A minimal mechanism for bacterial pattern formation.

Colonies of Escherichia coli or Salmonella typhimurium form geometrically complex patterns when exposed to, or feeding on, intermediates of the tricarboxylic acid (TCA) cycle. In response to the TCA cycle intermediate, the bacteria secrete aspartate, a potent chemo-attractant. As a result, the cells form high-density aggregates arranged in striking regular patterns. The simplest are temporary spots formed in a liquid medium by both E. coli and S. typhimurium. In semi-solid medium S. typhimurium forms concentric rings arising from a low-density bacterial lawn, which are either continuous or spotted, whereas E. coli forms complex patterns arising from a dense swarm ring, including interdigitated spots (also called sunflower spirals), radial spots, radial stripes and chevrons. We present a mathematical model that captures all three of the pattern-forming processes experimentally observed in both E. coli and S. typhimurium, using a minimum of assumptions.

Force and deformation on branching rudiments: cleaving between hypotheses.

A mathematical model of the forces and deformations of the tissues involved in branching morphogenesis is developed and solved. The epithelium and mesenchyme are modeled as Stokes fluids separated by an interface. Each fluid is assumed to have constant viscosity. An initially 3-lobed rudiment is deformed by three inwardly directed point forces. Relationships between the physical parameters of the model (tissue viscosity, clefting force, surface tension) and the time course and morphology are explored. We find that the surface tension, clefting force, and viscosity ratio of the two tissues have significant effects on the branching. We conclude that epithelial branching in soft gels is fundamentally different from epithelial branching in mesenchyme, because of the different mechanics. We propose that a complete understanding of branching morphogenesis requires measurements of the mechanical aspects.

Optimizing detection of tissue anisotropy by fluorescence recovery after photobleaching.

Fluorescence recovery after photobleaching (FRAP) has been widely used to measure fluid flow and diffusion in gels and tissues. It has not been widely used in detection of tissue anisotropy. This may be due to a lack of applicable theory, or due to inherent limitations of the method. We discuss theoretical aspects of the relationship between anisotropy of tissue structure and anisotropy of diffusion coefficients, with special regard to the size of the tracer molecule used. We derive a semi-mechanistic formula relating the fiber volume fraction and ratio of fiber and tracer molecule diameters to the expected anisotropy of the diffusion coefficients. This formula and others are tested on simulated random walks through random simulated and natural media. We determine bounds on the applicability of FRAP for detection of tissue anisotropy, and suggest minimum tracer sizes for detection of anisotropy in tissues of different composition (fiber volume fraction and fiber diameter). We find that it will be easier to detect anisotropy in monodisperse materials than in polydisperse materials. To detect mild anisotropy in a tissue, such as cartilage, which has a low fiber fraction would require a tracer molecule so large that it would be difficult to deliver to the tissue. We conclude that FRAP can be used to detect tissue anisotropy when the tracer molecule is sufficiently large relative to the fiber diameter, volume fraction, and degree of polydispersivity, and when the anisotropy is sufficiently pronounced.

Quantifying the significance of phage attack on starter cultures: a mechanistic model for population dynamics of phage and their hosts isolated from fermenting sauerkraut.

We investigated the possibility of using starter cultures in sauerkraut fermentation and thereby reducing the quantity of salt used in the process. This, in turn, would reduce the amount of waste salt that would enter in our water resources. Phage, naturally present in sauerkraut fermentation, could potentially affect the starter cultures introduced. Thus, a mechanistic mathematical model was developed to quantify the growth kinetics of the phage and starter cultures. The model was validated by independent experiments with two Leuconostoc mesenteroides strains isolated from sauerkraut and their corresponding phage. Model simulations and experimental evidence showed the presence of phage-resistant cell populations in starter cultures which replaced phage-sensitive cells, even when the initial phage density (P(0)) and multiplicity of infection (MOI) were low (P(0) < 1 x 10(3) PFU/ml; MOI < 10(-4)) in the MRS media. Based on the results of model simulation and parameter optimization, it was suggested that the kinetic parameters of phage-host interaction, especially the adsorption rate, vary with the initial phage and host densities and with time. The model was validated in MRS broth. Therefore, the effects of heterogeneity and other environmental factors, such as temperature and pH, should be considered to make the model applicable to commercial fermentations.

