Multimodality and flexibility of stochastic gene expression

We consider a general class of mathematical models for stochastic gene expression where the transcription rate is allowed to depend on a promoter state variable that can take an arbitrary (finite) number of values. We provide the solution of the master equations in the stationary limit, based on a factorization of the stochastic transition matrix that separates timescales and relative interaction strengths, and we express its entries in terms of parameters that have a natural physical and/or biological interpretation. The solution illustrates the capacity of multiple states promoters to generate multimodal distributions of gene products, without the need for feedback. Furthermore, using the example of a three states promoter operating at low, high, and intermediate expression levels, we show that using multiple states operons will typically lead to a significant reduction of noise in the system. The underlying mechanism is that a three-states promoter can change its level of expression from low to high by passing through an intermediate state with a much smaller increase of fluctuations than by means of a direct transition.

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.

Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input

A field of computational neuroscience develops mathematical models to describe neuronal systems. The aim is to better understand the nervous system. Historically, the integrate-and-fire model, developed by Lapique in 1907, was the first model describing a neuron. In 1952 Hodgkin and Huxley [8] described the so called Hodgkin-Huxley model in the article “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”. The Hodgkin-Huxley model is one of the most successful and widely-used biological neuron models. Based on experimental data from the squid giant axon, Hodgkin and Huxley developed their mathematical model as a four-dimensional system of first-order ordinary differential equations. One of these equations characterizes the membrane potential as a process in time, whereas the other three equations depict the opening and closing state of sodium and potassium ion channels. The membrane potential is proportional to the sum of ionic current flowing across the membrane and an externally applied current. For various types of external input the membrane potential behaves differently. This thesis considers the following three types of input: (i) Rinzel and Miller [15] calculated an interval of amplitudes for a constant applied current, where the membrane potential is repetitively spiking; (ii) Aihara, Matsumoto and Ikegaya [1] said that dependent on the amplitude and the frequency of a periodic applied current the membrane potential responds periodically; (iii) Izhikevich [12] stated that brief pulses of positive and negative current with different amplitudes and frequencies can lead to a periodic response of the membrane potential. In chapter 1 the Hodgkin-Huxley model is introduced according to Izhikevich [12]. Besides the definition of the model, several biological and physiological notes are made, and further concepts are described by examples. Moreover, the numerical methods to solve the equations of the Hodgkin-Huxley model are presented which were used for the computer simulations in chapter 2 and chapter 3. In chapter 2 the statements for the three different inputs (i), (ii) and (iii) will be verified, and periodic behavior for the inputs (ii) and (iii) will be investigated. In chapter 3 the inputs are embedded in an Ornstein-Uhlenbeck process to see the influence of noise on the results of chapter 2.

Mathematical Optimization Applied to Thermal and Electrical Energy Systems

This Thesis aims at building and discussing mathematical models applications focused on Energy problems, both on the thermal and electrical side. The objective is to show how mathematical programming techniques developed within Operational Research can give useful answers in the Energy Sector, how they can provide tools to support decision making processes of Companies operating in the Energy production and distribution and how they can be successfully used to make simulations and sensitivity analyses to better understand the state of the art and convenience of a particular technology by comparing it with the available alternatives. The first part discusses the fundamental mathematical background followed by a comprehensive literature review about mathematical modelling in the Energy Sector. The second part presents mathematical models for the District Heating strategic network design and incremental network design. The objective is the selection of an optimal set of new users to be connected to an existing thermal network, maximizing revenues, minimizing infrastructure and operational costs and taking into account the main technical requirements of the real world application. Results on real and randomly generated benchmark networks are discussed with particular attention to instances characterized by big networks dimensions. The third part is devoted to the development of linear programming models for optimal battery operation in off-grid solar power schemes, with consideration of battery degradation. The key contribution of this work is the inclusion of battery degradation costs in the optimisation models. As available data on relating degradation costs to the nature of charge/discharge cycles are limited, we concentrate on investigating the sensitivity of operational patterns to the degradation cost structure. The objective is to investigate the combination of battery costs and performance at which such systems become economic. We also investigate how the system design should change when battery degradation is taken into account.

Markers Models in Survival Analysis and Applications to Issues Associated with AIDS

Jewell and Kalbfleisch (1992) consider the use of marker processes for applications related to estimation of the survival distribution of time to failure. Marker processes were assumed to be stochastic processes that, at a given point in time, provide information about the current hazard and consequently on the remaining time to failure. Particular attention was paid to calculations based on a simple additive model for the relationship between the hazard function at time t and the history of the marker process up until time t. Specific applications to the analysis of AIDS data included the use of markers as surrogate responses for onset of AIDS with censored data and as predictors of the time elapsed since infection in prevalent individuals. Here we review recent work on the use of marker data to tackle these kinds of problems with AIDS data. The Poisson marker process with an additive model, introduced in Jewell and Kalbfleisch (1992) may be a useful "test" example for comparison of various procedures.

Hybrid stochastic simulations of intracellular reaction-diffusion systems.

