D Exchange Kinetic of Macromolecule

Article pubs.acs.org/ac

Cite This: Anal. Chem. 2018, 90, 5116−5121

Analytical Description of the H/D Exchange Kinetic of Macromolecule Yury Kostyukevich,†,‡,§ Alexey Kononikhin,‡,§ Igor Popov,§ and Eugene Nikolaev*,†,‡,§,⊥ †

Skolkovo Institute of Science and Technology, Novaya St., 100, Skolkovo 143025, Russian Federation Institute for Energy Problems of Chemical Physics Russian Academy of Sciences, Leninskij pr. 38 k.2, 119334 Moscow, Russia § Moscow Institute of Physics and Technology, 141700 Dolgoprudnyi, Moscow Region, Russia ⊥ Emanuel Institute for Biochemical Physics, Russian Academy of Sciences, Kosygina st. 4, 119334 Moscow, Russia ‡

S Supporting Information *

ABSTRACT: We present the accurate analytical solution obtained for the system of rate equations describing the isotope exchange process for molecules containing an arbitrary number of equivalent labile atoms. The exact solution was obtained using Mathematica 7.0 software, and this solution has the form of the time-dependent Gaussian distribution. For the case when forward exchange considerably overlaps the back exchange, it is possible to estimate the activation energy of the reaction by obtaining a temperature dependence of the reaction degree. Using a previously developed approach for performing H/D exchange directly in the ESI source, we have estimated the activation energies for ions with different functional groups and they were found to be in a range 0.04−0.3 eV. Since the value of the activation energy depends on the type of functional group, the developed approach can have potential analytical applications for determining types of functional groups in complex mixtures, such as petroleum, humic substances, bio-oil, and so on. he first isotopic exchange reactions between hydrogen/ deuterium and 16O/18O were observed by Lewis in 1933.1,2 The analytical potential of this reaction was quickly realized, and in a few years, Hans Ussing postulated that only the H atoms of the surface layer of the particles are able to exchange with the surrounding water.3 Based on this principle, H/D exchange reaction is currently widely used for the investigation of the conformational dynamic of biological molecules both in the solution4−9 and in the gas phase.10−14 Despite the wide use of this approach, there still exists a certain difficulty in the treatment of the H/D exchange reaction in terms of a system of rate equations. Simple systems, like

T

AX + BX* ↔ AX* + BX

Normally the isotopic distribution has a Gaussian form, though sometimes it unusually broadens21−23 or even becomes bimodal,24,25 which means that the target molecule can exist in different conformations.26,27 The shape of the isotopic distribution in the equilibrium obeys the binomial distribution and there are rapid methods to calculate it.28,29 Recent progress in the quantum chemistry and molecular dynamics made it possible to accurately describe the isotope exchange process even in the case of peptide.30−33 Khakinejad et al. demonstrated that gas phase hydrogen/deuterium of a peptide can be accurately simulated using molecular dynamics and hydrogen accessibility scoring-number of the effective collisions model.33 Such simulations are complicated and require the consideration of several hundreds of possible structures of peptide. At the same time, to our knowledge, no simple theory was developed to describe the evolution of the isotopic distribution when an isotopically pure molecule is subjected to the isotope exchange reaction. Here we present the analytical solution of the general system of rate equations describing the isotopic exchange of an arbitrary macromolecule. For simplicity, we will consider a case of hydrogen/deuterium exchange reaction.

