Dissertations / Theses on the topic 'Cell Division - Stochastic Simulation'

Modelling and simulation are essential to modern research in cell biology. This thesis follows a journey starting from the construction of new stochastic methods for discrete biochemical systems to using them to simulate a population of interacting haematopoietic stem cell lineages. The first part of this thesis is on discrete stochastic methods. We develop two new methods, the stochastic extrapolation framework and the Stochastic Bulirsch-Stoer methods. These are based on the Richardson extrapolation technique, which is widely used in ordinary differential equation solvers. We believed that it would also be useful in the stochastic regime, and this turned out to be true. The stochastic extrapolation framework is a scheme that admits any stochastic method with a fixed stepsize and known global error expansion. It can improve the weak order of the moments of these methods by cancelling the leading terms in the global error. Using numerical simulations, we demonstrate that this is the case up to second order, and postulate that this also follows for higher order. Our simulations show that extrapolation can greatly improve the accuracy of a numerical method. The Stochastic Bulirsch-Stoer method is another highly accurate stochastic solver. Furthermore, using numerical simulations we find that it is able to better retain its high accuracy for larger timesteps than competing methods, meaning it remains accurate even when simulation time is speeded up. This is a useful property for simulating the complex systems that researchers are often interested in today. The second part of the thesis is concerned with modelling a haematopoietic stem cell system, which consists of many interacting niche lineages. We use a vectorised tau-leap method to examine the differences between a deterministic and a stochastic model of the system, and investigate how coupling niche lineages affects the dynamics of the system at the homeostatic state as well as after a perturbation. We find that larger coupling allows the system to find the optimal steady state blood cell levels. In addition, when the perturbation is applied randomly to the entire system, larger coupling also results in smaller post-perturbation cell fluctuations compared to non-coupled cells. In brief, this thesis contains four main sets of contributions: two new high-accuracy discrete stochastic methods that have been numerically tested, an improvement that can be used with any leaping method that introduces vectorisation as well as how to use a common stepsize adapting scheme, and an investigation of the effects of coupling lineages in a heterogeneous population of haematopoietic stem cell niche lineages.

Numerous challenges arise in modeling and simulation as biochemical networks are discovered with increasing complexities and unknown mechanisms. With the improvement in experimental techniques, biologists are able to quantify genes and proteins and their dynamics in a single cell, which calls for quantitative stochastic models for gene and protein networks at cellular levels that match well with the data and account for cellular noise. This dissertation studies a stochastic spatiotemporal model of the Caulobacter crescentus cell cycle. A two-dimensional model based on a Turing mechanism is investigated to illustrate the bipolar localization of the protein PopZ. However, stochastic simulations are often impeded by expensive computational cost for large and complex biochemical networks. The hybrid stochastic simulation algorithm is a combination of differential equations for traditional deterministic models and Gillespie's algorithm (SSA) for stochastic models. The hybrid method can significantly improve the efficiency of stochastic simulations for biochemical networks with multiscale features, which contain both species populations and reaction rates with widely varying magnitude. The populations of some reactant species might be driven negative if they are involved in both deterministic and stochastic systems. This dissertation investigates the negativity problem of the hybrid method, proposes several remedies, and tests them with several models including a realistic biological system. As a key factor that affects the quality of biological models, parameter estimation in stochastic models is challenging because the amount of empirical data must be large enough to obtain statistically valid parameter estimates. To optimize system parameters, a quasi-Newton algorithm for stochastic optimization (QNSTOP) was studied and applied to a stochastic budding yeast cell cycle model by matching multivariate probability distributions between simulated results and empirical data. Furthermore, to reduce model complexity, this dissertation simplifies the fundamental cooperative binding mechanism by a stochastic Hill equation model with optimized system parameters. Considering that many parameter vectors generate similar system dynamics and results, this dissertation proposes a general α-β-γ rule to return an acceptable parameter region of the stochastic Hill equation based on QNSTOP. Different objective functions are explored targeting different features of the empirical data.
Doctor of Philosophy
Modeling and simulation of biochemical networks faces numerous challenges as biochemical networks are discovered with increased complexity and unknown mechanisms. With improvement in experimental techniques, biologists are able to quantify genes and proteins and their dynamics in a single cell, which calls for quantitative stochastic models, or numerical models based on probability distributions, for gene and protein networks at cellular levels that match well with the data and account for randomness. This dissertation studies a stochastic model in space and time of a bacterium’s life cycle— Caulobacter. A two-dimensional model based on a natural pattern mechanism is investigated to illustrate the changes in space and time of a key protein population. However, stochastic simulations are often complicated by the expensive computational cost for large and sophisticated biochemical networks. The hybrid stochastic simulation algorithm is a combination of traditional deterministic models, or analytical models with a single output for a given input, and stochastic models. The hybrid method can significantly improve the efficiency of stochastic simulations for biochemical networks that contain both species populations and reaction rates with widely varying magnitude. The populations of some species may become negative in the simulation under some circumstances. This dissertation investigates negative population estimates from the hybrid method, proposes several remedies, and tests them with several cases including a realistic biological system. As a key factor that affects the quality of biological models, parameter estimation in stochastic models is challenging because the amount of observed data must be large enough to obtain valid results. To optimize system parameters, the quasi-Newton algorithm for stochastic optimization (QNSTOP) was studied and applied to a stochastic (budding) yeast life cycle model by matching different distributions between simulated results and observed data. Furthermore, to reduce model complexity, this dissertation simplifies the fundamental molecular binding mechanism by the stochastic Hill equation model with optimized system parameters. Considering that many parameter vectors generate similar system dynamics and results, this dissertation proposes a general α-β-γ rule to return an acceptable parameter region of the stochastic Hill equation based on QNSTOP. Different optimization strategies are explored targeting different features of the observed data.

