On a bilateral birth-death process with alternating rates

We consider a bilateral birth-death process characterized by a constant transition rate $\lambda$ from even states and a possibly different transition rate $\mu$ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.Comment: 13 pages, 3 figure

A three-dimensional cyclic random motion with finite velocities driven by geometric counting processes

We consider a stochastic process $\{(\boldsymbol{X}(t),V(t)), t \geq 0\}$ which describes a particle performing a minimal cyclic random motion with finite velocities in $\mathbb{R}^3$. The particle can take four directions moving with different velocities $\vec{v}_j$, for $1 \leq j \leq4$, so that the diffusion region is a tetrahedron $\mathcal{T}(t)$. Moreover, we assume that the sequence of sojourn times along each velocity $\vec{v}_j$ follows a geometric counting process of intensity $\lambda_j$, $1 \leq j \leq4$. We first describe the direction vectors $\vec{v}_j$ and the domain $\mathcal{T}(t)$; then, we obtain the closed-form expressions of the initial and absolutely continuous components of the probability law of the process when the starting velocity is $\vec{v}_1$. We also investigate the limiting behavior of the probability density of the process when the intensities $\lambda_j$ tend to infinity. Finally, we introduce the first-passage time problem for the first component of $\boldsymbol{X}(t)$ through a constant positive boundary $\beta > 0$ providing the bases for future developments

Combining Pathway Identification and Breast Cancer Survival Prediction via Screening-Network Methods.

Breast cancer is one of the most common invasive tumors causing high mortality among women. It is characterized by high heterogeneity regarding its biological and clinical characteristics. Several high-throughput assays have been used to collect genome-wide information for many patients in large collaborative studies. This knowledge has improved our understanding of its biology and led to new methods of diagnosing and treating the disease. In particular, system biology has become a valid approach to obtain better insights into breast cancer biological mechanisms. A crucial component of current research lies in identifying novel biomarkers that can be predictive for breast cancer patient prognosis on the basis of the molecular signature of the tumor sample. However, the high dimension and low sample size of data greatly increase the difficulty of cancer survival analysis demanding for the development of ad-hoc statistical methods. In this work, we propose novel screening-network methods that predict patient survival outcome by screening key survival-related genes and we assess the capability of the proposed approaches using METABRIC dataset. In particular, we first identify a subset of genes by using variable screening techniques on gene expression data. Then, we perform Cox regression analysis by incorporating network information associated with the selected subset of genes. The novelty of this work consists in the improved prediction of survival responses due to the different types of screenings (i.e., a biomedical-driven, data-driven and a combination of the two) before building the network-penalized model. Indeed, the combination of the two screening approaches allows us to use the available biological knowledge on breast cancer and complement it with additional information emerging from the data used for the analysis. Moreover, we also illustrate how to extend the proposed approaches to integrate an additional omic layer, such as copy number aberrations, and we show that such strategies can further improve our prediction capabilities. In conclusion, our approaches allow to discriminate patients in high-and low-risk groups using few potential biomarkers and therefore, can help clinicians to provide more precise prognoses and to facilitate the subsequent clinical management of patients at risk of disease

Analysis of a birth and death process with alternating rates and of a telegraph process with underlying random walk

