The critical balance between dopamine D2 receptor and RGS for the sensitive detection of a transient decay in dopamine signal.

In behavioral learning, reward-related events are encoded into phasic dopamine (DA) signals in the brain. In particular, unexpected reward omission leads to a phasic decrease in DA (DA dip) in the striatum, which triggers long-term potentiation (LTP) in DA D2 receptor (D2R)-expressing spiny-projection neurons (D2 SPNs). While this LTP is required for reward discrimination, it is unclear how such a short DA-dip signal (0.5-2 s) is transferred through intracellular signaling to the coincidence detector, adenylate cyclase (AC). In the present study, we built a computational model of D2 signaling to determine conditions for the DA-dip detection. The DA dip can be detected only if the basal DA signal sufficiently inhibits AC, and the DA-dip signal sufficiently disinhibits AC. We found that those two requirements were simultaneously satisfied only if two key molecules, D2R and regulators of G protein signaling (RGS) were balanced within a certain range; this balance has indeed been observed in experimental studies. We also found that high level of RGS was required for the detection of a 0.5-s short DA dip, and the analytical solutions for these requirements confirmed their universality. The imbalance between D2R and RGS is associated with schizophrenia and DYT1 dystonia, both of which are accompanied by abnormal striatal LTP. Our simulations suggest that D2 SPNs in patients with schizophrenia and DYT1 dystonia cannot detect short DA dips. We finally discussed that such psychiatric and movement disorders can be understood in terms of the imbalance between D2R and RGS.

Introduction

In animals’ reward learning, phasic dopamine (DA) signal in the brain conveys important information, called reward prediction errors [1,2]. Unexpected reward causes a phasic increase in striatal DA level for 0.2–1 s (DA burst), whereas unexpected reward omission leads to a transient decrease in DA for 0.5–2 s (DA dip) [3,4]. Such phasic DA signals are decoded into striatal synaptic plasticity that refines animal behavior to obtain larger rewards [1,5-8]. In particular, we previously found that 0.5–2-s DA dips triggered long-term potentiation (LTP) in dopamine D2 receptor (D2R)-expressing spiny-projection neurons (D2 SPNs) of the striatum [1,9]. The LTP is induced only if the DA dip coincides with the presynaptic release of glutamate and postsynaptic burst firing of a D2 SPN (pre–post pairing), under the presence of adenosine [1,10]. The DA dip needs to occur together with pre–post pairing, and this LTP is required for reward-discrimination learning [1]. Intracellularly, the DA-dip signal leads to the deactivation of D2R and then the decrease in the GTP-bound form of inhibitory G protein (Gi-GTP). The 0.5-s DA dip is similar to the timescale of the G-protein signaling (0.1–1 s) [11], raising a question about how the DA-dip signal is so reliably encoded into these signaling molecules. The same issue might be extended to psychiatric/movement disorders, because the alteration in D2R signaling and LTP in the striatum have been implicated [2,12,13].

We thus previously demonstrated using computational modeling, how such a rapid signal is encoded in the D2R signaling [10]. In the D2 LTP model, DA-bound D2R produces Gi-GTP, and the Gi-GTP inhibits adenylate cyclase (AC), in particular, AC type 1 (AC1) as in our case [1,5]. In contrast, neuronal firing and adenosine signals elevate the levels of Ca2+ and the GTP form of stimulatory G protein (Golf-GTP), respectively, both of which jointly activate AC1 [14,15]. Together, Gi, Golf, and Ca2+ signals are integrated at AC1 that functions as a coincidence detector [10,15]. For this to occur, the DA-dip signal itself must be transmitted to the AC. The AC activity triggers the longer molecular dynamics, such as increase in cyclic adenosine monophosphate (cAMP), resulting in the occurrence of LTP [1,5].

The similar DA-dip detection seems to be realized in the other subtype of AC, AC type 5 (AC5), because AC5 is abundantly expressed in the striatum [16,17]. Golf-GTP alone sufficiently triggers the activity of AC5, while the AC5 activity is inhibited by Gi [18-20], and the DA-dip detection on AC5 should affect striatal synaptic plasticity [21] as well as the change of somatic neuronal excitability [22,23]. In particular, AC5 likely functions in the soma because somatic AC activity does not require the increase in Ca2+ level [5,19]. D2 signaling models have demonstrated that AC5 in D2 SPNs could detect short DA dips in conjunction with accompanying Golf signals, referred to as the “tone sensing” mode [19,20,24]. In summary, in either case of the type 1 or 5, DA-dip signal must be transmitted to AC if it is processed through Gi signaling for the coincidence detection.

The models of D2 signaling have demonstrated DA-dip detection by AC; however, these models consist of complicated signaling involving many parameters [10,19,24]. It is thus difficult to capture what components of the model parameters, i.e., molecular concentrations and kinetics constants, are essential for the transmission of DA-dip signal. Regarding our D2 LTP model [10], some of the parameters were well constrained by experimental measurements (e.g. Ca2+ signal [25]), while others were determined only based on order estimations or ratio constraints, and the DA-dip detection may also depend on remaining unconstrained factors. One way to examine the parameter dependence is a sensitivity analysis [26], i.e., exhaustive quantification of the changes in model output associated with changes in parameter values. However, this analysis is essentially phenomenological, which cannot address underlying principles. If the concentration dependence of the D2 model is analytically solved, it would provide precise parameter dependence as well as the underlying mechanisms simultaneously.

Further, molecular concentrations in D2 SPNs are altered depending on the age and health condition. In particular, the expression levels of D2R and its counteracting molecule, regulators of G protein signaling type 9–2 (RGS9–2), are both increased with age [27,28]. In schizophrenia, a psychiatric disorder, D2R shows supersensitivity [29,30], whereas the expression level of RGS9–2 is decreased [31,32]. In DYT1 dystonia, a movement disorder, the balance between D2R and RGS9–2 is conversely biased toward RGS9–2 [28]. Those psychiatric/movement disorders are also known to show abnormal LTP in D2 SPNs [1,2,33]. Thus, the D2 LTP model should be examined under the various concentrations of D2R and RGS9–2. However, it is still poorly understood how the concentrations of D2R and RGS9–2 are related to LTP and brain function/dysfunction.

In the present study, we selectively targeted the D2R–Gi–AC part (D2 model) from the previous D2-LTP model to clarify the requirements for the DA-dip detection. We first examined the concentration dependence of five representative molecules (), and revealed that D2R and RGS must be balanced within a narrow concentration range, which was consistent with their age-dependent co-increase observed in an experiment [28]. The balance requirement appeared to be valid under both non-competitive and competitive Gi inhibition of AC, and theoretical analyses further proved the universality of the balance requirements. If the balance was biased toward D2R as in schizophrenia [29-32], the increased D2R activity produced excess amount of Gi-GTP, disrupting the DA-dip detection for LTP. If the balance was biased toward RGS as in the case of DYT1 dystonia [28], the smaller amount of Gi-GTP also disturbed the DA-dip detection. We further discuss the relationship among the D2R–RGS balance, LTP, and disease mechanisms.

Methods

Overview of modeling

We selected the D2R–Gi–AC part from the previous kinetic model of LTP in striatal D2 SPNs (, gray shaded area) [10], and examined the “D2 model” to address whether DA-dip signal was transmitted to AC with a time resolution of 0.5~2 s [3,4]. In our primary target experiment [1], D2 SPNs accepted three types of input stimulation: and a continuous pharmacological activation of adenosine A2A receptors (A2AR; ), tonic DA signal and its pause (DA dip; ), and phasic pre–post pairing (). According to the knowledge of intracellular signaling [9], the tonic signal of DA activates D2R, leading to the GDP/GTP exchange in Gi. The produced Gi-GTP inhibits the activity of AC1, in particular, in the presence of Gβγ () [18]. The Gi-GTP is rapidly hydrolyzed due to GTPase activating proteins, especially, RGS9–2 in the striatum [27,34,35]. Thus, a pause of tonic DA increases the GDP-bound form of Gi (Gi-GDP), which results in the detachment of Gi-GDP from AC1, and the Gi-free AC1 is disinhibited. In contrast, pre–post pairing generates a transient signal of Ca2+/calmodulin (Ca2+-CaM) (), and A2AR activity produces a continuous signal of Golf-GTP (). The Ca2+-CaM and Golf-GTP both bind and activate AC1 in a synergistic manner () [14,15]. The AC1 activity produces cAMP, which activates cAMP-dependent protein kinase (PKA), and enhances the induction of LTP or other neuronal functions. The DA-dip also disinhibits another subtype of AC, AC5 [20,24]. The disinhibited AC5 is activated by the binding of Golf-GTP alone [18,36], and Ca2+-CaM is not required for the activation () [37]. We chose the shared part of both AC1 and AC5 as the D2 model to examine the DA-dip detectability in AC (see the subsection “Readouts”).

