Intrinsic Hydrogen–Deuterium Exchange Rates in H₂O/D₂O Mixtures

Abstract

Hydrogen–deuterium exchange (HDX) measurements are widely used to probe protein structural dynamics. Quantitative interpretation of HDX data relies on the concept of an intrinsic exchange rate, which is well characterized in isotopically pure H₂O or D₂O but does not explicitly account for the back exchange that necessarily occurs in H₂O/D₂O mixtures: in this case, both the approach-to-equilibrium rate and the equilibrium deuterium enrichment of amides depend nontrivially on solvent composition and acidity. A practical method is presented to predict intrinsic forward and reverse amide exchange rates in H₂O/D₂O mixtures. The approach combines known second-order reference rates measured in pure solvents with established empirical descriptions of H₂O/D₂O mixtures. The resulting framework yields explicit expressions for forward and back exchange rates as functions of solvent composition and acidity and correctly recovers the known limits in pure H₂O and pure D₂O. The model predicts composition-dependent kinetic isotope effects and an equilibrium amide fractionation factor of ϕ = 1.20 for unstructured peptides under base-catalyzed conditions, in close agreement with the experimental value 1.22 reported for poly-d,l-alanine. By providing a physically motivated description of exchange in mixed solvents, this method offers a practical starting point for quantitatively correcting back exchange in HDX–MS and HDX–NMR experiments.

Introduction

Biophysical techniques measuring amide hydrogen–deuterium exchange (HDX) are broadly used to fingerprint structural dynamics of proteins. − The Linderstro̷m-Lang model describes HDX of protein backbone amides according to the reaction

$${\mathrm{NH}}_{\mathrm{cl}}\underset{k_{\mathrm{cl}}}{\overset{k_{\mathrm{op}}}{\rightleftharpoons }}{\mathrm{NH}}_{\mathrm{op}}\stackrel{k_{\mathrm{int}}}{\to }\mathrm{exchanged}$$ 1

where the amide can assume either a closed (NH_cl) or open (NH_op) conformation, the former being exchange-incompetent. An amide in the open state exchanges with a pseudo-first-order rate k _int, called the intrinsic exchange rate (also referred to as the “chemical” exchange rate), which can be estimated as a function of sequence, temperature and pH (or pD) in pure H₂O (or D₂O). − The protection factor P = k _cl/k _op is the inverse opening equilibrium constant that one aims to determine to characterize structural dynamics. − When local dynamics is much faster than exchange (k _op + k _cl ≫ k _int), the observed exchange rate k _obs is related to intrinsic exchange rate and protection factor as k _obs = k _int/(1 + P).

HDX is typically detected using nuclear magnetic resonance (NMR) spectroscopy or mass spectrometry (MS). NMR experiments and the labeling step in HDX-MS are commonly conducted in buffers containing 80–95% D₂O. While mixed-solvent equilibrium and kinetic isotope effects are well established, , they are often treated approximately or neglected in practical analyses, despite a body of work that used HDX in H₂O/D₂O mixtures to quantify amide fractionation and probe hydrogen-bonded structure. − Mixed H₂O/D₂O solvents are also deliberately used in NMR protocols that quantify amide–solvent exchange, including CLEANEX-PM and SOLEXSY.

In H₂O/D₂O mixtures, exchange is intrinsically bidirectional and cannot be described by a single rate constant. In this work, this is made explicit by decomposing intrinsic exchange into forward and reverse components, enabling a quantitative treatment of kinetic and equilibrium isotope effects in mixed solvents that is not accessible within standard intrinsic-rate models. This work presents a practical method to predict forward and reverse amide exchange rates in H₂O/D₂O mixtures as functions of the solvent deuterium fraction and acidity. The approach combines published reference parameters measured in pure solvents with an operational acidity scale for mixtures, and a probabilistic treatment for reprotonation inserting H or D. The resulting expressions provide explicit mixture-dependent rates, agree with previously reported data on the amide fractionation of PDLA and recover pure-solvent limits. −

Methods

Proton Transfer Theory

The chemistry of HDX reactions is described by proton transfer theory , :

$$\mathrm{XH}+\mathrm{Y}\rightleftharpoons (\mathrm{XH}···\mathrm{Y}\rightleftharpoons \mathrm{X}···\mathrm{HY})\rightleftharpoons \mathrm{X}+\mathrm{YH}$$ 2