A complex network of genes and proteins governs the robust progression through cell cycles in the presence of inevitable noise. Stochastic modeling is viewed as a key paradigm to study the effects of intrinsic and extrinsic noise on the dynamics of biochemical networks. A detailed quantitative description of such complex and multiscale networks via stochastic modeling poses several challenges. First, stochastic models generally require extensive computations, particularly when applied to large networks. Second, the accuracy of stochastic models is highly dependent on the quality of the parameter estimation based on experimental observations. The goal of this dissertation is to address these problems by developing new efficient methods for modeling and simulation of stochastic effects in biochemical systems. Particularly, a hybrid stochastic model is developed to represent a detailed molecular mechanism of cell cycle control in budding yeast cells. In a single multiscale model, the proposed hybrid approach combines the advantages of two regimes: 1) the computational efficiency of a deterministic approach, and 2) the accuracy of stochastic simulations. The results show that this hybrid stochastic model achieves high computational efficiency while generating simulation results that match very well with published experimental measurements. Furthermore, a new hierarchical deep classification (HDC) algorithm is developed to address the parameter estimation problem in a monomolecular system. The HDC algorithm adopts a neural network that, via multiple hierarchical search steps, finds reasonably accurate ranges for the model parameters. To train the neural network in the presence of experimental data scarcity, the proposed method leverages the domain knowledge from stochastic simulations to generate labeled training data. The results show that the proposed HDC algorithm yields accurate ranges for the model parameters and highlight the potentials of model-free learning for parameter estimation in stochastic modeling of complex biochemical networks.
Doctor of Philosophy
Cell cycle is a process in which a growing cell replicates its DNA and divides into two cells. Progression through the cell cycle is regulated by complex interactions between networks of genes, transcripts, and proteins. These interactions inside the confined volume of a cell are subject to inherent noise. To provide a quantitative description of the cell cycle, several deterministic and stochastic models have been developed. However, deterministic models cannot capture the intrinsic noise. In addition, stochastic modeling poses the following challenges. First, stochastic models generally require extensive computations, particularly when applied to large networks. Second, the accuracy of stochastic models is highly dependent on the accuracy of the estimated model parameters. The goal of this dissertation is to address these challenges by developing new efficient methods for modeling and simulation of stochastic effects in biochemical networks. The results show that the proposed hybrid model that combines stochastic and deterministic modeling approaches can achieve high computational efficiency while generating accurate simulation results. Moreover, a new machine learning-based method is developed to address the parameter estimation problem in biochemical systems. The results show that the proposed method yields accurate ranges for the model parameters and highlight the potentials of model-free learning for parameter estimation in stochastic modeling of complex biochemical networks.

Over the past few years, it has been increasingly recognized that stochastic mechanisms play a key role in the dynamics of biological systems. Genetic networks are one example where molecular-level fluctuations are of particular importance. Here stochasticity in the expression of gene products can result in genetically identical cells in the same environment displaying significant variation in biochemical or physical attributes. This variation can influence individual and population-level fitness. In this thesis we first explore the background required to obtain analytical solutions and perform simulations of stochastic models of gene expression. Then we develop an algorithm for the stochastic simulation of gene expression and heterogeneous cell population dynamics. The algorithm combines an exact method to simulate molecular-level fluctuations in single cells and a constant-number Monte Carlo approach to simulate the statistical characteristics of growing cell populations. This approach permits biologically realistic and computationally feasible simulations of environment and time-dependent cell population dynamics. The algorithm is benchmarked against steady-state and time-dependent analytical solutions of gene expression models, including scenarios when cell growth, division, and DNA replication are incorporated into the modelling framework. Furthermore, using the algorithm we obtain the steady-state cell size distribution of a large cell population, grown from a small initial cell population undergoing stochastic and asymmetric division, to the size distribution of a small representative sample of this population simulated to steady-state. These comparisons demonstrate that the algorithm provides an accurate and efficient approach to modelling the effects of complex biological features on gene expression dynamics. The algorithm is also employed to simulate expression dynamics within 'bet-hedging' cell populations during their adaption to environmental stress. These simulations indicate that the cell population dynamics algorithm provides a framework suitable for simulating and analyzing realistic models of heterogeneous population dynamics combining molecular-level stochastic reaction kinetics, relevant physiological details, and phenotypic variability and fitness.