The D2 model performs the coincidence detection between DA dip and pre–post pairing.

(A) Three signaling cascades toward AC1 for LTP in D2 SPNs of the striatum [10]. First, adenosine stimulates A2AR, which produces Golf-GTP to activate AC1. Second, basal DA signal leads to the activity of D2R, and then produces a GTP from of Gi (Gi-GTP), which inhibits the AC1 activity. Third, pre–post pairing signal leads to the postsynaptic increase in Ca2+-CaM that stimulates AC1. Signaling in the gray shaded area is modeled in the present D2 model. (B-F) Coincidence detection between DA dip and pre–post pairing. (B) In Iino et al. (2020), A2AR were pharmacologically activated to give a continuous signal of Golf-GTP [1]. (C) DA fibers are optogenetically stimulated tonically at 5Hz (red lines). The tonic stimulation accompanies a 0.5-s pause (DA dip, tDA,delay = 0.5 s), as a representation of unexpected reward omission. (D) Sensory/action signals are represented by pre–post pairing at 10 times and 10 Hz (blue lines), which gives a transient Ca2+ signal. (E) The Golf, Gi, and Ca2+-CaM signals transiently activates AC1 (ACactive). (F) A timing window for DA-dip delay on the maximal amplitudes of ACactive. The D2 model is simulated under the non-competitive binding among Golf, Gi, and Ca2+-CaM (standard non-competitive model; see ).

In the following subsection, we described the basic principles of kinetic formulation, input signals, and readouts of the D2 model. The detailed description of the D2 model is provided in section A in and , and molecular concentrations and kinetic constants are summarized in , respectively. Compared to the previous D2 LTP model, some parameters were updated according to experimental evidence. We thus confirmed that the updated set of parameters provided a time window as shown in the previous D2 LTP model () [10], and it was used as a standard set of parameters. Computer simulation of the D2 model was carried out using MATLAB SimBiology (R2020a; MathWorks). The developed MATLAB code and its SBML-style files are available at the public repository GitHub (https://github.com/urakubo/ModelRP2.git).

Binding and enzymatic reactions

All molecular interactions in the D2 model were represented by binding and enzymatic reactions under the mass assumption [10,38,39]. In the formulation, “:” denotes non-covalent binding between molecules. Gβ:Gγ is exceptionally denoted by Gβγ, because Gβ:Gγ is known to form a tight complex [40]. GTP and GDP forms of GX are represented by GX-GTP and GX-GDP, respectively, and Ca2+/calmodulin is represented by Ca2+-CaM. [X] denotes time-varying molecular concentration, [X]tot denotes the total concentration, and [X]buff denotes the buffered concentration. [X]basal denotes molecular concentration at the basal state (d[X]/dt = 0 under [DA] = [DA]basal), [X]dip denotes molecular concentration at the DA-dip state (d[X]/dt = 0 under [DA] = [DA]dip). [DA]basal and [DA]dip are described in the subsection “Inputs.”

A binding reaction in which A binds to B to form A:B was expressed by the following equation: where kon and koff are the rate constants for the forward and backward reactions, respectively. Here, koff/kon is known as the dissociation constant, Kd. Enzymatic reactions were modeled based on the Michaelis-Menten (MM) equation: where S, E, and P denote substrate, enzyme, and product, respectively, and Km and kcat are the Michaelis constant and product turnover rate, respectively. We did not consider E–S complexes for simplicity, similarly to a previous study [41].

Inputs

In Iino et al. (2020), channelrhodopsin-2-expressed DA fibers were stimulated with 5-Hz light stimulation with a 0.4-s pause, and the pause signal was successfully observed as a DA-dip signal of extracellular DA dynamics [1]. However, the penetration of an observation probe (5~8 μm diameter) might interfere with the DA dynamics, because the cycle of DA release, diffusion, and uptake is known to occur only in a span of ~10 μm [42]. Recent DA observation using an ultrafast fluorescent probe shows the faster dynamics of DA (t1/2 ~ 0.1 s) [43,44]. Thus, based on a preceding model [45], we first simulated the rapid concentration dynamics of DA, [DA], as follows (Figs ): where Tdip is the duration of a DA pause, tDA,delay is the onset time of the DA pause, [DA]dip is the bottom concentration of produced DA dip, and [DA]opto is the amplitude of DA signal by a single light pulse. kcat,DAT[DAT] and [DA]opto were determined so as to give an average concentration of 0.5 μM and a half-valued period of 0.1 s [1,44].

We next characterized the DA-dip signal as a square wave dip of [DA] ():

Here, we set [DA]basal = 0.5 μM and [DA]dip = 0.05 μM unless otherwise stated (). Those levels of DA have been estimated based on a reference experiment () [1]. We finally gave a stepwise-decreasing signal of DA to quantify the DA-dip detection in the response of AC:

In Iino et al. (2020), A2AR was continuously activated to produce Golf-GTP with a constant rate [1]. The Golf-GTP is known to bind to a specific site of AC, while it is autonomously hydrolyzed and detached from the AC [46]. The dynamics of Golf-GTP was simply modeled as a constant concentration of buffered Golf, [Golf]buff ().

We also simulated a pre–post pairing-induced dynamics of Ca2+. The pre–post pairing was constituted of 10-consecutive elemental pairs at 10 Hz ():

Each pairing led to Ca2+ influx via voltage-gated Ca2+ channels (VGCC) and N-Methyl-D-aspartate receptors (NMDAR). The Ca2+ bound to CaM, while free Ca2+ was uptaken by a Ca2+ pump. The Ca2+-CaM further bound to a specific site of AC. Detailed scheme and parameters of the Ca2+ signal are described in Urakubo et al. (2020) [10], and the MATLAB code of the full D2 model is available at the public repository GitHub (https://github.com/urakubo/ModelRP2.git).

Readouts

Golf and Ca2+-CaM synergistically activate AC1 [14,15], and Golf alone sufficiently activate AC5 [37]. The Golf-dependent activity of AC5 is inhibited by Gi, and the Ca2+-CaM-dependent component of AC1 activity is also inhibited by Gi, while the Golf-dependent component of AC1 activity is only weakly inhibited by Gi [18]. We here assumed that the Golf- and Ca2+-CaM-dependent synergistic activity of AC1 was strongly inhibited by Gi. Golf, Gi, and Ca2+-CaM are known to have their specific binding sites at AC1/5 [36,47,48], and molecular dynamics simulation shows that AC5 forms an inactive ternary complex with Golf and Gi [20]. Those pieces of evidence support the idea that Gi non-competitively inhibits AC, i.e., Golf, Gi, and Ca2+-CaM independently bind to AC, and the Gi-binding itself inhibits the enzymatic activity of AC (non-competitive inhibition; ). In contrast, biochemical studies suggest that Gi competitively inhibits AC5 [36,49], i.e., Gi inhibits the AC activity by prohibiting the binding of activators, i.e., Golf and Ca2+-CaM (). In short, the mechanism of Gi inhibition is currently obscure even in the case of well-studied AC5. Considering this situation, we examined two extreme cases of the Gi inhibition: 100% non-competitive binding among Gi, Golf, and Ca2+-CaM, (standard non-competitive model) and 100% competitive binding between Gi and the other two molecules (competitive model), and ACprimed was introduced as a shared readout of AC1 and AC5.

Standard non-competitive model

The concentration of active-state AC, [ACactive], was defined by the following equation (): where , , and denote the binding sites of AC for Gi, Golf, and Ca2+-CaM, respectively, and i = 1,…,9 denote the states of Ca2+-CaM. Here, denotes the concentration of Gi-free AC, i.e., . We then selected the shared part of AC1 and AC5, as a primed state of AC for the activity: where [Golf]buff/{Kd,Golf+[Golf]buff} denotes the binding ratio of AC to Golf. If [Golf]buff is a constant,

Thus, ACprimed is simply under the constant [Golf]buff. All simulation and analyses were conducted on the standard non-competitive model unless otherwise stated.

