*3.1. Glycosidic Linkage Geometries*

In non-sulfated chondroitin 20-mer MD simulations, we found that all φ, ψ dihedrals sampled in GlcAβ1-3GalNAc linkages were centered about a global free energy minimum (Min I) while GalNAcβ1-4GlcA linkages showed more flexibility. In addition to a global minimum, ∆*G*(φ, ψ) for GalNAcβ1-4GlcA also has two local minima (Min II and Min II') (Figure 3 and Table 1). To validate these observed glycosidic linkage geometries, we looked at the free energy minima of non-sulfated chondroitin glycosidic linkage dihedrals from biased MD simulations of disaccharides (using dihedral definitions φ = O5-C1-O-C*<sup>n</sup>* and ψ = C1-O-C*n*-C(*n*+1) as opposed to the IUPAC ψ = C1-O-C*n*-C(*n*−1) used in our study) [54] (Table S1). We found that at each free energy minimum in β1-3 and β1-4 linkages, our φ dihedrals differed by no more than +/-2.5◦ and our ψ dihedrals differed by no more than +/-127.5◦ , which is in close agreement if we assume C1-O-C3-C<sup>2</sup> = C1-O-C3-C<sup>4</sup> + 120◦ and C1-O-C4-C<sup>3</sup> = C1-O-C4-C<sup>5</sup> - 120◦ . Additionally, our data were mostly in agreement with the most energetically-favorable glycosidic linkage dihedrals (i.e., at global minima) in non-sulfated chondroitin hexasaccharides from MD simulations (using dihedral definitions φ = O5-C1-O-C*<sup>n</sup>* and ψ

= C1-O-C*n*-C(*n*+1)) and validated by NMR [96] (Table S1). The biggest difference was in our β1-3 ψ dihedrals, which differed by about +100◦ (+120◦ difference expected). This study restrained pyranose rings to a <sup>4</sup>C<sup>1</sup> chair and did not use explicit solvent in simulations. Each of these factors may contribute to interactions between neighboring monosaccharides and thus glycosidic linkage conformation, which would explain the variation from our results.

 ‐ ‐ β ‐ β ‐ ‐ **Figure 3.** ∆*G*(φ, ψ) in non-sulfated chondroitin 20-mer ensembles for aggregated GlcAβ1-3GalNAc and GalNAcβ1-4GlcA glycosidic linkage data (**a**,**b**) MD-generated ensembles, (**c**,**d**) constructed ensembles before minimization, and (**e**,**f**) constructed ensembles after minimization; contour lines every 1 kcal/mol.



 ‐ ‐ <sup>1</sup> φ, ψ dihedral angles were sorted into 2.5◦ bins.

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For internal validation of the construction algorithm, we compared glycosidic linkage input and output data. If the algorithm is performing correctly, ∆*G*(φ, ψ) from the MD ensemble and the constructed ensemble *before minimization* will be nearly identical, and ∆*G*(φ, ψ) from the constructed ensemble *after minimization* will not be substantially different. Performing the comparison between the ensemble of 40,000 20-mer conformations from the MD and a constructed ensemble of the same size confirms this to be the case. Figure 3 demonstrates that ∆*G*(φ, ψ) for GlcAβ1-3GalNAc and for GalNAcβ1-4GlcA glycosidic linkages are qualitatively identical when comparing the MD-generated and constructed ensembles. Quantitative analysis (Table 1) shows that the global minima (Min I) for both types of linkages and the secondary local minima (Min II and Min II') for GalNAcβ1-4GlcA linkages are basically identical with 0◦ to 5◦ differences between the MD-generated input data and the constructed ensemble output data before minimization. The minimization in the construction algorithm, used to resolve any steric clashes, results in relatively minor changes in the location of the global minima, also ranging from 0◦ to 5◦ . Constructed ensemble glycosidic dihedral values after minimization are within 5◦ of the MD data, and ∆*G*(φ, ψ) values are within 0.1 kcal/mol.

As detailed in Methods and discussed below, the minimization portion of the construction algorithm not only relieves any steric clashes, but also is used to detect bond-strain energies indicative of ring piercing. The close similarity of ∆*G*(φ, ψ) for the constructed ensemble after minimization in comparison to the MD-generated ensemble suggests that few constructed conformations have large steric clashes, resulting in large conformational shifts after minimization. It also suggests that a few constructed conformations have ring-piercing events that necessitate their exclusion from the constructed ensemble altogether. Indeed, this is the case: during the creation of the 40,000-member constructed ensemble, only 18 conformations were excluded because they failed to meet the bond-strain energy criterion.
