Mathematical modelling of p53 signalling during DNA damage response: a survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage

Brain Dynamics From Mathematical Perspectives: A Study of Neural Patterning

The brain is the central hub regulating thought, memory, vision, and many other processes occurring within the body. Neural information transmission occurs through the firing of billions of connected neurons, giving rise to a rich variety of complex patterning. Mathematical models are used alongside direct experimental approaches in understanding the underlying mechanisms at play which drive neural activity, and ultimately, in understanding how the brain works. This thesis focuses on network and continuum models of neural activity, and computational methods used in understanding the rich patterning that arises due to the interplay between non-local coupling and local dynamics. It advances the understanding of patterning in both cortical and sub-cortical domains by utilising the neural field framework in the modelling and analysis of thalamic tissue – where cellular currents are important in shaping the tissue firing response through the post-inhibitory rebound phenomenon – and of cortical tissue. The rich variety of patterning exhibited by different neural field models is demonstrated through a mixture of direct numerical simulation, as well as via a numerical continuation approach and an analytical study of patterned states such as synchrony, spatially extended periodic orbits, bumps, and travelling waves. Linear instability theory about these patterns is developed and used to predict the points at which solutions destabilise and alternative emergent patterns arise. Models of thalamic tissue often exhibit lurching waves, where activity travels across the domain in a saltatory manner. Here, a direct mechanism, showing the birth of lurching waves at a Neimark-Sacker-type instability of the spatially synchronous periodic orbit, is presented. The construction and stability analyses carried out in this thesis employ techniques from non-smooth dynamical systems (such as saltation methods) to treat the Heaviside nature of models. This is often coupled with an Evans function approach to determine the linear stability of patterned states. With the ever-increasing complexity of neural models that are being studied, there is a need to develop ways of systematically studying the non-trivial patterns they exhibit. Computational continuation methods are developed, allowing for such a study of periodic solutions and their stability across different parameter regimes, through the use of Newton-Krylov solvers. These techniques are complementary to those outlined above. Using these methods, the relationship between the speed of synaptic transmission and the emergent properties of periodic and travelling periodic patterns such as standing waves and travelling breathers is studied. Many different dynamical systems models of physical phenomena are amenable to analysis using these general computational methods (as long as they have the property that they are sufficiently smooth), and as such, their domain of applicability extends beyond the realm of mathematical neuroscience

Brain Dynamics From Mathematical Perspectives: A Study of Neural Patterning

The brain is the central hub regulating thought, memory, vision, and many other processes occurring within the body. Neural information transmission occurs through the firing of billions of connected neurons, giving rise to a rich variety of complex patterning. Mathematical models are used alongside direct experimental approaches in understanding the underlying mechanisms at play which drive neural activity, and ultimately, in understanding how the brain works. This thesis focuses on network and continuum models of neural activity, and computational methods used in understanding the rich patterning that arises due to the interplay between non-local coupling and local dynamics. It advances the understanding of patterning in both cortical and sub-cortical domains by utilising the neural field framework in the modelling and analysis of thalamic tissue – where cellular currents are important in shaping the tissue firing response through the post-inhibitory rebound phenomenon – and of cortical tissue. The rich variety of patterning exhibited by different neural field models is demonstrated through a mixture of direct numerical simulation, as well as via a numerical continuation approach and an analytical study of patterned states such as synchrony, spatially extended periodic orbits, bumps, and travelling waves. Linear instability theory about these patterns is developed and used to predict the points at which solutions destabilise and alternative emergent patterns arise. Models of thalamic tissue often exhibit lurching waves, where activity travels across the domain in a saltatory manner. Here, a direct mechanism, showing the birth of lurching waves at a Neimark-Sacker-type instability of the spatially synchronous periodic orbit, is presented. The construction and stability analyses carried out in this thesis employ techniques from non-smooth dynamical systems (such as saltation methods) to treat the Heaviside nature of models. This is often coupled with an Evans function approach to determine the linear stability of patterned states. With the ever-increasing complexity of neural models that are being studied, there is a need to develop ways of systematically studying the non-trivial patterns they exhibit. Computational continuation methods are developed, allowing for such a study of periodic solutions and their stability across different parameter regimes, through the use of Newton-Krylov solvers. These techniques are complementary to those outlined above. Using these methods, the relationship between the speed of synaptic transmission and the emergent properties of periodic and travelling periodic patterns such as standing waves and travelling breathers is studied. Many different dynamical systems models of physical phenomena are amenable to analysis using these general computational methods (as long as they have the property that they are sufficiently smooth), and as such, their domain of applicability extends beyond the realm of mathematical neuroscience

