On the dynamics of the adenylate energy system: homeorhesis vs homeostasis.

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life

Using Delay-Differential Equations for Modeling Calcium Cycling in Cardiac Myocytes

The cycling of calcium at the intracellular level of cardiac cells plays a key role in the excitation-contraction process. The interplay between ionic currents, buffering agents, and calcium release from the sarcoplasmic reticulum (SR) is a complex system that has been shown experimentally to exhibit complex dynamics including period-2 states (alternans) and higher-order rhythms. Many of the calcium cycling activities involve the sensing, binding, or diffusion of calcium between intracellular compartments; these are physical processes that take time and typically are modeled by “relaxation” equations where the steady-state value and time course of a particular variable are specified through an ordinary differential equation (ODE) with a time constant. An alternative approach is to use delay-differential equations (DDEs), where the delays in the system correspond to non-instantaneous events. In this thesis, we present a thorough overview of results from calcium cycling experiments and proposed intracellular calcium cycling models, as well as the context of alternans and delay-differential equations in cardiac modeling. We utilize a DDE to model the diffusion of calcium through the SR by replacing the relaxation ODE typically used for this process. The relaxation time constant τa is replaced by a delay δj, which could also be interpreted as the refractoriness of ryanodine receptor channels after releasing calcium from the sarcoplasmic reticulum. This is the first application of delay-differential equations to modeling calcium cycling dynamics, and to modeling cardiac systems at the cellular level. We analyzed the dynamical behaviors of the system and focus on the factors that have been shown to produce alternans and irregular dynamics in experiments and models with cardiac myocytes. We found that chaotic calcium dynamics could occur even for a more physiologically revelant SR calcium release slope than comparable ODE models. Increasing the SR release slope did not affect the calcium dynamics, but only shifted behavior down to lower values of the delay, allowing alternans, higher-order behavior, and chaos to occur for smaller delays than in simulations with a normal SR release slope. For moderate values of the delay, solely alternans and 1:1 steady-state behavior were observed. Above a particular threshold value for the delay, chaos appeared in the dynamics and further increasing the delay caused the system to destabilize under broader ranges of periods. We also compare our results with other models of intracellular calcium cycling and suggest promising avenues for further development of our preliminary work

Mathematical modelling and brain dynamical networks

In this thesis, we study the dynamics of the Hindmarsh-Rose (HR) model which studies the spike-bursting behaviour of the membrane potential of a single neuron. We study the stability of the HR system and compute its Lyapunov exponents (LEs). We consider coupled general sections of the HR system to create an undirected brain dynamical network (BDN) of Nn neurons. Then, we study the concepts of upper bound of mutual information rate (MIR) and synchronisation measure and their dependence on the values of electrical and chemical couplings. We analyse the dynamics of neurons in various regions of parameter space plots for two elementary examples of 3 neurons with two different types of electrical and chemical couplings. We plot the upper bound Ic and the order parameter rho (the measure of synchronisation) and the two largest Lyapunov exponents LE1 and LE2 versus the chemical coupling gn and electrical coupling gl. We show that, even for small number of neurons, the dynamics of the system depends on the number of neurons and the type of coupling strength between them. Finally, we evolve a network of Hindmarsh-Rose neurons by increasing the entropy of the system. In particular, we choose the Kolmogorov-Sinai entropy: HKS (Pesin identity) as the evolution rule. First, we compute the HKS for a network of 4 HR neurons connected simultaneously by two undirected electrical and two undirected chemical links. We get different entropies with the use of different values for both the chemical and electrical couplings. If the entropy of the system is positive, the dynamics of the system is chaotic and if it is close to zero, the trajectory of the system converges to one of the fixed points and loses energy. Then, we evolve a network of 6 clusters of 10 neurons each. Neurons in each cluster are connected only by electrical links and their connections form small-world networks. The six clusters connect to each other only by chemical links. We compare between the combined effect of chemical and electrical couplings with the two concepts, the information flow capacity Ic and HKS in evolving the BDNs and show results that the brain networks might evolve based on the principle of the maximisation of their entropies

Quantification of signaling networks

Studies in living system in the past several decades have generated qualitative understanding of the molecular interactions resulting in large networks. These networks were essentially deciphered by breaking the components of a cell through a reductionist approach. Biological networks comprising of interactions between genes, proteins and metabolites co-ordinate in the regulation of cellular processes. However, understanding the cellular function also requires quantitative information including network dynamics, which results due to an inherent design principle embedded in the network. Interactions within the network are well organized to form a definite regulatory structure, which in turn exhibits different emergent properties. The property of the network helps the cell to achieve the desired phenotypic state in a controlled manner. The dynamics of the network or the relationship between network structure and cellular behavior cannot be understood intuitively from the interaction map of the network. Computational methods can now be employed to study these networks at system level. The field of systems biology looks at integrating the interaction maps obtained through molecular biological approach. Various studies at the system level have been reported for pathways namely chemotactic response in bacteria, cell cycle and osmotic signaling in yeast, growth factor stimulated signaling pathways in mammals. This review focuses on understanding signaling networks with the help of mathematical models

