A Mathematical Model of the Human Circadian System and Its Application to Jet Lag

A mathematical model of the circadian system is described that is appropriate for application to jet lag. The core of the model is a van der Pol equation with an external force. Approximate solutions of this equation in which the external force is composed of a constant and an oscillating term are investigated. They lead to analytical expressions for the amplitude and period of free-running rhythms and for the frequency limits of the entrainment region. The free-running period increases quadratically with stiffness. Both period and amplitude depend on the value of the constant external force. The width of the range of entrainment is mostly determined by the external force, whereas the relative position of this range follows the intrinsic period of the oscillator. Experiments with forced and spontaneous internal desynchronization were evaluated using these analytical expressions, and estimates were obtained for the intrinsic period of the oscillator, its stiffness, and the external force. A knowledge of these model parameters is essential for predictions about circadian dynamics, and there are practical implications for the assessment of the adaptation after rapid transmeridian travel.     

Circadian Rhythms, Jet Lag, and Chronobiotics: An Overview

This overview considers the origins of jet lag in terms of altered circadian rhythmicity. The properties required of a chronobiotic—an agent to cause phase adjustment of the body clock—are discussed, and an account is given of the major candidates at the present time: light, melatonin, activity, and benzodiazepines. It is concluded that current knowledge indicates that a combination of factors is likely to be most effective.     

A novel approach to the van der pol oscillator: Natural frequency and entrainment

Abstract

The physics of the van der Pol oscillator as realized by the Meissner circuit is discussed by analogy to the beat phenomenon and by a consequent analysis of current balance. The current balance method leads to a new, very accurate equation for the dependence of the oscillator frequency on the feedback parameter. Several aspects of entrainment (existence, limited frequency range, dependence on parameters, phase shift) can be explained, too. Numerical results are presented which have been obtained by solving the homogeneous and inhomogeneous van der Pol equation with a Runge‐Kutta method.     

Transition Between Advance and Delay Responses to Eastbound Transmeridian Flights

In response to eastbound transmeridian flights, which result in zeitgeber phase advance shifts, adaptation of the circadian system to the new time zone by phase delays and advances are observed. The delay response to an advance zeitgeber shift has been called an antidromic response. For the shift at which the transition from an advance to an antidromic response occurs, the term critical shift is introduced.

For the study of critical shifts, a flight experiment across nine time zones and numerical simulations of a van der Pol equation have been evaluated. The interest is focussed on the determination of a range for critical abrupt shifts. An abrupt shift means that the ensemble of zeitgebers including geophysical zeitgebers and the rest-activity cycle is shifted immediately in the new time zone. The range of critical advance shifts has been estimated to reach from + 7 to + 10 hr. In the literature, results were reported which would imply a much wider range. The discussion of these observations shows that the actual shifts were presumably not abrupt in the quoted experiments.

The consequences of critical shifts for jet lag symptoms are investigated. If reduced circadian amplitudes and long times taken for the resynchronization contribute to the feeling of jet lag, the symptoms will be worst for shifts close to the critical one, as numerical simulations revealed. Manipulations of such shifts with the aim to alleviate jet lag are discussed.     

Further Assessments of the Relationship Between Jet Lag and Some of Its Symptoms

The disruption of circadian rhythms following time‐zone transitions gives rise to the syndrome of jet lag. The power of some of the symptoms of jet lag to predict the amount of jet lag measured at the same and at different times of the day has been investigated. Eleven healthy subjects were studied in an Isolation Unit for two days after a simulated flight from the UK to Beijing (8 time zones to the east). At six time‐points (08:30, 11:00, 14:00, 17:00, 20:00, and 23:00 h), the subjects recorded their jet lag, and the differences from “normal” (that is, from days in which there is no jet lag) of alertness, hunger, indigestion, concentration, motivation, and irritability. They recorded at 08:30 h the type of food they had eaten since rising at 08:00 h and, at the other times, the type of food eaten in the last three hours. Assessments were made by visual analogue scales or, in the case of type of food, by a nominal scale. Following the time‐zone transition, the adjustment of meals appeared to be complete almost immediately. Jet lag and its symptoms were present during both experimental days. Jet lag tended to rise during the course of the daytime, accompanied by falls in alertness, motivation, and concentration. Correlation matrices between jet lag and each of the other variables were produced, using lags between the variable (from up to 5 time‐points before the assessment of jet lag to 5 time‐points afterwards) and pooling the results from both days. These matrices indicated that significant correlations existed only between jet lag and alertness, concentration, and motivation, and then only when these other variables were assessed at the same time as jet lag or 1 or 2 time‐points earlier. Jet lag was then treated as the dependent variable and the symptoms as covariates in analysis of covariances (ANCOVAs), with the days treated as a random effect. This analysis enabled the significance of potential predictors of jet lag, together with their β‐coefficients (the relationship between a unit change of each significant predictor and the change in jet lag), to be calculated. Falls in alertness and motivation were significant predictors of increased jet lag, provided that they were measured at the same time, when they accounted for about 50% of the jet lag; when measured at other time‐points, they did not act as significant predictors. It is concluded that the amount of jet lag varies during the course of the day and that it can be predicted from contemporaneous assessments of alertness and motivation—but not from assessments made at other times of the day, nor from other variables that are symptoms of jet lag, even though these other variables are significantly increased. In considering the results of this and our previous study, we reiterate the view that the exact meaning of “jet lag” is complex and that the particular combination of factors that contribute to it might vary with the time of day that the assessment is made. Inferences about any decrements due to time‐zone transitions cannot be made reliably at times of the day that differ from the time when jet lag is assessed.     

