Some Completely Monotonic Properties for the (p, g)-Gamma Function

MSC 2010: 33B15, 26A51, 26A48

Modeling perceived collision risk in port water navigation

An increase in the likelihood of navigational collisions in port waters has put focus on the collision avoidance process in port traffic safety. The most widely used on-board collision avoidance system is the automatic radar plotting aid which is a passive warning system that triggers an alert based on the pilot’s pre-defined indicators of distance and time proximities at the closest point of approaches in encounters with nearby vessels. To better help pilot in decision making in close quarter situations, collision risk should be considered as a continuous monotonic function of the proximities and risk perception should be considered probabilistically. This paper derives an ordered probit regression model to study perceived collision risks. To illustrate the procedure, the risks perceived by Singapore port pilots were obtained to calibrate the regression model. The results demonstrate that a framework based on the probabilistic risk assessment model can be used to give a better understanding of collision risk and to define a more appropriate level of evasive actions.

A Local Circuit Model of Learned Straitial and Dopamine Cell Responses under Probabilistic Schedules of Reward

Before choosing, it helps to know both the expected value signaled by a predictive cue and the associated uncertainty that the reward will be forthcoming. Recently, Fiorillo et al. (2003) found the dopamine (DA) neurons of the SNc exhibit sustained responses related to the uncertainty that a cure will be followed by reward, in addition to phasic responses related to reward prediction errors (RPEs). This suggests that cue-dependent anticipations of the timing, magnitude, and uncertainty of rewards are learned and reflected in components of the DA signals broadcast by SNc neurons. What is the minimal local circuit model that can explain such multifaceted reward-related learning? A new computational model shows how learned uncertainty responses emerge robustly on single trial along with phasic RPE responses, such that both types of DA responses exhibit the empirically observed dependence on conditional probability, expected value of reward, and time since onset of the reward-predicting cue. The model includes three major pathways for computing: immediate expected values of cures, timed predictions of reward magnitudes (and RPEs), and the uncertainty associated with these predictions. The first two model pathways refine those previously modeled by Brown et al. (1999). A third, newly modeled, pathway is formed by medium spiny projection neurons (MSPNs) of the matrix compartment of the striatum, whose axons co-release GABA and a neuropeptide, substance P, both at synapses with GABAergic neurons in the SNr and with the dendrites (in SNr) of DA neurons whose somas are in ventral SNc. Co-release enables efficient computation of sustained DA uncertainty responses that are a non-monotonic function of the conditonal probability that a reward will follow the cue. The new model's incorporation of a striatal microcircuit allowed it to reveals that variability in striatal cholinergic transmission can explain observed difference, between monkeys, in the amplitutude of the non-monotonic uncertainty function. Involvement of matriceal MSPNs and striatal cholinergic transmission implpies a relation between uncertainty in the cue-reward contigency and action-selection functions of the basal ganglia. The model synthesizes anatomical, electrophysiological and behavioral data regarding the midbrain DA system in a novel way, by relating the ability to compute uncertainty, in parallel with other aspects of reward contingencies, to the unique distribution of SP inputs in ventral SN.

A microscopic study of structural and electronic properties of functionalized silicon surfaces based on first-principles

