Turing structures in a system with regulated gap-junctions.

In two coupled cells, each containing a bisubstrate-kinetics reaction system, the two substrates can cross the membranes through gap-junction protein channels (gating). Passing through the gap-junctions is controlled by one of the substrates, thus resulting in non-linear diffusion. Linear stability analysis gives the conditions, under which the symmetric fixpoint becomes unstable, leading to spatial asymmetry (Turing structures). The implications for morphogenesis are discussed.

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.

Turing structures in an enzyme-induction system with gap junction-mediated non-linear diffusion.

Two cells, each containing a reaction system modeling genetic induction, are coupled by diffusion. The substrate is moving through gap junctions, the number of which is regulated by the adjacent cells. This leads to a non-linear substrate diffusion term in the rate equations. Stability analysis reveals the conditions for the emergence of stable asymmetric solutions (dissipative structures). Due to non-linear diffusion rigid restrictions on the ratio of the two diffusion constants no longer exist. We demonstrate that substances operating as regulators of intercellular communication and participating in cellular metabolism may exhibit morphogenetic functions.

Five simultaneous steady states in a flipping two-compartment-system with optical antipodes.

The previously given systems-theoretic model for the synthesis of optical antipodes (AD and AL) in strongly asymmetric yield, which shows mono-bistable behaviour depending on the degree of "openess" of the chemical reaction system is reconsidered for two equal compartments (subscripts 1 and 2 on A) with coupling by diffusion. In this configuration three threshold values, j1, is less than j2 is less than j3, for the influx j of the common precursor substance appear. For j is less than j1 only one steady state (s.s.) with no optical activity (ADi = ALi, i-1;2) and equal distribution of the antipodes in both compartments (AD1 = AD2, AL1 = AL2) exists. For j is greater than j 1, this totally symmetric s.s. becomes unstable and a pair of s.s. with optical activity (AD1 is less than AL1, AD2 is less than AL2 or AD1 is greater than AL1, AD2 is greater than AL2) but no spatial asymmetry emerges (parallel flipping), i.e. both compartments be8have as a whole, showing a preponderance of either the D- or the L-form. For j is greater than j2 in addition two new s.s. are possible with antiparallel flipping (AD1 is less than AL1, AD2 is greater than AL2 or AD1 is greater than AL1, AD2 is less than AL2), i.e. in one compartment the D-form has the majority and in the other one the L-form, but these are stable only beyond a third threshold value j3. A third thinkable pair with no optical activity, but different sum concentrations in both cells, does not exist in this special circuitry, but can be obtained in a slightly changed arrangement. So for j is greater than j2, 5 different (4 stable, 1 unstable) s.s., exist for the same set of parameters, one of which is chosen by the system.