Competitive model

We defined the concentration of active-state AC, [ACactive], under the situation of 100%-competitive binding between Gi and the other two molecules, Golf and Ca2+-CaM (): where [Golf: AC: Ca2+-CaM] denotes the concentration of AC that binds to both Golf and Ca2+-CaM, but not to Gi. The Golf binding and Gi unbinding are shared requirements for the activities of AC1 and AC5 (). We thus picked up the shared part of AC1 and AC5, as a primed state of AC for the activity, ACprimed, as:

In both non-competitive and competitive models, ACprimed was set to be a readout. ACprimed is a dimensionless value (0 ≤ ACprimed ≤ 1), and 0% ≤ ACprimed ≤ 100% was used in the description. We also introduced ACbasal and ACdip to represent two steady states of ACprimed, i.e., where [DA] = [DA]basal, and where [DA] = [DA]dip.

DA-dip duration dependence in ACprimed

Here, we quantified the DA-dip duration-dependent response of ACprimed using the following equation:

〈ACprimed〉Tdip represents the average increase in ACprimed during the DA-dip period [0, Tdip] in Eq (4).

Concentrations of D2R and RGS under healthy and pathologic conditions

Four pairs of [D2R]tot and [RGS]tot were set to represent a healthy adult, healthy infant, and patients with DYT1 dystonia and schizophrenia. [D2R]tot and [RGS]tot in the standard set of parameters were used for the healthy-adult model, and 0.5 × [D2R]tot and 0.5 × [RGS]tot, were set for the healthy-infant model, because Bonsi et al. have observed a ~0.4-fold simultaneous decrease in their expression levels in the striatum of infant mice (postnatal day 7) [28]. Such a decrease has been seen in other studies as well [50,51]. A parameter set of 4.0 × [D2R]tot and 0.5 × [RGS]tot was used for the schizophrenia model, because D2 blockers need to occupy 70~85% of the total D2R to give clinical effects while avoiding side effects [52,53], and the level of RGS seems to decrease by 10~75% in schizophrenia [31,32]. A parameter set of 0.5 × [D2R]tot and 2.0 × [RGS]tot was used for the dystonia model, because Bonsi et al. have observed a 0.7-fold D2R decrease and 1.6-fold RGS increase in the protein expression levels of the striatal detergent-resistant membranes (DRM) in a mouse model of DYT1 dystonia [28], and another study has shown a ~0.25-fold decrease in the activity of striatal D2R in another mouse model of DYT1 dystonia [54]. Note that those values were determined only for exemplifying purpose, and the actual concentrations depend on the subjects.

Results

Characterization of DA pause as a square wave dip

The updated D2 model was first simulated to confirm the occurrence of a time window for DA-dip delay on the activity of AC1 () [10]. In this simulation, we applied the D2 model to three types of inputs (): a constant signal of Golf (), optogenetically-evoked tonic DA signal with a 0.5-s pause (), and 1-s pre–post pairing (). Those inputs resulted in a transient activity of AC1, ACactive (), and the AC1 activity depended on the delay of DA pause, tDA,delay (), as shown in the previous D2 LTP model [10]. The activity of AC1 is known to produce cAMP, leading to neuronal functions such as synaptic plasticity ().

Among the three input signals, DA pause is particularly interesting. The rapid dynamics of DA resulted in a fluctuating DA signal even under the basal state, and the 0.5-s DA pause appeared to be a minor event (). We thus examined how the DA-pause signal was transmitted to AC, by characterizing it as a square wave dip of DA (), and the normalized level of Gi-free and Golf-bound AC, ACprimed, was observed as a readout (see Methods; ). In the dynamics of ACprimed, the optogenetically-evoked signals of DA were well characterized by the square wave dips of DA (), because the fluctuation in the basal DA signal was attenuated through the D2R−Gi−AC signaling pathway ().

The response of ACprimed was dependent not only on the DA dynamics, but also on the concentrations of other constituent molecules (). If the concentration of RGS, [RGS]tot, was set to be 10 times higher than the standard concentration (), ACprimed showed ~80% of the maximal activity even with the basal DA signal (, light gray line). In this case, there was only small room for further activation. By contrast, if [RGS]tot was set to be 10 times lower than the standard concentration, ACprimed did not show any activity even during the DA dip (, dark gray line). We thus raised a next question: what are requirements on the parameters, i.e., molecular concentration and kinetic constants, for the DA-dip detection?

Amplitudes of ACprimed for DA-dip detection

Then, using a stepwise decreasing signal of DA (Eq (4)), we examined the molecular concentrations required for DA-dip detection (, top). Here, the DA-dip detectability was quantified by two variables: ACbasal and ACdip (). ACbasal denotes the steady-state level of ACprimed under [DA] = [DA]basal (, dark blue; see Methods), and ACdip denotes the steady-state level of ACprimed under [DA] = [DA]dip (, light blue). We observed ACbasal and ACdip if the concentration of one of the five constituent molecules, [D2R]tot, [RGS]tot, [AC]tot, [Gi]tot, and [Golf]buff, was varied ranging from 0.1-fold to 10-fold, while the concentrations of other molecules were kept unchanged ( and ). First, we observed that neither ACbasal nor ACdip was sensitive to [Gi]tot and [Golf]buff if they were higher than 4% of the standard concentration (, second right and right, black lines), while ACbasal and ACdip were both highly sensitive to [D2R]tot and [RGS]tot (, left and second left, black lines). This is because the concentrations of D2R and RGS determined the available amount of Gi-free AC for the activity. Here, for convenience, the regions that satisfies ACbasal < 30% and ACdip > 70% were highlighted as the regions that enabled DA-dip detection (blue and light-blue shaded areas, respectively; ). In the cases of [D2R]tot, [RGS]tot, and [AC]tot, the regions that satisfied ACbasal < 30% (blue) and those that fulfilled ACdip > 70% (light blue) occupied the opposite ends of the concentrations, and the intersection of the two regions satisfying both of them were limited within narrow concentration ranges. Note that the requirement of higher [D2R]tot for smaller ACbasal has been shown as the requirement of higher [DA]basal in a previous study (, left, top) [19], and the requirement of lower [D2R] for higher ACdip has also been shown as the requirement of lower [DA]dip in another study [24].

Because the windows for DA detection in [D2R]tot and [RGS]tot were particularly narrow, we further plotted their two-way relationship in a 2D space (). The DA-dip detectable region in [D2R]tot had a clear positive relationship with that in [RGS]tot; the higher [D2R]tot requires the higher [RGS]tot for the detection of a DA dip (). Very interestingly, Bonsi et al. (2019) have shown that the expression level of striatal RGS9–2 has a similar positive relationship with that of D2R (, black points) in postnatal development of mice, during which both their expressions are increased [28]. They have further examined a DYT1 dystonia model (Tor1a+/−-knock-out mice), and it shows a simultaneous decrease in the gross expression levels of D2R and RGS9–2. However, because DYT1 dystonia alters protein trafficking, the expression level of RGS9–2 is selectively increased in the fraction of DRM where D2Rs is located [28,55], and the perpendicular change appeared in the space of D2R and RGS9–2 (, red points). In the scheme of the D2 model, the disruption of the D2R–RGS balance makes DA dip undetectable, thus DYT1 dystonia cannot show DA-dip dependence in striatal LTP.

The requirement of the D2R–RGS balance appeared under [DA]basal = 0.5 μM and [DA]dip = 0.05 μM. The concentrations of DA were determined based on experimental measurements () [1]; however, at least [DA]basal is known to depend on the specific situation [56,57]. We thus simulated multiple cases of [DA]basal and [DA]dip, and found that they affected the regions of ACbasal < 30% and ACdip > 70%, and the region that satisfied both of them disappeared depending on the pair of [DA]basal and [DA]dip (). Nevertheless, the DA-dip detectable region in [D2R]tot always had a positive relationship with that in [RGS]tot (; left), and the D2R–RGS balance was always required regardless of the pair of [DA]basal and [DA]dip, if we considered analytical solutions that explain these boundaries (, dotted lines; see the subsection “Analytical formulation”).

All the simulations so far were based on the D2 model under 100%-non-competitive binding between Golf and Gi (standard non-competitive model; and section A in ). It is known that Golf stimulates both AC1 and AC5, and Gi inhibits their activities [58]. However, even in well studied AC5, it is still under investigation whether AC is inhibited by Gi through non-competitive inhibition or the allosteric exclusion of Golf binding [20,59]. We thus simulated the D2 model under 100%-competitive binding between Golf and Gi (competitive model; and section A in ). The simulation results were similar to those in the standard non-competitive model (), and the requirements of the D2R–RGS balance also appeared only with a slight bias toward RGS (). The actual Gi inhibition should fall in between the 100%-competitive and non-competitive models. Thus, the requirements of the D2R–RGS balance was robust to the mechanisms of Gi inhibition.

