arXiv subjects

Karsten Kruse

Publications and source records attributed to Karsten Kruse.

At least 55 records · Page 3Linked to original sources

arXiv2020

Nonequilibrium Physics in Biology

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 unknown from 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 sub- and multicellular systems outside of living organisms, or in high throughput data acquisition have yielded an unprecedented wealth of data about 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, the landscape and flux theory as well as active hydrodynamic gel theory have proven very useful in this endeavour. Together with concepts and tools developed in other areas of nonequilibrium physics, significant progresses have 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 ageing, immune responses and evolution. Here, we review recent advances in nonequilibrium physics and survey their application to biological systems. We expect many of these results to be important cornerstones as the field continues to build our understanding of life.

physics.bio-ph↗

arXiv2020

Vector-valued holomorphic functions in several variables

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.

math.FA↗

arXiv2020

On the solvability of the matrix equation $(1+ae^{-\frac{\|X\|}{b}})X=Y$

The treated matrix equation $(1+ae^{-\frac{\|X\|}{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of $(1+ae^{-\frac{\|X\|}{b}})X=Y$ in general normed spaces $(E,\|\cdot\|)$ w.r.t. the parameters $a,b\in\mathbb{R}$, $b\neq 0$, and give an algorithm to numerically compute its solutions in $E=\mathbb{R}^{m\times n}$, $m,n\in\mathbb{N}$, $m,n\geq 2$, equipped with the Frobenius norm.

math.GM↗

arXiv2020

Integer topological defects of cell monolayers -- mechanics and flows

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.

cond-mat.soft↗

arXiv2020

Quantifying material properties of cell monolayers by analyzing integer topological defects

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.

cond-mat.soft↗

arXiv2020

On the nuclearity of weighted spaces of smooth functions

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 $\mathcal{EV}(Ω)$ of smooth functions on an open subset $Ω\subset\mathbb{R}^{d}$ whose topology is given by a family of weights $\mathcal{V}$. We derive sufficient conditions on the weights which make $\mathcal{EV}(Ω)$ a nuclear space.

math.FA↗

arXiv2019

Lost-customers approximation of semi-open queueing networks with backordering -- An application to minimise the number of robots in robotic mobile fulfilment systems

We consider a semi-open queueing network (SOQN), where a customer requires exactly one resource from the resource pool for service. If there is a resource available, the customer is immediately served and the resource enters an inner network. If there is no resource available, the new customer has to wait in an external queue until one becomes available ("backordering"). When a resource exits the inner network, it is returned to the resource pool and waits for another customer. In this paper, we present a new solution approach. To approximate the inner network with the resource pool of the SOQN, we consider a modification, where newly arriving customers will decide not to join the external queue and are lost if the resource pool is empty "lost customers". We prove that we can adjust the arrival rate of the modified system so that the throughputs in each node are pairwise identical to those in the original network. We also prove that the probabilities that the nodes with constant service rates are idling are pairwise identical too. Moreover, we provide a closed-form expression for these throughputs and probabilities of idle nodes. To approximate the external queue of the SOQN with backordering, we construct a reduced SOQN with backordering, where the inner network consists only of one node, by using Norton's theorem and results from the lost-customers modification. In a final step, we use the closed-form solution of this reduced SOQN, to estimate the performance of the original SOQN. We apply our results to 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, analyse its stability and determine the minimal number of robots for such systems using the results from the first part.

math.PR↗

arXiv2019

Accuracy of position determination in Ca$^{2+}$ signaling

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$α$ is larger than its phosphorylation rate.

q-bio.CB↗

arXiv2019

Positional information readout in $Ca^{2+}$ signaling

Living cells respond to spatial signals. Signal transmission to the cell interior often involves the release of second messengers like $Ca^{2+}$ . They will eventually trigger a physiological response 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 the kinases to the cell membrane prior to phosphorylation and by increasing the rate of $Ca^{2+}$ loss from the cell interior. These findings could explain some salient features of conventional protein kinases C.

q-bio.SC↗

arXiv2019

The sound of an axon's growth

Axons are linear processes 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 that depends on the spectral decomposition of the oscillatory signal to determine the axon length.

q-bio.SC↗

arXiv2019

Series representations in spaces of vector-valued functions via Schauder decompositions

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation. This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space over a field $\mathbb{K}$, $\mathcal{F}(Ω)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $Ω$ and $\mathcal{F}(Ω,E)$ be an $E$-valued counterpart of $\mathcal{F}(Ω)$ (where the term $E$-valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of $\mathcal{F}(Ω)$ to elements of $\mathcal{F}(Ω,E)$? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of functions $\mathcal{F}(Ω)$ having an equicontinuous Schauder basis.

math.FA↗

arXiv2019

The approximation property for weighted spaces of differentiable functions

We study spaces $\mathcal{CV}^{k}(Ω,E)$ of $k$-times continuously partially differentiable functions on an open set $Ω\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space $\mathcal{CV}^{k}(Ω,E)$ is given a weighted topology generated by a family of weights $\mathcal{V}^{k}$. For the space $\mathcal{CV}^{k}(Ω,E)$ and its subspace $\mathcal{CV}^{k}_{0}(Ω,E)$ of functions that vanish at infinity in the weighted topology we try to answer the question whether their elements can be approximated by functions with values in a finite dimensional subspace. We derive sufficient conditions for an affirmative answer to this question using the theory of tensor products.

