ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

2002 ◽  
Vol 67 (3) ◽  
pp. 897-909
Author(s):  
David M. Evans ◽  
M. E. Pantano
Keyword(s):  
Growth Rate ◽  
Automorphism Group ◽  
Finite Subset ◽  
Growth Rates ◽  
Algebraic Closure ◽  
Fast Growth ◽  
Permutation Groups ◽  
Natural Numbers ◽  
Categorical Structure ◽  
Image Position

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

scholarly journals Deciphering the Principles of Bacterial Nitrogen Dietary Preferences: a Strategy for Nutrient Containment

mBio ◽  
2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Jilong Wang ◽  
Dalai Yan ◽  
Ray Dixon ◽  
Yi-Ping Wang
Keyword(s):  
Amino Acids ◽  
Growth Rate ◽  
Growth Rates ◽  
Regulatory Networks ◽  
Metabolic Flux ◽  
Nitrogen Sources ◽  
Fast Growth ◽  
Ammonia Assimilation ◽  
Wild Type ◽  
Dietary Preferences

ABSTRACT A fundamental question in microbial physiology concerns why organisms prefer certain nutrients to others. For example, among different nitrogen sources, ammonium is the preferred nitrogen source, supporting fast growth, whereas alternative nitrogen sources, such as certain amino acids, are considered to be poor nitrogen sources, supporting much slower exponential growth. However, the physiological/regulatory logic behind such nitrogen dietary choices remains elusive. In this study, by engineering Escherichia coli , we switched the dietary preferences toward amino acids, with growth rates equivalent to that of the wild-type strain grown on ammonia. However, when the engineered strain was cultured together with wild-type E. coli , this growth advantage was diminished as a consequence of ammonium leakage from the transport-and-catabolism (TC)-enhanced (TCE) cells, which are preferentially utilized by wild-type bacteria. Our results reveal that the nitrogen regulatory (Ntr) system fine tunes the expression of amino acid transport and catabolism components to match the flux through the ammonia assimilation pathway such that essential nutrients are retained, but, as a consequence, the fast growth rate on amino acids is sacrificed. IMPORTANCE Bacteria exhibit different growth rates under various nutrient conditions. These environmentally related behaviors reflect the coordination between metabolism and the underlying regulatory networks. In the present study, we investigated the intertwined nitrogen metabolic and nitrogen regulatory systems to understand the growth differences between rich and poor nitrogen sources. Although maximal growth rate is considered to be evolutionarily advantageous for bacteria (as remarked by François Jacob, who said that the “dream” of every cell is to become two cells), we showed that negative-feedback loops in the regulatory system inhibit growth rates on amino acids. We demonstrated that in the absence of regulatory feedback, amino acids are capable of supporting fast growth rates, but this results in ammonia leaking out from cells as “waste,” benefiting the growth of competitors. These findings provide important insights into the regulatory logic that controls metabolic flux and ensures nutrient containment but consequently sacrifices growth rate.

On the asymptotic behaviour of sample spacings

1981 ◽  
Vol 90 (2) ◽  
pp. 293-303 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Growth Rate ◽  
Asymptotic Behaviour ◽  
Growth Rates ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Unit Interval ◽  
Sample Spacing ◽  
Image Position ◽  
Almost All ◽  
Sample Spacings

SummarySuppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.Theorem 2. With probability one we havefor almost all x.Theorem 3. With probability one we have

scholarly journals GROUPES FINS

2014 ◽  
Vol 79 (4) ◽  
pp. 1120-1132
Author(s):  
CÉDRIC MILLIET
Keyword(s):  
Topological Space ◽  
Finite Index ◽  
Finite Subset ◽  
Algebraic Closure ◽  
Infinite Field ◽  
Image Position ◽  
Stable Structures

AbstractWe investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left( {dc{l^{eq}}\left( A \right)} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left( {x,y} \right)$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left( A \right)$ofAof definable subgroups ofGover$acl\left( A \right)$which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.

scholarly journals XMAP215 promotes microtubule catastrophe by disrupting the growing microtubule end

2020 ◽  
Author(s):  
Veronica Farmer ◽  
Göker Arpağ ◽  
Sarah Hall ◽  
Marija Zanic
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Fast Growth ◽  
Fast Growing ◽  
Microtubule Stability ◽  
Large Fluctuations ◽  
Microtubule Growth ◽  
Enhanced Stability

ABSTRACTThe GTP-tubulin cap is widely accepted to protect microtubules against catastrophe. The GTP-cap size is thought to increase with the microtubule growth rate, presumably endowing fast-growing microtubules with enhanced stability. It is unknown what GTP-cap properties permit frequent microtubule catastrophe despite fast growth. Here, we investigate microtubules grown in vitro in the presence and absence of the microtubule polymerase XMAP215. Using EB1 as a GTP-cap marker, we find that GTP-cap size increases regardless of whether growth acceleration is achieved by increasing tubulin concentration or by XMAP215. In spite of the increased mean GTP-cap size, microtubules grown with XMAP215 display increased catastrophe frequency, in contrast to microtubules grown with more tubulin, for which catastrophe is abolished. However, microtubules polymerized with XMAP215 have large fluctuations in growth rate and EB1 intensity; display tapered and curled ends; and undergo catastrophe at faster growth rates and with higher EB1 end-localization. Our results underscore the role of growth irregularities in overall microtubule stability.

