Master Equation: Biological Applications and Thermodynamic Description

It is well known that many realistic mathematical models of biological systems, such as cell growth, cellular development and differentiation, gene expression, gene regulatory networks, enzyme cascades, synaptic plasticity, aging and population growth need to include stochasticity. These systems are not isolated, but rather subject to intrinsic and extrinsic fluctuations, which leads to a quasi equilibrium state (homeostasis). The natural framework is provided by Markov processes and the Master equation (ME) describes the temporal evolution of the probability of each state, specified by the number of units of each species. The ME is a relevant tool for modeling realistic biological systems and allow also to explore the behavior of open systems. These systems may exhibit not only the classical thermodynamic equilibrium states but also the nonequilibrium steady states (NESS). This thesis deals with biological problems that can be treat with the Master equation and also with its thermodynamic consequences. It is organized into six chapters with four new scientific works, which are grouped in two parts: (1) Biological applications of the Master equation: deals with the stochastic properties of a toggle switch, involving a protein compound and a miRNA cluster, known to control the eukaryotic cell cycle and possibly involved in oncogenesis and with the propose of a one parameter family of master equations for the evolution of a population having the logistic equation as mean field limit. (2) Nonequilibrium thermodynamics in terms of the Master equation: where we study the dynamical role of chemical fluxes that characterize the NESS of a chemical network and we propose a one parameter parametrization of BCM learning, that was originally proposed to describe plasticity processes, to study the differences between systems in DB and NESS.

Reproductive biology of male franciscanas (Pontoporia blainvillei) (Mammalia: Cetacea) from Rio Grande do Sul, southern Brazil

The reproductive biology of male franciscanas (Pontoporia blainvillei), based on 121 individuals collected in Rio Grande do Sul State, southern Brazil, was studied. Estimates on age, length, and weight at attainment of sexual maturity are presented. Data on the reproductive seasonality and on the relationship between some testicular characteristics and age, size, and maturity status are provided. Sexual maturity was assessed by histological examination of the testes. Seasonality was determined by changes in relative and total testis weight, and in seminiferous tubule diameters. Testis weight, testicular index of maturity, and seminiferous tubule diameters were reliable indicators of sexual maturity, whereas testis length, age, length, and weight of the dolphin were not. Sexual maturity was estimated to be attained at 3.6 years (CI 95% =2.7–4.5) with the DeMaster method and 3.0 years with the logistic equation. Length and weight at attainment of sexual maturity were 128.2 cm (CI 95%=125.3–131.1 cm) and 26.4 kg (CI 95% =24.7–28.1 kg), respectively. It could not be verified that there was any seasonal change in the testis weight and in the seminiferous tubule diameters in mature males. It is suggested that at least some mature males may remain reproductively active throughout the year. The extremely low relative testis weight indicates that sperm competition does not occur in the species. On the other hand, the absence of secondary sexual characteristics, the reversed sexual size dimorphism, and the small number of scars from intrassexual combats in males reinforce the hypothesis that male combats for female reproductive access may be rare for franciscana. It is hypothesized that P. blainvillei form temporary pairs (one male copulating with only one female) during the reproductive period.

The growth patterns of juvenile and precocious Chinese mitten crabs, Eriocheir sinensis (Decapoda, Grapsidae), stocked in freshwater lakes of China

Stocking experiments with Eriocheir sinensis were conducted in two small, shallow lakes to study its growth pattern in 1994-1997. For the initially immature crabs, carapace width (CW) increases from 21.2 +/- 0.4 mm (mean +/- s.e.) for females and 22.3 +/- 0.5 mm for males in January, to 65.4 +/- 0.5 mm for females and 66.9 +/- 0.6 mm for males in October. There is no significant difference in CW and carapace length (CL), although there is a large difference in body weight (BW) between sexes in every month from January to August when crabs are juvenile, however, there are significant differences in CW, CL. and BW between sexes after September when the crabs become sexually mature. The growth curve from January to October fits a logistic equation and may be expressed as CW = 75.7 (1 + exp (0.914 - 0.011t))(-1) for females, and CW = 77.5 (1 + exp (0.889 - 0.011t))-1 for males, where CW is in mm, t in days. For precocious crabs (reaching maturity by the first autumn, CW does not change much from January to July, which indicates that precocious crabs stop growing. Like juveniles, the precocious crabs show no differences in CW and CL, but do show a statistically significant difference in BW between sexes.

