Bulletin of Mathematical Biology

Hung, Hui-Chih; Chen, Yu-Hou; Liu, Guang-Yaw; Lee, Hwei-Jen; Chang, Gu-Gang

doi: 10.1016/S0092-8240(03)00024-7pmid: 12875334

Mathematical models for the protein folding-unfolding process involving multiple intermediates have been derived. Computer fitting of the experimental data to this model generates various thermodynamic parameters for the folding-unfolding process. In this way, the complex folding-unfolding process of the multi-domain proteins can be analysed in a quantitative way. The application of the folding-unfolding model involving seven stages in human placental alkaline phosphatase is described.

Nagata, Wayne; Harrison, Lionel; Wehner, Stephan

doi: 10.1016/S0092-8240(03)00025-9pmid: 12875335

We study two chemical models for pattern formation in growing plant tips. For hemisphere radius and parameter values together optimal for spherical surface harmonic patterns of index l = 3, the Brusselator model gives an 84% probability of dichotomous branching pattern and 16% of annular pattern, while the hyperchirality model gives 88% probability of dichotomous branching and 12% of annular pattern. The models are two-morphogen reaction-diffusion systems on the surface of a hemispherical shell, with Dirichlet boundary conditions. Bifurcation analysis shows that both models give possible mechanisms for dichotomous branching of the growing tips. Symmetries of the models are used in the analysis.

Breward, Christopher; Byrne, Helen; Lewis, Claire

doi: 10.1016/S0092-8240(03)00027-2pmid: 12875336

In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.

Reinker, Stefan; Puil, Ernest; Miura, Robert

doi: 10.1016/S0092-8240(03)00026-0pmid: 12875337

Thalamic neurons exhibit subthreshold resonance when stimulated with small sine wave signals of varying frequency and stochastic resonance when noise is added to these signals. We study a stochastic Hindmarsh-Rose model using Monte-Carlo simulations to investigate how noise, in conjunction with subthreshold resonance, leads to a preferred frequency in the firing pattern. The resulting stochastic resonance (SR) exhibits a preferred firing frequency that is approximately exponential in its dependence on the noise amplitude. In similar experiments, frequency dependent SR is found in the reliability of detection of alpha-function inputs under noise, which are more realistic inputs for neurons. A mathematical analysis of the equations reveals that the frequency preference arises from the dynamics of the slow variable. Noise can then transfer the resonance over the firing threshold because of the proximity of the fast subsystem to a Hopf bifurcation point. Our results may have implications for the behavior of thalamic neurons in a network, with noise switching the membrane potential between different resonance modes.

Marcano-Velázquez, M.; Layton, Harold

doi: 10.1016/S0092-8240(03)00029-6pmid: 12875338

A nonlinear optimization technique, in conjunction with a single-nephron, single-solute mathematical model of the quail urine concentrating mechanism, was used to estimate parameter sets that optimize a measure of concentrating mechanism efficiency, viz., the ratio of the free-water absorption rate to the total NaCl active transport rate. The optimization algorithm, which is independent of the numerical method used to solve the model equations, runs in a few minutes on a 1000 MHz desktop computer. The parameters varied were: tubular permeabilities to water and solute; maximum active solute transport rates of the ascending limb of Henle and the collecting duct (CD); length of the prebend enlargement (PBE) of the descending limb; fractional solute delivery to the CD; solute concentration of tubular fluid entering the CD at the cortico-medullary boundary; and rate of exponential CD population decrease along the medullary cone. Using a base-case parameter set and parameter bounds suggested by physiologic experiments, the optimization algorithm identified a maximum-efficiency set of parameter values that increased efficiency by 40% above base-case efficiency; a minimum-efficiency set reduced efficiency by about 41%. When maximum-efficiency parameter values were computed as medullary length varied over the physiologic range, the PBE was found to make up 88% of a short medullary cone but only 8% of a long medullary cone.

Luca, Magdalena; Chavez-Ross, Alexandra; Edelstein-Keshet, Leah; Mogilner, Alex

doi: 10.1016/S0092-8240(03)00030-2pmid: 12875339

Chemotactic cells known as microglia are involved in the inflammation associated with pathology in Alzheimer’s disease (AD). We investigate conditions that lead to aggregation of microglia and formation of local accumulations of chemicals observed in AD senile plaques. We develop a model for chemotaxis in response to a combination of chemoattractant and chemorepellent signaling chemicals. Linear stability analysis and numerical simulations of the model predict that periodic patterns in cell and chemical distributions can evolve under local attraction, long-ranged repulsion, and other constraints on concentrations and diffusion coefficients of the chemotactic signals. Using biological parameters from the literature, we compare and discuss the applicability of this model to actual processes in AD.

Till, S.; Till, J.; Milsom, P.; Rowlands, G.

doi: 10.1016/S0092-8240(03)00028-4pmid: 12875340

We describe a new model for laser-induced retinal damage. Our treatment is prompted by the failure of the traditional approach to accurately describe the image size dependence of laser-induced retinal injuries and by a recently reported study which demonstrated that laser injuries to the retina might not appear for up to 48 h post exposure.

Fahlman, Andreas; Kayar, Susan

doi: 10.1016/S0092-8240(03)00038-7pmid: 12875341

We modelled the kinetics of H2 flux during gas uptake and elimination in conscious pigs exposed to hyperbaric H2. The model used a physiological description of gas flux fitted to the observed decompression sickness (DCS) incidence in two groups of pigs: untreated controls, and animals that had received intestinal injections of H2-metabolizing microbes that biochemically eliminated some of the H2 stored in the pigs’ tissues. To analyse H2 flux during gas uptake, animals were compressed in a dry chamber to 24 atm (ca 88% H2, 9% He, 2% O2, 1% N2) for 30–1440 min and decompressed at 0.9 atm min−1 (n = 70). To analyse H2 flux during gas elimination, animals were compressed to 24 atm for 3 h and decompressed at 0.45–1.8 atm min1(n = 58). Animals were closely monitored for 1 h post-decompression for signs of DCS. Probabilistic modelling was used to estimate that the exponential time constant during H2 uptake (τ in) and H2 elimination (τ out) were 79 ± 25 min and 0.76 ± 0.14 min, respectively. Thus, the gas kinetics affecting DCS risk appeared to be substantially faster for elimination than uptake, which is contrary to customary assumptions of gas uptake and elimination kinetic symmetry. We discuss the possible reasons for this asymmetry, and why absolute values of H2 kinetics cannot be obtained with this approach.