Stochastic effects in cellular systems are usually modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), which follows the same theoretical derivation as the chemical master equation (CME), but the low efficiency of SSA limits its application to large chemical networks. To improve efficiency of stochastic simulations, Haseltine and Rawlings proposed a hybrid of ODE and SSA algorithm, which combines ordinary differential equations (ODEs) for traditional deterministic models and SSA for stochastic models. In this dissertation, accuracy analysis, efficient implementation strategies, and application of of Haseltine and Rawlings's hybrid method (HR) to a budding yeast cell cycle model are discussed. Accuracy of the hybrid method HR is studied based on a linear chain reaction system, motivated from the modeling practice used for the budding yeast cell cycle control mechanism. Mathematical analysis and numerical results both show that the hybrid method HR is accurate if either numbers of molecules of reactants in fast reactions are above certain thresholds, or rate constants of fast reactions are much larger than rate constants of slow reactions. Our analysis also shows that the hybrid method HR allows for a much greater region in system parameter space than those for the slow scale SSA (ssSSA) and the stochastic quasi steady state assumption (SQSSA) method. Implementation of the hybrid method HR requires a stiff ODE solver for numerical integration and an efficient event-handling strategy for slow reaction firings. In this dissertation, an event-handling strategy is developed based on inverse interpolation. Performances of five wildly used stiff ODE solvers are measured in three numerical experiments. Furthermore, inspired by the strategy of the hybrid method HR, a hybrid of ODE and SSA stochastic models for the budding yeast cell cycle is developed, based on a deterministic model in the literature. Simulation results of this hybrid model match very well with biological experimental data, and this model is the first to do so with these recently available experimental data. This study demonstrates that the hybrid method HR has great potential for stochastic modeling and simulation of large biochemical networks.
Ph. D.

Les processus markoviens déterministes par morceaux (PDMP) forment une vaste classe de processus stochastiques caractérisés par une évolution déterministe entre des sauts à mécanisme aléatoire. Ce sont des processus de type hybride, avec une composante discrète de mode et une composante d’état qui évolue dans un espace continu. Entre les sauts du processus, la composante continue évolue de façon déterministe, puis au moment du saut un noyau markovien sélectionne la nouvelle valeur des composantes discrète et continue. Dans cette thèse, nous construisons des PDMP évoluant dans des espaces de mesures (de dimension infinie), pour modéliser des population de cellules en tenant compte des caractéristiques individuelles de chaque cellule. Nous exposons notre construction des PDMP sur des espaces de mesure, et nous établissons leur caractère markovien. Sur ces processus à valeur mesure, nous étudions un problème d'arrêt optimal. Un problème d'arrêt optimal revient à choisir le meilleur temps d'arrêt pour optimiser l'espérance d'une certaine fonctionnelle de notre processus, ce qu'on appelle fonction valeur. On montre que cette fonction valeur est solution des équations de programmation dynamique et on construit une famille de temps d'arrêt $epsilon$-optimaux. Dans un second temps, nous nous intéressons à un PDMP en dimension finie, le TCP, pour lequel on construit un schéma d'Euler afin de l'approcher. Ce choix de modèle simple permet d'estimer différents types d'erreurs. Nous présentons des simulations numériques illustrant les résultats obtenus
Piecewise deterministic Markov processes (PDMP) form a large class of stochastic processes characterized by a deterministic evolution between random jumps. They fall into the class of hybrid processes with a discrete mode and an Euclidean component (called the state variable). Between the jumps, the continuous component evolves deterministically, then a jump occurs and a Markov kernel selects the new value of the discrete and continuous components. In this thesis, we extend the construction of PDMPs to state variables taking values in some measure spaces with infinite dimension. The aim is to model cells populations keeping track of the information about each cell. We study our measured-valued PDMP and we show their Markov property. With thoses processes, we study a optimal stopping problem. The goal of an optimal stopping problem is to find the best admissible stopping time in order to optimize some function of our process. We show that the value fonction can be recursively constructed using dynamic programming equations. We construct some $epsilon$-optimal stopping times for our optimal stopping problem. Then, we study a simple finite-dimension real-valued PDMP, the TCP process. We use Euler scheme to approximate it, and we estimate some types of errors. We illustrate the results with numerical simulations

