where during the first 25 min KT5720 alone was applied, while during the last 40 min both anisomycin and KT5720 was applied. B, Quantification of PKC Apl II translocation (bars) and modeling output (line with square points). 90 min 5HT with KT5720 and anisomycin (light blue bars) is similar to that shown in Figure 1C and 90 min 5HT with anisomycin (dark blue bars) is similar to that shown in Figure 4D. These data points, from Farah et al. (2009), are reproduced here for comparison purposes with the following newly acquired data. Error bars are SEM. Red bars represent quantification of eGFP-PKC Apl II translocation during experimental protocol from A (n = 9 cells) compared to line with triangles for the modeling prediction of this experimental protocol (25 min 5HT with KT5720 followed by 65 min with KT5720 and ansiomycin). F, Top panel represents S dynamics in response to experimental protocol from D. Black line represents the ratio of S_OFF and S_ON to total S and the red line the ratio of S_IN2 to total S, and green line represents the ratio of S_AD and S_ADON to total S. Bottom panel represents the amount of AD over time.

https://doi.org/10.1371/journal.pcbi.1002324.g007

Modeling increase in desensitization by Desensitizer (D) protein

During spaced training, the desensitization of PKC Apl II translocation was increased in control cells in comparison to when protein translation was inhibited. This increase in desensitization was observable only when both PKA activity and protein translation are allowed to proceed, meaning a translated protein is mediating this increase and its rate of translation is dependent on PKA activity. We name this hypothetical protein Desensitizer (D), and its effects on PKC Apl II translocation are modeled by the green component of the network (Figure 2). Seven data sets were used to estimate the parameters of this component of the model: one continuous 90 min application of 5HT, five pulses of 5HT each lasting 5 min with 15 min washes in between, and five experiments, each with two pulses of 5HT each lasting 5 min but with a different wash period length (5 min, 10 min, 15 min, 30 min, 45 min,) in between the pulses [13] (Figure 8A, C, E). The resulting model formed an excellent fit to the data (R² = 0.99 for massed training, 0.99 for spaced training, and 0.75 for two pulses of 5HT with varying wash intervals). One exception is the 5 min pulse followed by a 5 min wash, where there is an increase in PKC Apl II translocation compared to the initial translocation, while our model shows no increase in translocation. We believe fitting this increase would require a more detailed dissection of the pathway between the GPCR and its downstream targets and is beyond the scope of this study.

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Figure 8. Fitting complete model to PKC translocation measured during no pharmacological interventions.

A, PKC Apl II translocation in response to 5×5 min application of 5HT with 15 min washes in between (bars) and modeling output (line). B, Top panel represents S dynamics in response to experimental protocol from A. Black line represents the ratio of S_OFF and S_ON to total S, red line the ratio of S_IN2 to total S, green line represents the ratio of S_AD and S_ADON to total S, and blue line represents the ratio of S_AD and S_ADON to total S. Middle panel is the amount of AD over time and the bottom panel is the amount of D over time. C, PKC Apl II translocation in response to a 90 min application of 5HT (bars) and modeling output (line). D, Top panel represents S dynamics in response to experimental protocol from G, with line colours identical to in B. Middle panel represents the amount of AD over time and the bottom panel the amount of D over time. E, PKC Apl II translocation in response to 2×5 min applications of 5HT with varying wash periods in between (black line) and modeling output (red line).

https://doi.org/10.1371/journal.pcbi.1002324.g008

Model successfully predicts the response to new spacing protocols

As one of the rationales for generating this model was to gain insight into the role of spacing, our final confirmation of the model tested an alternate spacing protocol. We designed an experiment that would require the functioning of all the model components and that made a specific prediction that was not obvious and could be tested. Interestingly, we found that if 15 min pulses of 5HT were used, the model predicted that longer washes would lead to increased desensitization. While 15 min pulses produce both D and AD, the model predicts that longer washes will reduce the levels of AD compared to D and thus predicts greater desensitization by longer washes (Figure 9B and E). In particular note that the model predicts that with the shorter spacing (Figure 9C), the amount of S complex in S_AD is larger than in S_PKA immediately before the second pulse, while with longer spacing (Figure 9F), the model predicts that there is more S complex in S_PKA, than in S_AD. Thus, the second pulse of serotonin during the protocol with longer spacing should be less able to translocate PKC Apl II because of the conversion of S_AD to S_PKA. To test this prediction, we performed a new imaging experiment where sensory neurons were exposed to three 15 min pulses of 5HT with either 15 min or 25 min washes in between the 5HT pulses. The results of this protocol are also sensitive to the delay and rate of D translation (parameters that had not yet been validated in a separate experiment). Both protocols were applied to Aplysia sensory neurons expressing eGFP-PKC Apl II. The amount of PKC translocation during both protocols matched the modeling prediction (R²>0.99) (Figure 9A, B, D, and E) and thus validates this component of the model as well as the functioning of the completed model. In particular, to highlight the effect of the wash, we calculated the amount of desensitization during the 15 min or 25 min wash (e.g. the amount of translocation at the beginning of pulse 2 compared to the end of pulse 1, or the beginning of pulse 3 compared to the end of pulse 2). The model predicted more desensitization during the longer wash and this was confirmed by the imaging experiment (Figure 10).