In the simulation, the concentration of Golf affected ACprimed only in the competitive model (Figs , right). This is because the binding of Golf to AC decreased the availability of Golf-free AC for Gi inhibition, and the increase in [Golf]buff led to a simultaneous increase in ACbasal and ACdip, which decreased the dynamic range of ACprimed (, right). Conversely, if [Golf]buff was set to be low, the dynamic range was restored (, right); however, the maximal activity of AC became small. As a result, there appeared to be an optimal [Golf]buff for ΔACactive where (). It was consistent with the simulation results in a previous study [20].

Rapid response of ACprimed for DA-dip detection

The DA-dip detection depends not only on the steady-state levels of ACprimed, but also on its time constant. That is, the time constant of Gi unbinding must be shorter than the DA-dip duration, because otherwise the DA-dip signal would not appear in the change of ACprimed [19]. We thus evaluated it using a variable, T1/2, where T1/2 (> 0 s) denotes the half maximal time of ACprimed after a sudden DA decrease (). Note that exponential fits were not utilized to quantify the increasing time constant because the ACprimed response did not always grow in an exponential manner. We obtained T1/2 using the same set of molecular concentrations as in , and found that T1/2 became less than 0.5 s only if [RGS]tot exceeded a certain level (, red shaded areas). Then, we plotted the DA-dip detectable area in the 2D space of [D2R]tot and [RGS]tot and found a slight dependence on [D2R]tot (). We finally overlaid this plot with the requirements on ACbasal and ACdip (). [D2R]tot and [RGS]tot were needed to fall in the overlapping region (ACbasal < 30%, ACdip > 70%, and T1/2 < 0.5 s) for the DA-dip detection in LTP and/or the change of neuronal excitability (). As expected, this region depended on [DA]basal and [DA]dip (). The higher [DA]basal and lower [DA]dip were required for the dynamics range in the response of ACprimed, while the lower [DA]basal was better for the rapid response (, right). The requirements of the D2R–RGS balance and high [RGS] were preserved regardless of the concentrations of DA.

All the characteristics of T1/2 were also preserved in the competitive model (Figs ), and T1/2 in the competitive model further depended on the binding/unbinding reaction rate of Golf (). This is because ACprimed represented the state of Gi-free and Golf-bound AC, and the Golf binding rate became a time liming process if the Golf binding process was slower than the Gi unbinding process. The reaction rate of Golf did not affect the requirement of the D2R–RGS balance (, center), because it did not affect the steady-state level of Golf, but the dissociation constant, Kd,Golf, should affect it. Similar reaction-rate dependency has been examined by Bruce et al. (2019) [20].

Analytical formulation

The D2 model revealed that DA dip could be detected only in a restricted range in the space of [RGS]tot and [D2R]tot (Fig 3D). However, this requirement has been demonstrated only for the standard set of parameters (), while it should also depend on the other type of parameters, i.e., kinetic constants (). Similarly, we defined the DA-dip detectable region, i.e., ACbasal < 30%, ACdip > 70%, and T1/2 < 0.5 s, mainly for convenience, and they do not necessarily take exactly these values. We thus derived their analytical solutions to examine the overall parameter dependence of ACbasal and ACdip, and T1/2.

Fig 3

DA-dip detectable region appears between two increasing boundaries in the space of D2R and RGS.

(A) Introduction of two measures, ACbasal and ACdip, to quantify DA-dip detectability. ACprimed under the basal DA signal, ACbasal, should be low, whereas that during a DA dip, ACdip, should be high. (B) Concentration requirements for the DA-dip detection under the two measures. While D2R, RGS, AC, Gi, and Golf are targeted, ACbasal and ACdip are measured under the altered concentrations of one of the target molecules. Simulation results (black solid lines) and analytical solutions (blue and light blue dotted lines, Eq (23)) are plotted. ACbasal < 30% and ACdip > 70% are highlighted as the regions necessary for DA-dip detection. (C) ACbasal < 30% and ACdip > 70% in the space of [D2R]tot and [RGS]tot. Analytical isolines of ACbasal = 30% and ACdip = 70% are overlaid. (D) Same as panel C, but the D2 model is based on the competitive binding between Golf and Gi (See ). (E) Age-dependent expression levels of RGS9–2 and D2R proteins (black points) and their altered levels in a mouse model of DYT1 dystonia (Tor1a (+/−), red points) in Bonsi et al. [28]. Data are taken from the DRM fraction of the mouse striatum (Figs 1A and 2C of Bonsi et al. [28]; modified under Creative Commons Attribution 3.0). P7,…, P60 denote mouse postnatal days. Data are normalized by the expression levels in P60 or wild type (WT).

To enable it, we first made simplification on the enzymatic reactions based on the MM formulation. We introduced the catalyst-saturated approximation, d[P]/dt ~ [E]kcat if Km << [S], to the GTP/GDP exchange of Gi-GDP, i.e.,

(a) Km,exch,Gi << [Gi:Gβγ],

where Km,exch,Gi is the Michaelis constant, and [Gi:Gβγ] is the substrate concentration. The constraint (a) was based on the facts that Km,exch,Gi ~ 10 nM () and [Gi]tot ~ 10 μM (). Thus, in almost all the situations, Km,exch,Gi was much lower than [Gi·Gβγ]. Similarly, we introduced the first-order rate approximation, d[P]/dt ~ [E][S]kcat/Km if Km >> [S], to the GTP hydrolysis of Gi, i.e.,

(b) Km,hyd,Gi >> [Gi-GTP],

where Km,hyd,Gi is the Michaelis constant, and [Gi-GTP] is the substrate concentration. The constraint (b) was based on the parameters that Km,hyd,Gi ~ 12 μM and [Gi]tot ~ 10 μM (). Only a subpopulation of [Gi]tot forms [Gi-GTP]; thus, Km,hyd,Gi > [Gi]tot > [Gi-GTP], and Km,hyd,Gi > [Gi-GTP]. We further set the following constraints:

(c) [D2R]·[DA] / [D2R:DA] = Kd,DA, where Kd,DA = koff,DA/kon,DA,

(d) V7 = 0 where V7 = kon,GiGDP[ACisite][Gi-GDP] (see section A in ).

The constraint (c) was set because the binding of DA to D2R rapidly reaches equilibrium (t1/2 ~ 30 ms, ) [11,19], and the constraint (d) is the assumption that Gi-GDP never binds to AC, which is compatible with Kd,GiGTP << Kd,GiGDP () [18]. Note that the constraint (d) was also assumed in the other D2 model as the simultaneous occurrence of Gi-GTP hydrolysis at the time of its detachment from AC5 [19]. Simplification of the D2 model based on the constraints (a-d) is described in section A in .

Based on the constraints (a-d), we successfully obtained the steady-state ratio of Golf-bound and Gi-free forms of AC, ACbasal and ACdip, as follows (Eqs (S53) and (S61) in section B in ): where k = kcat,hyd,Gi/Km,hyd,Gi·[RGS]tot, k = kcat,exch [DA]basal / ([DA]basal + Kd,DA), and Golf = [Golf]buff/Kd,Golf (see ). ACdip was also obtained by replacing k with k. The analytical ACbasal and ACdip were both well fitted with the simulated ACbasal and ACdip, respectively (Figs ; blue and light-blue dotted lines), and the analytical ACbasal and ACdip were the functions of [RGS]tot, [D2R]tot, and [AC]tot, but not the function of [Gi]tot. This is because the constraint (a) simplified the Gi-dependent V1 (Eqs (5), (S1), and (S5), section A in ) into a Gi-independent form (Eqs (S43) and (S44), section A in ). The constraint (a), Km,exch,Gi << [Gi:Gβγ], was invalid in the small range of [Gi]tot; thus, the analytical and simulated ACbasal were mismatched under [Gi]tot < ~0.04 μM (, second right).