A proton donor XH collides with a proton acceptor Y, forming a short-lived encounter complex (bracketed in eq ) in which proton redistribution is faster than dissociation. Here, XH and Y are generic acid/base labels and may be neutral or charged depending on the specific mechanism, the corresponding conjugate-acid pairs being XH/X and YH/Y. Complex formation occurs with a second-order (diffusion-limited) rate constant k _d estimated as 10¹⁰ M^–1 s^–1. The complex is regarded as highly dynamical but always weakly populated. Focusing on the forward process XH + Y → X + HY, the proton transfer rate constant k _t can be expressed as

$$k_{\mathrm{t}}=\frac{k_{\mathrm{d}}}{1+{10}^{\mathrm{\Delta }\mathrm{p}{K}_{\mathrm{a}}}}$$ 3

where ΔpK _a = pK _a(XH) – pK _a(YH) is the difference between the acidity of the donor XH and that of the conjugate acid YH of the acceptor. The acidity constant of a generic acid HA is defined as

$$\mathrm{p}{K}_{\mathrm{a}}(\mathrm{HA})=-\log K_{\mathrm{a}}(\mathrm{HA})=-\log \left(\frac{[{\mathrm{A}}^{-}][{\mathrm{H}}^{+}]}{[\mathrm{HA}]}\right)$$ 4

with K _a(HA) being the acid dissociation constant.

Intrinsic Exchange Rates

In 100% D₂O, amide HDX involves removal of a proton (H⁺) from the amide group and the transfer of a deuteron (D⁺) from bulk solvent to the amide. The reaction can be acid-, base-, or water-catalyzed, and the pseudo-first-order HDX intrinsic exchange rate of eq can be written as

$$k_{\mathrm{int}}=k_{\mathrm{acid}}[{\mathrm{D}}^{+}]+k_{\mathrm{base}}[{\mathrm{OD}}^{-}]+k_{\mathrm{water}}[{\mathrm{D}}_2\mathrm{O}]$$ 5

where k _acid, k _base, and k _water are the corresponding second-order rate constants. Temperature and pH dependence of HDX second-order rate constants were measured by Englander’s group for poly-d,l-alanine (PDLA) and 3-alanine (3-Ala). −

Here, emphasis is put on the base-catalyzed mechanism of HDX, which dominates by orders of magnitude at near-neutral conditions. Extension to other mechanisms (acid- and water-catalyzed) is reported in the Appendix. For base-catalyzed HDX in pure D₂O, the kinetic bottleneck is the first step, i.e., abstraction of the amide proton by OD^–. The amide is rapidly deuterated by a solvent D₂O molecule, and a new OD^– ion is produced. In this case, the intrinsic rate can be estimated as

$$k_{\mathrm{int}}=k_{\mathrm{B},\mathrm{ref}}(T)(B_{\lambda }\times B_{\rho })[{\mathrm{OD}}^{-}]$$ 6

B _λ and B _ρ are tabulated factors that depend on amino acid type and position relative to the amide group (λ for left, ρ for right, cfr Table ). These account for steric and inductive effects from neighboring residues, and were demonstrated to be simply additive. k _B,ref(T) is the reference rate (i.e., the exchange rate for PDLA or 3-Ala) at temperature T, which exhibits an Arrhenius-type dependence:

$$k_{\mathrm{B},\mathrm{ref}}(T)=k_{\mathrm{B},\mathrm{ref}}(T_{\mathrm{ref}})\exp \left(-\frac{E_{\mathrm{B}}}{R}\left(\frac{1}{T}-\frac{1}{T_{\mathrm{ref}}}\right)\right)$$ 7

where E _B = 17 kcal mol^–1, R is the gas constant, and k _B,ref(T _ref) is the second-order rate constant measured. Values for k _B,ref(20 °C) for H and D in H₂O and H in D₂O are given in Table . −

1. Reference Rates (PDLA) for the Base-Catalyzed Exchange of Amides in Disordered Peptides, at 20 °C.

		log k B,ref (M–1 min–1)
NH in H2O	k HH	10.08
ND in H2O	k DH	10.00
NH in D2O	k HD	10.18

^aReference rates are specified by two pedices, the first indicating the amide-bound isotope being removed, the second indicating the isotope in the catalytic base abstracting the proton, cfr Forward and Back Exchange Rates Section. Rates for 3-Ala are obtained multiplying the reference rates by a factor 1.35.

