An Off-Lattice Hybrid Discrete-Continuum Model of Tumor Growth and Invasion

We have developed an off-lattice hybrid discrete-continuum (OLHDC) model of tumor growth and invasion. The continuum part of the OLHDC model describes microenvironmental components such as matrix-degrading enzymes, nutrients or oxygen, and extracellular matrix (ECM) concentrations, whereas the discrete portion represents individual cell behavior such as cell cycle, cell-cell, and cell-ECM interactions and cell motility by the often-used persistent random walk, which can be depicted by the Langevin equation. Using this framework of the OLHDC model, we develop a phenomenologically realistic and bio/physically relevant model that encompasses the experimentally observed superdiffusive behavior (at short times) of mammalian cells. When systemic simulations based on the OLHDC model are performed, tumor growth and its morphology are found to be strongly affected by cell-cell adhesion and haptotaxis. There is a combination of the strength of cell-cell adhesion and haptotaxis in which fingerlike shapes, characteristic of invasive tumor, are observed.     

Symmetry and entropy of biological patterns: discrete Walsh functions for 2D image analysis

To quantify symmetry and entropy inherent in the discrete patterns such as spatial self-organization in cell sorting and mussel bed ecosystems, we introduce the discrete Walsh analysis. This analysis enables us to estimate the degree of the complicated symmetry, and to extract the symmetry from the pattern that seems to be asymmetric. The results obtained in this paper are summarized as follows. (I) The geometrical patterns of the cell sorting become systematic with the predominance of the particular symmetry. This implies that not only the entropy but also the particular symmetry can decrease in the biological process. (II) The magnitude of the symmetry is related to the absolute value of the adhesion, and the type of the symmetry is related to the sign of the adhesion. That is, centro-symmetry dominates in the cell sorting pattern caused by large negative adhesion, and double symmetry dominates in the pattern caused by large positive adhesion. (III) Spatial self-organization in mussel bed is accompanied by the decreasing of the centro-symmetry. This implies that the positive "adhesion" between mussel individuals increases with time. (IV) In the biological process, the Curie symmetry breaking occurs at intervals.     

Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices

Recently there has been a great deal of interest within the ecological community about the interactions of local populations that are coupled only by dispersal. Models have been developed to consider such scenarios but the theory needed to validate model outcomes has been somewhat lacking. In this paper, we present theory which can be used to understand these types of interaction when population exhibit discrete time dynamics. In particular, we consider a spatial extension to discrete-time models, known as coupled map lattices (CMLs) which are discrete in space. We introduce a general form of the CML and link this to integro-difference equations via a special redistribution kernel. General conditions are then derived for dispersal-driven instabilities. We then apply this theory to two discrete-time models; a predator-prey model and a host-pathogen model.     

Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.     

Stochastic Hierarchical Systems: Excitable Dynamics

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.     

New insights into the nature of the wool fibre surface

Wool fibres have been treated to remove the covalently bound lipid and characterised using lipid analysis, wettability and scanning probe microscopy. A model substrate consisting of alternating stripes of hydrophobic (predominantly CH₃ terminated molecules) and hydrophilic (COOH terminated molecules) surfaces, micro-printed onto a gold-coated mica surface was assessed using the SPM techniques of adhesion, friction and phase imaging and showed that SPM can easily distinguish these surfaces. When KOH/methanol treated wool fibres were examined, SPM showed an increase in coefficient of friction and a decrease in adhesion as the lipid is removed. The increased friction is consistent with studies on the model surface and confirms the hypothesis that the lipid layer decreases the surface friction of fibres. The decreased adhesion is consistent with results in the literature on hair, but is at odds with the results on the model surface. The strong contrast shown between the methyl and carboxylic acid surfaces in the friction image of the micro-patterned surface, and the complete absence of any such contrast developing with time of treatment of the wool fibres strongly suggests that the surface lipid is not present as a discrete outer layer on the fibre. A new model is proposed in which the lipid is intimately associated with the surface proteins and allows for changes in lipid concentration at the surface in response to changes in environmental conditions.     

A model for rouleaux pattern formation of red blood cells

Human red blood cells (RBCs) in a solution form rouleaux patterns under various conditions. The degree of rouleaux formation depends on, for example, the concentration and molecular weight of added large molecules. We present a two-dimensional discrete cellular space model in which an RBC is represented by a rectangle and differential adhesion is assumed among the longer (a-site), the shorter (b-site) sides of the rectangle and the solvent. The total sum of the adhesion energy is assumed to guide the step-by-step change of the model cell configuration and also define absolutely stable patterns. We compare the set of absolutely stable patterns and cell aggregate patterns for both actual and computer-simulated cases to obtain the basic validity of our framework. Then we proceed to assess the effects of added high polymers to the adhesion parameters. We first note that under suitable conditions, decrease in a-site-solvent affinity is necessary to have complex patterns rather than increase of a-a affinity. The hypothesis that addition of high polymers reduce the a-site-solvent affinity is concomitant with a newly proposed osmotic stress theory. The parameter fitting results for the experimental phase change curves can also be interpreted as supporting more the new theory than existing traditional explanations.     

