Showing papers by "Frank Jülicher published in 2019"
TL;DR: The physics of phase separation and emulsions are discussed and it is shown how the concepts that govern such phenomena can be extended to capture the physics of active emulsion.
Abstract: Phase separating systems that are maintained away from thermodynamic equilibrium via molecular processes represent a class of active systems, which we call active emulsions. These systems are driven by external energy input, for example provided by an external fuel reservoir. The external energy input gives rise to novel phenomena that are not present in passive systems. For instance, concentration gradients can spatially organise emulsions and cause novel droplet size distributions. Another example are active droplets that are subject to chemical reactions such that their nucleation and size can be controlled, and they can divide spontaneously. In this review, we discuss the physics of phase separation and emulsions and show how the concepts that govern such phenomena can be extended to capture the physics of active emulsions. This physics is relevant to the spatial organisation of the biochemistry in living cells, for the development of novel applications in chemical engineering and models for the origin of life.
179 citations
TL;DR: It is shown that p120-dependent mechanosensitive E-cadherin turnover regulates viscoelastic behavior of epithelial tissues, which implies a lower viscosity of wing epithelium.
116 citations
TL;DR: This work studies the fully self-organized shape dynamics using the theory of active fluids on deforming surfaces and develops a numerical approach to solve the corresponding force and torque balance equations.
Abstract: Mechanochemical processes in thin biological structures, such as the cellular cortex or epithelial sheets, play a key role during the morphogenesis of cells and tissues. In particular, they are responsible for the dynamical organization of active stresses that lead to flows and deformations of the material. Consequently, advective transport redistributes force-generating molecules and thereby contributes to a complex mechanochemical feedback loop. It has been shown in fixed geometries that this mechanism enables patterning, but the interplay of these processes with shape changes of the material remains to be explored. In this work, we study the fully self-organized shape dynamics using the theory of active fluids on deforming surfaces and develop a numerical approach to solve the corresponding force and torque balance equations. We describe the spontaneous generation of nontrivial surface shapes, shape oscillations, and directed surface flows that resemble peristaltic waves from self-organized, mechanochemical processes on the deforming surface. Our approach provides opportunities to explore the dynamics of self-organized active surfaces and can help to understand the role of shape as an integral element of the mechanochemical organization of morphogenetic processes.
109 citations
TL;DR: A study of how single C. elegans cells establish the polarity required for cell division reveals a general principle for pattern formation in living systems controlled by biochemical cues, including a general criterion for controlling biological pattern-forming systems.
Abstract: Spontaneous pattern formation in Turing systems relies on feedback. But patterns in cells and tissues seldom form spontaneously—instead they are controlled by regulatory biochemical interactions that provide molecular guiding cues. The relationship between these guiding cues and feedback in controlled biological pattern formation remains unclear. Here, we explore this relationship during cell-polarity establishment in the one-cell-stage Caenorhabditis elegans embryo. We quantify the strength of two feedback systems that operate during polarity establishment: feedback between polarity proteins and the actomyosin cortex, and mutual antagonism among polarity proteins. We characterize how these feedback systems are modulated by guiding cues from the centrosome, an organelle regulating cell cycle progression. By coupling a mass-conserved Turing-like reaction–diffusion system for polarity proteins to an active-gel description of the actomyosin cortex, we reveal a transition point beyond which feedback ensures self-organized polarization, even when cues are removed. Notably, the system switches from a guide-dominated to a feedback-dominated regime well beyond this transition point, which ensures robustness. Together, these results reveal a general criterion for controlling biological pattern-forming systems: feedback remains subcritical to avoid unstable behaviour, and molecular guiding cues drive the system beyond a transition point for pattern formation. A study of how single C. elegans cells establish the polarity required for cell division reveals a general principle for pattern formation in living systems controlled by biochemical cues.
99 citations
TL;DR: It is shown that liquid droplets can act as fast and effective buffers for gene expression noise and suggest a novel role of phase separation in biological information processing.