(1)

can be described by the following formula: −ln(1 − F ) = R[(a + b)ab]t

(2)

Here F is the fraction (progress variable) of isotopic exchange, R is the rate of exchange, [AX] + [AX*] = a, and [BX] + [BX*] = b.15−18 In a case of complex mechanism,19,20 the exact equations describing the exchange can be complicated, but eq 2 remains valid to a certain extent. It is obvious that if a molecule containing several labile atoms is subjected to the isotope exchange reaction molecules with different degree of substitution will be present in a system. The regular “bulk” chemistry cannot distinguish such molecules, but use of mass spectrometry can effectively separate them and reveal a real form of isotopic distribution. © 2018 American Chemical Society

Received: December 11, 2017 Accepted: March 20, 2018 Published: March 20, 2018 5116

DOI: 10.1021/acs.analchem.7b05151 Anal. Chem. 2018, 90, 5116−5121

Article

Analytical Chemistry

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THEORY OF THE EXCHANGE KINETIC

In order to prove this hypothesis, one must show that, for any N, I, kd, kh, and t, the solution (5) satisfies the arbitrary equation of the system (3):

Let us consider a molecule M that contains N equivalent labile atoms. Molecule is placed in media filled with D2O and H2O. Molecules of media continuously interact with the target molecule. If the concentration of M is relatively low, than exchange reaction can be considered as a pseudo first-order reactions with kD and kH, constants of forward and back exchange reaction. Let us denote Mi, the concentration of the molecule in which i atoms of hydrogen are replaced for deuterium. Molecules of media continually interact with M, but only those collisions in which D2O interacts with labile hydrogen and H2O interacts with labile deuterium will lead to the exchange. The portion of such collisions are i/N and (N −

dM i ⎛ N − (i − 1) N−i i ⎞ i+1 = ⎜ − kD − kH ⎟Mi + kH M i + 1 + kD Mi − 1 ⎝ N N⎠ N N dt

(6)

This can be done using Mathematica 7.0 (see Supporting Information). The eq 5 can be rewritten as ⎛ kd ⎞i N! (1 − e1/ N(−kd − kh)t )i ⎜ ⎟ i! (N − i)! ⎝ kh + kd ⎠

Mi(t ) =

N−i ⎛ kd ⎛ e1/ N(−kh− kd)t ⎞⎞⎟ ⎜⎜1 − ⎜⎜1 − ⎟⎟⎟ kh + kd ⎝ kh + kd ⎠⎠ ⎝

(7)

We can see that eq 7 has the form of binomial distribution, which can be approximated with a Gaussian distribution: Mi ≈

2 2 1 e−(i − μ) /2σ ; σ 2π

μ=N

kd (1 − e1/ N(−kd − kh)t ); kd + kh

σ=N

kd(1 − e1/ N(−kd − kh)t ) ⎛ k (1 − e1/ N(−kd − kh)t ) ⎞ ⎜⎜1 − d ⎟⎟ kd + kh kd + kh ⎝ ⎠ (8)

We can see, that the evolution of the isotopic envelope is described by the time dependence of expected value (average number of exchanges) and variance (width of the isotopic envelope). The full code for obtaining (guessing and proving) the solution using Mathematica 7.0 is given in the Supporting Information. There are two important limiting cases: (1) t → ∞. This is the case of the equilibration of forward and backward exchange reactions. The equilibrium is described with following parameters:

Figure 1. Proposed model of H/D exchange reaction of macromolecule. The exchange occur only if D2O molecule interacts with the labile hydrogen or H2O interacts with labile deuterium. N is total number of labile atoms.

i)/N correspondingly (see Figure 1). So, the exchange reaction can be described as the following system of rate equations: dM 0 1 = − kDM 0 + kH M1 dt N dM i ⎛ N−i i ⎞ i+1 Mi + 1 = ⎜ − kD − kH ⎟Mi + kH ⎝ ⎠ dt N N N N − (i − 1) Mi − 1 + kD N dMN 1 = kD MN − 1 − kHMN dt N

Mi ≈

(3)

kd kd + kh

(9)

2 2 1 e−(i − μ) /2σ ; σ 2π

σ = N (1 − e−(kd / N )t )e−(kd / N )t

(10)

We can see that the isotopic envelope has a zero width in the beginning then broadens to the maximum width at the time: Tmaxσ =

