A PDE-DDE model for cell polarization in fission yeast

We consider a one-dimensional model of cell polarization in fission yeast consisting of a hybrid partial differential equation–delay differential equation system. The model describes bulk diffusion of the signaling molecule Rho GTPase Cdc42 in the cytoplasm, which is coupled to a pair of delay differential equations at the ends of the cell via boundary conditions. The latter represent the binding of Cdc42 to the cell membrane and rerelease into the cytoplasm via unbinding. The nontrivial nature of the dynamics arises from the fact that both the binding and unbinding rates at each end are taken to depend on the local membrane concentration of Cdc42. In particular, the association rate is regulated by positive feedback and the dissociation rate is regulated by delayed negative feedback. We use linear stability analysis and numerical simulations to investigate the onset of limit cycle oscillations at the end compartments for a cell of fixed length, distinguishing between symmetric solutions in which the mean concentration is identical at both ends and asymmetric solutions where the mean concentration at one end dominates. We find that the critical time delay for the onset of oscillations via a Hopf bifurcation increases as the diffusion coefficient D decreases. We then solve the diffusion equation on a growing domain under the additional assumption that the total amount Ctot of the signaling molecule increases as the cell length increases. We show that the system undergoes a transition from asymmetric to symmetric oscillations as the cell grows, consistent with experimental findings of “new-end-take-off” in fission yeast. (The latter refers to the switch from monopolar to bipolar growth as the cell grows.) The critical length where the switch occurs depends on D and the growth rate

Doctor of Philosophy

dissertationWe formulate and analyze three spatio-temporal models for cell polarization in budding yeast, fission yeast, and the neuronal growth cone, respectively. We focus on the roles of diffusion and active transport of cytosolic molecules along cytoskeletal filaments on the establishment of a polarized distribution of membrane-bound molecules. Our first model couples the diffusion equation on a finite interval to a pair of delay differential equations at the boundaries. The model is used to study the oscillatory dynamics of the signaling molecule Cdc42 in fission yeast. We explore the effect of diffusion by performing a bifurcation analysis and find that the critical time delay for the onset of oscillations increases as the diffusion coefficient decreases. We then extend the model to a growing domain and show that there is a transition from asymmetric to symmetric oscillations as the cell grows. This is consistent with the experimental findings of “new-end-takeoff†in fission yeast. In our second model, we study the active transport of signaling molecules along a two-dimensional microtubule (MT) network in the neuronal growth cone. We consider a Rac1-stathmin-MT pathway and use a modified Dogteromâ€"Leibler model for the microtubule growth. In the presence of a nonuniform Rac1 concentration, we derive the resulting nonuniform length distribution of MTs and couple it to the active transport model. We calculate the polarized distribution of signaling molecules at the membrane using perturbation analysis and numerical simulation. We find the distribution is sensitive to the explicit Rac1 distribution and the stahmin-MT pathway. Our third model is a stochastic active transport model for vesicles containing signaling molecules in a filament network. We first derive the corresponding advection-diffusion model by a quasi-steady-state analysis. We find the diffusion is anisotropic and depends on the local density of filaments. The stability of the homogeneous steady state is sensitive to the geometry of filaments. For a parallelMTnetwork, the homogeneous steady state is linearly stable. For a network with filaments nucleated from the membrane (actin cytoskeleton), the homogeneous steady state is linearly unstable and a polarized distribution can occur

A theory of synchrony for active compartments with delays coupled through bulk diffusion

