Peroxisomal beta-oxidation--a metabolic pathway with multiple functions.

Fatty acid degradation in most organisms occurs primarily via the beta-oxidation cycle. In mammals, beta-oxidation occurs in both mitochondria and peroxisomes, whereas plants and most fungi harbor the beta-oxidation cycle only in the peroxisomes. Although several of the enzymes participating in this pathway in both organelles are similar, some distinct physiological roles have been uncovered. Recent advances in the structural elucidation of numerous mammalian and yeast enzymes involved in beta-oxidation have shed light on the basis of the substrate specificity for several of them. Of particular interest is the structural organization and function of the type 1 and 2 multifunctional enzyme (MFE-1 and MFE-2), two enzymes evolutionarily distant yet catalyzing the same overall enzymatic reactions but via opposite stereochemistry. New data on the physiological roles of the various enzymes participating in beta-oxidation have been gathered through the analysis of knockout mutants in plants, yeast and animals, as well as by the use of polyhydroxyalkanoate synthesis from beta-oxidation intermediates as a tool to study carbon flux through the pathway. In plants, both forward and reverse genetics performed on the model plant Arabidopsis thaliana have revealed novel roles for beta-oxidation in the germination process that is independent of the generation of carbohydrates for growth, as well as in embryo and flower development, and the generation of the phytohormone indole-3-acetic acid and the signal molecule jasmonic acid.

Water as an essential nutrient: the physiological basis of hydration.

How much water we really need depends on water functions and the mechanisms of daily water balance regulation. The aim of this review is to describe the physiology of water balance and consequently to highlight the new recommendations with regard to water requirements. Water has numerous roles in the human body. It acts as a building material; as a solvent, reaction medium and reactant; as a carrier for nutrients and waste products; in thermoregulation; and as a lubricant and shock absorber. The regulation of water balance is very precise, as a loss of 1% of body water is usually compensated within 24 h. Both water intake and water losses are controlled to reach water balance. Minute changes in plasma osmolarity are the main factors that trigger these homeostatic mechanisms. Healthy adults regulate water balance with precision, but young infants and elderly people are at greater risk of dehydration. Dehydration can affect consciousness and can induce speech incoherence, extremity weakness, hypotonia of ocular globes, orthostatic hypotension and tachycardia. Human water requirements are not based on a minimal intake because it might lead to a water deficit due to numerous factors that modify water needs (climate, physical activity, diet and so on). Water needs are based on experimentally derived intake levels that are expected to meet the nutritional adequacy of a healthy population. The regulation of water balance is essential for the maintenance of health and life. On an average, a sedentary adult should drink 1.5 l of water per day, as water is the only liquid nutrient that is really essential for body hydration.

Classification of human astrocytic gliomas on the basis of gene expression: a correlated group of genes with angiogenic activity emerges as a strong predictor of subtypes.

The development of targeted treatment strategies adapted to individual patients requires identification of the different tumor classes according to their biology and prognosis. We focus here on the molecular aspects underlying these differences, in terms of sets of genes that control pathogenesis of the different subtypes of astrocytic glioma. By performing cDNA-array analysis of 53 patient biopsies, comprising low-grade astrocytoma, secondary glioblastoma (respective recurrent high-grade tumors), and newly diagnosed primary glioblastoma, we demonstrate that human gliomas can be differentiated according to their gene expression. We found that low-grade astrocytoma have the most specific and similar expression profiles, whereas primary glioblastoma exhibit much larger variation between tumors. Secondary glioblastoma display features of both other groups. We identified several sets of genes with relatively highly correlated expression within groups that: (a). can be associated with specific biological functions; and (b). effectively differentiate tumor class. One prominent gene cluster discriminating primary versus nonprimary glioblastoma comprises mostly genes involved in angiogenesis, including VEGF fms-related tyrosine kinase 1 but also IGFBP2, that has not yet been directly linked to angiogenesis. In situ hybridization demonstrating coexpression of IGFBP2 and VEGF in pseudopalisading cells surrounding tumor necrosis provided further evidence for a possible involvement of IGFBP2 in angiogenesis. The separating groups of genes were found by the unsupervised coupled two-way clustering method, and their classification power was validated by a supervised construction of a nearly perfect glioma classifier.

