Model for Chemotaxis

doi:10.1016/0022-5193(71)90050-6

Home
/
Papers
/
Model for Chemotaxis

Journal Article•DOI•

Model for Chemotaxis

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

Cornell University¹, Rensselaer Polytechnic Institute²

01 Feb 1971-Journal of Theoretical Biology (Academic Press)-Vol. 30, Iss: 2, pp 225-234

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.

read less

About: This article is published in Journal of Theoretical Biology.The article was published on 1971-02-01. It has received 1660 citations till now.

...read moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

Collective Motion

[...]

Tamás Vicsek, Anna Zafeiris

24 Oct 2010-arXiv: Statistical Mechanics

TL;DR: In this paper, the basic laws describing the essential aspects of collective motion are reviewed and a discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, are provided.

...read moreread less

Abstract: We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior Our aim is to provide a balanced discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, so that readers with a variety of background could get both the basics and a broader, more detailed picture of the field The observations we report on include systems consisting of units ranging from macromolecules through metallic rods and robots to groups of animals and people Some emphasis is put on models that are simple and realistic enough to reproduce the numerous related observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many simultaneously moving entities As such, these models allow the establishing of a few fundamental principles of flocking In particular, it is demonstrated, that in spite of considerable differences, a number of deep analogies exist between equilibrium statistical physics systems and those made of self-propelled (in most cases living) units In both cases only a few well defined macroscopic/collective states occur and the transitions between these states follow a similar scenario, involving discontinuity and algebraic divergences

...read moreread less

2,120 citations

Journal Article•DOI•

A user’s guide to PDE models for chemotaxis

[...]

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.

...read moreread less

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.

...read moreread less

1,532 citations

Cites background or methods from "Model for Chemotaxis"

...In by far the majority of applications a constant diffusion coefficient is assumed, yet it is far more likely that this term should depend nonlinearly on the signal concentration and/or the cell density, as can be seen from derivations of Keller–Segel type systems through the various approaches mentioned in the introduction [15,45,80,92,95,96]....
[...]
...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]....
[...]
...Briefly, these are (i) arguments based on Fourier’s law and Fick’s law [45], (ii) biased random walk approaches [78], (iii) interacting particle systems [97], (iv) transport equations [2] or [35], and (v) stochastic processes [86]....
[...]

From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

[...]

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.

...read moreread less

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.

...read moreread less

1,138 citations

Book•

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

[...]

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.

...read moreread less

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.

...read moreread less

1,015 citations

Cites background from "Model for Chemotaxis"

...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....
[...]

Journal Article•DOI•

Traveling bands of chemotactic bacteria: a theoretical analysis.

[...]

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

Cornell University¹, Rensselaer Polytechnic Institute²

01 Feb 1971-Journal of Theoretical Biology

TL;DR: A phenomenological theory of traveling bands is developed starting with partial differential equations which describes the consumption of the critical substrate and the change in bacterial density due to random motion and to chemotaxis and predicts the shapes of the graphs of bacterial density and substrate concentration in the traveling band.

...read moreread less

977 citations

1
2
3
4
…
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Stochastic problems in Physics and Astronomy

[...]

Subrahmanyan Chandrasekhar¹•Institutions (1)

University of Chicago¹

01 Jan 1943-Reviews of Modern Physics

6,674 citations

Stochastic Problems in Physics and Astronomy

[...]

Subrahmanyan Chandrasekhar¹•Institutions (1)

University of Chicago¹

01 Jan 1954

6,117 citations

"Model for Chemotaxis" refers background or methods in this paper

...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.)...
[...]
...…we have J(x) = ildf[c(S+frA)]~(s)ds- xjAf[c(s-&A)]b(s)ds. (1) x Using an approximation which is often employed in theoretical studies of Brownian motion (Chandrasekhar, 1943), we keep only the lowest order terms in A, so that equation (1) becomes J(x) rz A’( -f[c(x)]b’(x) +(a- l)f’[c(x)]b(x)c’(x)}....
[...]
...The boundary condition to be imposed must be determined by the experimental configuration.t Once the conditional probability function has i See Chandrasekhar (1943) for interesting special solutions of equations (9) and (10)....
[...]