Nonlinear dynamics and fluctuations in biological systems

The present habilitation thesis in theoretical biological physics addresses two central dynamical processes in cells and organisms: (i) active motility and motility control and (ii) self-organized pattern formation. The unifying theme is the nonlinear dynamics of biological function and its robustness in the presence of strong fluctuations, structural variations, and external perturbations. We theoretically investigate motility control at the cellular scale, using cilia and flagella as ideal model system. Cilia and flagella are highly conserved slender cell appendages that exhibit spontaneous bending waves. This flagellar beat represents a prime example of a chemo-mechanical oscillator, which is driven by the collective dynamics of molecular motors inside the flagellar axoneme. We study the nonlinear dynamics of flagellar swimming, steering, and synchronization, which encompasses shape control of the flagellar beat by chemical signals and mechanical forces. Mechanical forces can synchronize collections of flagella to beat at a common frequency, despite active motor noise that tends to randomize flagellar synchrony. In Chapter 2, we present a new physical mechanism for flagellar synchronization by mechanical self-stabilization that applies to free-swimming flagellated cells. This new mechanism is independent of direct hydrodynamic interactions between flagella. Comparison with experimental data provided by experimental collaboration partners in the laboratory of J. Howard (Yale, New Haven) confirmed our new mechanism in the model organism of the unicellular green alga Chlamydomonas. Further, we characterize the beating flagellum as a noisy oscillator. Using a minimal model of collective motor dynamics, we argue that measured non-equilibrium fluctuations of the flagellar beat result from stochastic motor dynamics at the molecular scale. Noise and mechanical coupling are antagonists for flagellar synchronization. In addition to the control of the flagellar beat by mechanical forces, we study the control of the flagellar beat by chemical signals in the context of sperm chemotaxis. We characterize a fundamental paradigm for navigation in external concentration gradients that relies on active swimming along helical paths. In this helical chemotaxis, the direction of a spatial concentration gradient becomes encoded in the phase of an oscillatory chemical signal. Helical chemotaxis represents a distinct gradient-sensing strategy, which is different from bacterial chemotaxis. Helical chemotaxis is employed, for example, by sperm cells from marine invertebrates with external fertilization. We present a theory of sensorimotor control, which combines hydrodynamic simulations of chiral flagellar swimming with a dynamic regulation of flagellar beat shape in response to chemical signals perceived by the cell. Our theory is compared to three-dimensional tracking experiments of sperm chemotaxis performed by the laboratory of U. B. Kaupp (CAESAR, Bonn). In addition to motility control, we investigate in Chapter 3 self-organized pattern formation in two selected biological systems at the cell and organism scale, respectively. On the cellular scale, we present a minimal physical mechanism for the spontaneous self-assembly of periodic cytoskeletal patterns, as observed in myofibrils in striated muscle cells. This minimal mechanism relies on the interplay of a passive coarsening process of crosslinked actin clusters and active cytoskeletal forces. This mechanism of cytoskeletal pattern formation exemplifies how local interactions can generate large-scale spatial order in active systems. On the organism scale, we present an extension of Turing’s framework for self-organized pattern formation that is capable of a proportionate scaling of steady-state patterns with system size. This new mechanism does not require any pre-pattering clues and can restore proportional patterns in regeneration scenarios. We analytically derive the hierarchy of steady-state patterns and analyze their stability and basins of attraction. We demonstrate that this scaling mechanism is structurally robust. Applications to the growth and regeneration dynamics in flatworms are discussed (experiments by J. Rink, MPI CBG, Dresden).:1 Introduction 10 1.1 Overview of the thesis 10 1.2 What is biological physics? 12 1.3 Nonlinear dynamics and control 14 1.3.1 Mechanisms of cell motility 16 1.3.2 Self-organized pattern formation in cells and tissues 28 1.4 Fluctuations and biological robustness 34 1.4.1 Sources of fluctuations in biological systems 34 1.4.2 Example of stochastic dynamics: synchronization of noisy oscillators 36 1.4.3 Cellular navigation strategies reveal adaptation to noise 39 2 Selected publications: Cell motility and motility control 56 2.1 “Flagellar synchronization independent of hydrodynamic interactions” 56 2.2 “Cell body rocking is a dominant mechanism for flagellar synchronization” 57 2.3 “Active phase and amplitude fluctuations of the flagellar beat” 58 2.4 “Sperm navigation in 3D chemoattractant landscapes” 59 3 Selected publications: Self-organized pattern formation in cells and tissues 60 3.1 “Sarcomeric pattern formation by actin cluster coalescence” 60 3.2 “Scaling and regeneration of self-organized patterns” 61 4 Contribution of the author in collaborative