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

The oculomotor system as a model for the study of neuronal interaction processes

Identified neurons that control eye movements offer an excellent experimental target for the study of Information coding and neuronal interaction processes wíthin the central nervous system. Here are presented some prelimínary results of the motoneuron behaviour during steady eye fíxation, obtained by regressíon and analysis of variance techniques. A flexible information system intended for the systematic acquisitíon and analysis of simultaneous records of neuronal activity and both eyes angular position in a great amount of cells, oriented to the defínition of mathematical models, is also briefly outlíned.

Models of immune memory: On the role of cross-reactive stimulation, competition, and homeostasis in maintaining immune memory

There has been much debate on the contribution of processes such as the persistence of antigens, cross-reactive stimulation, homeostasis, competition between different lineages of lymphocytes, and the rate of cell turnover on the duration of immune memory and the maintenance of the immune repertoire. We use simple mathematical models to investigate the contributions of these various processes to the longevity of immune memory (defined as the rate of decline of the population of antigen-specific memory cells). The models we develop incorporate a large repertoire of immune cells, each lineage having distinct antigenic specificities, and describe the dynamics of the individual lineages and total population of cells. Our results suggest that, if homeostatic control regulates the total population of memory cells, then, for a wide range of parameters, immune memory will be long-lived in the absence of persistent antigen (T1/2 > 1 year). We also show that the longevity of memory in this situation will be insensitive to the relative rates of cross-reactive stimulation, the rate of turnover of immune cells, and the functional form of the term for the maintenance of homeostasis.

Saint Mathew Law and Bonini Paradox in Textual Theory of Complex Models

The mathematical models of the complex reality are texts belonging to a certain literature that is written in a semi-formal language, denominated L(MT) by the authors whose laws linguistic mathematics have been previously defined. This text possesses linguistic entropy that is the reflection of the physical entropy of the processes of real world that said text describes. Through the temperature of information defined by Mandelbrot, the authors begin a text-reality thermodynamic theory that drives to the existence of information attractors, or highly structured point, settling down a heterogeneity of the space text, the same one that of ontologic space, completing the well-known law of Saint Mathew, of the General Theory of Systems and formulated by Margalef saying: “To the one that has more he will be given, and to the one that doesn't have he will even be removed it little that it possesses.

Modern mathematical methods and models; a book of experimental text materials.

v. 1. Multicomponent methods.--v. 2. Mathematical models.

Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Using the canonical modelling approach to simplify the simulation of function in functional-structural plant models

Functional-structural plant models that include detailed mechanistic representation of underlying physiological processes can be expensive to construct and the resulting models can also be extremely complicated. On the other hand, purely empirical models are not able to simulate plant adaptability and response to different conditions. In this paper, we present an intermediate approach to modelling plant function that can simulate plant response without requiring detailed knowledge of underlying physiology. Plant function is modelled using a 'canonical' modelling approach, which uses compartment models with flux functions of a standard mathematical form, while plant structure is modelled using L-systems. Two modelling examples are used to demonstrate that canonical modelling can be used in conjunction with L-systems to create functional-structural plant models where function is represented either in an accurate and descriptive way, or in a more mechanistic and explanatory way. We conclude that canonical modelling provides a useful, flexible and relatively simple approach to modelling plant function at an intermediate level of abstraction.

Estimating variable returns to scale production frontiers with alternative stochastic assumptions

Two stochastic production frontier models are formulated within the generalized production function framework popularized by Zellner and Revankar (Rev. Econ. Stud. 36 (1969) 241) and Zellner and Ryu (J. Appl. Econometrics 13 (1998) 101). This framework is convenient for parsimonious modeling of a production function with returns to scale specified as a function of output. Two alternatives for introducing the stochastic inefficiency term and the stochastic error are considered. In the first the errors are added to an equation of the form h(log y, theta) = log f (x, beta) where y denotes output, x is a vector of inputs and (theta, beta) are parameters. In the second the equation h(log y,theta) = log f(x, beta) is solved for log y to yield a solution of the form log y = g[theta, log f(x, beta)] and the errors are added to this equation. The latter alternative is novel, but it is needed to preserve the usual definition of firm efficiency. The two alternative stochastic assumptions are considered in conjunction with two returns to scale functions, making a total of four models that are considered. A Bayesian framework for estimating all four models is described. The techniques are applied to USDA state-level data on agricultural output and four inputs. Posterior distributions for all parameters, for firm efficiencies and for the efficiency rankings of firms are obtained. The sensitivity of the results to the returns to scale specification and to the stochastic specification is examined. (c) 2004 Elsevier B.V. All rights reserved.

Stochastic modelling and simulations for solute transport in porous media

A stochastic model for solute transport in aquifers is studied based on the concepts of stochastic velocity and stochastic diffusivity. By applying finite difference techniques to the spatial variables of the stochastic governing equation, a system of stiff stochastic ordinary differential equations is obtained. Both the semi-implicit Euler method and the balanced implicit method are used for solving this stochastic system. Based on the Karhunen-Loeve expansion, stochastic processes in time and space are calculated by means of a spatial correlation matrix. Four types of spatial correlation matrices are presented based on the hydraulic properties of physical parameters. Simulations with two types of correlation matrices are presented.