A mathematical model of medial collateral ligament repair: migration, fibroblast proliferation and collagen formation

The partial rupture of ligament fibres leads to an injury known as grade 2 sprain. Wound healing after injury consists of four general stages: swelling, release of platelet-derived growth factor (PDGF), fibroblast migration and proliferation and collagen production. The aim of this paper is to present a mathematical model based on reaction-diffusion equations for describing the repair of the medial collateral ligament when it has suffered a grade 2 sprain. We have used the finite element method to solve the equations of this. The results have simulated the tissue swelling at the time of injury, predicted PDGF influence, the concentration of fibroblasts migrating towards the place of injury and reproduced the random orientation of immature collagen fibres. These results agree with experimental data reported by other authors. The model describes wound healing during the 9 days following such injury.     

A mathematical model for photoreceptor interactions

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.     

Osteoblasts and collagen orientation

Summary Bone was removed from the calvaria of anaesthetized 70 g rats or freshly killed young monkeys and the fibrous periosteum dissected off the inner, formative surface under 0.15 M cacodylate buffer. The bone and undisturbed osteoblasts were fixed in 3% glutaraldehyde in the same buffer for 24 to 48 hours, critical point dried and coated with evaporated carbon and gold for scanning electron microscopy (SEM). Fields of osteoblasts were photographed and chosen cells dissected off the osteoid using a tungsten needle. The control of the dissection was made possible by the use of a system of real-time stereo TV-speed SEM. The fields were rephotographed and the orientations of the osteoblasts were compared with that of the underlying collagen fibres. 62% of all osteoblasts lay with their long axes within 15° of the collagen fibre orientation below and 80% within 30°. Montages of large areas of osteoblasts were also made, and then compared with ones of the same area after the cells had been stripped off on adhesive tape. In general, the orientation of the collagen tended to be the same as the cell that formed it. Collagen fibres below cells at the periphery of a domain sometimes had the orientation of the cells in the adjacent patch. It is not possible to determine whether the cells controlled the orientation of the collagen, or vice versa, from this experiment, but other SEM evidence suggests that the collagen orientation in hard tissue matrices depends on the freedom of cells to move with respect to the matrix surface.Acknowledgements. This work has been supported by generous grants from the Medical Research Council and the Science Research Council. We are grateful to Elaine Bailey and Mr. P. Reynolds for technical assistance.     

A mathematical model for dorsal closure

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates).We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied.A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC).We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.     

A mathematical model for calcium homeostasis

In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.     

A mathematical model for radiation-induced myelopoiesis

A model for damage, repair, killing, and repopulation of myelopoietic marrow is presented. Evaluation produces time and dose-rate profiles during and following any complex irradiation. Equations model variable dose rates, multiple exposures, different sources, and arbitrary intervals between treatments. If factors which dominate the control of biological processes can be demonstrated, an option is to set biological rate constants to experimentally determined values. Previously, knowledge did not permit identification of dominating biological processes and their temporal rates. But a unique feature of this study is that unspecified lesions for killing and injury of cells are evaluated from mortality data on the animal species of choice. "Unspecified" is used to indicate a condition of assumption-free modeling of molecular processes, whereby rate constants for cellular effects are simply computed directly from animal mortality data. Coefficients (estimated by maximum-likelihood methods for nonspecific processes) are compared with experimental values for specific processes. The model has many uses, including modeling of the myelopoietic potential as a function of time. Another option is to calculate the whole-body survival curve for cells that control myelopoiesis as a result of the treatment schedule. Also through simple extensions of the model, an extremely complex protocol can be identified with an equivalent prompt dose value--even for partial-body, fractionated exposures.     

A mathematical model for indirectly transmitted diseases

We consider a mathematical model for the indirect transmission via a contaminated environment of a microparasite between two spatially distributed host populations having non-coincident spatial domains. The parasite is benign in a first population and lethal in the second one. Global existence results are given for the resulting reaction-diffusion system coupled with an ordinary differential equation. Then, invasion and persistence of the parasite are studied. A simplified model for the transmission of a hantavirus from bank vole to human populations is then analysed.     

A mathematical model for prokaryotic protein synthesis

A kinetic model for the synthesis of proteins in prokaryotes is presented and analysed. This model is based on a Markov model for the state of the DNA strand encoding the protein. The states that the DNA strand can occupy are: ready, repressed, or having a mRNA chain of length i in the process of being completed. The case i = 0 corresponds to the RNA polymerase attached, but no nucleotides attached to the chain. The Markov model consists of differential equations for the rates of change of the probabilities. The rate of production of the mRNA molecules is equal to the probability that the chain is assembled to the penultimate nucleotide, times the rate at which that nucleotide is attached. Similarly, the mRNA molecules can also be in different states, including: ready and having an amino acid chain of length j attached. The rate of protein synthesis is the rate at which the chain is completed. A Michaelis-Menten type of analysis is done, assuming that the rate of protein degradation determines the ’slow’ time, and that all the other kinetic rates are ‘fast’. In the self-regulated case, this results in a single ordinary differential equation for the protein concentration.     

