2.1. Circadian rhythms of dopamine synthesis

We have previously created mathematical models of circadian rhythms in dopamine produced by regulation of tyrosine hydroxylase (TH) and monoamine oxidase (MAO), involved in synthesis and degradation respectively, by the molecular clock [23,24], having further extended the models to study reciprocal influences of dopamine and melatonin on circadian rhythms [29,41]. Animal studies have revealed circadian rhythms in TH levels across different brain regions [27,42,43] with REV-ERB circadian nuclear receptors repressing TH gene transcription [27]. The MAO enzyme also varies with circadian rhythms, with gene transcription activated by the Brain and muscle arnt-like (BMAL1) clock protein [25]. In the midbrain, the clock proteins BMAL1 and REV-ERB are antiphasic [27] and in our mathematical models [23,24] we found that these mechanisms generate a nearly antiphasic relationship between circadian rhythms of TH and MAO activity. As a result, in this study we assumed time-dependent circadian variation in TH activity and MAO-mediated catabolism, and , to study circadian time-dependent administration of drugs that target the dopaminergic system. We varied the enzyme activities sinusoidally, multiplying by and by . The phases were chosen so that the phase gap is 8 hours as predicted by our previous model [23]. Both curves were simplistically chosen to vary from 0.75 to 1.25, so that the activity varies 25% below and above. This amplitude is close to those measured in our detailed circadian-dopamine model with curves fit to animal data, where the amplitude of TH mRNA levels was about 0.5 of its peak value and the amplitude of MAO activity was about 0.2 of its peak value [23]. The phase shifts were chosen so that reaches its maximum 18 hours into a 24-hour cycle and peaks roughly 8 hours later, as observed in earlier simulations [23]. In the next section where we study dose timing, the timing will be relative to these choices in phase shift. We note, however, that C_MAO had relatively little effect on the model solutions, and periodic behaviors are effectively driven by C_TH. With time-dependent changes in TH and MAO activity, there are corresponding circadian rhythms in reaction velocities and variable concentrations; see Fig 2. The rate of conversion of ldopa to cda is 30 μM/hr while μM/hr. As a result, cda is packaged into vesicles much more quickly than it is synthesized. In addition, the nominal model has a constant release rate of , so vda is released as eda at a rate of about 80 μM/hr. Because the packaging and release steps occur at similar rates in the model ( μM/hr), the time profiles of vda and eda closely follow that of cda. Thus, while the peak in ldopa occurs 17.169 hours into the cycle, cda, vda, and eda all peak 1.55 hours later. In addition, it is known that enzyme expression varies by about 25% between individuals of the same species [44–46], and we find that the qualitative behavior is robust to 25% shifts in the baseline of circadian variation; see the dotted curves in Fig 2.

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Fig 2. Circadian rhythms of dopamine synthesis, release, and reuptake.

The variables and reaction rates in the model display circadian rhythms due to the influence of the molecular clock on TH and MAO activity. (A) Circadian rhythms of reaction rates (μM/hr). (B) Circadian rhythms of variable concentrations (μM). For both panels, solid curves correspond to time-dependent circadian variation 25% below and above nominal and . Dotted curves correspond to circadian variation relative to 0.75 and 1.25 of nominal and , demonstrating robustness of the model behavior to shifts in the baseline of variation. Dashed gray lines indicate nominal steady state values.

https://doi.org/10.1371/journal.pcbi.1013508.g002

2.2. Critical timing of dopamine reuptake inhibitors

In this section, we demonstrate that circadian rhythms in the enzyme activity in the model create time-dependent changes in the efficacy of reuptake inhibitors. A class of drugs called dopamine reuptake inhibitors (DRIs) increases extracellular dopamine concentrations by binding to dopamine transporters (DATs) and inhibiting the reuptake of dopamine into the cell. As detailed in the Methods, we modeled the effect of DRIs as a fraction 1−x_dose(t) multiplied to the velocity of the DAT reaction, . In the model, x_dose increases instantaneously at some administration time and decays exponentially. When x_dose = 0, there is no effect in the model, and when , the velocity of the reaction is reduced. We tested various experimental conditions, summarized in Table 3 in the Methods. To test for circadian variation in DRI efficacy, we introduced doses of 0.2 and 0.5 (reducing by 20% and 50%) administered at t =  6, 12, 18, and 24; see Fig 3. We assumed a half-life of 15 hours, close to the half-life of modafinil which is used to promote wakefulness [47]. Throughout this section, we assume modafinil-like kinetics, and in Sect 2.3 we experiment with half-lives. The administration of the DRI causes a large spike in eda, which falls back down over the course of the day.