Eq (24) well described the simulated ACbasal and ACdip. However, even with the constraints (a–d), Eq (24) was still too complicated to provide an intuitive understanding. We thus further simplified Eq (13) by considering its asymptotic functions under k << koff,GiGDP, koff,GiGTP,, i.e., and under k >> koff,GiGDP, koff,GiGTP,, i.e.,

Here, k << koff,GiGDP, koff,GiGTP denotes the situation that Gi-GTP hydrolysis is much slower than the Gi-dissociation rate from AC, and k >> koff,GiGDP, koff,GiGTP denotes the reversed situation. Eqs (14) and (15) denote the isolines of ACbasal, i.e., [D2R] = f([RGS]tot; ACbasal), and they are both linear functions of [RGS]tot (∝ k); thus, ACbasal can be characterized by a transition between the two linear functions (). The gradients and D2R-intercept depend on ACbasal, [DA]basal, [AC]tot, and kinetic constants. One of the asymptotic functions, Eq (15), appeared to be curved in the logarithmic space (, light-blue dashed line), but it is due to the D2R-intercept, and indeed linear in the linear space. The other asymptotic function (Eq (14); ) appeared to be linear, because it has no D2R- or RGS-intercepts. ACdip was also derived where k was replaced with k, and showed the same characteristics ().

DA-dip detectable regions are characterized by asymptotic functions.

(A, B) Analytical isolines of ACbasal and ACdip (ACbasal = 30%, blue dotted line; ACdip = 70%, light-blue dotted lines) are characterized by a transition between two linear functions, Eq (14) and Eq (15) (gray and light-blue dashed lines, respectively). (C) Analytical iso-T1/2 line (T1/2 = 0.5 s, red dotted line) is approximated by the function Eq (19) (gray dashed line) if it is located within ACbasal < 30% and ACdip > 70% (blue shaded area).

We next considered the half maximal time of ACprimed(t), T1/2, where ACprimed(t = T1/2) = (ACbasal + ACdip) / 2 after a sudden decrease of [DA] at t = 0 s (Eq (5); ). Unfortunately, the dynamics of ACprimed(t) was governed by exponentials of exponential functions, which has a complicated form (gamma functions) in the analytical solution. To avoid it, we further introduced three additional constraints to the D2 model:

where Kd,GiGTP = koff,GiGTP /kon,GiGTP,, Kd,GiGDP = koff,GiGDP /kon,GiGDP, and Kd,Golf = koff,Golf /kon,Golf. The constraints (e-g) assume that the AC-Gi and AC-Golf bindings rapidly reach equilibrium, and the dynamics of ACprimed(t) is governed by Gi-GTP hydrolysis. This operation is called the rapid equilibrium assumption in enzymology. We denoted ACprimed and T1/2 under the constraints (e-g) as and , respectively, and was obtained as: where , and and were given by:

(e) [ACisite]·[Gi-GTP]/[ACisite:Gi-GTP] = Kd,GiGTP,

(f) [ACisite]·[Gi-GDP]/[ ACisite:Gi-GDP] = Kd,GiGDP,

(g) [AColfsite]·[Golf-GTP]/[ACisite:Golf] = Kd,Golf (only for the competitive model),

The derivations of Eqs (16), (17), and (18) are described in section C in . The analytical was well fitted with the simulated T1/2 (red dotted line, Figs ), showing the validity of the analytical formulation. The constraints (e, f) are not valid if the hydrolysis rate of Gi-GTP, k, is much larger than the rate of AC-Gi unbinding, and this invalidity appeared in the high [RGS]tot (, second left). Similarly, the constraint (g) is not valid if k is much larger than the rate of AC-Golf unbinding, as it also appeared (, right).

We finally simplified Eq (27) by considering if :

Here, represents an ideal situation, i.e., ACbasal = 0% and ACdip = 100%, or ACbasal + ACdip = 100%. The asymptotic (Fig 4C, gray dashed line) was almost the same as the analytical one (, red dotted line) if ACbasal < 30% and ACdip > 70% (, blue shaded area). We thus obtained an analytically closed form of the isoline of T1/2 in the DA-dip detectable region. In Eq (19), is the decreasing function of [RGS]tot (∝ k). Thus, [RGS] must be higher than the isoline for the shorter DA-dip detection.

Fig 4

DA dip detection requires a certain concentration of RGS.

(A) The third measure to quantify DA-dip detectability. Sudden decrease in the basal DA signal disinhibits AC with a half-maximal time, T1/2, which needs to be small for detection of a short (~0.5 s) DA-dip. (B) Concentration dependence of T1/2. Five molecules were targeted, and the simulation results (black solid lines) and analytical solutions (red dotted line, Eq (24)) were plotted. T1/2 < 0.5 s were highlighted in pink as the region that allows detection of the DA dip. (C) T1/2 < 0.5 s (pink) in the space of [D2R] and [RGS]. (D) Same as panel C, but the DA model was based on the competitive binding between Golf and Gi (See ). (E) The area of T1/2 < 0.5 s (pink shaded area in (C)) was overlapped with the region that satisfies both ACbasal < 30% and ACdip > 70% (blue shaded area). The ~0.5-s DA dips can only be detected in the overlapping area.

In summary, we derived the analytical forms of ACbasal, ACdip, and T1/2. ACbasal and ACdip were both characterized by the transition between two asymptotic linear functions (Eqs (14) and (15); ), and T1/2 also had an approximate closed form (Eqs (16)–(18); ). The boundaries of ACbasal and ACdip were linear regardless of the model parameters as far as they satisfied the constraints (a–d), and such a linear relationship was also seen in developing mice () [28].

AC-concentration dependence

The DA-dip detectability also depended on [AC]tot (Figs , center). We thus examined two-way relationships between [AC]tot and [D2R]tot () as well as between [AC]tot and [RGS]tot (). The DA-dip detectable region in [AC]tot showed a positive relationship with that in [D2R]tot (), and a negative relationship with that in [RGS]tot (). These [AC]tot dependences originated from the sequestration of Gi-GTP for the inhibition of AC. If Gi-GTP was set not to be sequestrated by AC, i.e., V4, …, V7 were removed only from Eq (S3) (), the [AC]tot dependences were completely eliminated (), while the relationship between [RGS]tot and [D2R]tot was still preserved (). Furthermore, the lower [AC]tot was, the lower [AC]tot dependences appeared () because the smaller amount of Gi-GTP was sequestered by AC.

Dynamics of ACprimed under psychiatric/movement disorders

Healthy mice show an age-dependent simultaneous increase in the levels of striatal D2R and RGS9–2 during postnatal development () [28,50,51]. This D2R–RGS balance is known to be disrupted in psychiatric/movement disorders. Schizophrenia patients often show a supersensitivity of D2R and/or an increase in DA [29,60], and mice in a corresponding mouse model show the decrease in the gross expression level of RGS9–2 in the striatum [31,32]. The altered levels of D2R and RGS disturb the intracellular signaling of SPNs [61], and such abnormal signaling and subsequent striatal dysfunction is expected to cause these psychological symptoms [62]. By contrast, a mouse model of DYT1 dystonia shows the decrease and increase in D2R and RGS9–2, respectively, in the co-existent fraction of D2R and RGS9–2 (DRM; , red points) [28], and the increased GTP hydrolysis may be related to involuntary movements. Thus, we explored the DA-dip detectability under such D2R–RGS imbalances ().

Fig 2

Characterization of optogenetically-evoked DA dynamics as a square wave dip.

(A) Simulation of optogenetically-evoked DA dynamics (5 Hz) and their pauses for 0.5 s, 1.0 s, and 2.0 s (top, middle, and bottom, respectively). The tonic optogenetic stimulation is set to generate 0.5 μM DA on average, and the DA pause is set to give 0.05 μM DA at bottom levels (see Methods). (B) Characterization of the optogenetic DA signals as square wave dips. (C) Introduction of ACprimed as the Gi-free and Golf-bound state of AC. Further Ca2+ stimulation leads to the activity of AC1. In the case of AC5, ACprimed corresponds to a normalized activity of AC. (D) Dynamics of ACprimed in response to the optogenetically-evoked DA signals. (E) Dynamics of ACprimed in response to the square-wave DA dips. (F) Appropriate range of [RGS]tot is required for the response of ACprimed. Square-wave DA dips are given with a duration of 1 s. ACprimed reaches almost the maximal level even with the basal DA signal if [RGS]tot is set to be 10 times higher than the standard concentration (9 μM, light gray line), and ACprimed does not show any increase if [RGS]tot is 10 times lower than the standard concentration (0.09 μM, dark gray line).

