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In this chapter we briefly discuss the results of a mathematical model formulated in [22] that incorporates many processes associated with tumour growth. The deterministic model, a system of coupled non-linear partial differential equations, is a combination of two previous models that describe the tumour-host interactions in the initial stages of growth [11] and the tumour angiogenic process [6]. Combining these models enables us to investigate combination therapies that target different aspects of tumour growth. Numerical simulations show that the model captures both the avascular and vascular growth phases. Furthermore, we recover a number of characteristic features of vascular tumour growth such as the rate of growth of the tumour and invasion speed. We also show how our model can be used to investigate the effect of different anti-cancer therapies. [ABSTRACT FROM AUTHOR]

The analysis of gene expression profiles of cells and tissues, performed by DNA microarray technology, strongly relies on proper bioinformatical methods of data analysis. Due to the large number of analyzed variables (genes) and the usually low number of cases (arrays) in one experiment, limited by the high cost of the technology, the biological reasoning is difficult without previous analysis, leading to a reduction of the problem dimensionality. A wide variety of methods have been developed; the most useful, from a biological point of view, are methods of supervised gene selection withestimation of false discovery rate. However, supervised gene selection is not always satisfying for the user of microarray technology, as the complexity of biological systems analyzed by microarrays rarely can be explained by one variable. Among unsupervised methods of analysis, hierarchical clustering and principal component analysis (PCA) have gained wide biological application. In our opinion, singular value decomposition (SVD) analysis, which is similar to PCA, has additional advantages that are very essential for the interpretation of the biological data. In this chapter we shall present how to apply SVD to unsupervised analysis of transcriptome data obtained by oligonucleotide microarrays. These results have been derived from several experiments, carried out at the DNA oligonucleotide microarray Laboratory at the Institute of Oncology, Gliwice, and are currently analyzed from a biological point of view. [ABSTRACT FROM AUTHOR]

In this chapter we present the decomposition approach to the analysis of large gene expression profile data sets. We address the problem of analysis of transient time-course data of expression profiles. We accept the assumption that co-expression of genes can be related to their belonging to the same Gaussian component. We assume that parameters of Gaussian components, means and variances, can differ between time instants. However, the gene composition of components is unchanged between time instants. For such problem formulations we derive the appropriate version of expectation-maximization algorithm recursions for the estimation of model parameters.We apply the derived method to the data on gene expression profiles of human K562 erythroleukemic cells and we discuss the obtained gene clustering. [ABSTRACT FROM AUTHOR]

In vivo applications of biocompatible magnetic nanoparticles in a carrier liquid controlled by an external magnetic field from outside the body have recently been proposed for specific drug delivery such as in locoregional cancer therapies or the occlusion aneurysms. They can also be used as guided contrast agents in myocardial imaging after myocardial infarction. However, the choice of the optimal clinical setting still remains a challenge for all of the mentioned applications. A numerical heterogeneous multiscale model can be used for an optimal a priori determination of the free parameters and might help to overcome this problem. [ABSTRACT FROM AUTHOR]

We discuss a model for the interaction between the parasite Trypanosoma cruzi and the immune system in Chagas disease, by separately describing the intracellular and extracellular parasite stages. The solution of the case where two antibody species are active is worked out in detail, and a diagram showing the differents outcomes of the model is obtained. Our predictions accurately reproduce experimental data on the infection evolution during the acute phase of the disease and lead to an estimate of the damage generated by direct parasite action [ABSTRACT FROM AUTHOR]

The inference of regulatory networks from microarray data relies on expression measures to identify gene activity patterns. However, currently existing xpression measures are not the direct measurements of mRNA concentration one would ideally need for an accurate determination of gene regulation. If the development of expression measures is to advance to the point where absolute target concentrations can be estimated, it is essential to have an understanding of physical processes leading to observed microarray data. We survey here the performance of existing expression measures for oligonucleotide microarrays and describe recent progress in developing physical dynamic adsorption models relating measured fluorescent dye intensities to underlying target mRNA concentration. [ABSTRACT FROM AUTHOR]

