*2.1. ECs Were Detected Outside the Sequence Ranges with PDB Coordinates, but Visible ECs Reside on the Interfaces*

Gremlin identified 31 ECs in 7 complexes (Supplementary Tables S1 and S3). Intriguingly, for 9 of these co-varying residue pairs, at least one of the constituent residues (mostly the IDP residue) fell outside the segments with PDB coordinates (invisible ECs; marked with N/A in Supplementary Table S3). These invisible ECs make up a remarkably high percentage of all identified ECs. Based on their Gremlin scores and probabilities, they do not seem to be mistakenly predicted pairs as they often have higher scores and probabilities than interface residue pairs with several atomic contacts. Of the 10 invisible EC residues, 5 were present in the respective PDB sequences but had no atomic coordinates, while the other 5 were not included in the constructs used for structure determination. High occurrence of these invisible EC residues suggests that the IDP segments used for structure determination and interaction analyses are often too short, lacking residues that could still form important contacts with the partner. There are actually several instances where the contribution to binding of segments that are disordered in the partner-bound state has been proven, such as binding of CREB to CBP [48] and binding of SF1 to U2Af65 [49], which forms the basis of the concept of fuzzy interactions [50].

All except one of the 22 visible IDP EC residues (Table 1, Figure 1) are interface residues according to PDBe PISA, and 12 of them are assigned as forming a direct H-bond or salt bridge (or both) with the partner (Table 1), although not necessarily with their corresponding EC pair (Supplementary Table S3). These values already support the validity of the predicted ECs, nevertheless, we decided to investigate them in more detail.



not be analyzed as they reside outside the sequence ranges with PDB coordinates. The numbers of such invisibleinterface (IF) pairs, since they provide the explanation for the difference between the number of identified ECs

 EC pairs (inv) are indicated in brackets in the last column along with

 and interface ECs.

a

**Figure 1.** High-scoring interprotein ECs indicating IDP-partner co-evolution. The six protein complexes with multiple high-scoring interprotein ECs predicted by Gremlin are introduced: (**A**) PDB complex 3HPW; (**B**) 5CQX; (**C**) 3M91; (**D**) 2A7U; (**E**) 1SC5; (**F**) 3M4W. IDPs are depicted in teal cartoon style, while the partners are in light grey. Rectangles indicate the part of the complexes which are depicted as detailed interaction maps below. In the interaction maps, the EC side chains are depicted as sticks colored teal in the IDP and salmon in the partner. The interprotein ECs have their respective residue side chains labelled and are connected by dashed lines. For the few pairs forming direct H-bonds or salt bridges according to PISA, the connecting lines are doubled. The 183Q-226W IDP-partner EC pair on the F panel had a distance 17.5Å, thus it was handled as an outlier. None of the depicted complexes represent SLiM-mediated interactions, as we could not identify any ECs for those (see Section 2.5 for further details).