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Figure 9. Longer wash periods lead to greater desensitization of PKC Apl II translocation.

A, Representative confocal fluorescence images of sensory neurons expressing eGFP-PKC Apl II during a 3×15 min application of 5HT with 15 min washes in between. B, Quantification of PKC Apl II translocation (bars, n = 9 cells) and modeling output (line). Error bars are SEM. C, Top panel represents S dynamics in response to experimental protocol from A. Black line represents the ratio of S_OFF and S_ON to total S, red line the ratio of S_IN2 to total S, green line represents the ratio of S_AD and S_ADON to total S. Middle panel represents the amount of AD over time and the bottom panel the amount of D over time. Dotted line represents time of second and third 5HT application. D, Representative confocal fluorescence images of sensory neurons expressing eGFP-PKC Apl II during a 3×15 min applications of 5HT with 25 min washes in between. E, Quantification of PKC translocation (bars, n = 6 cells) and modeling output (line). Error bars are SEM. F, Top panel represents S dynamics in response to experimental protocol from E. Black line represents the ratio of S_OFF and S_ON to total S, red line the ratio of S_IN2 to total S, green line represents the ratio of S_AD and S_ADON to total S. Dotted line represents time of second and third 5HT application. Middle panel represents the amount of AD over time and the bottom panel the amount of D over time.

https://doi.org/10.1371/journal.pcbi.1002324.g009

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Figure 10. Quantifying the amount of desensitization of PKC Apl II translocation occurring during wash periods.

Left pair of bars represent the amount of PKC Apl II translocation at the 5 min point of the second pulse divided by the amount of PKC Apl II translocation at the 15 min point of the first pulse. Right pair of bars represents the amount of PKC Apl II translocation at the 5 min point of the third pulse divided by the amount of PKC Apl II translocation at the 15 min point of the second pulse. Red bars correspond to 15 min 5HT with 15 min washes (n = 9) and Orange to 15 min 5HT with 25 min washes (n = 6). Student's unpaired two-tailed T test conducted and statistical significance of p<0.05 illustrated by *.

https://doi.org/10.1371/journal.pcbi.1002324.g010

Sensitivity analysis

A parameter sensitivity analysis was performed on the completed model to investigate which parameters were most important in driving the results of the model. Each parameter was varied between +/−5% and +/−50% while holding the other parameters at their defined values. The model was then simulated using a 90 min application of 5HT and its resulting PKC Apl II translocation compared to that observed by Farah et al. (2009), which was initially used to fit the model. The sensitivity of a parameter was classified as High if either a +/−5% change in its value caused a change in the fit of the data of over 25%. Similarly, the sensitivity of a parameter was Medium if either a +/−50% change in value caused a change in the fit of the data of over 25%, and Low if the +/−50% change in value did not change the fit by more than 25%. This was then repeated using a spaced application of 5HT (5×5 min 5HT with 15 min washes). The complete sensitivity analysis is summarized in Table 1.

Of the 41 parameters, 5 were classified as High, 12 as Medium, and 6 as Low in both massed and spaced training sensitivity analysis. Interestingly, the majority of the parameters (3/5) with high sensitivity for both types of training were those associated with the initial component of the model responsible for activating PKC Apl II. The remaining two parameters involved how AD works (the synthesis rate and its ability to stop S from going into S_PKA). It is not surprising that changing parameters that affect the initial translocation of PKC Apl II by 5HT and its decay after 5HT is removed would have a large effect on the model output, since the model was built around this core. However, these parameters were chosen in a somewhat arbitrary fashion to fit the initial data since the actual rates of DAG synthesis and decay are not known in this system. To ensure that the set of values we chose for these parameters are not critical for the working of the model, we found another parameter set that could fit the initial translocation data (see Table 1). Reassuringly, the rest of the model still worked, suggesting that the model was not dependent on the actual values for these initial parameters, just the ability of the model to replicate the known rate of PKC Apl II translocation and dissociation by 5HT.