We address the following question: Can one sustain, on the basis of mathematicalmodels, that for cancer cells, the loss of control by circadian rhythm favours a faster populationgrowth? This question, which comes from the observation that tumour growth in mice isenhanced by experimental disruption of the circadian rhythm, may be tackled by mathematicalmodelling of the cell cycle. For this purpose we consider an age-structured population modelwith control of death (apoptosis)rates and phase transitions, and two eigenvalues: one for periodic control coefficients (via a variant of Floquet theory in infinite dimension)and one forconstant coefficients (taken as the time average of the periodic case). We show by a direct proofthat, surprisingly enough considering the above-mentioned observation, the periodic eigenvalue is always greater than the steady state eigenvalue when the sole apoptosis rate is concerned. We also show by numerical simulations when transition rates between the phases of the cell cycle are concerned, that, without further hypotheses, no natural hierarchy between the two eigenvalues exists. This at least shows that, if such models are to take into account the above-mentioned observation, control of death rates inside phases is not sufficient, and that transition rates between phases are a key target in proliferation control. [ABSTRACT FROM AUTHOR]

We consider the numerical simulation of a time-dependent taxis-diffusion-reaction model of fracture healing in mice using the method of lines. The partial differential equation problem has an axi-symmetric structure and this is employed to properly reduce the model to an equivalent problem in two-dimensional (2D) space leading subsequently to an efficient spatial discretisation. Special care is given to respect conservation of mass and the non-negativity of the solution. The numerical simulation results are contrasted to those obtained from a simplistic reduction of the axi-symmetric model to 2D space (at the same computational cost).We observe quantitative and qualitative differences. [ABSTRACT FROM AUTHOR]

We have shown for a dataset of computationally predicted alternative splice sites how inherent information can be utilized to validate the predictions by applying statistics on different features typical for splice sites. As a promising splice site feature we investigated the frequencies of binding motifs in the context of exonic and intronic splice site flanks and between the alternative and reference splice sites. We show that both partitions of splice sites can statistically be separated not only by their distance to the splice signal consensus but also via frequencies of splice regulatory protein (SRp) binding motifs in the splice site environment. [ABSTRACT FROM AUTHOR]

A mathematical model of water flow between dialysis fluid in the peritoneal cavity and blood through the capillary wall and homogeneous interstitium driven by high hydrostatic and osmotic pressure of dialysis fluid is formulated. The model is based on nonlinear equations of reaction-diffusion-convection type. Numerical simulations provide the distribution profiles for hydrostatic pressure, glucose concentration, and water flux in the tissue for different times from the infusion of dialysis fluid into the peritoneal cavity for different transport parameters that represent clinical treatments of peritoneal dialysis. [ABSTRACT FROM AUTHOR]

We introduce a score to classify the severity of patients by analysing the information content of clinical time series. [ABSTRACT FROM AUTHOR]

Atherosclerotic lesions are predominantly localised to arterial bifurcations and bends, and are highly correlated with areas of low wall shear stress (WSS), but the underlying reason for this localisation is not fully understood. A key role is played by the endothelial cells, which regulate the transport of materials from the bloodstream to the artery wall and secrete vasoactive agents that modulate vascular tone. A mathematical model is presented, exploring the link between arterial geometry, WSS and factors related to atherogenesis. The model simulates the cellular response to the fluid shear stress on the cell membrane and the binding of ligands to cell surface receptors. This is used to calculate the rate of production of nitric oxide (NO), which is a potent vasodilator and anti-atherogenic factor. It is hypothesised that the section of endothelium adjacent to a region of recirculating flow is most at risk of developing atherosclerotic plaque, due to reduced bioavailability of NO. [ABSTRACT FROM AUTHOR]

A quantitative understanding of the bone remodeling process is of considerable biomedical and practical biotechnological interest to support the application of layer manufacturing techniques to produce scaffolds for surgical applications. Osteoclasts and osteoblasts play a principal role in different models of the bone multicellular unit operating in bone and display a rich spectrum of behaviour. The goal of this work is to show that it is possible to capture the cyclic dynamics of operating cells. The central idea of the mathematical model is that the regulatory nature of osteocytes is the basis of the cyclic-like behaviour associated with the system (remodeling process)as a whole. We developed this model taking due account of the apoptosis of osteocytes as a possible regulation loop in bone remodeling control. By applying the ordinary differential equations technique to the model, we show cyclic modes over a wide range of constants that have clear biological relevance. Simulations show that for a particular range of constants the model exhibits a torus-like quasi-steady state. Further investigation into these simulations indicates the existence of a surface in the osteoclasts-osteoblasts-osteocytes-bone space, that could be interpreted as a conservative value confirming the substrate-energy regenerative capability of the bone remodeling system. It is suggested that the nature of this recovering potential is directed against both mechanical and biochemical damage to the bone. [ABSTRACT FROM AUTHOR]

