*3.3. Treating Glycosidic Linkage and Ring Pucker Geometries as Independent Variables*

 ‐ ‐ ‐ To determine if linkage geometries and ring deformations are interdependent, individual glycosidic linkage ∆*G*(φ, ψ) plots (Figure S3) were created (as opposed to the aggregate data ∆*G*(φ, ψ) plots in Figure 3) and no distinguishing patterns emerged. These per-linkage glycosidic linkage data were also examined in the context of C-P plots of adjacent rings (Figure S4), for which there were also no distinguishing patterns. Additionally, φ and ψ values in linkages flanking GlcA rings not in a <sup>4</sup>C<sup>1</sup> chair conformation were checked. For each linkage type, these conformations were all centered about the global ∆*G*(φ, ψ) minima for the aggregate data (Figure 3), and 99.96% of conformations fell within the basin extending to ∆*G*(φ, ψ) = +2 kcal/mol (Figure 7). Furthermore, different types of non-4C<sup>1</sup> chair conformers did not have unique flanking linkage geometries. As no connection between linkage and

ring conformations was observed in this analysis of the MD data, each linkage conformation and ring pucker was treated independently in the construction algorithm.

 ‐ ‐ ‐ ‐ β ‐ β ‐ **Figure 7.** ∆*G*(φ, ψ) plots for glycosidic linkages flanking non-4C<sup>1</sup> GlcA conformers in non-sulfated chondroitin 20-mer MD-generated ensembles: (**a**) GlcAβ1-3GalNAc and (**b**) GalNAcβ1-4GlcA.

### *‐ 3.4. Handling Non-physical Constructed Conformations*

‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ To determine a criterion to exclude non-physical conformations from constructed ensembles, energy minimization with restrained endocyclic ring and glycosidic linkage dihedrals was performed on each conformation, and post-minimization bond potential energy and bond length probability distributions were analyzed. Conformations with outlying bond energies and abnormally long bonds may point to ring piercing (Figure 8) that cannot be fixed by minimization. To confirm this possibility, the post-minimization conformations with bond energies greater than that of the fully-extended 20-mer conformation after minimization were visualized. As anticipated, among these conformations, most with outlying total bond energies contained pierced rings which were not resolved by minimization. For each of the 12 20-mer conformations with a pierced ring, the difference between the post-minimization bond energy and that of the fully-extended 20-mer conformation is greater than the predicted energy change caused by bond distortions of that pierced ring (Table S2). The six conformations with outlying bond energies that did not contain pierced rings had kinks that resulted in bond length and bond angle distortions in glycosidic linkages that were nearly overlapping even after minimization. Of note, those conformations with bond energies that were not outlying were fully extended. These findings motivated using a bond potential energy cutoff in the construction algorithm. As stated previously, applying this cutoff resulted in 18 conformations being excluded during creation of the 40,000-member constructed ensemble. The resulting constructed ensemble contains no outlying bond lengths (Figure S5).

Dihedral angles before and after energy minimization were compared by analyzing glycosidic linkage ∆*G*(φ, ψ) (Table 1 and Figure 3c–f), monosaccharide ring C-P parameters (Figure 4c–f), and the change in φ and ψ dihedral angles due to minimization. All ∆*G*(φ, ψ) and C-P plots after energy minimization match those from before energy minimization and 99.6% of all φ and ψ dihedral angle differences before and after minimization are within 4◦ (Figure S6). This confirms that dihedrals do not undergo any major changes during minimization. Additionally, differences in end-to-end distance before and after minimization were calculated and the maximum change is 2.13 Å with 99.9% of changes under 0.5 Å, confirming that overall backbone conformation does not change as a result of minimization. According to these results, the selected bond potential energy cutoff and restraint scheme during minimization give conformations with little deviation from the initial constructed conformations before minimization.

 

*‐*

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‐ ‐ β ‐ β ‐

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**Figure 8.** Constructed 20-mer conformation with a GlcA ring pierced by a GalNAc C-CT bond and a close up panel showing atoms involved in the ring pierce; *E*<sup>b</sup> = 787.7 kcal/mol, fully-extended 20-mer post-minimization *E*<sup>b</sup> = 29.6 kcal/mol, ∆*E*<sup>b</sup> = 758.1 kcal/mol (Table S2).

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### *3.5. Internal Validation on 10-mers ‐*

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To further validate the algorithm and test the use of conformational parameters from MD-generated 20-mer ensembles to construct polymers of variable length, we constructed non-sulfated chondroitin 10-mer ensembles and compared them to MD-generated 10-mer ensembles. All linkage and most ring conformations in MD-generated 10-mer (Figure 9 and Figure S7) and 20-mer ensembles matched (Figure 3a,b and Figure 4a,b), with the exception of non-4C<sup>1</sup> GlcA rings which were not sampled in 10-mer simulations. This finding, combined with the report from NMR and force-field studies that GlcA skew-boat and boat conformations are negligible in non-sulfated chondroitin mono- and oligosaccharides [50,98], suggests that these GlcA conformations may result from intramolecular interactions in longer GAG polymers. ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐

 ‐ ‐ ‐ β ‐ β ‐ **Figure 9.** ∆*G*(φ, ψ) plots for each glycosidic linkage in non-sulfated chondroitin 10-mer MD-generated ensembles: (**a**) GlcAβ1-3GalNAc and (**b**) GalNAcβ1-4GlcA.

‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ Additionally, the end-to-end distance distributions of constructed and MD-generated 10-mer ensembles matched with minimal difference in the most probable end-to-end distance (Figure 10 and Table 2). Further, the radius of gyration is highly correlated with end-to-end distance in both MD-generated and constructed ensembles (Figure S2c,d). Of note, the end-to-end distance distributions of MD-generated 20-mer ensembles more closely matched those of 20-mer ensembles constructed using MD-generated 20-mer conformations (Figure 6) than those of 20-mer ensembles constructed using MD-generated disaccharide conformations (Figures S8 and S9). Together, these findings suggest

that MD-generated 20-mer conformational parameters are ideal for constructing chondroitin polymers of different lengths.

‐ ‐ ‐ ‐ ‐ ‐ **Figure 10.** End-to-end distance probability distribution of MD-generated (blue dashed lines) and constructed (red solid lines) 10-mer ensembles; each type of ensemble includes four sets of 10,000 conformations; probabilities were calculated for end-to-end distances sorted into 0.25 Å bins.


**Table 2.** Most Probable End-to-End Distances (*d*) in MD-Generated and Constructed Ensembles <sup>1</sup> .

‐ ‐ ‐ ‐ ‐ ‐ <sup>1</sup> Probabilities were calculated for end-to-end distances sorted into 0.5 Å bins for the 20-mer ensembles and 0.25 Å bins for the 10-mer ensembles. <sup>2</sup> All = end-to-end distance distribution aggregated across all four runs.