N ln 2 kd

(11)

and then shrinks to the zero width. Using the equations in eq 10, it is possible to determine activation energy of the exchange reaction if the temperature

(− 1)i (− 1 + e1/ N(−kd − kh)t )i kdi(e1/ N(−kh− kd)t kd + kh)N − i (kd + kh)N

μ=N

μ = N (1 − e−(kd / N )t );

(4)

N! i! (N − i)! ×

kd ⎛ kd ⎞ kdkh ; ⎜1 − ⎟=N kd + kh ⎝ kd + kh ⎠ (kd + kh)2

Mi ≈

This system cannot be solved directly, but using the software Mathematica 7.0, that can process analytical formulas, it is possible to find an analytical solution for any N. Analyzing solutions found for several values of N (see Appendix 1, for example) we proposed that the general solution should be Mi(t ) =

σ=N

(2) kd ≫ kh. For this case, the solution has the following form:

with the following initial conditions: M 0|t = 0 = M 00 ; Mi|t = 0 = 0, 0 < i ≤ N

2 2 1 e−(i − μ) /2σ ; σ 2π

(5) 5117


Article


capillary and exchange labile hydrogens for deuterium.40,41 Variation of the temperature of the desolvating capillary allows varying the degree of the exchange reaction. Therefore, using eq 14, it is possible to estimate the activation energy. All our experiments were performed on an LTQ FT Ultra (Thermo Electron Corp., Bremen, Germany) mass-spectrometer equipped with a 7T superconducting magnet. Ions were generated by an IonMax Electrospray ion source (Thermo Electron Corp., Bremen, Germany) in positive and negative ESI mode. The temperature of the desolvating capillary was varied from 50 to 450 °C. The length of the capillary was 105 mm and its inner diameter was 0.5 mm. The infusion rate of the sample was 1 μL/min and the needle voltage was 4000 V. For in-ESI source isotope exchange, the atmosphere was saturated with the D2O (or H218O) vapors in the region between ESI tip and the inlet of desolvating capillary by placing 400 μL of D2O (or H218O) on a copper plate positioned approximately 7 mm below the ESI needle (see Figure 3).

dependence of the reaction degree is given. The second equation in eq 10 can be rewritten as kd = −

μ⎞ N ⎛⎜ ln 1 − ⎟ ⎝ t N⎠

(12)

Substituting the expression for k using Arrhenius equation:

kd = kd0e−Ea / kT

(13)

we obtain −

⎛N⎞ ⎛ N ⎛ ⎛ ⎛ Ea μ ⎞⎞ μ ⎞⎞ = ln⎜− 0 ln⎜1 − ⎟⎟ = ln⎜ 0 ⎟ + ln⎜ − ln⎜1 − ⎟⎟ ⎝ ⎝ ⎠ ⎝ kT N ⎠ N ⎠⎠ ⎝ kd t ⎠ ⎝ kd t (14)

So, the activation energy of the H/D exchange reaction can be obtained as an angle of the line: ⎛ 1 ⎛ ⎛ μ ⎞⎞⎞ ⎜ − ; ln⎜ −ln⎜1 − ⎟⎟⎟ ⎝ ⎝ N ⎠⎠⎠ ⎝ kT

(15)

The developed theory has following limitations: labile atoms on the surface of molecule are not always equivalent, the molecule can change its conformation, so additional atoms will become available for the exchange. Special approaches based on the molecular dynamics and hydrogen accessibility scoringnumber of effective collisions model32,33 should be used in order to accurately describe the isotope exchange of the large molecule taking into account conformer dynamics, effects of charge etc.

■

Figure 3. Experimental setup for investigation of the kinetic of the isotope exchange.

SIMULATION AND EXPERIMENT We have simulated the evolution dynamics of the deuterium distribution for a molecule that contains N = 100 equiv labile hydrogens and for kD = 1 and kH = 0. The results are presented in Figure 2. It can be seen that the isotopic distribution adopts a Gaussian shape, shifts, and then shrinks.