We first confirmed that the D2 model with the standard set of parameters (healthy-adult model; , gray circle) successfully detected short DA dips (~0.5 s; ). The ACprimed was sufficiently low under the steady-state Gi-GTP (; ACbasal = 19% where [G-GTP]basal = 0.24 μM), whereas the ACprimed was sufficiently and rapidly increased during DA dips (; ACdip = 78% where [G-GTP]dip = 0.016 μM). Then we set a condition for healthy infant that had 0.5 × [D2R]tot and 0.5 × [RGS]tot (, green circle), and found the similar response of ACprimed to the DA dips (; ACbasal = 17% where [G-GTP]basal = 0.23 μM; ACdip = 78% where [G-GTP]dip = 0.014 μM), although it showed lower sensitivity to the shorter dips. On the contrary, a schizophrenia model that had 4.0 × [D2R]tot and 0.5 × [RGS]tot (, blue circle) did not detect the DA dips because of the excessive amount of Gi-GTP. It caused the excessively low ACprimed at the basal state and its lower and slower increase (; ACbasal = 2% where [G-GTP]basal = 2.8 μM; ACdip = 21% where [G-GTP]dip = 0.19 μM). Similarly, a dystonia model that had 0.5 × [D2R] and 2.0 × [RGS]tot (, red circle) showed high ACbasal and weak responses to the DA dips (Fig 6E; ACbasal = 57% where [G-GTP]basal = 0.050 μM; ACdip = 94% where [G-GTP]dip = 0.0043 μM). In summary, the D2 models for healthy adult and healthy infant responded to DA dips with sufficiently large dynamic ranges (, gray and green lines, respectively), although the healthy-infant model was less sensitive to shorter DA dips. The schizophrenia model showed the slower and smaller responses of AC against DA dips (, blue line), and the dystonia model showed the rapid but small response (, red line).

Fig 6

DA dip is undetectable under the pathologic imbalance between D2R and RGS.

(A) Levels of D2R and RGS under healthy and pathologic conditions. Healthy infant mice show lower levels of D2R and RGS (green circle) than healthy adult mice do (standard model; gray circle, see Fig 2D) [28]. Schizophrenia patients are known to show higher and lower activities of D2R and RGS, respectively (blue circle) [29,31,32], and a mouse model of DYT1 dystonia shows the opposite changes in their levels (red circle, Fig 2D) [28]. Gray arrow denotes the hypothetical trajectory of changes in RGS and D2R levels if the levels of RGS and D2R are increased from the dystonic levels by the same ratio as in the increasing ratio under the development of healthy mice, i.e., ([D2R], [RGS]) = μ × ([D2R]tot, [RGS]tot) + ([D2R]Dystonia, [RGS]Dystonia) where [D2R]Dystonia and [RGS]Dystonia are the dystonic levels of D2R and RGS, and μ is the constant (> 0; see Discussion). (B–E) Dynamics of [G-GTP] and ACprimed against DA dips with a variety of durations (0.01 s, 0.02 s, …, 2.56 s). (F) Summary of the DA-dip duration dependences quantified using Tdip− ACbasal (see Methods).

Apparently, their DA-dip detectability was disrupted by the excessive or insufficient levels of Gi-GTP at the basal states (, center), and they can be understood using a schematic picture (). At the basal state, DA-dependent D2R activity works like a faucet (tap) that provides Gi-GTP with a constant rate (, left). The production rate, V1, is almost independent of [Gi:Gβγ] because [Gi]tot (~9 μM) is much higher than Km,exch,Gi (~0.01 μM, ; constraint (a)). The produced Gi-GTP is drained through RGS, where the draining rate, V2 + V8, is proportional to [G-GTP] + [AC:G-GTP] (Eqs (S40) and (S41), section A in ) because the GTP hydrolysis is unsaturated (constraint (b)). The difference in these rates makes a pool of Gi-GTP (, left). In the healthy-adult model, the pooled Gi-GTP almost completely inhibits AC, which is represented by a ball (). Here, the Gi-soaked part of AC is inhibited, and the Gi-free part is disinhibited (, inset). During the DA-dip, the production of Gi-GTP almost stops (, right), and the level of Gi-GTP is decreased for the activation of AC for LTP and/or the change of neuronal excitability. In the schizophrenia model, the larger [D2R]tot makes the production rate of Gi-GTP, V1, much higher than its draining rate, V2 + V8 (S8D Fig), leading to the complete inhibition of AC. In the dystonia model, the production rate is conversely lower than the draining rate (). These imbalances caused the decrease the levels of DA-dip detection ().

Time windows for ACactive under psychiatric/movement disorders

Finally, we examined the timing detection for AC1 in D2 SPNs in the healthy adult, healthy infant, schizophrenia, and DYT1 dystonia (). Compared to the healthy-adult model (the standard set of parameters, and ), the healthy-infant model showed the smaller level of DA-dip detection in ACacitve because the DA dips had a short period of 0.5 s (). The schizophrenia model did not show any increase in ACactive (), while the dystonia model detected the DA-dip, but the timing-independent component of ACacitve was high (). Because of the high timing-independent ACactive, the peak-to-basal ratio of ACactive in the DYT1 dystonia became the lowest (), indicating the small signal-to-noise ratio even if the activity of the downstream signaling can adapt to the high timing-independent levels of ACactive.

Decreased efficacy of the coincidence detection in pathologic/movement disorders.

(A-D) Time windows for DA-dip delay against pre–post pairing on the activity of AC1, [ACactive], in the models of healthy adult (A), healthy infant (B), schizophrenia (C), and DYT1 dystonia (D). For each DA-di delay, a 0.5-s DA dip is paired with 1-s pre-post pairing. Obtained maximal [ACactive] are normalized by the total concentrations of AC1, [AC]tot. (E) Maximal amplitudes of ACactive normalized by the basal level.

Following a previous study [24], we also simulated AC5 coincidence detection between a phasic burst of adenosine (duration: 1 s) and a DA dip/burst (duration: 2 s; ). The time windows for the delay of DA dip/burst were similar to those in AC1 (), and the dystonia model rather showed DA-burst detection, because the longer DA burst completely inhibited the increase in ACactive due to the adenosine burst (, thin lines).

Discussion

Here, we showed that a D2R–RGS balance was required for the detection of DA dips in D2 SPNs, where the DA-dip detection is important for the LTP and/or change of neuronal excitability. High-level RGS was further required for the detection of short (~0.5 s) DA dips. These requirements were satisfied in healthy development but disrupted in our models of schizophrenia and DYT1 dystonia.

The D2 model highlights the importance of RGS. In particular, a type of RGS, RGS9–2, is known to be specifically and abundantly expressed in the striatum [27,34,63]. The expression of RGS9–2 is co-regulated with that of D2R [28,64], and RGS9–2 itself is associated and colocalized with D2R [65]. The present study provides a reason why RGS9–2 is so tightly coupled with D2R. It is necessary for the reliable detection of short DA dips. The abundance of RGS9 is also seen in the retinal phototransduction processes [66], and the rapid hydrolysis of transduction via RGS9 is vital for the rapid visual processing both in rods and cones [67]. Although G-protein signaling is often discussed from its chronic aspects [68], it can also serve for sub-second information processing. In addition, a variety of RGS subtypes are expressed in the brain in a region-specific manner [63,69]. RGS8 and RGS10 are characteristically expressed in cerebellar Purkinje cells as well as in the stratum granulosum of the hippocampus, respectively [63]. They may also play roles for region-specific information processing.

Nair et al. have already demonstrated DA-dip detection by AC5 in the model of D2 SPNs [19,24], in which a set of model parameters were determined based on observed molecular activities. The D2 signaling model has shown the requirement of a longer DA dip for the AC5 response (c.f., ) [19]. It has also shown the requirement of higher concentration of basal DA for the AC5 response, which was shown as the region of ACbasal < 30% in the D2 model (, left). Further, the D2 signaling model has shown the requirement of lower concentration of DA during a DA dip [24], as shown in ACdip > 70% of the D2 model (, left). However, the D2 signaling model does not incorporate RGS, but the GTP hydrolysis of Gi is assumed to occur with a constant rate. Here, we extensively examined the concentration dependence of five target molecules (), and found the requirement of the D2R–RGS balance for the DA dip detection, together with the analytical solutions that prove the universality of the balance requirement. Nair et al. have also demonstrated a possible coincidence detection between DA durst and acetylcholine (ACh) dip, the latter of which leads to the deactivation of a Gi-coupled receptor (muscarinic M4 ACh receptor; M4R) [24]. RGS should also play a critical role in such other types of Gi inactivation-driven events.

