Mathematical modeling of planar polarity

Although it is well established that the Frizzled receptor is involved in the transmission of polarity information from cell to cell in the Drosophila cuticle, its precise role is still unclear. A recent paper by presents a mathematical model of a feedback loop-based mechanism for propagation of polarity between cells that can account for the known functions of Frizzled.     

Mathematical modeling of tumor-induced angiogenesis

Journal of Mathematical Biology -     

Mathematical modeling of cancer radiovirotherapy

Cancer virotherapy represents a dynamical system that requires mathematical modeling for complete understanding of the outcomes. The combination of virotherapy with radiation (radiovirotherapy) has been recently shown to successfully eliminate tumors when virotherapy alone failed. However, it introduces a new level of complexity. We have developed a mathematical model, based on population dynamics, that captures the essential elements of radiovirotherapy. The existence of corresponding equilibrium points related to complete cure, partial cure, and therapy failure is proved and discussed. The parameters of the model were estimated by fitting to experimental data. By using simulations we analyzed the influence of parameters that describe the interaction between virus and tumor cell on the outcome of the therapy. Furthermore, we evaluated relevant therapeutic scenarios for radiovirotherapy, and offered elements for optimization.     

Mathematical modeling of capillary density

Capillary density is important as a determinant for total oxygen transport to tissue. Because both capillary morphology and fiber composition vary considerably from muscle to muscle, measurement of capillary morphology and fiber composition vary considerably from muscle to muscle, measurement of capillary density alone cannot provide the detailed information necessary for analyzing physical phenomena. In this report we consider the capillary:fiber ratio, fiber types, fiber diameters, and fiber composition as components of a unit to express capillary density. We have applied the hexagonal fiber array model to calculate capillary density in cat and dog striated muscle and compared this with experimental data in the literature. The results indicate that this model may be useful for predicting capillary densities from simple biopsy procedures.     

Mathematical modeling of affinity ultrafiltration process

An affinity ultrafiltration process has been developed by exploiting affinity binding in conjunction with cross-flow filtration. The process was proven to possess high resolution, high recovery yield, and ease of scale-up. The process could purify trypsin from a trypsin-chymotrypsin mixture batchwise or continuously. Essential for applying this concept was the synthesis of a water-soluble high-molecular-weight polymer bearing m-aminobenzamidine, a strong and specific trypsin in hibitor. A mathematical model was also developed to describe the dynamic behavior of the newly developed purification process. The model was able to predict the profiles of enzyme concentrations in the process with high accuracy.     

Mathematical modeling of metabolism and hemodynamics

We provide a mathematical study of a model of energy metabolism and hemodynamics of glioma allowing a better understanding of metabolic modifications leading to anaplastic transformation from low grade glioma. This mathematical analysis allows ultimately to unveil the solution to a viability problem which seems quite pertinent for applications to medecine.     

Mathematical modeling and optimization of plasmadiafiltration

A mathematical model of mass transport of toxic substances with small, middle and large molecules weight in the body compartments and in the extracorporal system was worked out and used in the clinic for individual optimization and prediction of final results when treating patients with acute hepatic and renal failure in plasmadiafiltration. Permeability and the sieving coefficients were found "in vivo" in the plasma for 3 types of dialysers with different membranes. For practical use of this model a program was written by an interactive dialogue for the personal computer.     

Mathematical modeling of T-cell activation kinetic

T-cell activation is a crucial step in mounting of the immune response. The dynamics of T-cell receptor (TCR) specific recognition of peptide presented by major histocompatibility complex (MHC) molecule decides the fate of the T cell. Several biochemical interactions interfere resulting in a highly complex mechanism that would be difficult to understand without computer help. The aim of the present study was to define a mathematical model in order to approach the kinetics of monoclonal T-cell-specific activation. The reaction scheme was first described and the model was tested using experimental parameters from the published data. Simulations were concordant with experimental data showing proportional decrease of membrane TCR and production of interleukin-2 (IL-2). Agonist and antagonist peptides induce different levels of intracellular signal that could make the yes or no decision for entry to cell cycle. Different conditions (peptide concentrations, initial TCR density and exogenous IL-2 levels) can be tested. Several parameters are missing for parameters estimation and adjustment before it could be adapted for a polyclonal T-cell reaction model. However, the model should be of interest in setting experiments, simulation of clinical responses and optimization of preventive or therapeutic immunotherapy.     

