Mathematical and numerical methods for inverse problems using fundamental solutions techniques

Fundação para a Ciência e a Tecnologia - SFRH/BD/27914/2006

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

Further results on the inverse along an element in semigroups and rings

In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-regularity and left (right) $*$-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.

The inverse along a product and its applications

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix.

Development of a subject-specific musculoskeletal shoulder model : inverse and forward dynamics applications

A prevalência de pessoas que referem dor no complexo articular do ombro, com concomitante limitação na capacidade para realizar atividades da vida diária, é elevada. Estes níveis de prevalência sobrecarregam quer os utentes, como a própria sociedade. A evidência científica atual indicia a existência de uma relação entre as alterações da articulação escápulo-torácica e as patologias associadas à articulação gleno-umeral. A capacidade de quantificar, cinemática e cineticamente, as disfunções ao nível das articulações escápulo-torácica e gleno-umeral, é algo de enorme importância, quer para a comunidade biomecânica, como para a clínica. No decorrer dos trabalhos desta tese foi desenvolvido, através do software OpenSim, um modelo tridimensional músculo-esquelético do complexo articular do ombro que inclui a representação do tórax/coluna, clavícula, omoplata, úmero, rádio, cúbito e articulações que permitem os movimentos relativos desses segmentos, assim como, 16 músculos e 4 ligamentos. Com um total de 11 graus de liberdade, incluindo um novo modelo articular escápulo-torácico, os resultados demonstram que este é capaz de reconstruir de forma precisa e rápida os movimentos escápulo-torácicos e glenoumerais, recorrendo para tal, à cinemática inversa, e à dinâmica inversa e direta. Conta ainda com um método de transformação inovador para determinar, com base nas especificidades dos sujeitos, os locais de inserção muscular. As principais motivações subjacentes ao desenvolvimento desta tese foram contribuir para o aprofundar do atual conhecimento sobre as disfunções do complexo articular do ombro e, simultaneamente, proporcionar à comunidade clínica uma ferramenta biomecânica de livre acesso com o intuito de melhor suportar as decisões clínicas e dessa forma concorrer para uma prática mais efetiva.

Inverse Problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are nondegenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

A musculoskeletal shoulder model based on pseudo-inverse and null-space optimization.

The goal of the present work was assess the feasibility of using a pseudo-inverse and null-space optimization approach in the modeling of the shoulder biomechanics. The method was applied to a simplified musculoskeletal shoulder model. The mechanical system consisted in the arm, and the external forces were the arm weight, 6 scapulo-humeral muscles and the reaction at the glenohumeral joint, which was considered as a spherical joint. The muscle wrapping was considered around the humeral head assumed spherical. The dynamical equations were solved in a Lagrangian approach. The mathematical redundancy of the mechanical system was solved in two steps: a pseudo-inverse optimization to minimize the square of the muscle stress and a null-space optimization to restrict the muscle force to physiological limits. Several movements were simulated. The mathematical and numerical aspects of the constrained redundancy problem were efficiently solved by the proposed method. The prediction of muscle moment arms was consistent with cadaveric measurements and the joint reaction force was consistent with in vivo measurements. This preliminary work demonstrated that the developed algorithm has a great potential for more complex musculoskeletal modeling of the shoulder joint. In particular it could be further applied to a non-spherical joint model, allowing for the natural translation of the humeral head in the glenoid fossa.

Gene duplication and neofunctionalization: POLR3G and POLR3GL.

RNA polymerase III (Pol III) occurs in two versions, one containing the POLR3G subunit and the other the closely related POLR3GL subunit. It is not clear whether these two Pol III forms have the same function, in particular whether they recognize the same target genes. We show that the POLR3G and POLR3GL genes arose from a DNA-based gene duplication, probably in a common ancestor of vertebrates. POLR3G- as well as POLR3GL-containing Pol III are present in cultured cell lines and in normal mouse liver, although the relative amounts of the two forms vary, with the POLR3G-containing Pol III relatively more abundant in dividing cells. Genome-wide chromatin immunoprecipitations followed by high-throughput sequencing (ChIP-seq) reveal that both forms of Pol III occupy the same target genes, in very constant proportions within one cell line, suggesting that the two forms of Pol III have a similar function with regard to specificity for target genes. In contrast, the POLR3G promoter-not the POLR3GL promoter-binds the transcription factor MYC, as do all other promoters of genes encoding Pol III subunits. Thus, the POLR3G/POLR3GL duplication did not lead to neo-functionalization of the gene product (at least with regard to target gene specificity) but rather to neo-functionalization of the transcription units, which acquired different mechanisms of regulation, thus likely affording greater regulation potential to the cell.

Constitutive activity and inverse agonism at the α(₁a) and α(₁b) adrenergic receptor subtypes.

The α(1b)-adrenergic receptor (AR) was, after rhodopsin, the first G protein-coupled receptor (GPCR) in which point mutations were shown to trigger constitutive (agonist-independent) activity. Constitutively activating mutations have been found in other AR subtypes as well as in several GPCRs. This chapter briefly summarizes the main findings on constitutively active mutants of the α(1a)- and α(1b)-AR subtypes and the methods used to predict activating mutations, to measure constitutive activity of Gq-coupled receptors and to investigate inverse agonism. In addition, it highlights the implications of studies on constitutively active AR mutants on elucidating the molecular mechanisms of receptor activation and drug action.