H₂O/D₂O Mixtures

This section introduces an operational acidity scale for mixed H₂O/D₂O solvents and then analyzes the influence of solvent composition on hydroxide availability and reprotonation probabilities that dictate intrinsic exchange rates. Let L denote either protium (H) or deuterium (D) isotope. An isotopic exchange reaction in a solvent with D₂O mole fraction x (H₂O mole fraction 1 – x) involving a solute XL is

$$\mathrm{XH}+\mathrm{DOL}\stackrel{{\varphi }_{\mathrm{XL}}}{\rightleftharpoons }\mathrm{XD}+\mathrm{HOL}$$ 8

Here, XH and XD are the solute protiated and deuterated species, while HOL and DOL are solvent molecules containing at least one H or D atom, respectively. The equilibrium constant ϕ_XL of the reaction eq is called the fractionation factor of XL

$${\varphi }_{\mathrm{XL}}=\frac{(\mathrm{D}/\mathrm{H}{)}_{\mathrm{XL}}}{(\mathrm{D}/\mathrm{H}{)}_{{\mathrm{L}}_2\mathrm{O}}}=\frac{[\mathrm{XD}]}{[\mathrm{XH}]}\times \frac{1-x}{x}$$ 9

which quantifies the enrichment in D of solute XL with respect to the solvent.

The concentrations [H⁺] and [OH^–] in pure H₂O follow from the pH and the ionic product, K _w,H2O, through

$$\begin{array}{c}\mathrm{pH}=-\log [{\mathrm{H}}^{+}]\\ \mathrm{pOH}=\mathrm{p}{K}_{\mathrm{w},{\mathrm{H}}_2\mathrm{O}}-\mathrm{pH}=-\log [{\mathrm{OH}}^{-}]\end{array}$$

where pK _w,H2O = −log K _w,H2O. Analogous relations hold in pure D₂O defining [D⁺] and [OD^–] in terms of pD, pOD and K _w,D2O.

In H₂O/D₂O mixtures, multiple isotopologue ionization equilibria coexist (involving H₂O, HDO, D₂O and the corresponding ions). Rather than assigning separate equilibrium constants to each elementary process, it is convenient to work with an operational acidity scale based on the total concentration of [L⁺] = [H⁺] + [D⁺], introducing the pL

$$\mathrm{pL}=-\log [{\mathrm{L}}^{+}]$$ 10

as used in studies of mixed H₂O/D₂O solvents by glass electrodes. Similarly one can define the pOL via

$$\mathrm{pOL}=-\log [{\mathrm{OL}}^{-}]$$ 11

where [OL^–] = [OH^–] + [OD^–]. At 25 °C, the ionic product of D₂O (pK _w,D2O = 1.3 × 10^–15) is lower than that of H₂O (pK _w,H2O = 14). , In mixed solvents, the corresponding effective ionic product pK _w(x) = pL­(x) + pOL­(x) varies nonlinearly with the deuterium mole fraction x, and can be described empirically as

$$\begin{array}{c}\mathrm{\Delta }\mathrm{p}{K}_{\mathrm{w}}(x)=\mathrm{p}{K}_{\mathrm{w}}(x)-\mathrm{p}{K}_{\mathrm{w},{\mathrm{H}}_2\mathrm{O}}\\ =0.7282x+0.0512x^2+0.0826x^3\end{array}$$ 12

To relate the mixture pL­(x) to the glass electrode pH-meter reading, pH*, the following relation can be used:

$$\mathrm{pL}(x)={\mathrm{pH}}^{*}\left(1+\frac{\mathrm{\Delta }\mathrm{p}{K}_{\mathrm{w}}(x)}{\mathrm{p}{K}_{\mathrm{w},{\mathrm{H}}_2\mathrm{O}}}\right)$$ 13

For D₂O at 25 °C and pH* = 7, eq yields pD = pL(1) ≃ 7.43, i.e., the traditional “+0.4” correction widely used by HDX practitioners. Example computations of these quantities are shown in the upper panel of Figure .

1.

Acidity, concentration of ions, and intrinsic rates in mixtures, expressed as a function of the deuterium fraction in the mixture, in three different scenarios: constant pH read (left), constant pL (center), and constant pOL (right). Upper panels: mixture pH read, pL, and pOL. Middle panels: concentrations of the ions relevant for base-catalyzed HDX: [OH^–], [OD^–], and [OL^–] = [OH^–] + [OD^–]. Lower panels: intrinsic forward k _forw, back k _back, and mixture intrinsic k _int(x) = k _forw(x) + k _back(x) exchange rates of PDLA. Magnitude and direction of kinetic isotope effects depend on the quantity fixed while exploring different H₂O/D₂O compositions. Data for 25 °C.

The temperature dependence of pK _w,H2O has been empirically determined as

$$\mathrm{p}{K}_{\mathrm{w},{\mathrm{H}}_2\mathrm{O}}(T)=670.86-\frac{34,691.6}{T}-105.15\ln T+0.10757T+\frac{\mathrm{2,358,000}}{T^2}$$ 14

At given composition x, the pK(x) can be computed via eq . Once the mixture pH* is measured, the pL­(x) can be computed using eq and the pOL­(x) is given by subtraction of the two values as stated above. Known x and pL­(x), the concentrations [OH^–]­(x) and [OD^–]­(x) can be computed by , as illustrated in the middle panels of Figure , in the cases of fixed pH*, pL, and pOL.

