Quantification of Circadian Rhythms in Single Cells

Bioluminescence techniques allow accurate monitoring of the circadian clock in single cells. We have analyzed bioluminescence data of Per gene expression in mouse SCN neurons and fibroblasts. From these data, we extracted parameters such as damping rate and noise intensity using two simple mathematical models, one describing a damped oscillator driven by noise, and one describing a self-sustained noisy oscillator. Both models describe the data well and enabled us to quantitatively characterize both wild-type cells and several mutants. It has been suggested that the circadian clock is self-sustained at the single cell level, but we conclude that present data are not sufficient to determine whether the circadian clock of single SCN neurons and fibroblasts is a damped or a self-sustained oscillator. We show how to settle this question, however, by testing the models'' predictions of different phases and amplitudes in response to a periodic entrainment signal (zeitgeber).     

Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel

A single neuronal model incorporating distributed delay (memory)is proposed. The stochastic model has been formulated as a Stochastic Integro-Differential Equation (SIDE) which results in the underlying process being non-Markovian. A detailed analysis of the model when the distributed delay kernel has exponential form (weak delay) has been carried out. The selection of exponential kernel has enabled the transformation of the non-Markovian model to a Markovian model in an extended state space. For the study of First Passage Time (FPT) with exponential delay kernel, the model has been transformed to a system of coupled Stochastic Differential Equations (SDEs) in two-dimensional state space. Simulation studies of the SDEs provide insight into the effect of weak delay kernel on the Inter-Spike Interval(ISI) distribution. A measure based on Jensen-Shannon divergence is proposed which can be used to make a choice between two competing models viz. distributed delay model vis-á-vis LIF model. An interesting feature of the model is that the behavior of (CV(t))((ISI)) (Coefficient of Variation) of the ISI distribution with respect to memory kernel time constant parameter η reveals that neuron can switch from a bursting state to non-bursting state as the noise intensity parameter changes. The membrane potential exhibits decaying auto-correlation structure with or without damped oscillatory behavior depending on the choice of parameters. This behavior is in agreement with empirically observed pattern of spike count in a fixed time window. The power spectral density derived from the auto-correlation function is found to exhibit single and double peaks. The model is also examined for the case of strong delay with memory kernel having the form of Gamma distribution. In contrast to fast decay of damped oscillations of the ISI distribution for the model with weak delay kernel, the decay of damped oscillations is found to be slower for the model with strong delay kernel.     

A closed NPZ model with delayed nutrient recycling

We consider a closed Nutrient-Phytoplankton-Zooplankton (NPZ) model that allows for a delay in the nutrient recycling. A delay-dependent conservation law allows us to quantify the total biomass in the system. With this, we can investigate how a planktonic ecosystem is affected by the quantity of biomass it contains and by the properties of the delay distribution. The quantity of biomass and the length of the delay play a significant role in determining the existence of equilibrium solutions, since a sufficiently small amount of biomass or a long enough delay can lead to the extinction of a species. Furthermore, the quantity of biomass and length of delay are important since a small change in either can change the stability of an equilibrium solution. We explore these effects for a variety of delay distributions using both analytical and numerical techniques, and verify these results with simulations.     

Saccharification of Miscanthus x giganteus, incorporation of lignocellulosic by-product in cementitious matrix

Given the non competition of miscanthus with food and animal feed, this lignocellulosic species has attracted attention as a possible biofuel resource. However, sustainability of ethanol production from lignocelluloses biomass would imply reduction in the consumption of chemicals and/or energetic means, but also valorization of the lignocellulosic by-product remaining from enzymatic saccharification. Introduction of these by-products into a cementitious matrix could be used in manufacturing a lightweight composite. Miscanthus biomass was submitted to chemical pretreatments followed by saccharification using an enzymatic cocktail. Residues from saccharification were then mixed with a cementitious matrix. Given their mechanical properties and a good adherence between cement and by-product, the hardened materials could be used. However, the delay in the beginning of setting time is too long, which prevents the direct use of by-product into cementitious matrix. Preliminary experiments using a setting accelerator in the cementitious matrix permitted significant reduction in the setting time delay.     

