Morphogene adsorption as a Turing instability regulator: Theoretical analysis and possible applications in multicellular embryonic systems

The Turing instability in the reaction-diffusion system is a widely recognized mechanism of the morphogen gradient self-organization during the embryonic development. One of the essential conditions for such self-organization is sharp difference in the diffusion rates of the reacting substances (morphogens). In classical models this condition is satisfied only for significantly different values of diffusion coefficients which cannot hold for morphogens of similar molecular size. One of the most realistic explanations of the difference in diffusion rate is the difference between adsorption of morphogens to the extracellular matrix (ECM). Basing on this assumption we develop a novel mathematical model and demonstrate its effectiveness in describing several well-known examples of biological patterning. Our model consisting of three reaction-diffusion equations has the Turing-type instability and includes two elements with equal diffusivity and immobile binding sites as the third reaction substance. The model is an extension of the classical Gierer-Meinhardt two-components model and can be reduced to it under certain conditions. Incorporation of ECM in the model system allows us to validate the model for available experimental parameters. According to our model introduction of binding sites gradient, which is frequently observed in embryonic tissues, allows one to generate more types of different spatial patterns than can be obtained with two-components models. Thus, besides providing an essential condition for the Turing instability for the system of morphogen with close values of the diffusion coefficients, the morphogen adsorption on ECM may be important as a factor that increases the variability of self-organizing structures.

The necessary and sufficient conditions of the Turing instability arising in the systems of the reaction-diffusion equations are already known [1]. Let us consider the equation system has the following general form: Appendix S1.2. Turing instability analysis of the classical GM model Here we provide the analysis of Classical Gierer-Meinhardt model defining by following equation: Firstly, the stationary point of the pure-reaction system (u, v) is solitary. We have (S1.2.2) Reaction stability of the stationary point. The Jacobian matrix found at the stationary point, M, is: Eigenvalues of M are: This proves that the reaction system (S1.2.1) is stable iff µ v is bigger than µ u : It can be shown in the usual way, that the system will demonstrate oscillatory behavior when Finally, the first quarter of the parametric plane µ u , µ v is divided by three lines into the areas where different phase portraits are formed ( Fig S1).
Fig S1. Bifurcation diagram of the system (S1.2.1) in (µ u , µ v ) parametric space. The zone of stability of the stationary point is marked green, the opposite zone is marked red. Focus lost its stability when intersecting the diagonal line.
Reaction-diffusion saddle-type instability of the stationary point. The diffusion-augmented Jacobian matrix has the form: Clearly, its eigenvalues: Only one λ value have a positive real part iff Roots have the form The Turing condition requires the range of k 2 values to be positive and finite. Hence, we get and This inequality can be rewritten in the canonical form: Due to Eq (S1.2.7) the second multiplier is always less than zero. Finally, we get (S1.2.9) For rather small difference between µ v and µ u diffusion coefficients should differ in 6 times to supply Turing instability that is not possible for protein molecules of approximately same size. Appendix S1.3. Turing instability of the modified system Here we analyse the extended Gierer-Meinhardt model defined by equation: At first, let us find the stationary points of the reaction part of the system (S1.3.1): Stationary point of the modified system In this section we show that the modified system has the unique stationary point.
After the transformation: Next substitute u * = (u + w 0 − w) and reduce the system: The stationary points satisfy Eq (4) of the main text: (S1.3.4) As v is found, next we find the rest two variables by joining the first equation derived now and the third equation of (S1.3.3): After transformations: µ v and the equation could be written as: The discriminant could be written as: For the final solution we get: Let us introduce an auxiliary parameter β, γ: In these notations the stationary concentrations are of the form: Note, that the stationary value of v does not depend on any adsorption parameters (w 0 , k 1 , k −1 ).
Reaction stability of the stationary point. Let us build the Jacobian in the stationary point: After the manipulations with rows and columns in the matrix (M − λE) we get the simpler matrix with the same determinant: Now it become clear that M has three eigenvalues; at least one of them has a simple form and is always negative: Also, considering Eq (S1.3.7), one can prove that Substituting this expression together with Eq (S1.3.7) into Eq (S1.3.8), we conclude that the upper left minor have the same form as the characteristic matrix of the original system (S1.2.3). Since one eigenvalue is always negative and two other eigenvalues have the same expression as eigenvalues of the classical system, we see that the reaction stability condition also looks like Eq (S1.2.4).
Reaction-diffusion saddle-type instability of the stationary point. The diffusion-augmented Jacobian has the complicated form: After the same transformations as were performed in (S1.3.8) the sufficient condition can be written as the following: The determinant should have only one root with a positive real part for a finite positive range of k. Formulation of this condition in an explicit analytical form is out of question. However, one can note that the upper left minor of the matrix (S1.3.9) is precisely equal to the matrix (S1.2.5) of the classical system. It may help if one wants to add an adsorption to some other two-component reaction-diffusion system or to generalize the results of the present study.