Nous modélisons différents évènements aléatoires intervenant en biologie. Dans une première partie, on étudie certaines propriétés de populations de neurones. En utilisant un modèle de facilitation dépression synaptique, on étudie le phénomène de bursts synchrones observés à différentes échelles de populations. On étudie ensuite la transition entre état haut et bas de neurones induite par le bruit. Afin de comprendre le phénomène d’oscillations du temps de séjour dans l’état haut, onétudie le problème de premier temps de passage pour une classe de processus stochastiquesdans un domaine avec un attracteur situé près d’un cycle limite. On construit une classede systèmes conjugués à celui de Hopf que l’on étudie via les méthodes d’approximations WKB et couche limite. On s’intéresse enfin aux interactions entre neurones et astrocytes, et plus spécifiquement à l’intégration du potassium. En introduisant un modèle à trois compartiments, on simule les dynamiques du potassium pour différents protocoles de stimulations pour déterminer comment les canaux astrocytaires influencent la neurotransmission. Dans une deuxième partie, on s’intéresse au temps d’atteinte d’un seuil pour des réactions. Les méthodes introduites sont ensuite utilisées pour étudier le contrôle du fuseau mitotique et la régulation posttranscriptionnellede l’expression génétique dans le noyau et le cytoplasme. Dans la troisième partie, on modélise la dynamique stochastique des longueurs de télomères après chaque division cellulaire. En étudiant le processus de drift et saut associé, on prédit la distribution stationnaire ainsi que la longueur du télomère le plus court d’une cellule
In this PhD, we model specific stochastic events occurring in different biological contexts. In the first part, we study three different properties of neural networks. Using a mean field facilitation-depression synaptic model, we unravel the synchronous long lasting bursting observed at various scales of neural populations. Next, we study the neuronal noise induced transition between Up& Down states. To study the oscillatory peaks of the time spent in Up state, we consider the exit problem for a class of stochastic processes in a domain with an attractor located close to a limit cycle. We construct a class of systems conjugate to the Hopf bifurcation system that we study using WKB approximation and boundary layer analysis. We finally focus on neuroglial interactions and more specifically on astrocytic potassium. Using a tri-compartment model, we simulate the potassium dynamics for different stimulation protocols and we determine how astrocytic channels can influence neurotransmission. In the second part, we focus on the threshold activation for stochastic chemical reactions in cellular micro-domains. We compute the probability and the mean first time to reach a threshold for different reactions. The methods are applied to study the mitotic spindle checkpoint and the problem of gene expression and post-transcriptional regulation. The third part is finally dedicated to the stochastic dynamics of telomere length across cell divisions. We model the dynamics of telomere length as a drift and jump process, which allows predicting the distribution of telomere length and the length of the shortest telomere

Les structures d'acto-myosine sont impliquées dans de nombreuses fonctions cellulaires. Comprendre leur organisation et leur comportement collectif est toujours difficile. Nous avons étudié l'anneau cytokinétique dans les cellules de mammifères et dans les levures de fission, en orientant les cellules dans les microcavités, ce qui permet de voir l'anneau dans un seul plan focal. Avec cette configuration, nous révélons de nouvelles structures et des dynamiques distinctes pour les deux systèmes cellulaires. Dans les cellules de mammifères, nous trouvons des motifs réguliers de la myosine et la formine. Les caractéristiques de ces motifs sont stables tout au long de sa fermeture et leur apparition coïncide avec la constriction. Nous proposons que ce phénomène est une propriété inhérente du réseau d'acto-myosine et que la formation de ces motifs entraîne une augmentation du stress. Ces hypothèses sont confirmées par notre modèle en champ moyen. Par contraste, l'anneau de levure de fission montre des inhomogénéités tournantes de l'actine, de la myosine, des protéines de la construction de la paroi (Bgs) et d'autres protéines. La dynamique des inhomogénéités de myosine est inchangée, si la croissance de la paroi est inhibée. Cependant, l'inhibition du mouvement des inhomogénéités conduit à l'arrêt de la fermeture. Nous proposons que la fermeture de l'anneau est entraînée par la rotation de l'actine et de la myosine qui tirent des protéines Bgs, lesquelles construisent ainsi le septum. Cette hypothèse est confirmée par nos calculs et par des simulations numériques. Nous suggérons que la transition entre les états de différents ordres et dynamiques pourrait être une façon de réguler in vivo les systèmes d'acto-myosine
Actomyosin structures are involved in many cell functions. Understanding their organization and collective behavior is still challenging. We study the cytokinetic ring in mammalian cells and in fission yeasts, by orienting cells in microcavities. This allows seeing the ring in a single plane of focus. With this setup, we reveal new structures and distinct dynamics for both cellular systems. In mammalian cells we find a pattern of regular clusters of myosin and formin. The characteristics of this pattern are stable throughout closure and its formation coincides with the onset of constriction. We propose that its characteristic is an inherent property of the actomyosin network and that its formation leads to an increase in stress generation. These hypotheses are supported by our theoretical mean field model. In contrast, fission yeast rings show rotating inhomogeneities (speckles), i.e. rotations of actin, myosin, cell wall building proteins (Bgs) and other proteins. Myosin speckles dynamic is unchanged, if wall growth is inhibited. However, the inhibition of speckle motion leads to stalled closure. We propose that the ring closure is driven by the rotation of actin and myosin, which pull Bgs thereby building the septum. This model is supported by our calculations and by numerical simulations. We suggest that the transition between states of different orders and dynamics might be a way to regulate actomyosin systems in vivo

An object-oriented framework is presented for developing and simulating individual-based models of cell populations. The framework supplies classes to define objects called simulation channels that encapsulate the algorithms that make up a simulation model. These may govern state-updating events at the individual level, perform global state changes, or trigger cell division. Simulation engines control the scheduling and execution of collections of simulation channels, while a simulation manager coordinates the engines according to one of two scheduling protocols. When the ensemble of cells being simulated reaches a specified maximum size, a procedure is introduced whereby random cells are ejected from the simulation and replaced by newborn cells to keep the sample population size constant but representative in composition. The framework permits recording of population snapshot data and/or cell lineage histories. Use of the framework is demonstrated through validation benchmarks and two case studies based on experiments from the literature.