Free boundary problems associated with biological tissue growing under conditions of nutrient limitation are formulated. Analysis by linear-stability methods, clarifying the models' stability properties, is then described. [ABSTRACT FROM AUTHOR]

Pigmentation patterns in butterfly wings are one of the most spectacular and vivid examples of pattern formation in biology. In this chapter, we devote our attention to the mechanisms for generating global patterns. We focus on the relationship between pattern forming mechanisms for the fore- and hindwing patterns. Through mathematical modeling and computational analysis of Papilio dardanus and polytes, our results indicate that the patterns formed on the forewing need not correlate to those of the hindwing in the sense that the formation mechanism is the same for both patterns. The independence of pattern formation mechanisms means that the coordination of united patterns of fore- and hindwings is accidental. This is remarkable, because from Oudemans's principle [10], patterns appearing on the exposed surface of fore- and hindwing at the natural resting position are often integrated to form a composite and united adaptive pattern with their surrounding environment. [ABSTRACT FROM AUTHOR]

The theory of chemical organizations is employed as a novel method to analyze and understand biological network models. The method allows us to decompose a chemical reaction network into sub-networks that are (algebraically) closed and self-maintaining. Such sub-networks are termed organizations. Although only stoichiometry is considered to compute organizations, the analysis allows us to narrow down the potential dynamic behavior of the network: organizations represent potential steady state compositions of the system. When applied to a model of sugar metabolism in E. coli including gene expression, signal transduction, and enzymatic activities, some organizations are found to coincide with inducible biochemical pathways. [ABSTRACT FROM AUTHOR]

In our previous study [3] we described the method for a direct three-dimensional (3D) computer simulation of ferredoxin-dependent cyclic electron transport around the photosystem 1 pigment-protein complex. Simulations showed that the spatial organization of the system plays a significant role in shaping the kinetics of the redox turnover of P700 (the reaction center of a photosystem 1 pigment-protein complex). In this paper we develop the direct 3D model of cyclic electron transport and apply it to study the nature of fast and slow components of the P700+ dark reduction process. We demonstrate that the slow phase of this process is diffusion controlled and is determined by the diffusion of reduced plastoquinone and plastocyanin molecules from the granal to the stromal areas of the thylakoid membrane. [ABSTRACT FROM AUTHOR]

A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor-specific capillary network. The emerging morphology, characterized by the compartmentalization of the tumor into several regions differing in vessel density, diameter and degree of tumor necrosis, is in accordance with experimental data for human melanoma. Vessel collapse, due to a combination of severely reduced blood flow and solid stress exerted by the tumor, leads to a correlated percolation process that is driven towards criticality by the mechanism of hydrodynamic vessel stabilization. [ABSTRACT FROM AUTHOR]

Glioblastoma is the most malignant form of brain cancer. It is extremely invasive; the mechanisms that govern invasion are not well understood. To better understand the process of invasion, we conducted an in vitro experiment in which a 3D tumor spheroid is implanted into a collagen gel. The paths of individual invasive cells were tracked. These cells were modeled as radially biased, persistent random walkers. The radial velocity bias was found to be 19.6 μm/hr. [ABSTRACT FROM AUTHOR]

A simple mathematical model is proposed to study the influence of cell fission on transport. The model describes fractional tumor development, which is a one-dimensional continuous time random walk (CTRW). An answer to the question of how the malignant neoplasm cells appear at an arbitrary distance from the primary tumor is proposed. The model is a possible consideration for diffusive cancers as well. A chemotherapy influence on the CTRW is studied by an observation of stationary solutions. [ABSTRACT FROM AUTHOR]

We present a modified first-order backward Euler finite difference scheme to solve advection-reaction-diffusion systems on fixed and continuously deforming domains. We compare our scheme to the second-order semi-implicit backward finite differentiation formula and conclude that for the type of equations considered, the first-order scheme has a larger region of stability for the time step than that of the second-order scheme (at least by a factor of ten) and therefore the first-order scheme becomes a natural choice when solving advection-reactiondiffusion systems on growing domains. [ABSTRACT FROM AUTHOR]

