N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Computational Analysis of Microarray Data

Most of our knowledge of biodiversity and its causes in the deep-sea benthos derives from regional-scale sampling studies of the macrofauna. Improved sampling methods and the expansion of investigations into a wide variety of habitats have revolutionized our understanding of the deep sea. Local species diversity shows clear geographic variation on spatial scales of 100-1000 km. Recent sampling programs have revealed unexpected complexity in community structure at the landscape level that is associated with large-scale oceanographic processes and their environmental consequences. We review the relationships between variation in local species diversity and the regional-scale phenomena of boundary constraints, gradients of productivity, sediment heterogeneity, oxygen availability, hydrodynamic regimes, and catastrophic physical disturbance. We present a conceptual model of how these interdependent environmental factors shape regional-scale variation in local diversity. Local communities in the deep sea may be composed of species that exist as metapopulations whose regional distribution depends on a balance among global-scale, landscape-scale, and small-scale dynamics. Environmental gradients may form geographic patterns of diversity by influencing local processes such as predation, resource partitioning, competitive exclusion, and facilitation that determine species coexistence. The measurement of deep-sea species diversity remains a vital issue in comparing geographic patterns and evaluating their potential causes. Recent assessments of diversity using species accumulation curves with randomly pooled samples confirm the often-disputed claim that the deep sea supports higher diversity than the continental shelf. However, more intensive quantitative sampling is required to fully characterize the diversity of deep-sea sediments, the most extensive habitat on Earth. Once considered to be constant, spatially uniform, and isolated, deep-sea sediments are now recognized as a dynamic, richly textured environment that is inextricably linked to the global biosphere. Regional studies of the last two decades provide the empirical background necessary to formulate and test specific hypotheses of causality by controlled sampling designs and experimental approaches.

Environmental influences on regional deep-sea species diversity

Breast cancer is one of the leading causes of death in women in the world It is a complex disease with challenges to accurate diagnosis due to cancer subtypes that are difficult to distinguish The most common subtype is Invasive Ductal Carcinoma (IDC), a cancer in ductal cells that line the milk ducts in the breast In depth understanding of the genetic basis of IDC can help treat it more effectively Microarray based gene expression analysis is making new grounds in accurate diagnosis of diseases including cancer Microarray experiments are designed to measure the expression levels of thousands of genes in various cells/tissues of interest and they are analysed to decipher a small set of genes that constitutes the gene signature of a particular disease The few studies on breast cancer gene expression compare cancer subtypes but very few have compared gene expression between matched cancer and healthy tissues in the breast (Turashvili et al, 2007) The few studies that have compared different subtypes have little agreement on the gene signatures (Turashvili, 2007; Zhao et al, 2004, Sorlie, et al, 2001) Therefore, it is highly beneficial to further assess the validity of genes identified as differentially expressed, in order to boost confidence in the usefulness of the genes in various medical applications including diagnosis, prognosis and drug development In this study, the validity of differentially expressed genes pertaining to a carefully conducted experiment on breast tissues affected by Invasive Ductal Carcinoma (IDL) and matched healthy tissues is conducted using neural networks and statistical methods The data was obtained from NCBI database and deposited by Turashvili et al (2007) from their experiments on breast cancer The original authors extracted a 326 gene signature for IDC using statistical methods In our study, the ability of this gene set to discriminate the disease state from healthy state is investigated and validated using two sets of independent datasets Our visual and qualitative exploration using Self organizing maps (SOM) followed by statistical tests indicated that the validation data supported 80% of the original gene signature Another SOM results declared that the original gene set is able to classify patients as being healthy or having IDC Original gene set was optimally clustered into two classes based on correlation of expression patterns of genes by SOM /Ward clustering The two classes and genes in them were supported by 60% of the validation data As an alternative, PCA was used to determine genes with correlated expressions in the original gene signature and 4 PCs accounted for 86% of the variation in the data with the first 2 PCs accounting for around 70% Top most important 100 genes in PC1 and PC2 provided 52% support for the two SOM classes with PC1 dominating class 1 and PC2, class 2 Genes that were validated by independent data in the two SOM classes were used in conjunction with PC1 and PC2 to extract highly influential genes from the top 6%, 18% and 57% of the original genes represented by PC1 and PC2 These key genes may prove to be the most crucial in identifying ductal tumor from healthy tissues Four new genes were among key genes that may shed more light onto the disease mechanism The key genes as well as overall set of validated genes may provide further support to understand or refine genetic networks that these genes are part of in the next stage of our study