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Fig 3. Single-dose administration of a dopamine reuptake inhibitor (DRI) at different times of the day.

(A) The plotted doses initially block either 20% or 50% of the dopamine transporters and decay exponentially with a half-life of 15 hours. The time of administration (6, 12, 18, or 24 hours into the day) has a substantial influence on the eda curve, which is plotted relative to its nominal steady-state concentration in the absence of drug. A gray dotted line shows the time course of eda in the absence of drug and serves as a reference trajectory. Dose = 0.2 or 0.5 at a single administration time t = 6, 12, 18, or 24, going left to right. (B) Time-dependent efficacy of DRIs. During the 24 hours following a single dose of a DRI, the effects on mean eda are minimal. However, there are substantial effects of dose time on the median and standard deviation of eda. Solid curves correspond to an initial DAT occupancy of 20% and dashed curves correspond to 50% occupancy and the DRI half-life is taken to be 15 hours as in the previous panel. Dose = 0.2 or 0.5 at single administration times throughout the entire day.

https://doi.org/10.1371/journal.pcbi.1013508.g003

We find that the administration time of DRIs has a substantial effect on the dynamical behavior of dopamine in the model. The inhibition of dopamine reuptake causes eda to rapidly increase and ldopa, cda, and vda to decrease. When the DRI is administered at a time when all model variables are elevated (e.g. t = 18), even though extracellular dopamine levels go up rapidly, the rate of the TH reaction, , is on its way back down due to circadian rhythms. Because the production of intracellular dopamine is decreasing over the next few hours, eda levels are not sustained.

To summarize the time-dependent effects of DRIs, we computed the mean, median, and standard deviation of eda during the 24-hour period following each dose. These metrics capture different aspects of the system’s response: the mean reflects the overall magnitude of elevation, the median provides a measure of the typical eda level over the 24-hour window, and the standard deviation indicates variability in the response profile. In particular, a higher median suggests that eda remains elevated for a substantial portion of the 24-hour period. Together, these metrics offer a compact and interpretable summary of how DRI efficacy depends on the time of administration.

We found that the largest changes in mean or median eda over the next 24 hours do not necessarily come from large spikes in the concentration. We tested dose times every half-hour from t = 0 to t = 24 and found that the mean eda during the 24-hour period following administration did not change significantly; see Fig 3B. Interestingly, the median eda varied significantly from 12.9% to 26.8% above steady state for Dose = 0.5. The standard deviation of eda over the 24-hour period following administration was also sensitive to dose time. When DRIs are administered when eda is elevated and before dopamine synthesis naturally decreases due to circadian rhythms, the boost in eda is large but short-lived (e.g. <6 hours). When DRIs are administered before the natural increase in dopamine synthesis, the eda boost is relatively small but the circadian drive kicks in and eda stays elevated longer. Dopamine is essential for executive functions [48], and the time-of-day effects predicted by our model are consistent with clinical studies measuring cognitive performance. Modafinil administered during circadian troughs (early morning) had the most noticeable effect on alertness and performance on cognitive tasks [49,50], with efficacy lasting about 10 hours despite its half-life of 10-15 hours [49].

With repeated doses of Dose = 0.2 at the same time each day, we find that the time of administration has a substantial influence on the dynamical behavior of dopamine in the long run as well. Repeated daily doses increase the concentration of eda on average over time; see Fig 4A. As in our simulations with a single dose, administration at t = 18 quickly increased eda substantially (by more than 20%) after the first dose. With repeated doses, the spike in eda went up to 40% above the steady state value. We found that the time of administration did not have a significant influence on the long-term change in the moving mean over time, but that it influenced the shape of the eda curve like in our single dose simulations. The 24-hour moving median and moving standard deviation were both sensitive to dose time, a pattern we found across a wide range of doses; see Fig 4B. While we might not see noticeable changes in the concentration of eda on average, the dynamical behaviors are notably different across dose administration times, with larger daily fluctuations in eda for dose times between t = 12 and t = 24.

