A class of singular first order differential equations with applications in reaction-diffusion

Agências Financiadoras: FCT e MIUR

Constraint reasoning for differential models

The basic motivation of this work was the integration of biophysical models within the interval constraints framework for decision support. Comparing the major features of biophysical models with the expressive power of the existing interval constraints framework, it was clear that the most important inadequacy was related with the representation of differential equations. System dynamics is often modelled through differential equations but there was no way of expressing a differential equation as a constraint and integrate it within the constraints framework. Consequently, the goal of this work is focussed on the integration of ordinary differential equations within the interval constraints framework, which for this purpose is extended with the new formalism of Constraint Satisfaction Differential Problems. Such framework allows the specification of ordinary differential equations, together with related information, by means of constraints, and provides efficient propagation techniques for pruning the domains of their variables. This enabled the integration of all such information in a single constraint whose variables may subsequently be used in other constraints of the model. The specific method used for pruning its variable domains can then be combined with the pruning methods associated with the other constraints in an overall propagation algorithm for reducing the bounds of all model variables. The application of the constraint propagation algorithm for pruning the variable domains, that is, the enforcement of local-consistency, turned out to be insufficient to support decision in practical problems that include differential equations. The domain pruning achieved is not, in general, sufficient to allow safe decisions and the main reason derives from the non-linearity of the differential equations. Consequently, a complementary goal of this work proposes a new strong consistency criterion, Global Hull-consistency, particularly suited to decision support with differential models, by presenting an adequate trade-of between domain pruning and computational effort. Several alternative algorithms are proposed for enforcing Global Hull-consistency and, due to their complexity, an effort was made to provide implementations able to supply any-time pruning results. Since the consistency criterion is dependent on the existence of canonical solutions, it is proposed a local search approach that can be integrated with constraint propagation in continuous domains and, in particular, with the enforcing algorithms for anticipating the finding of canonical solutions. The last goal of this work is the validation of the approach as an important contribution for the integration of biophysical models within decision support. Consequently, a prototype application that integrated all the proposed extensions to the interval constraints framework is developed and used for solving problems in different biophysical domains.

Positive radial solutions of the Dirichlet problem for the Minkowski-Curvature equation in a ball

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation { -div(del upsilon/root 1-vertical bar del upsilon vertical bar(2)) in B-R, upsilon=0 on partial derivative B-R,B- where B-R is a ball in R-N (N >= 2). According to the behaviour off = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

Bioactive beads for local sensing of proteases in 3D engineered tissues

Dissertação para obtenção do Grau de Doutor em Bioengenharia (MIT)

Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

Nuclear receptor Rev-erb alpha (Nr1d1) functions in concert with Nr2e3 to regulate transcriptional networks in the retina.

The majority of diseases in the retina are caused by genetic mutations affecting the development and function of photoreceptor cells. The transcriptional networks directing these processes are regulated by genes such as nuclear hormone receptors. The nuclear hormone receptor gene Rev-erb alpha/Nr1d1 has been widely studied for its role in the circadian cycle and cell metabolism, however its role in the retina is unknown. In order to understand the role of Rev-erb alpha/Nr1d1 in the retina, we evaluated the effects of loss of Nr1d1 to the developing retina and its co-regulation with the photoreceptor-specific nuclear receptor gene Nr2e3 in the developing and mature retina. Knock-down of Nr1d1 expression in the developing retina results in pan-retinal spotting and reduced retinal function by electroretinogram. Our studies show that NR1D1 protein is co-expressed with NR2E3 in the outer neuroblastic layer of the developing mouse retina. In the adult retina, NR1D1 is expressed in the ganglion cell layer and is co-expressed with NR2E3 in the outer nuclear layer, within rods and cones. Several genes co-targeted by NR2E3 and NR1D1 were identified that include: Nr2c1, Recoverin, Rgr, Rarres2, Pde8a, and Nupr1. We examined the cyclic expression of Nr1d1 and Nr2e3 over a twenty-four hour period and observed that both nuclear receptors cycle in a similar manner. Taken together, these studies reveal a novel role for Nr1d1, in conjunction with its cofactor Nr2e3, in regulating transcriptional networks critical for photoreceptor development and function.

A decision-making Fokker-Planck model in computational neuroscience

Minimal models for the explanation of decision-making in computational neuroscience are based on the analysis of the evolution for the average firing rates of two interacting neuron populations. While these models typically lead to multi-stable scenario for the basic derived dynamical systems, noise is an important feature of the model taking into account finite-size effects and robustness of the decisions. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker-Planck partial differential equation. In particular, we discuss the existence, positivity and uniqueness for the solution of the stationary equation, as well as for the time evolving problem. Moreover, we prove convergence of the solution to the the stationary state representing the probability distribution of finding the neuron families in each of the decision states characterized by their average firing rates. Finally, we propose a numerical scheme allowing for simulations performed on the Fokker-Planck equation which are in agreement with those obtained recently by a moment method applied to the stochastic differential system. Our approach leads to a more detailed analytical and numerical study of this decision-making model in computational neuroscience.

