Overview of modeling

We selected the D2R–G_i–AC part from the previous kinetic model of LTP in striatal D2 SPNs (Fig 1A, 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; Fig 1B), tonic DA signal and its pause (DA dip; Fig 1C), and phasic pre–post pairing (Fig 1D). According to the knowledge of intracellular signaling [9], the tonic signal of DA activates D2R, leading to the GDP/GTP exchange in G_i. The produced G_i-GTP inhibits the activity of AC1, in particular, in the presence of G_βγ (S1 Table) [18]. The G_i-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 G_i (G_i-GDP), which results in the detachment of G_i-GDP from AC1, and the G_i-free AC1 is disinhibited. In contrast, pre–post pairing generates a transient signal of Ca²⁺/calmodulin (Ca²⁺-CaM) (Fig 1D), and A2AR activity produces a continuous signal of G_olf-GTP (Fig 1B). The Ca²⁺-CaM and G_olf-GTP both bind and activate AC1 in a synergistic manner (Fig 1E) [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 G_olf-GTP alone [18,36], and Ca²⁺-CaM is not required for the activation (S1B and S2B Figs) [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”).

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 S1 Appendix and S1 and S2 Figs, and molecular concentrations and kinetic constants are summarized in S1 and S2 Tables, 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 (Fig 1F) [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. G_β:G_γ is exceptionally denoted by G_βγ, because G_β:G_γ is known to form a tight complex [40]. GTP and GDP forms of G_X are represented by G_X-GTP and G_X-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: (1) where k_on and k_off are the rate constants for the forward and backward reactions, respectively. Here, k_off/k_on is known as the dissociation constant, K_d. Enzymatic reactions were modeled based on the Michaelis-Menten (MM) equation: (2) where S, E, and P denote substrate, enzyme, and product, respectively, and K_m and k_cat 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 (t_1/2 ~ 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 1C and 2A): (3) where T_dip is the duration of a DA pause, t_DA,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. k_cat,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] (Fig 2B): (4)

Here, we set [DA]_basal = 0.5 μM and [DA]_dip = 0.05 μM unless otherwise stated (Fig 2B). Those levels of DA have been estimated based on a reference experiment (S1 Table) [1]. We finally gave a stepwise-decreasing signal of DA to quantify the DA-dip detection in the response of AC: (5)

In Iino et al. (2020), A2AR was continuously activated to produce G_olf-GTP with a constant rate [1]. The G_olf-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 G_olf-GTP was simply modeled as a constant concentration of buffered G_olf, [G_olf]_buff (Fig 1B).

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 (Fig 1D): (6)

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

G_olf and Ca²⁺-CaM synergistically activate AC1 [14,15], and G_olf alone sufficiently activate AC5 [37]. The G_olf-dependent activity of AC5 is inhibited by G_i, and the Ca²⁺-CaM-dependent component of AC1 activity is also inhibited by G_i, while the G_olf-dependent component of AC1 activity is only weakly inhibited by G_i [18]. We here assumed that the G_olf- and Ca²⁺-CaM-dependent synergistic activity of AC1 was strongly inhibited by G_i. G_olf, G_i, 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 G_olf and G_i [20]. Those pieces of evidence support the idea that G_i non-competitively inhibits AC, i.e., G_olf, G_i, and Ca²⁺-CaM independently bind to AC, and the G_i-binding itself inhibits the enzymatic activity of AC (non-competitive inhibition; S1 Fig). In contrast, biochemical studies suggest that G_i competitively inhibits AC5 [36,49], i.e., G_i inhibits the AC activity by prohibiting the binding of activators, i.e., G_olf and Ca²⁺-CaM (S2 Fig). In short, the mechanism of G_i inhibition is currently obscure even in the case of well-studied AC5. Considering this situation, we examined two extreme cases of the G_i inhibition: 100% non-competitive binding among G_i, G_olf, and Ca²⁺-CaM, (standard non-competitive model) and 100% competitive binding between G_i and the other two molecules (competitive model), and AC_primed was introduced as a shared readout of AC1 and AC5.