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Fig 4. Repeated doses of DRIs at different times of the day.

(A) A DRI dose is given at the same time every day, 6 hours into the day (yellow curve) or 18 hours into the day (blue curve) for 7 days. Both dosing schedules elevate the 24-hour moving mean and median of eda relative to the nominal steady state concentration over the course of several days. Though repeated doses at 18 hours cause initial spikes in eda, the 24-hour moving median remains consistently lower over the following 7 days compared to dosing at 6 hours, indicating that eda stays elevated for a larger portion of each 24-hour period when doses are given at 6 hours. The 24-hour standard deviations indicate that eda is much more variable throughout a 24-hour period with the later administration time. Dose = 0.2 for administration times t_i = 6 + 24(i−1) (yellow curve) or t_i = 18 + 24(i−1) (blue curve) for . (B) Heat maps of mean, median, and standard deviation across 7 days of repeated doses for varying doses and dose times.

https://doi.org/10.1371/journal.pcbi.1013508.g004

2.3. Stable dosing regimes

We expected the influence of DRI administration in the model on extracellular dopamine concentrations to depend on the half-life of the administered drug. In Sect 2.2, we assumed a half-life of 15 hours for doses that initially inhibit 0.2 or 0.5 of the dopamine transporters. These doses had time-dependent effects on the temporal dynamics of eda, with some dose/dose-time combinations causing large spikes followed by a rapid decline and others allowing eda to be sustained for longer. In the differential equations, most of the extracellular dopamine is taken back up into the cell, at a rate of . With longer half-lives, repeated daily doses of DRIs can cause eda to accumulate rapidly. We computed mean eda over 7 days of repeated doses at t = 6 for half-lives between 1 and 24 hours and doses between 0 and 1 (0% and 100% initial inhibition of DAT). The average concentration of eda did not change more than two-fold for a large range of half-lives and doses; see Fig 5A.

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Fig 5. Stable dosing regimes and linear stability analysis.

(A) Mean eda over 7 days of repeated daily doses at t = 6. The change in mean eda is robust to a large range of half-lives and doses. The mean eda monotonically increases with both half-life and dose, with steep changes outside of the homeostatic plateau (large blue region). (B) Largest real part of the eigenvalues obtained from linear stability analysis over a full 24-hour circadian cycle, evaluated under varying levels of . The yellow shaded region represents the range of maximum real eigenvalues across circadian phases for each value of s_DAT. The blue line denotes the average of these eigenvalues at each activity level. The consistently negative values indicate that the system’s equilibrium remains locally stable across all circadian phases and drug inhibition levels . (C) Average equilibrium eda* across circadian phases for . The thick purple line shows the mean over 24 hrs, and the shaded region indicates circadian variation. Equilibrium eda* decreases monotonically with increasing activity, reflecting convergence to lower steady states as drug effects decay.

https://doi.org/10.1371/journal.pcbi.1013508.g005

Next, we performed a linear stability analysis of the nominal model described by Eqs (15)–(18), incorporating a scaling factor applied to to simulate the pharmacodynamic effect of DRIs at specific time points. A value of s_DAT = 1 corresponds to the absence of drug effect, while values less than 1 represent increasing degrees of transporter inhibition following drug administration. The time-dependent drug effect was previously modeled using an exponential decay, and here s_DAT reflects the instantaneous activity at any given time.

To assess how stability depends on dosing phase, we swept the circadian phase that modulates and . At each pair we treated the clock parameters as fixed (taking the quasi-static assumption that the clock varies more slowly than dopamine dynamics), solved for the biologically feasible equilibrium , and evaluated the Jacobian matrix at equilibrium. Numerical calculations indicate that this feasible equilibrium is unique for every . Details of calculations can be found in the Matlab code in the code repository. We then derived the characteristic polynomial , and solved for its roots to assess local stability.

