Utilização de quitosana obtida de resíduos de camarão para avaliar a capacidade de adsorção de íons Fe3+

A quitosana é produzida através de uma desacetilação alcalina da quitina, a qual é encontrada em exoesqueleto de crustáceos, parede celular de fungos e materiais biológicos. Calcula-se que os resíduos de camarão apresentam de 5 a 7% do seu peso total na forma de quitina, sugerindo que estes sejam utilizados para obtenção do biopolímero. Os processos para obtenção destes biopolímeros consiste nas seguintes etapas: desmineralização, desproteinização e desodorização, obtendo-se assim, a quitina úmida. Após seca, passa por uma desacetilação química para a conversão em quitosana úmida, sendo purificada e posteriormente seca. A quitosana, por apresentar grupamentos amino livres em sua estrutura, é uma molécula capaz de formar complexos estáveis com cátions metálicos. O objetivo geral deste trabalho foi obter quitina a partir de resíduos de camarão (Penaeus brasiliensis) com posterior produção de quitosana, e avaliar sua capacidade de complexação com íons Fe3+, em solução. A quitosana produzida foi caracterizada através do grau de desacetiliação e da massa molecular viscosimétrica, Para caracterização estrutural das amostras de quitosana, utilizaram-se espectrometria de infravermelho e espectrofotometria UV-Visível, bem como para o complexo formado de quitosana e ferro. Para analisar a eficiência da remoção deste íon, foram feitas análises em espectrometria de absorção atômica em chama e em espectrofotometria UV-Visível. Uma análise estatística foi realizada para avaliar a percentagem de remoção do íon ferro das soluções, sendo utilizado um planejamento fatorial em dois níveis, tendo como variáveis independentes o pH do meio, a quantidade de quitosana adicionada, a granulometria da mesma e o tempo de reação. A quitosana apresentou grau de desacetilação de 87±2% e massa molecular viscosimétrica de 196±4kDa, sendo esses valores, comparáveis à quitosana disponível comercialmente. Na melhor região de trabalho definida pela análise estatística, obteve-se uma remoção máxima de 85 % do íon ferro das soluções.

Análise histométrica do desenvolvimento testicular de cutias (Dasyprocta aguti) criadas em cativeiros

Foi estudado, por meio da histometria, o desenvolvimento testicular em 31 cutias da espécie Dasyprocta aguti desde o nascimento até 14 meses de idade. O diâmetro e a área, médios, foram obtidos a partir de 30 secções transversais de cordões e/ou túbulos seminíferos, em cada testículo, utilizando-se sistema de computadorizado de analises de imagem e uma ocular micrométrica Zeiss CPL 10X, acoplada a uma objetiva de 40X. As proporções volumétricas do testículo foram obtidas com o método estereométrico, segundo Elias, Henning e Schwartz¹. O diâmetro tubular médio apresentou crescimento lento desde o nascimento até os oito meses de idade, nas duas metodologias empregadas. Quando foi usada a ocular micrométrica observou-se que, a partir de nove meses, o diâmetro tubular teve um crescimento acelerado, chegando a duplicar o seu valor, se comparado com grupo etário que o antecedia. A proporção volumétrica dos cordões testiculares e túbulos seminíferos cresceu gradualmente, atingindo, aos nove meses, seu valor máximo (86,50%). As células de Leydig apresentaram proporção volumétrica decrescente, e seus maiores valores foram expressivos do nascimento até quatro meses de idade (7,00 ± 1,77% a 9,55 ± 0,64%) e mínimos a partir de nove meses, tendendo ainda a uma estabilização. O estroma diminuiu com a evolução da idade caindo bruscamente a partir da puberdade. Conclui-se que o diâmetro dos cordões testiculares e túbulos seminíferos apresentou maior crescimento, coincidindo com o início da puberdade e a proporção volumétrica das células de Leydig encontrou-se, respectivamente, mais alta e mais baixa no mesmo período.