DA-dip duration dependence in AC_primed

Here, we quantified the DA-dip duration-dependent response of AC_primed using the following equation: (12)

〈AC_primed〉_Tdip represents the average increase in AC_primed during the DA-dip period [0, T_dip] 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.

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 (Fig 1) [10]. In this simulation, we applied the D2 model to three types of inputs (Fig 1A): a constant signal of G_olf (Fig 1B), optogenetically-evoked tonic DA signal with a 0.5-s pause (Fig 1C), and 1-s pre–post pairing (Fig 1D). Those inputs resulted in a transient activity of AC1, AC_active (Fig 1E), and the AC1 activity depended on the delay of DA pause, t_DA,delay (Fig 1F), 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 (Fig 1A).

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 (Fig 1C). We thus examined how the DA-pause signal was transmitted to AC, by characterizing it as a square wave dip of DA (Fig 2A and 2B), and the normalized level of G_i-free and G_olf-bound AC, AC_primed, was observed as a readout (see Methods; Fig 2C). In the dynamics of AC_primed, the optogenetically-evoked signals of DA were well characterized by the square wave dips of DA (Fig 2D and 2E), because the fluctuation in the basal DA signal was attenuated through the D2R−G_i−AC signaling pathway (S3 Fig).

The response of AC_primed was dependent not only on the DA dynamics, but also on the concentrations of other constituent molecules (Fig 2F). If the concentration of RGS, [RGS]_tot, was set to be 10 times higher than the standard concentration (S1 Table), AC_primed showed ~80% of the maximal activity even with the basal DA signal (Fig 2F, 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, AC_primed did not show any activity even during the DA dip (Fig 2F, 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 AC_primed for DA-dip detection

Then, using a stepwise decreasing signal of DA (Eq (4)), we examined the molecular concentrations required for DA-dip detection (Fig 3A, top). Here, the DA-dip detectability was quantified by two variables: AC_basal and AC_dip (Fig 3A). AC_basal denotes the steady-state level of AC_primed under [DA] = [DA]_basal (Fig 3A, dark blue; see Methods), and AC_dip denotes the steady-state level of AC_primed under [DA] = [DA]_dip (Fig 3A, light blue). We observed AC_basal and AC_dip if the concentration of one of the five constituent molecules, [D2R]_tot, [RGS]_tot, [AC]_tot, [G_i]_tot, and [G_olf]_buff, was varied ranging from 0.1-fold to 10-fold, while the concentrations of other molecules were kept unchanged (Fig 3B and S1 Table). First, we observed that neither AC_basal nor AC_dip was sensitive to [G_i]_tot and [G_olf]_buff if they were higher than 4% of the standard concentration (Fig 3B, second right and right, black lines), while AC_basal and AC_dip were both highly sensitive to [D2R]_tot and [RGS]_tot (Fig 3B, left and second left, black lines). This is because the concentrations of D2R and RGS determined the available amount of G_i-free AC for the activity. Here, for convenience, the regions that satisfies AC_basal < 30% and AC_dip > 70% were highlighted as the regions that enabled DA-dip detection (blue and light-blue shaded areas, respectively; Fig 3B). In the cases of [D2R]_tot, [RGS]_tot, and [AC]_tot, the regions that satisfied AC_basal < 30% (blue) and those that fulfilled AC_dip > 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 AC_basal has been shown as the requirement of higher [DA]_basal in a previous study (Fig 3B, left, top) [19], and the requirement of lower [D2R]_tot for higher AC_dip 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 (Fig 3C). 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 (Fig 3C). 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 (Fig 3E, 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 (Fig 3D, 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 (S1 Table) [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 AC_basal < 30% and AC_dip > 70%, and the region that satisfied both of them disappeared depending on the pair of [DA]_basal and [DA]_dip (S5 Fig). Nevertheless, the DA-dip detectable region in [D2R]_tot always had a positive relationship with that in [RGS]_tot (S5A and S5B Fig; 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 (S5 Fig, dotted lines; see the subsection “Analytical formulation”).

All the simulations so far were based on the D2 model under 100%-non-competitive binding between G_olf and G_i (standard non-competitive model; S1 Fig and section A in S1 Appendix). It is known that G_olf stimulates both AC1 and AC5, and G_i inhibits their activities [58]. However, even in well studied AC5, it is still under investigation whether AC is inhibited by G_i through non-competitive inhibition or the allosteric exclusion of G_olf binding [20,59]. We thus simulated the D2 model under 100%-competitive binding between G_olf and G_i (competitive model; S2 Fig and section A in S1 Appendix). The simulation results were similar to those in the standard non-competitive model (S4A and S4B Fig), and the requirements of the D2R–RGS balance also appeared only with a slight bias toward RGS (Fig 3D). The actual G_i 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 G_i inhibition.

In the simulation, the concentration of G_olf affected AC_primed only in the competitive model (Figs 3B and S4B, right). This is because the binding of G_olf to AC decreased the availability of G_olf-free AC for G_i inhibition, and the increase in [G_olf]_buff led to a simultaneous increase in AC_basal and AC_dip, which decreased the dynamic range of AC_primed (S4B Fig, right). Conversely, if [G_olf]_buff was set to be low, the dynamic range was restored (S4B Fig, right); however, the maximal activity of AC became small. As a result, there appeared to be an optimal [G_olf]_buff for ΔAC_active where (S4C Fig). It was consistent with the simulation results in a previous study [20].

Rapid response of AC_primed for DA-dip detection

The DA-dip detection depends not only on the steady-state levels of AC_primed, but also on its time constant. That is, the time constant of G_i unbinding must be shorter than the DA-dip duration, because otherwise the DA-dip signal would not appear in the change of AC_primed [19]. We thus evaluated it using a variable, T_1/2, where T_1/2 (> 0 s) denotes the half maximal time of AC_primed after a sudden DA decrease (Fig 4A). Note that exponential fits were not utilized to quantify the increasing time constant because the AC_primed response did not always grow in an exponential manner. We obtained T_1/2 using the same set of molecular concentrations as in Fig 3B, and found that T_1/2 became less than 0.5 s only if [RGS]_tot exceeded a certain level (Fig 4B, 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 (Fig 4C). We finally overlaid this plot with the requirements on AC_basal and AC_dip (Fig 4E). [D2R]_tot and [RGS]_tot were needed to fall in the overlapping region (AC_basal < 30%, AC_dip > 70%, and T_1/2 < 0.5 s) for the DA-dip detection in LTP and/or the change of neuronal excitability (Fig 4E). As expected, this region depended on [DA]_basal and [DA]_dip (S5 Fig). The higher [DA]_basal and lower [DA]_dip were required for the dynamics range in the response of AC_primed, while the lower [DA]_basal was better for the rapid response (S5A Fig, right). The requirements of the D2R–RGS balance and high [RGS]_tot were preserved regardless of the concentrations of DA.

All the characteristics of T_1/2 were also preserved in the competitive model (Figs 4D and S4D), and T_1/2 in the competitive model further depended on the binding/unbinding reaction rate of G_olf (S6 Fig). This is because AC_primed represented the state of G_i-free and G_olf-bound AC, and the G_olf binding rate became a time liming process if the G_olf binding process was slower than the G_i unbinding process. The reaction rate of G_olf did not affect the requirement of the D2R–RGS balance (S6A and S6B Fig, center), because it did not affect the steady-state level of G_olf, but the dissociation constant, K_d,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 (S1 and S2 Tables), while it should also depend on the other type of parameters, i.e., kinetic constants (S2 Table). Similarly, we defined the DA-dip detectable region, i.e., AC_basal < 30%, AC_dip > 70%, and T_1/2 < 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 AC_basal and AC_dip, and T_1/2.