/pdf/validating-a-gene-expression-signature-of-invasive-ductal-1djmfgpn5p.pdf

Validating a gene expression signature of invasive ductal carcinoma of the breast and detecting key genes using neural networks

In the previous chapter we derived a stochastic solute transport model (Eq. 3.14); we developed the methods to estimate its parameters, and investigated its behaviour numerically. We see some promise to characterise the solute dispersion at different flow lengths, and there are some indications that Eq. (3.14) produce the behaviours that would be interpreted as capturing the scale-dependency of dispersivity.

A Generalized Mathematical Model in One-Dimension

Chapter 5 - Potential Theory Approach to SDEs

The main objective of this chapter is to investigate the performance of the \({\textit{ISP LOLAS}}\) algorithm on a large biochemical system. We analyse the state-space and probabilities of the species by implementing and integrating the mathematical model of the G1/S cell-cycle checkpoint involving the DNA-damage signal transduction pathway. The mathematical model is based on chemical kinetic principles for simulating pathways to show the dynamic behaviour of a cell cycle checkpoint pathway having reactive components. The cell cycle checkpoint pathways involve interactions between different enzymes and proteins in linked reactions. Most models developed for cell cycle checkpoints are quantitative, notably the G1/S checkpoint, which involves interactions between different proteins. They provide important information about the internal mechanisms and complex behaviour of cell cycle checkpoints.

A Large Model Case Study: Solving CME for G1/S Checkpoint Involving the DNA-Damage Signal Transduction Pathway

Recent experiments have shown that cellular senescence, a mechanism employed by cells for thwarting cell proliferation, plays an important role in protecting cells against cancer; therefore, a deeper understanding of cellular senescence can lead to effective cancer treatment. Inhibition of CDK2 is thought to be the critical trigger for cellular senescence. In this study, we first implement a mathematical model of G1/S transition involving the DNA-damage pathway and show that cellular senescence can be achieved by lowering CDK2. The robustness of CDK2 in triggering cellular senescence is determined from the probability (β) of DNA-damaged cells passing G1/S checkpoint for normal CDK2 and CDK2-deficient situations based on different thresholds of the peak time of two important biomarkers, CycE and E2F. The comparison of the values of β under the normal CDK2 and lower CDK2 levels reveals that reducing CDK2 levels can decrease the percentage of damaged cells passing G1/S checkpoint; more importantly, 50% reduction of CDK2 achieves 65% reduction in the percentage of damaged cells passing the G1/S checkpoint. These results point out that the developed model can highlight the possibility of lowering the bar for cellular senescence by reducing CDK2 levels. The results of investigation of β for the different thresholds of the peak times of other biomarkers show that β is insensitive to these perturbations of the peak time indicating that CDK2 activity is robust in lowering the senescence bar for low and high levels of DNA-damage. Furthermore, a mathematical formulation of robustness indicates that the robustness of CDK2 -triggered senescence increases with decreasing levels of CDK2, and is slightly greater for low-level DNA damage condition.

/pdf/robustness-of-cdk2-in-triggering-cellular-senescence-based-3l1cz3cox9.pdf

Don Kulasiri

Papers

Validating a gene expression signature of invasive ductal carcinoma of the breast and detecting key genes using neural networks

A Generalized Mathematical Model in One-Dimension

Chapter 5 - Potential Theory Approach to SDEs

A Large Model Case Study: Solving CME for G1/S Checkpoint Involving the DNA-Damage Signal Transduction Pathway

Robustness of CDK2 in triggering cellular senescence based on probability of DNA-damaged cells passing G1/S checkpoint