Long-term recovery of bracken (Pteridium aquilinum (L.) Kuhn) after asulam spraying

This paper investigates the effects of a single asulam application, sprayed from the air, on the rhizome biomass, bud density, fronds and carbohydrate reserves of bracken (Pteridium aquilinum (L.) Kuhn) using a time sequence approach. Regression models were used to investigate how these characteristics varied with time after spraying, and were used, where appropriate, to calculate the time taken for full recovery after treatment. Frond density and biomass recovered in approximately eight years, bud numbers in seven, but rhizome biomass and total carbohydrate reserves required 10 to 12 years to recover. The consequences of these results are compared with predictions from a computer model and discussed in relation to the best timing of re-treatment and the management needed for long term control.     

The Stochastic Theory of Cell Proliferation

A stochastic theory of cell kinetics has been developed based on a realistic model of cell proliferation. A characteristic transit time, _i, has been assigned to each of the four states (G₁, S, G₂, M) of the cell cycle. The actual transit time, t_i, for any cell is represented by a distribution around _i with a variance σ_i². Analytic and computer formulations have been used to describe the time development of such characteristics as age distribution, labeling experiments, and response to perturbations of the system by, for example, irradiation and temperature. The decay of synchrony is analyzed in detail and is shown to proceed as a damped wave. From the first few peaks of the synchrony decay one can obtain the distribution function for the cell cycle time. The later peaks decay exponentially with a characteristic decay constant, λ, which depends only on the average cell-cycle time, , and the associated variance. It is shown that the system, upon any sudden disturbance, approaches new “equilibrium” proliferation characteristics via damped periodic transients, the damping being characterized by λ. Thus, the response time of the system, /λ, is as basic a parameter of the system as the cell-cycle time.     

Division synchrony and the dynamics of microbial populations: A size-specific model

A discrete, environmentally coupled, size-specific model of microbial population dynamics in continuous culture is presented. It is mathematically simpler than other models based on similar assumptions and lends itself to numerical and analytic solutions. It displays several phenomena which have been reported in the experimental literature but which are not well understood; specifically, a loose relationship between biomass and numbers (i.e., a time lag between mass growth and cell division) and a critical damping of biomass while numbers continue to oscillate. In addition, the model provides several new predictions: The stable biomass distribution is independent of the environmental factors considered in the model and uniformly distributes the biomass among the size classes. The rate of approach to stability and the frequency of waves through the size distributions are a function of the flow rate and the variance in rate of growth and size at division. The model should provide a useful basis for studying the effects of size specificity on the dynamics of microbial populations cultured in chemostats.     

Responses of the Circadian Locomotor Activity Rhythm of Mus Booduga to Shifts in LD Schedules

The responses of the field mouse Mus booduga to shifts in schedules of LD cycles were monitored and the results were interpreted with the help of a PRC constructed for the same species. The results reveal that, M. booduga reentrained faster with a lesser number of transients after delay shifts than advance shifts, thus exhibiting “asymmetry effect.” A positive correlation was observed between the number of transients and the number of hours of shift. In most of the shifts, the sign of the transients (negative for delaying transients and positive for advancing transients) coincided with the direction of the shift. Interestingly, 11 and 12 h of advance shifting resulted in delaying transients. An 11-h advance shift can also be interpreted as a 13-h delay. Reentrainment through delaying transients is faster as compared to reentrainment through advancing transients. Thus, this animal might have taken a “shorter route,” as proved by the fact that an 11-h advance shift has evoked delaying transients. But a 13-h advance shift evoked only advancing transients. This prompts us to speculate that there may be a “phase jump” in M. booduga. Further, irrespective of whether L or D has been doubled in a 12-h shift, both evoked only delaying transients.     