Multiphase mechanics of capsule formation in tumors.

The presence of a capsule around a tumor is known to be correlated with benign status, and the absence of a capsule often has negative implications for patient prognosis. A mechanical description is presented of the growth of a tumor and the resulting deformations of surrounding normal tissue. A mathematical model of the mechanics is analyzed using physical parameters measured in vivo and in vitro. The model has only three dimensionless parameters, and its results are very robust with respect to parameter variation. We show that the presence of contractility in the surrounding tissue, corresponding to a host defense, can make an existing capsule denser and constrain the tumor better, but cannot be responsible for the observed pressure gradients in encapsulated tumors. Some implications for treatment are discussed.

A mechanism for early branching in lung morphogenesis.

The lung is a highly branched fluid-filled structure, that develops by repeated dichotomous branching of a single bud off the foregut, of epithelium invaginating into mesenchyme. Incorporating the known stress response of developing lung tissues, we model the developing embryonic lung in fluid mechanical terms. We suggest that the repeated branching of the early embryonic lung can be understood as the natural physical consequence of the interactions of two or more plastic substances with surface tension between them. The model makes qualitative and quantitative predictions, as well as suggesting an explanation for such observed phenomena as the asymmetric second branching of the embryonic bronchi.

Analytical model of corneal surgery.

We present a model of the human cornea in order to study the changes in its shape resulting from surgical operations (e.g., radial keratotomy). A simple closed-form solution is given for a thin linearly elastic spherical shell model of the cornea. We assume axisymmetry and isotropy in the shell surface. The surgery is modeled by permitting Young's modulus and shell thickness to depend on position. The analytical nature of the solution permits principal shell curvatures to be explicitly calculated. The model is used to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.

A thin-shell model of the cornea and its application to corneal surgery.

BACKGROUND: We present a thin-shell, analytical model of the vertebrate cornea to study the changes in shape resulting from surgical operations (eg, radial keratotomy). METHODS: A simple closed-form solution is derived for a thin linearly elastic spherical model of the cornea. We assume that the shell is symmetrical about a central axis and that the modulus of elasticity is the same in all directions. The surgery is modeled by allowing the modulus of elasticity (or equivalently the thickness) of the shell to depend upon position, measured as an angle from the axis of symmetry. RESULTS: The analytical nature of the solution allows us to compute the principal curvatures of the cornea explicitly. For example, for representative parameters, the model predicts the average corneal curvature changes from about 43 diopters before keratotomy to about 38 D after keratotomy. CONCLUSIONS: The model is used to estimate Young's modulus from experimental data reported previously by Thomas et al (Invest Ophthalmol Vis Sci 1991;32:1000), as well as to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.

Model and analysis of chemotactic bacterial patterns in a liquid medium.

A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals.

Theoretical analysis of conjugate localization in two-step cancer chemotherapy.

Considerable research has been aimed at improving the efficacy of chemotherapeutic agents for cancer therapy. A promising two-step approach that is designed to minimize systemic drug toxicity while maximizing activity in tumors employs monoclonal antibody-enzyme conjugates for the activation of anti-cancer prodrugs. A mathematical model based on the biology of human 3677 melanoma xenografts in nude mice is presented, analyzed, and numerically simulated to study the biodistribution, pharmacokinetics, and intratumoral localization properties of L49-beta-lactamase fusion proteins in solid tumor masses. The model predictions were compared with published experimental data and an excellent correlation was found to exist. Analytic expressions for the total concentration of conjugate in the tumor, the time at which the concentration is maximal, and the half life of conjugate in the tissue were derived. From these results, key parameters were isolated; and the effects of the tumor vasculature, binding kinetics, and administration schedule were investigated. The antibody-antigen dissociation ratio, the conjugate permeability, and the inter-capillary half distance within the tumor mass were found to strongly influence localization and retention in the tumor. The model was used to examine various dosing strategies in an attempt to determine which regimen would provide the best biodistribution results. The results of administering a uniform dose of conjugate via bolus injection, multiple injections, and continuous infusion were compared. The model predicts that when saturation of binding sites does not occur, dosing strategy has little effect on the amount of conjugate that localizes in the tumor.

