arXiv subjects

Erwin Frey

Publications and source records attributed to Erwin Frey.

At least 73 records · Page 4Linked to original sources

arXiv2018

Topologically robust zero-sum games and Pfaffian orientation -- How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation

To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is the replicator equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio $φ= 1.6180...$, cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.

cond-mat.stat-mech↗

arXiv2018

Crowding and pausing strongly affect dynamics of kinesin-1 motors along microtubules

Molecular motors of the kinesin-1 family move in a directed and processive fashion along microtubules (MTs). It is generally accepted that steric hindrance of motors leads to crowding effects; however, little is known about the specific interactions involved. We employ an agent-based lattice gas model to study the impact of interactions which enhance the detachment of motors from crowded filaments on their collective dynamics. The predictions of our model quantitatively agree with the experimentally observed concentration dependence of key motor characteristics including their run length, dwell time, velocity, and landing rate. From the anomalous stepping statistics of individual motors which exhibit relatively long pauses we infer that kinesin-1 motors sometimes lapse into an inactive state. Hereby, the formation of traffic jams amplifies the impact of single inactive motors and leads to a crowding dependence of the frequencies and durations of the resulting periods of no or slow motion. We interpret these findings and conclude that kinesin-1 spends a significant fraction of its stepping cycle in a weakly bound state in which only one of its heads is bound to the MT.

physics.bio-ph↗

arXiv2018

CsrA and its regulators control the time-point of ColicinE2 release in Escherichia coli

The bacterial SOS response is a cellular reaction to DNA damage, that, among other actions, triggers the expression of colicin - toxic bacteriocins in Escherichia coli that are released to kill close relatives competing for resources. However, it is largely unknown, how the complex network regulating toxin expression controls the time-point of toxin release to prevent premature release of inefficient protein concentrations. Here, we study how different regulatory mechanisms affect production and release of the bacteriocin ColicinE2 in Escherichia coli. Combining experimental and theoretical approaches, we demonstrate that the global carbon storage regulator CsrA controls the duration of the delay between toxin production and release and emphasize the importance of CsrA sequestering elements for the timing of ColicinE2 release. In particular, we show that ssDNA originating from rolling-circle replication of the toxin-producing plasmid represents a yet unknown additional CsrA sequestering element, which is essential in the ColicinE2-producing strain to enable toxin release by reducing the amount of free CsrA molecules in the bacterial cell. Taken together, our findings show that CsrA times ColicinE2 release and reveal a dual function for CsrA as an ssDNA and mRNA-binding protein, introducing ssDNA as an important post-transcriptional gene regulatory element.

q-bio.MN↗

arXiv2018

Limited Resources Induce Bistability in Microtubule Length Regulation

The availability of protein is an important factor for the determination of the size of the mitotic spindle. Involved in spindle-size regulation is kinesin-8, a molecular motor and microtubule (MT) depolymerase, which is known to tightly control MT length. Here, we propose and analyze a theoretical model in which kinesin-induced MT depolymerization competes with spontaneous polymerization while supplies of both tubulin and kinesin are limited. In contrast to previous studies where resources were unconstrained, we find that, for a wide range of concentrations, MT length regulation is bistable. We test our predictions by conducting in vitro experiments, and find that the bistable behavior manifests in a bimodal MT length distribution.

physics.bio-ph↗

arXiv2018

Interactions mediated by a public good transiently increase cooperativity in growing Pseudomonas putida metapopulations

Bacterial communities have rich social lives. A well-established interaction involves the exchange of a public good in Pseudomonas populations, where the iron-scavenging compound pyoverdine, synthesized by some cells, is shared with the rest. Pyoverdine thus mediates interactions between producers and non-producers and can constitute a public good. This interaction is often used to test game theoretical predictions on the "social dilemma" of producers. Such an approach, however, underestimates the impact of specific properties of the public good, for example consequences of its accumulation in the environment. Here, we experimentally quantify costs and benefits of pyoverdine production in a specific environment, and build a model of population dynamics that explicitly accounts for the changing significance of accumulating pyoverdine as chemical mediator of social interactions. The model predicts that, in an ensemble of growing populations (metapopulation) with different initial producer fractions (and consequently pyoverdine contents), the global producer fraction initially increases. Because the benefit of pyoverdine declines at saturating concentrations, the increase need only be transient. Confirmed by experiments on metapopulations, our results show how a changing benefit of a public good can shape social interactions in a bacterial population.

