Atmospheric deposition is an important pathway for inputting microplastics: Insight into the spatiotemporal distribution and deposition flux in a mega city.

Microplastics (MPs) refer to plastic particles with a size less than 5 mm, which attracted widespread attention as an emerging pollutant. The monitoring of atmospheric microplastics (AMPs) in a megacity was carried out to study the characteristics and spatiotemporal distribution of AMPs, explore the sources and estimate the deposition flux. The results showed that the annual average abundance of AMPs in Wuhan was 82.85 ± 57.66 n·m-2·day-1. The spatiotemporal distribution characteristics of AMPs show that spring was the highest season, followed by autumn, winter, and summer; the city center was higher than the suburbs. Fiber was the main type of AMPs in Wuhan, followed by fragment, film and pellet. The proportion of AMPs were mainly small (<0.5 mm) and medium (0.5-1.0 mm). Transparent and white were the main colors of AMPs, followed by red, brown. A total of 10 types polymers were detected, polyethylene terephthalate (PET) was dominant. There are positive correlations between AMPs and SO2, NO2 in the atmosphere, indicating that they might be influenced by intense human activity. The polycyclic aromatic hydrocarbons (PAHs) and AMPs in spring showed an extremely significant positive correlation (p < 0.05). AMPs might mainly originate from the wear and tear shedding of textiles, the aging of agricultural films and plastic waste based on their polymer types and main uses. The potential geographical sources of AMPs were mainly the surrounding cities. The annual deposition flux of AMPs was about 308 tons if there were no remove processes, which highlighted the importance of atmospheric transport and deposition of MPs. The analysis of the abundance, morphological characteristics and sources of AMPs can provide data support and reference for mega-cities with high global population activities, or cities in global mid-latitude regions.

A mathematical model for the robust blood glucose tracking.

In this paper, we study the problem of the robust blood glucose tracking. Tracking here means that the error between a state variable of a system under control and its desired time-varying reference converges to zero over time. Robustness here means that a controller designed for a system can tolerate a small variation of the system parameters. Since the parameters in the blood glucose regulation system differ in people, such a robust controller is useful in the insulin pump technology: an insulin pump equipped with such a robust controller could be used in a group of people. Thus, in our study, parameter uncertainties are introduced into a mathematical model of the blood glucose regulation system. Using an actual blood glucose level as feedback and an exogenous glucose input and a desired glucose reference as feedforward, we design a robust feedback and feedforward controller, which drives the blood glucose to track the desired time-varying glucose reference for any small uncertainties. Numerical simulations with published experimental blood glucose data are conducted to further confirm our theoretical results.

Asymptotic tracking and disturbance rejection of the blood glucose regulation system.

Type 1 diabetes patients need external insulin to maintain blood glucose within a narrow range from 65 to 108 mg/dl (3.6 to 6.0 mmol/l). A mathematical model for the blood glucose regulation is required for integrating a glucose monitoring system into insulin pump technology to form a closed-loop insulin delivery system on the feedback of the blood glucose, the so-called "artificial pancreas". The objective of this paper is to treat the exogenous glucose from food as a glucose disturbance and then develop a closed-loop feedback and feedforward control system for the blood glucose regulation system subject to the exogenous glucose disturbance. For this, a mathematical model for the glucose disturbance is proposed on the basis of experimental data, and then incorporated into an existing blood glucose regulation model. Because all the eigenvalues of the disturbance model have zero real parts, the center manifold theory is used to establish blood glucose regulator equations. We then use their solutions to synthesize a required feedback and feedforward controller to reject the disturbance and asymptotically track a constant glucose reference of 90  mg/dl. Since the regulator equations are nonlinear partial differential equations and usually impossible to solve analytically, a linear approximation solution is obtained. Our numerical simulations show that, under the linear approximate feedback and feedforward controller, the blood glucose asymptotically tracks its desired level of 90 mg/dl approximately.

Store-operated calcium entry could prevent continuous spiking of membrane potential to sustain normal intracellular calcium oscillations and normal potential bursting in pancreatic ß-cells.