Thermokinetic investigation of effect of temperature on optimum NaCl concentration for petroleum bacterial growth

The power-time curves of growth of three strains of petroleum bacteria at different NaCl concentrations at 40.0 and 50.0 degreesC have been determined by using a 2277 Thermometric Thermal Activity Analyser. An equation of a power-time curve, ln[alphaP(K)/P(t) - 1] = ln[(alphaK - N-0)/N-0] - alphakt, was established based on the generalized logistic equation, where P(t) is the thermal power at time t, K the carrying capacity, P-K = P0K, P-0 the thermal power of one cell, N-0 the bacterial population at time zero, alpha = (k - D)/k. The method of four observed points with the same time interval was used to calculate the value of P-K. The growth rate constant k and the death rate constant D were calculated. The NaCl concentration of optimum growth rate of petroleum bacteria at 40.0 and 50.0 degreesC, respectively, have been obtained according to the curves k - D versus NaCl concentration, which are 0.26, 0.54 and 0.57 mol l(-1) for B-1, B-2 and B-3, respectively, at 50.0 degreesC, 0.26, 0.55 and 0.56 mol l(-1) for B-1, B-2 and B-3, respectively, at 40.0 degreesC. The results indicated that the effect of temperature on NaCl concentration of optimum growth rate was small. (C) 2002 Elsevier Science B.V. All rights reserved.

Use of a predictive model for food insecurity estimates in Brazil

In 2004 the National Household Survey (Pesquisa Nacional par Amostras de Domicilios - PNAD) estimated the prevalence of food and nutrition insecurity in Brazil. However, PNAD data cannot be disaggregated at the municipal level. The objective of this study was to build a statistical model to predict severe food insecurity for Brazilian municipalities based on the PNAD dataset. Exclusion criteria were: incomplete food security data (19.30%); informants younger than 18 years old (0.07%); collective households (0.05%); households headed by indigenous persons (0.19%). The modeling was carried out in three stages, beginning with the selection of variables related to food insecurity using univariate logistic regression. The variables chosen to construct the municipal estimates were selected from those included in PNAD as well as the 2000 Census. Multivariate logistic regression was then initiated, removing the non-significant variables with odds ratios adjusted by multiple logistic regression. The Wald Test was applied to check the significance of the coefficients in the logistic equation. The final model included the variables: per capita income; years of schooling; race and gender of the household head; urban or rural residence; access to public water supply; presence of children; total number of household inhabitants and state of residence. The adequacy of the model was tested using the Hosmer-Lemeshow test (p=0.561) and ROC curve (area=0.823). Tests indicated that the model has strong predictive power and can be used to determine household food insecurity in Brazilian municipalities, suggesting that similar predictive models may be useful tools in other Latin American countries.

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u equivalent to 1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain. (C) 2009 Elsevier Inc. All rights reserved.

Allometry of the secondary sexual characters of the shore crab Pachygrapsus transversus (Gibbes, 1850) (Brachyura, Grapsidae)

The allometric growth of secondary sexual characters in Pachygrapsus transversus is investigated from the 2(nd) crab stage onward. Clear sexual dimorphism is restricted to abdominal morphology, but ANCOVA analyses showed that chelae become larger in males and the carapace becomes wider in females. Size at the puberty moult in both sexes was estimated using Somerton's computer techniques. Mature II analyses applied to bi-log gonopod length vs, carapace length relationships indicated a puberty moult at 5.0 mm in males.In females, Mature I analyses detected the overlapping growth phase lines in bi-log carapace length vs. abdomen width scatterplots. Fitting the logistic equation provided an estimate of 50% maturity at 5.5 mm. The regression lines separate young and resting individuals from the potentially reproductive females, but they do not separate young from adult crabs. Year-round monthly samples showed that the proportion of small adult-like females is higher during the breeding season. After breeding, females may moult to a young-like morphotype, as observed in controlled laboratory conditions. Moulting to a resting condition splits smaller mature females into different growth phase lines. Therefore, estimates of female size at sexual maturity by means of abdomen allometric growth analyses are inadequate in this species.

