Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods

Stochastic analysis of nonlinear dynamics and feedback control for gene regulatory networks with applications to synthetic biology

The focus of the thesis is the investigation of the generalized repressilator model (repressing genes ordered in a ring structure). Using nonlinear bifurcation analysis stable and quasi-stable periodic orbits in this genetic network are characterized and a design for a switchable and controllable genetic oscillator is proposed. The oscillator operates around a quasi-stable periodic orbit using the classical engineering idea of read-out based control. Previous genetic oscillators have been designed around stable periodic orbits, however we explore the possibility of quasi-stable periodic orbit expecting better controllability. The ring topology of the generalized repressilator model has spatio-temporal symmetries that can be understood as propagating perturbations in discrete lattices. Network topology is a universal cross-discipline transferable concept and based on it analytical conditions for the emergence of stable and quasi-stable periodic orbits are derived. Also the length and distribution of quasi-stable oscillations are obtained. The findings suggest that long-lived transient dynamics due to feedback loops can dominate gene network dynamics. Taking the stochastic nature of gene expression into account a master equation for the generalized repressilator is derived. The stochasticity is shown to influence the onset of bifurcations and quality of oscillations. Internal noise is shown to have an overall stabilizing effect on the oscillating transients emerging from the quasi-stable periodic orbits. The insights from the read-out based control scheme for the genetic oscillator lead us to the idea to implement an algorithmic controller, which would direct any genetic circuit to a desired state. The algorithm operates model-free, i.e. in principle it is applicable to any genetic network and the input information is a data matrix of measured time series from the network dynamics. The application areas for readout-based control in genetic networks range from classical tissue engineering to stem cells specification, whenever a quantitatively and temporarily targeted intervention is required

Engineering Dynamics and Life Sciences

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Diversity and evolution of ecosystems: From genomes to the biosphere

In this thesis, I present research on biological systems at three different scales of space and time: biodiversity of ecological systems, the dynamics of repetitive elements and their diversity in the genome, and the development of phylogenetic trees in evolution. The unifying theme is the interplay between ecology and evolution, expressed within an ecosystem, within genomes, and over the evolutionary history of life. Part I concerns biodiversity on the ecological scale. I study the “Kill the Winner” (KtW) hypothesis, a proposed solution to the biodiversity paradox questioning why many competitors can coexist in a single niche. The original KtW model is deterministic and expressed in terms of continuous biomass concentrations, and appears to predict the coexistence of species. Here I present a stochastic individual-level model for the KtW paradigm, representing populations as finite integers. We find an extinction cascade and a monotonic loss of diversity in time due to the stochasticity, thus failing to explain diversity in the presence of stochasticity. To solve this problem, we couple the coevolution of predators and prey with the KtW model, and show that the diversity of the stochastic system can arise from the constant population flux induced by the emergence of new mutants, although there are undoubtedly contributions from the spatial variations in populations too. Our results suggest that diversity reflects the dynamical interplay between ecological and evolutionary processes, and is driven by how far the system is from an equilibrium state. Part II consists of three projects on the dynamics and diversity of repetitive DNA elements on the genomic scale. The first project is to develop a statistical mechanical model for the interaction between two types of DNA transposons, known as LINE and SINE. These mobile genetic elements are respectively autonomous and non-autonomous: SINE steals the machinery of LINE to complete its migration, and thus acts as a parasite. We have found that the demographic noise due to the discreteness of element copy numbers leads to noisy oscillations on the evolutionary time scale, in a similar way to that resulting in the predator-prey quasi-cycles in ecology. By viewing these DNA elements as predators and prey, we have shown that the dynamics in the genome can fruitfully be analyzed using the analogy to ecological models. In the second project, we look for the predicted quasi-cycles of LINE and SINE in the genomic history of the ancient fish coelacanth. We analyze the periodicity of the age distribution recorded in the genome by the molecular clock, and also develop a theoretical model to examine under what conditions can the cycles be recorded. Our analyses provide a procedure for future research work, but the conclusion is that the rapid deterioration of DNA transposons due to mutations means that the observational window is restricted to the last 50 million years, which is not long enough to conclude that the predicted oscillations are present. In the third project, we further explore the analogy of a genome to an ecosystem and DNA elements to organisms. We use the metric known as the rank-abundance distribution (RAD) from ecology to study the diversity of junk DNA “species”. We have found that the RADs for all the 46 examined species can be reasonably fit by a power law, with very similar exponents. This universal RAD can help identify the underlying microscopic evolutionary processes of these DNA species. Our work demonstrates that applying ecological methods to study genomic elements may provide novel insights for genome functions and evolution. Part III focuses on the development of phylogenetic trees on the evolutionary scale. The topology of phylogenetic trees has been found to obey a universal scaling law. The exponent lies in between the two extreme cases of completely balanced binary trees and completely imbalanced ones. We seek evolutionary processes that can generate the observed topology, and here study in particular the effect of niche construction on the large-scale structure of phylogenetic trees. In contrast to the conventional natural selection framework, which treats the environment independently of the organisms under selection, the niche construction theory views the feedback of organisms on their environment as a crucial and explicit process in evolution. We present a coarse-grained statistical model of niche construction coupled to simple models of speciation, and show that the resultant phylogenetic tree topology can exhibit a scale-invariant structure, through a singularity arising from large niche construction fluctuations. These results show in principle how the scaling laws of phylogenetic tree topology can emerge from rather general assumptions about the interplay between ecological and evolutionary processes