Forward and Back Exchange Rates

The reaction considered for HDX of an open amide in a H₂O/D₂O mixture is

$$\mathrm{NH}+\mathrm{DOL}\underset{{\tilde{k}}_{\mathrm{back}}}{\stackrel{{\tilde{k}}_{\mathrm{forw}}}{\rightleftharpoons }}\mathrm{ND}+\mathrm{HOL}$$ 15

where k̃ _forw and k̃ _back are the base-catalyzed reactions rate constants. Reference rates in pure solvents (Table ) are denoted k _HH, k _DH, k _HD, k _DD respectively. Here, the first index indicates the isotope initially bound to the amide (NH or ND) and the second indicates the isotope of the catalytic base (OH^– or OD^–) performing proton abstraction. These correspond to the base-catalyzed rate removal of the amide-bound proton, followed by rapid reprotonation from the bulk solvent. The unmeasured rate k _DD was estimated as

$$k_{\mathrm{DD}}=\frac{k_{\mathrm{HD}}{k}_{\mathrm{DH}}}{k_{\mathrm{HH}}}$$ 16

that is, assuming that reaction rates are governed entirely by the zero-point energies of the participating species.

The rates of eq are obtained as a sum of terms, cfr eq , each having the same structure as in eq :

$$\begin{array}{ll}{\tilde{k}}_{\mathrm{forw}}(x)& =k_{\mathrm{HH}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{HH}}[{\mathrm{OH}}^{-}](x)+k_{\mathrm{HD}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{HD}}[{\mathrm{OD}}^{-}](x)\\ {\tilde{k}}_{\mathrm{back}}(x)& =k_{\mathrm{DH}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{DH}}[{\mathrm{OH}}^{-}](x)+k_{\mathrm{DD}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{DD}}[{\mathrm{OD}}^{-}](x)\end{array}$$ 17

The coefficients of the products (B _λ × B _ρ) are tabulated, cfr Appendix.

The scheme of eq can be simplified to a pseudo-first order reaction

$$\mathrm{NH}\underset{k_{\mathrm{back}}}{\stackrel{k_{\mathrm{forw}}}{\rightleftharpoons }}\mathrm{ND}$$ 18

where the pseudo-first order forward and back exchange rates are weighted by the probability of encountering a reactive solvent molecule:

$$\begin{array}{ll}{k}_{\mathrm{forw}}(x)& =x{\tilde{k}}_{\mathrm{forw}}(x)\\ k_{\mathrm{back}}(x)& =(1-x){\tilde{k}}_{\mathrm{back}}(x)\end{array}$$ 19

that is, upon removal of the proton, in the forward reaction reprotonation occurs by DOL with probability x = [DOL]/[L₂O], and by HOL with probability 1 – x = [HOL]/[L₂O] in the reverse.

The solution to the kinetics in eq , with the constraint d­([NH] + [ND])/dt = 0 and calling D(t) the normalized concentration of deuterated amides over time, is

$$D(t)=D_{\mathrm{eq}}(x)-(D_0-D_{\mathrm{eq}}(x)){\mathrm{e}}^{-k_{\mathrm{int}}(x)t}$$ 20

where D ₀ is the initial condition,

$$D_{\mathrm{eq}}(x)=\frac{k_{\mathrm{forw}}(x)}{k_{\mathrm{forw}}(x)+k_{\mathrm{back}}(x)}$$ 21

is the fraction of deuterated amides at equilibrium, and the intrinsic rate in the mixture is defined as

$$k_{\mathrm{int}}(x)=k_{\mathrm{forw}}(x)+k_{\mathrm{back}}(x)$$ 22

Results and Discussion

The rates k _forw and k _back are generally different (for a given residue at fixed sequence, temperature and pH) and depend on the fraction x of D₂O in the solvent. The stationary state (eq ), as well as the approach-to-equilibrium rate (eq ), are functions of x through k _forw(x) and k _back(x), cfr eq . In other words, kinetic and equilibrium isotope effects are present.

Kinetic Isotope Effects

Kinetic isotope effects cause the reaction rate k _int(x) to vary as a function of the fraction x of D₂O in the solvent. These are due not only to the different second order rates of Table , but also to x influencing the mixture ion product pK _w(x) and acidity pL­(x), cfr eqs and .

