**4. Discussion**

A common pattern on the maps presented in Figure 4 and the ones presented in Figure 2 in [31] indicates that the proximity of the HSA protein does not prevent the HA chain from taking the shape of a helix (cf., Figure 5 in [31]). In the case of CS6, positions of preferred (Φ,Ψ) angles regions have been recently reported in [46]. The location of the angles presented in this study for HSA-CS6 complexes overlaps with those for the HA chain in [31]. Therefore, it can be concluded that CS6 keeps its *α*-helix structure. However, distributions of the (Φ,Ψ) angles (Figures 3 and 4) show that GAGs can show helix structures and the random coil conformations at various ratios. Bear in mind that the frequency distributions of the (Φ,Ψ) angles are created from all NG units over the whole time; thus, we cannot say anything about how the conformations of the chains evolved. Instead, we see the static characteristic maps, which can tell us about the stability of the conformations in the simulation process.

We have demonstrated that conformational entropy is a parameter that enables us to characterize the structure of GAGs in interaction with HSA globally. We can see that it is slightly dependent on the pair of angles but rather for the CS6 case. In the case of CS6, the lower entropy value of the (Φ1−3,Ψ1−3) angles compared to the entropy value of (Φ1−4,Ψ1−4) angles indicate that mathematical features of the histograms for the angle pair 1-4 relative to the angle pair 1–3 must be more uniform. We understand this idea that (Φ1−4,Ψ1−4) angles during simulations appear more uniform in the space of its value than angles (Φ1−3,Ψ1−3) in the space of its value. This property is not visible for the angles in the case of HA.

Numerically derived histograms have various numbers of maxima, but these local maxima may be wider or narrower, which is essential in the entropy calculated here. Therefore, our work builds upon [46], which focuses on a sole CS6 and shows that a smaller number of regions are preferred to occupy by (Φ,Ψ) angles than our work that shows a larger number where HSA was added to the solution.

As the entropy is one of the features of above-mentioned histograms, we may expect other features of these histograms to be also informative about particular GAG, ion, or monomer. Hence, further analysis concerning pattern recognition machine learning techniques like SVM (Support Vector Machine) [47] and its modification for colored image processing [48] can be performed to extend the research. Another approach to this problem may be the colored image segmentation by a random walk [49], Sub-Markov random walk [50] or the Hurst Exponent image processing in [51,52]. Alternatively, one can analyze angles as multivariate series and process them with dedicated tools such as higher-order multivariate cumulants, see [53]. The goal of deep analysis mentioned above would be to search for more sensitive features and analyze whether entropy is a stable feature compared to others. If such features can be determined, their utility may appear in analytical, chemical medical applications. The secondary GAGs structure resembles flat bands transformed into a helix or twisted into a sheet originating from intermolecular hydrogen bonds. In diluted GAG solutions, the macromolecules have semi-rigid coiled chains and could form helix bands and even helical rings. Due to formation of a rigid helix, the macromolecules of GAG attract a great quantity of water and organize the broad domains of the tertiary polymer structure [54]. Binding to HSA reduces degrees of freedom of polymer and thus mechanical properties. On the other hand, HA and CS6 at higher concentrations, external force or other factors can change HSA's tertiary structure, forming material of different properties.

#### **5. Conclusions**

Macromolecular complexes are building blocks in the functioning of physiological processes. When optimal conditions are fulfilled for given pairs, the system can function efficiently. Protein–ligand interactions are critical to optimal biochemical, biological, or biophysical results. Often, a given complex can serve several functions, as in the case presented in this study, where HA-HSA complexes decrease friction and can be used in drug delivery systems. Our results show how crucial components of synovial fluid interplay with each other at equilibrium. We have shown that HA and CS6 can form stable complexes with HSA.

Moreover, the binding sites for both molecules overlap, which indicates that they both can induce a similar effect on HSA while functioning. The molecular mass used in this study is one limitation of the presented results, as interactions (and mechanical and biological properties) between GAGs and proteins strongly depend on their molecular mass and concentration. This fact emerges from the chemistry of polymers of interest. GAGs chain's amount of expansion is enormous for a semi-flexible polymer. The polymer configuration is constantly in a state of motion and change. However, the water increases the effective size of each hyaluronic acid because of its hydrophilic nature. The mass increase results in the average density decrease because the increase in mass is slower than in the volume. Thus, GAGs chains with a high molecular weight (more than 1, 000 kDa) occupy a substantial volume. However, adsorption at HSA strongly influences the local mechanical properties of GAGs resulting in efficient lubrication.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/e24060811/s1, Figure S1: Normalized histograms for different realizations for HSA-CS6 complexes.; Figure S2: Normalized histograms for different realizations for HSA-HA complexes.; Table S1: Binding energies in kcal/mol for 10 realizations of HSA-CS6 and HSA-HA complexes after docking.

**Author Contributions:** Conceptualization, P.S. and P.B.; methodology, P.B., N.K., K.D. and P.S.; software, P.S., K.D. and N.K.; validation, N.K., P.B., P.W. and B.M.; formal analysis, K.D., P.S., P.B., N.K. and P.W.; investigation, P.B., N.K., K.D., P.S., P.W. and B.M.; resources, P.S., P.B., P.W., N.K., K.D. and B.M.; data curation, P.B., N.K. and P.S.; writing—original draft preparation, P.B., P.W., N.K. and K.D.; writing—review and editing, P.B., N.K. and K.D.; visualization, P.B., N.K. and P.S.; supervision, K.D. and P.B.; project administration, P.B.; funding acquisition, B.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data and processing code is available in the public repository at [41], https://github.com/iitis/polymer\_entropy accessed on 7 June 2022.

**Acknowledgments:** The work is supported by BN-WTiICh-11/2022 of the Bydgoszcz University of Science and Technology. Calculations were carried out at the Academic Computer Centre in Gda ´nsk.

**Conflicts of Interest:** The authors declare no conflict of interest.
