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 
. 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
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 
. While this pathway consists of multiple steps, such as G-protein activation of phospholipase C and phospholipase D 
, 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 
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 
. Washing off 5HT prompts the complete dissociation of PKC Apl II within one minute 
. To replicate these findings, we used a simple network architecture, whereby in the presence of 5HT, SOFF
, which then transforms to SIN1
represents the inactivated receptor that can become activated by 5HT, turning SOFF
, which then produces DAG allowing for the translocation of PKC Apl II. SIN1
is an inactivated receptor that needs to be recycled before it can become activated by 5HT again. At a biochemical level, the transitions from SON
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 
. For simplicity, we have reduced these multiple steps into the two steps (SON
) 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 SON
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, SIN1
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 
(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 . 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 
. 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 SIN1
, passing through SIN2
(; black network only, equations can be found in the Materials and Methods
section). Here SIN2
acts as a secondary inactivated state that requires a longer processing time than SIN1
before recycling back to SOFF
. 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 
PKA, which is activated by 5HT, has been shown to increase desensitization of PKC Apl II translocation in the absence of protein translation 
. 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 
. Our modifications to this PKA model are described in the next sections. Activity of PKA is capable of converting SOFF
directly into SPKA
, where SON
is not immediately attainable and PKC Apl II cannot be activated (; 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 SIN1
, probably representing a regulated recycling endosome 
. It is important to note that since PKA can convert SOFF
, conversion to SPKA
does not require S to go through the active state, SON
, such as the desensitization mediated by SIN2
. 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 
and is consistent with data on heterologous desensitization of GPCRs in other systems 
. This consideration also removed the alternate topology where SPKA
would represent alternate sorting from SIN1
, since the receptor is only in the SIN1
state when the receptor goes through the active state.
The recycling of SPKA back into SOFF 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 SPKA 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 
. 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 SOFF
, a form of S preserved from the desensitization pathways leading to SIN2
, but similar to SOFF
in its ability to become activated by 5HT and cause the translocation of PKC Apl II (, 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 
. 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 
including Aplysia 
, 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 
. 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 SOFF
and inhibiting its recycling back to SOFF
(, complete network; equations can be found in the Materials and Methods
section). Another possible architecture would have been to generate another state of S (SD
), 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 
, 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 
. 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 
. To diminish the role of D during massed training, we introduced two additional effects of the AD protein. First, AD inhibited the transition from SOFF
, and second, it could transform not only SOFF
but also SPKA
(; 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 SOFF 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 SIN2 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 
. Thus, there must be a homologous desensitization pathway (; 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 
. Several parameter estimation methods were used, and surprisingly, all of them yielded recycling rates of SIN2
back to SOFF
) that were near zero (parameter values can be found in ), resulting in an excellent fit to the data as can be seen in (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 SIN2
during massed training in the absence of protein translation and PKA activity (). 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 
. However, these experiments were not done in the presence of a PKA inhibitor.
Modeling and experimental validation of homologous desensitization pathway.
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 SIN2 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 SIN2 (). 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) (). 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 ().
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 
, 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 SIN2
, 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 
. In order to model PKA mediated desensitization, we included a reduced and modified version of a previous model of PKA activity 
. 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 
. This revision was necessary since PKA activity during the wash period is required for desensitization 
. The new PKA dynamics to massed and spaced training can be seen in . 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 
The black and blue networks () 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 
. The parameters were estimated to fit both data sets. The conversion of SOFF
is modeled using mass action kinetics. The recycling of SPKA
back into SOFF
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) (, ). It was not possible to replicate both the massed training and spaced training data sets without including the PKA inhibition of SPKA
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.
Modeling and experimental validation of desensitization mediated by PKA pathway.
Massed training in the absence of protein synthesis leads to more desensitization of PKC Apl II translocation when PKA is active 
. However, the model predicts that soon after 5HT is washed away, PKA becomes inactive and SPKA
can recycle back to SOFF
. This recycling suggests that unlike SIN2
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 (). This recovery happens because during the 90 min stimulation, the majority of S is held in SPKA
, and during the wash most of SPKA
recycles back to SOFF
. This recycling allows for a greater amount of PKC translocation compared to when PKA was inhibited and the majority of S is found in SIN1
(). 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 (). 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 (). 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 .
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 .
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 SPKA). 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 ). 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.