math.FA↗

arXiv2019

Weighted vector-valued functions and the $\varepsilon$-product

We introduce a new class $\mathcal{FV}(Ω,E)$ of spaces of weighted functions on a set $Ω$ 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 $\mathcal{FV}(Ω,E)$ to derive sufficient conditions such that $\mathcal{FV}(Ω,E)$ can be linearised, i.e. that $\mathcal{FV}(Ω,E)$ is topologically isomorphic to the $\varepsilon$-product $\mathcal{FV}(Ω)\varepsilon E$ where $\mathcal{FV}(Ω):=\mathcal{FV}(Ω,\mathbb{K})$ and $\mathbb{K}$ is the scalar field of $E$.

math.FA↗

arXiv2019

Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions

We study the inhomogeneous Cauchy-Riemann equation on spaces $\mathcal{EV}(Ω,E)$ of weighted $\mathcal{C}^{\infty}$-smooth $E$-valued functions on an open set $Ω\subset\mathbb{R}^{2}$ whose growth on strips along the real axis is determined by a family of continuous weights $\mathcal{V}$ where $E$ is a locally convex Hausdorff space over $\mathbb{C}$. We derive sufficient conditions on the weights $\mathcal{V}$ such that the kernel $\operatorname{ker}\overline{\partial}$ of the Cauchy-Riemann operator $\overline{\partial}$ in $\mathcal{EV}(Ω):=\mathcal{EV}(Ω,\mathbb{C})$ has the property $(Ω)$ of Vogt. Then we use previous results and conditions on the surjectivity of the Cauchy-Riemann operator $\overline{\partial}\colon\mathcal{EV}(Ω)\to\mathcal{EV}(Ω)$ and the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for (PLS)-spaces to deduce the surjectivity of the Cauchy-Riemann operator on the space $\mathcal{EV}(Ω,E)$ if $E:=F_{b}'$ where $F$ is a Fréchet space satisfying the condition $(DN)$ or if $E$ is an ultrabornological (PLS)-space having the property $(PA)$. As a consequence, for every family of right-hand sides $(f_λ)_{λ\in U}$ in $\mathcal{EV}(Ω)$ which depends smoothly, holomorphically or distributionally on a parameter $λ$ there is a family $(u_λ)_{λ\in U}$ in $\mathcal{EV}(Ω)$ with the same kind of parameter dependence which solves the Cauchy-Riemann equation $\overline{\partial}u_λ=f_λ$ for all $λ\in U$.

math.FA↗

arXiv2014

Response to Halatek and Frey: Effective two-dimensional model does account for geometry sensing by self-organized proteins patterns

The Min proteins from Escherichia coli can self-organize into traveling waves on supported lipid bilayers. In Proc. Natl. Acad. Sci. USA 109, 15283 (2012) we showed that these waves are guided along the boundaries of membrane patches. We introduced an effective two-dimensional model reproducing the observed patterns. In arXiv:1403.5934v1, Jacob Halatek and Erwin Frey contest the ability of our effective two-dimensional model to describe the dynamics of Min proteins on patterned supported lipid bilayers. We thank Halatek and Frey for their interest in our work and for again highlighting the importance of dimensionality and geometry for pattern formation by the Min proteins. Here we reply in detail to the objections by Halatek and Frey and show that (1) our effective two-dimensional model reproduces the observed patterns on isolated patches and that (2) a three-dimensional version of our model produces similar patterns on square patches.

q-bio.SC↗

arXiv2013

The actin cortex as an active wetting layer

Using active gel theory we study theoretically the properties of the cortical actin layer of animal cells. The cortical layer is described as a non-equilibrium wetting film on the cell membrane. The actin density is approximately constant in the layer and jumps to zero at its edge. The layer thickness is determined by the ratio of the polymerization velocity and the depolymerization rate of actin.

cond-mat.soft↗

arXiv2012

Spontaneous sarcomere dynamics

Sarcomeres are the basic force generating units of striated muscles and consist of an interdigitating arrangement of actin and myosin filaments. While muscle contraction is usually triggered by neural signals, which eventually set myosin motors into motion, isolated sarcomeres can oscillate spontaneously between a contracted and a relaxed state. We analyze a model for sarcomere dynamics, which is based on a force-dependent detachment rate of myosin from actin. Our numerical bifurcation analysis of the spontaneous sarcomere dynamics reveals notably Hopf bifurcations, canard explosions, and gluing bifurcations. We discuss possible implications for experiments.

nlin.AO↗

arXiv2010

Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length

Self-organization of proteins in space and time is of crucial importance for the functioning of cellular processes. Often, this organization takes place in the presence of strong random fluctuations due to the small number of molecules involved. We report on stochastic switching of the Min-protein distributions between the two cell halves in short Escherichia coli cells. A computational model provides strong evidence that the macroscopic switching is rooted in microscopic noise on the molecular scale. In longer bacteria, the switching turns into regular oscillations that are required for positioning of the division plane. As the pattern becomes more regular, cell-to-cell variability also lessens, indicating cell length-dependent regulation of Min-protein activity.

q-bio.SC↗