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 starvation1–6. Some genes and environmental conditions affecting starvation survival have been identified7–13, but despite almost a century of study14–16, 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 cells16,17 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.
Microbes exhibit an astounding phenotypic diversity, including large variations in growth rates and their ability to adapt to sudden changes in conditions. Understanding such fundamental traits based on molecular mechanisms has largely remained elusive due to the complexity of the underlying metabolic and regulatory network. Here, we study the two major opposing flux configurations of central carbon metabolism, glycolysis and gluconeogenesis using a coarse-grained kinetic model. Our model captures a remarkable self-organization of metabolism in response to nutrient availability: key regulatory metabolites respond to the directionality of flux and adjust activity and expression levels of metabolic enzymes to efficiently guide flux through the metabolic network. The model recapitulates experimentally observed temporal dynamics of metabolite concentrations, enzyme abundances and growth rates during metabolic shifts. In addition, it reveals a fundamental limitation of flux based sensing: after nutrient shifts, metabolite levels collapse and the cell becomes "blind" to direction of flux. The cell can partially overcome this limitation at the cost of three trade-offs between lag times, growth rates and metabolic futile cycling that constrain the efficiency of self-organization after nutrient shifts. We show that these trade-offs impose a preferential flux direction and can explain the glycolysis preference observed for Escherichia coli, Saccharomyces cerevisiae and Bacillus subtilis, which only shift fast to glycolysis, but slow to gluconeogenisis Remarkably, as predicted from the model, we experimentally confirmed this preference could also be reversed in different species. Indeed, P. aeruginosa shows precisely the opposite phenotypic patterns, switching very quickly to gluconeogenesis, but showing multi-hour lag times that sharply increase with pre-shift growth rate in shifts to glycolysis. These trade-offs between opposing flux directions can explain specialization of microorganisms for either glycolytic or gluconeogenic substrates and can help elucidate the complex phenotypic 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).
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