Surface modification of silicon with organic monolayers tethered to the surface by different linkers is an important process in realizing future (opto-)electronic devices. Understanding the role played by the nature of the linking group and the chain length on the adsorption structures and electronic properties of these assemblies is vital to advance this technology. This Thesis is a study of such properties and contributes in particular to a microscopic understanding of induced changes in the work function of experimentally studied functionalized silicon surfaces. Using first-principles density functional theory (DFT), at the first step, we provide predictions for chemical trends in the work function of hydrogenated silicon (111) surfaces modified with various terminations. For nonpolar terminating atomic species such as F, Cl, Br, and I, the change in the work function is directly proportional to the amount of charge transferred from the surface, thus relating to the difference in electronegativity of the adsorbate and silicon atoms. The change is a monotonic function of coverage in this case, and the work function increases with increasing electronegativity. Polar species such as −TeH, −SeH, −SH, −OH, −NH2, −CH3, and −BH2 do not follow this trend due to the interaction of their dipole with the induced electric field at the surface. In this case, the magnitude and sign of the surface dipole moment need to be considered in addition to the bond dipole to generally describe the change in work function. Compared to hydrogenated surfaces, there is slight increase in the work function of H:Si(111)-XH, where X = Te, Se, and S, whereas reduction is observed for surfaces covered with −OH, −CH3, and −NH2. Next, we study the hydrogen passivated Si(111) surface modified with alkyl chains of the general formula H:Si–(CH2)n–CH2 and H:Si–X–(CH2)n–CH3, where X = NH, O, S and n = (0, 1, 3, 5, 7, 9, 11), at half coverage. For (X)–Hexyl and (X)–Dodecyl functionalization, we also examined various coverages up to full monolayer grafting in order to validate the result of half covered surface and the linker effect on the coverage. We find that it is necessary to take into account the van der Waals interaction between the alkyl chains. The strongest binding is for the oxygen linker, followed by S, N, and C, irrespective of chain length. The result revealed that the sequence of the stability is independent of coverage; however, linkers other than carbon can shift the optimum coverage considerably and allow further packing density. For all linkers apart from sulfur, structural properties, in particular, surface-linker-chain angles, saturate to a single value once n > 3. For sulfur, we identify three regimes, namely, n = 0–3, n = 5–7, and n = 9–11, each with its own characteristic adsorption structures. Where possible, our computational results are shown to be consistent with the available experimental data and show how the fundamental structural properties of modified Si surfaces can be controlled by the choice of linking group and chain length. Later we continue by examining the work function tuning of H:Si(111) over a range of 1.73 eV through adsorption of alkyl monolayers with general formula -[Xhead-group]-(CnH2n)-[Xtail-group], X = O(H), S(H), NH(2). The work function is practically converged at 4 carbons (8 for oxygen), for head-group functionalization. For tail-group functionalization and with both head- and tail-groups, there is an odd-even effect in the behavior of the work function, with peak-to-peak amplitudes of up to 1.7 eV in the oscillations. This behavior is explained through the orientation of the terminal-group's dipole. The shift in the work function is largest for NH2-linked and smallest for SH-linked chains and is rationalized in terms of interface dipoles. Our study reveals that the choice of the head- and/or tail-groups effectively changes the impact of the alkyl chain length on the work function tuning using self-assembled monolayers and this is an important advance in utilizing hybrid functionalized Si surfaces. Bringing together the understanding gained from studying single type functionalization of H:Si(111) with different alkyl chains and bearing in mind how to utilize head-group, tail-group or both as well as monolayer coverage, in the final part of this Thesis we study functionalized H:Si(111) with binary SAMs. Aiming at enhancing work function adjustment together with SAM stability and coverage we choose a range of terminations and linker-chains denoted as –X–(Alkyl) with X = CH3, O(H), S(H), NH(2) and investigate the stability and work function of various binary components grafted onto H:Si(111) surface. Using binary functionalization with -[NH(2)/O(H)/S(H)]-[Hexyl/Dodecyl] we show that work function can be tuned within the interval of 3.65-4.94 eV and furthermore, enhance the SAM’s stability. Although direct Si-C grafted SAMs are less favourable compared to their counterparts with O, N or S linkage, regardless of the ratio, binary functionalized alkyl monolayers with X-alkyl (X = NH, O) is always more stable than single type alkyl functionalization with the same coverage. Our results indicate that it is possible to go beyond the optimum coverage of pure alkyl functionalized SAMs (50%) by adding a linker with the correct choice of the linker. This is very important since dense packed monolayers have fewer defects and deliver higher efficiency. Our results indicate that binary anchoring can modify the charge injection and therefore bond stability while preserving the interface electronic structure.