In the present study, we derived analytical solutions for the DA-dip detection. They were valid in a wide range of parameters, as far as the D2 model satisfied the constraints (a–g). In particular, the constraints (a, b) were related to famous simplifications of the MM formulation. The constraint (a) enabled the catalyst-saturated approximation, and the constraint (b) enabled the first-order rate approximation. They contributed not only to the simplification but also to stable DA-dip detection. For example, if the Gi-GTP hydrolysis were conversely saturated, the GTP hydrolysis rate, V2 + V8 (Eq (S65) in section C in ), would be independent of [GTP] (= [Gi-GTP] + [AC:Gi-GTP]), but only dependent on [RGS]. Then, [GTP] would become an increasing or decreasing function depending only on [RGS], and Gi took only one of two stable states, i.e., fully hydrolyzed or non-hydrolyzed states. ACprimed could not show a rapid increase from the fully non-hydrolyzed state of Gi (i.e., 100% Gi-GTP), nor detect a short DA-dip. The kinetics constants and molecular concentrations seem to be arranged for effective DA-dip detection, which may be a design principle in the biological system, as often quoted in the field of systems biology [70].

In schizophrenia, the balance between D2R and RGS activities is considered to be biased toward D2R (), because D2R blockers works as antipsychotic drugs for schizophrenia [52,53], and schizophrenia shows show a supersensitivity of D2R and/or an increase in striatal DA [29,60] as well as a decrease in striatal RGS9–2 [31,32]. Our D2 model predicts that the excessive activity of postsynaptic D2R completely inhibits AC, disabling the detection of any DA dips as well as subsequent LTP (). Correspondingly, chronic upregulation of D2R is known to result in the rewiring of connections first from the striatum to the globus pallidus (GPe) [71] and next within the frontal cortex [61,62]. Such rewiring may lead to the persistent symptoms of schizophrenia. D2R blockers are widely used for the medication of schizophrenia [52], but the present study predicts that the excess blockage of D2R again disables DA-dip detection. Indeed, excessive treatment with the D2R blockers is known to induce extrapyramidal side effects including tardive dyskinesia [31,72], a movement disorder that shares the similar symptoms with DYT1 dystonia [73] whose mouse model shows the D2R–RGS imbalance toward RGS (Fig 2D).

DYT1 dystonia is a hyperkinetic movement disorder, which originates from a Δgag mutation in the TOR1A gene. It decreases the gross expression levels of both D2R and RGS9–2 [73,74], but the level of RGS9-2 is selectively increased in the DRM where D2R is located, and the D2R–RGS balance is biased toward RGS. In our D2 model, the excess RGS first decreases the level of Gi-GTP, and then increases the AC activity for cAMP production. Excess cAMP is expected to be a cause of abnormal LTP [33,54,75,76]. Such synaptic abnormality leads to the abnormal neuronal activities in GPe [77] and thalamus [78], and they may cause muscle contractions and irregular tremors [79]. Notwithstanding with these reports, DA drugs do not provide clinical benefit [80], and the viral overexpression of RGS9–2 rather restores normal neuronal electrophysiology [28]. These inconsistent observations are reconciled in the scheme of the D2 model as follows. First, DA drugs should not work because they cannot accurately control the level of D2R activity to counteract with RGS (). Next, the overexpression of RGS9–2 also rescues the decreased expression level of D2R [28], and the increased levels of RGS and D2R can be balanced within the DA-dip detectable region. Indeed, this rebalancing would be realized if the levels of RGS and D2R were increased from the dystonic levels by the same ratio as in normal development (, gray arrow). The overexpression of RGS9–2 thus can restore normal neuronal activity [28]. Together, the D2 model explains many aspects of DYT1 dystonia in the space of D2R and RGS, including the aspects of healthy development and schizophrenia.

The balance requirement between D2R and RGS itself has been recognized in the field of psychiatry [32,53], and here we re-formalized it as requirements for the DA-dip detection for LTP in D2 SPNs. Note that pairs of [D2R]tot and [RGS]tot for the healthy and pathologic conditions were determined only for exemplifying purpose (Fig 6A), and their absolute values are case-dependent even in mice experiments (see Methods). In particular, schizophrenia is caused by both environmental and genetic factors, and many combinations of the genes affect schizophrenia [81]. Thus, the pairs of effective [D2R]tot and [RGS]tot are expected to span a wide range depending on the patients, even if it is caused by excessive D2R activity.

The other major movement disorder, L-DOPA induced dyskinesia (LID), could also be related to the D2R–RGS imbalance. The depletion of DA is associated with Parkinson’s disease, and the DA depletion is restored by administration of a DA precursor, L-DOPA [82]. The L-DOPA medication leads to varying levels of basal DA, which is followed by the co-adaptation of D2R and RGS9–2 [28,65]. The incompleteness of the adaptation may be a cause of LID. Indeed, RGS9–2 knock-out (KO) develops LID [65], and the overexpression of RGS9–2 diminishes its involuntary movements [35]. LID also shows abnormal LTP [13,83]. However, other recent studies have shown that LID are primarily linked to the abnormal activity of D1 SPNs [84-86], and the dissection of mixed effects is necessary to understand this currently untreatable disease.

In the present study, we showed the D2R–RGS balance as a shared requirement of both AC1 and AC5. As it is known, AC5 constitutes ~80% of the total cAMP levels in SPNs [17], and contributes to synaptic plasticity [21]. Considering its abundance, AC5 should also perform the coincidence detection in some forms of synaptic plasticity as well as the plastic changes in neuronal excitability [22,23]. The AC5 coincidence detection is predicted to occur between Golf and Gi () [20,24]. However, in our primary target experiment, an AC1-specific inhibitor fully suppressed AC-dependent cAMP signaling, and DA signal alone did not activate the D1R–Golf–AC–cAMP signaling, but further pre–post pairing (presumably Ca2+) was required for the cAMP signal [5]. These observations support the role of Ca2+-sensitive AC1, and further, the AC1-based D2 LTP model successfully predicted the narrow time window (~2 s) () [10]. Such Ca2+-sensitive ACs have been known as a coincidence detector for the classical conditioning in Aplysia and Drosophila [15,87-89]; thus, the Ca2+ requirement may be evolutionarily conserved [90]. The cause of missing AC5 signal in the target experiment is unknown [5], while an AC5-based model predicts much longer time window (~10 s) [91]. In general, the expression of AC1 in the striatum is known to be decreased with the development, while the Ca2+-CaM-dependent activity of AC persists [92].

AC1 and AC5 are both expressed in the cortex and hippocampus [16]. However, these regions only show weak DA signal because of the small numbers of DA fibers [93,94], and the DA signal likely plays more modulatory roles with longer time constants (~10 min) [6,95]. Rather, in these brain regions, adenosine can show persistent and transient signals with a time range of ~5 s [96,97]. Adenosine A1 receptors (A1AR) then produce Gi/o-GTP, while A2AR produce Gs/olf-GTP [98,99]. Thus, adenosine dip may lead to the reduction of Gi/o-GTP via A1AR, similarly to the DA-dip signal through D2R. In the striatum, cholinergic interneurons also show tonic and ~0.1-s phasic activities [100], and the ACh release stimulates postsynaptic M4R in D1 SPNs for Gi-GTP [20,24,101]. The produced Gi-GTP may also activate AC, similar to DA-dip signal. Further, DA and ACh signals are known to interact with each other. The activity of DA fibers modulates the activity of cholinergic interneurons via co-released glutamate [102-104], and ACh conversely stimulates the DA fibers [105]. The ACh–DA interaction may enhance the coincidence detection in vivo [106].

An important limitation of the current D2 model is the mass assumption. In addition to well-known dimerization of D2R and A2AR [107], recent experiments have suggested that AC5, Gi, and Golf also participate in the formation of a macromolecular complex that accelerated reactions among the constituents [108,109]. Such a macromolecular complex no longer obeys the mass assumption; thus, their reactions cannot be formulated by Eqs (1) and (2), but should be described in a more mechanistic manner. In the macromolecular complex, the DA signal would be more rapidly and directly transferred to the activity of AC, the short DA-dip might be more easily detected by AC. This is an attractive scenario and an important direction for future studies. Nevertheless, the D2 model should first be built based on the mass assumption, especially before the validation of its dominance in the physiological condition. As an example, the D2R:Gi:Gβγ pre-coupled complex has been discovered and examined [110-112], but it seems not to be dominant in the physiological condition, from the viewpoint of the affinity of D2R for DA (Kd,DA in pre-coupled complex, 25 nM; Kd,DA in the physiological condition, ~10 μM) [1,19,43,113,114]. In addition, even if such a macromolecular complex is found to work dominantly, the current D2 model will serve as a good reference.