RNA-based gene duplication sheds new light on mammalian sex chromosome evolution

ABSTRACT : Gene duplication is a fundamental source of raw material for the origin of genetic novelty. It has been assumed for a long time that DNA-based gene duplication was the only source of new genes. Recently however, RNA-based gene duplication (retroposition) was shown in multiple organisms to contribute significantly to their genetic diversity. This mechanism produces intronless gene copies (retrocopies) that are inserted in random genomic position, independent of the position of the parental source genes. In human, mouse and fruit fly, it was demonstrated that the X-linked genes spawned an excess of functional retroposed gene copies (retrogenes). In human and mouse, the X chromosome also recruited an excess of retrogenes. Here we further characterized these interesting biases related to the X chromosome in mammals. Firstly, we have confirmed presence of the aforementioned biases in dog and opossum genome. Then based on the expression profile of retrogenes during various spermatogenetic stages, we have provided solid evidence that meiotic sex chromosome inactivation (MSCI) is responsible for an excess of retrogenes stemming from the X chromosome. Moreover, we showed that the X-linked genes started to export an excess of retrogenes just after the split of eutherian and marsupial mammalian lineages. This suggests that MSCI has originated around this time as well. More fundamentally, as MSCI reflects the spread of recombination barrier between the X and Y chromosomes during their evolution, our observation allowed us to re-estimate the age of mammalian sex chromosomes. Previous estimates suggested that they emerged in the common ancestor of all mammals (before the split of monotreme lineage); whereas, here we showed that they originated around the split of marsupial and eutherian lineages, after the divergence of monotremes. Thus, the therian (marsupial and eutherian) sex chromosomes are younger than previously thought. Thereafter, we have characterized the bias related to the recruitment of genes to the X chromosome. Sexually antagonistic forces are most likely driving this pattern. Using our limited retrogenes expression data, it is difficult to determine the exact nature of these forces but some conclusions have been made. Lastly, we looked at the history of this biased recruitment: it commenced around the split of marsupial and eutherian lineages (akin to the biased export of genes out of the X). In fact, the sexually antagonistic forces are predicted to appear just around that time as well. Thereby, the history of the recruitment of genes to the X, provides an indirect evidence that these forces are responsible for this bias.

Age-dependent gain of alternative splice forms and biased duplication explain the relation between splicing and duplication.

We analyze here the relation between alternative splicing and gene duplication in light of recent genomic data, with a focus on the human genome. We show that the previously reported negative correlation between level of alternative splicing and family size no longer holds true. We clarify this pattern and show that it is sufficiently explained by two factors. First, genes progressively gain new splice variants with time. The gain is consistent with a selectively relaxed regime, until purifying selection slows it down as aging genes accumulate a large number of variants. Second, we show that duplication does not lead to a loss of splice forms, but rather that genes with low levels of alternative splicing tend to duplicate more frequently. This leads us to reconsider the role of alternative splicing in duplicate retention.

Mutation screening of patients with Alzheimer disease identifies APP locus duplication in a Swedish patient.

BACKGROUND: Missense mutations in three different genes encoding amyloid-β precursor protein, presenilin 1 and presenilin 2 are recognized to cause familial early-onset Alzheimer disease. Also duplications of the amyloid precursor protein gene have been shown to cause the disease. At the Dept. of Geriatric Medicine, Karolinska University Hospital, Sweden, patients are referred for mutation screening for the identification of nucleotide variations and for determining copy-number of the APP locus. METHODS: We combined the method of microsatellite marker genotyping with a quantitative real-time PCR analysis to detect duplications in patients with Alzheimer disease. RESULTS: In 22 DNA samples from individuals diagnosed with clinical Alzheimer disease, we identified one patient carrying a duplication on chromosome 21 which included the APP locus. Further mapping of the chromosomal region by array-comparative genome hybridization showed that the duplication spanned a maximal region of 1.09 Mb. CONCLUSIONS: This is the first report of an APP duplication in a Swedish Alzheimer patient and describes the use of quantitative real-time PCR as a tool for determining copy-number of the APP locus.

Une posture inversée: le yoga global

L'article expose un comparatisme reposant sur la pratique des regards croisés dont nous postulons la pertinence pour comprendre l'échange d'idées et de pratiques entre l'Inde et l'«Occident». Durant les XIXe et XXe siècles naît l'un des phénomènes de mondialisation les plus intéressants, à savoir l'exportation et la globalisation du yoga indien. Pendant que l'«Occident» s'éveille au yoga spirituel inauguré par Vivekananda (dès 1893), des yogis indiens soumettent leur tradition à l'expérimentation scientifique moderne (années 20). Dans leur ouvrage commun Sport et Yoga (1941/48), Selvarajan Yesudian (le yogi chrétien indien) et Elisabeth Haich (l'ésotériste hongroise) illustrent d'une manière paradigmatique les synthèses créatives qui peuvent s'opérer dans les processus d'échanges pluri-dimensionels et pluri-directionnels entre les traditions indiennes et européennes que seule une posture comparative, capable de faire le va-et-vient entre les deux traditions, est à même de saisir.