Dynamical Models of biological networks

In der Molekularbiologie sind mathematische Modelle von regulatorischen und metabolischen Netzwerken essentiell, um von einer Betrachtung isolierter Komponenten und Interaktionen zu einer systemischen Betrachtungsweise zu kommen. Genregulatorische Systeme eignen sich besonders gut zur Modellierung, da sie experimentell leicht zugänglich und manipulierbar sind. In dieser Arbeit werden verschiedene genregulatorische Netzwerke unter Zuhilfenahme von mathematischen Modellen analysiert. Weiteres wird ein Modell einer in silico Zelle vorgestellt und diskutiert. Zunächst werden zwei zyklische genregulatorische Netzwerke - der klassische Repressilator und ein Repressilator mit zusätzlicher Autoaktivierung – im Detail mit analytischen Methoden untersucht. Um den Einfluß zufällig schwankender Molekülzahlen auf die Dynamik der beiden Systeme zu untersuchen, werden stochastische Modelle erstellt und die beiden oszillierenden Systeme verglichen. Weiteres werden mögliche Auswirkungen von Genduplikationen auf ein einfaches genregulatorisches Netzwerk untersucht. Dazu wird zunächst ein kleines Netzwerk von GATA Transkriptionsfaktoren, das eine zentrale Rolle in der Regulation des Stickstoffmetabolismus in Hefe spielt, modelliert und das Modell mit experimentellen Daten verglichen, um Parameterregionen einschränken zu können. Außerdem werden potentielle Topologien genregulatorischer Netzwerke von GATA Transkriptionsfaktoren in verwandten Fungi mittels sequenzbasierender Methoden gesucht und verglichen. Im letzten Teil der Arbeit wird MiniCellSim vorgestellt, ein Modell einer selbständigen in silico Zelle. Es erlaubt ein dynamisches System, das eine Protozelle mit einem genregulatorischen Netzwerk, einem einfachen Metabolismus und einer Zellmembran beschreibt, aus einer Sequenz abzuleiten. Nachdem alle Parameter, die zur Berechnung des dynamischen Systems benötigt werden, ohne zusätzliche Eingabe nur aus der Sequenzinformation abgeleitet werden, kann das Modell für Studien zur Evolution von genregulatorischen Netzwerken verwendet werden.In this thesis different types of gene regulatory networks are analysed using mathematical models. Further a computational framework of a novel, self-contained in silico cell model is described and discussed. At first the behaviour of two cyclic gene regulatory systems - the classical repressilator and a repressilator with additional auto-activation - are inspected in detail using analytical bifurcation analysis. To examine the behaviour under random fluctuations, stochastic versions of the systems are created. Using the analytical results sustained oscillations in the stochastic versions are obtained, and the two oscillating systems compared. In the second part of the thesis possible implications of gene duplication on a simple gene regulatory system are inspected. A model of a small network formed by GATA-type transcription factors, central in nitrogen catabolite repression in yeast, is created and validated against experimental data to obtain approximate parameter values. Further, topologies of potential gene regulatory networks and modules consisting of GATA-type transcription factors in other fungi are derived using sequence-based approaches and compared. The last part describes MiniCellSim, a model of a self-contained in silico cell. In this framework a dynamical system describing a protocell with a gene regulatory network, a simple metabolism, and a cell membrane is derived from a string representing a genome. All the relevant parameters required to compute the time evolution of the dynamical system are calculated from within the model, allowing the system to be used in studies of evolution of gene regulatory and metabolic networks

Limit-cycle oscillatory coexpression of cross-inhibitory transcription factors: a model mechanism for lineage promiscuity

Lineage switches are genetic regulatory motifs that govern and maintain the commitment of a developing cell to a particular cell fate. A canonical example of a lineage switch is the pair of transcription factors PU.1 and GATA-1, of which the former is affiliated with the myeloid and the latter with the erythroid lineage within the haematopoietic system. On a molecular level, PU.1 and GATA-1 positively regulate themselves and antagonize each other via direct protein–protein interactions. Here we use mathematical modelling to identify a novel type of dynamic behaviour that can be supported by such a regulatory architecture. Guided by the specifics of the PU.1–GATA-1 interaction, we formulate, using the law of mass action, a system of differential equations for the key molecular concentrations. After a series of systematic approximations, the system is reduced to a simpler one, which is tractable to phase-plane and linearization methods. The reduced system formally resembles, and generalizes, a well-known model for competitive species from mathematical ecology. However, in addition to the qualitative regimes exhibited by a pair of competitive species (exclusivity, bistable exclusivity, stable-node coexpression) it also allows for oscillatory limit-cycle coexpression. A key outcome of the model is that, in the context of cell-fate choice, such oscillations could be harnessed by a differentiating cell to prime alternately for opposite outcomes; a bifurcation-theory approach is adopted to characterize this possibility

Nonlinear oscillations and chaos in chemical cardiorespiratory control

We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work

Mathematical Modeling of Biological Systems

Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine