Benjamin M Friedrich

Abstract: Adherent cells exert active forces and elastically deform their substrate. We consider a simple model geometry of a spread cell adhered to a thick substrate layer. We show how anisotropic cell shapes and active cell contractility induce cytoskeletal shear within the cell in a substrate-stiness dependent manner. This cytoskeletal shear represents a possible mechanical guidance cue for cell force polarization, and could, for example, trigger initial nematic alignment of nascent stress bers at early stages of cytoskeletal organization. Cell induced substrate strains propagate a depth into the substrate that is comparable to the linear dimension of the spread cell. As a consequence, cellular strains depend on the lateral dimensions of the spread cell. We employ Fourier techniques and a mean-eld coupling approximation, which allows both for analytical progress and qualitative insight.

Abstract: In many bacterial species, the protein FtsZ forms a cytoskeletal ring that marks the future division site and scaolds the division machinery. In rodshaped bacteria, most frequently membrane-attached FtsZ rings or ring fragments are reported and occasionally helices. By contrast, axial FtsZ clusters have never been reported. In this paper, we investigate theoretically how dynamic FtsZ aggregates align in rod-shaped bacteria. We study systematically dierent physical mechanisms that aect the alignment of FtsZ polymers using a computational model that relies on autocatalytic aggregation of FtsZ laments at the membrane. Our study identies a general tool kit of physical and geometrical mechanisms by which rod-shaped cells align biopolymer aggregates. Our analysis compares the relative impact of each mechanism on the circumferential alignment of FtsZ as observed in rod-shaped bacteria. We determine spontaneous curvature of FtsZ polymers and axial connement of FtsZ on the membrane as the strongest factors. Including Min oscillations in our model, we nd that these stabilize axial and helical clusters on short time scales, but promote the formation of an FtsZ ring at the cell middle at longer times. This eect could provide an explanation to the long standing puzzle of transiently observed oscillating FtsZ helices in E.coli cells prior to cell division.

Abstract:

Abstract: During chemotaxis and phototaxis, sperm, algae, marine zooplankton, and other microswimmers move on helical paths or drifting circles by rhythmically bending cell protrusions called motile cilia or flagella. Sperm of marine invertebrates navigate in a chemoattractant gradient by adjusting the flagellar waveform and, thereby, the swimming path. The waveform is periodically modulated by Ca2+ oscillations. How Ca2+ signals elicit steering responses and shape the path is unknown. We unveil the signal transfer between the changes in intracellular Ca2+ concentration ([Ca2+]i) and path curvature (κ). We show that κ is modulated by the time derivative d[Ca2+]i/dt rather than the absolute [Ca2+]i. Furthermore, simulation of swimming paths using various Ca2+ waveforms reproduces the wealth of swimming paths observed for sperm of marine invertebrates. We propose a cellular mechanism for a chemical differentiator that computes a time derivative. The cytoskeleton of cilia, the axoneme, is highly conserved. Thus, motile ciliated cells in general might use a similar cellular computation to translate changes of [Ca2+]i into motion.

Abstract: Contractile function of striated muscle cells depends crucially on the almost crystalline order of actin and myosin laments in myobrils, but the physical mechanisms that lead to myobril assembly remains ill-dened. Passive diusive sorting of actin laments into sarcomeric order is kinetically impossible, suggesting a pivotal role of active processes in sarcomeric pattern formation. Using a one-dimensional computational model of an initially unstriated actin bundle, we show that actin lament treadmilling in the presence of processive plus-end crosslinking provides a simple and robust mechanism for the polarity sorting of actin filaments as well as for the correct localization of myosin filaments. We propose that the coalescence of crosslinked actin clusters could be key for sarcomeric pattern formation. In our simulations, sarcomere spacing is set by filament length prompting tight length control already at early stages of pattern formation. The proposed mechanism could be generic and apply both to premyobrils and nascent myobrils in developing muscle cells as well as possibly to striated stress-bers in non-muscle cells.

Abstract: Angetrieben von regelmä�igen Wellenbewegungen ihrer Gei�el schwimmen Seeigelspermien entlang von gekrümmten Schwimmbahnen in Spirallinien. Dies erlaubt es den Spermien, eine chemische Fährte aufzunehmen und die Eizelle zu orten. Dabei verarbeitet das Spermium ein chemisches Signal der Eizelle mit einer ähnlichen �Hardware� wie sie Nervenzellen verwenden. Entsprechend steuert das Spermium seine Schwimmbahn. Ein einfaches geometrisches Prinzip ermöglicht eine robuste Navigation. Auch viele andere Mikroschwimmer verwenden es.

