Does It Bind? A Method to Determine the Affinity of Calcium and Magnesium Ions for Polymers Using 1H NMR Spectroscopy

The binding of calcium and magnesium ions (M2+) by polymers and other macromolecules in aqueous solution is ubiquitous across chemistry and biology. At present, it is difficult to assess the binding affinity of macromolecules for M2+ without recourse to potentiometric titrations and/or isothermal titration calorimetry. Both of these techniques require specialized equipment, and the measurements can be difficult to perform and interpret. Here, we present a new method based on 1H NMR chemical shift imaging (CSI) that enables the binding affinity of polymers to be assessed in a single experiment on standard high-field NMR equipment. In our method, M2+ acetate salt is weighed into a standard 5 mm NMR tube and a solution of polymer layered on top. Dissolution and diffusion of the salt carry the M2+ and acetate ions up through the solution. The concentrations of acetate, [Ac], and free (unbound) M2+, [M2+]f, are measured at different positions along the sample by CSI. Binding of M2+ to the polymer reduces [M2+]f and hinders the upward diffusion of M2+. A discrepancy is thus observed between [Ac] and [M2+]f from which the binding affinity of the polymer can be assessed. For systems which form insoluble complexes with M2+, such as sodium polyacrylate or carboxylate-functionalized nanocellulose (CNC), we can determine the concentration of M2+ at which the polymer will precipitate. We can also predict [M2+]f when a solution of polymer is mixed homogeneously with M2+ salt. We assess the binding properties of sodium polyacrylate, alginate, polystyrene sulfonate, CNC, polyethyleneimine, ethylenediamenetetraacetic acid, and maleate.


S3
where I0 is the ionic strength of the sample prior to addition of M 2+  where  indicates the uncertainty in the variable.
Differentiating Equation S1 with respect to each variable: To determine whether a measurement of [M 2+ ]f is acceptable, the following filter is applied: 1. If the chemical shifts of glycolate and/or sulfoacetate are < (a -0.0005 ppm), the measurement is invalid and is set to -999.

S14
The maximum B, Bmax, is then estimated as: where is the percentage error of [Ac] from integration of the 1 H NMR resonance of acetate. Bmin is obtained as:

S16
The uncertainty in B, B, is obtained as: [Ac] was determined by integration of the 1 H resonances of acetate against either DMSO or tert-Butanol using Equation S18: [Ac] = kA/R S18 where A and R denote the acetate and reference signals, respectively. To determine (Equation S15, S16) and the conversion factor, k, the concentration of acetate determined by NMR was compared with the known concentration of acetate in the homogeneous titrations of NaCl, EDTA, PAA, PSS, PAA, PEI, and alginate. Averaging across all samples, k was determined as 2.92 and 3.83 for DMSO and tert-Butanol respectively, with a maximum deviation of 6% for all datasets, based on the slope of a plot of the measured versus known concentration of acetate. The random error in integration within a CSI dataset was determined as 5% by comparing the integrals of DMSO and tert-butanol in the NaCl and maleate samples of Figure 1b. is thus estimated as 8%. The determination of [Ac] by integration against the resonances of DMSO or tert-Butanol, or by lineshape deconvolution were found to give equivalent results (Figure S-1).