Fig 5B presents the average of the largest real parts of the eigenvalues across all for s_DAT values ranging from 0.01 to 1. The shaded yellow region indicates the range of these maximum real eigenvalues over the circadian cycle for each level of activity. At very low values of s_DAT, particularly when the value approaches zero, certain circadian phases such as produce positive real eigenvalues. For instance, when s_DAT = 0 or s_DAT = 0.0001, the system exhibits local instability at this specific phase. However, despite these outliers, the dominant eigenvalue remains negative across most circadian phases, suggesting that the timing of DRI dosing has a limited impact on overall system stability. Importantly, all eigenvalues remain negative across the full range of simulated conditions in the majority of cases, confirming that the equilibrium state (see Fig 5C) is locally stable under varying degrees of activity and circadian modulation. Thus, although these equilibria represent transient approximations under constant effect of circadian phase, they provide valuable insight into the dynamics and stability characteristics of the system under circadian modulation.

Fig 5C shows the average equilibrium eda*, similarly across all for s_DAT values ranging from 0.01 to 1. The shaded light purple region illustrates the extent of circadian variation. We observe that the average equilibrium eda* decreases monotonically with increasing activity. Although the time-dependent drug effect is analytically complex, Fig 5C demonstrates that as the drug effect exponentially decays according to the drug’s half-life, eda progressively converges towards its equilibrium state, which steadily decreases. This behavior aligns with findings presented in Fig 3, showing that following a single dose, eda smoothly converges to its equilibrium.

While our primary linear stability analysis was performed at equilibrium points for fixed s_DAT and ϕ, this approach approximates the behavior of the system along actual drug decay trajectories. This is because as the drug effect wanes over time, the system’s state evolves along a sequence of slowly changing equilibria determined by the instantaneous value of s_DAT. In this quasi-static regime, evaluating the Jacobian at each of these equilibria effectively captures the system’s local stability along the drug decay trajectory. Therefore, although we do not explicitly compute the Jacobian along a full time-course trajectory (Fig 3A and Fig 4A), our analysis in Fig 5C implicitly reflects this evolution, as it shows how the system tracks the moving equilibrium as s_DAT decays. These findings are consistent with the results shown in Fig 5A, which demonstrate robust system output across a wide spectrum of dosing regimens and drug half-lives.

2.4. Generation of ultradian oscillations via coupling to dopaminergic network

We extended our reduced dopamine model by adding two additional Eqs (20) and (21) to explore the potential for intrinsic ultradian dopamine oscillations driven by neuronal population-level feedback. This Dopamine Ultradian Oscillator (DUO) framework is motivated by experimental observations of ultradian rhythms in behavior and physiology [39,51], which are hypothesized to originate from collective dopaminergic activity in neuronal populations.

In the DUO model, extracellular dopamine (eda) diffuses from multiple dopaminergic neuron terminals into a shared extracellular dopamine pool (edapool), representing local dopaminergic tone; see Fig 6A. Elevated edapool levels activate autoreceptor-mediated signaling (D2), which provides delayed negative feedback on dopamine release. This model structure enables intrinsic ultradian rhythms independent of external circadian input. Using a representative parameter set (see details in Sect 4.5), we observed stable oscillations in eda with an ultradian period of approximately 4.6 hours in the absence of circadian modulation (Fig 6B).

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Fig 6. Schematic representation and simulation of Dopamine Ultradian Oscillator (DUO) demonstrating ultradian rhythms with and without circadian modulation.

(A) Schematic illustration of the DUO mechanism. eda diffuses locally from dopaminergic neuron terminals into a collective dopamine pool (edapool). Elevated dopaminergic tone subsequently activates autoreceptor signaling (via D2 autoreceptors), leading to negative feedback inhibition of further eda release. (B) Simulation depicting pure ultradian rhythms of eda levels over 48 hours in the absence of circadian modulation. The rhythmic pattern is consistent and stable. (C) Simulation illustrating ultradian rhythms of eda levels within a circadian framework, showing fluctuations where dopamine concentrations are relatively lower during the sleep and relatively higher during wakefulness. Parameters for simulations in (B) and (C) were , , , , , , , m = 1.1.