An accelerated pace of discovery in biological sciences is made possible by a new generation of computational biology and bioinformatics tools. In this dissertation we develop novel computational, analytical, and high performance simulation techniques for biological problems, with applications to the yeast cell division cycle, and to the RNA-Sequencing of the yellow fever mosquito. Cell cycle system evolves stochastic effects when there are a small number of molecules react each other. Consequently, the stochastic effects of the cell cycle are important, and the evolution of cells is best described statistically. Stochastic simulation algorithm (SSA), the standard stochastic method for chemical kinetics, is often slow because it accounts for every individual reaction event. This work develops a stochastic version of a deterministic cell cycle model, in order to capture the stochastic aspects of the evolution of the budding yeast wild-type and mutant strain cells. In order to efficiently run large ensembles to compute statistics of cell evolution, the dissertation investigates parallel simulation strategies, and presents a new probabilistic framework to analyze the performance of dynamic load balancing algorithms. This work also proposes new accelerated stochastic simulation algorithms based on a fully implicit approach and on stochastic Taylor expansions. Next Generation RNA-Sequencing, a high-throughput technology to sequence cDNA in order to get information about a sample's RNA content, is becoming an efficient genomic approach to uncover new genes and to study gene expression and alternative splicing. This dissertation develops efficient algorithms and strategies to find new genes in Aedes aegypti, which is the most important vector of dengue fever and yellow fever. We report the discovery of a large number of new gene transcripts, and the identification and characterization of genes that showed male-biased expression profiles. This basic information may open important avenues to control mosquito borne infectious diseases.
Ph. D.

Fluctuations, or "noise", can play a key role in determining the behaviour of living systems. The molecular-level fluctuations that occur in genetic networks are of particular importance. Here, noisy gene expression can result in genetically identical cells displaying significant variation in phenotype, even in identical environments. This variation can act as a basis for natural selection and provide a fitness benefit to cell populations under stress. This thesis focuses on the development of new conceptual knowledge about how gene expression noise and gene network topology influence drug resistance, as well as new simulation techniques to better understand cell population dynamics. Network topology may at first seem disconnected from expression noise, but genes in a network regulate each other through their expression products. The topology of a genetic network can thus amplify or attenuate noisy inputs from the environment and influence the expression characteristics of genes serving as outputs to the network. The main body of the thesis consists of five chapters: 1. A published review article on the physical basis of cellular individuality. 2. A published article presenting a novel method for simulating the dynamics of cell populations. 3. A chapter on modeling and simulating replicative aging and competition using an object-oriented framework. 4. A published research article establishing that noise in gene expression can facilitate adaptation and drug resistance independent of mutation. 5. An article submitted for publication demonstrating that gene network topology can affect the development of drug resistance. These chapters are preceded by a comprehensive introduction that covers essential concepts and theories relevant to the work presented.

From the developing embryo to the evacuation of football stadiums, the migration and movement of populations of individuals is a vital part of human life. Such movement often occurs in crowded conditions, where the space occupied by each individual impacts on the freedom of others. This thesis aims to analyse and understand the effects of occupied volume (volume exclusion) on the movement of the individual and the population. We consider, as a motivating system, the rearrangement of individuals required to turn a clump of cells into a functioning embryo. Specifically, we consider the migration of cranial neural crest cells in the developing chick embryo. Working closely with experimental collaborators we construct a hybrid model of the system, consisting of a continuum chemoattractant and individual-based cell description and find that multiple cell phenotypes are required for successful migration. In the crowded environment of the migratory system, volume exclusion is highly important and significantly enhances the speed of cell migration in our model, whilst reducing the numbers of individuals that can enter the domain. The developed model is used to make experimental predictions, that are tested in vivo, using cycles of modelling and experimental work to give greater insight into the biological system. Our formulated model is computational, and is thus difficult to analyse whilst considering different parameter regimes. The second part of the thesis is driven by the wish to systematically analyse our model. As such, it concentrates on developing new techniques to derive continuum equations from diffusive and chemotactic individual-based and hybrid models in one and two spatial dimensions with the incorporation of volume exclusion. We demonstrate the accuracy of our techniques under different parameter regimes and using different mechanisms of movement. In particular, we show that our derived continuum equations almost always compare better to data averaged over multiple simulations than the equivalent equations without volume exclusion. Thus we establish that volume exclusion has a substantial effect on the evolution of a migrating population.

Inductive plasmas are simulated by using a one-dimensional particle-in-cell simulation including Monte Carlo collision techniques (pic/mcc). To model inductive heating, a non-uniform radio-frequency (rf) electric field, perpendicular to the electron motion is included into the classical particle-in-cell scheme. The inductive plasma pic simulation is used to confirm recent experimental results that electric double layers can form in current-free plasmas. These results differ from previous experimental or simulation systems where the double layers are driven by a current or by imposed potential differences. The formation of a super-sonic ion beam, resulting from the ions accelerated through the potential drop of the double layer and predicted by the pic simulation is confirmed with nonperturbative laser-induced fluorescence measurements of ion flow. It is shown that at low pressure, where the electron mean free path is of the order of, or greater than the system length, the electron energy distribution function (eedf) is close to Maxwellian, except for its tail which is depleted at energies higher than the plasma potential. Evidence supporting that this depletion is mostly due to the high-energy electrons escaping to the walls is given. ¶ A new hybrid simulation scheme (particle ions and Boltzmann/particle electrons), accounting for non-Maxwellian eedf and self-consistently simulating low-pressure high-density plasmas at low computational cost is proposed. Results obtained with the “improved” hybrid model are in much better agreement with the full pic simulation than the classical non self-consistent hybrid model. This model is used to simulate electronegative plasmas and to provide evidence supporting the fact that propagating double layers may spontaneously form in electronegative plasmas. It is shown that critical parameters of the simulation were very much aligned with critical parameters of the experiment.