Fonctionnalisation covalente de monocouches et bicouches de graphène

Le graphène est une nanostructure de carbone hybridé sp2 dont les propriétés électroniques et optiques en font un matériau novateur avec un très large potentiel d’application. Cependant, la production à large échelle de ce matériau reste encore un défi et de nombreuses propriétés physiques et chimiques doivent être étudiées plus en profondeur pour mieux les exploiter. La fonctionnalisation covalente est une réaction chimique qui a un impact important dans l’étude de ces propriétés, car celle-ci a pour conséquence une perte de la structure cristalline des carbones sp2. Néanmoins, la réaction a été très peu explorée pour ce qui est du graphène déposé sur des surfaces, car la réactivité chimique de ce dernier est grandement dépendante de l’environnement chimique. Il est donc important d’étudier la fonctionnalisation de ce type de graphène pour bien comprendre à la fois la réactivité chimique et la modification des propriétés électroniques et optiques pour pouvoir exploiter les retombées. D’un autre côté, les bicouches de graphène sont connues pour avoir des propriétés très différentes comparées à la monocouche à cause d’un empilement des structures électroniques, mais la croissance contrôlée de ceux-ci est encore très difficile, car la cinétique de croissance n’est pas encore maîtrisée. Ainsi, ce mémoire de maîtrise va porter sur l’étude de la réactivité chimique du graphène à la fonctionnalisation covalente et de l’étude des propriétés optiques du graphène. Dans un premier temps, nous avons effectué des croissances de graphène en utilisant la technique de dépôt chimique en phase vapeur. Après avoir réussi à obtenir du graphène monocouche, nous faisons varier les paramètres de croissance et nous nous rendons compte que les bicouches apparaissent lorsque le gaz carboné nécessaire à la croissance reste présent durant l’étape de refroidissement. À partir de cette observation, nous proposons un modèle cinétique de croissance des bicouches. Ensuite, nous effectuons une étude approfondie de la fonctionnalisation du graphène monocouche et bicouche. Tout d’abord, nous démontrons qu’il y a une interaction avec le substrat qui inhibe grandement le greffage covalent sur la surface du graphène. Cet effet peut cependant être contré de plusieurs façons différentes : 1) en dopant chimiquement le graphène avec des molécules réductrices, il est possible de modifier le potentiel électrochimique afin de favoriser la réaction; 2) en utilisant un substrat affectant peu les propriétés électroniques du graphène; 3) en utilisant la méthode d’électrogreffage avec une cellule électrochimique, car elle permet une modulation contrôlée du potentiel électrochimique du graphène. De plus, nous nous rendons compte que la réactivité chimique des bicouches est moindre dû à la rigidité de structure due à l’interaction entre les couches. En dernier lieu, nous démontrons la pertinence de la spectroscopie infrarouge pour étudier l’effet de la fonctionnalisation et l’effet des bicouches sur les propriétés optiques du graphène. Nous réussissons à observer des bandes du graphène bicouche dans la région du moyen infrarouge qui dépendent du dopage. Normalement interdites selon les règles de sélection pour la monocouche, ces bandes apparaissent néanmoins lorsque fonctionnalisée et changent grandement en amplitude dépendamment des niveaux de dopage et de fonctionnalisation.

Tonsils of the soft palate do not mediate the response of pigs to oral vaccination with heat-inactivated Mycobacterium bovis

Mycobacterium bovis causes animal tuberculosis (TB) in cattle, humans, and other mammalian species, including pigs. The goal of this study was to experimentally assess the responses of pigs with and without a history of tonsillectomy to oral vaccination with heat-inactivated M. bovis and challenge with a virulent M. bovis field strain, to compare pig and wild boar responses using the same vaccination model as previously used in the Eurasian wild boar (Sus scrofa), to evaluate the use of several enzyme-linked immunosorbent assays (ELISAs) and lateral flow tests for in vivo TB diagnosis in pigs, and to verify if these tests are influenced by oral vaccination with inactivated M. bovis. At necropsy, the lesion and culture scores were 20% to 43% higher in the controls than those in the vaccinated pigs. Massive M. bovis growth from thoracic tissue samples was observed in 4 out of 9 controls but in none of the 10 vaccinated pigs. No effect of the presence or absence of tonsils was observed on these scores, suggesting that tonsils are not involved in the protective response to this vaccine in pigs. The serum antibody levels increased significantly only after challenge. At necropsy, the estimated sensitivities of the ELISAs and dual path platform (DPP) assays ranged from 89% to 94%. In the oral mucosa, no differences in gene expression were observed in the control group between the pigs with and without tonsils. In the vaccinated group, the mRNA levels for chemokine (C-C motif) receptor 7 (CCR7), interferon beta (IFN-β), and methylmalonyl coenzyme A mutase (MUT) were higher in pigs with tonsils. Complement component 3 mRNA levels in peripheral blood mononuclear cells (PBMC) increased with vaccination and decreased after M. bovis challenge. This information is relevant for pig production in regions that are endemic for M. bovis and for TB vaccine research.