Endocannabinoid system and psychiatry: in search of a neurobiological basis for detrimental and potential therapeutic effects.

Public concern on mental health has noticeably increased given the high prevalence of neuropsychiatric disorders. Cognition and emotionality are the most affected functions in neuropsychiatric disorders, i.e., anxiety disorders, depression, and schizophrenia. In this review, most relevant literature on the role of the endocannabinoid (eCB) system in neuropsychiatric disorders will be presented. Evidence from clinical and animal studies is provided for the participation of CB1 and CB2 receptors (CB1R and CB2R) in the above mentioned neuropsychiatric disorders. CBRs are crucial in some of the emotional and cognitive impairments reported, although more research is required to understand the specific role of the eCB system in neuropsychiatric disorders. Cannabidiol (CBD), the main non-psychotropic component of the Cannabis sativa plant, has shown therapeutic potential in several neuropsychiatric disorders. Although further studies are needed, recent studies indicate that CBD therapeutic effects may partially depend on facilitation of eCB-mediated neurotransmission. Last but not least, this review includes recent findings on the role of the eCB system in eating disorders. A deregulation of the eCB system has been proposed to be in the bases of several neuropsychiatric disorders, including eating disorders. Cannabis consumption has been related to the appearance of psychotic symptoms and schizophrenia. In contrast, the pharmacological manipulation of this eCB system has been proposed as a potential strategy for the treatment of anxiety disorders, depression, and anorexia nervosa. In conclusion, the eCB system plays a critical role in psychiatry; however, detrimental consequences of manipulating this endogenous system cannot be underestimated over the potential and promising perspectives of its therapeutic manipulation.

Molecular basis of interaction between Na,K-ATPase and FXYD proteins.

