Benjamin M. Friedrich

High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory.

SUMMARY 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.

Chemotaxis of sperm cells

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.

Reactivating head regrowth in a regeneration-deficient planarian species

Species capable of regenerating lost body parts occur throughout the animal kingdom, yet close relatives are often regeneration incompetent. Why in the face of ‘survival of the fittest’ some animals regenerate but others do not remains a fascinating question. Planarian flatworms are well known and studied for their ability to regenerate from minute tissue pieces, yet species with limited regeneration abilities have been described even amongst planarians. Here we report the characterization of the regeneration defect in the planarian Dendrocoelum lacteum and its successful rescue. Tissue fragments cut from the posterior half of the body of this species are unable to regenerate a head and ultimately die. We find that this defect originates during the early stages of head specification, which require inhibition of canonical Wnt signalling in other planarian species. Notably, RNA interference (RNAi)-mediated knockdown of Dlac-β-catenin-1, the Wnt signal transducer, restored the regeneration of fully functional heads on tail pieces, rescuing D. lacteum’s regeneration defect. Our results demonstrate the utility of comparative studies towards the reactivation of regenerative abilities in regeneration-deficient animals. Furthermore, the availability of D. lacteum as a regeneration-impaired planarian model species provides a first step towards elucidating the evolutionary mechanisms that ultimately determine why some animals regenerate and others do not.

The rate of change in Ca2+ concentration controls sperm chemotaxis

Sperm navigate in a chemoattractant gradient by translating changes in intracellular calcium concentration over time into changes in curvature of the swimming path.

The computational sperm cell

Sperm are guided to the egg by a gradient of chemical attractants - a process called chemotaxis. The binding of the chemoattractant to receptors on the surface of the flagellum triggers a cascade of signaling events that eventually lead to an influx of Ca(2+) ions. Based on these Ca(2+) surges, which control the waveform of the flagellar beat, sperm adjust their swimming path toward the egg. In past years, many components of chemotactic signaling have been identified. Moreover, kinetic spectroscopy and imaging techniques unraveled the sequence of cellular events controlling swimming behavior. During navigation in a chemical gradient, sperm perform a surprising variety of computational operations. Here we discuss theoretical concepts of navigation strategies and the cellular underpinnings.

Steering chiral swimmers along noisy helical paths

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 and discuss the chemotaxis index.

Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga

Significance The eukaryotic flagellum is a best-seller of nature: These slender cell appendages propel sperm and many other microswimmers, including disease-causing protists. In mammalian airways or the oviduct, collections of flagella beat in synchrony to pump fluids efficiently. Flagellar synchronization was proposed to rely on mechanical feedback by hydrodynamic forces, but the details are not well understood. Here, we used theory and experiment to elucidate a mechanism of synchronization in the model organism Chlamydomonas, a green algal cell that swims with two flagella like a breaststroke swimmer. Our analysis shows how synchronization arises by a coupling of swimming and flagellar beating and characterizes an exemplary force–velocity relationship of the flagellar beat. The unicellular green alga Chlamydomonas swims with two flagella that can synchronize their beat. Synchronized beating is required to swim both fast and straight. A long-standing hypothesis proposes that synchronization of flagella results from hydrodynamic coupling, but the details are not understood. Here, we present realistic hydrodynamic computations and high-speed tracking experiments of swimming cells that show how a perturbation from the synchronized state causes rotational motion of the cell body. This rotation feeds back on the flagellar dynamics via hydrodynamic friction forces and rapidly restores the synchronized state in our theory. We calculate that this “cell-body rocking” provides the dominant contribution to synchronization in swimming cells, whereas direct hydrodynamic interactions between the flagella contribute negligibly. We experimentally confirmed the two-way coupling between flagellar beating and cell-body rocking predicted by our theory.

Striated Acto-Myosin Fibers Can Reorganize and Register in Response to Elastic Interactions with the Matrix

The remarkable striation of muscle has fascinated many for centuries. In developing muscle cells, as well as in many adherent, nonmuscle cell types, striated, stress fiberlike structures with sarcomere-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, to our knowledge, 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 interfiber registry. Our physical theory predicts a dependence of interfiber 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.

Scaling and Regeneration of Self-Organized Patterns

Biological patterns generated during development and regeneration often scale with organism size. Some organisms, e.g., flatworms, can regenerate a rescaled body plan from tissue fragments of varying sizes. Inspired by these examples, we introduce a generalization of Turing patterns that is self-organized and self-scaling. A feedback loop involving diffusing expander molecules regulates the reaction rates of a Turing system, thereby adjusting pattern length scales proportional to system size. Our model captures essential features of body plan regeneration in flatworms as observed in experiments.

Active Phase and Amplitude Fluctuations of Flagellar Beating

The eukaryotic flagellum beats periodically, driven by the oscillatory dynamics of molecular motors, to propel cells and pump fluids. Small but perceivable fluctuations in the beat of individual flagella have physiological implications for synchronization in collections of flagella as well as for hydrodynamic interactions between flagellated swimmers. Here, we characterize phase and amplitude fluctuations of flagellar bending waves using shape mode analysis and limit-cycle reconstruction. We report a quality factor of flagellar oscillations Q = 38.0 ± 16.7 (mean ± s.e.). Our analysis shows that flagellar fluctuations are dominantly of active origin. Using a minimal model of collective motor oscillations, we demonstrate how the stochastic dynamics of individual motors can give rise to active small-number fluctuations in motor-cytoskeleton systems.