We extend recent work on the analysis of synchronization in a pair of biochemical oscillators coupled by linear bulk diffusion, in order to explore the effects of discrete delays. More specifically, we consider two well-mixed, identical compartments located at either end of a bounded, one-dimensional domain. The compartments can exchange signaling molecules with the bulk domain, within which the signaling molecules undergo diffusion. The concentration of signaling molecules in each compartment is modeled by a delay differential equation (DDE), while the concentration in the bulk medium is modeled by a partial differential equation (PDE) for diffusion. Coupling in the resulting PDE–DDE system is via flux terms at the boundaries. Using linear stability analysis, numerical simulations and bifurcation analysis, we investigate the effect of diffusion on the onset of a supercritical Hopf bifurcation. The direction of the Hopf bifurcation is determined by numerical simulations and a winding number argument. Near a Hopf bifurcation point, we find that there are oscillations with two possible modes: in-phase and anti-phase. Moreover, the critical delay for oscillations to occur increases with the diffusion coefficient. Our numerical results suggest that the selection of the in-phase or anti-phase oscillation is sensitive to the diffusion coefficient, time delay and coupling strength. For slow diffusion and weak coupling both modes can coexist, while for fast diffusion and strong coupling, only one of the modes is dominant, depending on the explicit choice of DDE

Dynamically active compartments coupled by a stochastically gated gap junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate

