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Control of microtubule abundance, stability, and length is crucial to regulate intracellular transport as well as cell polarity and division. How microtubule stability depends on tubulin addition or removal at the dynamic ends is well studied. However, microtubule rescue, the event when a microtubule switches from shrinking to growing, occurs at tubulin exchange sites along the shaft. Molecular motors have recently been shown to promote such exchanges. Using a stochastic theoretical description, we study how microtubule stability and length depends on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in presence of kinesin-1 molecular motors. Although the average microtubule dynamics can be captured by an effective rescue rate, the dynamics of individual microtubules differs dramatically when rescue occurs only at exchange sites. Furthermore, we study in detail a transition from bounded to unbounded microtubule growth. Our results provide novel insights into how molecular motors imprint information of microtubule stability on the microtubule network.SIGNIFICANCEThe microtubule network is essential for vital cellular processes like the organization of intracellular transport and division. Although microtubule assembly occurs at its tips, it has recently been reported that tubulin is exchanged along the microtubule shaft. Tubulin exchange plays an essential role in regulating microtubule dynamics and can be induced by molecular motors. Here, we provide the first systematic study of the impact of shaft dynamics on the regulation of rescue events, where a microtubule switches from shrinking to growing. Our results illustrate how the usage of microtubules as tracks for intracellular transport regulates the microtubule network and thus offers a novel perspective on intracellular organization.

We consider exponential single server queues with state-dependent arrival and service rates that evolve under influences of external environments. The transitions of the queues are influenced by the environment’s state and the movements of the environment depend on the status of the queues (bidirectional interaction). The environment is constructed in a way to encompass various models from the recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process, we prove separability for a large class of such interactive systems; that is, the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state. For nonseparable systems, we develop ergodicity and exponential ergodicity criteria via Lyapunov functions. By examples we explain principles for bounding departure rates of served customers (throughputs) of nonseparable systems by throughputs of related separable systems as upper and lower bound.

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time $$T>0$$ T > 0 by taking into account the orbit of the initial value under the semigroup for $$t\in [0,T]$$ t ∈ [ 0 , T ] , measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauß–Weierstraß semigroup and the Ornstein–Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the case of Banach spaces.

For physiological function, organs must organise constituent cell types in precise spatial and orientational order. While in single-cell type epithelia, morphogenesis is driven by forces exerted at cell junctions 1-3, it is not known whether forces can drive the acquisition of patterns and polarity in organs comprising multiple cell types. We investigate the process of global alignment in the avian auditory epithelium. This consists of two cell types, hair cells (HCs), with an asymmetric mechanosensory hair bundle on its apex, and supporting cells (SCs) 4. During development, these two cell types form a regular pattern, and the hair bundle of each HC aligns with the tissue axis. Through a combination of experiments and theory, we find that tissue-wide force patterning generates both positional and orientational order. Positional order is established by differences between the mechanical activity of cell-cell junctions. These differences are encoded by cell-specific genetic programmes restricting the di-phosphorylated form of non-muscle myosin-II regulatory light chain (RLC) to specific junctions. Importantly, the hair bundle organises supracellular patterns of myosin activity through the local attenuation of RLC phosphorylation, generating global polarity across the entire tissue. Our findings show that junctional asymmetries couple cell rearrangements to polarity, potentially a fundamental feature of development and repair in tissues consisting of different cell types.

This paper is dedicated to e-assessment in mathematics and the development of the corresponding electronic tasks. Due to its electronic nature e-assessment can be used to efficiently provide a huge number of tasks for training. This makes it a valuable tool for large classes, which are typical for mathematics as a service subject. In order to develop good tasks, one can address different aspects, e.g. the randomization of problems under the restriction that they are comparable with regard to fairness, providing open-ended tasks to demonstrate understanding rather than guesses and the handling of errors, i.e. how to cope with mistakes made by students. When it comes to demonstrate understanding, mathematical proofs play a prominent role and we present a way how to transform them to an electronic task in form of a proof puzzle. A further aspect of tasks is their authenticity with regard to today’s work environment, which goes along with the application of specialized software. In this contribution we describe how to deal with each of the mentioned aspects of tasks.

