DelayAndPeriodicity

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be splitted into two parts: pseudo-continuous and strongly unstable. The pseudo-continuous part of the spectrum mediates destabilization of periodic solutions.Comment: 24 pages, 9 figure

Optimum ground states for spin-32\frac{3}{2} chains

We present a set of {\em optimum ground states} for a large class of spin-$\frac{3}{2}$ chains. Such global ground states are simultaneously ground states of the local Hamiltonian, i.e. the nearest neighbour interaction in the present case. They are constructed in the form of a matrix product. We find three types of phases, namely a {\em weak antiferromagnet}, a {\em weak ferromagnet}, and a {\em dimerized antiferromagnet}. The main physical properties of these phases are calculated exactly by using a transfer matrix technique, in particular magnetization and two spin correlations. Depending on the model parameters, they show a surprisingly rich structure.Comment: LaTeX, 22 pages, 6 embedded Postscript figure

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).Comment: minor revision, correcting mistakes in formulation of Lemma 2.3 and A.5 (which are also present in the Journal paper): center of neighborhood must be in $C^1$, which is the case for the main theore

Modeling the Basal Dynamics of P53 System

The tumor suppressor p53 has become one of most investigated genes. Once activated by stress, p53 leads to cellular responses such as cell cycle arrest and apoptosis.Most previous models have ignored the basal dynamics of p53 under nonstressed conditions. To explore the basal dynamics of p53, we constructed a stochastic delay model by incorporating two negative feedback loops. We found that protein distribution of p53 under nonstressed condition is highly skewed with a fraction of cells showing high p53 levels comparable to those observed under stressed conditions. Under nonstressed conditions, asynchronous and spontaneous p53 pulses are triggered by basal DNA double strand breaks produced during normal cell cycle progression. The first peaking times show a predominant G1 distribution while the second ones are more widely distributed. The spontaneous pulses are triggered by an excitable mechanism. Once initiated, the amplitude and duration of pulses remain unchanged. Furthermore, the spontaneous pulses are filtered by ataxia telangiectasia mutated protein mediated posttranslational modifications and do not result in substantial p21 transcription. If challenged by externally severe DNA damage, cells generate synchronous p53 pulses and induce significantly high levels of p21. The high expression of p21 can also be partially induced by lowering the deacetylation rate.Our results demonstrated that the dynamics of p53 under nonstressed conditions is initiated by an excitable mechanism and cells become fully responsive only when cells are confronted with severe damage. These findings advance our understanding of the mechanism of p53 pulses and unlock many opportunities to p53-based therapy

State-dependent distributed-delay model of orthogonal cutting

In this paper we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chat- ter and describe the dynamics of the tool-workpiece sys- tem during cutting by delay-diferential equations. We model the cutting-force as the resultant of a force sys- tem distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative de- lay. According to the literature on stress distribution along the rake face, the length of the chip-tool inter- face, where the distributed cutting-force system is act- ing, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative efect. Therefore, the additional short de- lay is state-dependent. It is shown that involving state- dependent delay in the model does not afect linear sta- bility properties, but does afect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bi- furcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions

A mathematical model of the human metabolic system and metabolic flexibility

In healthy subjects some tissues in the human body display metabolic flexibility, by this we mean the ability for the tissue to switch its fuel source between predominantly carbohydrates in the post prandial state and predominantly fats in the fasted state. Many of the pathways involved with human metabolism are controlled by insulin, and insulin- resistant states such as obesity and type-2 diabetes are characterised by a loss or impairment of metabolic flexibility. In this paper we derive a system of 12 first-order coupled differential equations that describe the transport between and storage in different tissues of the human body. We find steady state solutions to these equations and use these results to nondimensionalise the model. We then solve the model numerically to simulate a healthy balanced meal and a high fat meal and we discuss and compare these results. Our numerical results show good agreement with experimental data where we have data available to us and the results show behaviour that agrees with intuition where we currently have no data with which to compare

Genome-wide association reveals genetic effects on human Aβ₄₂and τ protein levels in cerebrospinal fluids: a case control study

<p>Abstract</p> <p>Background</p> <p>Alzheimer's disease (AD) is common and highly heritable with many genes and gene variants associated with AD in one or more studies, including APOE ε2/ε3/ε4. However, the genetic backgrounds for normal cognition, mild cognitive impairment (MCI) and AD in terms of changes in cerebrospinal fluid (CSF) levels of Aβ_1-42, T-tau, and P-tau_181P, have not been clearly delineated. We carried out a genome-wide association study (GWAS) in order to better define the genetic backgrounds to these three states in relation to CSF levels.</p> <p>Methods</p> <p>Subjects were participants in the Alzheimer's Disease Neuroimaging Initiative (ADNI). The GWAS dataset consisted of 818 participants (mainly Caucasian) genotyped using the Illumina Human Genome 610 Quad BeadChips. This sample included 410 subjects (119 Normal, 115 MCI and 176 AD) with measurements of CSF Aβ_1-42, T-tau, and P-tau_181PLevels. We used PLINK to find genetic associations with the three CSF biomarker levels. Association of each of the 498,205 SNPs was tested using additive, dominant, and general association models while considering APOE genotype and age. Finally, an effort was made to better identify relevant biochemical pathways for associated genes using the ALIGATOR software.</p> <p>Results</p> <p>We found that there were some associations with APOE genotype although CSF levels were about the same for each subject group; CSF Aβ_1-42levels decreased with APOE gene dose for each subject group. T-tau levels tended to be higher among AD cases than among normal subjects. From adjusted result using APOE genotype and age as covariates, no SNP was associated with CSF levels among AD subjects. <it>CYP19A1 </it>'aromatase' (rs2899472), <it>NCAM2</it>, and multiple SNPs located on chromosome 10 near the <it>ARL5B </it>gene demonstrated the strongest associations with Aβ_1-42in normal subjects. Two genes found to be near the top SNPs, <it>CYP19A1 </it>(rs2899472, p = 1.90 × 10^-7) and <it>NCAM2 </it>(rs1022442, p = 2.75 × 10^-7) have been reported as genetic factors related to the progression of AD from previous studies. In AD subjects, APOE ε2/ε3 and ε2/ε4 genotypes were associated with elevated T-tau levels and ε4/ε4 genotype was associated with elevated T-tau and P-tau_181Plevels. Pathway analysis detected several biological pathways implicated in Normal with CSF β-amyloid peptide (Aβ_1-42).</p> <p>Conclusions</p> <p>Our genome-wide association analysis identified several SNPs as important factors for CSF biomarker. We also provide new evidence for additional candidate genetic risk factors from pathway analysis that can be tested in further studies.</p