Swarming pattern formation of self-propelled entities is a prominent example of collective behavior in biology. Here we focus on bacterial swarming and show that the rod shape of self-propelled individuals is able to drive swarm formation without any kind of signaling. The underlying mechanism we propose is purely mechanical and is evidenced through two different mathematical approaches: an on-lattice and an off-lattice individual-based model. The intensities of swarm formation we obtain in both approaches uncover that the length-width aspect ratio controls swarm formation. Moreover we show that there is an optimal aspect ratio that favors swarming. [ABSTRACT FROM AUTHOR]

Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the formation of regularly spaced "islands" of cartilage analogous to the cartilage primordia of the developing limb skeleton. In this chapter a discrete, multiscale agent-based stochastic model is described, which is based on an extended cell representation coupled with biologically motivated reaction-diffusion processes and cell-matrix adhesion, for studying the behavior of limb bud precartilage mesenchymal cells. The model is calibrated using experimental data, and the sensitivity of key parameters is studied. [ABSTRACT FROM AUTHOR]

The adaptation of the growth of Escherichia coli to the availability of a carbon source is controlled by a complex genetic regulatory network whose functioning is still little understood. Using a qualitative method based on piecewise-linear differential equations, which is able to overcome the current lack of quantitative data on kinetic parameters and molecular concentrations, we model the carbon starvation response network and simulate the response of E. coli cells to carbon deprivation. This allows us to identify essential features of the transition between exponential and stationary phase and to make new predictions on the qualitative system behavior, following a carbon upshift. [ABSTRACT FROM AUTHOR]

A transition network (TN) is a graph-theoretical concept describing the transitions between (meta)stable states of dynamical systems. Here, we review methods to generate and analyze a TN for molecular systems. The appropriate identification of states and transitions from the potential energy surface of the molecule is discussed. We describe a formalism transforming a TN on a static energy surface into a time-dependent dynamic TN that yields the population probabilities for each system state and the interstate transition rates. Three analysis methods that help in understanding the dynamics of the molecular system based on the TN are discussed: (1) Disconnectivity graphs allow important features of the energy surface captured in a static TN to be visualized, (2) graph-theoretical methods enable the computation of the best transition paths between two predefined states of the TN, and (3) statistical methods from complex network analysis identify important features of the TN topology. A broad review of the literature is given, and some open research directions are discussed. [ABSTRACT FROM AUTHOR]

Gene expression microarrays have become a popular high-throughput technique in functional genomics. By enabling the monitoring of thousands of genes simultaneously, this technique holds enormous potential to extend our understanding of various biological processes. However, the large amount of data poses a challenge when interpreting the results. Moreover, microarray data often contain frequent missing values, which may drastically affect the performance of different data analysis methods. Therefore, it is essential to effectively exploit additional biological information when analyzing and interpreting the data. In the present study, we investigate the relationship between gene expression profile and promoter sequence profile in the context of missing value imputation. In particular, we demonstrate that the selection of predictive genes for expression value estimation can be considerably improved by the incorporation of transcription factor binding information. [ABSTRACT FROM AUTHOR]

The chemical master equation in combination with chemical rate equations is used as a tool to study Markovian models of genetic regulatory networks in prokaryotes. States of the master equation represent the binding and unbinding of protein complexes to DNA, resulting in a gene being expressed or not expressed in a cell, while protein and substrate concentrations are represented by continuum variables which evolve via differential equations. The model is applied to a moderately complex biological system, the switching mechanism of the Bacteriophage driven by competition between production of CI and Cro proteins. Numerical simulations of the model successfully move between lysogenic and lytic states as the host bacterium is stressed by the application of ultraviolet light. [ABSTRACT FROM AUTHOR]

Formal kinetic methods to analyze biocatalytic systems are traditionally based on the law of mass action. This law involves the assumption that each molecular state has an exponentially distributed lifetime. We regard this assumption as unduly restrictive and advocate a more general, service theory based approach (termed mass service kinetics or, briefly, service kinetics). In service-theoretic terms biocatalysts are servers and their ligands are customers. The time intervals between arrivals of ligand molecules at special service loci (active or binding sites) as well as the service periods at these loci need not be exponentially distributed; rather, they may adopt any distribution (e.g., Erlangian, hyperexponential, variomorphic). We exemplify the impact of nonexponential time distributions on a performance measure of wide interest: the steady-state throughput. For its computation we use matrix-analytic methods. Specifically, we show that nonexponential interarrival times convert hyperbolic mass action systems (whose characteristic is a hyperbolic velocity-concentration or dose-response curve) into nonhyperbolic mass service systems and that the type and extent of their nonhyperbolicity are determined by the type and parameters of the interarrival time distribution. Furthermore, we analyze the combined effect of a non-Poissonian arrival process and a waiting site near the catalyst's active site on the throughput of the system. A major conclusion of our and other studies is that it is a questionable practice to routinely and exclusively use mass action kinetics for the interpretation and performance evaluation of biocatalytic systems. [ABSTRACT FROM AUTHOR]

