A mathematical model of melatonin synthesis and interactions with the circadian clock.

A new mathematical model of melatonin synthesis in pineal cells is created and connected to a slightly modified previously created model of the circadian clock in the suprachiasmatic nucleus (SCN). The SCN influences the production of melatonin by upregulating two key enzymes in the pineal. The melatonin produced enters the blood and the cerebrospinal fluid and thus the SCN, influencing the circadian clock. We show that the model of melatonin synthesis corresponds well with extant experimental data and responds similarly to clinical experiments on bright light in the middle of the night. Melatonin is widely used to treat jet lag and sleep disorders. We show how the feedback from the pineal to the SCN causes phase resetting of the circadian clock. Melatonin doses early in the evening advance the clock and doses late at night delay the clock with a dead zone in between where the phase of the clock does not change.

Mathematical insights into the role of dopamine signaling in circadian entrainment.

The circadian clock in the mammalian brain comprises interlocked molecular feedback loops that have downstream effects on important physiological functions such as the sleep-wake cycle and hormone regulation. Experiments have shown that the circadian clock also modulates the synthesis and breakdown of the neurotransmitter dopamine. Imbalances in dopamine are linked to a host of neurological conditions including Parkinson's disease, attention-deficit/hyperactivity disorder, and mood disorders, and these conditions are often accompanied by circadian disruptions. We have previously created a mathematical model using nonlinear ordinary differential equations to describe the influences of the circadian clock on dopamine at the molecular level. Recent experiments suggest that dopamine reciprocally influences the circadian clock. Dopamine receptor D1 (DRD1) signaling has been shown to aid in the entrainment of the clock to the 24-hour light-dark cycle, but the underlying mechanisms are not well understood. In this paper, we use our mathematical model to support the experimental hypothesis that DRD1 signaling promotes circadian entrainment by modulating the clock's response to light. We model the effects of a phase advance or delay, as well as the therapeutic potential of a REV-ERB agonist. In addition to phase shifts, we study the influences of photoperiod, or day length, in the mathematical model, connect our findings with the experimental and clinical literature, and determine the parameter that affects the critical photoperiod that signals seasonal changes to physiology.

Uncovering the dynamics of a circadian-dopamine model influenced by the light-dark cycle.

The neurotransmitter dopamine (DA) is known to be influenced by the circadian timekeeping system in the mammalian brain. We have previously created a single-cell differential equations model to understand the mechanisms behind circadian rhythms of extracellular DA. In this paper, we investigate the dynamics in our model and study different behaviors such as entrainment to the 24-hour light-dark cycle and robust periodicity versus decoupling, quasiperiodicity, and chaos. Imbalances in DA are often accompanied by disrupted circadian rhythms, such as in Parkinson's disease, hyperactivity, and mood disorders. Our model provides new insights into the links between the circadian clock and DA. We show that the daily rhythmicity of DA can be disrupted by decoupling between interlocked loops of the clock circuitry or by quasiperiodic clock behaviors caused by misalignment with the light-dark cycle. The model can be used to further study how the circadian clock affects the dopaminergic system, and to help develop therapeutic strategies for disrupted DA rhythms.

One-carbon metabolism during the menstrual cycle and pregnancy.

Many enzymes in one-carbon metabolism (OCM) are up- or down-regulated by the sex hormones which vary diurnally and throughout the menstrual cycle. During pregnancy, estradiol and progesterone levels increase tremendously to modulate physiological changes in the reproductive system. In this work, we extend and improve an existing mathematical model of hepatic OCM to understand the dynamic metabolic changes that happen during the menstrual cycle and pregnancy due to estradiol variation. In particular, we add the polyamine drain on S-adenosyl methionine and the direct effects of estradiol on the enzymes cystathionine ß-synthase (CBS), thymidylate synthase (TS), and dihydrofolate reductase (DHFR). We show that the homocysteine concentration varies inversely with estradiol concentration, discuss the fluctuations in 14 other one-carbon metabolites and velocities throughout the menstrual cycle, and draw comparisons with the literature. We then use the model to study the effects of vitamin B12, vitamin B6, and folate deficiencies and explain why homocysteine is not a good biomarker for vitamin deficiencies. Additionally, we compute homocysteine throughout pregnancy, and compare the results with experimental data. Our mathematical model explains how numerous homeostatic mechanisms in OCM function and provides new insights into how homocysteine and its deleterious effects are influenced by estradiol. The mathematical model can be used by others for further in silico experiments on changes in one-carbon metabolism during the menstrual cycle and pregnancy.

A mathematical model of circadian rhythms and dopamine.