The Relationship Between Assessments of Jet Lag and Some of Its Symptoms

The power of the symptoms of jet lag in predicting the amount of jet lag measured at the same and different times of the day has been investigated. A total of 85 subjects was studied for 6 days after a flight from the UK to Australia (10 time zones to the east). At 08:00, 12:00, 16:00, 20:00, and 24:00h, the subjects recorded their jet lag and fatigue. At 08:00h, they also assessed their sleep. At 12:00 and 16:00h, they assessed their attitude to a meal, as well as their motivation, commitment, and irritability. On retiring, they recorded bowel activity. Assessments were by visual analog scales. Jet lag was treated as the dependent variable and the symptoms as covariates in ANCOVAs. Fatigue was a powerful predictor of jet lag, provided it was measured at the same time, and some aspects of sleep predicted jet lag measured on retiring or rising. The other symptoms predicted jet lag less powerfully and/or at a wider range of times. It is concluded that, even though jet lag at any time of the day can be predicted from contemporaneous assessments of fatigue and that it can be predicted on retiring or rising from some aspects of changed sleep, jet lag is predicted less reliably from other symptoms, including aspects of mental performance. These findings question exactly what causes jet lag at a particular time of day, and so are relevant to studies which use this measurement to investigate the problems associated with time‐zone transitions, and ways to ameliorate them.     

医用数学模型的格式化与显示

本文介绍医用数学模型格式化的方法,为了显示模型连续解的图象和计算机处理过程,文中提供了一些典型的格式化模型.采用数学推理手段将数学模型变换为可被模拟电路及数字电路所接受的模式,就成了格式化模型．实验结果表明,该方法是行之有效的．     

Comparative Study to Determine the Optimal Melatonin Dosage form for the Alleviation of Jet Lag

To compare the impact of various dosage forms of melatonin and placebo on jet lag symptoms, 320 volunteers who had flights over 6 to 8 time zones were recruited for a double-blind, randomized, placebo-controlled study. The volunteers received either melatonin 0.5-mg fast-release (FR) formulation, melatonin 5-mg FR formulation, melatonin 2-mg controlled-release (CR) formulation, or placebo. The study medication was taken once daily at bedtime during 4 days after an eastward flight. The volunteers completed the Profile of Mood States (POMS), sleep log, and symptoms questionnaires once daily and the Karolinska Sleepiness Scale (KSS) three times daily prior to departure and during the 4 days of medication intake postflight. A total of 234 (73.1%) participants were compliant and completed the study. The FR melatonin formulations were more effective than the slow-release formulation. The 5-mg FR formulation significantly improved the self-rated sleep quality (p <. 05), shortened sleep latency (p <. 05), and reduced fatigue and daytime sleepiness (p <. 05) after intercontinental flight. The lower physiological dose of 0.5 mg was almost as effective as the pharmacological dose of 5.0 mg. Only the hypnotic properties of melatonin, sleep quality and sleep latency, were significantly greater with the 5.0-mg dose. (Chronobiology International, 15(6), 655-666, 1998)     

The Endogenous Melatonin (MT) Signal Facilitates Reentrainment of the Circadian System to Light-Induced Phase Advances by Acting Upon MT2 Receptors