Rhyme and Reason: Analyses of Dual Retrieval Cues

If and only if each single cue uniquely defines its target, a independence model based on fragment theory can predict the strength of a combined dual cue from the strengths of its single cue components. If the single cues do not each uniquely define their target, no single monotonic function can predict the strength of the dual cue from its components; rather, what matters is the number of possible targets. The probability of generating a target word was .19 for rhyme cues, .14 for category cues, and .97 for rhyme-plus-category dual cues. Moreover, some pairs of cues had probabilities of producing their targets of .03 when used individually and 1.00 when used together, whereas other pairs had moderate probabilities individually and together. The results, which are interpreted in terms of multiple constraints limiting the number of responses, show why rhymes, which play a minimal role in laboratory studies of memory, are common in real-world mnemonics.

Estigma e percepções dos médicos de Clínica Geral sobre a doença mental e o suicídio

RESUMO: INTRODUÇÃO: A OMS (2001) revela que cerca de 450 milhões de pessoas sofrem de perturbações mentais ou comportamentais em todo o mundo, mas apenas uma pequena minoria tem tratamento, ainda que elementar. Transformam-se em vítimas por causa da sua doença e convertem-se em alvos de estigma e discriminação. O suicídio é considerado como um grande problema de saúde pública em todo o mundo, é uma das principais causas de morte de jovens adultos e situa-se entre as três maiores causas de morte na população entre 15-34 anos (OMS, 2001). As perturbações mentais aumentam o risco de suicídio. A depressão, esquizofrenia, e a utilização de substâncias incrementam o risco de suicídio. Estudos (Sartorius, 2002; Magliano et al., 2012) mostram que os profissionais de saúde, tal como o público em geral, podem ter atitudes negativas e estigma em relação às pessoas com perturbações mentais, podendo agir em conformidade, uma vez feito e conhecido o diagnóstico psiquiátrico. Os clínicos gerais são os receptores das perturbações mentais e tentativas de suicídio nas principais portas de entrada no acesso a cuidados de saúde. As crenças, conhecimentos e contacto com a doença mental e o suicídio, podem influenciar a atenção clínica. OBJECTIVOS: Avaliar o estigma e as percepções dos médicos de clínica geral em relação às tentativas de suicídio, o suicídio e perturbações mentais bem como os possíveis factores associados a estes fenómenos. MATERIAIS E MÉTODOS: Estudo do tipo transversal, combinando métodos quantitativos e qualitativos. A amostra é constituída por 125 sujeitos, médicos de clínica geral. Utilizaram-se as versões adaptadas dos seguintes instrumentos: Questionário sobre Percepções e Estigma em Relação à Saúde Mental e ao Suicídio (Liz Macmin e SOQ, Domino, 2005) e a Escala de Atitudes sobre a Doença Mental (Amanha Hahn, 2002). Para o tratamento estatístico dos dados usou-se a estatística 1) descritiva e 2) Análise estatística das hipóteses formuladas (Qui Quadrado - 2) a correlação entre variáveis (Spearman: ρ, rho). Os dados conectados foram limpos de inconsistências com base no pacote informático e estatístico SPSS versão 20. Para a aferição da consistência interna foi usado o teste de Alfa de Cronbach. RESULTADOS: Uma boa parte da amostra (46.4%) refere que não teve formação formal ou informal em saúde mental e (69.35%) rejeitam a ideia de que “grupos profissionais como médicos, dentistas e psicólogos são mais susceptíveis a cometer o suicídio”. Já (28.0%) têm uma perspectiva pessimista quanto a possibilidade de recuperação total dos sujeitos com perturbação mental. Sessenta e oito(54.4%) associa sujeitos com perturbação mental, a comportamentos estranhos e imprevisíveis, 115 (92.0%) a um baixo QI e 35 (26.7%) a poderem ser violentas e e perigosas. Os dados mostram uma associação estatisticamente significativa (p0.001) entre as variáveis: tempo de serviço no SNS, recear estar perto de sujeitos com doença mental e achar que os sujeitos com doença mental são mais perigosos que outros. Em termos estatísticos, existe uma associação estatitisticamente significativa entre as duas variáveis(X2=9,522; p0.05): percepção de que “é vergonhoso ter uma doença mental” e os conhecimentos em relação à doença mental. Existe uma correlação positiva, fraca e estatisticamente significativa entre os conhecimentos dos clínicos gerais(beneficiar-se de formação em saúde mental) e a percepção sobre os factores de risco (0,187; P0,039). DISCUSSÃO E CONCLUSÕES: A falta de conhecimento sobre as causas e factores de risco para os comportamentos suicidários, opções de intervenção e tratamento, particularmente no âmbito da doença mental, podem limitar a procura de ajuda individual ou dos próximos. Percepções negativas como o facto de não merecerem prioridade nos serviços, mitos (frágeis e cobarde, sempre impulsivo, chamadas de atenção, problemas espirituais) podem constituir-se como um indicador de que os clínicos gerais podem sofrer do mesmo sistema de estigma e crenças, de que sofre o público em geral, podendo agir em conformidade (atitudes de afastamento ereceio). As atitudes são influenciadas por factores como a formação, cultura e sistema de crenças. Sujeitos com boa formação na área da saúde mental têm uma percepção positiva e optimista sobre os factores de risco e uma atitude positiva em relação aos sujeitos com doença mental e comportamentos suicidários.