Initiation of slime mold aggregation viewed as an instability.

doi:10.1016/0022-5193(70)90092-5

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Initiation of slime mold aggregation viewed as an instability.

Journal Article•DOI•

Initiation of slime mold aggregation viewed as an instability.

Evelyn Fox Keller¹, Lee A. Segel¹•Institutions (1)

Cornell University¹

01 Mar 1970-Journal of Theoretical Biology (J Theor Biol)-Vol. 26, Iss: 3, pp 399-415

TL;DR: A mathematical formulation of the general interaction of amoebae, as mediated by acrasin is presented, and a detailed analysis of the aggregation process is provided.

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About: This article is published in Journal of Theoretical Biology.The article was published on 1970-03-01. It has received 3125 citations till now. The article focuses on the topics: Cell aggregation & Population.

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Journal Article•DOI•

Model for Chemotaxis

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Evelyn Fox Keller¹, Lee A. Segel²•Institutions (2)

Cornell University¹, Rensselaer Polytechnic Institute²

01 Feb 1971-Journal of Theoretical Biology

TL;DR: The chemotactic response of unicellular microscopic organisms is viewed as analogous to Brownian motion, and a macroscopic flux is derived which is proportional to the chemical gradient.

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1,660 citations

Cites background or methods from "Initiation of slime mold aggregatio..."

...The dependence of cell density b(x, t) on position and time is described by the differential equation abjat = -w (7) where the vector flux J would be given by J = - j.Nb +xbVc. (8) (See Keller & Segel, 1970.)...
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...As in Keller & Segel (1970) one can adopt a phenomenological approach and proceed without waiting for the unfolding of microscopic detail, particularly as much of this detail does not affect the macroscopic phenomenon....
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...On the basis of general macroscopic arguments, we have previously formulated an equation for the macroscopic flux of cellular slime mold amebae and have used it to describe the initiation of aggregation in that system (Keller & Segel, 1970)....
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Journal Article•DOI•

A user’s guide to PDE models for chemotaxis

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Thomas Hillen¹, Kevin J. Painter²•Institutions (2)

University of Alberta¹, Heriot-Watt University²

01 Jan 2009-Journal of Mathematical Biology

TL;DR: This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.

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Abstract: Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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1,532 citations

Cites background or methods from "Initiation of slime mold aggregatio..."

...A number of models have been developed based on systems of equations similar to (1) that successfully capture many key features of the lifecycle [38,44]....
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...Theoretical and mathematical modelling of chemotaxis dates to the pioneering works of Patlak in the 1950s [86] and Keller and Segel in the 1970s [44,45]....
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From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

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Dirk Horstmann¹•Institutions (1)

University of Cologne¹

10 Jan 2003

TL;DR: This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models and offers possible generalizations of these results to more universal models.

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Abstract: This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

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1,138 citations

Cites background or methods from "Initiation of slime mold aggregatio..."

...Even though Keller’s and Segel’s stability analysis of the uniform state in [69] and the presented instability criterion is valid for a very general formulation of the system, the next “landmark” in the studies of the Keller-Segel model was the paper by V....
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...To model the aggregation of a cellular slime population they assume in [69] the following basic processes that take place during the aggregation phase:...
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...It is possible that the reduction to two equations that was done in [69] was too restrictive to cover all observable generated patterns during the aggregation of mobile species....
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...4) the uniform distribution (u0, v0) becomes unstable if [69] k2(u0,v0)v0) k1(u0,v0)u0 + u0f ′(v0) k3(v0)+v0k′ 3(v0) > 1....
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Journal Article•DOI•

Dispersion and Population Interactions

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Simon A. Levin

01 Mar 1974-The American Naturalist

TL;DR: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns).