Evaluation of feline leukemia virus control measures.

A susceptible-infected-recovered-susceptible (SIRS) model of the epidemiology of feline leukemia virus is formulated and analysed. The dynamics of the disease are dramatically different in no-risk, low-risk and high-risk subpopulations of asocial, free roaming, and multiple cat household cats. Among low risk (<1% prevalence) free roaming cats, the model predicts that an effective immunization rate of 4% year-1, or an effective removal rate of 8% year-1 are adequate to control the disease completely. Under higher risk (10% prevalence) conditions, an effective immunization rate of 23-72% year-1 or a removal rate of 69-145% year-1 are required for control. At very high (30%) prevalence rates, even heroic measures may not suffice to substantially reduce disease prevalence: a vaccination rate of 100% year-1 even if attainable, would only slightly reduce disease prevalence from 30% to 29%. We conclude that the current estimated effective feline leukemia virus immunization rate of 11-19% of the general population is inadequate to provide herd immunity in the subpopulation of cats which are genuinely at risk of infection. A substantial increase in the vaccination rate and/or intensification of test and removal efforts in the at risk population would be required to attain an effective level of protection.

A mechanical model for the formation of vascular networks in vitro.

Endothelial cells, when cultured on gelled basement membrane matrix exert forces of tension through which they deform the matrix and at the same time they aggregate into clusters. The cells eventually form a network of cord-like structures connecting cell aggregates. In this network, almost all of the matrix has been pulled underneath the cell cords and cell clusters. This phenomenon has been proposed as a possible model for the growth and development of planar vascular systems in vitro. Our hypothesis is that the matrix is reorganized and the cellular networks form as a result of traction forces exerted by the cells on the matrix and the latter's elasticity. We construct and analyze a mathematical model based on this hypothesis and examine conditions necessary for the formation of the pattern. We show cell migration is not necessary for pattern formation and that isotropic, strain-stimulated traction is sufficient to form the observed patterns.

Simple versus sophisticated models of breath alcohol exhalation profiles.

For medicolegal purposes, breath alcohol content is typically determined from an end-expiratory sample. Measurements obtained by this method necessarily underestimate the alveolar breath alcohol content, and therefore underestimate the blood alcohol content. We suggest and analyse an improved paradigm which uses the entire time-series of breath alcohol measurements during exhalation, not simply the last recorded value. We present two mathematical models for the exhaling lung, and discuss the implications of each for more accurate and therefore more reliable breath alcohol measurement.

Mathematical and experimental analysis of localization of anti-tumour antibody-enzyme conjugates.

Considerable research has been aimed at improving the efficacy of chemotherapeutic agents for cancer therapy. A promising two-step approach that is designed to minimize systemic drug toxicity while maximizing activity in tumours employs monoclonal antibody (mAb)-enzyme conjugates for the activation of anticancer prodrugs. We present, analyse and numerically simulate a mathematical model based on the biology of the system to study the biodistribution, pharmacokinetics and localization properties of mAb-enzyme conjugates in tumour tissue. The model predictions were compared with experimental observations and an excellent correlation was found to exist. In addition, the critical parameters affecting conjugate half-life were determined to be the inter-capillary half-distance and the antibody-antigen binding affinity. An approximation is presented relating the per cent injected dose per gram to inter-capillary half-distance and time. Finally, the model was used to examine various dosing strategies in an attempt to determine which regimen would provide the best biodistribution results. We compared the results of administering a uniform dose of fusion protein via bolus injection, multiple injections and continuous infusion. The model predicts that dosing strategy has little effect on the amount of conjugate that localizes in the tumour.