By construction, the measured reference rates are recovered in pure H₂O and D₂O: by eqs and , for x = 0 (pure H₂O),

$$\begin{array}{l}{k}_{\mathrm{forw}}(0)=0,\\ k_{\mathrm{back}}(0)=k_{\mathrm{DH}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{DH}}[{\mathrm{OH}}^{-}]\end{array}$$

while for x = 1 (pure D₂O),

$$\begin{array}{l}{k}_{\mathrm{forw}}(1)=k_{\mathrm{HD}}(B_{\lambda }\times B_{\rho }{)}_{\mathrm{HD}}[{\mathrm{OD}}^{-}],\\ k_{\mathrm{back}}(1)=0\end{array}$$

The question “how does k _int(x) vary with x?” has no unique answer: it depends on the acidity measure held constant while varying x. At fixed temperature, the ionic product of the mixture pK _w(x) is assumed to depend on x only, cfr eq . The observed exchange rate depends on the acidity of the solution through the concentration of ions OH^– and OD^–, cfr eq . However, different definitions of acidity are possible in a mixture, i.e., based on pH*, pL or pOL. The behavior of k _int(x) in these scenarios is illustrated in the lower panels of Figure and commented below. These results extend the observations made for pure solvents to mixtures.

In the first column of Figure , mixtures of different compositions that yield same pH* are considered. Starting from neutral pure H₂O at 25 °C (pH = 7), for x > 0 one finds pK _w(x) > pK _w,H2O (eq ). For varying composition, the solution remains neutral, thus both pL and pOL increase. A higher pOL implies fewer available OL^– ions for base-catalyzed HDX, hence k _int(x) decreases for increasing x. As a result, the intrinsic rate of H → D in pure D₂O is about 2-fold slower than D →H in pure H₂O.

In the second example, the effective acidity of the mixture, pL, was fixed to 7.43 (because pL(1) = 7.43 at pH* = 7 and T = 25 °C). Here, decreasing x causes pK _w(x) to decrease. Because pL is fixed, pOL decreases too, and a more substantial kinetic isotope effect is observed. D →H in H₂O results about 5-fold faster than H → D in D₂O on a pL scale.

The third scenario evaluates the effect of x at fixed pOL. In this case, k _int(x) increases with x. This is because the concentration of catalysts is fixed hence the HDX rates depends on the rates of Table . Since k _HH > k _DH and k _HD > k _DD, that is, it is easier to extract H than D from the amide group, the HDX rate is higher for increasing D₂O content. In this case, the extent of the isotope effect (moving from pure D₂O to H₂O) is smaller than the other cases (1.5 times faster in D₂O).

As a visual example of how the intrinsic exchange rate k _int(x) varies with x, Figure displays the approach-to-equilibrium rates (eq ) for two short sequences at fixed pH*, which is the experimentally controlled parameter, and varying D₂O content in the mixture. Kinetic isotope effects lead to a decrease of the reaction rate constant when the D₂O content is increased, as discussed above.

2.

Intrinsic exchange rates k _int(x) for short polypeptide sequences ‘AAAAAAA’ (left) and ‘PEPTIDE’ (right), calculated for varying D₂O fraction in the mixture x, T = 25 °C, pH* = 7, and using PDLA reference rates.

Equilibrium Isotope Effects

An equilibrium isotope effect is described by the fractionation factor (eq ), which for reaction eq is simply ϕ = k̃ _forw(x)/k̃ _back(x), cfr Appendix. Within the assumptions of the base-catalyzed model developed above, the fractionation factor reduces to

$$\varphi =\frac{{\tilde{k}}_{\mathrm{forw}}(x)}{{\tilde{k}}_{\mathrm{back}}(x)}=\frac{k_{\mathrm{HH}}}{k_{\mathrm{DH}}}=1.20$$ 23

with k _HH and k _DH derived from Table .

This value is unaffected by neighboring side chains, because the local sequence multiplicative factors (B _λ × B _ρ) cancel in the ratio, cfr Appendix. A corollary of eq is that, for random coil peptides in conditions where base catalysis dominates, the equilibrium deuterium enrichment of amides will exceed the bulk D₂O content, i.e., D _eq(x) > x. This is shown in Figure . A physical basis of this difference is given by the difference in the pK _a between amide deuterated (ND) and protiated (NH) forms: pK _a(ND) > pK _a(NH), thus D tends to be bound more tightly to the amide, and the amide is enriched with D at equilibrium. The value ϕ = 1.20, as well as the predicted deuterium enrichment, is insensitive to temperature and pH (provided that the base-catalyzed mechanism is the dominant one).

3.

Equilibrium isotope effects. At equilibrium, the amide is enriched in D with respect to solvent (D _eq(x) > x, as ϕ = 1.20 > 1).