The indolamine melatonin is an important rhythmic endocrine signal in the circadian system. Exogenous melatonin can entrain circadian rhythms in physiology and behavior, but the role of endogenous melatonin and the two membrane-bound melatonin receptor types, MT1 and MT2, in reentrainment of daily rhythms to light-induced phase shifts is not understood. The present study analyzed locomotor activity rhythms and clock protein levels in the suprachiasmatic nuclei (SCN) of melatonin-deficient (C57BL/6J) and melatonin-proficient (C3H/HeN) mice, as well as in melatonin-proficient (C3H/HeN) mice with targeted deletion of the MT1, MT2, or both receptors, to determine effects associated with phase delays or phase advances of the light/dark (LD) cycle. In all mouse strains and genotypes, reentrainment of locomotor activity rhythms was significantly faster after a 6-h phase delay than a 6-h phase advance. Reentrainment after the phase advance was, however, significantly slower than in melatonin-deficient animals and in mice lacking functional MT2 receptors than melatonin-proficient animals with intact MT2 receptors. To investigate whether these behavioral differences coincide with differences in reentrainment of clock protein levels in the SCN, mPER1, mCRY1 immunoreactions were compared between control mice kept under the original LD cycle and killed at zeitgeber time 04 (ZT04) or at ZT10, respectively, and experimental mice subjected to a 6-h phase advance of the LD cycle and sacrificed at ZT10 on the third day after phase advance. This ZT corresponds to ZT04 of the original LD cycle. Under the original LD cycle, the numbers of mPER1- and mCRY1-immunoreactive cell nuclei were low at ZT04 and high at ZT10 in the SCN of all mouse strains and genotypes investigated. Notably, mouse strains with intact melatonin signaling and functional MT2 receptors showed a significant increase in the number of mPER1- and mCRY1-immunoreactive cell nuclei at the new ZT10 as compared to the former ZT04. These data suggest the endogenous melatonin signal facilitates reentrainment of the circadian system to phase advances on the level of the SCN molecular clockwork by acting upon MT2 receptors. (Author correspondence: M.Pfeffer@em.uni-frankfurt.de)     

Jet Riemann-Lagrange几何分析静脉注射吸毒人员HIV/AIDS模型(英文)

介绍并应用Jet Riemann-Lagrange几何,通过解一类Kaldor流方程去分析静脉注射吸毒人员HIV/AIDS模型,我们得到几个有意义的结果.     

A Mathematical Model of Uterine Dynamics and its Application to Human Parturition

We have developed a simple mathematical model with three physiologically significant states to describe the changes in intrauterine pressure associated with a contraction during human parturition. The myometrium is modelled as a set of smooth muscle cells, each of which is in one of three states (quiescent, contracted, refractory) at a given time. These states are occupied according to a cycle governed by three temporal parameters. The solutions of the equations describing the model show an oscillatory behavior for particular values of these parameters, which is very similar to the time dependant development of intrauterine pressure during labor. Due to its non-linear terms, our model could lead to chaotic oscillations (in the mathematical sense), whose clinical counterpart may occur in cases of dystocia. Despite its simplicity, this model appears to be a useful guide to further investigations of the oscillatory behavior of the myometrium, or other smooth muscles, in normal and pathological situations.     

A Mathematical Model for Leaf Photosynthesis of Mulberry

A mathematical model of leaf photosynthesis has been established. In this model, the processes of photosynthesis are divided into two parts, ie., the carboxylation process driven by light which is dependent on temperature and CO2 concentration, and the diffusion of CO2 from atmosphere to the carboxylation site. Finatly, CO2 uptake by the leaf is understood as dependent on 1), the CO2 response curve of the leaf mesophyll and 2). the CO2 partial pressure in the intercellular space in leaf. The COs response curve of the leaf photosynthesis is described mathematically in terms of carboxylation efficiency (Ca) or its initial slope and the photosynthetic capacity (Pm) or the CO2-saturated uptake rate of CO2 uptake, and dark respiration (Rd). The dependency of photosynthesis on leaf temperature and incident light intensity is incorporated into variations of those parameters which establish an appropriate response to internal CO2 pressure for particular light and temperature conditions prevailing at any time. Secondly the interactiion of stomata with photosynthesis is represented as an empirical relation between stomatal conductance and a combined environmental physiological index, APn·Hx/CaThe parameters used in the modelwere estimated with Marquardt-Newton method for non-linear function. Field measurements of mulberry leaf photosynthesis provided a data set for model testing. The resuks show that the simulated values of the model agree well with observed data. The model was used to analyse the response surface of leaf conductance and photosynthesis to environmental factors—Applications and limitations of the model are discussed     