We propose a dynamical store-operated calcium entry (SOCE) model to analyze the complex role of SOCE in modulating calcium oscillations and electrical activity in pancreatic ß-cells and provide a new mathematical insight. Using this model, we simulate the SOCE role in a number of cases with different SOCE conductances. When the SOCE conductance is set to 0 or very small (5 pS), our numerical simulation conforms to the experimental observation that endoplasmic reticulum (ER) calcium can sustain normal calcium oscillations and the depletion of ER calcium transforms the normal calcium oscillations into a sustained calcium increase with oscillations of much higher frequency and much smaller amplitude, and transforms the normal membrane potential oscillations to a pattern of continuous spiking. When the SOCE conductance is increased to 20 pS and the ER calcium is depleted, our numerical simulation conforms to the other experimental observation that the normal calcium and potential oscillations are sustained and augmented a little bit. Moreover, the oscillation frequency is increased a very little bit. A further increase of the conductance to 35 pS slows down the oscillation a little bit. This numerical evidence suggests that a sufficiently large SOCE can prevent the continuous spiking of membrane potential to sustain the normal calcium oscillations and the normal membrane potential bursting. A careful examination of our simulated dynamics of the ATP/ADP ratio, the ATP-sensitive outward K(+) current, and the voltage-gated inward Ca(2+) current reveals that intracellular periodic Ca(2+) peaks perhaps resulted from SOCE might play a role in stabilizing the membrane potential at its resting level (avoiding the continuous spiking) for a certain period of time by accelerating ATP consumption, reducing the ratio ATP/ADP, opening the ATP-sensitive potassium channel, and repolarizing the membrane potential.

Designing dynamical output feedback controllers for store-operated Ca²+ entry.

Store-operated calcium entry (SOCE) has been proposed as the main process controlling Ca²+ entry in non-excitable cells. Although recent breakthroughs in experimental studies of SOCE have been made, its mathematical modeling has not been developed. In the present work, SOCE is viewed as a feedback control system subject to an extracellular agonist disturbance and an extracellular calcium input. We then design a dynamic output feedback controller to reject the disturbance and track Ca²+ resting levels in the cytosol and the endoplasmic reticulum (ER). The constructed feedback control system is validated by published experimental data and its global asymptotic stability is proved by using the LaSalle's invariance principle. We then simulate the dynamic responses of STIM1 and Orai1, two major components in the operation of the store-operated channels, to the depletion of Ca²+ in the ER with thapsigargin, which show that: (1) Upon the depletion of Ca²+ in the ER, the concentrations of activated STIM1 and STIM1-Orai1 cluster are elevated gradually, indicating that STIM1 is accumulating in the ER-PM junctions and that the cytosolic portion of the active STIM1 is binding to Orai1 and driving the opening of CRAC channels for Ca²+ entry; (2) after the extracellular Ca²+ addition, the concentrations of both STIM1 and STIM1-Orai1 cluster decrease but still much higher than the original levels. We also simulate the system responses to the agonist disturbance, which show that, when a sequence of periodic agonist pulses is applied, the system returns to its equilibrium after each pulse. This indicates that the designed feedback controller can reject the disturbance and track the equilibrium.

An age-dependent feedback control model of calcium dynamics in yeast cells.

The functional decline of selected proteins or organelles leads to aging at the intracellular level. Identification of these proteins or organelles is usually challenging to traditional single-factor approaches since these factors are inter-connected via feedback or feedforward controls. Establishing a feedback control model to simulate the interactions of multiple factors is an insightful approach to guide the search for proteins involved in aging. However, there are only a few mathematical models describing the age-dependent accumulation of DNA mutations, which are directly or indirectly induced by deterioration of the intracellular environment including alteration of calcium homeostasis, a contributor of aging. Thus, based on Cui and Kaandorp's model, we develop an age-dependent mathematical model for the calcium homeostasis in budding yeast Saccharomyces cerevisiae. Our model contains cell cycle-dependent aging factors and can qualitatively reproduce calcium shocks and calcium accumulations in cells observed in experiments. Using this model, we predict calcium oscillations in wild type, pmc1 Delta, and pmr1 Delta cells. This prediction suggests that Pmr1p plays a major role in regulating cytosolic calcium. Combining the model with our experimental lifespan data, we predict an upper-limit of cytosolic calcium tolerance for cell survival. This prediction indicates that, for aged cells (>35 generations), no pmr1 Delta can tolerate the cytosolic calcium concentration of 0.1 microM while a very small fraction (1%) of aged wild type cells (>50 generations) can tolerate a high cytosolic calcium concentration of 0.5 microM.

A molecular mathematical model of glucose mobilization and uptake.