BACKGROUND: The superchiasmatic nucleus (SCN) serves as the primary circadian (24hr) clock in mammals and is known to control important physiological functions such as the sleep-wake cycle, hormonal rhythms, and neurotransmitter regulation. Experimental results suggest that some of these functions reciprocally influence circadian rhythms, creating a highly complex network. Among the clock's downstream products, orphan nuclear receptors REV-ERB and ROR are particularly interesting because they coordinately modulate the core clock circuitry. Recent experimental evidence shows that REV-ERB and ROR are not only crucial for lipid metabolism but are also involved in dopamine (DA) synthesis and degradation, which could have meaningful clinical implications for conditions such as Parkinson's disease and mood disorders. METHODS: We create a mathematical model consisting of differential equations that express how the circadian variables are influenced by light, how REV-ERB and ROR feedback to the clock, and how REV-ERB, ROR, and BMAL1-CLOCK affect the dopaminergic system. The structure of the model is based on the findings of experimentalists. RESULTS: We compare our model predictions to experimental data on clock components in different light-dark conditions and in the presence of genetic perturbations. Our model results are consistent with experimental results on REV-ERB and ROR and allow us to predict the circadian variations in tyrosine hydroxylase and monoamine oxidase seen in experiments. By connecting our model to an extant model of dopamine synthesis, release, and reuptake, we are able to predict circadian oscillations in extracellular DA and homovanillic acid that correspond well with experimental observations. CONCLUSIONS: The predictions of the mathematical model are consistent with a wide variety of experimental observations. Our calculations show that the mechanisms proposed by experimentalists by which REV-ERB, ROR, and BMAL1-CLOCK influence the DA system are sufficient to explain the circadian oscillations observed in dopaminergic variables. Our mathematical model can be used for further investigations of the effects of the mammalian circadian clock on the dopaminergic system. The model can also be used to predict how perturbations in the circadian clock disrupt the dopaminergic system and could potentially be used to find drug targets that ameliorate these disruptions.

Injectate volumes needed to reach specific landmarks in s1 transforaminal epidural injections.

OBJECTIVES: We identify the contrast volumes needed to reach specific landmarks during S1 transforaminal epidural injections (S1-TFEIs). DESIGN: Prospective, nonrandomized, observational human study. Setting. Academic/private pain management practice. Subjects. Forty-two patients undergoing S1-TFEIs were investigated. Thirty-seven patients were included in this study. Interventions. S1-TFEIs were performed using contrast-enhanced fluoroscopic visualization. MAIN OUTCOME MEASUREMENTS: After confirming appropriate spinal needle position, up to 5 mL of nonionic contrast was slowly injected. Under biplanar fluoroscopic guidance, contrast volumes were recorded as flow reached specific anatomic landmarks: the ipsilateral S1 pedicle, the superior aspect of the L5-S1 disc space, and across the midline of the spinous process. RESULTS: After injecting 2 mL of contrast, 100% of S1-TFEIs spread to the medial aspect of the ipsilateral superior pedicle of S1. After injecting 3.0 mL of contrast, 92% of S1-TFEIs spread to the superior aspect of the L5-S1 intervertebral disc. After injecting 4 mL of contrast, 27% of S1-TFEIs spread beyond the midline of the spinous process, but by only a few millimeters. CONCLUSIONS: This study demonstrates injectate volumes needed to reach specific anatomic landmarks in S1-TFEIs. A volume of 3.0 mL of contrast reaches the superior aspect of the L5-S1 intervertebral disc 92% of the time.

Injectate volumes needed to reach specific landmarks in lumbar transforaminal epidural injections.

OBJECTIVES: To identify the volumes of contrast material needed to reach specific landmarks during lumbar transforaminal epidural injections (L-TFEIs). DESIGN: Prospective, nonrandomized, observational human study. SETTING: Academic/private pain management practice. PATIENTS: Sixty-nine patients undergoing L-TFEIs were investigated. Sixty patients were included in this study. INTERVENTIONS: L-TFEIs were performed with the use of contrast-enhanced fluoroscopic visualization. MAIN OUTCOME MEASUREMENTS: After the appropriate spinal needle position was confirmed, up to 5.0 mL of nonionic contrast material was slowly injected. Under biplanar fluoroscopic guidance, contrast volumes were recorded as flow reached specific anatomic landmarks: ipsilateral neural foramen, ipsilateral disks superior and inferior to the injected level, and across the midline of the spinous process. RESULTS: After 1.1 mL of contrast was injected, 100% of L-TFEIs spread to the medial aspect of the superior pedicle (PED) of the corresponding level of injection. After 2.8 mL of contrast was injected, 95% of L-TFEIs spread to the superior aspect of the superior intervertebral disk (IVD) of the corresponding level of injection. After 3.6 mL of contrast was injected, 95% of L-TFEIs spread to the inferior aspect of the inferior IVD of the corresponding level of injection. After 3 mL of contrast was injected, 88% of L-TFEIs spread to cover both the superior and inferior IVDs of the corresponding level of injection. After 4 mL of contrast was injected, 93% of L-TFEIs spread to cover both the superior and inferior IVDs of the corresponding injection. After 4 ml of contrast was injected, 55% of L-TFEIs spread beyond the midline of the spinous process, but barely. CONCLUSION: This study demonstrates injectate volumes needed to reach specific anatomic landmarks in L-TFEIs. A volume of 4.0 mL of injectate reaches both the superior aspect of the superior IVD and the inferior aspect of the inferior IVD 93% of the time.