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

Given a Banach space $X$ and an additional coarser Hausdorff locally convex topology $\tau$ on $X$ we characterise the generators of $\tau$-bi-continuous semigroups in the spirit of the Lumer--Phillips theorem, i.e. by means of dissipativity w.r.t.~a directed system of seminorms and a range condition.

We consider a semi-open queueing network (SOQN), where one resource from a resource pool is needed to serve a customer. If on arrival of a customer some resource is available, the resource is forwarded to an inner network to complete the customer’s order. If no resource is available, the new customer waits in an external queue until one becomes available (“backordering”). When a resource exits the inner network, it is returned to the resource pool. We develop a new solution approach. In a first step we modify the system such that new arrivals are lost if the resource pool is empty (“lost customers”). We adjust the arrival rate of the modified system such that the throughputs in all nodes of the inner network are pairwise identical to those in the original network. Using queueing theoretical methods, in a second step we reduce this inner network to a two-station system including the resource pool. For this two-station systems, we invert the first step and obtain a standard SOQN which can be solved analytically. We apply our results to storage and delivering systems with robotic mobile fulfilment systems (RMFSs). Instead of sending pickers to the storage area to search for the ordered items and pick them, robots carry shelves with ordered items from the storage area to picking stations. We model the RMFS as an SOQN to determine the minimal number of robots. Ecopti GmbH (Paderborn, Germany) Beijing Hanning Tech Co., Ltd. (Beijing, China)

We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, B\"aumer, Lumer and Neubrander and Langenbruch.

We introduce a new class $$\mathcal {FV}(\Omega ,E)$$ FV ( Ω , E ) of weighted spaces of functions on a set $$\Omega$$ Ω with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of $$\mathcal {FV}(\Omega ,E)$$ FV ( Ω , E ) to derive sufficient conditions such that $$\mathcal {FV}(\Omega ,E)$$ FV ( Ω , E ) can be linearised, i.e. that $$\mathcal {FV}(\Omega ,E)$$ FV ( Ω , E ) is topologically isomorphic to the $$\varepsilon$$ ε -product $$\mathcal {FV}(\Omega )\varepsilon E$$ FV ( Ω ) ε E where $$\mathcal {FV}(\Omega ):=\mathcal {FV}(\Omega ,\mathbb {K})$$ FV ( Ω ) : = FV ( Ω , K ) and $$\mathbb {K}$$ K is the scalar field of E.

Nuclearity plays an important role for the Schwartz kernel theorem to hold and in transferring the surjectivity of a linear partial differential operator from scalar-valued to vector-valued functions via tensor product theory. In this paper we study weighted spaces $\mathcal{EV}(\Omega)$ of smooth functions on an open subset $\Omega\subset\mathbb{R}^{d}$ whose topology is given by a family of weights $\mathcal{V}$. We derive sufficient conditions on the weights which make $\mathcal{EV}(\Omega)$ a nuclear space.

Topological defects are one of the most conspicuous features of liquid crystals. In two dimensional nematics, they have been shown to behave effectively as particles with both, charge and orientation, which dictate their interactions. Here, we study "twisted" defects that have a radially dependent orientation. We find that twist can be partially relaxed through the creation and annihilation of defect pairs. By solving the equations for defect motion and calculating the forces on defects, we identify four distinct elements that govern the relative relaxational motion of interacting topological defects, namely attraction, repulsion, co-rotation and co-translation. The interaction of these effects can lead to intricate defect trajectories, which can be controlled by setting relevant timescales., 8 pages, 6 figures

Monolayers of anisotropic cells exhibit long-ranged orientational order and topological defects. During the development of organisms, orientational order often influences morphogenetic events. However, the linkage between the mechanics of cell monolayers and topological defects remains largely unexplored. This holds specifically at the time scales relevant for tissue morphogenesis. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. In particular, we use a hydrodynamical description of an active polar fluid to study the steady-state mechanical patterns at integer topological defects. Our description includes three distinct sources of activity: traction forces accounting for cell-substrate interactions as well as anisotropic and isotropic active nematic stresses accounting for cell-cell interactions. We apply our approach to C2C12 cell monolayers in small circular confinements, which form isolated aster or spiral topological defects. By analyzing the velocity and orientational order fields in spirals as well as the forces and cell number density fields in asters, we determine mechanical parameters of C2C12 cell monolayers. Our work shows how topological defects can be used to fully characterize the mechanical properties of biological active matter., Comment: 41 pages, 11 figures