A new molecular mathematical model is developed by considering the kinetics of GLUT2, GLUT3, and GLUT4, the process of glucose mobilization by glycogen phosphorylase and glycogen synthase in liver, and the dynamics of the insulin signaling pathway. The new model can qualitatively reproduce the experimental glucose and insulin data. It also enables us to use the Bendixson criterion about the existence of periodic orbits of a two-dimensional dynamical system to mathematically predict that the oscillations of glucose and insulin are not caused by liver, instead they would be caused by the mechanism of insulin secretion from pancreatic beta cells. Furthermore it enables us to conduct a parametric sensitivity analysis. The analysis shows that both glucose and insulin are most sensitive to the rate constant for conversion of PI(3,4,5)P(3) to PI(4,5)P(2), the multiplicative factor modulating the rate constant for conversion of PI(3,4,5)P(3) to PI(4,5)P(2), the multiplicative factor that modulates insulin receptor dephosphorylation rate, and the maximum velocity of GLUT4. Moreover, the sensitivity analysis predicts that an increase of the apparent velocity of GLUT4, a combination of elevated mobilization rate of GLUT4 to the plasma membrane and an extended duration of GLUT4 on the plasma membrane, will result in a decrease in the needs of plasma insulin. On the other hand, an increase of the GLUT4 internalization rate results in an elevated demand of insulin to stimulate the mobilization of GLUT4 from the intracellular store to the plasma membrane.

Identification of longevity genes with systems biology approaches.

Identification of genes involved in the aging process is critical for understanding the mechanisms of age-dependent diseases such as cancer and diabetes. Measuring the mutant gene lifespan, each missing one gene, is traditionally employed to identify longevity genes. While such screening is impractical for the whole genome due to the time-consuming nature of lifespan assays, it can be achieved by in silico genetic manipulations with systems biology approaches. In this review, we will introduce pilot explorations applying two approaches of systems biology in aging studies. One approach is to predict the role of a specific gene in the aging process by comparing its expression profile and protein-protein interaction pattern with those of known longevity genes (top-down systems biology). The other approach is to construct mathematical models from previous kinetics data and predict how a specific protein contributes to aging and antiaging processes (bottom-up systems biology). These approaches allow researchers to simulate the effect of each gene's product in aging by in silico genetic manipulations such as deletion or over-expression. Since simulation-based approaches are not as widely used as the other approaches, we will focus our review on this effort in more detail. A combination of hypothesis from data-mining, in silico experimentation from simulations, and wet laboratory validation will make the systematic identification of all longevity genes possible.

Modeling a simplified regulatory system of blood glucose at molecular levels.

In this paper, we propose a new mathematical control system for a simplified regulatory system of blood glucose by taking into account the dynamics of glucose and glycogen in liver and the dynamics of insulin and glucagon receptors at the molecular level. Numerical simulations show that the proposed feedback control system agrees approximately with published experimental data. Sensitivity analysis predicts that feedback control gains of insulin receptors and glucagon receptors are robust. Using the model, we develop a new formula to compute the insulin sensitivity. The formula shows that the insulin sensitivity depends on various parameters that determine the insulin influence on insulin-dependent glucose utilization and reflect the efficiency of binding of insulin to its receptors. Using Lyapunov indirect method, we prove that the new control system is input-output stable. The stability result provides theoretical evidence for the phenomenon that the blood glucose fluctuates within a narrow range in response to the exogenous glucose input from food. We also show that the regulatory system is controllable and observable. These structural system properties could explain why the glucose level can be regulated.

A theoretic control approach in signal-controlled metabolic pathways.

Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, the regulatory mechanism is analyzed mathematically. For this analysis, a mathematical model for the pathways is first established using a system of differential equations. Then the linear stability, controllability, and observability of the system are investigated. We show that the linearized system is controllable and observable, and that the real parts of all eigenvalues of the linearized system are nonpositive using Routh's stability criterion. Controllability and observability are structural properties of a dynamical system. Thus our results may explain why the metabolic path ways can be controlled and regulated. Finally observer-based and proportional output feedback controllers are designed to regulate the end product to its de sired level. Applications to the regulation of blood glucose levels are discussed.

Does a fast mixer really exist?

In this paper, we are concerned about the limit behavior of the decay rate of variance of a passive and diffusive scalar in a flow field as the diffusivity of the scalar goes to zero. Motivated by the concept of the fast dynamo in the dynamo theory, we term a flow as fast mixer if the decay rate remains away from zero as the diffusivity goes to zero. We first repeat numerical simulations with flow maps and velocity fields used in the existing literature, including the lattice map, the 1D baker's map, and the sinusoidal shear flow. Our simulations shows that, in all cases, the decay rate tends to zero as the diffusivity goes to zero. For the closed flows in a bounded domain, we then theoretically proved this result under certain plausible conditions on the flows. For the open flows in the whole space, we show that the effective diffusivity matrix tends to zero in the limit without the conditions for the closed flow. In conclusion, although a fast mixer might exist, it could be very difficult to find one.