The formation of new blood vessels in vivo is a multistep process in which sprouting endothelial cells (ECs) form tubes with lumina, these tubes being additionally organized as capillary networks. In vitro models of tubulogenesis have been developed to investigate this highly regulated multifactorial process, with special attention paid to the determinant role of mechanical interactions between ECs and the extracellular matrix (ECM). In agreement with experimental results obtained when culturing endothelial EAhy926 cells on fibrin gels, we defined theoretical thresholds between cellular traction and active cell migration along ECM strain fields above which tubulogenesis is induced.We additionally illustrated how mechanical factors may provide long-ranged positional information signals leading to localized network formation, thus providing an alternative view to the classical approach of morphogenesis based on gradients of diffusible morphogens. [ABSTRACT FROM AUTHOR]

We evaluate different mechanisms for spatial domain formation of guanosine triphosphatases (GTPases) on cellular membranes. A kinetic model of the basic guanine-nucleotide cycle common to all GTPases is developed and coupled along a one-dimensional axis by diffusion of inactive and activated GTPases. We ask whether a parameter set exists such that domain formation is possible by Turing's mechanism, i.e., purely by reactions and diffusion, and show that the Turing instability does not occur in this model for any parameter combination. But, as revealed by stability and bifurcation analysis, domain formation is reproduced after augmenting the model with combinations of two spatial interaction mechanisms: 1. attraction and 2. adhesion among active GTPases. These interactions can be mediated by effector proteins that bind active GTPases, and the model therefore predicts domains to disintegrate if effector binding is inhibited. [ABSTRACT FROM AUTHOR]

Many eukaryotic cell types share the ability to migrate directionally in response to external chemoattractant gradients. The binding of chemoattractants to specific receptors leads to a wide range of biochemical responses that become highly localized as cells polarize and migrate by chemotaxis. This ability is central in the development of complex organisms, and is the result of a billion years of evolution. Cells exposed to shallow gradients in chemoattractant concentration respond with strongly asymmetric accumulation of several factors, including the phosphoinositides PIP3 and PIP2, the PI 3-kinase PI3K, and phosphatase PTEN. This early symmetry-breaking stage is believed to trigger effector pathways leading to cell movement. Although many signaling factors implied in directional sensing have been recently discovered the physical mechanism of signal amplification is not yet well understood. We propose that directional sensing is the consequence of a phase ordering process mediated by phosphoinositide diffusion and driven by the distribution of the chemotactic signal. By studying a realistic reaction-diffusion lattice model that describes PI3K and PTEN enzymatic activity, recruitment to the plasma membrane, and diffusion of their phosphoinositide products, we have shown that the effective enzyme-enzyme interaction induced by catalysis and diffusion introduces an instability of the system towards phase separation for realistic values of physical parameters. In this framework, large reversible amplification of shallow chemotactic gradients, selective localization of chemical factors, macroscopic response timescales, and spontaneous polarization arise naturally. [ABSTRACT FROM AUTHOR]

Calcium oscillations play an important role in intracellular signal transduction. As a second messenger, Ca2+ represents a link between several input signals and several target processes in the cell. Whereas the frequency of simple Ca2+ oscillations enables a selective activation of a specific protein and herewith a particular process, the question arises of how at the same time two or more classes of proteins can be specifically regulated. The question is general and concerns the problem of how one second messenger can transmit more than one signal simultaneously (bow-tie structure of signalling). To investigate whether a complex Ca2+ signal like bursting, a succession of low-peak and high-peak oscillatory phases, could selectively activate different proteins, several bursting patterns with simplified square pulses were applied in a theoretical model. The results indicate that bursting Ca2+ oscillations allow a differential regulation of two different calcium-binding proteins, and hence, perform the desired function. [ABSTRACT FROM AUTHOR]