Novel approaches to study the design principles of turing patterns

A fundamental concern in biology is the origins of, and the mechanisms responsible for the structures and patterns observed within, organisms [1]. Turing patterns and the Turing mechanism may explain the processes behind biological pattern formation. Theoretical studies of the Turing mechanism show that it is highly sensitive to fluctuations and variations in kinetic parameters. Various experiments have shown that biochemical processes in living cells are inherently noisy systems, they are subjected to a diverse range of fluctuations. This ‘robustness problem’ raises the question of how such a seemingly sensitive mechanism could produce robust patterns amidst noise [2]. Recent computational advances allow for large-scale explorations of the design space of regulatory networks underpinning pattern production. Such explorations generate insights into the Turing mechanism’s robustness and sensitivity. Part 1 of the thesis performs a large-scale exploration within a discrete modelling framework, identifying the same pattern producing network types identified within previous studies. The equivalence we find across modelling frameworks suggests that a deeper underlying principle of these Turing mechanisms exist. In contrast to the continuous case , networks appear to be more robust in the discrete framework we explore here, suggesting that these networks might be more robust than previously thought. Part 2 of the thesis focuses on Turing patterns as a inverse problem: is it possible to infer the parameters that most likely produced a given pattern? Here, we distill the information of a pattern into a one-dimensional representation based on resistance distances, a concept from electrical networks [3]. We shown this representation to be robust against fluctuations in the pattern stemming from random initial conditions, or stochasticity of the model, and therefore permits the application of machine learning methods such as neural networks and support vector regression for parameter inference. We apply this method to infer one and three parameters for both deterministic and stochastic models of the Gierer-Meinhardt system. We show that the ’resistance distance histogram’ method is more robust to noise, and performs better for limited number of data samples than a vanilla convolutional neural network approach. Robustness of parameter inference with respect to noise and limited data samples is of particular importance when considering real experimental data. Overall, this thesis advances our understanding on the design principles of pattern formation, and provides insight into possible methods for inferring details of regulatory networks behind experimental evidence of Turing patterns.Open Acces

Phase Transitions Induced by Diversity and Examples in Biological Systems

Tesis leída en l'Universitat de les Illes Balears en diciembre de 2010The present thesis covers various topics that range over di erent aspects of scientific research. On one end there is the specific analysis of a precise form that models some experimental observations. A good theoretical understanding of the mathematics that describe the observations can be a guide to the experimentalist and help estimate the validity of the measurements. On the other end there are abstract models whose relation to physical systems seem far but they are prototypic for a broad range of di erent systems and the drawn conclusions tend to be quite general. Depending on the abstraction and on the simplifications in use the distinction between both ends might not be sharp. The ordering of the research results presented in part II of this thesis somehow reflects the seamless transition from one end to the other. To introduce the reader into the context of the genuine results we provide introductory material in the chapters of the present part I.Peer reviewe