Abstract: A central problem in cellular control is how cells cope with the inherent noise in gene expression. Although transcriptional and posttranscriptional feedback mechanisms can suppress noise, they are often slow, and cannot explain how cells buffer acute fluctuations. Here, by using a physical model that links fluctuations in protein concentration to the theory of phase separation, we show that liquid droplets can act as fast and effective buffers for gene expression noise. We confirm our theory experimentally using an engineered phase separating protein that forms liquid-like compartments in mammalian cells. These data suggest a novel role of phase separation in biological information processing.
80 citations
TL;DR: The results reveal the physiological origins of Kleiber’s law in planarians and have general implications for understanding a fundamental scaling law in biology.
Abstract: Kleiber's law, or the 3/4 -power law scaling of the metabolic rate with body mass, is considered one of the few quantitative laws in biology, yet its physiological basis remains unknown. Here, we report Kleiber's law scaling in the planarian Schmidtea mediterranea. Its reversible and life history-independent changes in adult body mass over 3 orders of magnitude reveal that Kleiber's law does not emerge from the size-dependent decrease in cellular metabolic rate, but from a size-dependent increase in mass per cell. Through a combination of experiment and theoretical analysis of the organismal energy balance, we further show that the mass allometry is caused by body size dependent energy storage. Our results reveal the physiological origins of Kleiber's law in planarians and have general implications for understanding a fundamental scaling law in biology.
56 citations
TL;DR: This work computationally reconstructed 3D tissue geometry from microscopy images of mouse liver tissue and analyzed it applying soft-condensed-matter-physics concepts to suggest that bi-directional communication between hepatocytes and sinusoids underlies the self-organization of liver tissue.
Abstract: Functional tissue architecture originates by self-assembly of distinct cell types, following tissue-specific rules of cell-cell interactions. In the liver, a structural model of the lobule was pioneered by Elias in 1949. This model, however, is in contrast with the apparent random 3D arrangement of hepatocytes. Since then, no significant progress has been made to derive the organizing principles of liver tissue. To solve this outstanding problem, we computationally reconstructed 3D tissue geometry from microscopy images of mouse liver tissue and analyzed it applying soft-condensed-matter-physics concepts. Surprisingly, analysis of the spatial organization of cell polarity revealed that hepatocytes are not randomly oriented but follow a long-range liquid-crystal order. This does not depend exclusively on hepatocytes receiving instructive signals by endothelial cells, since silencing Integrin-β1 disrupted both liquid-crystal order and organization of the sinusoidal network. Our results suggest that bi-directional communication between hepatocytes and sinusoids underlies the self-organization of liver tissue.
40 citations
TL;DR: In this article, a model of the self-organization of the cell cortex is presented based on a hydrodynamic theory of curved active surfaces, where active stresses on this surface are regulated by diffusing molecular species.
Abstract: The cell cortex, a thin film of active material assembled below the cell membrane, plays a key role in cellular symmetry-breaking processes such as cell polarity establishment and cell division. Here, we present a minimal model of the self-organization of the cell cortex that is based on a hydrodynamic theory of curved active surfaces. Active stresses on this surface are regulated by a diffusing molecular species. We show that coupling of the active surface to a passive bulk fluid enables spontaneous polarization and the formation of a contractile ring on the surface via mechanochemical instabilities. We discuss the role of external fields in guiding such pattern formation. Our work reveals that key features of cellular symmetry breaking and cell division can emerge in a minimal model via general dynamic instabilities.
36 citations
TL;DR: In this article, the experimental average values of the minima of stochastic entropy production lie above -kB, where kB is the Boltzmann constant, in agreement with recent theoretical predictions for nonequilibrium steady states.
Abstract: We experimentally study negative fluctuations of stochastic entropy production in an electronic double dot operating in nonequilibrium steady-state conditions. We record millions of random electron tunneling events at different bias points, thus collecting extensive statistics. We show that for all bias voltages, the experimental average values of the minima of stochastic entropy production lie above -kB, where kB is the Boltzmann constant, in agreement with recent theoretical predictions for nonequilibrium steady states. Furthermore, we also demonstrate that the experimental cumulative distribution of the entropy production minima is bounded, at all times and for all bias voltages, by a universal expression predicted by the theory. We also extend our theory by deriving a general bound for the average value of the maximum heat absorbed by a mesoscopic system from the environment and compare this result with experimental data. Finally, we show by numerical simulations that these results are not necessarily valid under nonstationary conditions. (Less)
29 citations
TL;DR: It is shown that fluid pumping and tissue flexoelectricity play a crucial role in lumen formation, and the large variety of long-time states that are accessible for the cell aggregate and its lumen are explored.
