Activator-inhibitor Model Research Articles

Binding affinity of bicarbonate and formate in herbicide-resistant D1 mutants of Synechococcus sp. PCC 7942

We examined the effects of mutations at amino acid residues S264 and F255 in the D1 protein on the binding affinity of the stimulatory anion bicarbonate and inhibitory anion formate in Photosystem II (PS II) in Synechococcus sp. PCC 7942. Measurements on the rates of oxygen evolution in the wild type and mutant cells in the presence of different concentrations of formate with a fixed bicarbonate concentration and vice versa, analyzed in terms of an equilibrium activator-inhibitor model, led to the conclusion that the equilibrium dissociation constant for bicarbonate is increased in the mutants, while that of the formate remains unchanged (11±0.5 mM). The hierarchy of the equilibrium dissociation constant for bicarbonate (highest to lowest, ±2 μM) was: D1-F255L/S264A (46 μM)>D1-F255Y/ S264A (31 μM)≈D1-S264A (34 μM)≈D1-F255Y (33 μM)>wild type (25 μM). The data suggest the importance of D1-S264 and D1-F255 in the bicarbonate binding niche. A possible involvement of bicarbonate and these two residues in the protonation of QB (-), the reduced secondary plastoquinone of PS II, in the D1 protein is discussed.

Global Existence of Weak Solutions for Interface Equations Coupled with Diffusion Equations

A weak formulation for an interface dynamics coupled with a diffusion equation is introduced. A global-in-time weak solution is constructed for an arbitrary initial data under a periodic boundary condition. The result applies to the interface equation obtained as a certain singular limit of some reaction-diffusion systems including the activator-inhibitor model.

Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type

We consider a bistable reaction-diffusion equation coupled with a time-dependent constrained condition $$\left\{ \begin{gathered} u_t = \varepsilon ^2 u_{xx} + \sigma ^{ - 2} [u(1 - u^2 ) - \xi ] \hfill \\ x \in I = (0,1), t > 0, \hfill \\ \xi _t = \int_I {udx} - \gamma \xi \hfill \\ \end{gathered} \right.$$ where γ, δ and e are positive constants. This equation lies in a framework of activator-inhibitor models which arise in biology. When e is sufficiently small, it is found that internal layers of widthO(e) appear in theu-component under the zero-flux boundary conditions, and that these layers propagate very slowly with velocityO(e−A/e) for some positive constantA.

Kinetically cooperative models: boundary movement in optical resolution, phase transitions, and biological morphogenesis

Computer simulations are described for simple models of kinetically cooperative competition between two or more a priori equally matched antagonists. These might be optically enantiomeric molecules, crystal defects in two phases, differentiation states of biological cells, etc. The models give a perspective on the relationship between the reaction–diffusion theory of pattern formation, homeogenetic induction between biological cells, and the currently popular "cellular automaton" computer programmes. The models express self-activation without inhibition. Therefore, they do not have all that is needed for indefinitely long stabilization of pattern by reaction–diffusion, as first established in the Turing activator–inhibitor model and used later in most reaction–diffusion models. But by the same token, boundaries between disparate regions do not reach stable positions, but must move until they finally reach the edges of the system and disappear. Therefore, these models are convenient for studying aspects of boundary movement in the régime of small-number statistics. To see this without artefacts from array geometry, we use hexagonal arrays in place of the more popular square ones.We conclude that the behaviour of these models in one respect contrasts with and in another resembles the expected deterministic behaviour in large-number statistics. Regions totally surrounded by an antagonist often grow, while deterministically they must shrink. On the other hand, boundaries tend to straighten, as they would deterministically. In a system of overall rectangular habit, with or without periodic edge conditions making it a cylinder or a torus, boundaries tend to align with the shorter dimension, making a pattern of "square stripes".

Oscillations in theoretical models of induction

A two-variable model for the genetic regulatory mechanism of induction is proposed. In a feedforward step an autocatalytically accumulated substrate induces the transcription of its own degrading enzyme. The differential equations for enzyme and substrate are treated analytically and it is found that in a defined parameter range the system becomes unstable and shows structurally stable limit cycle oscillations. The system behaves like an activator-inhibitor model and instability is likely to arise if the transcription process is slow. In a slightly modified system oscillations inside a cell are generated if an external parameter (extracellular substrate concentration) exceeds a certain threshold and all other parameters are unchanged. Possible biological implications of these results are destabilization of metabolic units by transport processes and feedforward catalysis.

A local activator-inhibitor model of vertebrate skin patterns

A model for vertebrate skin patterns is presented in which the differentiated (colored) pigment cells produce two diffusible morphogens, an activator and an inhibitor. The concentrations of these two substances at any point on the skin determine whether a pigment cell at that point will be colored or not. Computer simulations with this model show many realistic features of spot and stripe patterns found in vertebrates.

A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt

A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt

Morphogen pattern formation and development in growth.

The dependence of the spatial concentration profiles of morphogens on a characteristic dimension is obtained by continuation techniques for Gierer and Meinhardt's activator-inhibitor model of morphogenesis. The study of the behaviour of the system during growth, where the linear and exponential increase of the characteristic dimension is considered, revealed that more complex patterns of morphogen spatial concentrations appear regularly in a reproducible way.