Dynamic simulation of regulatory networks using SQUAD.

BACKGROUND: The ambition of most molecular biologists is the understanding of the intricate network of molecular interactions that control biological systems. As scientists uncover the components and the connectivity of these networks, it becomes possible to study their dynamical behavior as a whole and discover what is the specific role of each of their components. Since the behavior of a network is by no means intuitive, it becomes necessary to use computational models to understand its behavior and to be able to make predictions about it. Unfortunately, most current computational models describe small networks due to the scarcity of kinetic data available. To overcome this problem, we previously published a methodology to convert a signaling network into a dynamical system, even in the total absence of kinetic information. In this paper we present a software implementation of such methodology. RESULTS: We developed SQUAD, a software for the dynamic simulation of signaling networks using the standardized qualitative dynamical systems approach. SQUAD converts the network into a discrete dynamical system, and it uses a binary decision diagram algorithm to identify all the steady states of the system. Then, the software creates a continuous dynamical system and localizes its steady states which are located near the steady states of the discrete system. The software permits to make simulations on the continuous system, allowing for the modification of several parameters. Importantly, SQUAD includes a framework for perturbing networks in a manner similar to what is performed in experimental laboratory protocols, for example by activating receptors or knocking out molecular components. Using this software we have been able to successfully reproduce the behavior of the regulatory network implicated in T-helper cell differentiation. CONCLUSION: The simulation of regulatory networks aims at predicting the behavior of a whole system when subject to stimuli, such as drugs, or determine the role of specific components within the network. The predictions can then be used to interpret and/or drive laboratory experiments. SQUAD provides a user-friendly graphical interface, accessible to both computational and experimental biologists for the fast qualitative simulation of large regulatory networks for which kinetic data is not necessarily available.

Malignant transformation of DMBA/TPA-induced papillomas and nevi in the skin of mice selectively lacking retinoid-X-receptor alpha in epidermal keratinocytes.

Retinoid-X-receptor alpha (RXRalpha), a member of the nuclear receptor (NR) superfamily, is a ligand-dependent transcriptional regulatory factor. It plays a crucial role in NR signalling through heterodimerization with some 15 NRs. We investigated the role of RXRalpha and its partners on mouse skin tumor formation and malignant progression upon topical DMBA/TPA treatment. In mutants selectively ablated for RXRalpha in keratinocytes, epidermal tumors increased in size and number, and frequently progressed to carcinomas. As keratinocyte-selective peroxisome proliferator-activated receptor gamma (PPARgamma) ablation had similar effects, RXRalpha/PPARgamma heterodimers most probably mediate epidermal tumor suppression. Keratinocyte-selective RXRalpha-null and vitamin-D-receptor null mice also exhibited more numerous dermal melanocytic growths (nevi) than control mice, but only nevi from RXRalpha mutant mice progressed to invasive human-melanoma-like tumors. Distinct RXRalpha-mediated molecular events appear therefore to be involved, in keratinocytes, in cell-autonomous suppression of epidermal tumorigenesis and malignant progression, and in non-cell-autonomous suppression of nevi formation and progression. Our study emphasizes the crucial role of keratinocytes in chemically induced epidermal and melanocytic tumorigenesis, and raises the possibility that they could play a similar role in UV-induced tumorigenesis, notably in nevi formation and progression to melanoma.

On the analysis of travelling waves to a nonlinear flux limited reaction-diffusion equation

In this paper we study the existence and qualitative properties of travelling waves associated to a nonlinear flux limited partial differential equation coupled to a Fisher-Kolmogorov-Petrovskii-Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C2 classical regularity, but also the existence of discontinuous entropy travelling wave solutions.

The loss of circadian PAR bZip transcription factors results in epilepsy.

DBP (albumin D-site-binding protein), HLF (hepatic leukemia factor), and TEF (thyrotroph embryonic factor) are the three members of the PAR bZip (proline and acidic amino acid-rich basic leucine zipper) transcription factor family. All three of these transcriptional regulatory proteins accumulate with robust circadian rhythms in tissues with high amplitudes of clock gene expression, such as the suprachiasmatic nucleus (SCN) and the liver. However, they are expressed at nearly invariable levels in most brain regions, in which clock gene expression only cycles with low amplitude. Here we show that mice deficient for all three PAR bZip proteins are highly susceptible to generalized spontaneous and audiogenic epilepsies that frequently are lethal. Transcriptome profiling revealed pyridoxal kinase (Pdxk) as a target gene of PAR bZip proteins in both liver and brain. Pyridoxal kinase converts vitamin B6 derivatives into pyridoxal phosphate (PLP), the coenzyme of many enzymes involved in amino acid and neurotransmitter metabolism. PAR bZip-deficient mice show decreased brain levels of PLP, serotonin, and dopamine, and such changes have previously been reported to cause epilepsies in other systems. Hence, the expression of some clock-controlled genes, such as Pdxk, may have to remain within narrow limits in the brain. This could explain why the circadian oscillator has evolved to generate only low-amplitude cycles in most brain regions.