Delayed reproduction in Arabidopsis thaliana improves fitness in soil with suboptimal phosphorus availability

Low phosphorus availability (low P) often delays flowering and maturity in annual plants, while abiotic stress generally accelerates flowering and maturity. The utility of this response is unknown. We hypothesize that phenological delay in low P is beneficial by permitting more time for phosphorus acquisition and utilization. We grew seven genotypes of Arabidopsis thaliana with contrasting phenology in high and low P. Low P delayed bolting and maturity in all genotypes. Low P decreased root length, but not root-length duration (the integral of root length over time), because phenological delay allowed low-P plants to compensate for shorter root length. Root-length duration was correlated with phosphorus accumulation. Leaf phosphorus duration (the integral of leaf phosphorus over time) was correlated with reproductive biomass, indicating the utility of increased phosphorus utilization. Phenological delays accounted for up to 30% of biomass production when low-P plants were compared to models of plants with no delays. These results support the hypothesis that phenological delay in low P is adaptive and leads to increased phosphorus acquisition and utilization. Because low P conditions are prevalent, understanding the utility of this response could be useful in crop breeding and in predicting plant responses to global climate change.     

Mathematical models for some types of chemical information systems

Ant species on a high evolutionary level have evolved chemical recruitment systems such as mass recruitment or quality recruitment. The recruitment process from the nest to a food source may be damped by crowding effects at the source. For four patterns of behavior (mass/quality recruitment; with/without damping) we study mathematical models for the time development of the quantity of food at the source. Each of the models can be reduced to a second order time-delayed differential equation which will be studied in the equivalent form of a first order (nonlinear) functional differential equation. We discuss the complete exploitation of a given source. In case of mass recruitment there possibly remains a threshold quantity of food not worth exploiting. However, every source will be exploited completely (in finite time) provided that the volatility of the trail pheromone is small compared with the exploitation activities of the colony and the distance from the nest to the source. In addition, for the damped models the capacity of the crowded source must be large compared with the initial quantity of food offered. The efficiency of the exploitation activities of some species allows conclusions on their evolutionary development.     

Decreasing motion artifacts in calcium-dependent fluorescence transients from the perfused mouse heart using frequency filtering

A strategy has been developed for the removal of motion artifact and noise in calcium-dependent fluorescence transients from the perfused mouse heart using frequency filtering. An analytical model indicates that the spectral removal of motion artifacts is independent of the phase shift of the motion waveform in the frequency domain, and thus to the time shift (or delay) of motion in the time domain. This is based on the "shift theorem" of Fourier analysis, which avoids erroneous correction of motion artifact when using the motion signal obtained using reflectance from the heart. Several major steps are adopted to implement this model for elimination of motion as well as detection noise from the fluorescence transient signals from the calcium-sensitive probe Rhod-2. These include (1) extracting the fluorescence calcium transient signal from the raw data by using power spectrum density (PSD) in the frequency domain by subtracting the motion recorded using the reflectance of excitation light, (2) digitally filtering out the random noise using multiple bandpass filters centralized at harmonic frequencies of the transients, and (3) extracting high frequency noise with a Gaussian Kernel filter method. The processed signal of transients acquired with excessive motion artifact is comparable to transients acquired with minimal motion obtained by immobilizing the heart against the detection window, demonstrating the usefulness of this technique.     

Boolean network model predicts cell cycle sequence of fission yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.     

Growth dynamics of a methylotroph (Methylomonas L3) in continuous cultures. II. Growth inhibition and comparison against an unstructured model

