The ability to survive starvation is an integral part of bacterial fitness and determines composition, turnover and biodiversity in microbial ecosystems. Starving bacteria enter a state known as plasmolysis in which their cytoplasm contracts from the cell wall. Plasmolysis is often thought to be a pathological, passive condition, arising automatically from the lack of ATP. Here we show that contrary to this notion, maintaining plasmolysis is an active, ATP-consuming state that is essential for starvation survival. We show that ion homeostasis to maintain plasmolysis consumes the largest part of the energy budget of starving cells and directly determines death rates in starvation. Our mathematical model accurately predicts death rates for various starvation conditions and perturbations. This enabled the development of an optimized starvation medium that would be ideally suited for preserving and transplanting natural microbial communities by maintaining viability but preventing outgrowth of a subset of the species. When bacteria starve, their cytoplasm detaches from the cell wall. A model now shows that this process determines bacterial death rates and can be controlled to keep bacteria viable in a starved state.

We investigate the constitutive equations when the functionally graded materials (FGMs) are considered under the strain gradient elasticity theory (SGET). The SGET considered includes both the Helmholtz and bi-Helmholtz types. The constitutive equations become more complicated and contain more terms due to the interaction between the material gradation and the nonlocal effect of the strain gradient. We also derive the governing par-tial differential equations (PDEs) by the balance law of linear momentum, and the PDEs also appear to be highly complicated. Under the assumption that material gradation is of the exponential form, a simpler set of governing PDEs can be reached. Solutions to those PDEs under the formulation of different modes of crack problems are discussed.Published by Elsevier Inc.

Central carbon metabolism is highly conserved across microbial species, but can catalyze very different pathways depending on the organism and their ecological niche. Here, we study the dynamic reorganization of central metabolism after switches between the two major opposing pathway configurations of central carbon metabolism, glycolysis, and gluconeogenesis in Escherichia coli, Pseudomonas aeruginosa, and Pseudomonas putida. We combined growth dynamics and dynamic changes in intracellular metabolite levels with a coarse-grained model that integrates fluxes, regulation, protein synthesis, and growth and uncovered fundamental limitations of the regulatory network: After nutrient shifts, metabolite concentrations collapse to their equilibrium, rendering the cell unable to sense which direction the flux is supposed to flow through the metabolic network. The cell can partially alleviate this by picking a preferred direction of regulation at the expense of increasing lag times in the opposite direction. Moreover, decreasing both lag times simultaneously comes at the cost of reduced growth rate or higher futile cycling between metabolic enzymes. These three trade-offs can explain why microorganisms specialize for either glycolytic or gluconeogenic substrates and can help elucidate the complex growth patterns exhibited by different microbial species.

The majority of microbes on earth, whether they live in the ocean, the soil or in animals, are not growing, but instead struggling to survive starvation 1–6 . Some genes and environmental conditions affecting starvation survival have been identified 7–13 , but despite almost a century of study 14–16 , we do not know which processes lead to irreversible loss of viability, which maintenance processes counteract them and how lifespan is determined from the balance of these opposing processes. Here, we used time-lapse microscopy to capture and characterize the cell death process of E. coli during carbon starvation for the first time. We found that a lack of nutrients results in the collapse of ion homeostasis, triggering a positive-feedback cascade of osmotic swelling and membrane permeabilization that ultimately results in lysis. Based on these findings, we hypothesized that ion transport is the major energetic requirement for starving cells and the primary determinant of the timing of lysis. We therefore developed a mathematical model that integrates ion homeostasis and ‘cannibalistic’ nutrient recycling from perished cells 16,17 to predict lifespan changes under diverse conditions, such as changes of cell size, medium composition, and prior growth conditions. Guided by model predictions, we found that cell death during starvation could be dramatically slowed by replacing inorganic ions from the medium with a non-permeating osmoprotectant, removing the cost of ion homeostasis and preventing lysis. Our quantitative and predictive model explains how survival kinetics are determined in starvation and elucidates the mechanistic underpinnings of starvation survival.

Central carbon metabolism is highly conserved across microbial species, but operates in very different ways depending on the organism and their ecological niche. Here, we study the dynamic re-organization of central metabolism after switches between the two major opposing pathway configurations of central carbon metabolism, glycolysis and gluconeogenesis in different bacteria. We combined growth dynamics and dynamic changes of intracellular metabolite levels with a coarse-grained model that integrates fluxes, regulation, protein synthesis and growth and uncovered fundamental limitations of the regulatory network: after nutrient shifts, metabolite concentrations collapse to their equilibrium, turning the cell blind to which direction the flux is supposed to flow through the metabolic network. The cell can partially alleviate this blindness by picking a preferred direction of regulation, at the expense of increasing lag times in the opposite direction. Moreover, decreasing both lag times simultaneously comes at the cost of reduced growth rate or higher futile cycling between metabolic enzymes. These three trade-offs can explain why microorganisms specialize for either glycolytic or gluconeogenic substrates and can help elucidate the complex growth patterns exhibited by different microbial species.

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).