In Figure 4, the kinetic of the following exchange reactions is presented: 16O/18O exchange42 for [IO3]− ion, 16O/18O exchange for [I3O13 ]−, H/D exchange reaction43,44 of ubiquitin6+, angiotensin+, dopamine+, resorcinol+, oligonucleotide45 A206−, and oligosaccharide dextran+.46 It can be seen that the isotopic distribution adopts a Gaussian shape for all samples, except dextran and resorcinol, for both H/D and 18 O/16O exchange reactions. Total numbers of exchangeable atoms are as follows: 3 for [IO3]−, 13 for [I3O13]−, 150 for [ubiquitin + 6H]6+, 155 for [ubiquitin + 11H]11+, 15 for [angiotensin + H]+, 5 for [dopamine + H]+, 55 for [A20 − 6H] 6− , 14 for [dextran5 + Na]+, and 6 for [resorcinol + H]+. We can see that the deuterium distribution in the case of dextran becomes bimodal. Previously, we have explained this effect by the assumption that ions of oligosaccharides can be formed in two conformations.47,48 One of the conformations is folded and the exchange goes slow, the other one is unfolded and the exchange goes fast. Molecule of resorcinol has two −OH groups, nevertheless, protonated ion demonstrates six exchanges. This can be explained by keto−enol tautomerism that facilitates the exchange in adjacent −CH positions. Previously, such effects were observed for several organic molecules.46,49 The mechanism of the H/D exchange involves the formation of the hydrogen bond between deuterium of the D2O and lone electron pair of the heteroatom (nitrogen or oxygen) of the molecule. Schematically, the reaction is represented in Scheme 1:50 This is why the exchange occurs almost instantly in O−H, Cl−H, Br−H, and I−H, and does not occur in Si−H, P−H, and C−H bonds. In neutral N−H groups, the exchange is fast but protonation of nitrogen makes electron pair shared with a substituent and the rate of exchange considerably decrease.

Figure 2. Simulation of the evolution of the isotopic distribution for a molecule with 100 equiv labile atoms and kD = 1, different moments of time.

Using eq 14, it is possible to estimate the activation energy if the temperature dependence of the isotope exchange reaction degree is obtained. Recently we have proposed a simple approach for performing of the H/D exchange directly in the ESI source.34−39 If one infuse D2O into the ESI source, then molecular ions interact with D2O in the heated desolvating 5118


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Figure 4. Experimental investigation of the H/D and 16O/18O exchange kinetic; a1, a2: [IO3]−; b1, b2: [I3O13]−; c1, c2: [ubiquitin]6; d1, d2: [dopamine]+; e1, e2: [angiotensin]+; f1, f2: [A20]6; g1−g4: [dextran5 + Na]+; h1−h4: [resorcinol]+.

see that for many compounds the maximum percentage of exchange is about only 50%. This can be explained by the slow rate of the exchange in certain groups (for example, in −NH2) and by the conformational effects that protect certain hydrogens from the contact with D2O. The effect of back exchange is negligible because the atmosphere in the ESI source is saturated with D2O. Increasing the reaction time by increasing the length of the desolvating capillary can help to increase the degree of the exchange.51,52 The determination of the activation energy is shown in the Figure 5B. We can see that for [IO3]− and [I3O13]− ions the charts obtained using eq 15 have the form of almost perfect lines. It means that the developed theory is valid and that oxygen atoms in those ions are equivalent. For other ions the labile atoms cannot be considered equivalent and that can explain the deviation of the curves in Figure 5B from straight

Scheme 1. H/D Reaction in the Complex Formed by Hydrogen Bond

Oxygen exchange occurs via nucleophilic substitution. For the exchange to take place, the central atom must be able to increase its coordination number and must carry positive charge. For example, the in the HClO4 the Cl atom cannot increase its coordination number, but iodine in [IO4]− easily increases it from 4 to 6.50 We have summarized the results of the observation of the kinetic of isotope exchange reactions for many compounds in the Figure 5. In the Figure 5A is represented the temperature dependence of the isotopic exchange reaction degree. We can