High methanol concentrations have a negative effect on the growth rate and the biomass yield of growth transients induced by methanol pulses in continuous cultures of Methylomonas L3. The physiological basis of this effect is investigated by measuring the effect of the methanol pulse on the cell energy charge (EC) and ATP, ADP, and AMP concentrations, and by comparing the results of the pulse transients against an unstructured model. The methanol pulse is shown to lead to increased values of the cell EC and ATP concentration, and thus, inhibition and reduced availability of biosynthetic energy are excluded as causes of inhibition. When the biomass and methanol profiles of the transient experiments are compared in phase-plane diagrams against computer simulations based on the model, satisfactory agreement between experimental data and model predictions is found in single-substrate, high-dilution-rate experiments. Conversely, poor agreement between experimental data and simulation results indicates a more severe growth inhibition than the model predicts at low dilution rates and a less severe one in mixed-substrate experiments. Based on these findings and other relevant physiological information, we propose that the large variations in the negative effect of methanol on growth result from the fact that cells accumulate methanol to widely different concentrations depending on their physiological state. In their effort to detoxify from the high intracellular methanol and formaldehyde concentrations, cells oxidize considerably more methanol than they can incorporate into biomass. This leads to a useless ATP surplus, which the cells must hydrolyze without doing any useful biosynthetic work, and this results in lower biomass yields.     

High-Frequency Quasi-Potential Waves in the Plasma Formed under Tunnel Ionization of Atoms

The dispersion law and collisionless damping rate of quasi-potential waves in the plasma formed upon tunnel ionization of gas atoms in the field of a short pulse of circularly or linearly polarized radiation are found. It is shown how the frequency and damping rate of quasi-potential waves depend on the wave propagation direction relative to the symmetry axis of the photoelectron distribution. It is established that, in plasma with a toroidal photoelectron velocity distribution, weakly damped waves with a linear dispersion law and frequency above the electron plasma frequency can propagate in a wide range of angles. In the case of a bi-Maxwellian photoelectron distribution, the frequency of weakly damped waves is comparable with the electron plasma frequency and the anisotropy of electron motion manifests itself in relatively small corrections to the dispersion law.     

Biennial cycling caused by demographic delays in a fire-adapted annual plant

We explored models explaining population cycling in the annual Warea carteri. We modeled the life cycle of W. carteri and compared projected trajectories to independently observed trajectories (up to 16 years) of plants in 74 patches in three populations. We built matrix models with an annual time step for two populations, including four stages, (recently produced seeds, seeds in the seed bank, seedlings, and adults) and five vital rates, summarized in seven transitions. Fluctuations of both observed and modeled populations were evaluated using power spectra, autocorrelation, amplitude, and damping. Observed populations had two point cycling. Observed amplitude was higher in frequently burned populations, reached its maximum 1 year after fire, and then dampened. Asymptotic transition and vital rate elasticities showed that seedling survival was the most important factor for long-term population growth, but transient elasticities showed that recruitment from the seed bank was important during the first years post-fire. Deterministic modeling and elasticity analyses indicated that delayed germination (for 1 year) may explain biennial population cycling. Stochastic models created similar cycling with slower damping than deterministic models, but still had lower amplitudes (especially 1–3 years post-fire) than observed populations. The biennial cycle in W. carteri is likely caused by the delay in seed germination, which creates two overlapping cohorts of plants, much like a strict biennial. Fire initiates the cycle by killing aboveground individuals and promoting seedling recruitment in the first post-fire year.     

Modification of spastic gait through mechanical damping

The effect of dissipative mechanical loads on spastic gait has been studied, to evaluate the feasibility of using mechanically damped orthoses to effect functional improvements in the gait of spastic patients. This concept is based on a hypothesis citing uninhibited, velocity-dependent stretch reflexes as a possible causal factor in spastic gait abnormalities, such as equinus and back-kneeing. In order to screen potential experimental subjects and to quantify velocity-dependent reflex behaviour, ankle rotation experiments and filmed gait analysis were performed. The results supported the existence of a velocity threshold. Orthosis simulation experiments were performed with one spastic subject, using a wearable, computer-controlled, electromechanical, below-knee orthosis simulator to apply a variety of damping loads to the ankle as the subject walked. Results indicated that appropriate damping can improve local joint kinematics. The damping causes a reduction in muscle stretch velocity which apparently results in reduced spastic reflex activity.     