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]k_cat if K_m << [S], to the GTP/GDP exchange of G_i-GDP, i.e.,

1. (a) K_m,exch,Gi << [G_i:G_βγ],

where K_m,exch,Gi is the Michaelis constant, and [G_i:G_βγ] is the substrate concentration. The constraint (a) was based on the facts that K_m,exch,Gi ~ 10 nM (S2 Table) and [G_i]_tot ~ 10 μM (S1 Table). Thus, in almost all the situations, K_m,exch,Gi was much lower than [G_i·G_βγ]. Similarly, we introduced the first-order rate approximation, d[P]/dt ~ [E][S]k_cat/K_m if K_m >> [S], to the GTP hydrolysis of G_i, i.e.,

1. (b) K_m,hyd,Gi >> [G_i-GTP],

where K_m,hyd,Gi is the Michaelis constant, and [G_i-GTP] is the substrate concentration. The constraint (b) was based on the parameters that K_m,hyd,Gi ~ 12 μM and [G_i]_tot ~ 10 μM (S1 and S2 Tables). Only a subpopulation of [G_i]_tot forms [G_i-GTP]; thus, K_m,hyd,Gi > [G_i]_tot > [G_i-GTP], and K_m,hyd,Gi > [G_i-GTP]. We further set the following constraints:

1. (c) [D2R]·[DA] / [D2R:DA] = K_d,DA, where K_d,DA = k_off,DA/k_on,DA,
2. (d) V₇ = 0 where V₇ = k_on,GiGDP[AC_i^site][G_i-GDP] (see section A in S1 Appendix).

The constraint (c) was set because the binding of DA to D2R rapidly reaches equilibrium (t_1/2 ~ 30 ms, S2 Table) [11,19], and the constraint (d) is the assumption that G_i-GDP never binds to AC, which is compatible with K_d,GiGTP << K_d,GiGDP (S2 Table) [18]. Note that the constraint (d) was also assumed in the other D2 model as the simultaneous occurrence of G_i-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 S1 Appendix.

Based on the constraints (a-d), we successfully obtained the steady-state ratio of G_olf-bound and G_i-free forms of AC, AC_basal and AC_dip, as follows (Eqs (S53) and (S61) in section B in S1 Appendix): (13) where k_RGS = k_cat,hyd,Gi/K_m,hyd,Gi·[RGS]_tot, k_DA,basal = k_cat,exch [DA]_basal / ([DA]_basal + K_d,DA), and G_olf = [G_olf]_buff/K_d,Golf (see S1 and S2 Tables and section B in S1 Appendix). AC_dip was also obtained by replacing k_DA,basal with k_DA,dip. The analytical AC_basal and AC_dip were both well fitted with the simulated AC_basal and AC_dip, respectively (Figs 3B, 3C, 3D and S4B; blue and light-blue dotted lines), and the analytical AC_basal and AC_dip were the functions of [RGS]_tot, [D2R]_tot, and [AC]_tot, but not the function of [G_i]_tot. This is because the constraint (a) simplified the G_i-dependent V₁ (Eqs (5), (S1), and (S5), section A in S1 Appendix) into a G_i-independent form (Eqs (S43) and (S44), section A in S1 Appendix). The constraint (a), K_m,exch,Gi << [G_i:G_βγ], was invalid in the small range of [G_i]_tot; thus, the analytical and simulated AC_basal were mismatched under [G_i]_tot < ~0.04 μM (Fig 3B, second right).

Eq (24) well described the simulated AC_basal and AC_dip. 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_RGS << k_off,GiGDP, k_off,GiGTP,, i.e., (14) and under k_RGS >> k_off,GiGDP, k_off,GiGTP,, i.e., (15)

Here, k_RGS << k_off,GiGDP, k_off,GiGTP denotes the situation that G_i-GTP hydrolysis is much slower than the G_i-dissociation rate from AC, and k_RGS >> k_off,GiGDP, k_off,GiGTP denotes the reversed situation. Eqs (14) and (15) denote the isolines of AC_basal, i.e., [D2R]_tot = f([RGS]_tot; AC_basal), and they are both linear functions of [RGS]_tot (∝ k_RGS); thus, AC_basal can be characterized by a transition between the two linear functions (Fig 5A). The gradients and D2R-intercept depend on AC_basal, [DA]_basal, [AC]_tot, and kinetic constants. One of the asymptotic functions, Eq (15), appeared to be curved in the logarithmic space (Fig 5A, 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); Fig 5A) appeared to be linear, because it has no D2R- or RGS-intercepts. AC_dip was also derived where k_DA,basal was replaced with k_DA,dip, and showed the same characteristics (Fig 5B).