Fonctionnalisation covalente de monocouches et bicouches de graphène

Le graphène est une nanostructure de carbone hybridé sp2 dont les propriétés électroniques et optiques en font un matériau novateur avec un très large potentiel d’application. Cependant, la production à large échelle de ce matériau reste encore un défi et de nombreuses propriétés physiques et chimiques doivent être étudiées plus en profondeur pour mieux les exploiter. La fonctionnalisation covalente est une réaction chimique qui a un impact important dans l’étude de ces propriétés, car celle-ci a pour conséquence une perte de la structure cristalline des carbones sp2. Néanmoins, la réaction a été très peu explorée pour ce qui est du graphène déposé sur des surfaces, car la réactivité chimique de ce dernier est grandement dépendante de l’environnement chimique. Il est donc important d’étudier la fonctionnalisation de ce type de graphène pour bien comprendre à la fois la réactivité chimique et la modification des propriétés électroniques et optiques pour pouvoir exploiter les retombées. D’un autre côté, les bicouches de graphène sont connues pour avoir des propriétés très différentes comparées à la monocouche à cause d’un empilement des structures électroniques, mais la croissance contrôlée de ceux-ci est encore très difficile, car la cinétique de croissance n’est pas encore maîtrisée. Ainsi, ce mémoire de maîtrise va porter sur l’étude de la réactivité chimique du graphène à la fonctionnalisation covalente et de l’étude des propriétés optiques du graphène. Dans un premier temps, nous avons effectué des croissances de graphène en utilisant la technique de dépôt chimique en phase vapeur. Après avoir réussi à obtenir du graphène monocouche, nous faisons varier les paramètres de croissance et nous nous rendons compte que les bicouches apparaissent lorsque le gaz carboné nécessaire à la croissance reste présent durant l’étape de refroidissement. À partir de cette observation, nous proposons un modèle cinétique de croissance des bicouches. Ensuite, nous effectuons une étude approfondie de la fonctionnalisation du graphène monocouche et bicouche. Tout d’abord, nous démontrons qu’il y a une interaction avec le substrat qui inhibe grandement le greffage covalent sur la surface du graphène. Cet effet peut cependant être contré de plusieurs façons différentes : 1) en dopant chimiquement le graphène avec des molécules réductrices, il est possible de modifier le potentiel électrochimique afin de favoriser la réaction; 2) en utilisant un substrat affectant peu les propriétés électroniques du graphène; 3) en utilisant la méthode d’électrogreffage avec une cellule électrochimique, car elle permet une modulation contrôlée du potentiel électrochimique du graphène. De plus, nous nous rendons compte que la réactivité chimique des bicouches est moindre dû à la rigidité de structure due à l’interaction entre les couches. En dernier lieu, nous démontrons la pertinence de la spectroscopie infrarouge pour étudier l’effet de la fonctionnalisation et l’effet des bicouches sur les propriétés optiques du graphène. Nous réussissons à observer des bandes du graphène bicouche dans la région du moyen infrarouge qui dépendent du dopage. Normalement interdites selon les règles de sélection pour la monocouche, ces bandes apparaissent néanmoins lorsque fonctionnalisée et changent grandement en amplitude dépendamment des niveaux de dopage et de fonctionnalisation.

A importância da adaptação da linguagem do visual merchandising para o varejo de baixa renda