Der Band enthält die Beiträge des GDM-Monats 2021, der statt der alljährlichen GDM-Tagung online vom 1.-25. März 2021 stattfand. Dabei handelt es sich um die Hauptvorträge und die Beiträge aus den Minisymposien bzw. die Zusammenfassung einer Podiumsdiskussion.

The cell cortex is a contractile actin meshwork, which determines cell shape and is essential for cell mechanics, migration and division. Because the cortical thickness is below optical resolution, it has been generally considered as a thin uniform two-dimensional layer. Using two mutually attracted magnetic beads, one inside the cell and the other in the extracellular medium, we pinch the cortex of dendritic cells and provide an accurate and time resolved measure of its thickness. Our observations draw a new picture of the cell cortex as a highly dynamic layer, harboring large fluctuations in its third dimension due to actomyosin contractility. We propose that the cortex dynamics might be responsible for the fast shape changing capacity of highly contractile cells that use amoeboid-like migration.

SUMMARYTubulin dimers assemble into dynamic microtubules which are used by molecular motors as tracks for intracellular transport. Organization and dynamics of the microtubule network is commonly thought to be regulated at the polymer ends, where tubulin-dimers can be added or removed. Here we show that molecular motors running on microtubules cause exchange of dimers along the shaft. These sites of dimer exchange act as rescue sites where depolymerising microtubules stop shrinking and start re-growing. Consequently, the average length of microtubules increases depending on how frequently they are used as motor tracks. An increase of motor activity densifies the cellular microtubule network and enhances cell polarity. Running motors leave marks in the shaft serving as traces of microtubule usage to organize the polarity landscape of the cell.

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefully clarify on mutual relations between previously studied variants of these notions. The abstract results -- exploiting techniques from topological vector spaces -- are thoroughly discussed by means of several example classes, such as semigroups on spaces of bounded continuous functions.

Amoeboid cell migration is characterized by frequent changes of the direction of motion and resembles a persistent random walk on long time scales. Although it is well known that cell migration is typically driven by the actin cytoskeleton, the cause of this migratory behavior remains poorly understood. We analyze the spontaneous dynamics of actin assembly due to nucleation promoting factors, where actin filaments lead to an inactivation of the nucleators. We show that this system exhibits excitable dynamics and can spontaneously generate waves, which we analyse in detail. By using a phase-field approach, we show that these waves can generate cellular random walks. We explore how the characteristics of these persistent random walks depend on the parameters governing the actin-nucleator dynamics. In particular, we find that the effective diffusion constant and the persistence time depend strongly on the speed of filament assembly and the rate of nucleator inactivation. Our findings point to a deterministic origin of the random walk behavior and suggest that cells could adapt their migration pattern by modifying the pool of available actin., 29 pages, 7 figures, 6 movies

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Summary Many organs are formed through folding of an epithelium. This change in shape is usually attributed to tissue heterogeneities, for example, local apical contraction. In contrast, compressive stresses have been proposed to fold a homogeneous epithelium by buckling. While buckling is an appealing mechanism, demonstrating that it underlies folding requires measurement of the stress field and the material properties of the tissue, which are currently inaccessible in vivo. Here, we show that monolayers of identical cells proliferating on the inner surface of elastic spherical shells can spontaneously fold. By measuring the elastic deformation of the shell, we infer the forces acting within the monolayer and its elastic modulus. Using analytical and numerical theories linking forces to shape, we find that buckling quantitatively accounts for the shape changes of our monolayers. Our study shows that forces arising from epithelial growth in three-dimensional confinement are sufficient to drive folding by buckling.