q-bio.PE↗

arXiv2018

Protein Pattern Formation

Protein pattern formation is essential for the spatial organization of many intracellular processes like cell division, flagellum positioning, and chemotaxis. A prominent example of intracellular patterns are the oscillatory pole-to-pole oscillations of Min proteins in \textit{E. coli} whose biological function is to ensure precise cell division. Cell polarization, a prerequisite for processes such as stem cell differentiation and cell polarity in yeast, is also mediated by a diffusion-reaction process. More generally, these functional modules of cells serve as model systems for self-organization, one of the core principles of life. Under which conditions spatio-temporal patterns emerge, and how these patterns are regulated by biochemical and geometrical factors are major aspects of current research. Here we review recent theoretical and experimental advances in the field of intracellular pattern formation, focusing on general design principles and fundamental physical mechanisms.

physics.bio-ph↗

arXiv2018

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatiotemporal regulation of many important intracellular processes in procaryotes, eucaryotes, and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article we review quantitative models for intracellular Min protein patterns in E. coli, Cdc42 polarization in S. cerevisiae, and the bipolar PAR protein patterns found in C. elegans. By analyzing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as "activators"', "inhibitors", or "substrate-depletion". Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction-diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

physics.bio-ph↗

arXiv2018

Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations

We prove a mean-field equation for the dynamics of quorum-sensing microbial populations. In the stochastic many-particle process, individuals of a population produce public good molecules to different degrees. Individual production is metabolically costly such that non-producers replicate faster than producers. In addition, individuals sense the average production level in the well-mixed population and adjust their production in response ("quorum sensing"). Here we prove that the temporal evolution of such quorum-sensing populations converges to a macroscopic mean-field equation for increasing population sizes. To prove convergence, we introduce an auxiliary stochastic mean-field process that mimics the dynamics of the mean-field equation and that samples independently the individual's production degrees between consecutive update steps. This way, the law of large numbers is separated from the propagation of errors due to correlations. Our developed method of an auxiliary stochastic mean-field process may help to prove mean-field equations for other stochastic many-particle processes.

math-ph↗

arXiv2018

Disentangling Entanglements in Biopolymer Solutions

Reptation theory has been highly successful in explaining the unusual material properties of entangled polymer solutions. It reduces the complex many-body dynamics to a single-polymer description where each polymer is envisaged to be confined to a tube through which it moves in a snake-like fashion. For flexible polymers, reptation theory has been amply confirmed by both experiments and simulations. In contrast, for semiflexible polymers experimental and numerical tests are either limited to the onset of reptation, or were performed for tracer polymers in a fixed, static matrix. Here we report Brownian dynamics simulations of entangled solutions of semiflexible polymers, which show that curvilinear motion along a tube (reptation) is no longer the dominant mode of dynamics. Instead, we find that polymers disentangle due to correlated constraint release which leads to equilibration of internal bending modes before polymers diffuse the full tube length. The physical mechanism underlying terminal stress relaxation is rotational diffusion mediated by disentanglement rather than curvilinear motion along a tube.

cond-mat.soft↗

arXiv2018

On the role of multiple scales in metapopulations of public good producers

Multiple scales in metapopulations can give rise to paradoxical behaviour: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilised in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilises this species due to stochastic effects, and that the formation of spatial patterns is not crucial for this stabilisation.

physics.bio-ph↗

arXiv2018

Exploiting ecology in drug pulse sequences in favour of population reduction

A deterministic population dynamics model involving birth and death for a two-species system, comprising a wild-type and more resistant species competing via logistic growth, is subjected to two distinct stress environments designed to mimic those that would typically be induced by temporal variation in the concentration of a drug (antibiotic or chemotherapeutic) as it permeates through the population and is progressively degraded. Different treatment regimes, involving single or periodical doses, are evaluated in terms of the minimal population size (a measure of the extinction probability), and the population composition (a measure of the selection pressure for resistance or tolerance during the treatment). We show that there exist timescales over which the low-stress regime is as effective as the high-stress regime, due to the competition between the two species. For multiple periodic treatments, competition can ensure that the minimal population size is attained during the first pulse when the high-stress regime is short, which implies that a single short pulse can be more effective than a more protracted regime. Our results suggest that when the duration of the high-stress environment is restricted, a treatment with one or multiple shorter pulses can produce better outcomes than a single long treatment. If ecological competition is to be exploited for treatments, it is crucial to determine these timescales, and estimate for the minimal population threshold that suffices for extinction. These parameters can be quantified by experiment.

physics.bio-ph↗

arXiv2018

The PomXYZ Proteins Self-Organize on the Bacterial Nucleoid to Stimulate Cell Division

