Interaction between the otolithic organ and semicircular canals of the vestibular apparatus in the system of directional orientation of man in space

A mathematical model is suggested of the biological system of angular stabilization in space relative to the local vertical using measures whose parameters are near to the parameters of the semicircular canals and otolithic organ of the man. On the basis of computer simulation for physiologically adequate external stimulations contribution of each of measures to the total process of angular stabilization is evaluated.     

Modelling adsorption and biological degradation of nutrients on peat

A dynamic mathematical and numerical model of adsorption and biological degradation of nutrients in an organic perfusion column with recycle has been developed. This model has applicability to industrial applications of biodegradation of nutrients in wastewater such as biofilters and biotrickling filters where concentrations are dilute and solid surface coverage is low. It successfully predicts that adsorption has the effect of masking a ‘true' rate of biological degradation behind an ‘observed' rate of degradation in the liquid phase. This is due to the adsorptive capacity of peat which provides a buffer for surges in loading and makes peat a useful carrier for engineered biological systems. Four dimensionless parameters were identified to totally describe the physical system without biological activity and a further two were identified for the system with biological activity. Analytical solutions to simplifications of the model were justified by showing that the assumption of a negligible concentration gradient in the column was valid after an initial perturbation in nutrient concentration had passed.     

A mathematical model for selective differentiation of neural progenitor cells on micropatterned polymer substrates

The biological hypothesis that the astrocyte-secreted cytokine, interleukin-6 (IL6), stimulates differentiation of adult rat hippocampal progenitor cells (AHPCs) is considered from a mathematical perspective. The proposed mathematical model includes two different mechanisms for stimulation and is based on mass-action kinetics. Both biological mechanisms involve sequential binding, with one pathway solely utilizing surface receptors while the other pathway also involves soluble receptors. Choosing biologically-reasonable values for parameters, simulations of the mathematical model show good agreement with experimental results. A global sensitivity analysis is also conducted to determine both the most influential and non-influential parameters on cellular differentiation, providing additional insights into the biological mechanisms.     

The significance of specific distributions and functions in models of quantitative inheritance

The main purpose of this paper is to attract attention to the fact that a mathematical model of quantitative inheritance can lead to qualitatively different results, if distributions and functions which in the model represent biological processes are assumed to take analytically different (even though qualitatively similar) forms. A simple model of the effect of environmental variation on the post-selection genotypic variance in a population under phenotypic stabilizing selection is considered. It is demonstrated that this model leads to qualitatively different conclusions depending on whether the phenotypic fitness function is assumed in a Gaussian or in a quadratic form.     

Competitive exclusion and persistence in models of resource and sexual competition

 We study a combined mathematical model of resource and sexual competition. The population dynamics in this model is analyzed through a coupled system of reaction-diffusion equations. It is shown that strong sexual competition and low birth rate lead to competitive exclusion of the biological species. If sexual competition is weak, then the persistence of the species is possible, depending on the initial density functions and the growth rates of the species. When sexual competition affects both species, persistence and competitive exclusion results are also obtained in terms of the ecological data in the model. Received 1 November 1995; received in revised form 13 January 1996     

Mathematical model for the homeostasis of alpha-macroglobulins in the rat

Alpha-macroglobulins (AM) are proteins that inactivate proteinases. Sodium monofluorophosphate (MFP) binds to AM and transiently changes AM plasma levels. As a consequence MFP is useful to modify AM homeostasis. A mathematical model to study the homeostasis of AM is proposed in this paper. The model describes changes in plasma concentration of AM, MFP concentration in the gastrointestinal tract, MFP plasma concentration, plasma concentration of AMMFP and includes rate constants of the processes involved in AM homeostasis. Estimation of the rate constants values was achieved using experimental and mathematical resources. The homeostasis of AM after an oral dose of 80 μmol of MFP was analyzed with a simulation tool. Experimental conditions that modify the homeostasis of AM had been simulated and validated using specific drugs that change some parameter of the system.The mathematical model describes accurately the behavior of the biological model. The results allow concluding that the simplifications made did not underestimate the main processes involved in the homeostasis and, also that the assumptions made were correct.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