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Abstract: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity. This arises due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns), effects of which are magnified by species interactions. In this way, homogeneous environments may become heterogeneous and heterogeneous environments even more so. In patchy environments, distinct patches are likely to be colonized initially by different species, and thereby a kind of founder effect results whereby individual patches evolve along different paths simply as a consequence of initial colonization patterns. Species which would be unable to invade may nevertheless survive by establishing themselves early and will moreover be found in lower densities in other areas as overflow from their "safe" areas. Spatially continuous environments may evolve toward es...

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1,097 citations

Book•

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

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Filippo Santambrogio

21 Oct 2015

TL;DR: In this paper, the primal and dual problems of one-dimensional problems are considered. But they do not consider the dual problems in L^1 and L^infinity theory.

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Abstract: Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

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1,015 citations

Cites background from "Initiation of slime mold aggregatio..."

...Keller-Segel An interesting model in mathematical biology (see [202, 203] for the original modeling) is the following: a population ρ of bacteria evolves in time, following diffusion and advection by a potential....
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References

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Book•

Hydrodynamic and Hydromagnetic Stability

[...]

Subrahmanyan Chandrasekhar

01 Jan 1961

11,419 citations

Journal Article•DOI•

The Chemical Basis of Morphogenesis

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A. M. Turing¹•Institutions (1)

University of Manchester¹

14 Aug 1952-Philosophical Transactions of the Royal Society B

TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.

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Abstract: It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

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9,015 citations

"Initiation of slime mold aggregatio..." refers background in this paper

...Instabilities of the kind which we shall consider have recently received some attention as possible mechanisms for certain kinds of structure or pattern formation in the biological world (Turing, 1952; Prigogine & Nicolis, 1967)....
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Journal Article•DOI•

On symmetry-breaking instabilities in dissipative systems

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Ilya Prigogine, Grégoire Nicolis

01 May 1967-Journal of Chemical Physics

TL;DR: In this paper, the authors investigate the possibility of an instability in purely dissipative systems involving chemical reactions and transport processes such as diffusion, but no hydrodynamic motion, and demonstrate that for well defined values of the constraints such as the chemical affinities of the over-all reactions and the constants involved, such systems can indeed become unstable.

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Abstract: The theory of hydrodynamic instability has always been an important part of fluid dynamics [see, e.g., Chandrasekhar, in Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, England, 1961) and Non‐Equilibrium Thermodynamics, Variation Techniques, and Stability, R. J. Donnelly, R. Herman, and I. Prigogine, Eds. (University of Chicago Press, Chicago, Ill., 1966)]. Such instabilities involve both convective processes (such as mechanical flow) and dissipative processes (such as viscous dissipation). We investigate the possibility of an instability in purely dissipative systems involving chemical reactions and transport processes such as diffusion, but no hydrodynamic motion. We demonstrate that for well‐defined values of the constraints such as the chemical affinities of the over‐all reactions and the constants involved, such systems can indeed become unstable. Such an instability is investigated following an example of autocatalytic reactions first proposed by Turing. The major feature of this ...

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485 citations

"Initiation of slime mold aggregatio..." refers background in this paper

...Instabilities of the kind which we shall consider have recently received some attention as possible mechanisms for certain kinds of structure or pattern formation in the biological world (Turing, 1952; Prigogine & Nicolis, 1967)....
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Journal Article•DOI•

The Cellular Slime Molds.

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Eugene H. Varney, John Tyler Bonner

01 Nov 1967-Bulletin of the Torrey Botanical Club

365 citations

Journal Article•DOI•

On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics☆

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F.G. Heineken¹, H. M. Tsuchiya¹, Rutherford Aris¹•Institutions (1)

University of Minnesota¹

01 Mar 1967-Bellman Prize in Mathematical Biosciences

TL;DR: Pseudosteady state hypothesis for enzyme reaction kinetics justified mathematically with asymptotic expansion of Michaelis-Menten kinetic equations solution as discussed by the authors, which is shown to be the same as the hypothesis for protein kinetics.

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Abstract: Pseudosteady state hypothesis for enzyme reaction kinetics justified mathematically with asymptotic expansion of Michaelis-Menten kinetic equations solution

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258 citations