Mathematical modeling of production of microbial lipids

In the microbial lipid production system using the yeast Rhodotorula gracilis, CFR-1, kinetics of lipid accumulation and substrate utilisation at initial substrate concentrations in the range of 20–100 kg/m³ were investigated using shake flask experiments. A mathematical representation based on logistic model for biomass and Luedeking-Piret model for lipid accumulation gave reasonably good agreement between the theoretical and experimental values for substrate concentration less than 60 kg/m³. The kinetic expressions and parameters obtained through shake flask studies were directly applied to experiments in the laboratory fermentors also and the models were found to hold good for the prediction of the change of biomass, product as well as substrate with time. The attainment of a saturation in the intracellular lipid accumulation with time, however, was not predicted by the model which was shown to be an inherent feature of the Luedeking-Piret model.List of Symbols S ₀, P ₀ kg/m³ Initial concentrations of sugar and lipid respectively - S, S(t) kg/m³ Concentrations of sugar and lipid respeclively at any timet - p,p(t) L kg/m³ Maximum concentration of lipid produced - E % Maximum sugar utilised - dP/dt kg/(m³ · h) Rate of lipid production - -dS/dt kg/(m³ · h) Rate of sugar utilisation - _max h^–1 Maximum specific growth rate - X _max kg/m³ Maximum biomass reached in a run - P _max kg/m³ Maximum product concentration - m, n Constants used in Luedeking-Piret model in eq. (7) - , Constants used to predict residual sugar - k _e maintainance coefficient - Y _x g/g Biomass yield based on sugar consumed - Y _p g/g Lipid yield based on sugar consumed - (dP/d t)_stat kg/(m³ · h) Rate of lipid production at stationary phase - (dS/dt)_stat kg/(m³ · h) Rate of sugar utilisation at stationary phase     

Mathematical modeling of pathogenicity of Cryptococcus neoformans

Cryptococcus neoformans (Cn) is the most common cause of fungal meningitis worldwide. In infected patients, growth of the fungus can occur within the phagolysosome of phagocytic cells, especially in non‐activated macrophages of immunocompromised subjects. Since this environment is characteristically acidic, Cn must adapt to low pH to survive and efficiently cause disease. In the present work, we designed, tested, and experimentally validated a theoretical model of the sphingolipid biochemical pathway in Cn under acidic conditions. Simulations of metabolic fluxes and enzyme deletions or downregulation led to predictions that show good agreement with experimental results generated post hoc and reconcile intuitively puzzling results. This study demonstrates how biochemical modeling can yield testable predictions and aid our understanding of fungal pathogenesis through the design and computational simulation of hypothetical experiments.     

Mathematical foundations for modeling poikilotherm mortality

A mathematical basis for incorporating mortality into models of insect development is presented. There has been much attention given to modeling insect development, but the quantitative treatment of mortality, especially of immature stages, is often overlooked. This paper investigates mortality as it affects immature-organism development, proceeding from the simple to the complex. The methods presented are illustrated with examples taken from boll-weevil immature mortality within cotton floral buds.     

Mathematical framework for modeling tissue growth

A phenomenological continuum mechanics framework for modeling growth of living tissues is proposed. Tissue is considered as an open system where mass is not conserved. The momentum balance is completed with the full-scale mass balance. Constitutive equations define simple growing materials. 'Thermoelastic' formulation of a simple growing material is specified. Within this framework traction free growth of a cylinder is considered. It is shown that the theory accommodates the case where stresses are not generated in uniform volumetric growth. It is also found that surface growth corresponds to a boundary layer solution of the governing equations.     