scholarly journals Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield

2017 ◽  
Author(s):  
Meike T. Wortel ◽  
Elad Noor ◽  
Michael Ferris ◽  
Frank J. Bruggeman ◽  
Wolfram Liebermeister
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Metabolic Pathways ◽  
Biomass Yield ◽  
Cost Minimization ◽  
Fast Growth ◽  
Basic Question ◽  
Model Parameters ◽  
Trade Off ◽  
Trade Offs

AbstractMicrobes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism inE. coli, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.Author SummaryWhen cells compete for nutrients, those that grow faster and produce more offspring per time are favored by natural selection. In contrast, when cells need to maximize the cell number at a limited nutrient supply, fast growth does not matter and an efficient use of nutrients (i.e. high biomass yield) is essential. This raises a basic question about metabolism: can cells achieve high growth rates and yields simultaneously, or is there a conflict between the two goals? Using a new modeling method called Enzymatic Flux Cost Minimization (EFCM), we predict cellular growth rates and find that growth rate/yield trade-offs and the ensuing preference for enzyme-efficient or substrate-efficient metabolic pathways are not universal, but depend on growth conditions such as external glucose and oxygen concentrations.

Jordan permutation groups and limits of 𝐷-relations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asma Ibrahim Almazaydeh ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Categorical Structure

Abstract We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

scholarly journals The Absence of (p)ppGpp Renders Initiation of Escherichia coli Chromosomal DNA Synthesis Independent of Growth Rates

mBio ◽  
2020 ◽  
Vol 11 (2) ◽  
Author(s):  
Llorenç Fernández-Coll ◽  
Monika Maciag-Dorszynska ◽  
Krishma Tailor ◽  
Stephen Vadia ◽  
Petra Anne Levin ◽  
...  
Keyword(s):  
Escherichia Coli ◽  
Growth Rate ◽  
Dna Replication ◽  
Dna Synthesis ◽  
Growth Rates ◽  
Exponential Growth ◽  
Replication Initiation ◽  
Fast Growth ◽  
Chromosomal Dna ◽  
Wide Range

ABSTRACT The initiation of Escherichia coli chromosomal DNA replication starts with the oligomerization of the DnaA protein at repeat sequences within the origin (ori) region. The amount of ori DNA per cell directly correlates with the growth rate. During fast growth, the cell generation time is shorter than the time required for complete DNA replication; therefore, overlapping rounds of chromosome replication are required. Under these circumstances, the ori region DNA abundance exceeds the DNA abundance in the termination (ter) region. Here, high ori/ter ratios are found to persist in (p)ppGpp-deficient [(p)ppGpp0] cells over a wide range of balanced exponential growth rates determined by medium composition. Evidently, (p)ppGpp is necessary to maintain the usual correlation of slow DNA replication initiation with a low growth rate. Conversely, ori/ter ratios are lowered when cell growth is slowed by incrementally increasing even low constitutive basal levels of (p)ppGpp without stress, as if (p)ppGpp alone is sufficient for this response. There are several previous reports of (p)ppGpp inhibition of chromosomal DNA synthesis initiation that occurs with very high levels of (p)ppGpp that stop growth, as during the stringent starvation response or during serine hydroxamate treatment. This work suggests that low physiological levels of (p)ppGpp have significant functions in growing cells without stress through a mechanism involving negative supercoiling, which is likely mediated by (p)ppGpp regulation of DNA gyrase. IMPORTANCE Bacterial cells regulate their own chromosomal DNA synthesis and cell division depending on the growth conditions, producing more DNA when growing in nutritionally rich media than in poor media (i.e., human gut versus water reservoir). The accumulation of the nucleotide analog (p)ppGpp is usually viewed as serving to warn cells of impending peril due to otherwise lethal sources of stress, which stops growth and inhibits DNA, RNA, and protein synthesis. This work importantly finds that small physiological changes in (p)ppGpp basal levels associated with slow balanced exponential growth incrementally inhibit the intricate process of initiation of chromosomal DNA synthesis. Without (p)ppGpp, initiations mimic the high rates present during fast growth. Here, we report that the effect of (p)ppGpp may be due to the regulation of the expression of gyrase, an important enzyme for the replication of DNA that is a current target of several antibiotics.

scholarly journals XMAP215 promotes microtubule catastrophe by disrupting the growing microtubule end

2021 ◽  
Vol 220 (10) ◽  
Author(s):  
Veronica Farmer ◽  
Göker Arpağ ◽  
Sarah L. Hall ◽  
Marija Zanic
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Fast Growth ◽  
Protective Effects ◽  
Fast Growing ◽  
Large Fluctuations ◽  
Structural Perturbations ◽  
Microtubule Growth ◽  
Presence And Absence ◽  
Enhanced Stability

The GTP-tubulin cap is widely accepted to protect microtubules against catastrophe. The GTP-cap size is thought to increase with the microtubule growth rate, presumably endowing fast-growing microtubules with enhanced stability. It is unknown what GTP-cap properties permit frequent microtubule catastrophe despite fast growth. Here, we investigate microtubules growing in the presence and absence of the polymerase XMAP215. Using EB1 as a GTP-cap marker, we find that GTP-cap size increases regardless of whether growth acceleration is achieved by increasing tubulin concentration or by XMAP215. Despite increased mean GTP-cap size, microtubules grown with XMAP215 display increased catastrophe frequency, in contrast to microtubules grown with more tubulin, for which catastrophe is abolished. However, microtubules polymerized with XMAP215 have large fluctuations in growth rate; display tapered and curled ends; and undergo catastrophe at faster growth rates and with higher EB1 end-localization. Our results suggest that structural perturbations induced by XMAP215 override the protective effects of the GTP-cap, ultimately driving microtubule catastrophe.

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