-------------ABSTRACT: INTRODUCTION: The WHO (2001) reveals that about 450 million people suffer from mental or behavioral disorders worldwide, but only a small minority have access to treatment, though elementary. They become victims because of their disease and they become the targets of stigma and discrimination. Suicide is seen as a major public health problem worldwide, is a leading cause of death for young adults and is included among the three major causes of death in the population aged 15-34 years (WHO, 2001). Mental disorders increase the risk of suicide. Depression, schizophrenia, and the substances misuse increase the risk of suicide. Studies (Sartorius, 2002; Magliano et al, 2012) show that health professionals, such as the general public, may have negative attitudes and stigma towards people with mental disorders, and can act accordingly after psychiatric diagnosis is known. General practitioners are the main entry points of mental disorders and suicide attempts in the health sistem. Beliefs, knowledge and contact with mental illness and suicide, may influence clinical care. OBJECTIVES: To assess stigma and perceptions of general practitioners in relation to suicide attempts, suicide and mental disorders as well as possible factors associated with these phenomena. MATERIAL AND METHODS: This was a descriptive cross-sectional study, combining quantitative and qualitative methods. The sample consisted of 125 subjects, general practitioners. We used adapted versions of the following instruments: Questionnaire of Perceptions and Stigma in Relation to Mental Health and Suicide (Liz Macmin and SOQ, Domino, 2005) and the Scale of Attitudes on Mental Illness (Tomorrow Hahn, 2002). For the statistical treatment of the data we used: 1) descriptive (Data distribution by absolute and relative frequencies for each of the variables under study (including mean and standard deviation measures of central tendency and deviation), 2) statistical analysis of hypotheses using (Chi Square - 2, a hypothesis test that is intended to find a value of dispersion for two nominal variables, evaluating the association between qualitative variables) and the correlation between variables (Spearman ρ, rho), a measure of non-parametric correlation, which evaluates an arbitrary monotonic function can be the description of the relationship between two variables, without making any assumptions about the frequency distribution of the variables). For statistical analysis of the correlations were eliminated subjects who did not respond to questions. The collected data were cleaned for inconsistencies based on computer and statistical package SPSS version 20. To measure the internal consistency was used the Cronbach's alpha test. RESULTS: A significant part of the sample 64 (46.4%) reported no formal or informal training in mental health and 86 (69.35%) reject the idea that "professional groups such as doctors, dentists and psychologists are more likely to commit suicide." On the other hand, 42 (28.0%) have a pessimistic view of the possibility of full recovery of individuals with mental disorder. Sixty-eight ( 54.4 % ) of them associates subjects with mental disorder to strange and unpredictable behavior, 115 ( 92.0 % ), to low IQ, 35 ( 26.7 % ) and even to violent and dangerous behavior, 78 ( 62.4 % ) The data show a statistically significant (p = 0.001) relationship between the following variables: length of service in the NHS, fear of being close to individuals with mental illness and considering individuals with mental illness more dangerous than others. In statistical terms, there is a dependency between the two variables (X2 = 9.522, p> 0.05): the perception that "it is shameful to have a mental illness" and knowledge regarding mental illness. There is a positive and statistically significant weak correlation between knowledge of general practitioners (benefit from mental health training) and the perception of the risk factors (0,187; P0,039). DISCUSSION AND CONCLUSIONS: The lack of knowledge about the causes and risk factors for suicidal behavior, intervention and treatment, particularly in the context of mental illness options, may decreaseseeking for help by individual and their relatives. Negative perceptions such as considering that they dont deserve priority in services, myths (weak and cowards, always impulsive, seeking for attentions, spirituals problems) may indicate that general practitioners, may suffer the same stigma and beliefs systems as the general public, and can act accordingly (withdrawal and fear attitudes). Attitudes are influenced by factors such as education, culture and belief system. Subjects with good training in mental health have a positive and optimistic perception of the risk factors and a positiveattitude towards individuals with mental illness and suicidal behaviour.