We next considered the half maximal time of AC_primed(t), T_1/2, where AC_primed(t = T_1/2) = (AC_basal + AC_dip) / 2 after a sudden decrease of [DA] at t = 0 s (Eq (5); Fig 3A). Unfortunately, the dynamics of AC_primed(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:

1. (e) [AC_i^site]·[G_i-GTP]/[AC_i^site:G_i-GTP] = K_d,GiGTP,
2. (f) [AC_i^site]·[G_i-GDP]/[ AC_i^site:G_i-GDP] = K_d,GiGDP,
3. (g) [AC_olf^site]·[G_olf-GTP]/[AC_i^site:G_olf] = K_d,Golf (only for the competitive model),

where K_d,GiGTP = k_off,GiGTP /k_on,GiGTP,, K_d,GiGDP = k_off,GiGDP /k_on,GiGDP, and K_d,Golf = k_off,Golf /k_on,Golf. The constraints (e-g) assume that the AC-G_i and AC-G_olf bindings rapidly reach equilibrium, and the dynamics of AC_primed(t) is governed by G_i-GTP hydrolysis. This operation is called the rapid equilibrium assumption in enzymology. We denoted AC_primed and T_1/2 under the constraints (e-g) as and , respectively, and was obtained as: (16) where , and and were given by: (17) (18)

The derivations of Eqs (16), (17), and (18) are described in section C in S1 Appendix. The analytical was well fitted with the simulated T_1/2 (red dotted line, Figs 4B, 4C, 4D and S4D), showing the validity of the analytical formulation. The constraints (e, f) are not valid if the hydrolysis rate of G_i-GTP, k_RGS, is much larger than the rate of AC-G_i unbinding, and this invalidity appeared in the high [RGS]_tot (Fig 4B, second left). Similarly, the constraint (g) is not valid if k_RGS is much larger than the rate of AC-G_olf unbinding, as it also appeared (S6B Fig, right).

We finally simplified Eq (27) by considering if : (19)

Here, represents an ideal situation, i.e., AC_basal = 0% and AC_dip = 100%, or AC_basal + AC_dip = 100%. The asymptotic (Fig 4C, gray dashed line) was almost the same as the analytical one (Fig 5C, red dotted line) if AC_basal < 30% and AC_dip > 70% (Fig 5C, blue shaded area). We thus obtained an analytically closed form of the isoline of T_1/2 in the DA-dip detectable region. In Eq (19), is the decreasing function of [RGS]_tot (∝ k_RGS). Thus, [RGS]_tot must be higher than the isoline for the shorter DA-dip detection.

In summary, we derived the analytical forms of AC_basal, AC_dip, and T_1/2. AC_basal and AC_dip were both characterized by the transition between two asymptotic linear functions (Eqs (14) and (15); Fig 5A and 5B), and T_1/2 also had an approximate closed form (Eqs (16)–(18); Fig 4B−4E). The boundaries of AC_basal and AC_dip 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 (Fig 3E) [28].

AC-concentration dependence

The DA-dip detectability also depended on [AC]_tot (Figs 3B and 4B, center). We thus examined two-way relationships between [AC]_tot and [D2R]_tot (S7C and S7F Fig) as well as between [AC]_tot and [RGS]_tot (S7D and S7G Fig). The DA-dip detectable region in [AC]_tot showed a positive relationship with that in [D2R]_tot (S7C Fig), and a negative relationship with that in [RGS]_tot (S7D and S7G Fig). These [AC]_tot dependences originated from the sequestration of G_i-GTP for the inhibition of AC. If G_i-GTP was set not to be sequestrated by AC, i.e., V₄, …, V₇ were removed only from Eq (S3) (S7A and S7H Fig), the [AC]_tot dependences were completely eliminated (S7J, S7K and S7N Fig), while the relationship between [RGS]_tot and [D2R]_tot was still preserved (S7I and S7L Fig). Furthermore, the lower [AC]_tot was, the lower [AC]_tot dependences appeared (S7C, S7D and S7G Fig) because the smaller amount of G_i-GTP was sequestered by AC.

