Atomistic simulations methods

The protocol for all simulations in this study is the same as the one described in the section 4.1.1, except the application of the force-fields and the corresponding compatible atomic partial charges. Mamatkulov [217] and Kirkwood–Buff Integral (KBI) (Naleem et al. [201]) force-fields are specifically designed for ions, hence the parametrization for dendritic polyglycerol (dPG) part in dPGS, i.e., the core scaffold of the dPGS without the terminal sulphate groups, has been kept intact according to the GAFF force-field.

Electronic continuum correction (ECCR) essentially rescales the atomic partial charges of dPGS and the surrounding ions, therefore GAFF force-field parametrization has been retained for this simulation as well.

Mg²⁺, Na⁺, and Cl ^- ions in the system are referred to with subscripts ++, +and - , respectively. The G2-dPGS is accompanied by the corresponding number of Na⁺ coun-terions N_\mathrm{s} = 24 electrically neutralizing the macromolecule. The number of salt ions i

(i = ++,+, - ) in a simulation box volume V is denoted as n_i, while the corresponding

total salt concentrations are denoted asc⁰_i =n_i/V. Bulk concentrationsc^\mathrm{b}_i (i= ++,+, - ) are calculated from the radial density distributions in far-field after the equilibration. Ma-matkulov, KBI and ECCR force-fields have been applied to both the cases ofc⁰₊₊ = 15mM and 30 mM, while OPLS and Mg(H2O)²⁺6 force-fields are employed only for the case of c⁰₊₊ = 30 mM. c⁰ _- is fixed close to 150 mM for all simulations, in order to mimic physio-logical ionic composition, and c⁰₊ is adjusted in order to maintain electroneutrality in the simulation box. All simulations have box sizes \sim 10 nm and are performed till \sim 80–

100 ns.

4.3.6 Density distribution functions

Fig. 4.10 shows the radial density distributions dPGS sulphur atoms (representing the COM of the terminal sulphate groups), Mg²⁺ and Na⁺ ions with respect to the dPGS-COM, and as calculated according to several force-fields. Similar to the case of the monovalent salt in Fig.4.2(a), the radial density of the terminal sulphur is inhomogeneous with pronounced peaks as a result of the excluded volume interactions along with bond

0

Figure 4.10: The quantitative comparison (in nm ^{- 3}) of radial density distributions of dPGS sulphur, Mg²⁺ and Na⁺ between charge rescaled (ECCR [218]) and non-polarizable force-fields (Mamatkulov et al. [217], KBI [201], OPLS [357] and Mg(H2O)²⁺6 [208]). The reference used for the distributions is the dPGS-COM. The profiles are shown for the Mg²⁺ salt concentrations c⁰₊₊ = 15mM and 30 mM.

constraints and charge repulsion, however, the peak location varies depending on the force-field. According to ECCR approach, the rescaled charges on the sulphate groups

are - 0.75e, which indicates lower intramolecular sulphate–sulphate repulsion as compared

to non-rescaled charges ( - 1e), leading to enhanced backfolding effect and shrinking of the dPGS size. It can be seen that this peak location does not change for ECCR, KBI and Mamatkulov force-fields, on going from c⁰₊₊ = 15 mM to 30 mM. Looking at the c⁰₊₊ = 30 mM case, the overall peak location for sulphur distribution ranges \sim 0.7 - 1.2nm, which is somewhat lower than that for the case of monovalent salt (1.2nm). This could be attributed to the introduction of Mg²⁺ ions bridging the oppositely charged sulphate groups and increasing the inter-sulphate group attraction. As discussed in the section 4.1.2.1, the major peak position of the sulphur distribution is used to define the intrinsic dPGS radius r_\mathrm{d}. The values of r_\mathrm{d} according to different force-fields are given in Table 4.6.