Abstract: The remarkable striation of muscle has fascinated many for centuries. In developing muscle cells, as well as in many adherent, non-muscle cell types, striated, stress fiber-like structures with sarcomer-periodicity tend to register: Based on several studies, neighboring, parallel fibers at the basal membrane of cultured cells establish registry of their respective periodic sarcomeric architecture, but the mechanism has not yet been identified. Here, we propose for cells plated on an elastic substrate or adhered to a neighboring cell, that acto-myosin contractility in striated fibers close to the basal membrane induces substrate strain that gives rise to an elastic interaction between neighboring striated fibers, which in turn favors inter-fiber registry. Our physical theory predicts a dependence of inter-fiber registry on externally controllable elastic properties of the substrate. In developing muscle cells, registry of striated fibers (premyofibrils and nascent myofibrils) has been suggested as one major pathway of myofibrillogenesis, where it precedes the fusion of neighboring fibers. This suggests a mechanical basis for the optimal myofibrillogenesis on muscle-mimetic elastic substrates that was recently observed by several groups in cultures of mouse, human, and chick derived muscle cells.

Abstract: We predict spontaneous nematic order in an ensemble of active force generators with elastic interactions as a minimal model for early nematic alignment of short stress �bers in non-motile, adhered cells. Mean-�eld theory is formally equivalent to Maier-Saupe theory for a nematic liquid. However, the elastic interactions are long-ranged (and thus depend on cell shape and matrix elasticity) and originate in cell activity. Depending on the density of force generators, we �nd two regimes of cellular rigidity sensing for which orientational, nematic order of stress �bers depends on matrix rigidity either in a step-like manner or with a maximum at an optimal rigidity.

Abstract: The shape of the flagellar beat determines the path along which a sperm cell swims. If the flagellum bends periodically about a curved mean shape then the sperm will follow a path with non-zero curvature. To test a simple hydrodynamic theory of flagellar propulsion known as resistive force theory, we conducted high-precision measurements of the head and flagellum motions during circular swimming of bull spermatozoa near a surface. We found that the fine structure of sperm swimming represented by the rapid wiggling of the sperm head around an averaged path is, to high accuracy, accounted for by resistive force theory and results from balancing forces and torques generated by the beating flagellum. We determined the anisotropy ratio between the normal and tangential hydrodynamic friction coefficients of the flagellum to be 1.81±0.07 (mean±s.d.). On time scales longer than the flagellar beat cycle, sperm cells followed circular paths of non-zero curvature. Our data show that path curvature is approximately equal to twice the average curvature of the flagellum, consistent with quantitative predictions of resistive force theory. Hence, this theory accurately predicts the complex trajectories of sperm cells from the detailed shape of their flagellar beat across different time scales.

Abstract: Chemotaxis along helical paths towards a target releasing a chemoattractant is found in sperm cells and many microorganisms. We discuss the stochastic differential geometry of the noisy helical swimming path of a chiral swimmer. A chiral swimmer equipped with a simple feedback system can navigate in a concentration gradient of chemoattractant. We derive an effective equation for the alignment of helical paths with a concentration gradient which is related to the alignment of a dipole in an external field. We discuss the chemotaxis index in the presence of fluctuations.

Abstract: Optimal search strategies and their implementations in biological systems are a subject of active research. Here we study a search problem which is motivated by the hunt of sperm cells for the egg. We ask for the probability for an active swimmer to find a target under the condition that the swimmer starts at a certain distance from the target. We find that success probability is maximal for a certain level of fluctuations characterized by the persistence length of the swimming path of the swimmer. We derive a scaling law for the optimal persistence length as a function of the initial target distance and search time by mapping the search on a polymer physics problem.

Abstract: Biological systems such as single cells must function in the presence of fluctuations. It has been shown in a two-dimensional experimental setup that sea urchin sperm cells move toward a source of chemoattractant along planar trochoidal swimming paths, i.e. drifting circles. In these experiments, a pronounced variability of the swimming paths is observed. We present a theoretical description of sperm chemotaxis in two dimensions which takes fluctuations into account. We derive a coarse-grained theory of stochastic sperm swimming paths in a concentration field of chemoattractant. Fluctuations enter as multiplicative noise in the equations for the sperm swimming path. We discuss the stochastic properties of sperm swimming and predict a concentration-dependence of the effective diffusion constant of sperm swimming which could be tested in experiments.