Tissues acquire function and shape via differentiation and morphogenesis. Both processes are driven by coordinating cellular forces and shapes at the tissue scale, but general principles governing this interplay remain to be discovered. Here we report that self-organization of myoblasts around integer topological defects, namely spirals and asters, suffices to establish complex multicellular architectures. In particular, these arrangements can trigger localized cell differentiation or, alternatively, when differentiation is inhibited, they can drive the growth of swirling protrusions. Both localized differentiation and growth of cellular vortices require specific stress patterns. By analysing the experimental velocity and orientational fields through active gel theory, we show that integer topological defects can generate force gradients that concentrate compressive stresses. We reveal these gradients by assessing spatial changes in nuclear volume and deformations of elastic pillars. We propose integer topological defects as mechanical organizing centres controlling differentiation and morphogenesis.

Let $$\varOmega $$ Ω be an open subset of $$\mathbb {R}^{2}$$ R 2 and E a complete complex locally convex Hausdorff space. The purpose of this paper is to find conditions on certain weighted Fréchet spaces $$\mathcal {EV}(\varOmega )$$ EV ( Ω ) of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator $${\overline{\partial }}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E)$$ ∂ ¯ : EV ( Ω , E ) → EV ( Ω , E ) is surjective. This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy–Riemann operator.

In developing organisms, internal cellular processes generate mechanical stresses at the tissue scale. The resulting deformations depend on the material properties of the tissue, which can exhibit long-ranged orientational order and topological defects. It remains a challenge to determine these properties on the time scales relevant for developmental processes. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. Specifically, we use a hydrodynamic description to characterize the stationary states of compressible active polar fluids around defects. We illustrate our approach by analyzing monolayers of C2C12 cells in small circular confinements, where they form a single topological defect with integer charge. We find that such monolayers exert compressive stresses at the defect centers, where localized cell differentiation and formation of three-dimensional shapes is observed., Comment: 5 pages, 4 figures

Tubulin dimers assemble into dynamic microtubules, which are used by molecular motors as tracks for intracellular transport. Organization and dynamics of the microtubule network are commonly thought to be regulated at the polymer ends, where tubulin dimers can be added or removed. Here, we show that molecular motors running on microtubules cause exchange of dimers along the shaft in vitro and in cells. These sites of dimer exchange act as rescue sites where depolymerizing microtubules stop shrinking and start re-growing. Consequently, the average length of microtubules increases depending on how frequently they are used as motor tracks. An increase of motor activity densifies the cellular microtubule network and enhances cell polarity. Running motors leave marks in the shaft, serving as traces of microtubule usage to organize the polarity landscape of the cell.

International audience; Dendritic cells "patrol" the human body to detect pathogens. In their search, dendritic cells perform a random walk by amoeboid migration. The efficiency of pathogen detection depends on the properties of the random walk. It is not known how the dendritic cells control these properties. Here, we quantify dendritic cell migration under well-defined 2-dimensional confinement and in a 3-dimensional collagen matrix through recording their long-term trajectories. We find 2 different migration states: persistent migration, during which the dendritic cells move along curved paths, and diffusive migration, which is characterized by successive sharp turns. These states exhibit differences in the actin distributions. Our theoretical and experimental analyses indicate that this kind of motion can be generated by spontaneous actin polymerization waves that contribute to dendritic cell polarization and migration. The relative distributions of persistent and diffusive migration can be changed by modification of the molecular actin filament nucleation and assembly rates. Thus, dendritic cells can control their migration patterns and adapt to specific environments. Our study offers an additional perspective on how dendritic cells tune their searches for pathogens.

We study the Cauchy���Riemann operator ...

Dynamic processes in living cells are highly organized in space and time. Unraveling the underlying molecular mechanisms of spatiotemporal pattern formation remains one of the outstanding challenges at the interface between physics and biology. A fundamental recurrent pattern found in many different cell types is that of self-sustained oscillations. They are involved in a wide range of cellular functions, including second messenger signaling, gene expression, and cytoskeletal dynamics. Here, we review recent developments in the field of cellular oscillations and focus on cases where concepts from physics have been instrumental for understanding the underlying mechanisms. We consider biochemical and genetic oscillators as well as oscillations that arise from chemo-mechanical coupling. Finally, we highlight recent studies of intracellular waves that have increasingly moved into the focus of this research field.