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O objetivo principal deste trabalho é adicionar e analisar uma equação que repre senta o volume no modelo dinâmico do ciclo celular de mamíferos proposto por Gérard e Goldbeter (2011). A divisão celular ocorre quando o complexo ciclinaB/Cdk1(quínase dependente de ciclina) é totalmente degradado atingindo um valor mínimo. Neste ponto, a célula é divida em duas novas células filhas e cada uma irá conter a metade do conteúdo citoplasmático da célula mãe. As equações do modelo de base são válidas apenas se o volume celular, onde as reações ocorrem, é constante. Quando o volume celular não é constante, isto é, a taxa de variação do volume em relação ao tempo é explicitamente levada em consideração no modelo matemático, então as equações do modelo original não são mais válidas. Portanto, todas as equações foram modificadas a partir do princípio de conservação das massas para considerar um volume que varia ao longo do tempo. Por meio desta abordagem, o volume celular afeta todas as variáveis do modelo. Dois méto dos diferentes de simulação foram efetuados: determinista e estocástico. Na simulação estocástica, o volume afeta todos os parâmetros do modelo que possuem de alguma forma unidade molar, enquanto que no determinista, ele é incorporado nas equações diferen ciais. Na simulação determinista, as espécies bioquímicas podem estar em unidades de concentração, enquanto na simulação estocástica tais espécies devem ser convertidas para número de moléculas que são diretamente proporcional ao volume celular. Em um esforço para entender a influência da nova equação sobre o modelo uma análise de estabilidade foi feita. Isso esclarece como o novo parâmetro µ, fator de crescimento do volume celular, impacta na estabilidade do ciclo limite do modelo. Para encontrar a solução aproximada do modelo determinista, o método Runge Kutta de quarta ordem foi implementado. Já para o modelo estocástico, o método direto de Gillespie foi usado. Para concluir, um modelo mais preciso, em comparação ao modelo de base, foi desenvolvido ao levar em consideração a influência da taxa de variação do volume celular sobre o ciclo celular.
The main goal of this work is to add and analyse an equation that represents the volume in a dynamical model of the mammalian cell cycle proposed by Gérard and Gold beter (2011). The cell division occurs when the cyclinB/Cdk1 (cyclin-dependent kinase) complex is totally degraded and it reaches a minimum value. At this point, the cell is divided into two newborn daughter cells and each one will contain the half of the cyto plasmic content of the mother cell. The equations of our base model are valid only if the cell volume, where the reactions occur, is constant. Whether the cell volume is not constant, that is, the rate of change of its volume with respect to time is explicitly taken into account in the mathematical model, then the equations of the original model are no longer valid. Therefore, every equations were modified from the mass conservation prin ciple for considering a volume that changes with time. Through this approach, the cell volume affects all model variables. Two different dynamic simulation methods were ac complished: deterministic and stochastic. In the stochastic simulation, the volume affects every model’s parameters which have molar unit, whereas in the deterministic one, it is incorporated into the differential equations. In deterministic simulation, the biochemical species may be in concentration units, while in stochastic simulation such species must be converted to number of molecules which are directly proportional to the cell volume. In an effort to understand the influence of the new equation over the model an stability analysis was performed. This elucidates how the new parameter µ, cell volume growth factor, impacts the stability of the model’s limit cycle. In order to find the approximated solution of the deterministic model, the fourth order Runge Kutta method was implemen ted. As for the stochastic model, the Gillespie’s Direct Method was used. In conclusion, a more precise model, in comparison to the base model, was created for the cell cycle as it now takes into consideration the rate of change of the cell volume.

Durante as últimas décadas, pesquisas em biologia do tumor com a utilização de novas técnicas de biologia molecular produziram informações em profusão, motivando e dando condições para que fossem criados novos modelos matemáticos dedicados à análise de vários aspectos de crescimento e proliferação da população celular. Alguns desses modelos têm sido dedicados à descrição e análise do regime estacionário do processo de desenvolvimento de uma população celular sob condições químicas que se consideram favorecer a aceleração ou desaceleração do crescimento da população de células tumorais. Todavia, a dinâmica temporal do crescimento de uma população de células tumorais ainda não foi analisada nesses trabalhos. Uma das dificuldades é o estabelecimento da interação entre células de múltiplos tipos que sirvam como descrição para essa dinâmica. Nosso trabalho vem preencher essa lacuna e a presente dissertação tem como objetivo a apresentação do modelo, desenvolvido por nós, de simulação da dinâmica do crescimento e proliferação celular do melanoma (câncer de baixa incidência, mas de letalidade extremamente alta) e também dos resultados obtidos através das simulações deste modelo computacional
During the last decades, tumor biology research with the use of new techniques in molecular biology resulted in a profusion of information that have given conditions and motivated the development of new mathematical models dedicated to analyzing various aspects of growth and proliferation of the cell population. Some of these models have been devoted to the description and analysis of the steady state of the development process of a cell population under chemical conditions that, in theory, promote the acceleration or deceleration of the growth of tumor cell population. However, these studies have not yet analyzed the temporal dynamics of growth of a tumor cell population. One of the difficulties is the establishment of the interaction between cells of multiple types that serve as the description for this dynamic. Our work fills this gap and this dissertation aims to present the model, developed by us, to simulate the growth dynamics and cellular proliferation of melanoma (cancer of low incidence but of extremely high lethality) and the results obtained through the simulations of this computational model