A evolução do varejo no Brasil, também atribuída ao crescimento econômico das últimas décadas, foi marcada pelo surgimento de novos formatos e estratégias comerciais e por um surpreendente processo de transformação social. Observa-se que a referida transformação social foi seguida pelo aumento do poder aquisitivo das diferentes classes sociais que compõem o cenário econômico. O crescimento econômico, proporcionado pela instituição de políticas econômicas e de inclusão social, fez despontar um nicho de mercado, que até o momento apresentava-se com acesso restrito ao consumo, formado pela classe social de baixa renda. O surgimento de um novo cenário mercadológico constituído por indivíduos pertencentes à classe social de baixa renda e a concorrência saturada dos mercados consolidados despertaram o interesse de muitas empresas em incrementar suas atividades investindo no emergente mercado de consumo. Para consolidar suas atuações em um mercado pouco conhecido, algumas empresas perceberam a necessidade de compatibilizar sua estratégias comerciais e de marketing ao novo perfil e comportamento dos respectivos consumidores. Entretanto, ainda é possível observar que muitas das referidas estratégias são desenvolvidas a partir de mitos acerca das motivações e comportamentos de consumo da classe social de baixa renda e, desta forma, não correspondente ao estilo de vida e formação cultural do referido público alvo. O presente estudo pretende contribuir para o aperfeiçoamento do varejo de baixa renda e, desta forma, apresentar a importância de planejar ações comerciais e de convencimento de consumo coordenadas com estratégias de marketing, notadamente do visual merchandising, adaptadas ao referido público alvo. Para alcançar a proposta deste estudo fez-se necessária a compreensão do estilo de vida do respectivo público alvo através do desenvolvimento de métodos de pesquisa investigativos, para uma análise detalhada e assertiva das respectivas motivações de consumo.

Prediction of a low-risk concentration of diflubenzuron to aquatic organisms and evaluation of clay and gravel in reducing the toxicity.

Resumo: Predição da concentração de baixo risco de diflubenzuron para organismos aquáticos e avaliação da argila e brita na redução da toxicidade. O diflubenzuron é um inseticida que além de ser usado agricultura, tem sido amplamente empregado na piscicultura, apesar do seu uso ser proibido nesta atividade. Este composto não consta na lista da legislação brasileira que estabelece limites máximos permissíveis em corpos de água para a proteção das comunidades aquáticas. No presente trabalho, a partir da toxicidade do diflubenzuron em organismos não-alvo, foi calculada a concentração de risco para somente 5% das espécies (HC5). O valor deste parâmetro foi estimado em aproximadamente 7 x 10-6 mg L-1 . Este baixo valor é devido à extremamente alta toxicidade do diflubenzuron para dafnídeos e à grande variação de sensibilidade entre as espécies testadas. Dois matérias de relativamente baixo custo se mostraram eficientes na remoção da toxicidade do diflubenzuron de soluções contendo este composto. Dentre esses materiais, a argila expandida promoveu a redução em aproximadamente 50% da toxicidade de uma solução contendo diflubenzuron. Os resultados podem contribuir para políticas públicas no Brasil relacionadas ao estabelecimento de limites máximos permissíveis de xenobióticos no compartimento aquático. Também, para a pesquisa de matérias inertes e de baixo custo com potencial de remoção de xenobióticos presentes em efluentes da aquicultura ou da agricultura. Abstract: Diflubenzuron is an insecticide that, besides being used in the agriculture, has been widely used in fish farming. However, its use is prohibited in this activity. Diflubenzuron is not in the list of Brazilian legislation establishing maximum permissible limits in water bodies for the protection of aquatic communities. In this paper, according toxicity data of diflubenzuron in non-target organisms, it was calculated an hazardous concentration for only 5% of the species (HC5) of the aquatic community. This parameter value was estimated to be about 7 x 10 -6 mg L -1 . The low value is due to the extreme high toxicity of diflubenzuron to daphnids and to the large variation in sensitivity among the species tested. Two relatively low cost and inert materials were efficient in removing the diflubenzuron from solutions containing this compound. Among these materials, expanded clay shown to promote reduction of approximately 50% of the toxicity of a solution containing diflubenzuron. The results may contribute to the establishment of public policies in Brazil associated to the definition of maximum permissible limits of xenobiotics in the aquatic compartment. This study is also relevant to the search of low cost and inert materials for xenobiotics removal from aquaculture or agricultural effluents.

Life Cycle Thinking for sustainability assessment and decision-making in selected food supply chains in Costa Rica