The theoretical value ϕ = 1.20 (eq ) agrees closely with the measured one for PDLA (ϕ_PDLA = 1.22). The larger fractionation factor reported for poly-d,l-lysine (PDLK) within the same study (ϕ_PDLK = 1.46) may be attributed to a residual helical propensity of PDLA. It is thus possible that the equilibrium fractionation expected for unstructured peptides is underestimated by eq , since structured amides generally display lower fractionation (usually, ϕ ≲ 1) than unstructured ones. , Residual uncertainties remain also because acid- and water-catalyzed exchange, solvent–side-chain interactions, and buffer composition can all shift the apparent fractionation. Notably, the equilibrium fractionation factor emerges directly from the kinetic framework without introducing additional adjustable parameters or sequence-specific assumptions. This indicates that the observed fractionation is an inherent consequence of intrinsic exchange chemistry in mixed solvents rather than an independent thermodynamic input.

Conclusions

This work provides a practical method to estimate intrinsic exchange rates in H₂O/D₂O mixtures as a function of solvent D₂O content and acidity, recovering known limits for pure solvents. − The model yields closed expressions for forward and back exchange rates, that for base-catalyzed exchange incorporate mixture-dependent catalyst availability and probabilistic insertion of H or D. For completeness, the corresponding acid- and water-catalyzed contributions are described in the Appendix.

Kinetic and equilibrium isotope effects emerge as natural consequences of the chemistry of the mixture and are not determined by composition alone: both the magnitude and the direction of the kinetic isotope effect depend on how acidity is specified or experimentally controlled (e.g., fixed pH* versus fixed operational acidity). Under base-catalyzed conditions, the framework predicts equilibrium amide enrichment in D, in good agreement with measurements performed on PDLA (ϕ ≃ 1.20). Mismatch with PDLK (ϕ ≃ 1.46) suggests that residual structure or other unmodeled factors can affect fractionation and should be assessed by systematic benchmarking against random-coil standards.

Beyond providing explicit intrinsic exchange rates in mixtures, this framework emphasizes that HDX is bidirectional and that should be described by both forward and back exchange rates to capture kinetic and equilibrium isotope effects. The same framework can be applied to HDX-NMR data, where mixture-dependent kinetics and final enrichment must be accounted for to obtain unbiased estimates of local stability (in terms of protection factors). This is also relevant to HDX-MS workflows, where back exchange can occur both during labeling (due to incomplete buffer deuteration) and in subsequent handling steps (e.g., quenching, usually performed at intermediate D₂O content, low temperature and pH). A consistent treatment of exchange in mixture is the first step toward physically grounded back exchange corrections. In this case, the chemistry of the mixture defines baseline effects, that must be integrated with the environment (e.g., salts, cosolvents, denaturants, and whatever additive that can alter exchange kinetics).

Appendix

Multiplicative Factors for Intrinsic Exchange Rates

Coefficients A _λ, A _ρ, B _λ, B _ρ are given in Table in both H₂O and D₂O for all residues except Asp (D), Glu (E) and His (H). In these cases, the coefficients A _ζ,L, B _ζ,L(X) of residue X, where ζ is λ or ρ, and L is H (for D → H exchange) or D (H → D), are given by

$$\begin{array}{c}{A}_{\zeta ,\mathrm{L}}(\mathrm{X})=\frac{{10}^{A_{\zeta }(X+)-\mathrm{p}{K}_{\mathrm{c},\mathrm{L}}(\mathrm{X})}+{10}^{A_{\zeta }(\mathrm{X0})-\mathrm{pL}(x)}}{{10}^{-\mathrm{p}{K}_{\mathrm{c},\mathrm{L}}(\mathrm{X})}+{10}^{-\mathrm{pL}(x)}},\\ B_{\zeta ,\mathrm{L}}(\mathrm{X})=\frac{{10}^{B_{\zeta }(X+)-\mathrm{p}{K}_{\mathrm{c},\mathrm{L}}(\mathrm{X})}+{10}^{B_{\zeta }(\mathrm{X0})-\mathrm{pL}(x)}}{{10}^{-\mathrm{p}{K}_{\mathrm{c},\mathrm{L}}(\mathrm{X})}+{10}^{-\mathrm{pL}(x)}}\end{array}$$ A1

where

$$\mathrm{p}{K}_{\mathrm{c},\mathrm{L}}(\mathrm{X})=-\log \left({10}^{C_{\mathrm{L}}(\mathrm{X})}\exp \left(-\frac{E_{\mathrm{a},\mathrm{L}}(\mathrm{X})}{R}\left(\frac{1}{T}-\frac{1}{T_{\mathrm{ref}}}\right)\right)\right)$$ A2

and parameters C _L(X), E _a,L(X), T _ref given in Table .