https://doi.org/10.1371/journal.pcbi.1013508.g006

We then examined the behavior of the DUO under circadian regulation of TH and MAO activity as in Sect 2.1. The ultradian rhythm persisted, maintaining a similar period of 4.6 hours, but was now modulated by the slower 24-hour circadian cycle (Fig 6C). Both rhythms were clearly evident: the circadian rhythm shaped the underlying time course of dopamine, with lower eda concentrations during simulated sleep intervals and higher levels during wakefulness, while the ultradian oscillations emerged as modulations around the primary circadian signal. Notably, the amplitude of ultradian oscillations varied over the circadian cycle, becoming more pronounced when dopamine synthesis was elevated, illustrating a dynamic interaction between the two rhythmic processes.

In previous sections, we assumed that the 24-hour variation in eda was solely governed by circadian regulation of TH and MAO, and thus did not account for ultradian influences. However, incorporating ultradian rhythms introduces additional complexity, as both the amplitude and period of these rhythms vary considerably across individuals—a variability observed in behavioral and dopaminergic ultradian oscillations, which lack the intrinsic stability of circadian rhythms and are highly responsive to changes in dopamine tone [39,51]. To reduce this complexity and focus on core dynamical features, we opted to use a fixed activity level, s_DAT, rather than modeling a time-dependent drug concentration. This simplification allows us to systematically explore how different steady-state levels of reuptake inhibition affect ultradian dynamics, without the confounding variability introduced by dose timing and individual-specific ultradian profiles. While a time-dependent dosing model may be appropriate in some contexts, it offers limited interpretability here due to the intrinsic variability of ultradian rhythms.

Building on this approach, we next explored how DRIs influence ultradian periodicity by varying the activity level (s_DAT from 0.3 to 1) and calculating the resulting ultradian periods (Fig 7A). As reuptake was increasingly inhibited, the ultradian period lengthened from close to 4 hours towards 12 hours, consistent with experimental findings that DRIs slow ultradian behavioral rhythms [39]. We also examined how ultradian-circadian interactions changed under different DRI levels (Fig 7B). At lower inhibition (s_DAT = 1), dopamine oscillations retained a shorter, more regular ultradian rhythm. As s_DAT decreased (i.e., stronger inhibition), the oscillations became broader in period and more irregular in amplitude. These changes reflect a complex interplay between the intrinsic dynamics of the DUO and the entraining influence of the circadian rhythm. Overall, our results suggest that DRIs modulate both the frequency and amplitude of dopamine ultradian rhythms, and that these effects are further shaped by the phase and strength of circadian inputs.

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Fig 7. Influence of drug-induced inhibition on the period of ultradian rhythms and their modulation within circadian cycles.

(A) Dependence of the ultradian rhythm period on varying drug activity levels (s_DAT) ranging from 0.3 to 1. The period monotonically decreases as the activity level increases. (B) Simulation of ultradian rhythms modulated by circadian rhythms at selected activity levels: s_DAT = 1, s_DAT = 0.5, and s_DAT = 0.3. These examples illustrate distinct rhythmic dynamics over a 96-hour timeframe. Parameters used in simulations for (A) and (B) were , , , , , , , m = 0.01.

https://doi.org/10.1371/journal.pcbi.1013508.g007

2.5. Bifurcation analysis of the Dopamine Ultradian Oscillator (DUO)

In this section, we characterize the limit-cycle dynamics of the DUO model through linear stability analysis in the absence of circadian modulation. Initially, we numerically computed the steady-state solution, denoted as , from the system described by Eqs (15)–(21). Subsequently, the Jacobian matrix, , was analytically derived using Mathematica (details of the derivation can be found in the Mathematica notebook in the code repository). Given that the steady-state z* explicitly depends on the parameter s_DAT, we substituted back into for each selected value of s_DAT.