Résumé au large public Notre corps est constitué de différents types de cellules. La condition minimale ou primordiale pour la survie des cellules est d'avoir de l'énergie. Cette tâche est assumée en partie par une protéine qui se situe dans la membrane de chaque cellule. Nommé Na, K¬ATPase ou pompe à sodium, c'est une protéine pressente dans toutes les cellules chez les mammifères est composée de deux sous-unités, α et β. En transportant 3 ions de sodium hors de la cellule et 2 ions de potassium à l'intérieur de la cellule, elle transforme l'énergie chimique sous forme de l'ATP en énergie motrice, qui permet aux cellules par la suite d'échanger des matériaux entre l'espace intracellulaire et extracellulaire ainsi que d'ingérer des nutriments provenant de son environnement. Le manque de cette protéine chez la souris entraîne la mort de l'embryon. Des défauts fonctionnels de cette protéine sont responsables de plusieurs maladies humaines comme par exemple, un type de migraine. En dehors de sa fonction vitale, cette protéine est également engagée dans diverses activités physiologiques comme la contractilité musculaire, l'activité nerveuse et la régulation du volume sanguin. Vue l'importance de cette protéine, sa découverte par Jens C. Skou en 1957 a été honorée d'un Prix Noble de chimie quarante ans plus tard. Depuis lors, nous connaissons de mieux en mieux les mécanismes de fonctionnement de la Na, K-ATPase. Entre autre, sa régulation par une famille de protéines appelées protéines FXYD. Cette famille contient 7 membres (FXYD 1-7). L'un d'entre eux nommé FXYD 2 est lié à une maladie héréditaire connue sous le nom de hypomagnesemia. Nous disposons actuellement d'informations concernant les conséquences de la régulation par les protéines FXYD sur activité de la Na, K-ATPase, mais nous savons très peu sur le mode d'interaction entre les protéines FXYD et la Na, K-ATPase. Dans ce travail de thèse, nous avons réussi à localiser des zones d'interaction dans la sous- unité a de la Na, K-ATPase et dans FXYD 7. En même temps, nous avons déterminé un 3ème site de liaison spécifique au sodium de la Na, K-ATPase. Une partie de ce site se situe à l'intérieur d'un domaine protéique qui interagit avec les protéines FXYD. De plus, ce site a été démontré comme responsable d'un mécanisme de transport de la Na, K-ATPase caractérisé par un influx ionique. En conclusion, les résultats de ce travail de thèse fournissent de nouvelles preuves sur les régions d'interaction entre la Na, K-ATPase et les protéines FXYD. La détermination d'un 3ème site spécifique au sodium et sa relation avec un influx ionique offrent la possibilité 1) d'explorer les mécanismes avec lesquels les protéines FXYD régulent l'activité de la Na, ATPase et 2) de localiser un site à sodium qui est essentielle pour mieux comprendre l'organisation et le fonctionnement de la Na, K-ATPase. Résumé Les gradients de concentration de Na+ et de K+ à travers la membrane plasmatique des cellules animales sont cruciaux pour la survie et l'homéostasie de cellules. De plus, des fonctions cellulaires spécifiques telles que la reabsorption de Na dans le rein et le côlon, la contraction musculaire et l'excitabilité nerveuse dépendent de ces gradients. La Na, K¬ATPase ou pompe à sodium est une protéine membranaire ubiquitaire. Elle crée et maintient ces gradients en utilisant l'énergie obtenu par l'hydrolyse de l'adénosine triphosphate. L'unité fonctionnelle minimale de cette protéine se compose d'une sous-unité catalytique α et d'une sous-unité régulatrice β. Récemment, il a été montré que des membres de la famille FXYD, sont des régulateurs tissu-spécifiques de la Na, K-ATPase qui influencent ses propriétés de transport. Cependant, on connaît peu de chose au sujet de la nature moléculaire de l'interaction entre les protéines FXYD et la Na, K-ATPase. Dans cette étude, nous fournissons, pour la première fois, l'évidence directe que des résidus du domaine transmembranaire (TM) 9 de la sous-unité α de la Na, K-ATPase sont impliqués dans l'interaction fonctionnelle et structurale avec les protéines FXYD. De plus nous avons identifié des régions dans le domaine transmembranaire de FXYD 7 qui sont importantes pour l'association stable avec la Na, K-ATPase et une série de résidus responsables des régulations fonctionnelles. Nous avons aussi montré les contributions fonctionnelles du TM 9 de la Na, K-ATPase à la translocation de Na + en déterminant un 3ème site spécifique au Na+. Ce site se situe probablement dans un espace entre TM 9, TM 6 et TM 5 de la sous-unité α de la pompe à sodium. De plus, nous avons constaté que le 3ème site de Na + est fonctionnellement lié à un courant entrant de la pompe sensible à l'ouabaïne et activé par le pH acide. En conclusion, ce travail donne de nouvelles perspectives de l'interaction structurale et fonctionnelle entre les protéines FXYD et la Na, K-ATPase. En outre, les contributions fonctionnelles de TM 9 offrent de nouvelles possibilités pour explorer le mécanisme par lequel les protéines FXYD régulent les propriétés fonctionnelles de la Na, K-ATPase. La détermination du 3ème site au Na + fournit une compréhension avancée du site spécifique au Na + de la Na, K-ATPase et du mécanisme de transport de la Na, K-ATPase. Summary The Na+ and K+ gradients across the plasma membrane of animal cells are crucial for cell survival and homeostasis. Moreover, specific tissue functions such as Na+ reabsorption in kidney and colon, muscle contraction and nerve excitability depend on the maintenance of these gradients. Na, K-ATPase or sodium pump, an ubiquitous membrane protein, creates and maintains these gradients by using the energy from the hydrolysis of ATP. The minimal functional unit of this protein is composed of a catalytic α subunit and a regulatory β subunit. Recently, members of the FXYD family, have been reported to be tissue-specific regulators of Na, K-ATPase by influencing its transport properties. However, little is known about the molecular nature of the interaction between FXYD proteins and Na, K-ATPase. In this study, we provide, for the first time, direct evidence that residues from the transmembrane (TM) domain 9 of the α subunit of Na, K-ATPase are implicated in the functional and structural interaction with FXYD proteins. Moreover, we have identified regions in the TM domain of FXYD 7 important for the stable association with Na, K-ATPase and a stretch of residues responsible for the functional regulations. We have further revealed the functional contributions of TM 9 of the Na, K-ATPase α subunit to the Na+ translocation by determining a 3rd Na+-specific cation binding site. This site is likely in a space between TM 9, TM 6 and TM 5 of the a subunit of the sodium pump. Moreover, we have found that the 3rd Na+ binding site is functionally linked to an acidic pH- activated ouabain-sensitive inward pump current. In conclusion, this work gives new insights into the structural and functional interaction between FXYD proteins and Na, K-ATPase. Functional contributions of TM 9 offer new possibilities to explore the mechanism by which FXYD proteins regulate functional properties of Na, K-ATPase. The determination of the 3rd Na+ binding site provides an advanced understanding concerning the Na+ -specific binding site of Na, K-ATPase and the 3rd Na+ site related transport mechanism.