2010 - 2011My thesis for the Doctoral Programme in Mathematics (November 1, 2008 - October 31, 2011) at University of Salerno, Italy, has been oriented to the analysis of two stochastic models, with particular emphasis on the de- termination of their probability laws and related properties. The discussion of the doctoral dissertation will be given in 20 March 2012. The first part of the thesis is devoted to the analysis of a birth and death process with alternating rates. We recall that an extensive survey on birth- death processes (BDP) has been provided by Parthasarathy and Lenin [3]. In this work the authors adopt standard methods of analysis (such as power series technique and Laplace transforms) to find explicit expressions for the transient and stationary distributions of BDPs and provide applications of such results to specific fields (communication systems, chemical and biolog- ical models). In particular, in Section 9 they use BDPs to describe the time changes in the concentrations of the components of a chemical reaction and discuss the role of BDPs in the study of diatomic molecular chains. More- over, the paper by StockMayer et al. [4] gives an example of application of stochastic processes in the study of chain molecular diffusion. In this work a molecule is modeled as a freely-joined chain of two regularly alternating kinds of atoms. All bonds have the same length but the two kinds of atoms have alternating jump rates, i.e. the forward and backward jump rates for even labeled beads are α and β, respectively, and these rates are reversed for odd labeled beads. The authors obtain the exact timedependent aver- age length of bond vectors. Inspired by this works, Conolly [1] studied an infinitely long chain of atoms joined by links of equal length. The links are assumed to be subject to random shocks, that force the atoms to move and the molecule to diffuse. The shock mechanism is different according to whether the atom occupies an odd or an even position on the chain. The originating stochastic model is a randomized random walk on the integers with an unusual exponential pattern for the inter-step time intervals. The authors analyze some features of this process and investigate also its queue counterpart, where the walk is confined to the non negative integers. Stimulated by the above researches, a birth and death process N(t) on the integers with a transition rate λ from even states and a possibly different rate μ from odd states has been studied in the first part of the thesis. A de- tailed description of the model is performed, and the Chapman-Kolmogorov equations are introduced. Then, the probability generating functions of even and odd states are then obtained. These allow to evaluate the transition probabilities of the process for arbitrary integer initial state. Certain sym- metry properties of the transition probabilities are also pinpointed. Then, the birth and death process obtained by superimposing a reecting bound- ary in the zero-state is analyzed. In particular, by making use of a Laplace transform approach, the probability of a transition from state 0 or state 1 to the zero-state is obtained. Formulas for mean and variance of both processes are finally provided. The second part of the thesis is devoted to the analysis of a generalized telegraph process with an underlying random walk. The classical telegraph process describes a random motion on the real line characterized by two _nite velocities with opposite directions, where the velocity changes are governed by a time-homogeneous Poisson process (see Orsingher [2]). The novelty in the proposed model consists in the use of new rules for velocity changes, which are now governed by a sequence of Bernoulli trials. This implies that the random times separating consecutive changes of direction of the mov- ing particle have a general distribution and form a non-regular alternating renewal process. Starting from the origin, the running particle performs an alternating motion with velocities c and -v (c; v > 0). The direction of the motion (forward and backward) is determined by the velocity sign. The particle changes the direction according to the outcome of a Bernoulli trial. Hence, this defines a (possibly asymmetric) random walk governing the choice of the velocity at any epoch. By adopting techniques based on renewal theory, the general form of probability law is determined as well as the mean of the process. Furthermore, two instances are investigated in detail, in which the random intertimes between consecutive velocity changes are exponentially distributed with (i) constant rates and with (ii) linearly increasing rates. In the first case, explicit expressions of the transition den- sity and of the conditional mean of the process are expressed as series of Gauss hypergeometric functions. The second case leads to a damped ran- dom motion, for which we obtain the transition density in closed form. It is interesting to note that the latter case yields a logistic stationary density. References [1] Conolly B.W. (1971) On randomized random walks. SIAM Review, 13, 81-99. [2] Orsingher, E. (1990) Probability law, flow functions, maximum distri- bution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl., 34, 49-66. [3] Parthasarathy P.R. and Lenin R.B. (2004) Birth and death process (BDP) models with applications-queueing, communication systems, chemical models, biological models: the state-of the- art with a time- dependent perspective. American Series in Mathematical and Manage- ment Sciences, vol. 51, American Sciences Press, Columbus (2004) [4] Stockmayer W.H., Gobush W. and Norvich R. (1971) Local-jump mod- els for chain dynamics. Pure Appl. Chem., 26, 555-561. NOTE The thesis consists of four chapters: Chapter 1. Some definitions and properties of stochastic processes. Chapter 2. Analysis of birth-death processes on the set of integers, char- acterized by alternating rates. Chapter 3. Results on the standard telegraph process. Chapter 4. Study of the telegraph process with an underlying random walk governing the velocity changes. [edited by author]X n.s

Some Results on Brownian Motion Perturbed by Alternating Jumps in Biological Modeling

We consider the model of random evolution on the real line consisting in a Brownian motion perturbed by alternating jumps. We give the probability density of the process and pinpoint a connection with the limit density of a telegraph process subject to alternating jumps. We study the first-crossing-time probability in two special cases, in the presence of a constant upper boundary

Cancer Markers Selection Using Network-Based Cox Regression: A Methodological and Computational Practice.