The remaining 18 parameters had sensitivities dependent on the type of 5HT application profile. Interestingly, about the same number of parameters had specific high sensitivity for massed (5) vs spaced (4). For spaced, two of these are again from the initial model and the others concern the synthesis rate of D and AD. Similarly, for massed, two of these are for the initial model and the others concern the synthesis rate for D. The sensitivity analysis suggests, similar to the experiments, that the critical parameters that determine the model are involved in the synthesis of D and AD.

Biological plausibility of the model

The architecture of our model is based on the fact that desensitization and resensitization of GPCRs is due to endocytosis followed by exocytosis, the simplest biological substantiation of the states of S are distinct endocytic recycling pathways of the receptor or distinct states of the receptor after association with GPCR binding-proteins [29], [39]. Below we review these processes and the plausibility of the parameters we have assigned to these steps. We also review other aspects of the model and whether the parameters and architecture are biologically plausible.

Comparison to other models that explain timing during plasticity

Spaced vs. massed training occurs in a number of different time scales, thus molecular mechanisms are required that act in different time domains. For example, induction of LTP by spaced stimuli requires PKA, but not when massed stimulation is used [58], and the spaced stimuli were required to recruit protein synthesis-dependent mechanism [59]. A recent model explains this finding based on the differential effects of calcium on PKA and CAMKII. This model depends on inter-trial intervals that range between seconds and 5 minutes [60]. The frequency dependent activation of CAMKII is sensitive to timing intervals in this period and is proposed to be the mechanism for sensing the spacing between stimuli [61]. In mice object recognition was enhanced by spacing of 15 min, compared to 5 min or massed training [62]. In mice that lacked Protein Phosphatase 1 (PP1), 5 min spacing was sufficient for learning. In this case the rate-limiting step for learning was activation of CREB, and spacing was required in order for PP1 to be deactivated before the next training trial allowing for CREB activation [62]. CREB activation is also the proposed difference between spaced and massed learning in Drosophila odor avoidance [63]. In Drosophila, spacing is regulated by waves of MAP kinase activation where both the activation and decay kinetics appear critical for the spacing interval [64]. In Aplysia, it has recently been demonstrated that for long-term facilitation, only two spaced trials are required, 45 minutes apart, but neither 30 nor 60 minute spacing is adequate [65]. Again, in this case the spacing corresponds to a wave of MAP kinase activation [65]. None of these cases directly implicate the rates of protein synthesis or degradation as critical for timing, although it is possible that the induction of MAP kinase activation at later times may require protein synthesis. Interestingly, the activation of CREB in mammals appears to require removal of CREB repressor, which requires both blockade of translation through eIF2a dephosphorylation [66] and increased degradation due to proteosomal activation [67]. Thus, in this case regulating the level of a protein also may mediate differences between spaced and massed trained determining whether or not transcription is activated. It will be interesting in the future to determine how generally neurons sense time through measuring the half-life of newly synthesized proteins.

Mathematical modeling

A mathematical model of the desensitization of PKC Apl II translocation in Aplysia californica sensory neurons was constructed in the MATLAB programming environment. The model consists of a system of integro-differential equations with delays, where each equation describes the change in concentration of the proteins PKC Apl II, PKA, Desensitizer (D), Anti-Desensitizer (AD), and of each instance of the signaling complex (S). Since we are only interested in PKC Apl II translocation occurring between the cytosol and plasma membrane [18] a single compartment model was used.

The complete model is depicted in Figure 2. The colours of this figure correspond to the components of the model. The model was constructed in a sequential manner. First, the components outlined in black and maroon were fit to data [13] at which point its parameters were specified and not allowed to change. Following these component's completion, the component outlined in blue was similarly constructed, then the red component and finally the green component. In order to illustrate this sequential construction within the model equations, we have named the parameters according to which component they reside in: A for the black component, B for blue, C for red, and D for green.