Abstract: We discuss the physical mechanisms that promote or suppress the nucleation of a fluid-filled lumen inside a cell assembly or a tissue. We discuss lumen formation in a continuum theory of tissue material properties in which the tissue is described as a 2-fluid system to account for its permeation by the interstitial fluid, and we include fluid pumping as well as active electric effects. Considering a spherical geometry and a polarized tissue, our work shows that fluid pumping and tissue flexoelectricity play a crucial role in lumen formation. We furthermore explore the large variety of long-time states that are accessible for the cell aggregate and its lumen. Our work reveals a role of the coupling of mechanical, electrical, and hydraulic phenomena in tissue lumen formation.
29 citations
TL;DR: The amount of dimeric and polymeric tubulin at mitotic centrosomes in C. elegans is measured to suggest that centrosomal microtubule nucleation may be driven in part by concentrating tubulin.
Abstract: During mitosis, the centrosome expands its capacity to nucleate microtubules. Understanding the mechanisms of centrosomal microtubule nucleation is, however, constrained by a lack of knowledge of the amount of soluble and polymeric tubulin at mitotic centrosomes. Here we combined light microscopy and serial-section electron tomography to measure the amount of dimeric and polymeric tubulin at mitotic centrosomes in early C. elegans embryos. We show that a C. elegans one-cell stage centrosome at metaphase contains >10,000 microtubules with a total polymer concentration of 230 µM. Centrosomes concentrate soluble α/β tubulin by about 10-fold over the cytoplasm, reaching peak values of 470 µM, giving a combined total monomer and polymer tubulin concentration at centrosomes of up to 660 µM. These findings support in vitro data suggesting that microtubule nucleation in C. elegans centrosomes is driven in part by concentrating soluble tubulin.
TL;DR: In this article, the second law of thermodynamics at stopping times has been shown to imply a bound on the average amount of heat and work a system can extract from its environment when stopped at a random time.
Abstract: A stopping time $T$ is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation $\langle e^{-S_{\rm tot}(T)}\rangle=1$ for the stochastic entropy production $S_{\rm tot}$ in a stationary physical system at stochastic stopping times $T$. This fluctuation relation implies the law $\langle S_{\rm tot}(T)\rangle\geq 0$, which states that it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time, and which we call the second law of thermodynamics at stopping times. This law implies bounds on the average amount of heat and work a system can extract from its environment when stopped at a random time. Furthermore, the integral fluctuation relation implies that certain fluctuations of entropy production are universal or are bounded by universal functions. These universal properties descend from the integral fluctuation relation by selecting appropriate stopping times: for example, when $T$ is a first-passage time for entropy production, then we obtain a bound on the statistics of negative records of entropy production. We illustrate these results on simple models of nonequilibrium systems described by Langevin equations and reveal two interesting phenomena. First, we demonstrate that isothermal mesoscopic systems can extract on average heat from their environment when stopped at a cleverly chosen moment and the second law at stopping times provides a bound on the average extracted heat. Second, we demonstrate that the average efficiency at stopping times of an autonomous stochastic heat engines, such as Feymann's ratchet, can be larger than the Carnot efficiency and the second law of thermodynamics at stopping times provides a bound on the average efficiency at stopping times.
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TL;DR: It is shown that coupling of the active surface to a passive bulk fluid enables spontaneous polarization and the formation of a contractile ring on the surface via mechanochemical instabilities.
Abstract: The cell cortex, a thin film of active material assembled below the cell membrane, plays a key role in cellular symmetry breaking processes such as cell polarity establishment and cell division. Here, we present a minimal model of the self-organization of the cell cortex that is based on a hydrodynamic theory of curved active surfaces. Active stresses on this surface are regulated by a diffusing molecular species. We show that coupling of the active surface to a passive bulk fluid enables spontaneous polarization and the formation of a contractile ring on the surface via mechano-chemical instabilities. We discuss the role of external fields in guiding such pattern formation. Our work reveals that key features of cellular symmetry breaking and cell division can emerge in a minimal model via general dynamic instabilities.