Translocation of cytosolic cPKC to the plasma membrane is a key event in their activation process but its exact nature is still unclear with particular dispute whether sole diffusion or additional active transport along the cell’s cytoskeleton contributes to cPKC’s dynamics. This was addressed by analyzing the recruitment behavior of PKCα while manipulating the cytoskeleton. Photolytic Ca2+ uncaging allowed us to quantify the kinetics of PKCα redistribution to the plasma membrane when fused to monomeric, dimeric and tetrameric fluorescence proteins. Results indicated that translocation kinetics were modulated by the state of oligomerization as expected for varying Stokes’ radii of the participating proteins. Following depolymerization of the microtubules and the actin filaments we found that Ca2+ induced membrane accumulation of PKCα was independent of the filamentous state of the cytoskeleton. Fusion of PKCα to the photo-convertible fluorescent protein Dendra2 enabled the investigation of PKCα-cytoskeleton interactions under resting conditions. Redistribution following spatially restricted photoconversion showed that the mobility of the fusion protein was independent of the state of the cytoskeleton. Our data demonstrated that in living cells neither actin filaments nor microtubules contribute to PKCα’s cytosolic mobility or Ca2+-induced translocation to the plasma membrane. Instead translocation is a solely diffusion-driven process.

Living cells respond to spatially confined signals. Intracellular signal transmission often involves the release of second messengers like ${\mathrm{Ca}}^{2+}$. They eventually trigger a physiological response, for example, by activating kinases that in turn activate target proteins through phosphorylation. Here, we investigate theoretically how positional information can be accurately read out by protein phosphorylation in spite of rapid second messenger diffusion. We find that accuracy is increased by binding of kinases to the cell membrane prior to phosphorylation and by increasing the rate of ${\mathrm{Ca}}^{2+}$ loss from the cell interior. These findings could explain some salient features of the conventional protein kinase $\mathrm{C}\ensuremath{\alpha}$.

A living cell senses its environment and responds to external signals. In this work, we study theoretically, the precision at which cells can determine the position of a spatially localized transient extracellular signal. To this end, we focus on the case, where the stimulus is converted into the release of a small molecule that acts as a second messenger, for example, Ca$^{2+}$, and activates kinases that change the activity of enzymes by phosphorylating them. We analyze the spatial distribution of phosphorylation events using stochastic simulations as well as a mean-field approach. Kinases that need to bind to the cell membrane for getting activated provide more accurate estimates than cytosolic kinases. Our results could explain why the rate of Ca$^{2+}$ detachment from the membrane-binding conventional Protein Kinase C$\alpha$ is larger than its phosphorylation rate.

Axons are linear structures of nerve cells that can range from a few tens of micrometers up to meters in length. In addition to external cues, the length of an axon is also regulated by unknown internal mechanisms. Molecular motors have been suggested to generate oscillations with an axon-length-dependent frequency that could be used to measure an axon's extension. Here, we present a mechanism for determining the axon length that couples the mechanical properties of an axon to the spectral decomposition of the oscillatory signal.

Life is characterized by a myriad of complex dynamic processes allowing organisms to grow, reproduce, and evolve. Physical approaches for describing systems out of thermodynamic equilibrium have been increasingly applied to living systems, which often exhibit phenomena not found in those traditionally studied in physics. Spectacular advances in experimentation during the last decade or two, for example, in microscopy, single-cell dynamics, in the reconstruction of subcellular and multicellular systems outside of living organisms, and in high throughput data acquisition, have yielded an unprecedented wealth of data on cell dynamics, genetic regulation, and organismal development. These data have motivated the development and refinement of concepts and tools to dissect the physical mechanisms underlying biological processes. Notably, landscape and flux theory as well as active hydrodynamic gel theory have proven useful in this endeavor. Together with concepts and tools developed in other areas of nonequilibrium physics, significant progress has been made in unraveling the principles underlying efficient energy transport in photosynthesis, cellular regulatory networks, cellular movements and organization, embryonic development and cancer, neural network dynamics, population dynamics and ecology, as well as aging, immune responses, and evolution. Here recent advances in nonequilibrium physics are reviewd and their application to biological systems is surveyed. Many of these results are expected to be important cornerstones as the field continues to build our understanding of life.