The present study has addressed the DA-dip detection in D2 SPNs. Similarly, phasic DA bursts are known to trigger LTP in D1 SPNs [5]. Yet, do the similar requirements exist in D1 SPNs? In D1 SPNs, DA binds to D1R, and the DA-bound D1R produces Golf-GTP. However, it is known that the hydrolysis of Golf-GTP is not mediated by any of the RGS, but occurs in an autocatalytic manner [46]; thus, we cannot ask the same question about D1 SPNs. Instead, we can raise another important question: how can the phasic bursts of DA trigger LTP in D1 SPNs, despite the fact that the basal DA signal alone does not trigger the LTP [1]? This question implies the existence of an adaptation mechanism for basal DA signal. A possible candidate is a feedback loop that involves a specific phosphorylation of D1R [115]. The DA-bound D1R produces Golf-GTP, which activates AC. The activated AC produces cAMP and then the active form of PKA. The PKA in turn phosphorylates D1R, leading to a ~100-fold decrease in the activity of D1R, with a time constant of ~10 min [115]. Thus, the D1R–AC–PKA signaling constitutes a negative feedback loop in which D1R responds solely to the phasic DA signal. Simulation of such a feedback loop will address the stability of D1 SPNs against the fluctuation of basal DA signal. However, it requires a different model with a different level of abstraction. Thus, this topic should be addressed in other future studies.

Schematic of the D2 model (standard non-competitive model).

Arrows denote the first order reactions or enzymatic reactions, and their fluxes are denoted by V1,…, V12. (A) Reactions of the D2R–Gi–AC part. The input DA (circled) first regulates a Gi-protein cycle (top), then Gi-AC binding cycle (bottom). Gi-GTP/ Gi-GDP binds to a specific site of AC (ACisite). (B) Under the scheme of non-competitive binding, Golf, Gi, and Ca2+-CaM independently interact with their specific sites of AC (AColfsite, ACisite, and ACCaMsite, respectively). Simultaneous binding of Golf and Ca2+-CaM is required for the activity of AC1 (top), and the binding of Golf alone leads to the activity of AC5 (bottom). The binding of Gi inhibits the activities of both AC1 and AC5. Reactions with dashed arrows are the same as the reactions V4,…, V7 indicated in panel A.

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Schematic of the D2 model (Competitive model).

Arrows denote the first order reactions or enzymatic reactions, and their fluxes are denoted by V1,…, V20. (A) Reactions of the D2R–Gi–AC part. Same as S1A Fig, but the Gi binding to AC disables the binding of Golf and Ca2+-CaM. (B) If AC is free from Gi, the AC can interact with Golf and Ca2+-CaM. Simultaneous binding of Golf and Ca2+-CaM is required for the activity of AC1 (top), and the binding of Golf alone sufficiently activates AC5 (bottom). Under the competitive binding, ACprimed corresponds to the levels of yellow-bordered states. Reactions with dashed arrows are the same as the reactions V4,…, V7 indicated in panel A.

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Dynamics of molecular activities in response to a DA dip.

Red arrows denote the onset of a 1-s DA dip. (A) Observed molecules. (B) Optogenetically-evoked dynamics of DA. (C–I) Subsequent molecular activities. (J) Square-drop signal as a representative of DA dip. (K-Q) Subsequent molecular activities.

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DA-dip detectable region in the competitive model.

(A) ACprimed is observed under the stepwise decreasing signal of DA. (B) Concentration dependence of the steady state levels of ACprimed, ACbasal and ACdip in the competitive model. (C) Golf-dependence of ΔACactivity in the non-competitive and competitive models (see Eqs (7) and (10)). where and denote the activities of AC5 under [DA]basal and [DA]dip, respectively. The biphasic Golf concentration dependence appears only in the competitive model, as shown in Bruce et al. (2019) [20]. (D) Parameter dependence of T1/2 in the competitive model.

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Requirement of D2R–RGS balance under various levels of [DA]basal and [DA]dip.

(A) [DA]basal is set to be 0.125, 0.25, …, 4 μM (left), and the areas of ACbasal < 30% (top) and T1/2 < 0.5 s (bottom) are plotted in the space of [RGS]tot and [DA]tot. Dotted lines denote the analytical solutions, and colored areas denote the simulation results. (B) [DA]dip is set to be 0.0125, 0.025, …, 0.4 μM (left), and the areas of ACdip > 70% (top) and T1/2 < 0.5 s (bottom) are plotted.

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Reaction rate of Golf affects T1/2 only in the competitive model.

Here, the binding/unbinding reaction rate of Golf, τ = 1/(kon,Golf[Golf]buff+koff,Golf), is subjected to change, whereas Kd,Golf (= koff,Golf / kon,Golf) and [Golf]buff are kept constant. (A, B) ACbasal, ACdip, and T1/2 in the standard non-competitive model (A), and competitive model (B). Green dotted line in panel B denotes τ = 0.5/log 2 (T1/2 = 0.5 s).

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AC-concentration dependence is based on the sequestration of Gi by AC.

(A–G) Areas of ACbasal < 30% (blue), ACdip > 70% (light blue), and T1/2 < 0.5 s (pink) in the standard non-competitive model (A). These areas are plotted in the spaces of [D2R]tot versus [RGS]tot (B, E), [D2R]tot versus [AC]tot (C, F), and [RGS]tot versus [AC]tot (D, G). Dotted lines denote their analytical solutions. (H–N) Same as panels A–G, but the standard D2 model is modified so that AC sequesters Gi, i.e., V4, V5, V6, and V7 are removed only from Eq (S3) but not from Eq (S12).

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Schematic of the D2-model dynamics.

(A) At the steady state (left), DA-bound D2R provides a constant flux of Gi-GTP (V1; see section A in ), while the Gi-GTP is drained through RGS with a speed (V2 + V8) that is proportional to the amount of Gi-GTP. Difference in the fluxes generates a pool of Gi-GTP that inhibits AC, which is represented by a sunk ball (AC). During the period of a DA dip (right), the supply of Gi-GTP nearly completely stops, and the pooled Gi-GTP is drained rapidly. AC is then disinhibited to be activated by Golf and Ca2+-CaM. (B-E) The Gi inhibition of AC in the healthy adult, healthy infant, schizophrenia, and dystonia, all of which are described in . (C) In the healthy-infant model, the influx and efflux of Gi-GTP are both small, and the smaller efflux causes delayed disinhibition of AC during a DA dip. (D) In the schizophrenia model, hyperactive D2R provides a larger amount of Gi-GTP, thus AC is fully inhibited and its disinhibition is delayed. (E) The larger efflux of Gi-GTP in the dystonia model results in the chronic activity of AC.

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AC5 performs the coincidence detection between Golf and Gi.

A transient burst in adenosine (thus Golf) coincides with a DA dip/burst with a variety of delay times. (A) Schematic of AC5 signaling. (B) Adenosine burst as a square wave of Golf (basal, 0.16 μM; burst, 0.8 μM; duration, 1 s). (C) Square-wave dip and burst of DA (basal, 0.5 μM; dip, 0.05 μM; burst, 2 μM; duration, 2 s). (D-G) Example traces of the AC5 activities (ACactive; top), and DA-dip/burst delay dependence of the maximal ACactive (bottom) in the healthy-adult, healthy-infant, schizophrenia, and dystonia models. In the example traces, adenosine and DA signals were given during the periods indicated by blue and red bars, respectively. Similar simulation (adenosine versus transient DA dips) has been conducted by Nair et al. (2015) [24].

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Molecular concentrations.

Densities of membrane molecules should have the unit of membrane area (/μm2), but not volume (μM). However, many of the referenced experiments/simulations have described them under homogenate conditions; therefore, we also adopted volume concentration for consistency. Note that Xarea /μm2 ~ Xvolume μM × 20, because a spherical spine with a radius rspine ~ 0.1 μm has the number of surface molecules Xarea × (4πrspine2), and the number of cytosolic molecules is Xvolume × 10−6NA × [4πrspine3/3 × 10−15] where NA = 6.02 ×1023 (Avogadro constant).

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Reaction rate constants.

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Model definition and analytical derivations.

(A) Definition of the D2 model. (B) Derivation of analytical ACbasal and ACdip. (C) Derivation of analytical T1/2.

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