Principles and theory of protein-based pattern formation

Biological systems perform functions by the orchestrated interplay of many small components without a "conductor." Such self-organization pervades life on many scales, from the subcellular level to populations of many organisms and whole ecosystems. On the intracellular level, protein-based pattern formation coordinates and instructs functions like cell division, differentiation and motility. A key feature of protein-based pattern formation is that the total numbers of the involved proteins remain constant on the timescale of pattern formation. The overarching theme of this thesis is the profound impact of this mass-conservation property on pattern formation and how one can harness mass conservation to understand the underlying physical principles. The central insight is that changes in local densities shift local reactive equilibria, and thus induce concentration gradients which, in turn, drive diffusive transport of mass. For two-component systems, this dynamic interplay can be captured by simple geometric objects in the (low-dimensional) phase space of chemical concentrations. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. Moreover, we introduce the notion of regional (in)stabilities, which allows one to characterize the dynamics in the highly nonlinear regime reveals an inherent connection between Turing instability and stimulus-induced pattern formation. The insights gained for conceptual two-component systems can be generalized to systems with more components and several conserved masses. In the minimal setting of two diffusively coupled "reactors," the full dynamics can be embedded in the phase-space of redistributed masses where the phase space flow is organized by surfaces of local reactive equilibria. Building on the phase-space analysis for two component systems, we develop a new approach to the important open problem of wavelength selection in the highly nonlinear regime. We show that two-component reaction–diffusion systems always exhibit uninterrupted coarsening (the continual growth of the characteristic length scale) of patterns if they are strictly mass conserving. Selection of a finite wavelength emerges due to weakly broken mass-conservation, or coupling to additional components, which counteract and stop the competition instability that drives coarsening. For complex dynamical phenomena like wave patterns and the transition to spatiotemporal chaos, an analysis in terms of local equilibria and their stability properties provides a powerful tool to interpret data from numerical simulations and experiments, and to reveal the underlying physical mechanisms. In collaborations with different experimental labs, we studied the Min system of Escherichia coli. A central insight from these investigations is that bulk-surface coupling imparts a strong dependence of pattern formation on the geometry of the spatial confinement, which explains the qualitatively different dynamics observed inside cells compared to in vitro reconstitutions. By theoretically studying the polarization machinery in budding yeast and testing predictions in collaboration with experimentalists, we found that this functional module implements several redundant polarization mechanisms that depend on different subsets of proteins. Taken together, our work reveals unifying principles underlying biological self-organization and elucidates how microscopic interaction rules and physical constraints collectively lead to specific biological functions.Biologische Systeme führen Funktionen durch das orchestrierte Zusammenspiel vieler kleiner Komponenten ohne einen "Dirigenten" aus. Solche Selbstorganisation durchdringt das Leben auf vielen Skalen, von der subzellulären Ebene bis zu Populationen vieler Organismen und ganzen Ökosystemen. Auf der intrazellulären Ebene koordiniert und instruieren proteinbasierte Muster Funktionen wie Zellteilung, Differenzierung und Motilität. Ein wesentliches Merkmal der proteinbasierten Musterbildung ist, dass die Gesamtzahl der beteiligten Proteine auf der Zeitskala der Musterbildung konstant bleibt. Das übergreifende Thema dieser Arbeit ist es, den tiefgreifenden Einfluss dieser Massenerhaltung auf die Musterbildung zu untersuchen und Methoden zu entwickeln, die Massenerhaltung nutzen, um die zugrunde liegenden physikalischen Prinzipien von proteinbasierter Musterbildung zu verstehen. Die zentrale Erkenntnis ist, dass Änderungen der lokalen Dichten lokale reaktive Gleichgewichte verschieben und somit Konzentrationsgradienten induzieren, die wiederum den diffusiven Transport von Masse antreiben. Für Zweikomponentensysteme kann dieses dynamische Wechselspiel durch einfache geometrische Objekte im (niedrigdimensionalen) Phasenraum der chemischen Konzentrationen erfasst werden. Auf dieser Phasenraumebene können physikalische Erkenntnisse durch geometrische Kriterien und grafische Konstruktionen gewonnen werden. Darüber hinaus führen wir den Begriff der regionalen (In-)stabilität ein, der es erlaubt, die Dynamik im hochgradig nichtlinearen Regime zu charakterisieren und einen inhärenten Zusammenhang zwischen Turing-Instabilität und stimulusinduzierter Musterbildung aufzuzeigen. Die für konzeptionelle Zweikomponentensysteme gewonnenen Erkenntnisse können auf Systeme mit mehr Komponenten und mehreren erhaltenen Massen verallgemeinert werden. In der minimalen Fassung von zwei diffusiv gekoppelten "Reaktoren" kann die gesamte Dynamik in den Phasenraum umverteilter Massen eingebettet werden, wobei der Phasenraumfluss durch Flächen lokaler reaktiver Gleichgewichte organisiert wird. Aufbauend auf der Phasenraumanalyse für Zweikomponentensysteme entwickeln wir einen neuen Ansatz für die wichtige offene Fragestellung der Wellenängenselektion im hochgradig nichtlinearen Regime. Wir zeigen, dass "coarsening" (das stetige wachsen der charakteristischen Längenskala) von Mustern in Zweikomponentensystemen nie stoppt, wenn sie exakt massenerhaltend sind. Die Selektion einer endlichen Wellenlänge entsteht durch schwach gebrochene Massenerhaltung oder durch Kopplung an zusätzliche Komponenten. Diese Prozesse wirken der Masseumverteilung, die coarsening treibt, entgegen und stoppen so das coarsening. Bei komplexen dynamischen Phänomenen wie Wellenmustern und dem Übergang zu raumzeitlichen Chaos bietet eine Analyse in Bezug auf lokale Gleichgewichte und deren Stabilitätseigenschaften ein leistungsstarkes Werkzeug, um Daten aus numerischen Simulationen und Experimenten zu interpretieren und die zugrunde liegenden physikalischen Mechanismen aufzudecken. In Zusammenarbeit mit verschiedenen experimentellen Labors haben wir das Min-System von Escherichia coli untersucht. Eine zentrale Erkenntnis aus diesen Untersuchungen ist, dass die Kopplung zwischen Volumen und Oberfläche zu einer starken Abhängigkeit der Musterbildung von der räumlichen Geometrie führt. Das erklärt die qualitativ unterschiedliche Dynamik, die in Zellen im Vergleich zu in vitro Rekonstitutionen beobachtet wird. Durch die theoretische Untersuchung der Polarisationsmaschinerie in Hefezellen, kombiniert mit experimentellen Tests theoretischer Vorhersagen, haben wir herausgefunden, dass dieses Funktionsmodul mehrere redundante Polarisationsmechanismen implementiert, die von verschiedenen Untergruppen von Proteinen abhängen. Zusammengenommen beleuchtet unsere Arbeit die vereinheitlichenden Prinzipien, die der intrazellulären Selbstorganisation zugrunde liegen, und zeigt, wie mikroskopische Interaktionsregeln und physikalische Bedingungen gemeinsam zu spezifischen biologischen Funktionen führen