The dynamics of single-substrate continuous cultures: the role of transport enzymes

A chemostat limited by a single growth-limiting substrate displays a rich spectrum of dynamics. Depending on the flow rate and feed concentration, the chemostat settles into a steady state or executes sustained oscillations. The transients in response to abrupt increases in the flow rate or the feed concentration are also quite complex. For example, if the increase in the flow rate is small, there is no perceptible change in the substrate concentration. If the increase in the flow rate is large, there is a large increase in the substrate concentration lasting several hours or days before the culture adjusts to a new steady state. In the latter case, the substrate concentration and cell density frequently undergo damped oscillations during their approach to the steady state. In this work, we formulate a simple structured model containing the inducible transport enzyme as the key intracellular variable. The model displays the foregoing dynamics under conditions similar to those employed in the experiments. The model suggests that long recovery times (on the order of several hours to several days) can occur because the initial transport enzyme level is too small to cope with the increased substrate supply. The substrate concentration, therefore, increases until the enzyme level is built up to a sufficiently high level by the slow process of enzyme induction. Damped and sustained oscillations can occur because transport enzyme synthesis is autocatalytic, and hence, destabilizing. At low dilution rates, the response of stabilizing processes, such as enzyme dilution and substrate consumption, becomes very slow, leading to damped and sustained oscillations.     

Biosorption of Cr (VI) with Trichoderma viride immobilized fungal biomass and cell free Ca-alginate beads

Ability of Cr (VI) biosorption with immobilized Trichoderma viride biomass and cell free Ca-alginate beads was studied in the present study. Biosorption efficiency in the powdered fungal biomass entrapped in polymeric matric of calcium alginate compared with cell free calcium alginate beads. Effect of pH, initial metal ion concentration, time and biomass dose on the Cr (VI) removal by immobilized and cell free Ca-alginate beads were also determined. Biosorption of Cr (VI) was pH dependent and the maximum adsorption was observed at pH 2.0. The adsorption equilibrium was reached in 90 min. The maximum adsorption capacity of 16.075 mgg(-1) was observed at dose 0.2 mg in 100 ml of Cr (VI) solution. The high value of kinetics rate constant Kad (3.73 x 10(-2)) with immobilized fungal biomass and (3.75 x 10(-2)) with cell free Ca- alginate beads showed that the sorption of Cr (VI) ions on immobilized biomass and cell free Ca-alginate beads followed pseudo first order kinetics. The experimental results were fitted satisfactory to the Langmuir and Freundlich isotherm models. The hydroxyl (-OH) and amino (-NH) functional groups were responsible in biosorption of Cr (VI) with fungal biomass spp. Trichoderma viride analysed using Fourier Transform Infrared (FTIR) Spectrometer.     

Modelling of continuous microbial cultivation taking into account the memory effects

The problem of chemostat dynamics modelling for the purpose of control is considered. The "memory" of the culture is explicitly taken into account. Two possibilities for improving the quality of the proposed modelling approaches are discussed. A general model that accounts for the culture `memory' by means of different `memory' functions in the expressions of the specific growth rate and of the specific consumption rate and a polynomial function of the substrate concentration for the yield factor is proposed. The case where the maintenance energy is taken into account is also discussed. Two modifications of the general model (w-type and S-type) are presented. A zero-order `memory' function and a i-function with delay are applied in order to describe the `memory' effects. Continuous growth of the strain Saccharomyces cerevisiae on a glucose limited medium is considered as a case study. Detailed investigations of the variety of models, derived from the general model by applying different `memory' functions and different assumptions are carried out. The results are compared with those previously reported for the same process. It is shown that a significant improvement in predicting the substrate dynamics (not accompanied by any decrease in the quality of the model with respect to the biomass concentration) could be achieved, involving a first- or second-order polynomial function for the yield factor. It is also shown that the quality of the model mainly depends on the way that `memory' function is incorporated. The detailed investigations give priority to the w-type models. In this case past values of both biomass and substrate variables are considered. The time delay models with pure (constant) delay and those which account for the culture `memory' by zero-order `memory' function (adaptability parameter) are compared with respect to their utilization for the purpose of model-based control.