Radial transport of nutrients: the plant root as a polarized epithelium.

In higher plants, roots acquire water and soil nutrients and transport them upward to their aerial parts. These functions are closely related to their anatomical structure; water and nutrients entering the root first move radially through several concentric layers of the epidermis, cortex, and endodermis before entering the central cylinder. The endodermis is the innermost cortical cell layer that features rings of hydrophobic cell wall material called the Casparian strips, which functionally resemble tight junctions in animal epithelia. Nutrient uptake from the soil can occur through three different routes that can be interconnected in various ways: the apoplastic route (through the cell wall), the symplastic route (through cellular connections), and a coupled trans-cellular route (involving polarized influx and efflux carriers). This Update presents recent advances in the radial transport of nutrients highlighting the coupled trans-cellular pathway and the roles played by the endodermis as a barrier.

A chemical Hamiltonian approach study of the basis set superposition error changes on electron densities and one- and two-center energy components

A comparision of the local effects of the basis set superposition error (BSSE) on the electron densities and energy components of three representative H-bonded complexes was carried out. The electron densities were obtained with Hartee-Fock and density functional theory versions of the chemical Hamiltonian approach (CHA) methodology. It was shown that the effects of the BSSE were common for all complexes studied. The electron density difference maps and the chemical energy component analysis (CECA) analysis confirmed that the local effects of the BSSE were different when diffuse functions were present in the calculations

C-H···O H-bonded complexes: how does basis set superposition error change their potential-energy surfaces?

Geometries, vibrational frequencies, and interaction energies of the CNH⋯O3 and HCCH⋯O3 complexes are calculated in a counterpoise-corrected (CP-corrected) potential-energy surface (PES) that corrects for the basis set superposition error (BSSE). Ab initio calculations are performed at the Hartree-Fock (HF) and second-order Møller-Plesset (MP2) levels, using the 6-31G(d,p) and D95++(d,p) basis sets. Interaction energies are presented including corrections for zero-point vibrational energy (ZPVE) and thermal correction to enthalpy at 298 K. The CP-corrected and conventional PES are compared; the unconnected PES obtained using the larger basis set including diffuse functions exhibits a double well shape, whereas use of the 6-31G(d,p) basis set leads to a flat single-well profile. The CP-corrected PES has always a multiple-well shape. In particular, it is shown that the CP-corrected PES using the smaller basis set is qualitatively analogous to that obtained with the larger basis sets, so the CP method becomes useful to correctly describe large systems, where the use of small basis sets may be necessary

Ab initio benchmark study for the oxidative addition of CH4 to Pd: importance of basis-set flexibility and polarization

