From St\"{a}ckel systems to integrable hierarchies of PDE's: Benenti class of separation relations

We propose a general scheme of constructing of soliton hierarchies from finite dimensional St\"{a}ckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in momenta integrals of motion.Comment: 24 page

Regulation of ykrL (htpX) by Rok and YkrK, a Novel Type of Regulator in Bacillus subtilis

Expression of ykrL of Bacillus subtilis, encoding a close homologue of the Escherichia coli membrane protein quality control protease HtpX, was shown to be upregulated under membrane protein overproduction stress. Using DNA affinity chromatography, two proteins were found to bind to the promoter region of ykrL: Rok, known as a repressor of competence and genes for extracytoplasmic functions, and YkrK, a novel type of regulator encoded by the gene adjacent to ykrL but divergently transcribed. Electrophoretic mobility shift assays showed Rok and YkrK binding to the ykrL promoter region as well as YkrK binding to the ykrK promoter region. Comparative bioinformatic analysis of the ykrL promoter regions in related Bacillus species revealed a consensus motif, which was demonstrated to be the binding site of YkrK. Deletion of rok and ykrK in a PykrL-gfp reporter strain showed that both proteins are repressors of ykrL expression. In addition, conditions which activated PykrL (membrane protein overproduction, dissipation of the membrane potential, and salt and phenol stress) point to the involvement of YkrL in membrane protein quality control

Separation of variables in quasi-potential systems of bi-cofactor form

We perform variable separation in the quasi-potential systems of equations of the form $\ddot{q}=-A^{-1}\nabla k=-\tilde{A}^{-1}\nabla\tilde{k}${}, where $A$ and $\tilde{A}$ are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux-Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions.Comment: LaTex, 30 pages, to appear in J. Phys. A: Math. Ge

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed

A model for stem cell population dynamics with regulated maturation delay

We develop a structured population model for the maturation process of stem cells in the form of a state-dependent delay differential equation. Moreover, results on existence, uniqueness and positivity of solutions as well as conditions of existence for equilibria and representations of these are established. We give biological interpretations for the conditions of existence of equilibria

Stability analysis of multi-compartment models for cell production systems

We study two-and three-compartment models of a hierarchical cell production system with cell division regulated by the level of mature cells. We investigate the structure of equilibria with respect to parameters as well as local stability properties for the equilibria. To interpret the results we adapt the concept of reproduction numbers, which is well known in ecology, to stem cell population dynamics. In the two-compartment model, the positive equilibrium is stable wherever it exists. In the three-compartment model, we find that the intermediate stage of differentiation is responsible for the emergence of an instability region in the parameter plane. Moreover, we prove that this region shrinks as the mortality rate for mature cells increases and discuss this result

Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new.Comment: 15 page

Exploring the impact of participative place-based community archaeology in rural Europe: Community archaeology in rural environments meeting societal challenges

This paper reviews the aims, context, approach and early outcomes of a new transnational participative archaeology project focussed on rural village communities. ‘Community Archaeology in Rural Environments Meeting Societal Challenges’ (CARE-MSoC) includes three European countries where participative community archaeology is new- the Czech Republic, the Netherlands and Poland. CARE-MSoC aims to explore the feasibility, value and impact of excavation by rural residents within their home communities by using a method which can be deployed anywhere and which in the UK has been shown to advance knowledge of the past while also delivering a wide range of social and heritage benefits: multiple test pit excavation within inhabited villages. Data presented here from the Czech Republic, the Netherlands and Poland show the activity to be popular and effective here in benefitting people while also attracting, sustaining and growing local interest in heritage participation in all three countries