*4.2. Structural Order–Disorder Prediction and Secondary Structure Predictions*

The structural order–disorder propensity of each protein was predicted using IUPred2A (https: //iupred2a.elte.hu/) [75] using the option for long disordered regions. This prediction had values ranging from 0 (strong propensity for an ordered structure) to 1 (strong propensity for a disordered structure), with 0.5 as the cut-off between the propensity for order and disorder. The results for each site of each protein were mapped onto its sequence alignment and taxon position in the phylogenetic tree using iTOL (https://itol.embl.de/) [76].

### *4.3. Rate of Evolution per Site*

We calculated the rate of evolution per site of human CDKL5, FOXG1, and MeCP2 relative to their orthologs using Rate4site (https://m.tau.ac.il/~{}itaymay/cp/rate4site.html) [77]. The aligned sequences of each protein dataset were calculated using the empirical Bayesian principle with the JTT model and 16 discrete categories of the prior gamma distribution. Gaps were treated as missing data, and outputs were standardized as Z scores. The results of the rate of evolution of each residue were then integrated with the structural order–disorder prediction result, and the distribution of the rate of evolution in the structural order and disorder of each protein was evaluated with the Mann–Whitney U-test using R software.