The Agenda 2030 contains 17 integrated Sustainable Development Goals (SDGs). SDG 12 for Sustainable Consumption and Production (SCP) promotes the efficient use of resources through a systemic change that decouples economic growth from environmental degradation. The Food Systems (FS) pillar in SDG 12 entails paramount relevance due to its interconnection to many other SDGs, and even when being a crucial world food supplier, the Latin American and Caribbean (LAC) Region struggles with environmental and social externalities, low investment in agriculture, inequity, food insecurity, poverty, and migration. Life Cycle Thinking (LCT) was regarded as a pertinent approach to identify hotspots and trade-offs, and support decision-making process to aid LAC Region countries as Costa Rica to diagnose sustainability and overcome certain challenges. This thesis aimed to ‘evaluate the sustainability of selected products from food supply chains in Costa Rica, to provide inputs for further sustainable decision-making, through the application of Life Cycle Thinking’. To do this, Life Cycle Assessment (LCA), Life Cycle Costing (LCC), and Social Life Cycle Assessment (S-LCA) evaluated the sustainability of food-waste-to-energy alternatives, and the production of green coffee, raw milk and leafy vegetables, and identified environmental, social and cost hotspots. This approach also proved to be a useful component of decision-making and policy-making processes together with other methods. LCT scientific literature led by LAC or Costa Rican researchers is still scarce; therefore, this research contributed to improve capacities in the use of LCT in this context, while offering potential replicability of the developed frameworks in similar cases. Main limitations related to the representativeness and availability of primary data; however, future research and extension activities are foreseen to increase local data availability, capacity building, and the discussion of potential integration through Life Cycle Sustainability Assessment (LCSA).

Data collapse, scaling functions, and analytical solutions of generalized growth models

We consider a nontrivial one-species population dynamics model with finite and infinite carrying capacities. Time-dependent intrinsic and extrinsic growth rates are considered in these models. Through the model per capita growth rate we obtain a heuristic general procedure to generate scaling functions to collapse data into a simple linear behavior even if an extrinsic growth rate is included. With this data collapse, all the models studied become independent from the parameters and initial condition. Analytical solutions are found when time-dependent coefficients are considered. These solutions allow us to perceive nontrivial transitions between species extinction and survival and to calculate the transition's critical exponents. Considering an extrinsic growth rate as a cancer treatment, we show that the relevant quantity depends not only on the intensity of the treatment, but also on when the cancerous cell growth is maximum.

Controlled nerve growth factor release from multi-ply alginate/chitosan-based nerve conduits.

The delivery kinetics of growth factors has been suggested to play an important role in the regeneration of peripheral nerves following axotomy. In this context, we designed a nerve conduit (NC) with adjustable release kinetics of nerve growth factor (NGF). A multi-ply system was designed where NC consisting of a polyelectrolyte alginate/chitosan complex was coated with layers of poly(lactide-co-glycolide) (PLGA) to control the release of embedded NGF. Prior to assessing the in vitro NGF release from NC, various release test media, with and without stabilizers for NGF, were evaluated to ensure adequate quantification of NGF by ELISA. Citrate (pH 5.0) and acetate (pH 5.5) buffered saline solutions containing 0.05% Tween 20 yielded the most reliable results for ELISA active NGF. The in vitro release experiments revealed that the best results in terms of reproducibility and release control were achieved when the NGF was embedded between two PLGA layers and the ends of the NC tightly sealed by the PLGA coatings. The release kinetics could be efficiently adjusted by accommodating NGF at different radial locations within the NC. A sustained release of bioactive NGF in the low nanogram per day range was obtained for at least 15days. In conclusion, the developed multi-ply NGF loaded NC is considered a suitable candidate for future implantation studies to gain insight into the relationship between local growth factor availability and nerve regeneration.

Kinetic study of the cobalt electrodeposition onto glassy carbon electrode from ammonium sulfate solutions

We carried out an electrochemical study of the cobalt electrodeposition onto glassy carbon electrode from an aqueous solution containing 10-2 M of CoSO4 + 1 M (NH4)2SO4 at natural pH 4.5. The potentiostatic study indicated a progressive 3D nucleation and growth during the deposition process. The average diffusion coefficient calculated for this system was 2.65 X 10-6 cm² s-1 while the ΔG for the formation of stable nucleus was 6.50 X 10-20 J/nuclei. The scanning electron microscopy images indicated the formation of small and homogeneous nucleus onto GCE of approximately 300 nm.

An infection control protocol: effectiveness of immersion solutions to reduce the microbial growth on dental prostheses

This investigation evaluated the effectiveness of an infection control protocol for cleansing and disinfecting removable dental prostheses. Sixty-four dentures were rubbed with sterile cotton swab immediately after they had been taken from patients' mouths. Samples were individually placed in the culture medium and immediately incubated at 37 +/- 2 degreesC. The dentures were scrubbed for 1 min with 4% chlorhexidine, rinsed for 1 min in sterile water and placed for 10 min in one of the following immersion solutions: 4% chlorhexidine gluconate, 1% sodium hypochlorite, Biocide (iodophors) and Amosan (alkaline peroxide). After the disinfection procedures, the dentures were immersed in sterile water for 3 min, reswabbed and the samples were incubated. All samples obtained in the initial culture were contaminated with micro-organisms. All the lower dentures immersed in Biocide showed positive growth, and the upper dentures were positive for growth in six of eight dentures. The 4% chlorhexidine gluconate, 1% sodium hypochlorite and Amosan solutions have been proved effective to reduce the growth of the micro-organisms in the 10 min immersion period. The protocol evaluated in this study seems to be a viable method to prevent cross-contamination between dental personnel and patients.