TL;DR: In this article, the second law of thermodynamics at stopping times has been shown to imply a bound on the average amount of heat and work a system can extract from its environment when stopped at a random time.
Abstract: A stopping time $T$ is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation $\langle e^{-S_{\rm tot}(T)}\rangle=1$ for the stochastic entropy production $S_{\rm tot}$ in a stationary physical system at stochastic stopping times $T$. This fluctuation relation implies the law $\langle S_{\rm tot}(T)\rangle\geq 0$, which states that it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time, and which we call the second law of thermodynamics at stopping times. This law implies bounds on the average amount of heat and work a system can extract from its environment when stopped at a random time. Furthermore, the integral fluctuation relation implies that certain fluctuations of entropy production are universal or are bounded by universal functions. These universal properties descend from the integral fluctuation relation by selecting appropriate stopping times: for example, when $T$ is a first-passage time for entropy production, then we obtain a bound on the statistics of negative records of entropy production. We illustrate these results on simple models of nonequilibrium systems described by Langevin equations and reveal two interesting phenomena. First, we demonstrate that isothermal mesoscopic systems can extract on average heat from their environment when stopped at a cleverly chosen moment and the second law at stopping times provides a bound on the average extracted heat. Second, we demonstrate that the average efficiency at stopping times of an autonomous stochastic heat engines, such as Feymann's ratchet, can be larger than the Carnot efficiency and the second law of thermodynamics at stopping times provides a bound on the average efficiency at stopping times.
TL;DR: It is found that finite thickness tissue slabs exist only in a restricted region of phase space and that relatively modest electric fields or imposed external flows can induce either proliferation or death.
TL;DR: In this article, the authors derived exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks.
Abstract: We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. Our results generalize the infimum law for entropy production and reveal a symmetry of the distribution of its maxima and minima. We also show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Mar{c}enko-Pastur distribution of random-matrix theory. Using this result, we obtain estimates for the extreme-value statistics of stochastic transport from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results by numerical simulations of stochastic models of molecular motors and discuss as illustrative example our theory in the context of sports.
TL;DR: Stochastic transport properties of colloidal beads in antiparallel networks of overlapping actin filaments are investigated using micropatterns of actin polymerization in vitro, finding that beads coated with myosin motors sensed the net polarity of the actin network, resulting in active bead positioning to regions of neutral polarity with a precision depending on the motor type.
Abstract: Cytoskeletal filaments assemble into dense parallel, antiparallel, or disordered networks, providing a complex environment for active cargo transport and positioning by molecular motors. The interplay between the network architecture and intrinsic motor properties clearly affects transport properties but remains poorly understood. Here, by using surface micropatterns of actin polymerization, we investigate stochastic transport properties of colloidal beads in antiparallel networks of overlapping actin filaments. We found that 200-nm beads coated with myosin Va motors displayed directed movements toward positions where the net polarity of the actin network vanished, accumulating there. The bead distribution was dictated by the spatial profiles of local bead velocity and diffusion coefficient, indicating that a diffusion-drift process was at work. Remarkably, beads coated with heavy-mero-myosin II motors showed a similar behavior. However, although velocity gradients were steeper with myosin II, the much larger bead diffusion observed with this motor resulted in less precise positioning. Our observations are well described by a 3-state model, in which active beads locally sense the net polarity of the network by frequently detaching from and reattaching to the filaments. A stochastic sequence of processive runs and diffusive searches results in a biased random walk. The precision of bead positioning is set by the gradient of net actin polarity in the network and by the run length of the cargo in an attached state. Our results unveiled physical rules for cargo transport and positioning in networks of mixed polarity.
TL;DR: Light microscopy and serial-section electron tomography are combined to measure the amount of dimer and polymer at mitotic centrosomes in early C. elegans embryos and support in vitro data suggesting that microtubule nucleation in C. aristans centrosome is driven in part by concentrating soluble tubulin.