Méthode de simulation avec les variables antithétiques

Les fichiers qui accompagnent mon document ont été réalisés avec le logiciel Latex et les simulations ont été réalisés par Splus(R).

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model

In this paper we have proposed and analyzed a simple mathematical model consisting of four variables, viz., nutrient concentration, toxin producing phytoplankton (TPP), non-toxic phytoplankton (NTP), and toxin concentration. Limitation in the concentration of the extracellular nutrient has been incorporated as an environmental stress condition for the plankton population, and the liberation of toxic chemicals has been described by a monotonic function of extracellular nutrient. The model is analyzed and simulated to reproduce the experimental findings of Graneli and Johansson [Graneli, E., Johansson, N., 2003. Increase in the production of allelopathic Prymnesium parvum cells grown under N- or P-deficient conditions. Harmful Algae 2, 135–145]. The robustness of the numerical experiments are tested by a formal parameter sensitivity analysis. As the first theoretical model consistent with the experiment of Graneli and Johansson (2003), our results demonstrate that, when nutrient-deficient conditions are favorable for the TPP population to release toxic chemicals, the TPP species control the bloom of other phytoplankton species which are non-toxic. Consistent with the observations made by Graneli and Johansson (2003), our model overcomes the limitation of not incorporating the effect of nutrient-limited toxic production in several other models developed on plankton dynamics.

Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Tetragonal to monoclinic phase transition observed during Zr anodisation

Plasma electrolytic oxidation (PEO) is a coating procedure that utilises anodic oxidation in aqueous electrolytes above the dielectric breakdown voltage to produce oxide coatings that have specific properties. These conditions facilitate oxide formation under localised high temperatures and pressures that originate from short-lived microdischarges at sites over the metal surface and have fast oxide volume expansion. Anodic ZrO2 films were prepared by subjecting metallic zirconium to PEO in acid solutions (H2C 2O4 and H3PO4) using a galvanostatic DC regime. The ZrO2 microstructure was investigated in films that were prepared at different charge densities. During the anodic breakdown, an important change in the amplitude of the voltage oscillations at a specific charge density was observed (i.e., the transition charge density (Q T)). We verified that this transition charge is a monotonic function of both the current density and temperature applied during the anodisation, which indicated that Q T is an intrinsic response of this system. The oxide morphology and microstructure were characterised using SEM and X-ray diffraction experiments (XRD) techniques. X-ray diffraction analysis revealed that the change in voltage oscillation was correlated with oxide microstructure changes during the breakdown process. © 2012 Springer-Verlag Berlin Heidelberg.