International initiatives such as the Cancer Genome Atlas (TCGA) and the International Cancer Genome Consortium (ICGC) are collecting multiple datasets at different genome-scales with the aim of identifying novel cancer biomarkers and predicting survival of patients. To analyze such data, several statistical methods have been applied, among them Cox regression models. Although these models provide a good statistical framework to analyze omic data, there is still a lack of studies that illustrate advantages and drawbacks in integrating biological information and selecting groups of biomarkers. In fact, classical Cox regression algorithms focus on the selection of a single biomarker, without taking into account the strong correlation between genes. Even though network-based Cox regression algorithms overcome such drawbacks, such network-based approaches are less widely used within the life science community. In this article, we aim to provide a clear methodological framework on the use of such approaches in order to turn cancer research results into clinical applications. Therefore, we first discuss the rationale and the practical usage of three recently proposed network-based Cox regression algorithms (i.e., Net-Cox, AdaLnet, and fastcox). Then, we show how to combine existing biological knowledge and available data with such algorithms to identify networks of cancer biomarkers and to estimate survival of patients. Finally, we describe in detail a new permutation-based approach to better validate the significance of the selection in terms of cancer gene signatures and pathway/networks identification. We illustrate the proposed methodology by means of both simulations and real case studies. Overall, the aim of our work is two-fold. Firstly, to show how network-based Cox regression models can be used to integrate biological knowledge (e.g., multi-omics data) for the analysis of survival data. Secondly, to provide a clear methodological and computational approach for investigating cancers regulatory networks

Network-based survival analysis methods for pathway detection in cancer

We compare three penalized Cox regression methods for high-dimensional survival data in order to identify the pathways involved into cancer occurrence and pro- gression. We analyze each method with three gene expression datasets including breast, lung and ovarian cancer. More precisely, we focus on cancer survival prediction and on top signature genes. The goal of this study is to gain a deeper insight of the benefits and drawbacks of the regression techniques in order to find the pathways involved in a specific type of cancer and identify cancer biomarkers useful for prognosis, diagnosis and treatment

Heme Oxygenase-1 and Brain Oxysterols Metabolism Are Linked to Egr-1 Expression in Aged Mice Cortex, but Not in Hippocampus

Throughout life, stress stimuli act upon the brain leading to morphological and functional changes in advanced age, when it is likely to develop neurodegenerative disorders. There is an increasing need to unveil the molecular mechanisms underlying aging, in a world where populations are getting older. Egr-1 (early growth response 1), a transcriptional factor involved in cell survival, proliferation and differentiation – with a role also in memory, cognition and synaptic plasticity, can be implicated in the molecular mechanism of the aging process. Moreover, Heme Oxygenase-1a (HO), a 32 kDa heat-shock protein that converts heme to iron, carbon monoxide and biliverdin, is a key enzyme with neuroprotective properties. Several in vitro and in vivo studies reported that HO-1 could regulate the metabolism of oxysterols, oxidation products of cholesterol that include markers of oxidative stress. Recently, a link between Egr-1 and HO-1 has been demonstrated in mouse lung cells exposed to cigarette smoke. In view of these data, we wanted to investigate whether Egr-1 can be implicated also in the oxysterol metabolism during brain aging. Our results show that Egr-1 expression is differently expressed in the cortex and hippocampus of old mice, as well as the oxysterol profile between these two brain areas. In particular, we show that the cortex experiences in an age-dependent fashion increasing levels of the Egr-1 protein, and that these correlate with the level of HO-1 expression and oxysterol abundance. Such a situation was not observed in the hippocampus. These results are further strenghtened by our observations made with Egr-1 KO mice, confirming our hypothesis concerning the influence of Egr-1 on oxysterol production and accumulation via regulation of the expression of HO-1 in the cortex, but not the hippocampus, of old mice. It is important to notice that most of the oxysterols involved in this process are those usually stimulated by oxidative stress, which would then represent the triggering factor for this mechanism

Liver gene therapy with intein-mediated F8 trans-splicing corrects mouse haemophilia A

: Liver gene therapy with adeno-associated viral (AAV) vectors is under clinical investigation for haemophilia A (HemA), the most common inherited X-linked bleeding disorder. Major limitations are the large size of the F8 transgene, which makes packaging in a single AAV vector a challenge, as well as the development of circulating anti-F8 antibodies which neutralise F8 activity. Taking advantage of split-intein-mediated protein trans-splicing, we divided the coding sequence of the large and highly secreted F8-N6 variant in two separate AAV-intein vectors whose co-administration to HemA mice results in the expression of therapeutic levels of F8 over time. This occurred without eliciting circulating anti-F8 antibodies unlike animals treated with the single oversized AAV-F8 vector under clinical development. Therefore, liver gene therapy with AAV-F8-N6 intein should be considered as a potential therapeutic strategy for HemA