Abstract: During mitosis, the centrosome expands its capacity to nucleate microtubules. Understanding the mechanisms of centrosomal microtubule nucleation is, however, constrained by a lack of knowledge of the amount of soluble and polymer tubulin at mitotic centrosomes. Here we combined light microscopy and serial-section electron tomography to measure the amount of dimer and polymer at mitotic centrosomes in early C. elegans embryos. We show that a C. elegans one-cell stage centrosome at metaphase contains more than ten thousand microtubules with a total polymer concentration of 230 M. Centrosomes concentrate soluble /{beta} tubulin by about tenfold over the cytoplasm, reaching peak values of 470 M, giving a combined total monomer and polymer tubulin concentration at centrosomes of up to 660 M. These findings support in vitro data suggesting that microtubule nucleation in C. elegans centrosomes is driven in part by concentrating soluble tubulin.
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TL;DR: In this paper, the authors derived exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks capturing key features of molecular motor motion along linear filaments.
Abstract: We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks capturing key features of molecular motor motion along linear filaments. Our results generalize the infimum law for entropy production and reveal a symmetry of the distribution of its maxima and minima, which are confirmed by numerical simulations of stochastic models of molecular motors. We also show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marcenko-Pastur distribution of random-matrix theory. Using this result, we obtain estimates for the extreme-value statistics of molecular motors from the eigenvalue distributions of suitable Wishart and Laguerre random matrices.
TL;DR: In vitro experiments and numerical simulations are used to characterize the spontaneous behavior of a growing cell colony in vitro and it is shown that in vitro tissue elongation arises from anisotropy in the average cell elongation, which sets the direction along which boundary cells migrate radially resulting in a non-isotropic elongation.
Abstract: Tissue elongation is a central morphogenetic event occurring in all organisms in development1,2. During this process, symmetry of cells and tissues is broken by different mechanisms, such as neighbor exchange3,4, cell elongation5,6 and oriented cell division7. While the phenomenon is known to involve remodeling of adherens junctions4 and acto-myosin4,8 at the molecular level, mesoscopic mechanisms leading to distinct morphogenesis processes are poorly understood. This is partly because inputs from morphogen gradients9 or from neighboring tissues10,11 can affect tissue autonomous self-organization in vivo. It is therefore difficult to disentangle cell intrinsic from externally mediated behaviors. Here we use in vitro experiments and numerical simulations to characterize the spontaneous behavior of a growing cell colony in vitro. We show that in vitro tissue elongation arises from anisotropy in the average cell elongation. This anisotropy sets the direction along which boundary cells migrate radially resulting in a non-isotropic elongation that arises primarily through cell elongation. For colonies submitted to a time periodic stretch, the axis of global symmetry breaking can be imposed by external force, and tissue elongation arises through oriented neighbor exchange. Emergence of radially migrating cells and the interplay between cell elongation and cell rearrangements are confirmed by numerical simulations based on a vertex model. Our results suggest that spontaneous shape deformation is related to the mean orientation of the nematic cell elongation field in the absence of any external input. This provides a framework to explain autonomous tissue elongation and how contributions from different mesoscopic mechanisms can be modulated by external forces.
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TL;DR: A cell-based model of morphogen spreading that combines secretion in a local source, extracellular diffusion and cellular trafficking is presented and the slower mode defines the effective diffusion and degradation as well as the shape of the concentration profile in steady state.
Abstract: Morphogens are secreted signaling molecules that mediate tissue patterning and growth of embryonic tissues. They are secreted in a localized region and spread through the tissue to form a graded concentration profile. We present a cell-based model of morphogen spreading that combines secretion in a local source, extracellular diffusion and cellular trafficking. We introduce hydrodynamic modes of morphogen transport and characterize the dynamics of transport by dispersion relations of these dynamic eigenmodes. These dispersion relations specify the characteristic relaxation time of a mode as a function of its wavelength. In a simple model we distinguish two distinct dynamic modes characterized by different timescales. We find that the slower mode defines the effective diffusion and degradation as well as the shape of the concentration profile in steady state. Using our approach we discuss mechanisms of morphogen transport in the developing wing imaginal disc of the fruit fly extit{Drosophila}, distinguishing three transport regimes: transport by extracellular diffusion, transport by transcytosis and a regime where both transport mechanisms are combined.
TL;DR: Stochastic transport properties of colloidal beads in antiparallel networks of overlapping actin filaments are investigated using surface micro-patterns of actin polymerization in-vitro, finding that beads coated with myosin motors sensed the net polarity of the actin network, resulting in active bead positioning to regions of neutral polarity with a precision depending on the motor type.