A1. Multiplicative Factors for Intrinsic Exchange Rates − .

	log A λ	log A ρ	log B λ	log B ρ
A	0.00	0.00	0.00	0.00
C	–0.54	–0.46	0.62	0.55
C2	–0.74	–0.58	0.55	0.46
D0	0.90	0.58	0.10	–0.18
D+	–0.90	–0.12	0.69	0.60
E0	–0.90	0.31	–0.11	–0.15
E+	–0.60	–0.27	0.24	0.39
F	–0.52	–0.43	–0.24	0.06
G	–0.22	0.22	–0.03	0.17
H0			–0.10	0.14
H+	–0.80	–0.51	0.80	0.83
I	–0.91	–0.59	–0.73	–0.23
K	–0.56	–0.29	–0.04	0.12
L	–0.57	–0.13	–0.58	–0.21
M	–0.64	–0.28	–0.01	0.11
N	–0.58	–0.13	0.49	0.32
P		–0.19		–0.24
Pc		–0.85		0.60
Q	–0.47	–0.27	0.06	0.20
R	–0.59	–0.32	0.08	0.22
S	–0.44	–0.39	0.37	0.30
T	–0.79	–0.47	–0.07	0.20
V	–0.74	–0.30	–0.70	–0.14
W	–0.40	–0.44	–0.41	–0.11
Y	–0.41	–0.37	–0.27	0.05
NT		–1.32		1.62
CT	0.96		–1.80

A2. Parameters for the Computation of B _λ and B _ρ Coefficients for Asp, Glu, and His .

residue	E a,H (kcal/mol)	C H	E a,D (kcal/mol)	C D	T ref (K)
D	0.96	–3.87	1	–4.48	278
E	1.083	–4.33	1.083	–4.93	278
H	7.5	–7	7.5	–7.42	278

Calculation of ϕ

From definitions of and , one has that (B _λ × B _ρ)_HH = (B _λ × B _ρ)_DH and (B _λ × B _ρ)_DD = (B _λ × B _ρ)_HD. Calling ξ = (B _λ × B _ρ)_HD/(B _λ × B _ρ)_HH (ξ = 1 if Asp, Glu or His are not neighboring to the amide group), the rates k̃ _forw and k̃ _back of eq are written

$$\begin{array}{l}{\tilde{k}}_{\mathrm{forw}}(x)=(B_{\lambda }\times B_{\rho }{)}_{\mathrm{HH}}(k_{\mathrm{HH}}[{\mathrm{OH}}^{-}](x)+\xi k_{\mathrm{HD}}[{\mathrm{OD}}^{-}](x))\\ {\tilde{k}}_{\mathrm{back}}(x)=(B_{\lambda }\times B_{\rho }{)}_{\mathrm{HH}}(k_{\mathrm{DH}}[{\mathrm{OH}}^{-}](x)+\xi k_{\mathrm{DD}}[{\mathrm{OD}}^{-}](x))\end{array}$$ A3

­[OH^–]­(x) and [OD^–]­(x) can be written in terms of [OL^–]­(x) using the definition of fractional abundance of OD^– in OL^– :

$$f_{\mathrm{OL}}(x)=\frac{[{\mathrm{OD}}^{-}](x)}{[{\mathrm{OL}}^{-}](x)}=\frac{x{\varphi }_{\mathrm{OL}}}{1-x+x{\varphi }_{\mathrm{OL}}}$$ A4

where we used the fractionation parameter ϕ_OL = 0.43. Hence becomes

$$\begin{array}{l}{\tilde{k}}_{\mathrm{forw}}(x)=(B_{\lambda }\times B_{\rho })(k_{\mathrm{HH}}(1-f_{\mathrm{OL}})+\xi k_{\mathrm{HD}}{f}_{\mathrm{OL}})[{\mathrm{OL}}^{-}](x)\\ {\tilde{k}}_{\mathrm{back}}(x)=(B_{\lambda }\times B_{\rho })(k_{\mathrm{DH}}(1-f_{\mathrm{OL}})+\xi k_{\mathrm{DD}}{f}_{\mathrm{OL}})[{\mathrm{OL}}^{-}](x)\end{array}$$ A5

where the subscript “HH” of (B _λ × B _ρ) has been dropped to ease the notation. Recalling eq , and defining ψ = k _HD/k _HH = k _DD/k _DH,

$$\begin{array}{l}{\tilde{k}}_{\mathrm{forw}}(x)=k_{\mathrm{HH}}(B_{\lambda }\times B_{\rho })(1-f_{\mathrm{OL}}+\xi \psi f_{\mathrm{OL}})[{\mathrm{OL}}^{-}](x)\\ {\tilde{k}}_{\mathrm{back}}(x)=k_{\mathrm{DH}}(B_{\lambda }\times B_{\rho })(1-f_{\mathrm{OL}}+\xi \psi f_{\mathrm{OL}})[{\mathrm{OL}}^{-}](x)\end{array}$$ A6

from which one has

$$\frac{{\tilde{k}}_{\mathrm{forw}}}{{\tilde{k}}_{\mathrm{back}}}=\frac{k_{\mathrm{HH}}}{k_{\mathrm{DH}}}(=1.20)$$ A7