Stochastic models for X chromosome inactivation

Summary A multivariate Gaussian model for mammalian development is presented with the associated biological and mathematical assumptions. Many biological investigations use the female mammal X chromosome to test hypotheses and to estimate parameters of the developmental system. In particular, Lyon's (1961) hypotheses are used as a basis of the mathematical model. Experimental mouse data and three sets of human experimental data are analyzed using the hypothesized Gaussian model. The estimated biological parameters are consistent with some current biological theories.     

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.     

A model for the initiation of replication in Escherichia coli

The role of the protein DnaA as the principal control of replication initiation is investigated by a mathematical model. Data showing that DnaA is growth rate regulated suggest that its concentration alone is not the only factor determining the timing of initiation. A mathematical model with stochastic and deterministic components is constructed from known experimental evidence and subdivides the total pool of DnaA protein into four forms. The active form, DnaA.ATP, can be bound to the origin of replication, oriC, where it is assumed that a critical level of these bound molecules is needed to initiate replication. The active form can also exist in a reserve pool bound to the chromosome or a free pool in the cytoplasm. Finally, a large inactive pool of DnaA protein completes the state variables and provides an explanation for how the DnaA.ATP form could be the principal controlling element in the timing of initiation. The fact that DnaA protein is an autorepressor is used to derive its synthesis rate. The model studies a single exponentially growing cell through a series of cell divisions. Computer simulations are performed, and the results compare favorably to data for different cell cycle times. The model shows synchrony of initiation events in agreement with experimental results.     

Bubbles induced fluctuations of some properties of aqueous solutions

A mathematical model of the air diffusion process through the water surface taking into account the reverse flow of bubbles was proposed. The numerical solving gives self-oscillation of the dissolved air concentration. Spectral density of these oscillations has the 1/f-noise form, which explains the fluctuating nature of the observed properties of water and biological effects, which depend on the amount of dissolved air. A mechanism of spontaneous water luminescence observed after laser irradiation was suggested. The calculated luminescence self-oscillations have the form of a fractal with an infinite sequence of periods, which evolve in time and that is qualitatively consistent with the experiment.     

Effect of ferrous ion on the biological activity in a UASB reactor: mathematical modeling and verification

The effect of ferrous ion on the biological activity in a upflow anaerobic sludge blanket (UASB) reactor was studied. A mathematical model was developed and validated in order to simulate the dynamic behavior of a UASB reactor. This model took into consideration of all the biological and physicochemical reactions. The model was able to simulate the accumulation of iron in the sludge bed and its effect on the biological activity of the anaerobic sludge. A significant increase in the maximum uptake rate of methanogens and acidogens was revealed when iron was supplemented to the UASB reactor. The addition of ferrous iron induced a stable and excellent COD conversion rate. The model can be a useful tool for the prediction of process performance in the future and can be used to assist in the operation of biogas plants. The ferrous ion addition proved to enhance the biological activity of UASB sludge. Thus, the model can be used for the design of treatment plants that will take advantage of the benefits of iron addition.     

A mathematical model for the freezing process in biological tissue

A mathematical model has been developed to study the process of freezing in biological organs. The model consists of a repetitive unit structure comprising a cylinder of tissue with an axial blood vessel (Krogh cylinder) and it is analysed by the methods of irreversible thermodynamics. The mathematical simulation of the freezing process in liver tissue compares remarkably well with experimental data on the structure of tissue frozen under controlled thermal conditions and the response of liver cells to changes in cooling rate. The study also supports the proposal that the damage mechanism responsible for the lack of success in attempts to preserve tissue in a frozen state, under conditions in which cells in suspension survive freezing, is direct mechanical damage caused by the formation of ice in the vascular system.     

A mathematical analysis of a model for tumour angiogenesis

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.     