Abstract: Cytoskeletal filaments assemble into dense parallel, antiparallel or disordered networks, providing a complex environment for active cargo transport and positioning by molecular motors. The interplay between the network architecture and intrinsic motor properties clearly affects transport properties but remains poorly understood. Here, by using surface micro-patterns of actin polymerization, we investigate stochastic transport properties of colloidal beads in antiparallel networks of overlapping actin filaments. We found that 200-nm beads coated with myosin-Va motors displayed directed movements towards positions where the net polarity of the actin network vanished, accumulating there. The bead distribution was dictated by the spatial profiles of local bead velocity and diffusion coefficient, indicating that a diffusion-drift process was at work. Remarkably, beads coated with heavy mero-myosin-II motors showed a similar behavior. However, although velocity gradients were steeper with myosin II, the much larger bead diffusion observed with this motor resulted in less precise positioning. Our observations are well described by a three-state model, in which active beads locally sense the net polarity of the network by frequently detaching from and reattaching to the filaments. A stochastic sequence of processive runs and diffusive searches results in a biased random walk. The precision of bead positioning is set by the gradient of net actin polarity in the network and by the run length of the cargo in an attached state. Our results unveiled physical rules for cargo transport and positioning in networks of mixed polarity. Significance statement Cellular functions rely on small groups of molecular motors to transport their cargoes throughout the cell along polar filaments of the cytoskeleton. Cytoskeletal filaments self-assemble into dense networks comprising intersections and filaments of mixed polarity, challenging directed motor-based transport. Using micro-patterns of actin polymerization in-vitro, we investigated stochastic transport of colloidal beads in antiparallel networks of overlapping actin filaments. We found that beads coated with myosin motors sensed the net polarity of the actin network, resulting in active bead positioning to regions of neutral polarity with a precision depending on the motor type. A theoretical description of our experimental results provides the key physical rules for cargo transport and positioning in filament networks of mixed polarity.
TL;DR: In this paper, a stochastic hydrodynamic approach for an active fluid layer of finite thickness was used to generalize the Casimir stress for nematic liquid crystals in thermal equilibrium to active systems.
Abstract: We calculate the Casimir stresses in a thin layer of active fluid with nematic order. By using a stochastic hydrodynamic approach for an active fluid layer of finite thickness $L$, we generalize the Casimir stress for nematic liquid crystals in thermal equilibrium to active systems. We show that the active Casimir stress differs significantly from its equilibrium counterpart. For contractile activity, the active Casimir stress, although attractive like its equilibrium counterpart, diverges logarithmically as $L$ approaches a threshold of the spontaneous flow instability from below. In contrast, for small extensile activity, it is repulsive, has no divergence at any $L$ and has a scaling with $L$ different from its equilibrium counterpart.
TL;DR: Frank Jülicher looks back on the life of Suzanne Eaton and reflects on how her work on Drosophila tissue morphogenesis contributed to the fields of cell and developmental biology.
Abstract: Suzanne Eaton, Professor at the Technical University Dresden and Group Leader at the Max Planck Institute of Molecular Cell Biology and Genetics in Dresden, tragically died on 2 July 2019. Suzanne was a remarkable person, both as a scientist and as a human being. Having worked closely with Suzanne for many years, I remember here some of her key scientific contributions.
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TL;DR: In this paper, a stochastic hydrodynamic approach for an active fluid layer of finite thickness was used to generalize the Casimir stress for nematic liquid crystals in thermal equilibrium to active systems.
Abstract: We calculate the Casimir stresses in a thin layer of active fluid with nematic order. By using a stochastic hydrodynamic approach for an active fluid layer of finite thickness $L$, we generalize the Casimir stress for nematic liquid crystals in thermal equilibrium to active systems. We show that the active Casimir stress differs significantly from its equilibrium counterpart. For contractile activity, the active Casimir stress, although attractive like its equilibrium counterpart, diverges logarithmically as $L$ approaches a threshold of the spontaneous flow instability from below. In contrast, for small extensile activity, it is repulsive, has no divergence at any $L$ and has a scaling with $L$ different from its equilibrium counterpart.