Dynamics of AC_primed under psychiatric/movement disorders

Healthy mice show an age-dependent simultaneous increase in the levels of striatal D2R and RGS9–2 during postnatal development (Fig 3E) [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; Fig 3E, 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 6).

We first confirmed that the D2 model with the standard set of parameters (healthy-adult model; Fig 6A, gray circle) successfully detected short DA dips (~0.5 s; Fig 6B). The AC_primed was sufficiently low under the steady-state G_i-GTP (Fig 6B; AC_basal = 19% where [G_i-GTP]_basal = 0.24 μM), whereas the AC_primed was sufficiently and rapidly increased during DA dips (Fig 6B; AC_dip = 78% where [G_i-GTP]_dip = 0.016 μM). Then we set a condition for healthy infant that had 0.5 × [D2R]_tot and 0.5 × [RGS]_tot (Fig 6A, green circle), and found the similar response of AC_primed to the DA dips (Fig 6C; AC_basal = 17% where [G_i-GTP]_basal = 0.23 μM; AC_dip = 78% where [G_i-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 (Fig 6A, blue circle) did not detect the DA dips because of the excessive amount of G_i-GTP. It caused the excessively low AC_primed at the basal state and its lower and slower increase (Fig 6D; AC_basal = 2% where [G_i-GTP]_basal = 2.8 μM; AC_dip = 21% where [G_i-GTP]_dip = 0.19 μM). Similarly, a dystonia model that had 0.5 × [D2R]_tot and 2.0 × [RGS]_tot (Fig 6A, red circle) showed high AC_basal and weak responses to the DA dips (Fig 6E; AC_basal = 57% where [G_i-GTP]_basal = 0.050 μM; AC_dip = 94% where [G_i-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 (Fig 6F, 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 (Fig 6F, blue line), and the dystonia model showed the rapid but small response (Fig 6F, red line).

Apparently, their DA-dip detectability was disrupted by the excessive or insufficient levels of G_i-GTP at the basal states (Fig 6B−6E, center), and they can be understood using a schematic picture (S8A Fig). At the basal state, DA-dependent D2R activity works like a faucet (tap) that provides G_i-GTP with a constant rate (S8A Fig, left). The production rate, V₁, is almost independent of [G_i:G_βγ] because [G_i]_tot (~9 μM) is much higher than K_m,exch,Gi (~0.01 μM, S2 Table; constraint (a)). The produced G_i-GTP is drained through RGS, where the draining rate, V₂ + V₈, is proportional to [G_i-GTP] + [AC:G_i-GTP] (Eqs (S40) and (S41), section A in S1 Appendix) because the GTP hydrolysis is unsaturated (constraint (b)). The difference in these rates makes a pool of G_i-GTP (S8A Fig, left). In the healthy-adult model, the pooled G_i-GTP almost completely inhibits AC, which is represented by a ball (S8A Fig). Here, the G_i-soaked part of AC is inhibited, and the G_i-free part is disinhibited (S8A Fig, inset). During the DA-dip, the production of G_i-GTP almost stops (S8A Fig, right), and the level of G_i-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 G_i-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 (S8E Fig). These imbalances caused the decrease the levels of DA-dip detection (Fig 6F).

Time windows for AC_active 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 (Fig 7). Compared to the healthy-adult model (the standard set of parameters, S1 and S2 Tables and Fig 7A), the healthy-infant model showed the smaller level of DA-dip detection in AC_acitve because the DA dips had a short period of 0.5 s (Fig 7B). The schizophrenia model did not show any increase in AC_active (Fig 7C), while the dystonia model detected the DA-dip, but the timing-independent component of AC_acitve was high (Fig 7D). Because of the high timing-independent AC_active, the peak-to-basal ratio of AC_active in the DYT1 dystonia became the lowest (Fig 7E), indicating the small signal-to-noise ratio even if the activity of the downstream signaling can adapt to the high timing-independent levels of AC_active.

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; S9A−S9C Fig). The time windows for the delay of DA dip/burst were similar to those in AC1 (S9D−S9G Fig), and the dystonia model rather showed DA-burst detection, because the longer DA burst completely inhibited the increase in AC_active due to the adenosine burst (S9G Fig, thin lines).