S2. Determination of pH from 1 H chemical shift of 2-methylimidazole
The pH of the solution at each position along the M 2+ gradient is determined from the observed 1 H chemical shift, obs, of 2-methylimidazole using Equation S19: pH = pK a,0 + 0.51√I 1 + √I − 0.1I + log 10 ( δ H − δ obs δ obs − δ L ) S19 where I is the ionic strength of the solution. H and L are the chemical shifts of the fully protonated and deprotonated forms, respectively. The pKa at I = 0, pKa,0, is taken as 7.96 from our previous work. 51 For calculation of pH in this work using Equation S19, we use the ionic strength of the solution prior to diffusion of M 2+ acetate (I0, Section S-1) rather than explicitly calculate the ionic strength at each point along our gradient. According to Equation S19, this approximation introduces an error of <0.2 units in the measured pH. It has been demonstrated that H and L of imidazole are not significantly affected by the presence of Ca 2+ or Mg 2+ . 52 H and L for 2-methylimidazole are provided in           To determine DAc and DM (Equation 3), samples were prepared containing 50 mM NaCl, 1 mM 2MI, 1 mM glycolate and sulfoacetate, 0.2 mM DSS, 0.01 vol% DMSO and 0.01 vol% tert-butanol. Three experiments were performed for Ca 2+ and Mg 2+ to assess the reproducibility of the concentration gradients (Figure S-9, above).
[Ac] can be assumed to follow a Gaussian profile (Equation S20): [Ac] = 2m where m is the mass of M 2+ acetate salt weighed into the tube, r the tube radius (2.1 mm), Mr the molecular weight of the salt (180 g/mol for calcium acetate hydrate, 214.5 g/mol for magnesium acetate tetrahydrate), DAc the diffusion coefficient of acetate, t the time since sample preparation, Z the vertical position from the absolute bottom of the NMR tube and h the thickness of the acetate when weighed out (2 mm). DAc of 1 x 10 -9 m 2 s -1 was chosen, based on visual inspection of the data ( Figure S-9). This value is in good agreement with results presented elsewhere for diffusion of acetate. 54 The poor agreement with Equation S20 at t < 3 hours is attributable to unavoidable upward mixing of the acetate salt during preparation of the sample.

S14
To estimate DM, N was calculated for the datasets on Figure Assuming acetate and M 2+ follow independent Gaussian profiles (Equation S20), we obtain the ratio of the two at any point from Equation S22 :    The interaction of M 2+ with the polymer reduces DM below the value measured in 50 mM NaCl, DM,NaCl (Section S-5). Assuming DM is constant throughout the sample, the total concentration of M 2+ , [M 2+ ]total, at any vertical position along the sample is given by Equation S23 : ]total, we may calculate B as: where [Ac] is calculated using Equation S20, and N using Equation 3. In real polymer systems, B thus contains a contribution from the reduction in DM, as well as M 2+ associated with the polymer. Therefore    Figure S-15 below. The exchange between free and bound M 2+ means that DM < DM,NaCl, even when the binding sites are saturated.  Without 50 mM NaCl, Mg 2+ exhibits apparent strong binding to 2 mg/mL alginate by the CSI method ( Figure S-17b). This strong binding is not observed by homogeneous mixing of alginate with Mg 2+ acetate, or when 50 mM NaCl is present (Figure 4b). The apparent strong binding is attributable to the role of M 2+ in balancing the negative charge of the alginate in the absence of the excess Na + , as well as the exclusion of acetate ions. 56 Assuming the diffusion coefficient of an M 2+ ion balancing the charge of an anionic polymer is 0 (Section S-7), DM is given by Equation S27 : where Cp is the concentration of singly charged groups on the polymer and [Na + ]total = Cp + [NaCl]. The factor 2 assumes that each M 2+ ion balances the charge of two charged groups on the polymer. Plots of DM/DNaCl for 2 mg/mL alginate (Cp = 8 mM) calculated using Equation S27 are provided below on Figure S-18. Without 50 mM NaCl, a significant reduction in DM is predicted that will give a positive value of B ( Figure S-14), even in the absence of a significant binding effect. S24 S10. Expanded plots from CSI datasets of citrate-functionalised CNC and 4 mg/mL sodium alginate, 1 H spectra of 0.5 wt% CNC in tap water       (Figure 4k). Spectra referenced to 0.1 mM DSS (0 ppm). Spectra were acquired off-lock in 64 scans using the perfect echo WATERGATE sequence of Adams et al. 57 , incorporating the double echo W5 sequence of Liu et al. 58 The delay between successive pulses in the selective pulse train was set at 333 μs. The 90° pulse was set at 12 μs. The signal acquisition time and relaxation delay were 4.37 s and 1.0 s, respectively. S11. Optical transmittance of CNC samples at 600 nm