We then derived the characteristic polynomial of the Jacobian matrix:

Solving for roots numerically, we identified the eigenvalues corresponding to each s_DAT. Our analysis shows that the eigenvalues possess nonzero imaginary parts across the entire range of , indicating oscillatory behavior. Importantly, the largest real part of the eigenvalues transitions from negative to positive as s_DAT decreases through the critical value s_DAT = 0.26, indicating a Hopf bifurcation point (Fig 8A). Biologically, this result implies that large-enough inhibition of DAT activity (lower s_DAT values) can disrupt the ultradian rhythmic behavior inherent to the system.

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Fig 8. Bifurcation analysis of the Dopamine Ultradian Oscillator (DUO) with respect to s_DAT.

(A) Maximum real part of eigenvalues of the Jacobian matrix as a function of s_DAT demonstrate the presence of a Hopf bifurcation at s_DAT = 0.26. (B) Non-monotonic relationship between s_DAT and the amplitude of eda oscillations. As s_DAT decreases, amplitude initially increases, peaking at s_DAT = 0.42. Further reductions in s_DAT sharply diminish oscillation amplitude, ultimately abolishing ultradian behavior at sufficiently low DAT activity. (C) Sensitivity analysis showing stable ultradian rhythms across individual parameter variations (0.75–1.25 × baseline), demonstrating DUO model robustness. The baseline parameters used in simulations were the same as in Fig 7.

https://doi.org/10.1371/journal.pcbi.1013508.g008

To further elucidate the impact of dopamine reuptake inhibition on the oscillatory dynamics, we examined the amplitude of eda oscillations as a function of s_DAT. As illustrated in Fig 8B, the oscillation amplitude exhibits a non-monotonic relationship with s_DAT. Specifically, the amplitude increases with decreasing s_DAT until it reaches a maximum at approximately s_DAT = 0.39. Beyond this point, further reduction of s_DAT sharply decreases the amplitude, ultimately abolishing ultradian rhythms altogether. Thus, our model predicts that while moderate inhibition of DAT can increase the amplitude of ultradian fluctuations, excessive inhibition destabilizes and suppresses these rhythmic patterns.

Besides evaluating drug effects, we also examined the sensitivity of the DUO model to variations in all other parameters. To quantify this sensitivity, we introduced a relative magnitude as a scalar multiplier for each parameter in the DUO model. Each parameter was individually varied from 0.75 to 1.25 times its original baseline value. For each adjusted parameter set, we numerically simulated the DUO model over a duration of 100 days. The system’s behavior during the final 4 days of each simulation was recorded to represent the steady-state response.

We generated ridge plots to visualize the amplitude and period of the eda oscillations for parameter sets corresponding to Fig 7. Similar plots for parameter sets shown in Fig 6 are provided in S1 Fig. As illustrated in Fig 8C, varying each parameter within the 0.75 to 1.25 range consistently resulted in non-zero amplitudes and periods, confirming the robustness of ultradian rhythmic behavior in the model. Additionally, both amplitude and period did not change dramatically, suggesting that ultradian rhythms in our model can persist despite natural parameter fluctuations.

4.1. Model reduction

The model of dopamine synthesis, release, and reuptake by Best et al. [9] consists of 9 ordinary differential equations describing changes in the concentrations of variables: dihydrobiopterin (bh₂), tetrahydrobiopterin (bh₄), tyrosine (tyr), l-3,4-dihydroxyphenylalanine (ldopa), cytosolic dopamine (cda), vesicular dopamine (vda), extracellular dopamine (eda), homovanillic acid (hva), and tyrosine pool (tyrpool). The equations are provided below.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

The reaction rates are given by

(10)

(11)

(12)

(13)

(14)

where NADPH = 330 and NADP = 26 are cofactors for the DRR reaction and all other parameter values are provided in Table 1. The steady state concentrations of the full model are and .

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Table 1. Parameter values in full model ( = μM, = μM/hr, and = 1/hr).

https://doi.org/10.1371/journal.pcbi.1013508.t001

We reduced the model from 9 to 4 state variables by making several simplifications to the biological assumptions. First, we assume the substrate tyrosine is relatively stable in the cell (at steady-state, M in [9]). It is known that tyrosine availability in the blood has a minimal influence on dopamine synthesis, suggesting homeostatic mechanisms that maintain stable intracellular tyrosine concentrations [58]. In the full model, tyr enters into the neuron at a constant rate and exchanges linearly with the tyrosine pool. We take , removing Eqs (3) and (9) from the full model and taking the steady state tyr = 126. Note that tyrpool does not appear in any of the other equations in the full model.

