Fig 1.
Schematic diagram of full and reduced mathematical models.
(A) Schematic diagram of full model of dopamine synthesis, release, and reuptake by Best et al. [9]. Figure modified from [9]. Abbreviations: btyr, blood tyrosine; bh2, dihydrobiopterin; bh4, tetrahydrobiopterin; tyr, tyrosine; l-dopa, l-3,4-dihydroxyphenylalanine; cda, cytosolic dopamine; vda, vesicular dopamine; eda, extracellular dopamine; hva, homovanillic acid; trypool, the tyrosine pool; vTyr, neutral amino acid transporter; DRR, dihydrobiopterin reductase; TH, tyrosine hydroxylase; AADC, aromatic amino acid decarboxylase; MAT, vesicular monoamine transporter; DAT, dopamine transporter; auto, D2 dopamine auto receptors; MAO monoamine oxidase; COMT, catecholamine O-methyl transferase; NADP, nicotinamide adenine dinucleotide phosphate; NADPH, nicotinamide adenine dinucleotide phosphate, reduced form. (B) Tyrosine (tyr) is converted into levodopa (ldopa), which is decarboxylated to make cytosolic dopamine (cda). Cytosolic dopamine is then packaged into vesicles (vesicular dopamine, vda) and released into the extracellular space as extracellular dopamine (eda). The state variables of the model equations are in rectangles, and enzymes that determine reaction rates are in ellipses. Enzymes that are influenced by the molecular clock in the model are highlighted in yellow.
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.
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.
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 ti = 6 + 24(i−1) (yellow curve) or ti = 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.
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 sDAT. 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.
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.
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 (sDAT) 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: sDAT = 1, sDAT = 0.5, and sDAT = 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.
Fig 8.
Bifurcation analysis of the Dopamine Ultradian Oscillator (DUO) with respect to sDAT.
(A) Maximum real part of eigenvalues of the Jacobian matrix as a function of sDAT demonstrate the presence of a Hopf bifurcation at sDAT = 0.26. (B) Non-monotonic relationship between sDAT and the amplitude of eda oscillations. As sDAT decreases, amplitude initially increases, peaking at sDAT = 0.42. Further reductions in sDAT 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.
Table 1.
Parameter values in full model ( = μM,
= μM/hr, and
= 1/hr).
Table 2.
Parameter values in reduced model ( = μM,
= μM/hr, and
= 1/hr).
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.
Table 3.
Simulation conditions for the Circadian Time (CT) of DRI administration.
In our study, t = 0 coincides with CT0.