Das transformadas integrais ao cálculo fracionário aplicado à equação logística

Pós-graduação em Biometria - IBB

A prelude to the fractional calculus applied to tumor dynamic

In order to refine the solution given by the classical logistic equation and extend its range of applications in the study of tumor dynamics, we propose and solve a generalization of this equation, using the so-called Fractional Calculus, i.e., we replace the ordinary derivative of order 1, in one version of the usual equation, by a non-integer derivative of order 0 < α < 1, and recover the classical solution as a particular case. Finally, we analyze the applicability of this model to describe the growth of cancer tumors.

Cifrado de imágenes digitales basado en teoría del caos: mapas logísticos

Las nuevas tendencias de compartir archivos multimedia a través de redes abiertas, demanda el uso de mejores técnicas de encriptación que garanticen la integridad, disponibilidad y confidencialidad, manteniendo y/o mejorando la eficiencia del proceso de cifrado sobre estos archivos. Hoy en día es frecuente la transferencia de imágenes a través de medios tecnológicos, siendo necesario la actualización de las técnicas de encriptación existentes y mejor aún, la búsqueda de nuevas alternativas. Actualmente los algoritmos criptográficos clásicos son altamente conocidos en medio de la sociedad informática lo que provoca mayor vulnerabilidad, sin contar los altos tiempos de procesamiento al momento de ser utilizados, elevando la probabilidad de ser descifrados y minimizando la disponibilidad inmediata de los recursos. Para disminuir estas probabilidades, el uso de la teoría de caos surge como una buena opción para ser aplicada en un algoritmo que tome partida del comportamiento caótico de los sistemas dinámicos, y aproveche las propiedades de los mapas logísticos para elevar el nivel de robustez en el cifrado. Es por eso que este trabajo propone la creación de un sistema criptográfico basado sobre una arquitectura dividida en dos etapas de confusión y difusión. Cada una de ellas utiliza una ecuación logística para generar números pseudoaleatorios que permitan desordenar la posición del píxel y cambiar su intensidad en la escala de grises. Este proceso iterativo es determinado por la cantidad total de píxeles de una imagen. Finalmente, toda la lógica de cifrado es ejecutada sobre la tecnología CUDA que permite el procesamiento en paralelo. Como aporte sustancial, se propone una nueva técnica de encriptación vanguardista de alta sensibilidad ante ruidos externos manteniendo no solo la confidencialidad de la imagen, sino también la disponibilidad y la eficiencia en los tiempos de proceso.---ABSTRACT---New trends to share multimedia files over open networks, demand the best use of encryption techniques to ensure the integrity, availability and confidentiality, keeping and/or improving the efficiency of the encryption process on these files. Today it is common to transfer pictures through technological networks, thus, it is necessary to update existing techniques encryption, and even better, the searching of new alternatives. Nowadays, classic cryptographic algorithms are highly known in the midst of the information society which not only causes greater vulnerability, but high processing times when this algorithms are used. It raise the probability of being deciphered and minimizes the immediate availability of resources. To reduce these odds, the use of chaos theory emerged as a good option to be applied on an algorithm that takes advantage of chaotic behavior of dynamic systems, and take logistic maps’ properties to raise the level of robustness in the encryption. That is why this paper proposes the creation of a cryptographic system based on an architecture divided into two stages: confusion and diffusion. Each stage uses a logistic equation to generate pseudorandom numbers that allow mess pixel position and change their intensity in grayscale. This iterative process is determined by the total number of pixels of an image. Finally, the entire encryption logic is executed on the CUDA technology that enables parallel processing. As a substantial contribution, it propose a new encryption technique with high sensitivity on external noise not only keeping the confidentiality of the image, but also the availability and efficiency in processing times.

Structural equation model of construction contract dispute potential

This paper presents the results of a structural equation model (SEM) for describing and quantifying the fundamental factors that affect contract disputes between owners and contractors in the construction industry. Through this example, the potential impact of SEM analysis in construction engineering and management research is illustrated. The purpose of the specific model developed in this research is to explain how and why contract related construction problems occur. This study builds upon earlier work, which developed a disputes potential index, and the likelihood of construction disputes was modeled using logistic regression. In this earlier study, questionnaires were completed on 159 construction projects, which measured both qualitative and quantitative aspects of contract disputes, management ability, financial planning, risk allocation, and project scope definition for both owners and contractors. The SEM approach offers several advantages over the previously employed logistic regression methodology. The final set of structural equations provides insight into the interaction of the variables that was not apparent in the original logistic regression modeling methodology.