The subject of this thesis is the generation of spatial temporal structures in living cells. Specifically, we studied the Min-system in the bacterium Escherichia coli. It consists of the MinC, the MinD, and the MinE proteins, which play an important role in the correct selection of the cell division site. The Min-proteins oscillate between the two cell poles and thereby prevent division at these locations. In this way, E. coli divides at the center, producing two daughter cells of equal size, providing them with the complete genetic patrimony. Our goal is to perform a quantitative study, both theoretical and experimental, in order to reveal the mechanism underlying the Min-oscillations. Experimentally, we characterize theMin-system, measuring the temporal period of the oscillations as a function of the cell length, the time-averaged protein distributions, and the in vivo Min-protein mobility by means of different fluorescence microscopy techniques. Theoretically, we discuss a deterministic description based on the exchange of Minproteins between the cytoplasm and the cytoplasmic membrane and on the aggregation current induced by the interaction between membrane-bound proteins. Oscillatory solutions appear via a dynamic instability of the homogenous protein distributions. Moreover, we perform stochastic simulations based on a microscopic description, whereby the probability for each event is calculated according to the corresponding probability in the master equation. Starting from this microscopic description, we derive Langevin equations for the fluctuating protein densities which correspond to the deterministic equations in the limit of vanishing noise. Stochastic simulations justify this deterministic model, showing that oscillations are resistant to the perturbations induced by the stochastic reactions and diffusion. Predictions and assumptions of our theoretical model are compatible with our experimental findings. Altogether, these results enable us to propose further experiments in order to quantitatively compare the different models proposed so far and to test our model with even higher precision. They also point to the necessity of performing such an analysis through single cell measurements.

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)
Das 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

The subject of this thesis is the generation of spatial temporal structures in living cells. Specifically, we studied the Min-system in the bacterium Escherichia coli. It consists of the MinC, the MinD, and the MinE proteins, which play an important role in the correct selection of the cell division site. The Min-proteins oscillate between the two cell poles and thereby prevent division at these locations. In this way, E. coli divides at the center, producing two daughter cells of equal size, providing them with the complete genetic patrimony. Our goal is to perform a quantitative study, both theoretical and experimental, in order to reveal the mechanism underlying the Min-oscillations. Experimentally, we characterize theMin-system, measuring the temporal period of the oscillations as a function of the cell length, the time-averaged protein distributions, and the in vivo Min-protein mobility by means of different fluorescence microscopy techniques. Theoretically, we discuss a deterministic description based on the exchange of Minproteins between the cytoplasm and the cytoplasmic membrane and on the aggregation current induced by the interaction between membrane-bound proteins. Oscillatory solutions appear via a dynamic instability of the homogenous protein distributions. Moreover, we perform stochastic simulations based on a microscopic description, whereby the probability for each event is calculated according to the corresponding probability in the master equation. Starting from this microscopic description, we derive Langevin equations for the fluctuating protein densities which correspond to the deterministic equations in the limit of vanishing noise. Stochastic simulations justify this deterministic model, showing that oscillations are resistant to the perturbations induced by the stochastic reactions and diffusion. Predictions and assumptions of our theoretical model are compatible with our experimental findings. Altogether, these results enable us to propose further experiments in order to quantitatively compare the different models proposed so far and to test our model with even higher precision. They also point to the necessity of performing such an analysis through single cell measurements.

In recent years, there has been increased awareness that stochasticity in chemical reactions and diffusion of molecules can have significant effects on the outcomes of intracellular processes, particularly given the low copy numbers of many proteins and mRNAs present in a cell. For such molecular species, the number and locations of molecules can provide a more accurate and detailed description than local concentration. In addition to diffusion, drift in the movements of molecules can play a key role in the dynamics of intracellular processes, and can often be modeled as arising from potential fields. Examples of sources of drift include active transport, variations in chemical potential, material heterogeneities in the cytoplasm, and local interactions with subcellular structures. This dissertation presents a new numerical method for simulating the stochastically varying numbers and locations of molecular species undergoing chemical reactions and drift-diffusion. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for reaction-diffusion systems and the Wang—Peskin—Elston lattice discretization of the Fokker—Planck equation that describes drift-diffusion processes in which the drift arises from potential fields. In the FPKMC method, each molecule is enclosed within a "protective domain," either by itself or with a small number of other molecules. To sample when a molecule leaves its protective domain or a reaction occurs, the original FPKMC method relies on analytic solutions of one- and two-body diffusion equations within the protective domains, and therefore cannot be used in situations with non-constant drift. To allow for such drift in our new method (hereafter Dynamic Lattice FPKMC or DL-FPKMC), each molecule undergoes a continuous-time random walk on a lattice within its protective domain, and the lattices change adaptively over time. One of the most commonly used spatial models for stochastic reaction-diffusion systems is the Smoluchowski diffusion-limited reaction (SDLR) model. The DL-FPKMC method generates convergent realizations of an extension of the SDLR model that includes drift from potentials. We present detailed numerical results demonstrating the convergence and accuracy of our method for various types of potentials (smooth, discontinuous, and constant). We also present several illustrative applications of DL-FPKMC, including examples motivated by cell biology.