Functional Variants in NOS1 and NOS2A Are Not Associated with Progressive Hearing Loss in Ménière's Disease in a European Caucasian Population

Hearing loss in Meniere's disease (MD) is associated with loss of spiral ganglion neurons and hair cells. In a guinea pig model of endolymphatic hydrops, nitric oxide synthases (NOS) and oxidative stress mediate loss of spiral ganglion neurons. To test the hypothesis that functional variants of NOS1 and NOS2A are associated with MD, wed genotyped three functional variants of NOS1 (rs41279104,rs2682826, and a cytosine-adenosine microsatellite repeat in exon 1f) and the CCTTT repeat in the promoter of NOS2A gene (rs3833912) in two independent MD sets(273 patients in total) and 550 controls. A third cohort of American patients was genotyped as replication cohort for the CCTTT repeat. Neither allele nor genotype frequencies of rs41279104 and rs2682826 were associated with MD, although longer alleles of the cytosine-adenosine microsatellite repeat were marginally significant (corrected p = 0.05) in the Mediterranean cohort but not in a second Galicia cohort. Shorter numbers of the CCTTT repeat in NOS2A were significantly more frequent in Galicia controls (OR = 0.37 [CI, 0.18-0.76], corrected p =0.04), but this finding could not be replicated in Mediterranean or American case-control populations. Meta-analysis did not support an association between CCTTT repeats and risk for MD. Severe hearing loss (>75 dB) was also not associated with any functional variants studied. Functional variants of NOS1 and and NOS2A do not confer susceptibility for MD.

Repression of flagellar genes in exponential phase by CsgD and CpxR, two crucial modulators of Escherichia coli biofilm formation.

Escherichia coli adapts its lifestyle to the variations of environmental growth conditions, swapping between swimming motility or biofilm formation. The stationary-phase sigma factor RpoS is an important regulator of this switch, since it stimulates adhesion and represses flagellar biosynthesis. By measuring the dynamics of gene expression, we show that RpoS inhibits the transcription of the flagellar sigma factor, FliA, in exponential growth phase. RpoS also partially controls the expression of CsgD and CpxR, two transcription factors important for bacterial adhesion. We demonstrate that these two regulators repress the transcription of fliA, flgM, and tar and that this regulation is dependent on the growth medium. CsgD binds to the flgM and fliA promoters around their -10 promoter element, strongly suggesting direct repression. We show that CsgD and CpxR also affect the expression of other known modulators of cell motility. We propose an updated structure of the regulatory network controlling the choice between adhesion and motility.

Bristle pattern formation in the thorax formation of Drosophila: a systems level approach

A fundamental question in developmental biology is how tissues are patterned to give rise to differentiated body structures with distinct morphologies. The Drosophila wing disc offers an accessible model to understand epithelial spatial patterning. It has been studied extensively using genetic and molecular approaches. Bristle patterns on the thorax, which arise from the medial part of the wing disc, are a classical model of pattern formation, dependent on a pre-pattern of trans-activators and –repressors. Despite of decades of molecular studies, we still only know a subset of the factors that determine the pre-pattern. We are applying a novel and interdisciplinary approach to predict regulatory interactions in this system. It is based on the description of expression patterns by simple logical relations (addition, subtraction, intersection and union) between simple shapes (graphical primitives). Similarities and relations between primitives have been shown to be predictive of regulatory relationships between the corresponding regulatory factors in other Systems, such as the Drosophila egg. Furthermore, they provide the basis for dynamical models of the bristle-patterning network, which enable us to make even more detailed predictions on gene regulation and expression dynamics. We have obtained a data-set of wing disc expression patterns which we are now processing to obtain average expression patterns for each gene. Through triangulation of the images we can transform the expression patterns into vectors which can easily be analysed by Standard clustering methods. These analyses will allow us to identify primitives and regulatory interactions. We expect to identify new regulatory interactions and to understand the basic Dynamics of the regulatory network responsible for thorax patterning. These results will provide us with a better understanding of the rules governing gene regulatory networks in general, and provide the basis for future studies of the evolution of the thorax-patterning network in particular.