Then, we fix tetrahydrobiopterin (at steady state, M in [9]) and dihydrobiopterin (at steady state, M in [9]). In the full model, the inhibition of TH by cda competes with the activation of TH by bh₄. The cofactor bh₄ is typically tightly regulated and alterations in concentration are associated with neurological diseases like Parkinson’s and neuroinflammation [59]. Taking further reduces our system. This reduction closely approximates the original model since Eqs (1) and (2) couple to the rest of the system via (Eq (10)) and the effect of bh₄ on quickly saturates. Finally, since homovanillic acid (hva) is downstream of the system, we chose to omit it in this study. Thus, we were able to reduce the system by a total of 5 state variables. The 4 remaining variables in our reduced model of dopamine synthesis are l-3,4-dihydroxyphenylalanine (ldopa), cytosolic dopamine (cda), vesicular dopamine (vda), and extracellular dopamine (eda). Schematic diagrams of both the full and reduced mathematical models are provided in Fig 1 for comparison.

4.2. Reduced model equations

The simplifications described in Sect 4.1 allowed for a much smaller, more analytically tractable system of equations. The 4 ordinary differential equations in the reduced model are given by Eqs (15)–(18),

(15)

(16)

(17)

(18)

with given by Eq (10) with fixed tyr = 126 and bh₄ = 319. Thus, the first factor in Eq (10) is effectively a constant. The second factor models the effect of autoreceptors via eda feedback, where at steady-state (M) and if eda goes above or below the steady state the factor will reduce or increase , respectively. With tyr and bh₄ as constants, the third factor models the inhibition by cda. Based on the parameter values, this effect is relatively small.

The remaining reaction rates are kept the same as in the full model [9] and are provided below:

All parameter values for the reduced model were kept the same as in [9] and are provided in Table 2. The parameter choices are explained in detail by Best et al. [9], justified by experimental measurements and observations. Thus, we felt it was appropriate to use them as the baseline for our model. Numerical solutions of the reduced model were computed using ode23s in MATLAB. The MATLAB code is provided at https://github.com/rubyshkim/YaoKim_DA.

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Table 2. Parameter values in reduced model ( = μM, = μM/hr, and = 1/hr).

https://doi.org/10.1371/journal.pcbi.1013508.t002

In Sect 2.1, we multiply by and by to model the circadian variation in enzyme activity, where the phase shifts and amplitudes were chosen to match detailed models of circadian control of dopamine synthesis [23,24].

4.3. Reduced model preserves important dynamical behaviors

We investigated numerical solutions to confirm that important dynamical features related to autoregulation were preserved. In the full model [9], equations and parameters were carefully determined based on known kinetics and experimental measurements. A schematic diagram of the model is provided in Fig 1A. In our reduced model (Fig 1B), we used the same functional forms and parameters and expected to get similar steady state concentrations. The steady state concentrations in μM of the variables in the reduced model, and eda = 0.002, are identical to the concentrations reported in the full model with the exception of vda which is 99.9% of the full model’s concentration. These values are also biologically plausible. Experimental measurements suggest that extracellular dopamine in the midbrain is typically in the low nanomolar range [60–62]. At steady state, the model concentration of vesicular dopamine is about 4.6 orders of magnitude larger than eda. About 97% of intracellular dopamine (cda  +  ) is stored in the vesicles in the model, consistent with experimental studies [63,64].

The full model constructed by Best et al. [9] displays dopamine homeostasis to firing rate and enzyme activity due to autoreceptor feedback, modeled via the second term in , Eq (10). Our reduced model almost exactly reproduces the homeostatic behavior studied in [9] in response to changes in firing rate. In the simulations, firing rate is adjusted by multiplying a function fire(t) in the equations for vda and eda. In the nominal model, . In the absence of autoreceptors in both the full and reduced models, eda is sensitive to firing rate, where a 2-fold increase in firing rate causes a 2-fold increase in extracellular dopamine concentration. With autoreceptors, the eda curve is much more robust to changes in firing rate; see Fig 9A, which is almost identical to Fig 9 (lower panel) in [9].