Extension to Other Mechanisms

In H₂O/D₂O mixtures (D₂O mole fraction x), the intrinsic exchange rates k _forw(x), k _back(x) are written as a sum of contributions (of acid-, base-, and water-catalyzed exchange):

$$\begin{array}{c}{k}_{\mathrm{forw}}(x)=k_{\mathrm{A},\mathrm{forw}}(x)+k_{\mathrm{B},\mathrm{forw}}(x)+k_{\mathrm{W},\mathrm{forw}}(x),\\ k_{\mathrm{back}}(x)=k_{\mathrm{A},\mathrm{back}}(x)+k_{\mathrm{B},\mathrm{back}}(x)+k_{\mathrm{W},\mathrm{back}}(x)\end{array}$$

In the main text, the derivation focused on the base-catalyzed contribution, which dominates near-neutral conditions and therefore governs most HDX measurements. For completeness, this appendix outlines how the other two pathways can be incorporated into the framework presented in the main text for the case of the base-catalyzed pathway.

Water-Catalyzed Exchange

In the water-catalyzed pathway, a solvent water molecule acts as a base, abstracting the amide H or D. This contribution is essentially pH-independent and usually negligible compared to the other terms, except in the region of the minimum. Sequence effects are addressed by the coefficients of Table , as k _W = k _W,ref(B _λ × B _ρ), with reference rates given in Table . Literature tabulates the rates of H → D exchange in D₂O (k _W ) and the rate of D → H in H₂O (k _W ). In a mixed solvent with deuterium mole fraction x, it is assumed that the rate of abstraction depends only on the isotope being removed, while the isotope incorporated upon reprotonation is determined solely by the solvent composition. In this case, the forward and back water-catalyzed intrinsic exchange rates are

$$\begin{array}{ll}{k}_{\mathrm{W},\mathrm{forw}}(x)=x{k}_{\mathrm{W}}^{\mathrm{H}\to \mathrm{D}},& k_{\mathrm{W},\mathrm{back}}(x)=(1-x)k_{\mathrm{W}}^{\mathrm{D}\to \mathrm{H}}\end{array}$$ A8

A3. Reference Rates for Acid- and Water-Catalyzed Exchange of at 20°C .

		log k A,ref (M–1 min–1)		log k W,ref (M–1 min–1)
NH in D2O	k A	1.62	k W	–1.50
ND in H2O	k A	1.40	k W	–1.60

Acid-Catalyzed Exchange

In the acid-catalyzed pathway, exchange proceeds through protonation of the amide by solvated hydronium species before loss of the bound isotope. In this case, the isotopic identity of the proton-donating species, i.e., D⁺ or H⁺, determines whether the event contributes to H → D or D → H exchange. Acid-catalyzed second-order rates k _A , k _A can be calculated from reference values (Table ) and accounting for the sequence dependence with the corresponding multiplicative factors A _λ, A _ρ from Table , e.g., k _A = k _A,ref (A _λ × A _ρ)_HD. Since the key isotopic selectivity lies in the isotope donated to the amide, it is assumed that the intrinsic rate remains the same as in pure solvent, but the relevant concentrations [H⁺]­(x) and [D⁺]­(x) should be used:

$$\begin{array}{ll}{k}_{\mathrm{A},\mathrm{forw}}(x)=k_{\mathrm{A}}^{\mathrm{H}\to \mathrm{D}}[{\mathrm{D}}^{+}](x),& k_{\mathrm{A},\mathrm{back}}(x)=k_{\mathrm{A}}^{\mathrm{D}\to \mathrm{H}}[{\mathrm{H}}^{+}](x)\end{array}$$ A9

The relative abundance of [D⁺] is given, analogously to , as

$$f_{\mathrm{L}}(x)=\frac{[{\mathrm{D}}^{+}](x)}{[{\mathrm{L}}^{+}](x)}=\frac{x\mathcal{l}}{1+x-x\mathcal{l}}$$ A10

where $\mathcal{l}=0.69$ is the fractionation parameter for L₃O⁺ and [L⁺]­(x) = [H⁺]­(x) + [D⁺]­(x), cfr main text.

A Python implementation of the method presented, allowing to compute k _forw and k _back for arbitrary sequences as a function of solvent temperature, glass electrode pH read, and deuterium fraction, is available at https://github.com/pacilab/hdx-rates-mixtures.

The authors declare no competing financial interest.