In situ and simultaneous nanostructural and spectroscopic studies of ZnO nanoparticle and Zn-HDS formations from hydrolysis of ethanolic zinc acetate solutions induced by water

The simultaneous formation of nanometer sized zinc oxide (ZnO), and acetate zinc hydroxide double salt (Zn-HDS) is described. These phases, obtained using the sol-gel synthesis route based on zinc acetate salt in alcoholic media, were identified by direct characterization of the reaction products in solution using complementary techniques: nephelometry, in situ Small-Angle X-ray Scattering (SAXS), UV-Vis spectroscopy and Extended X-ray Absorption Fine Structures (EXAFS). In particular, the hydrolytic pathway of ethanolic zinc acetate precursor solutions promoted by addition of water with the molar ratio N = [H2O]/[Zn2+] = 0.05 was investigated in this paper. The aim was to understand the formation mechanism of ZnO colloidal suspension and to reveal the factors responsible for the formation of Zn-HDS in the final precipitates. The growth mechanism of ZnO nanoparticles is based on primary particle (radius approximate to 1.5 nm) rotation inside the primary aggregate (radius < 3.5 nm) giving rise to an epitaxial attachment of particles and then subsequent coalescence. The growth of second ZnO aggregates is not associated with the Otswald ripening, and could be associated with changes in equilibrium between solute species induced by the superficial etching of Zn-HDS particles at the advanced stage of kinetic.

Monodispersed spindle-type goethite nanoparticles from Fe-III solutions

A new synthetic route for producing monodispersed and single crystal acicular goethite particles with small particle size and a high axial ratio adequate for use as a high density magnetic recording media precursor is reported. It essentially consists of the hydrolysis of alkaline Fe-III suspensions in the presence of carbonate by a three-step procedure, the formation of ferrihydrite primary particles, the ferrihydrite dissolution and nucleation of goethite, and the growth of the goethite nuclei. Changing the temperature of heating during ageing achieved a separation of the two last stages. X-Ray diffraction, transmission electron microscopy, infrared spectroscopy and surface area data have been used to determine the mechanism responsible for the formation of goethite particles with controlled size and shape. The best conditions to prepare monodispersed goethite particles have been established. The results show that uniform goethite particles of (a) 60 nm length with an axial ratio of 6 and (b) 230 nm length with a high axial ratio of 10, can be obtained by using an [OH]/[Fe] molar ratio of 0.35 in the initial suspensions with carbonate or sodium hydroxide, respectively. The [OH]/[Fe] molar ratio determines the particle size and elongation by controlling the hydrolysis reaction rate, while the carbonate ions promote a constant [OH] in the solution, keeping the pH around 10 during the entire synthesis process. This procedure, associated with the appropriate temperature control, leads, under certain conditions, to highly homogeneous goethite particles with sizes smaller than those obtained using sodium hydroxide with the same [OH]/[Fe] ratio.

Deterministic Solutions for a Step-growth Polymerization

Chain topology, including branch node, chain link and cross-link dynamics that contribute to the number of elastically active strands and junctions, are calculated using purely deterministic derivations. Solutions are not coupled to population density distributions. An eigenzeit transformation assists in the conversion of expressions derived by chemical reaction principles from time to conversion space, yielding transport phenomena type expressions where the rate of change in the molar concentrations of branch nodes with respect to conversion is expressed as functions of the fraction of reactive sites on precursors and reactants. Analogies are hypothesized to exist in cross-linking space that effectively distribute branch nodes with i reacted moieties between cross-links having j bonds extending to the gel. To obtain solutions, reacted sites on nodes or links with finite chain extensions are examined in terms of stoichiometry associated with covalent bonding. Solutions replicate published results based on Miller and Macosko’s recursive procedure and results obtained from truncated weighted sums of population density distributions as suggested by Flory.