Nutrient-responsive mTOR signalling grows on Sterile ground

The control of cell growth, that is cell size, is largely controlled by mTOR (the mammalian target of rapamycin), a large serine/threonine protein kinase that regulates ribosome biogenesis and protein translation. mTOR activity is regulated both by the availability of growth factors, such as insulin/IGF-1 (insulin-like growth factor 1), and by nutrients, notably the supply of certain key amino acids. The last few years have seen a remarkable increase in our understanding of the canonical, growth factor-regulated pathway for mTOR activation, which is mediated by the class I PI3Ks (phosphoinositide 3-kinases), PKB (protein kinase B), TSC1/2 (the tuberous sclerosis complex) and the small GTPase, Rheb. However, the nutrient-responsive input into mTOR is important in its own right and is also required for maximal activation of mTOR signalling by growth factors. Despite this, the details of the nutrient-responsive signalling pathway(s) controlling mTOR have remained elusive, although recent studies have suggested a role for the class III PI3K hVps34. In this issue of the Biochemical Journal, Findlay et al. demonstrate that the protein kinase MAP4K3 [mitogen-activated protein kinase kinase kinase kinase-3, a Ste20 family protein kinase also known as GLK (germinal centre-like kinase)] is a new component of the nutrient-responsive pathway. MAP4K3 activity is stimulated by administration of amino acids, but not growth factors, and this is insensitive to rapamycin, most likely placing MAP4K3 upstream of mTOR. Indeed, MAP4K3 is required for phosphorylation of known mTOR targets such as S6K1 (S6 kinase 1), and overexpression of MAP4K3 promotes the rapamycin-sensitive phosphorylation of these same targets. Finally, knockdown of MAP4K3 levels causes a decrease in cell size. The results suggest that MAP4K3 is a new component in the nutrient-responsive pathway for mTOR activation and reveal a completely new function for MAP4K3 in promoting cell growth. Given that mTOR activity is frequently deregulated in cancer, there is much interest in new strategies for inhibition of this pathway. In this context, MAP4K3 looks like an attractive drug target since inhibitors of this enzyme should switch off mTOR, thereby inhibiting cell growth and proliferation, and promoting apoptosis.

Spatial and temporal variability in somatic growth of green sea turtles (Chelonia mydas) resident in the Hawaiian Archipelago

The somatic growth dynamics of green turtles ( Chelonia mydas) resident in five separate foraging grounds within the Hawaiian Archipelago were assessed using a robust non-parametric regression modelling approach. The foraging grounds range from coral reef habitats at the north-western end of the archipelago, to coastal habitats around the main islands at the southeastern end of the archipelago. Pelagic juveniles recruit to these neritic foraging grounds from ca. 35 cm SCL or 5 kg ( similar to 6 years of age), but grow at foraging-ground-specific rates, which results in quite different size- and age-specific growth rate functions. Growth rates were estimated for the five populations as change in straight carapace length ( cm SCL year) 1) and, for two of the populations, also as change in body mass ( kg year) 1). Expected growth rates varied from ca. 0 - 2.5 cm SCL year) 1, depending on the foraging-ground population, which is indicative of slow growth and decades to sexual maturity, since expected size of first-time nesters is greater than or equal to 80 cm SCL. The expected size- specific growth rate functions for four populations sampled in the southeastern archipelago displayed a non-monotonic function, with an immature growth spurt at ca. 50 - 53 cm SCL ( similar to 18 - 23 kg) or ca. 13 - 19 years of age. The growth spurt for the Midway atoll population in the northwestern archipelago occurs at a much larger size ( ca. 65 cm SCL or 36 kg), because of slower immature growth rates that might be due to a limited food stock and cooler sea surface temperature. Expected age-at-maturity was estimated to be ca. 35 - 40 years for the four populations sampled at the south-eastern end of the archipelago, but it might well be > 50 years for the Midway population. The Hawaiian stock comprises mainly the same mtDNA haplotype, with no differences in mtDNA stock composition between foraging-ground populations, so that the geographic variability in somatic growth rates within the archipelago is more likely due to local environmental factors rather than genetic factors. Significant temporal variability was also evident, with expected growth rates declining over the last 10 - 20 years, while green turtle abundance within the archipelago has increased significantly since the mid-1970s. This inverse relationship between somatic growth rates and population abundance suggests a density-dependent effect on somatic growth dynamics that has also been reported recently for a Caribbean green turtle stock. The Hawaiian green turtle stock is characterised by slow growth rates displaying significant spatial and temporal variation and an immature growth spurt. This is consistent with similar findings for a Great Barrier Reef green turtle stock that also comprises many foraging-ground populations spanning a wide geographic range.