在环境制约情况下的人口数学模型

在文［１］Ｙｕｌｅ模型的基础上加上环境允许的极限人口数Ｎ_ｍ，用偏微分方程的特征理论进行数学研究．     

用蚀斑法滴定痘苗病毒的准确性及其精确数学模型的探讨

用蚀斑法滴定病毒是确定感染病毒颗粒存在数量的一种较准确方法。本实验表明,痘苗病毒吸附4h后仍有大量病毒粒子未能吸附到细胞单层,进而测定出病毒接种量、维持液加量和所测病毒滴度间具有一种互为消长的非线性相关性。因而设计了几种检测方法,其准确性均优于常规痘苗病毒蚀斑测定法。利用装配有Mathematic软件包的计算机在痘苗病毒接种量、维持液加量和所测病毒滴度间建立了曲线拟合模型和曲面拟合模型。通过曲线拟合模型推断病毒感染滴度为常规法滴定值的近5倍。     

高等植物磷酸烯醇式丙酮酸羧化酶聚合态可逆变化的数学模型及其定性分析

磷酸烯醇式丙酮酸羧化酶（PEPCase）是高等植物生理代谢中一个极为重要的酶类,其一个重要特征是PEPCase的聚合呈现解离─聚合的周期性振荡行为。本文提出了一个PEPCase可逆解离─聚合反应的数学模型。通过对该模型的定性分析给出了极限环解存在的条件,并由此指出了目前生理学研究中应注意的若干问题。     

A Mathematical Model of the Human Respiratory Control System

The respiratory system exhibits the properties of a control system of the regulator type. Equations describing this biological control system have been derived. Transient and steady-state solutions for various CO₂ and O₂ step input disturbances were obtained utilizing a digital computer and are compared with experimental results. The effectiveness of the respiratory system as a regulator is investigated. Further extensions of the model are suggested.     

A Mathematical Model of the Pancreatic Ductal Epithelium

A mathematical model of the HCO⁻ ₃-secreting pancreatic ductal epithelium was developed using network thermodynamics. With a minimal set of assumptions, the model accurately reproduced the experimentally measured membrane potentials, voltage divider ratio, transepithelial resistance and short-circuit current of nonstimulated ducts that were microperfused and bathed with a CO₂/HCO⁻ ₃-free, HEPES-buffered solution, and also the intracellular pH of duct cells bathed in a CO₂/HCO⁻ ₃-buffered solution. The model also accurately simulated: (i) the effect of step changes in basolateral K⁺ concentration, and the effect of K⁺ channel blockers on basolateral membrane potential; (ii) the intracellular acidification caused by a Na⁺-free extracellular solution and the effect of amiloride on this acidification; and (iii) the intracellular alkalinization caused by a Cl⁻-free extracellular solution and the effect of DIDS on this alkalinization. In addition, the model predicted that the luminal Cl⁻ conductance plays a key role in controlling both the HCO⁻ ₃ secretory rate and intracellular pH during HCO⁻ ₃ secretion. We believe that the model will be helpful in the analysis of experimental data and improve our understanding of HCO⁻ ₃-transporting mechanisms in pancreatic duct cells. Received: 18 October 1995/Revised: 5 July 1996     

种子萌发与休眠的自磷酸化蛋白激酶控制机制的数学模型及其分析

自磷酸化蛋白激酶对代谢的开关作用可能控制着种子的萌发与休眠。本文提出了自磷酸化蛋白激酶控制机理的数学模型。通过对该模型定性分析指出，自磷酸化蛋白激酶对种子萌发与休眠的控制机理具有双稳态的突变性质，得出了产生双稳态的临界条件。     

A Mathematical Model of the Human Metabolic System and Metabolic Flexibility

In healthy subjects some tissues in the human body display metabolic flexibility, by this we mean the ability for the tissue to switch its fuel source between predominantly carbohydrates in the postprandial state and predominantly fats in the fasted state. Many of the pathways involved with human metabolism are controlled by insulin and insulin-resistant states such as obesity and type-2 diabetes are characterised by a loss or impairment of metabolic flexibility. In this paper we derive a system of 12 first-order coupled differential equations that describe the transport between and storage in different tissues of the human body. We find steady state solutions to these equations and use these results to nondimensionalise the model. We then solve the model numerically to simulate a healthy balanced meal and a high fat meal and we discuss and compare these results. Our numerical results show good agreement with experimental data where we have data available to us and the results show behaviour that agrees with intuition where we currently have no data with which to compare.     

特定雌性动物群周期性地获得稳定收获量的数学模型

利用Ｌｅｓｌｉｅ矩阵,给出了特定雌性动物群周期性地获得稳定收获量的数学模型,并对模型适用于集约化饲养家畜的特殊情况进行了讨论。