Flexibility of the protein itself is not affected with any of those linkers, unless you have interaction between the linker and your protein (unlikely) or between protein and tag (possible, depending on the Tag (peptide? fusion protein?) and your protein structure and properties). All the linkers you cite are very flexible. The end result also depends by "n", keep in mind that longer tags introduce more hydrophobicity and could alter solubility/stability of low Mw proteins. You can find more details in one of the many reviews on the topic, e.g.:
The Role of Protein Loops and Linkers in Conformational Dynamics and Allostery
Chem. Rev., 2016, 116 (11), pp 6391–6423
DOI: 10.1021/acs.chemrev.5b00623
Fusion protein linkers: property, design and functionality.
Flexibility (or disorder) increases with the number of glycine residues (as serine conformation is a bit more constraint due to its side chain content). Among the three, the third will be the more flexible.
I will tag a 211aa transcription factor with a 3xFLAG epitope to perform a ChIP-seq experiment.
I know that the FLAG epitope is highly hydrophilic, which reduces its chance of interacting with the tagged protein. However, I need to be sure that the tag will not interact in any way with my protein. Any conformational changes induced by the tag could potentially change its DNA-binding properties.
I have read somewhere that Glycine linkers help to ensure proper folding of tagged proteins. Therefore, a physical spacer between the tag and my protein (such as a Gly-Ser linker) sounds like a good strategy to address this issue.
I have designed a 6aa linker (PROTEIN-Gly-Gly-Gly-Ser-Gly-Gly-FLAG) to reduce any detrimental effects introduced by the tag (Initially I added just six Gly residues, but after some reading it looks like Ser residues are important because they give a hydrophilic characteristic to the linker).
So, my concern is if this configuration is okay, if the linker will be flexible enough to allow proper folding of my protein regardless of the tag.
If you have any other thoughts I'd be glad to hear it!