Figure 5. Determination of activation energy: (A) dependence of the isotope exchange reaction degree on the temperature; (B) estimation of the activation energy. 5119


Article

Analytical Chemistry lines, moreover, for large ions (proteins, oligosaccharides, oligonucleotides) we must take into account possible conformational changes that can occur during the ionization and transport.21,22,33,53,54 In addition, the important question is the variation of the gas dynamics in the desolvating capillary with the temperature. Changes in the gas velocity leads to the variation of t, and changes in pressure leads to the variation of k0d in eq 14. But previously performed computer simulation52,55,56 of the gas dynamics in the inlet capillary and the fact that logarithm is slow changing function allows to neglect this effect. The activation energy was estimated for several ions and was found to be in a range 0.04−0.3 eV. We can see that the value of the activation energy differ for compounds belonging to different classes. Indeed, the activation energies for H/D exchange reaction for dopamine and resorcinol were found to be 0.058 and 0.048 eV; correspondingly, angiotensin, ubiquitin, and oligonucleotide demonstrate activation energies in the range 0.072−0.085 eV. Oligosaccharide has an activation energy of ∼0.17 eV. Based on this result, it will be possible using in-ESI source H/D exchange to understand the nature of functional groups for unknown compounds and relate those compounds to most probable class.

■

M 0(t ) = M1(t ) = M 2(t ) = M3(t ) = M4(t ) = M5(t ) =

■

(kd + kh)5

;

− 5(− 1 + e1/5(−kd − kh)t )kd(e1/5(−kh− kd)t kd + kh)4 (kd + kh)5

;

10(− 1 + e1/5(−kd − kh)t )2 kd2(e1/5(−kh− kd)t kd + kh)3 (kd + kh)5

;

− 10(− 1 + e1/5(−kd − kh)t )3 kd3(e1/5(−kh− kd)t kd + kh)2 (kd + kh)5 5(− 1 + e1/5(−kd − kh)t )4 kd4(e1/5(−kh− kd)t kd + kh)1 (kd + kh)5

;

;

− (− 1 + e1/5(−kd − kh)t )5 kd5 (kd + kh)5 (16)

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.7b05151. Additional information (PDF).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

CONCLUSION

ORCID

We have obtained the analytical solution for the system of rate equations describing the isotope exchange process for molecule. We treat the molecule as a sphere in which all labile atoms are equivalent and placed on the surface. The exchange occurs only in those collisions of molecules of media with target molecule in which participate different isotopes. For example, the exchange does not occur when D2O interacts with already deuterated part of the molecule. The exact solution of the system (eq 3) was obtained using Mathematica 7.0 software and this solution has the form of the time-dependent Gaussian distribution. For the case when forward exchange considerably overlaps the back exchange it is possible to estimate the activation energy of the reaction by obtaining a temperature dependence of the reaction degree. The activation energy was estimated for several ions and was found to be in a range 0.04− 0.3 eV. The value of activation energy can indicate the presence of certain functional groups in the molecule. The deviation of the relationship eq 15 from the straight line can indicate the presence in the molecule nonequivalent labile atoms. This can have potential analytical applications for determining of types of functional groups in complex mixtures, such as petroleum, humic substances, bio-oil, and so on.

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(e1/5(−kh− kd)t kd + kh)5

Yury Kostyukevich: 0000-0002-1955-9336 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

The research was supported by the Russian Scientific Foundation Grant No. 14-24-00114. Notes

The authors declare no competing financial interest.

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REFERENCES

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APPENDIX 1

The exact solution of the system eq 3 for N = 5, obtained using Mathematica 7.0: 5120


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D Exchange Kinetic of Macromolecule

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