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where ( α 2 , α 5 , α 4 , α 12 , α 13 ) are additional terms that can be used to change the saturation of the sigmoids. In this example they are set to one.

Appendix 5—figure 2
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Left: Parameters used in Equation 35 to simulate the synthetic circuits shown in Figure 5 . Right: One-dimensional simulation using the sigmoid regulation terms.

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the terms with k 9 and k 16 are not sigmoid regulation terms, since they represent first order decays. Similar to linear models, numerical simulations confirm that this system can form a stable spatial pattern. We also observe that changing the saturation of different regulatory sigmoid has interesting effects on the final aspect of the pattern. These numerical simulations are left for future theoretical analysis and are beyond the focus of this study that deals with realistic reaction-diffusion topologies and the analytical conditions that lead to pattern formation.

The simulations in the main text were performed by deriving systems of partial differential equations with linear reaction terms and negative cubic saturation terms from the networks, similar to previous approaches ( SHIRTS Blouses Four Friends Shopping Online High Quality Clearance Genuine Clearance Enjoy Free Shipping Wide Range Of Recommend For Sale DwKBM
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). Systems of partial differential equations can be derived from the Jacobian matrix 𝐉 and the diffusion matrix 𝐃 as shown in the following example ( Appendix 6—figure 1 ):

where the matrix 𝐒 contains cubic saturation terms only for the nodes that are part of the destabilizing positive feedback (green arrows) and is zero otherwise.

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