The most basic component of the model is the translocation of PKC Apl II from the cytosol to the plasma membrane (maroon component). This translocation is proportional to the concentration of diacylglycerol (DAG) on the membrane and thus the translocation is given by the following equation:(1)where k_DAGp is the rate of PKC Apl II translocation to the membrane, and k_DAGd the rate of PKC Apl II removal from the membrane. S_ON represents the proportion of S currently in the active state, which is capable of translocating PKC Apl II to the membrane. The inactive state of S is given by S_OFF, and S_IN1 is a transition state between S_ON and S_OFF. S can be transformed into 3 other states, as will be described next. We require the total amount of S to remain constant by employing the following restriction: , where i = ON, OFF, IN1, IN2, PKA, AD, and D. We scale each S variable by 1/S_TOT, such that all parameters k_i, i = A1–A5, B1–B2, C1–C2, and D1–D3 will have units min⁻¹. We have set S_TOT = 1, where we refrain from assigning units to the S variables since we cannot measure the concentrations of PKA or PKC in Aplysia neurons in order to accurately define a unit of measure. Furthermore, units are not assigned to any variable with concentration as a possible dimension. This simplification is justified since we have developed a single compartment model of Aplysia sensory neurons to qualitatively describe the dynamics of PKC desensitization. Using non-dimensional variables and parameters allows us to observe important dynamics, such as relative magnitudes of proteins and the time course of S recycling, which allow us to gain insight into the molecular regulatory mechanism involved in the desensitization of PKC translocation.

The following equations describe the rates of change of concentration of the first four S states:(2)(3)(4)(5)where [5HT] represents the concentration of 5HT being applied to the system and is given a standard value of 10 µM during any application of 5HT, k_A1 represents the transformation of S_OFF into S_ON, k_A2 of S_ON into S_IN1, k_A3 of S_IN1 into S_OFF, k_A4 of S_IN1 into S_IN2, and k_A5 of S_IN2 into S_OFF. The additional terms in equation (3) refer to the further transformations that S_OFF can undergo. Without these additional terms these equations describe the black model in Figure 3A. The first additional transformation of S_OFF is mediated by the catalytic subunit of PKA, where S_OFF is converted to S_PKA, which has the following equation:(6)where k_B1 is rate constant of the transformation from S_OFF into S_PKA, which is brought about by the activity of the catalytic subunit of PKA or the protein D. However, the protein AD, through a Hill function with coefficient k_D1b and half saturation k_D1a, can inhibit this conversion. The recycling of S_PKA into S_OFF occurs with rate constant k_B2, but is inhibited by PKA and D through inhibiting Hill functions with coefficients k_B2b and k_D2b, respectively, and half saturations k_B2a and k_D2a, respectively. Also, activity of AD can convert S_PKA into S_AD with a rate constant k_D3 and a Hill function with coefficient k_D3b and half saturation k_D3a.

The dynamics of the catalytic and regulatory subunits of PKA are adapted from a model presented by Pettigrew et al. (2005), where the changes to this model are described in the results. PKA dynamics are given in the following equations:(7)(8)(9)(10)Where V_m is the cAMP synthesis rate constant, and K_5HT is the half saturation of the Hill function associated with cAMP synthesis. K_fpka is the rate constant associated with the dissociation of the catalytic and regulatory subunits, while the reassociation rate is given by K_bpka. The amount of PKA activity is set equal to the amount of the free catalytic subunit (C); PKA = C. The parameters associated with protein synthesis are given the subscript S to differentiate them from signaling complex dynamics. The synthesis of AD and D are given by the following equations:(11)(12)(13)(14)

AD synthesis depends on the total activity of PKC Apl II over a previous time window of duration given by intPKC in equation (12). The integration of PKC Apl II activity leads to the synthesis of AD through a Hill function with coefficient k_S1b and half saturation k_S1a. k_S2 represents the AD degradation constant. Similarly, D synthesis depends on an integration of PKA activity over a time period of intPKA in equation (14), which leads to the synthesis of D with a rate constant of k_S3 and through a Hill function with coefficient k_S3b and half saturation k_S3a. The degradation of D is given by rate constant k_S4. D leads to the transformation of S_OFF into S_PKA, which was described above in equation (6), while AD transforms S_OFF into S_AD, whose dynamics are modeled with the following equation:(15)where k_C1 is the rate constant associated with the transformation of S_OFF into S_AD, k_C2 the rate constant of the recycling of S_AD into S_OFF, which can be inhibited by AD through a Hill function with coefficient k_C2b and half saturation k_C2a. Also, S_AD can become activated when 5HT transforms it into S_ADON, whose dynamics are given in the following equation:(16)whose transformation and recycling rate constants are identical to those of S_OFF into S_ON. S_ADON activates PKC Apl II in an identical fashion to S_ON in equation (1).

