THE LANGEVIN OR KRAMERS APPROACH TO BIOLOGICAL MODELING

2004 ◽  
Vol 14 (10) ◽  
pp. 1561-1583 ◽  
Author(s):  
KARL P. HADELER ◽  
THOMAS HILLEN ◽  
FRITHJOF LUTSCHER
Keyword(s):  
Wave Equations ◽  
Space Variable ◽  
Linear Partial Differential Equation ◽  
Reaction Diffusion ◽  
Nonlinear Problems ◽  
Kramers Equation ◽  
Moment Approach ◽  
The Moment ◽  
Nonlinear Damped Wave Equations ◽  
And Diffusion

In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

scholarly journals Reaction-Diffusion Systems: Self-Balancing Diffusion and the Use of the Extent of Reaction as a Descriptor of Reaction Kinetics

2019 ◽  
Author(s):  
Miloslav Pekař
Keyword(s):  
Reaction Kinetics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
And Diffusion ◽  
Reaction And Diffusion

Self-balancing diffusion is a concept which restricts the introduction of extents of reactions. This concept is analyzed in detail for mass- and molar-based balances of reaction-diffusion mixtures, in relation to non-self-balancing cases, and with respect to its practical consequences. A note on a recent generalization of the concept of reaction and diffusion extents is also included.<br>

The Chemical Basis of Morphogenesis (1952)

2004 ◽  
Author(s):  
Alan Turing
Keyword(s):  
Reaction System ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Stationary Waves ◽  
Chemical Substances ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Physical Laws ◽  
And Diffusion

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 homogenous, 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. In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge. The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points.

Role of facilitated diffusion of calcium by calbindin in intestinal calcium absorption

1992 ◽  
Vol 262 (2) ◽  
pp. C517-C526 ◽  
Author(s):  
J. J. Feher ◽  
C. S. Fullmer ◽  
R. H. Wasserman
Keyword(s):  
Negative Feedback ◽  
Calcium Absorption ◽  
Feedback Inhibition ◽  
Basolateral Membrane ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Facilitated Diffusion ◽  
Calbindin D9k ◽  
And Diffusion

Computer simulations of transcellular Ca2+ transport in enterocytes were carried out using the simulation program SPICE. The program incorporated a negative-feedback entry of Ca2+ at the brush-border membrane that was characterized by an inhibitor constant of 0.5 microM cytosolic Ca2+ concentration ([Ca2+]). The basolateral Ca(2+)-ATPase was simulated by a four-step mechanism that resulted in Michaelis-Menten kinetics with a Michaelis constant of 0.24 microM [Ca2+]. The cytosolic diffusion of Ca2+ was simulated by dividing the cytosol into 10 slabs of equal width. Ca2+ binding to calbindin-D9K was simulated in each slab, and diffusion of free Ca2+, free calbindin, and Ca(2+)-laden calbindin was simulated between each slab. The cytosolic [Ca2+] of the simulated cells was regulated within the physiological range. Calbindin-D9K reduced the cytosolic [Ca2+] gradient, increased Ca2+ entry into the cell by removing the negative-feedback inhibition of Ca2+ entry, increased cytosolic Ca2+ flow, and increased the efflux of Ca2+ across the basolateral membrane by increasing the free [Ca2+] immediately adjacent to the pump. The enhancement of transcellular Ca2+ transport was nearly linearly dependent on calbindin-D9K concentration. The values of the dissociation constant (Kd) for calbindin-D9K were previously obtained experimentally in the presence and absence of KCl. Calbindin with the Kd obtained in the presence of KCl enhanced the simulated Ca2+ transport more than with the Kd obtained in the absence of KCl. This result suggests that the physiological Kd of calbindin is optimal for the enhancement of transcellular Ca2+ transport. The simulated Ca2+ flow was less than that predicted from the "near-equilibrium" analytic solution of the reaction-diffusion problem.

Programmable reactions and diffusion using DNA for pattern formation in hydrogel medium

2019 ◽  
Vol 4 (3) ◽  
pp. 639-643 ◽  
Author(s):  
Keita Abe ◽  
Ibuki Kawamata ◽  
Shin-ichiro M. Nomura ◽  
Satoshi Murata
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
And Diffusion

We demonstrate a method of pattern formation based on an artificial reaction diffusion system in hydrogel medium.

Modeling of Phase Competition and Diffusion Zone Morphology Evolution at Initial Stages of Reaction Diffusion

2005 ◽  
Vol 237-240 ◽  
pp. 1193-1198 ◽  
Author(s):  
Mykola Pasichnyy ◽  
Andriy Gusak
Keyword(s):  
Monte Carlo ◽  
Strong Dependence ◽  
Reaction Diffusion ◽  
Interatomic Interaction ◽  
Reactive Diffusion ◽  
Morphology Evolution ◽  
Interaction Energies ◽  
Phase Competition ◽  
Intermediate Phases ◽  
And Diffusion

The initial stages of reactive diffusion have been studied by Monte-Carlo simulations. New MC-scheme describing the competition of two ordered intermediate phases is proposed. Main peculiarity of presented MC-model is a strong dependence of interatomic interaction energies on the local atomic surrounding, enabling us to distinguish new phases more distinctly.

The reaction-diffusion (RD) theory of wool (hair) follicle initiation and development. II. Original secondary follicles

1995 ◽  
Vol 46 (2) ◽  
pp. 357 ◽  
Author(s):  
BN Nagorcka
Keyword(s):  
Early Stage ◽  
Fibre Diameter ◽  
Reaction Diffusion ◽  
Future Generations ◽  
Chemical Components ◽  
Full Account ◽  
System Equations ◽  
And Diffusion ◽  
Total P ◽  
Good Agreement

In an accompanying paper it was shown that a spatial prepattern mechanism based on a biochemical reaction referred to as a reaction-diffusion (RD) system is able to account for many aspects of the initiation and development of primary (P) wool follicles. In this paper the same RD system is applied to the initiation and development of original secondary (SO) follicles. Prepatterns are generated by solving the equations describing the reaction and diffusion of the chemical components of the RD system in early stage follicles. It is demonstrated that the prepattern mechanism can account for the loss of a sweat gland causing a change from P follicle initiation to SO follicle initiation. The RD system equations are also solved in the epidermis. The time sequence of prepatterns obtained in the epidermis account for the tendency of SO follicles to group with P follicles, by initiating in-between members of the trio group of P follicles as well as in between existing SO follicles. The prepatterns obtained did not account for the tendency of secondary follicles to initiate on the posterior side of the trio group. Good agreement was obtained between the predicted increase in total follicle density and the increase in follicle density observed during follicle initiation by Carter and Hardy (1947), provided full account was taken of the interaction between existing follicles and each new future generations of follicles. The prepattern mechanism provides a fundamental basis for an inverse genetic correlation between total P and SO follicle density and fibre diameter.

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