Abstract: We develop a theoretical description of sperm chemotaxis. Sperm cells of many species are guided to the egg by chemoattractants, a process called chemotaxis. Motor proteins in the flagellum of the sperm generate a regular beat of the flagellum, which propels the sperm in a fluid. In the absence of a chemoattractant, sperm swim in circles in two dimensions and along helical paths in three dimensions. Chemoattractants stimulate a signaling system in the flagellum, which regulates the motors to control sperm swimming. Our theoretical description of sperm chemotaxis in two and three dimensions is based on a generic signaling module that regulates the curvature and torsion of the swimming path. In the presence of a chemoattractant, swimming paths are drifting circles in two dimensions and deformed helices in three dimensions. The swimming paths can be described by a dynamical system that exhibits different dynamic regimes, which correspond to different chemotactic behaviours. We conclude that sampling a concentration field of chemoattractant along circular and helical swimming paths is a robust strategy for chemotaxis that works reliably for a vast range of parameters.

Abstract: Filaments of bacterial flagella are perfect tubular stackings polymerized out of just one kind of building block: the flagellin protein. Surprisingly, they do not form straight tubes, but exhibit a symmetry-breaking coiling into helical shapes which is essential for their biological function as cell ``propeller''. The co-existence of two conformational states for flagellin within the filament is believed to be responsible for the helical shapes by producing local misfit which results in curvature and twist. In this paper, we present a coarse-grained description with an elastic energy functional for the filament derived from its microscopic structure. By minimising this functional we can answer the question of spatial distribution of flagellin states which is crucial for the observed coupling of curvature and twist. Our approach extends a classical theory of Calladine, which had to assume this spatial distribution from the outset.

Abstract:

Abstract: It is known that the algebraic \deRham cohomology group $\hDR{i}(X_0/\Q)$ of a nonsingular variety $X_0/\Q$ has the same rank as the rational singular cohomology group $\h^i\sing(\Xh;\Q)$ of the complex manifold $\Xh$ associated to the base change $X_0\times_{\Q}\C$. However, we do not have a natural isomorphism $\hDR{i}(X_0/\Q)\iso\h^i\sing(\Xh;\Q)$. Any choice of such an isomorphism produces certain integrals, so called periods, which reveal valuable information about $X_0$. The aim of this thesis is to explain these classical facts in detail. Based on an approach of Kontsevich, different definitions of a period are compared and their properties discussed. Finally, the theory is applied to some examples. These examples include a representation of $\zeta(2)$ as a period and a variation of mixed Hodge structures used by Goncharov.

Abstract: Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification of rational double points in terms of Dynkin diagrams.

Abstract: Sperm cells are guided to the egg by chemoattractants in many species. The sperm cells are propelled in a liquid by the regular beat of their flagellum. In the presence of a concentration gradient of a chemoattractant, they can steer upwards the concentration gradient, a process called chemotaxis. Eggs release chemoattractants to guide the sperm cells to the egg. Sperm chemotaxis is best studied experimentally in the sea urchin. There, specic receptors in the flagellar membrane of the sperm cells are activated upon binding of chemoattractant molecules and trigger a signaling cascade which ultimately changes the activity of the molecular motors which drive the flagellar beat and result in a swimming response. Sea urchin sperm cells swim along circular and helical paths. Sperm cells of the sea urchin and several other species swim along helical paths far from boundary surfaces in the absence of chemoattractant. In a two-dimensional experimental geometry, sperm swimming paths are planar circles. The non-zero curvature of their swimming paths is a direct consequence of an asymmetry of their flagellar beat. In a concentration gradient of chemoattractant, sperm swimming path are drifting circles in two dimensions and bend helices in three dimensions. What is the working mechanism of sperm chemotaxis? In this thesis, we develop a theoretical description of sperm chemotaxis which can be subsumed as follows: While swimming along an approximately circular path in a concentration gradient a sperm cell traces a periodic concentration stimulus from the concentration field that has the frequency of circular swimming. The chemotactic signaling system processes this stimulus and causes a periodic modulation of the curvature of the swimming path which then gives rise to a swimming path which is a drifting circle. The relative direction of the drift with respect to the gradient direction is determined by the phase shift between the stimulus and the curvature oscillations. This picture is in perfect agreement with recent experimental ndings. The mechanism is more general and also works in three dimensions for swimming along helical paths. Our results. Our theoretical description of sperm chemotaxis claries the concepts underlying sperm chemotaxis. In particular, we derive the role of internal timing of the chemotactic signaling system for sperm chemotaxis. We conclude that sampling a concentration eld along circular and helical paths is a robust strategy for chemotaxis that does not require fine-tuning of parameters and which works reliable also in the presence of fluctuations. In a last chapter of this thesis, we discuss sperm chemotaxis in the more general context of an abstract search problem.