Cell division site positioning is precisely regulated to generate correctly sized and shaped daughters. We uncover a novel strategy to position the FtsZ cytokinetic ring at midcell in the social bacterium Myxococcus xanthus. PomX, PomY and the nucleoid-binding ParA/MinD ATPase PomZ self-assemble forming a large nucleoid-associated complex that localizes at the division site before FtsZ to directly guide and stimulate division. PomXYZ localization is generated through self-organized biased random motion on the nucleoid towards midcell and constrained motion at midcell. Experiments and theory show that PomXYZ motion is produced by diffusive PomZ fluxes on the nucleoid into the complex. Flux differences scale with the intracellular asymmetry of the complex and are converted into a local PomZ concentration gradient across the complex with translocation towards the higher PomZ concentration. At midcell, fluxes equalize resulting in constrained motion. Flux-based mechanisms may represent a general paradigm for positioning of macromolecular structures in bacteria.

q-bio.SC↗

arXiv2018

Regulation of Pom cluster dynamics in Myxococcus xanthus

Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ dimers interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and the PomZ protein. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter's midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ dimers on the nucleoid is slow relative to the cluster's velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.

q-bio.SC↗

arXiv2017

Evolution of a Fluctuating Population in a Randomly Switching Environment

Environment plays a fundamental role in the competition for resources, and hence in the evolution of populations. Here, we study a well-mixed, finite population consisting of two strains competing for the limited resources provided by an environment that randomly switches between states of abundance and scarcity. Assuming that one strain grows slightly faster than the other, we consider two scenarios--one of pure resource competition, and one in which one strain provides a public good--and investigate how environmental randomness (external noise) coupled to demographic (internal) noise determines the population's fixation properties and size distribution. By analytical means and simulations, we show that these coupled sources of noise can significantly enhance the fixation probability of the slower-growing species. We also show that the population size distribution can be unimodal, bimodal or multimodal and undergoes noise-induced transitions between these regimes when the rate of switching matches the population's growth rate.

q-bio.PE↗

arXiv2017

Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers

Autoinducers are small signaling molecules that mediate intercellular communication in microbial populations and trigger coordinated gene expression via "quorum sensing". Elucidating the mechanisms that control autoinducer production is, thus, pertinent to understanding collective microbial behavior, such as virulence and bioluminescence. Recent experiments have shown a heterogeneous promoter activity of autoinducer synthase genes, suggesting that some of the isogenic cells in a population might produce autoinducers, whereas others might not. However, the mechanism underlying this phenotypic heterogeneity in quorum-sensing microbial populations has remained elusive. In our theoretical model, cells synthesize and secrete autoinducers into the environment, up-regulate their production in this self-shaped environment, and non-producers replicate faster than producers. We show that the coupling between ecological and population dynamics through quorum sensing can induce phenotypic heterogeneity in microbial populations, suggesting an alternative mechanism to stochastic gene expression in bistable gene regulatory circuits.

q-bio.PE↗

arXiv2017

Master equations and the theory of stochastic path integrals

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approximation of rare event probabilities and derive path integral representations of Fokker-Planck equations. To make our review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

cond-mat.stat-mech↗

arXiv2017

Generic transport mechanisms for molecular traffic in cellular protrusions

Transport of molecular motors along protein filaments in a half-closed geometry is a common feature of biologically relevant processes in cellular protrusions. Using a lattice gas model we study how the interplay between active and diffusive transport and mass conservation leads to localised domain walls and tip localisation of the motors. We identify a mechanism for task sharing between the active motors (maintaining a gradient) and the diffusive motion (transport to the tip), which ensures that energy consumption is low and motor exchange mostly happens at the tip. These features are attributed to strong nearest-neighbour correlations that lead to a strong reduction of active currents, which we calculate analytically using an exact moment-identity, and might prove useful for the understanding of correlations and active transport also in more elaborate systems.

physics.bio-ph↗

arXiv2016

A nonequilibrium diffusion and capture mechanism ensures tip-localization of regulating proteins on dynamic filaments

Diffusive motion of regulatory enzymes on biopolymers with eventual capture at a reaction site is a common feature in cell biology. Using a lattice gas model we study the impact of diffusion and capture for a microtubule polymerase and a depolymerase. Our results show that the capture mechanism localizes the proteins and creates large-scale spatial correlations. We develop an analytic approximation that globally accounts for relevant correlations and yields results that are in excellent agreement with experimental data. Our results show that diffusion and capture operates most efficiently at cellular enzyme concentrations which points to in vivo relevance.

physics.bio-ph↗