publications 62 5 Eidesstattliche Versicherung 64 6 Appendix: Reprints of publications 66Das Thema der vorliegenden Habilitationsschrift in Theoretischer Biologischer Physik ist die nichtlineare Dynamik funktionaler biologischer Systeme und deren Robustheit gegenüber Fluktuationen und äußeren Störungen. Wir entwickeln hierzu theoretische Beschreibungen für zwei grundlegende biologische Prozesse: (i) die zell-autonome Kontrolle aktiver Bewegung, sowie (ii) selbstorganisierte Musterbildung in Zellen und Organismen. In Kapitel 2, untersuchen wir Bewegungskontrolle auf zellulärer Ebene am Modelsystem von Zilien und Geißeln. Spontane Biegewellen dieser dünnen Zellfortsätze ermöglichen es eukaryotischen Zellen, in einer Flüssigkeit zu schwimmen. Wir beschreiben einen neuen physikalischen Mechanismus für die Synchronisation zweier schlagender Geißeln, unabhängig von direkten hydrodynamischen Wechselwirkungen. Der Vergleich mit experimentellen Daten, zur Verfügung gestellt von unseren experimentellen Kooperationspartnern im Labor von J. Howard (Yale, New Haven), bestätigt diesen neuen Mechanismus im Modellorganismus der einzelligen Grünalge Chlamydomonas. Der Gegenspieler dieser Synchronisation durch mechanische Kopplung sind Fluktuationen. Wir bestimmen erstmals Nichtgleichgewichts-Fluktuationen des Geißel-Schlags direkt, wofür wir eine neue Analyse-Methode der Grenzzykel-Rekonstruktion entwickeln. Die von uns gemessenen Fluktuationen entstehen mutmaßlich durch die stochastische Dynamik molekularen Motoren im Innern der Geißeln, welche auch den Geißelschlag antreiben. Um die statistische Physik dieser Nichtgleichgewichts-Fluktuationen zu verstehen, entwickeln wir eine analytische Theorie der Fluktuationen in einem minimalen Modell kollektiver Motor-Dynamik. Zusätzlich zur Regulation des Geißelschlags durch mechanische Kräfte untersuchen wir dessen Regulation durch chemische Signale am Modell der Chemotaxis von Spermien-Zellen. Dabei charakterisieren wir einen grundlegenden Mechanismus für die Navigation in externen Konzentrationsgradienten. Dieser Mechanismus beruht auf dem aktiven Schwimmen entlang von Spiralbahnen, wodurch ein räumlicher Konzentrationsgradient in der Phase eines oszillierenden chemischen Signals kodiert wird. Dieser Chemotaxis-Mechanismus unterscheidet sich grundlegend vom bekannten Chemotaxis-Mechanismus von Bakterien. Wir entwickeln eine Theorie der senso-motorischen Steuerung des Geißelschlags während der Spermien-Chemotaxis. Vorhersagen dieser Theorie werden durch Experimente der Gruppe von U.B. Kaupp (CAESAR, Bonn) quantitativ bestätigt. In Kapitel 3, untersuchen wir selbstorganisierte Strukturbildung in zwei ausgewählten biologischen Systemen. Auf zellulärer Ebene schlagen wir einen einfachen physikalischen Mechanismus vor für die spontane Selbstorganisation von periodischen Zellskelett-Strukturen, wie sie sich z.B. in den Myofibrillen gestreifter Muskelzellen finden. Dieser Mechanismus zeigt exemplarisch auf, wie allein durch lokale Wechselwirkungen räumliche Ordnung auf größeren Längenskalen in einem Nichtgleichgewichtssystem entstehen kann. Auf der Ebene des Organismus stellen wir eine Erweiterung der Turingschen Theorie für selbstorganisierte Musterbildung vor. Wir beschreiben eine neue Klasse von Musterbildungssystemen, welche selbst-organisierte Muster erzeugt, die mit der Systemgröße skalieren. Dieser neue Mechanismus erfordert weder eine vorgegebene Kompartimentalisierung des Systems noch spezielle Randbedingungen. Insbesondere kann dieser Mechanismus proportionale Muster wiederherstellen, wenn Teile des Systems amputiert werden. Wir bestimmen analytisch die Hierarchie aller stationären Muster und analysieren deren Stabilität und Einzugsgebiete. Damit können wir zeigen, dass dieser Skalierungs-Mechanismus strukturell robust ist bezüglich Variationen von Parametern und sogar funktionalen Beziehungen zwischen dynamischen Variablen. Zusammen mit Kollaborationspartnern im Labor von J. Rink (MPI CBG, Dresden) diskutieren wir Anwendungen auf das Wachstum von Plattwürmern und deren Regeneration in Amputations-Experimenten.:1 Introduction 10 1.1 Overview of the thesis 10 1.2 What is biological physics? 12 1.3 Nonlinear dynamics and control 14 1.3.1 Mechanisms of cell motility 16 1.3.2 Self-organized pattern formation in cells and tissues 28 1.4 Fluctuations and biological robustness 34 1.4.1 Sources of fluctuations in biological systems 34 1.4.2 Example of stochastic dynamics: synchronization of noisy oscillators 36 1.4.3 Cellular navigation strategies reveal adaptation to noise 39 2 Selected publications: Cell motility and motility control 56 2.1 “Flagellar synchronization independent of hydrodynamic interactions” 56 2.2 “Cell body rocking is a dominant mechanism for flagellar synchronization” 57 2.3 “Active phase and amplitude fluctuations of the flagellar beat” 58 2.4 “Sperm navigation in 3D chemoattractant landscapes” 59 3 Selected publications: Self-organized pattern formation in cells and tissues 60 3.1 “Sarcomeric pattern formation by actin cluster coalescence” 60 3.2 “Scaling and regeneration of self-organized patterns” 61 4 Contribution of the author in collaborative publications 62 5 Eidesstattliche Versicherung 64 6 Appendix: Reprints of publications 6

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”

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described