Binomial Fitting for Deuterium Exchange Analysis

Modules for binomial fitting were developed and implemented with Visual Basic scripts as follows. Isotopic peaks in the spectra were converted to integer values by subtracting the monoisotopic peak m/z and multiplying by the charge state (z). The peptide natural abundance isotopic distribution was either read directly from the undeuterated spectra or calculated from the amino acid (and carbohydrate) composition [5 (link)]. The number of slow-exchanging amides was based on the peptide sequence and used as the number of events (n) in calculation of the binomial distribution function (eq. 1). n was initially estimated as the number of amino acids minus the number of prolines, minus 1 for the N-terminal residue. In some cases peptides contained fast exchanging residues, which back-exchange rapidly [23 (link)] and therefore n was set slightly lower. For glycopeptides the glycan groups need to be taken into account as some carbohydrate groups (primarily N-Acetyl hexosamines) will also retain deuterium [24 (link)]. Examination of the fit to a fully deuterated standard was used to assess whether the value of n generated the correct envelope width, and with the majority of peptides the initial estimate was accurate. For a few highly protected peptides, better fits were achieved with a slightly lower n for the early time points, presumably because some amides have yet to exchange by then.
The centroid shift in each spectrum relative to the undeuterated profile served as an initial estimate of the binomial distribution probability (“p”). Each theoretical peak (I_m^calc) was reconstructed by applying the natural abundance profile to each peak in the binomial distribution with up to 3 points of zero padding on both sides of the mass envelope as described by Chik et al [19 (link)]. The peak intensities were scaled by a weighting term (A), using the intensity of the highest data point as an initial guess. Least squares regression was performed using the Gauss-Newton algorithm implemented within the Excel Solver module (Microsoft, Redmond WA) to minimize the discrepancy (χ²) between the isotopic peaks and the calculated binomial profile by varying p and A (eq. 2). For data sets showing overlap with an interfering ion, the asymmetry term (λ) was user defined (typically between 2 and 10, based on visual assessment) and applied to points where the fit exceeded the data. The resulting degree of deuteration was calculated as either the percentage relative to the undeuterated (p_UN) and fully deuterated (p_TD) values (p_t-p_UN)/(p_TD-p_UN) or average deuterium uptake (p_t·n), and was plotted in the summary page for each time point (t) and condition. $${I_m}^{\mathit{\text{calc}}}=A·\frac{n!}{m!(n-m)!}{p}^x{(1-p)}^{n-m}$$
$$\begin{gathered} {\chi }^2=\sum _{m=0}^n(\lambda )·{({I_m}^{\mathit{\text{calc}}}-{I_m}^{\text{exp}})}^2 \\ \lambda =1\text{if}{I_m}^{\mathit{\text{calc}}}<{I_m}^{\mathit{\text{exp}}} \\ \lambda >1\text{if}{I_m}^{\mathit{\text{calc}}}<{I_m}^{\mathit{\text{exp}}} \end{gathered}$$