Bulletin of Mathematical Biology

Hu, Kai

doi: 10.1016/j.bulm.2004.05.008pmid: 15710179

This paper considers the three-dimensional structure of B-form DNA. The molecule may be open or covalently closed. For the former, its two ends are not allowed to move or rotate freely in space unless the molecule is under the influence of rigid body motions of the ambient space. Implied by the elastic rod model for DNA, the molecule writhes immediately when subject to a terminal twist as long as its axis is none of the following curves: lines, circular arcs, circular helices. This result is remarkably different from well-known results about DNA of other conformations. For example, if a DNA is regarded as an elastic rod whose axis is a circle, then it has no induced writhe when subject to a terminal twist until the latter meets a critical extent.

Zheng, X.; Wise, S.; Cristini, V.

doi: 10.1016/j.bulm.2004.08.001pmid: 15710180

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.

Willson, Stephen

doi: 10.1016/j.bulm.2004.07.007pmid: 15710181

The method of minimum evolution reconstructs a phylogenetic tree T for n taxa given dissimilarity data d. In principle, for every tree W with these n leaves an estimate for the total length of W is made, and T is selected as the W that yields the minimum total length. Suppose that the ordinary least-squares formula S W (d) is used to estimate the total length of W. A theorem of Rzhetsky and Nei shows that when d is positively additive on a completely resolved tree T, then for all W ≠ T it will be true that S W (d) > S T (d). The same will be true if d is merely sufficiently close to an additive dissimilarity function. This paper proves that as n grows large, even if the shortest branch length in the true tree T remains constant and d is additive on T, then the difference S W (d)-S T (d) can go to zero. It is also proved that, as n grows large, there is a tree T with n leaves, an additive distance function d T on T with shortest edge ε, a distance function d, and a tree W with the same n leaves such that d differs from d T by only approximately ε/4, yet minimum evolution incorrectly selects the tree W over the tree T. This result contrasts with the method of neighbor-joining, for which Atteson showed that incorrect selection of W required a deviation at least ε/2. It follows that, for large n, minimum evolution with ordinary least-squares can be only half as robust as neighbor-joining.

Mistro, Diomar; Rodrigues, Luiz; Ferreira, Wilson

doi: 10.1016/j.bulm.2004.07.006pmid: 15710182

A general mathematical model for population dispersal featuring long range taxis is presented and exemplified by the dispersal episode of the Africanized honey bees (Apis mellifera adansonii) throughout the American Continent. The mathematical model is a discrete-time and nonlocal model represented by an integrodifference recursion. A newtaxis concept is defined and introduced into the mathematical model by an appropriate modification of the redistribution kernel. The model is capable of predicting the natural barrier for the expansion of the Africanized honey bees in the southern part of the Continent due to low winter temperatures. It also describes a sensitive expansion velocity with respect to the quality of resources, which can explain the AHB’s astounding spread rate, by using two different kinds of population dynamics strategies, one for a resourceful environment and the other for poor regions.

Sun, Shuyu; Wheeler, Mary; Obeyesekere, Mandri; Patrick, Charles

doi: 10.1016/j.bulm.2004.07.004pmid: 15710183

Understanding the formation and structure of a capillary network is critical for any reparative strategy since the capillary network dictates tissue survival, hemodynamics, and mass transport. Vascular assembly and patterning has largely been studied using a reductionist approach where a particular endothelial cell molecular pathway or cellular mechanism is investigated as a relatively closed system. This trend of research has yielded a staggering wealth of genes, proteins, and cells that play critical roles in angiogenesis and some have resulted in successful targeted angiogenic therapies. However, these genes, proteins, and cells do not exist in discrete closed systems, rather they are intimately coupled across spatial and temporal dimensions. Designing experiments to study a single or group of perturbations is fraught with confounding complications. An integrative tool is required that incorporates gene, protein, and cell information and appropriately describes the complex systems behavior of vascular assembly and patterning.

Wang, Ruiqi; Jing, Zhujun; Chen, Luonan

doi: 10.1016/j.bulm.2004.07.005pmid: 15710184

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.

Woodroffe, P.; King, J.; Varley, C.; Southgate, J.

doi: 10.1016/j.bulm.2004.08.006pmid: 15710185

Urothelial cells line the bladder. If the urothelium is damaged, it is vital that it repairs itself quickly. Experimental results shedding light on how this repair process works are presented, revealing in particular the dependence of the response on the length of time for which the drug Troglitazone (TZ) is applied. A simple mathematical model for the basic mechanism (comprising ordinary differential equations) is then developed and analysed, seeking specifically to clarify and quantify the mechanisms governing the dependence of the cell differentiation response on the TZ administration time, rather than providing a comprehensive model of differentiation. Through biologically justified simplifications, analysis reveals that the model gives results in accord with the experimental observations, and suggests new experiments that may aid further understanding. Directions in which this preliminary modelling of the PPARγ (peroxisome proliferator activated receptor γ) pathway could be usefully extended are also indicated.

Sánchez-Garduño, Faustino

doi: 10.1016/j.bulm.2004.09.001pmid: N/A