Mathematical modeling of a single-cell enzyme assay

A quantitative assay of beta-galactosidase activity in single cells of Saccharomyces cerevisiae has been developed using a fluorogenic substrate and flow cytometry [reported in Wittrup & Bailey, Cytometry, 9,394 (1988)]. The beta-galactosidase activity is expressed in yeast from the Escherichia coli lacZ gene under the control of the yeast GAL10 promoter, and is used as a marker for multicopy plasmid content. A nonfluorescent fluorogenic substrate is enzymatically cleaved by intracellular beta-galactosidase to form a fluorescent product. The accumulation of fluorescent product in single cells was found to depend on bulk substrate concentration and single-cell enzyme activity in a fashion that could not be described by a Michaelis-Menten kinetic rate form. It has been demonstrated that diffusion limitation rather than enzyme activity can determine the level of single-cell fluorescence under certain assay conditions, and a mathematical model has; been formulated which accounts for substrate and product diffusion. Guided by the mathematical model, the assay conditions were modified to allow measurement of single-cell enzyme activity rather than diffusion rates.     

Mathematical modeling and simulation of seated stability

Various methods have been used to quantify the kinematic variability or stability of the human spine. However, each of these methods evaluates dynamic behavior within the stable region of state space. In contrast, our goal was to determine the extent of the stable region. A 2D mathematical model was developed for a human sitting on an unstable seat apparatus (i.e., the “wobble chair”). Forward dynamic simulations were used to compute trajectories based on the initial state. From these trajectories, a scalar field of trajectory divergence was calculated, specifically a finite time Lyapunov exponent (FTLE) field. Theoretically, ridges of local maxima within this field are expected to partition the state space into regions of qualitatively different behavior. We found that ridges formed at the boundary between regions of stability and failure (i.e., falling). The location of the basin of stability found using the FTLE field matched well with the basin of stability determined by an alternative method. In addition, an equilibrium manifold was found, which describes a set of equilibrium configurations that act as a low dimensional attractor in the controlled system. These simulations are a first step in developing a method to locate state space boundaries for torso stability. Identifying these boundaries may provide a framework for assessing factors that contribute to health risks associated with spinal injury and poor balance recovery (e.g., age, fatigue, load/weight, and distribution). Furthermore, an approach is presented that can be adapted to find state space boundaries in other biomechanical applications.     

Mathematical modeling of mitochondrial adenine nucleotide translocase

We have developed a mathematical model of adenine nucleotide translocase (ANT) function on the basis of the structural and kinetic properties of the transporter. The model takes into account the effect of membrane potential, pH, and magnesium concentration on ATP and ADP exchange velocity. The parameters of the model have been estimated from experimental data. A satisfactory model should take into account the influence of the electric potential difference on both ternary complex formation and translocation processes. To describe the dependence of translocation constants on electric potential we have supposed that ANT molecules carry charged groups. These groups are shifted during the translocation. Using the model we have evaluated the translocator efficiency and predicted the behavior of ANT under physiological conditions.     

Mathematical modeling of the hypothalamic-pituitary-adrenal system activity

Mathematical modeling has proven to be valuable in understanding of the complex biological systems dynamics. In the present report we have developed an initial model of the hypothalamic-pituitary-adrenal system self-regulatory activity. A four-dimensional non-linear differential equation model of the hormone secretion was formulated and used to analyze plasma cortisol levels in humans. The aim of this work was to explore in greater detail the role of this system in normal, homeostatic, conditions, since it is the first and unavoidable step in further understanding of the role of this complex neuroendocrine system in pathophysiological conditions. Neither the underlying mechanisms nor the physiological significance of this system are fully understood yet.     

Mathematical modeling of differentiation in dictyostelium discoideum

Methods for the dynamic analysis of biochemical differentiation are presented. These are demonstrated in the analysis of biochemical differentiation of the carbohydrate system in D. discoideum. Procedures for simplification which are presented are projection and contraction of the system trajectory in state space and the generation of reduced equivalent dynamic metabolic networks. The importance of the hierarchical structure of differentiating systems is discussed and the concept of a dynamic embedding diagram is introduced. It is shown that complex systems must be analyzed on an epoch by epoch basis, each epoch being a period of time characterized by a constant dynamic embedding diagram, and that widely different time scales and state space scales may be necessary in different epochs. In particular there is no a priori lower limit to the time scale which may be necessary during the analysis. Some problems in mathematically defining differentiation are discussed.