In this dissertation, we consider a complex biological system known as cortical microtubule (CMT) system, where stochastic dynamics of the components (i.e., the CMTs) are defined in both space and time. CMTs have an inherent spatial dimension of their own, as their length changes over time in addition to their location. As a result of their dynamics in a confined space, they run into and interact with each other according to simple stochastic rules. Over time, CMTs acquire an ordered structure that is achieved without any centralized control beginning with a completely disorganized system. It is also observed that this organization might be distorted, when parameters of dynamicity and interactions change due to genetic mutation or environmental conditions. The main question of interest is to explore the characteristics of this system and the drivers of its self-organization, which is not feasible relying solely on biological experiments. For this, we replicate the system dynamics and interactions using computer simulations. As the simulations successfully mimic the organization seen in plant cells, we conduct an extensive analysis to discover the effects of dynamics and interactions on system characteristics by experimenting with different input parameters. To compare simulation results, we characterize system properties and quantify organization level using metrics based on entropy, average length and number of CMTs in the system. Based on our findings and conjectures from simulations, we develop analytical models for more generalized conclusions and efficient computation of system metrics. As a fist step, we formulate a mean-field model, which we use to derive sufficient conditions for organization to occur in terms of input parameters. Next, considering the parameter ranges that satisfy these conditions, we develop predictive methodologies for estimation of expected average length and number of CMTs over time, using a fluid model, transient analysis, and approximation algorithms tailored to our problem. Overall, we build a comprehensive framework for analysis and control of microtubule organization in plant cells using a wide range of models and methodologies in conjunction. This research also has broader impacts related to the fields of bio-energy, healthcare, and nanotechnology; in addition to its methodological contribution to stochastic modeling of systems with high-level spatial and temporal complexity.

For cell-biological processes, it is the complex interaction of their biochemical components, affected by both stochastic and spatial considerations, that create the overall picture. Formal modeling provides a method to overcome the limits of experimental observation in the wet-lab by moving to the abstract world of the computer. The limits of the abstract world again depend on the expressiveness of the modeling language used to formally describe the system under study. In this thesis, reaction constraints for the pi-calculus are proposed as a language for the stochastic and spatial modeling of cell-biological processes. The goal is to develop a language with sufficient expressive power to model dynamic cell structures, like fusing compartments. To this end, reaction constraints are augmented with two language constructs: priority and a global imperative store, yielding two different modeling languages, including non-deterministic and stochastic semantics. By several modeling examples, e.g. of Euglena's phototaxis, and extensive expressiveness studies, e.g. an encoding of the spatial modeling language BioAmbients, including a prove of its correctness, the usefulness of reaction constraints, priority, and a global imperative store for the modeling of cell-biological processes is shown. Thereby, besides dynamic cell structures, different modeling styles, e.g. individual-based vs. population-based modeling, and different abstraction levels, as e.g. provided by reaction kinetics following the law of Mass action or the Michaelis-Menten theory, are considered.

Inductive plasmas are simulated by using a one-dimensional particle-in-cell simulation including Monte Carlo collision techniques (pic/mcc). To model inductive heating, a non-uniform radio-frequency (rf) electric field, perpendicular to the electron motion is included into the classical particle-in-cell scheme. The inductive plasma pic simulation is used to confirm recent experimental results that electric double layers can form in current-free plasmas. These results differ from previous experimental or simulation systems where the double layers are driven by a current or by imposed potential differences. The formation of a super-sonic ion beam, resulting from the ions accelerated through the potential drop of the double layer and predicted by the pic simulation is confirmed with nonperturbative laser-induced fluorescence measurements of ion flow. It is shown that at low pressure, where the electron mean free path is of the order of, or greater than the system length, the electron energy distribution function (eedf) is close to Maxwellian, except for its tail which is depleted at energies higher than the plasma potential. Evidence supporting that this depletion is mostly due to the high-energy electrons escaping to the walls is given. ¶ A new hybrid simulation scheme (particle ions and Boltzmann/particle electrons), accounting for non-Maxwellian eedf and self-consistently simulating low-pressure high-density plasmas at low computational cost is proposed. Results obtained with the “improved” hybrid model are in much better agreement with the full pic simulation than the classical non self-consistent hybrid model. This model is used to simulate electronegative plasmas and to provide evidence supporting the fact that propagating double layers may spontaneously form in electronegative plasmas. It is shown that critical parameters of the simulation were very much aligned with critical parameters of the experiment.

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 66
Das 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 66