Proteomic analysis of the reactive phenotype of astrocytes following endothelin-1 exposure

Reactive gliosis is an invariant feature of the pathology of central nervous system (CNS) injury and a major determinant of neuronal survival and regeneration. To begin to understand the alterations in astrocyte protein expression that drive glial changes that occur following injury, we used an established model system (endothelin-1 stimulation of hypertrophy) and proteomic analysis to define a discrete set of differentially expressed proteins and post-translational modifications that occur as the astrocytes change from a quiescent to a reactive state. This orchestrated set of changes included proteins involved in cytoskeletal reorganization (caldesmon, calponin, alpha B-crystallin, stathmin, collapsing response mediator protein-2), cell adhesion (vinculin, galectin-1), signal transduction (RACK-1) and astrocyte differentiation (glutamine synthetase). Using proteomic analysis to understand what drives astrocyte expression of these functionally divergent molecules may offer insight into the mechanisms by which astrocytes can exhibit both pro-regenerative and anti-regenerative activities following CNS injury.     

Formal modeling and analysis of the MAL-associated biological regulatory network: insight into cerebral malaria

The discrete modeling formalism of René Thomas is a well known approach for the modeling and analysis of Biological Regulatory Networks (BRNs). This formalism uses a set of parameters which reflect the dynamics of the BRN under study. These parameters are initially unknown but may be deduced from the appropriately chosen observed dynamics of a BRN. The discrete model can be further enriched by using the model checking tool HyTech along with delay parameters. This paves the way to accurately analyse a BRN and to make predictions about critical trajectories which lead to a normal or diseased response. In this paper, we apply the formal discrete and hybrid (discrete and continuous) modeling approaches to characterize behavior of the BRN associated with MyD88-adapter-like (MAL)--a key protein involved with innate immune response to infections. In order to demonstrate the practical effectiveness of our current work, different trajectories and corresponding conditions that may lead to the development of cerebral malaria (CM) are identified. Our results suggest that the system converges towards hyperinflammation if Bruton's tyrosine kinase (BTK) remains constitutively active along with pre-existing high cytokine levels which may play an important role in CM pathogenesis.     

A Delay Differential Model for Pandemic Influenza with Antiviral Treatment

The use of antiviral drugs has been recognized as the primary public health strategy for mitigating the severity of a new influenza pandemic strain. However, the success of this strategy requires the prompt onset of therapy within 48 hours of the appearance of clinical symptoms. This requirement may be captured by a compartmental model that monitors the density of infected individuals in terms of the time elapsed since the onset of symptoms. We show that such a model can be expressed by a system of delay differential equations with both discrete and distributed delays. The model is analyzed to derive the criterion for disease control based on two critical factors: (i) the profile of treatment rate; and (ii) the level of treatment as a function of time lag in commencing therapy. Numerical results are also obtained to illustrate the feasible region of disease control. Our findings show that due to uncertainty in the attack rate of a pandemic strain, initiating therapy immediately upon diagnosis can significantly increase the likelihood of disease control and substantially reduce the required community-level of treatment. This suggests that reliable diagnostic methods for influenza cases should be rapidly implemented within an antiviral treatment strategy.     

Genomic analysis of C-type lectins

Many biological effects of complex carbohydrates are mediated by lectins that contain discrete carbohydrate-recognition domains. At least seven structurally distinct families of carbohydrate-recognition domains are found in lectins that are involved in intracellular trafficking, cell adhesion, cell-cell signalling, glycoprotein turnover and innate immunity. Genome-wide analysis of potential carbohydrate-binding domains is now possible. Two classes of intracellular lectins involved in glycoprotein trafficking are present in yeast, model invertebrates and vertebrates, and two other classes are present in vertebrates only. At the cell surface, calcium-dependent (C-type) lectins and galectins are found in model invertebrates and vertebrates, but not in yeast; immunoglobulin superfamily (I-type) lectins are only found in vertebrates. The evolutionary appearance of different classes of sugar-binding protein modules parallels a development towards more complex oligosaccharides that provide increased opportunities for specific recognition phenomena. An overall picture of the lectins present in humans can now be proposed. Based on our knowledge of the structures of several of the C-type carbohydrate-recognition domains, it is possible to suggest ligand-binding activity that may be associated with novel C-type lectin-like domains identified in a systematic screen of the human genome. Further analysis of the sequences of proteins containing these domains can be used as a basis for proposing potential biological functions.     

Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model

We develop a discrete model of malignant invasion using a thermodynamic argument. An extension of the Potts model is used to simulate a population of malignant cells experiencing interactions due to both homotypic and heterotypic adhesion while also secreting proteolytic enzymes and experiencing a haptotactic gradient. In this way we investigate the influence of changes in cell-cell adhesion on the invasion process. We demonstrate that the morphology of the invading front is influenced by changes in the adhesiveness parameters, and detail how the invasiveness of the tumour is related to adhesion. We show that cell-cell adhesion has less of an influence on invasion compared with cell-medium adhesion, and that increases in both proteolytic enzyme secretion rate and the coefficient of haptotaxis act in synergy to promote invasion. We extend the simulation by including proliferation, and, following experimental evidence, develop an algorithm for cell division in which the mitotic rate is explicitly related to changes in the relative magnitudes of homotypic and heterotypic adhesiveness. We show that although an increased proliferation rate usually results in an increased depth of invasion into the extracellular matrix, it does not invariably do so, and may, indeed, cause invasiveness to be reduced.     

Regulatory mechanisms required for DE-cadherin function in cell migration and other types of adhesion

Cadherin-mediated adhesion can be regulated at many levels, as demonstrated by detailed analysis in cell lines. We have investigated the requirements for Drosophila melanogaster epithelial (DE) cadherin regulation in vivo. Investigating D. melanogaster oogenesis as a model system allowed the dissection of DE-cadherin function in several types of adhesion: cell sorting, cell positioning, epithelial integrity, and the cadherin-dependent process of border cell migration. We generated multiple fusions between DE-cadherin and alpha-catenin as well as point-mutated beta-catenin and analyzed their ability to support these types of adhesion. We found that (1) although linking DE-cadherin to alpha-catenin is essential, regulation of the link is not required in any of these types of adhesion; (2) beta-catenin is required only to link DE-cadherin to alpha-catenin; and (3) the cytoplasmic domain of DE-cadherin has an additional specific function for the invasive migration of border cells, which is conserved to other cadherins. The nature of this additional function is discussed.     

Detailed mechanics of membrane-membrane adhesion and separation. I. Continuum of molecular cross-bridges

The mechanics of membrane-membrane adhesion are developed for the approximation that the molecular cross-bridging forces are continuously distributed as a normal stress (force per unit area). The significance of the analysis is that the finite range of the cross-bridging forces and the microscopic contact angle are not assumed negligible. Since the cross-bridging and adhesion forces are finite range interactions, there are two membrane regions: a free zone where the membranes are not subject to attractive forces; and an adherent zone where the membranes are held together by attractive stresses. The membrane is treated as an elastic continuum. The approach is to analyze the mechanics for each zone separately and then to require continuity of the solutions at the interface between the zones. Final solution yields the membrane contour and stresses proximal to and within the contact zone as well as the microscopic contact angle at the edge of the contact zone. It is demonstrated that the classical Young equation is consistent with this model. The results show that the microscopic contact angle becomes appreciable when the strength of adhesion is large or the length of the cross-bridge is large; however, the microscopic contact angle approaches zero as the membrane elastic stiffness increases. The solution predicts the width of the contact zone over which molecular bonds are stretched. It is this boundary region where increased biochemical activity is expected. In the classical model presented here, the level of tension necessary to oppose spreading of the contact is equal to the minimal level of tension required to separate the adherent membranes. This behavior is in contrast with that derived for the case of discrete molecular cross-bridges where the possibility of different levels of tension associated with adhesion and separation is introduced. The discrete cross-bridge case is the subject of a companion paper.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

Generalized linear mixture models for handling nonignorable dropouts in longitudinal studies

This paper presents a method for analysing longitudinal data when there are dropouts. In particular, we develop a simple method based on generalized linear mixture models for handling nonignorable dropouts for a variety of discrete and continuous outcomes. Statistical inference for the model parameters is based on a generalized estimating equations (GEE) approach (Liang and Zeger, 1986). The proposed method yields estimates of the model parameters that are valid when nonresponse is nonignorable under a variety of assumptions concerning the dropout process. Furthermore, the proposed method can be implemented using widely available statistical software. Finally, an example using data from a clinical trial of contracepting women is used to illustrate the methodology.