Logistic regression analysis of potential prognostic factors for pulmonary thromboembolism

Objective: To identify potential prognostic factors for pulmonary thromboembolism (PTE), establishing a mathematical model to predict the risk for fatal PTE and nonfatal PTE.Method: the reports on 4,813 consecutive autopsies performed from 1979 to 1998 in a Brazilian tertiary referral medical school were reviewed for a retrospective study. From the medical records and autopsy reports of the 512 patients found with macroscopically and/or microscopically,documented PTE, data on demographics, underlying diseases, and probable PTE site of origin were gathered and studied by multiple logistic regression. Thereafter, the jackknife method, a statistical cross-validation technique that uses the original study patients to validate a clinical prediction rule, was performed.Results: the autopsy rate was 50.2%, and PTE prevalence was 10.6%. In 212 cases, PTE was the main cause of death (fatal PTE). The independent variables selected by the regression significance criteria that were more likely to be associated with fatal PTE were age (odds ratio [OR], 1.02; 95% confidence interval [CI], 1.00 to 1.03), trauma (OR, 8.5; 95% CI, 2.20 to 32.81), right-sided cardiac thrombi (OR, 1.96; 95% CI, 1.02 to 3.77), pelvic vein thrombi (OR, 3.46; 95% CI, 1.19 to 10.05); those most likely to be associated with nonfatal PTE were systemic arterial hypertension (OR, 0.51; 95% CI, 0.33 to 0.80), pneumonia (OR, 0.46; 95% CI, 0.30 to 0.71), and sepsis (OR, 0.16; 95% CI, 0.06 to 0.40). The results obtained from the application of the equation in the 512 cases studied using logistic regression analysis suggest the range in which logit p > 0.336 favors the occurrence of fatal PTE, logit p < - 1.142 favors nonfatal PTE, and logit P with intermediate values is not conclusive. The cross-validation prediction misclassification rate was 25.6%, meaning that the prediction equation correctly classified the majority of the cases (74.4%).Conclusions: Although the usefulness of this method in everyday medical practice needs to be confirmed by a prospective study, for the time being our results suggest that concerning prevention, diagnosis, and treatment of PTE, strict attention should be given to those patients presenting the variables that are significant in the logistic regression model.

New Equation for Prediction of Reverse Remodeling after Cardiac Resynchronization Therapy

Objectives: To integrate data from two-dimensional echocardiography (2D ECHO), three-dimensional echocardiography (3D ECHO), and tissue Doppler imaging (TDI) for prediction of left ventricular (LV) reverse remodeling (LVRR) after cardiac resynchronization therapy (CRT). It was also compared the evaluation of cardiac dyssynchrony by TDI and 3D ECHO. Methods: Twenty-four consecutive patients with heart failure, sinus rhythm, QRS = 120 msec, functional class III or IV and LV ejection fraction (LVEF) = 0.35 underwent CRT. 2D ECHO, 3D ECHO with systolic dyssynchrony index (SDI) analysis, and TDI were performed before, 3 and 6 months after CRT. Cardiac dyssynchrony analyses by TDI and SDI were compared with the Pearson's correlation test. Before CRT, a univariate analysis of baseline characteristics was performed for the construction of a logistic regression model to identify the best predictors of LVRR. Results: After 3 months of CRT, there was a moderate correlation between TDI and SDI (r = 0.52). At other time points, there was no strong correlation. Nine of twenty-four (38%) patients presented with LVRR 6 months after CRT. After logistic regression analysis, SDI (SDI > 11%) was the only independent factor in the prediction of LVRR 6 months of CRT (sensitivity = 0.89 and specificity = 0.73). After construction of receiver operator characteristic (ROC) curves, an equation was established to predict LVRR: LVRR =-0.4LVDD (mm) + 0.5LVEF (%) + 1.1SDI (%), with responders presenting values >0 (sensitivity = 0.67 and specificity = 0.87). Conclusions: In this study, there was no strong correlation between TDI and SDI. An equation is proposed for the prediction of LVRR after CRT. Although larger trials are needed to validate these findings, this equation may be useful to candidates for CRT. (Echocardiography 2012;29:678-687)

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.