TY - JOUR
T1 - On the stationary state analysis of reaction-diffusion mechanisms for biological pattern formation
AU - Newman, Stuart A.
AU - Frisch, H. L.
AU - Percus, J. K.
N1 - Funding Information:
We thank two anonymous reviewers for suggesting improvements in the manuscript. This work was supported by grants from the NSF (DCB-8609106) and NIH (HD22564) (S.A.N.),
Funding Information:
donors of the Petroleum Research Fund of the American Chemical Society (H.L.F.), and DOE contract DE-ACO2-7603077 (J.K.P.).
PY - 1988/9/21
Y1 - 1988/9/21
N2 - We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.
AB - We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.
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U2 - 10.1016/S0022-5193(88)80201-7
DO - 10.1016/S0022-5193(88)80201-7
M3 - Article
C2 - 3244280
AN - SCOPUS:0024290623
SN - 0022-5193
VL - 134
SP - 183
EP - 197
JO - Journal of Theoretical Biology
JF - Journal of Theoretical Biology
IS - 2
ER -