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Fig 9. Homeostasis in reduced mathematical model.

The reduced mathematical model displays the same homeostatic features as in the full model. (A) The autoreceptors allow extracellular dopamine (eda) to be relatively robust to changes in firing rate. The eda concentration is plotted as a percentage of the nominal steady state value. (B) eda is homeostatic to changes in enzyme activity, that is, the of the reactions catalyzed by TH and DAT, within the 75–125% homeostatic region (contour lines). Outside this band, particularly at low activity levels, changes in eda become much more drastic. The white dot corresponds to the nominal model. Values are plotted as percentages of nominal model values.

https://doi.org/10.1371/journal.pcbi.1013508.g009

In addition, our reduced model displays similar behaviors as in the full model [9] in response to changes in enzyme activity. TH is the rate-limiting enzyme in dopamine synthesis and autoreceptors reduce fluctuations in eda caused by changes in TH activity [9]. In simulations, we modify the maximum rates of the TH and DAT reactions by 0-150% of their nominal values and calculate the corresponding steady-state value of eda; see Fig 9B compared to Fig 11 in [11].

4.4. Dopamine reuptake inhibitors

We modeled the effects of dopamine reuptake inhibitors (DRIs) by adding an additional variable for the concentration, x_dose(t). The effect of x_dose is that it inhibits the reuptake of dopamine back into the cell. We model this inhibition simplistically by multiplying by in Eqs (16) and (18), so that when x_dose = 1 there is no reuptake and when x_dose = 0 reuptake happens at the normal rate. In addition, x_dose decays exponentially,

(19)

with r chosen appropriately based on measured half-lives of DRIs. In our simulations in Fig 3 and Fig 4A, we chose to model a DRI with a half-life of 15 hours, close to the half-life of the drug modafinil [47], so that . We introduce instantaneous doses of Dose at dose times t = t_i where is the dirac delta function. In simulations, this is equivalent to updating the initial condition for x_dose at the administration times. In the Results, we demonstrate the effects of DRI timing in the model in several different conditions, summarized in Table 3.

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Table 3. Simulation conditions for the Circadian Time (CT) of DRI administration.

In our study, t = 0 coincides with CT0.

https://doi.org/10.1371/journal.pcbi.1013508.t003

4.5. Coupling of model to eda pool

We extend our model from the perspective that the DUO is composed of a population of oscillators rather than individual neurons [39,51]. We couple the model to a larger pool of eda in the projection region, so that the neuron senses dopaminergic tone. The local eda diffuses out, elevating dopaminergic tone, which then promotes autoreceptor signaling (D2). The two new ODEs are provided below, where edapool and D2 are dimensionless, latent variables representing dopaminergic tone and autoreceptor signaling.

(20)

(21)

We couple from the original reduced model to the eda pool by modifying the second term so that it depends on D2. The term in Eq (10) is replaced by

(22)

which is a sigmoidal function that decreases with D2. In the model, when the s_DAT fraction is reduced from 1, eda will build up and D2 will also go up, so adapts by shifting to the right. In total, 2 new ODEs and 8 new parameters are added to describe coupling of the dopamine model to dopaminergic tone.

The precise mechanisms of dopamine ultradian oscillations remain elusive [51]. As a result, edapool and D2 are dimensionless and do not represent precise quantities, but are latent variables accounting for feedback from dopaminergic tone. We chose parameter values using Latin hypercube sampling with several assumptions. First, we took , accounting for contributions of dopamine from nearby dopaminergic terminals. Dopamine concentrations vary widely across brain regions [65,66], so we allowed for changes across three orders of magnitude, . The remaining for were chosen to be small in comparison, within the range [0.1,40], and the parameters α and β were chosen to keep qualitatively close to the autoreceptor term in the original model (Eq (10)), in the range [0.5,4.5]. The parameter was chosen to shift to have effects in the solution range of D2, and adjusted the steepness of . Within these conditions, parameters leading to stable limit cycle solutions were considered.