The system was solved numerically by employing a 4^th order Runga-Kutta scheme to solve the differential equations and the Trapezoid Rule to solve the integrals [68]. Parameter estimation was conducted with the help of the MATLAB Optimization Toolbox and Global Optimization Toolbox, specifically the functions lsqcurvefit, ga, and fmincon. These functions were used to minimize the least squares distance between the modeling output and experimental data. Values of individual parameters are given in Table 1.

Aplysia cell culture preparation

Adult Aplysia californica (76 to 100 g; University of Miami Aplysia Resource Facility, RSMAS, FL) organisms were anesthetized by an injection of 50 to 100 ml of 400 mM (isotonic) MgCl2. Pleuropedal ganglia were removed and digested in L15 medium containing 1% protease type IX (Sigma). L15 medium was purchased from Sigma and supplemented with 0.2 M NaCl, 26 mM MgSO4·7H2O, 35 mM dextrose, 27 mM MgCl2·6H2O, 4.7 mM KCl, 2 mM NaHCO3, 9.7 mM CaCl2·2H2O, 15 mM HEPES, and the pH was adjusted to 7.4. Following digestion, tail sensory neurons were isolated and plated in L15 medium containing 50% Aplysia hemolymph on MatTek glassbottom culture dishes (MatTek Corporation, Ashland, MA) with a glass surface of 14 mm and a coverslip thickness of 0.085 to 0.13 mm. The dishes were pretreated with poly-L-lysine (molecular weight, >300,000; Sigma).

Plasmid construction and microinjection of plasmid vectors

The pNEX3 enhanced green fluorescent protein (eGFP) PKC Apl II has been described previously [13], [19], [69]. On day 1 after isolation, solutions of plasmids in distilled water containing 0.25% fast green were microinjected into neurons from back-filled glass micropipettes. The tip of the micropipette was inserted into the cell nucleus, and short pressure pulses (10–50 ms duration; 20 lb/in2) were delivered until the nucleus became uniformly green. The cells were incubated for 4–5 hrs at room temperature and then kept at 4°C until use.

Confocal microscopy of Aplysia neurons

Neurons expressing eGFP-PKC Apl II were imaged on a Zeiss laser-scanning microscope (Zeiss, Oberkochen, Germany) with an Axiovert 200 and a ×40 or ×63 oil immersion objective with a 25-mW argon laser with 25% laser output. The laser line was attenuated to 4% transmission output prior to live imaging. 5HT (10 µM) was added to the dish in L15 medium containing 50% hemolymph. 5HT was washed away with artificial seawater (ASW; 10 mM HEPES, pH 7.5, 0.46 M NaCl, 10 mM KCl, 11.2 mM CaCl2·2H2O, 55 mM MgCl2·6H2O). For spaced training, neurons received five applications of 10 µM 5HT (5 min each) at an intertrial interval (ITI) of 20 min. For massed training, neurons received a single continuous application of 10 µM 5HT for 90 min. All experiments were performed at room temperature (20 to 23°C).

Drug treatment

Anisomycin (Sigma-Aldrich), and KT5720 (Calbiochem) were used at concentrations of 50 µM, and were present in the media throughout spaced or massed training. There was no pre-incubation with these drugs prior to 5HT treatment and because of this we erred on the high side of the concentrations that have been used previously. The controls used in all experiments were always performed from the same batch of animals when the drugs were used. The translation inhibitor anisomycin was purchased from Sigma-Aldrich.

Image analysis

The level of PKC Apl II translocation for each cell was determined by tracing three rectangles at random locations at the plasma membrane and three rectangles at random locations in the cytosol. The width of the membrane rectangles was three-five pixels wide to avoid cytoplasmic contamination, but otherwise the size of the rectangles was not constrained. The average intensity at the membrane (Im) and the average intensity in the cytosol (Ic) was then calculated and the Im/Ic ratio is the degree of membrane association. In all figures, the control used to normalize the translocations in the presence of a drug is the post 5HT #1 in the presence of that drug.