Modulation of acridine mutagen ICR191 intercalation to DNA by methylxanthines—Analysis with mathematical models

Caffeine (CAF) and other methylxanthines (MTX) may interact directly with several aromatic, intercalating ligands through mixed stacking aggregation. Formation of such stacking hetero-complexes may decrease their free form concentration and, in consequence, diminish their biological activity, which is often related to their direct interaction with DNA. In this paper interactions of acridine mutagen (ICR191) with DNA in the presence of three MTX: caffeine (CAF), pentoxifylline (PTX) and theophylline (TH) are investigated. Several mathematical models are used to calculate all association constant values and every component concentration in each analyzed mixture. Model McGhee–von Hippel is used to analyze ligand–DNA interaction, and model Zdunek et al.—to analyze ligand–MTX interactions. Finally, two distinct mathematical models are employed to analyze three-component mixture containing ligand, MTX and DNA molecules. The first model describes possible interactions of ligand with DNA and MTX, and rejects direct MTX interactions with DNA. The second model describes all interactions mentioned above and, additionally, allows MTX to interact directly with DNA. Results obtained using these models are similar. However, correspondence of theoretical results to experimental data is better for the first model than the second one. In this paper possible interactions of ICR191 with eukaryotic cell chromatin are also analyzed, showing that CAF reduces acridine mutagen potential to interact directly with cell chromatin. Additionally, it is demonstrated that MTX inhibit mutagenic activity of ICR191 in a dose-dependent manner. Furthermore, biological activity of ICR191–MTX mixtures corresponds with concentration of free mutagen form calculated using appropriate mathematical model.     

Population modeling for a captive squirrel monkey colony

Population modeling for a squirrel monkey colony breeding in a captive laboratory environment was approached with the use of two different mathematical modeling techniques. Deterministic modeling was used initially on a spreadsheet to estimate future census figures for animals in various age/sex classes. Historical data were taken as input parameters for the model, combined with harvesting policies to calculate future population figures in the colony. This was followed by a more sophisticated stochastic model that is capable of accommodating random variations in biological phenomena, as well as smoothing out measurement errors. Point estimates (means) for input parameters used in the deterministic model are replaced by probability distributions fitted into historical data from colony records. With the use of Crystal Ball (Decisioneering, Inc., Denver, CO) software, user-selected distributions are embedded in appropriate cells in the spreadsheet model. A Monte Carlo simulation scheme running within the spreadsheet draws (on each cycle) random values for input parameters from the distribution embedded in each relevant cell, and thus generates output values for forecast variables. After several thousand runs, a distribution is formed at the output end representing estimates for population figures (forecast variables) in the form of probability distributions. Such distributions provide the decision-maker with a mathematical habitat for statistical analysis in a stochastic setting. In addition to providing standard statistical measures (e.g., mean, variance, and range) that describe the location and shape of the distribution, this approach offers the potential for investigating crucial issues such as conditions surrounding the plausibility of extinction.     

A mathematical model of the sleep/wake cycle

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous “flip–flop” conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip–flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip–flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip–flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.     

Heterogeneous proliferation within engineered cartilaginous tissue: the role of oxygen tension

This article investigates heterogeneous proliferation within a seeded three-dimensional scaffold structure with the purpose of improving protocols for engineered tissue growth. A simple mathematical model is developed to examine the very strong interaction between evolving oxygen profiles and cell distributions within cartilaginous constructs. A comparison between predictions based on the model and experimental evidence is given for both spatial and temporal evolution of the oxygen tension and cell number density, showing that behaviour for the first 14 days can be explained well by the mathematical model. The dependency of the cellular proliferation rate on the oxygen tension is examined and shown to be similar in size to previous work but linear in form. The results show that cell-scaffold constructs that rely solely on diffusion for their supply of nutrients will inevitably produce proliferation-dominated regions near the outer edge of the scaffold in situations when the cell number density and oxygen consumption rate exceed a critical level. Possible strategies for reducing such non-uniform proliferation, including the conventional methods of enhancing oxygen transport, are outlined based on the model predictions.