Fig. 4.10 further shows the radial number density distributions of Mg²⁺ ions c++(r)

c⁰₊₊ c⁰₊ c^\mathrm{b}₊₊ c^\mathrm{b}₊ n++ r_\mathrm{d} r\mathrm{e}ff

ECCR[218] 14.98 162.17 11.77 142.37 9 1.01 1.44

30.45 132.77 23.55 114.12 18 1.00 1.52

Mamatkulov[217] 15.87 168.34 7.87 149.02 9 1.40 1.93

28.23 163.07 17.35 143.46 16 1.37 1.93

KBI[201] 15.10 163.75 5.23 148.32 9 1.10 1.75

31.33 132.54 15.50 127.87 18 1.08 1.75

OPLS[357] 33.34 142.53 24.80 113.35 18 1.07 1.62 Mg(H2O)²⁺6 [208] 30.55 131.28 22.25 116.91 18 1.12 1.43

Table 4.6: The summary of structural and electrostatic parameters of G2-dPGS evaluated from the simulations employing non-polarizable and charge rescaled (ECCR) force-fields. c⁰_i (i = + + (\mathrm{M}\mathrm{g}²⁺), +(\mathrm{N}\mathrm{a}⁺)and - (\mathrm{C}\mathrm{l} ^- )) denotes total salt concentration whilec^\mathrm{b}_i denotes the bulk concentration measured from the density distribution of the respective species in far-field. n₊₊ is the total number of Mg²⁺ ions in the simulation box and r_\mathrm{d} is the dPGS bare radius calculated as radius of gyration of the sulphur atoms with respect to dPGS-COM. r\mathrm{e}ff and Z\mathrm{e}ff correspond to the effective radius and the effective charge valency of G2-dPGS respectively, calculated using the Alexander prescription (cf. Sec.3.1.2.4.1).

Concentrations and radii are expressed in mM and nm, respectively.

according to different force-fields. While the distributions in the short-range differ based on a force-field, all distribution decay in an exponential (DH-like or Yukawa) fashion.

Looking at both c⁰₊₊ = 15 mM and 30 mM cases, it can be clearly seen that the KBI profiles largely overestimate the magnitude, compared to ECCR and Mamatkulov ones.

In c⁰₊₊ = 15 mM case, KBI shows a clear single-peak distribution with the peak loca-tion coincident with that of the corresponding sulphur distribuloca-tion, which indicates that dPGS–Mg²⁺ interaction is largely dominated by the interaction of Mg²⁺ cations with the dPGS sulphate groups, implying electrostatics playing a major role in the interaction.

ECCR and Mamatkulov profiles, on the other hand, display much lower magnitude, indi-cating overall lower Mg²⁺ intake by dPGS. In the case of ECCR, Mg²⁺ with its rescaled charge of 1.5e represents enhanced electronic polarization of surrounding water, decreas-ing its interaction strength with sulphate groups. The case of c⁰₊₊ = 30 mM shows the diversity of the Mg²⁺ density distribution around G2-dPGS, according to different force-fields. This further highlights the earlier discussed issue regarding the parametrization of divalent ions in the molecular simulations. Except OPLS, all the force-fields show mostly a single-peak distributions. The double-peak OPLS profile shows the attractive interac-tion between Mg²⁺ and the bonded oxygen atoms in the dPGS core, which is also slightly echoed by ECCR profile.

Na⁺ density distribution c+(r) is further shown in Fig. 4.10. For both the cases of c⁰₊₊ = 15 mM and 30 mM, ECCR and KBI profiles show enhanced Na⁺ penetration in the dPGS interior. In the ECCR scaling of Na⁺ ions, apart from charge rescaling, their ionic radii are also reduced, compared with the original full charges model, in order to retain proper Na⁺–water distance. The reduced sizes enable Na⁺ ions to penetrate the dPGS core. Another reason for this profile according to ECCR approach is the significant backfolding of the sulphate groups in the dPGS core, attracting more Na⁺ ions in that region.