A large-deviations approach to gelation

A large-deviations principle (LDP) is derived for the state at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. The proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected

A large-deviations principle for all the cluster sizes of a sparse Erdős–Rényi graph

Let (Formula presented.) be the Erdős–Rényi graph with connection probability (Formula presented.) as N → ∞ for a fixed t ∈ (0, ∞). We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of (Formula presented.), registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order N), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at t = 1 as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus–Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition

A large-deviations approach to gelation

A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erd\H{o}s-R\'enyi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erd\H{o}s-R\'enyi graphs are connected

A large-deviations principle for all the components in a sparse inhomogeneous random graph

We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07]

Large deviations for Markov jump processes with uniformly diminishing rates

We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further discuss the optimality of our assumptions on the decay of the jump rates. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of mass action kinetics

Large deviations for Markov jump processes with uniformly diminishing rates

We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics

Meniscus maturation in the swine model: role of endostatin in cellular differentiation

The development of an engineered meniscus derives from the need to regenerate a tissue which is largely unable to self-repair with consequent loss of functionality. Hence a deeper knowledge of the native meniscus morphology and biomechanics in its different regions, including molecules involved in regulation of the maturation process, is essential. The meniscus is a complex tissue, displaying great regional variation in extracellular matrix components and in vascularization, as a result of several biomechanical stimuli. Its biochemical composition is modulated to adapt the tissue to the different functions that are required throughout growth, until a \u201cmature\u201d phase is reached in adulthood. The aim of this work is to evaluate the biological role of Endostatin in the regulation of angiogenesis as in the fibro-chondrogenic differentiation of neonatal meniscal cells in the pig. The swine is an attractive model for meniscal repair studies, as its knee joint is closely comparable to the human one in terms of anatomical structure, vascularization, and healing potential. Our preliminary data show that Endostatin contributes to the acquisition of chondrocyte phenotype in an undifferentiated but committed cellular population. Thus, a better understanding of the role of Endostatin in cell metabolism might lead to a deeper knowledge of the events regulating meniscus maturation. These findings may be crucial for the development of an engineered scaffold able to induce meniscal cell differentiation by releasing Endostatin-rich microspheres

Phase II Study of Dehydroepiandrosterone in Androgen Receptor-Positive Metastatic Breast Cancer.

LESSONS LEARNED: The androgen receptor (AR) is present in most breast cancers (BC), but its exploitation as a therapeutic target has been limited.This study explored the activity of dehydroepiandrosterone (DHEA), a precursor being transformed into androgens within BC cells, in combination with an aromatase inhibitor (to block DHEA conversion into estrogens), in a two-stage phase II study in patients with AR-positive/estrogen receptor-positive/human epidermal growth receptor 2-negative metastatic BC.Although well tolerated, only 1 of 12 patients obtained a prolonged clinical benefit, and the study was closed after its first stage for poor activity. BACKGROUND: Androgen receptors (AR) are expressed in most breast cancers, and AR-agonists have some activity in these neoplasms. We investigated the safety and activity of the androgen precursor dehydroepiandrosterone (DHEA) in combination with an aromatase inhibitor (AI) in patients with AR-positive metastatic breast cancer (MBC). METHODS: A two-stage phase II study was conducted in two patient cohorts, one with estrogen receptor (ER)-positive (resistant to AIs) and the other with triple-negative MBC. DHEA 100 mg/day was administered orally. The combination with an AI aimed to prevent the conversion of DHEA into estrogens. The main endpoint was the clinical benefit rate. The triple-negative cohort was closed early. RESULTS: Twelve patients with ER-positive MBC were enrolled. DHEA-related adverse events, reported in four patients, included grade 2 fatigue, erythema, and transaminitis, and grade 1 drowsiness and musculoskeletal pain. Clinical benefit was observed in one patient with ER-positive disease whose tumor had AR gene amplification. There was wide inter- and intra-patient variation in serum levels of DHEA and its metabolites. CONCLUSION: DHEA showed excellent safety but poor activity in MBC. Although dose and patient selection could be improved, high serum level variability may hamper further DHEA development in this setting