A mathematical model for vasogenic brain edema

Animals reproducing sexually are faced with a dilemma. They can mate in the local area where they were born, but must then take the risk that they will mate with a close relative and produce offspring suffering from inbreeding depression. Or they can migrate to a new area and mate with an unrelated animal, but then there is a chance that they will die or lose valuable reproductive time during the migration.This dilemma is here described by some simple models. The condition for spreading of a new reproductive strategy in a population is given. The results are discussed using data from the Great Tit, Man and the Japanese Quail on the effect of inbreeding.     

A mathematical model for the immunosenescence.

We propose a model for the dynamics of the immune system by considering the subpopulations of virgin and memory T lymphocytes on a time scale corresponding to the human life span. In the deterministic balance equation we introduce a fluctuating term in order to take into account the chronic antigenic stress. Starting from the hypothesis that the depletion of virgin cells with cytotoxic properties (CD8+) is a mortality marker, the model provides survival curves quite similar to the demographic curves.     

A mathematical model for experimental gene evolution

The purpose of this paper is to determine the optimal mutation rate for random mutagenesis procedures used to make mutant libraries for subsequent screening. When the mutation rate is low, the probability of achieving a rare beneficial mutation is low. When the mutation rate is high, the probability of producing lethal mutations which result in loss of function is also high. We demonstrate that between these two extremes, an optimal mutation rate exists for experimental gene improvement. This rate depends strongly on the number of simultaneous mutations required for a beneficial change of the gene, but only weakly on the number of possible lethal mutations. This model predicts that when mutagenesis is performed at the optimum mutation rate, at least 63% (1--e(-1)) of the cloned genes in a mutant library will be non-functional.     

Relative orientation of collagen molecules within a fibril: a homology model for homo sapiens type I collagen

Type I collagen is an essential extracellular protein that plays an important structural role in tissues that require high tensile strength. However, owing to the molecule’s size, to date no experimental structural data are available for the Homo sapiens species. Therefore, there is a real need to develop a reliable homology model and a method to study the packing of the collagen molecules within the fibril. Through the use of the homology model and implementation of a novel simulation technique, we have ascertained the orientations of the collagen molecules within a fibril, which is currently below the resolution limit of experimental techniques. The longitudinal orientation of collagen molecules within a fibril has a significant effect on the mechanical and biological properties of the fibril, owing to the different amino acid side chains available at the interface between the molecules.     

A quantitative assay for collagen synthesis in microwell fibroblast cultures.

A new method for estimating collagen synthesis in microwell cultures of fibroblasts is presented, ³H-Labeled-proline-collagen was purified by successive salt precipitations at acid and neutral pH in the presence of carrier collagen. Variability between replicates was less than 10% (standard deviation) and recovery of labeled collage internal standards was greater than 90%. More than 90% of recoverable radioactivity was in collagen as demonstrated by polyacrylamide gel electrophoresis and carboxymethyl cellulose chromatography. The culture system is highly reproducible and allows use of a large number of separate cultures with uniformity of culture conditions and economy of reagents.     

A mathematical model for neutrophil gradient sensing and polarization

Directed cell migration in response to chemical cues, also known as chemotaxis, is an important physiological process involved in wound healing, foraging, and the immune response. Cell migration requires the simultaneous formation of actin polymers at the leading edge and actomyosin complexes at the sides and back of the cell. An unresolved question in eukaryotic chemotaxis is how the same chemoattractant signal determines both the cell's front and back. Recent experimental studies have begun to reveal the biochemical mechanisms necessary for this polarized cellular response. We propose a mathematical model of neutrophil gradient sensing and polarization based on experimentally characterized biochemical mechanisms. The model demonstrates that the known dynamics for Rho GTPase and phosphatidylinositol-3-kinase (PI3K) activation are sufficient for both gradient sensing and polarization. In particular, the model demonstrates that these mechanisms can correctly localize the “front” and “rear” pathways in response to both uniform concentrations and gradients of chemical attractants, including in actin-inhibited cells. Furthermore, the model predictions are robust to the values of many parameters. A key result of the model is the proposed coincidence circuit involving PI3K and Ras that obviates the need for the “global inhibitors” proposed, though never experimentally verified, in many previous mathematical models of eukaryotic chemotaxis. Finally, experiments are proposed to (in)validate this model and further our understanding of neutrophil chemotaxis.