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

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.

...read moreread less

3,125 citations

"Model for Chemotaxis" refers background or methods in this paper

...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.)...
[...]
...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....
[...]
...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)....
[...]

Journal Article•DOI•

Chemotaxis in bacteria.

[...]

Julius Adler¹•Institutions (1)

University of Wisconsin-Madison¹

12 Aug 1966-Science

TL;DR: These results show that E. coli is chemotactic toward oxygen and energy sources such as galactose, glucose, aspartic acid, threonine, or serine and that chemotaxis allows bacteria to find that environment which provides them with the greatest supply of energy.

...read moreread less

Abstract: Motile Escherichia coli placed at one end of a capillary tube containing an energy source and oxygen migrate out into the tube in one or two bands, which are clearly visible to the naked eye and can also be demonstrated by photography, microscopy, and densitometry and by assaying for bacteria throughout the tube. The formation of two bands is not due to heterogeneity among the bacteria, since the bacteria in each band, when reused, will form two more bands. If an anaerobically utilizable energy source such as galactose is present in excess over the oxygen, the first band consumes all the oxygen and a part of the sugar and the second band uses the residual sugar anaerobically. On the other hand, if oxygen is present in excess over the sugar, the first band oxidizes all the sugar and leaves behind unused oxygen, and the second band uses up the residual oxygen to oxidize an endogenous energy source. The essence of the matter is that the bacteria create a gradient of oxygen or of an energy source, and then they move preferentially in the direction of the higher concentration of the chemical. As a consequence, bands of bacteria (or rings of bacteria in the case of agar plates) form and move out. These results show that E. coli is chemotactic toward oxygen and energy sources such as galactose, glucose, aspartic acid, threonine, or serine. The full repertoire of chemotactic responses by E. coli is no doubt greater than this, and a more complete list remains to be compiled. The studies reported here demonstrate that chemotaxis allows bacteria to find that environment which provides them with the greatest supply of energy. It is clearly an advantage for bacteria to be able to carry out chemotaxis, since by this means they can avoid unfavorable conditions and seek optimum surroundings. Finally, it is necessary to acknowledge the pioneering work of Englemann, Pfeffer, and the other late-19thcentury biologists who discovered chemotaxis in bacteria, and to point out that the studies reported here fully confirm the earlier reports of Beijerinck (4) and Sherris and his collaborators (5,6) on a band of bacteria chemotactic toward oxygen. By using a chemically defined medium instead of a complex broth, it has been possible to study this band more closely and to demonstrate in addition the occurrence of a second band of bacteria chemotactic toward an energy source. Beijerinck (4) did, in fact, sometimes observe a second band, but he did not offer an explanation for it.

...read moreread less

1,116 citations

"Model for Chemotaxis" refers background in this paper

...For organisms such as flagellated cells which t In the experiments of Adler (1966a), for example, typical concentrations of the critical substrate O2 are of the order of lo-* M....
[...]
...Introduction The chemotactic sensitivity of such one-celled organisms as Escherichia coli (Adler, 1966a,b) and myxamebae (see, e.g. Bonner, 1967) has been well documented, but the ability of an organism of microscopic dimensions to sense and respond to macroscopic chemical gradients has often been…...
[...]

Journal Article•DOI•

Traveling bands of chemotactic bacteria: a theoretical analysis.

[...]

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

Cornell University¹, Rensselaer Polytechnic Institute²

01 Feb 1971-Journal of Theoretical Biology

...read moreread less

977 citations

"Model for Chemotaxis" refers background in this paper

...In a study of traveling bands in E. co& (Keller & Segel, 1971) based on a flux equation in the form of equation (3), the authors were able to find solutions which seem to be in reasonable accord with observation under the assumptions p = constant and x = 6c-l, 6 constant....
[...]
...On the other hand, the fact that the phenomenological analysis of Keller & Segel (1971) seems to capture the principal features of travelling bands does not demonstrate the validity of the particular assumptions they made concerning the functions p and x....
[...]