Investigating the Structure of FtsZ to Understand its Functional Role in Bacterial Cell Division

FtsZ, a bacterial tubulin homologue, is a cytoskeleton protein that plays key roles in cytokinesis of almost all prokaryotes. FtsZ assembles into protofilaments (pfs), one subunit thick, and these pfs assemble further to form a “Z ring” at the center of prokaryotic cells. The Z ring generates a constriction force on the inner membrane, and also serves as a scaffold to recruit cell-wall remodeling proteins for complete cell division in vivo. FtsZ can be subdivided into 3 main functional regions: globular domain, C terminal (Ct) linker, and Ct peptide. The globular domain binds GTP to assembles the pfs. The extreme Ct peptide binds membrane proteins to allow cytoplasmic FtsZ to function at the inner membrane. The Ct linker connects the globular domain and Ct peptide. In the present studies, we used genetic and structural approaches to investigate the function of Escherichia coli (E. coli) FtsZ. We sought to examine three questions: (1) Are lateral bonds between pfs essential for the Z ring? (2) Can we improve direct visualization of FtsZ in vivo by engineering an FtsZ-FP fusion that can function as the sole source of FtsZ for cell division? (3) Is the divergent Ct linker of FtsZ an intrinsically disordered peptide (IDP)?

One model of the Z ring proposes that pfs associate via lateral bonds to form ribbons; however, lateral bonds are still only hypothetical. To explore potential lateral bonding sites, we probed the surface of E. coli FtsZ by inserting either small peptides or whole FPs. Of the four lateral surfaces on FtsZ pfs, we obtained inserts on the front and back surfaces that were functional for cell division. We concluded that these faces are not sites of essential interactions. Inserts at two sites, G124 and R174 located on the left and right surfaces, completely blocked function, and were identified as possible sites for essential lateral interactions. Another goal was to find a location within FtsZ that supported fusion of FP reporter proteins, while allowing the FtsZ-FP to function as the sole source of FtsZ. We discovered one internal site, G55-Q56, where several different FPs could be inserted without impairing function. These FtsZ-FPs may provide advances for imaging Z-ring structure by super-resolution techniques.

The Ct linker is the most divergent region of FtsZ in both sequence and length. In E. coli FtsZ the Ct linker is 50 amino acids (aa), but for other FtsZ it can be as short as 37 aa or as long as 250 aa. The Ct linker has been hypothesized to be an IDP. In the present study, circular dichroism confirmed that isolated Ct linkers of E. coli (50 aa) and C. crescentus (175 aa) are IDPs. Limited trypsin proteolysis followed by mass spectrometry (LC-MS/MS) confirmed Ct linkers of E. coli (50 aa) and B. subtilis (47 aa) as IDPs even when still attached to the globular domain. In addition, we made chimeras, swapping the E. coli Ct linker for other peptides and proteins. Most chimeras allowed for normal cell division in E. coli, suggesting that IDPs with a length of 43 to 95 aa are tolerated, sequence has little importance, and electrostatic charge is unimportant. Several chimeras were purified to confirm the effect they had on pf assembly. We concluded that the Ct linker functions as a flexible tether allowing for force to be transferred from the FtsZ pf to the membrane to constrict the septum for division.

Series representations of the remainders in the expansions for certain functions with applications

We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1)2/(et+1) , sech t and coth t and establish simple bounds for these remainders when t>0t>0 . Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers Bn.