To obtain a state-of-the-art benchmark potential energy surface (PES) for the archetypal oxidative addition of the methane C-H bond to the palladium atom, we have explored this PES using a hierarchical series of ab initio methods (Hartree-Fock, second-order Møller-Plesset perturbation theory, fourth-order Møller-Plesset perturbation theory with single, double and quadruple excitations, coupled cluster theory with single and double excitations (CCSD), and with triple excitations treated perturbatively [CCSD(T)]) and hybrid density functional theory using the B3LYP functional, in combination with a hierarchical series of ten Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account either through a relativistic effective core potential for palladium or through a full four-component all-electron approach. Counterpoise corrected relative energies of stationary points are converged to within 0.1-0.2 kcal/mol as a function of the basis-set size. Our best estimate of kinetic and thermodynamic parameters is -8.1 (-8.3) kcal/mol for the formation of the reactant complex, 5.8 (3.1) kcal/mol for the activation energy relative to the separate reactants, and 0.8 (-1.2) kcal/mol for the reaction energy (zero-point vibrational energy-corrected values in parentheses). This agrees well with available experimental data. Our work highlights the importance of sufficient higher angular momentum polarization functions, f and g, for correctly describing metal-d-electron correlation and, thus, for obtaining reliable relative energies. We show that standard basis sets, such as LANL2DZ+ 1f for palladium, are not sufficiently polarized for this purpose and lead to erroneous CCSD(T) results. B3LYP is associated with smaller basis set superposition errors and shows faster convergence with basis-set size but yields relative energies (in particular, a reaction barrier) that are ca. 3.5 kcal/mol higher than the corresponding CCSD(T) values

Accurate calculations of intermolecular interaction energies using explicitly correlated coupled cluster wave functions and a dispersion-weighted MP2 method.

Explicitly correlated coupled-cluster calculations of intermolecular interaction energies for the S22 benchmark set of Jurecka, Sponer, Cerny, and Hobza (Chem. Phys. Phys. Chem. 2006, 8, 1985) are presented. Results obtained with the recently proposed CCSD(T)-F12a method and augmented double-zeta basis sets are found to be in very close agreement with basis set extrapolated conventional CCSD(T) results. Furthermore, we propose a dispersion-weighted MP2 (DW-MP2) approximation that combines the good accuracy of MP2 for complexes with predominately electrostatic bonding and SCS-MP2 for dispersion-dominated ones. The MP2-F12 and SCS-MP2-F12 correlation energies are weighted by a switching function that depends on the relative HF and correlation contributions to the interaction energy. For the S22 set, this yields a mean absolute deviation of 0.2 kcal/mol from the CCSD(T)-F12a results. The method, which allows obtaining accurate results at low cost, is also tested for a number of dimers that are not in the training set.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

Epithelial sodium channel/degenerin family of ion channels: a variety of functions for a shared structure.

The recently discovered epithelial sodium channel (ENaC)/degenerin (DEG) gene family encodes sodium channels involved in various cell functions in metazoans. Subfamilies found in invertebrates or mammals are functionally distinct. The degenerins in Caenorhabditis elegans participate in mechanotransduction in neuronal cells, FaNaC in snails is a ligand-gated channel activated by neuropeptides, and the Drosophila subfamily is expressed in gonads and neurons. In mammals, ENaC mediates Na+ transport in epithelia and is essential for sodium homeostasis. The ASIC genes encode proton-gated cation channels in both the central and peripheral nervous system that could be involved in pain transduction. This review summarizes the physiological roles of the different channels belonging to this family, their biophysical and pharmacological characteristics, and the emerging knowledge of their molecular structure. Although functionally different, the ENaC/DEG family members share functional domains that are involved in the control of channel activity and in the formation of the pore. The functional heterogeneity among the members of the ENaC/DEG channel family provides a unique opportunity to address the molecular basis of basic channel functions such as activation by ligands, mechanotransduction, ionic selectivity, or block by pharmacological ligands.

Bernoulli correction to viscous losses: Radial flow between two parallel discs

For a massless fluid (density = 0), the steady flow along a duct is governed exclusively by viscous losses. In this paper, we show that the velocity profile obtained in this limit can be used to calculate the pressure drop up to the first order in density. This method has been applied to the particular case of a duct, defined by two plane-parallel discs. For this case, the first-order approximation results in a simple analytical solution which has been favorably checked against numerical simulations. Finally, an experiment has been carried out with water flowing between the discs. The experimental results show good agreement with the approximate solution

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.