D Exchange Kinetic of Macromolecule

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Analytical description of the H/D exchange kinetic of macromolecule. Yury I. Kostyukevich, Alexey S. Kononikhin, Igor A. Popov, and Eugene N. Nikolaev Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b05151 • Publication Date (Web): 20 Mar 2018 Downloaded from http://pubs.acs.org on March 21, 2018

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Analytical Chemistry

1

Analytical description of the H/D exchange kinetic

2

of macromolecule.

3 4

AUTHOR NAMES

5

Yury Kostyukevicha,b,c, Alexey Kononikhinb,c, Igor Popovc and Eugene Nikolaev*a,b,c

6

AUTHOR ADDRESS

7

a

8

Federation

9

b

Skolkovo Institute of Science and Technology Novaya St., 100, Skolkovo 143025 Russian

Institute for Energy Problems of Chemical Physics Russian Academy of Sciences Leninskij pr.

10

38 k.2, 119334 Moscow, Russia;

11

c

Moscow Institute of Physics and Technology, 141700 Dolgoprudnyi, Moscow Region, Russia

12 13

KEYWORDS

14

Isotopic exchange, Kinetic, ESI, Proteins, FT ICR.

15

ABSTRACT

ACS Paragon Plus Environment

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Analytical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 23

1

We present the accurate analytical solution obtained for the system of rate equations describing

2

the isotope exchange process for molecule containing arbitrary number of equivalent labile

3

atoms. The exact solution was obtained using Mathematica 7.0 software and this solution has the

4

form of the time-dependent Gaussian distribution. For the case when forward exchange

5

considerable overlaps the back exchange it is possible to estimate the activation energy of the

6

reaction by obtaining a temperature dependence of the reaction degree. Using previously

7

developed approach for performing H/D exchange directly in the ESI source, we have estimated

8

the activation energies for ions with different functional groups and they were found to be in a

9

range 0.04 eV - 0.3 eV. Since the value of the activation energy depends on the type of the

10

functional group, the developed approach can have potential analytical applications for

11

determining of types of functional groups in complex mixtures, such as petroleum, humic

12

substances, bio-oil etc.

13 14 15

INTRODUCTION The first isotopic exchange reactions between Hydrogen/Deuterium and

16

O/18O were

16

observed by Lewis in 19331,2. The analytical potential of this reaction was quickly realized and in

17

few years Hans Ussing postulated that only the H-atoms of the surface layer of the particles are

18

able to exchange with the surrounding water 3. Based on this principle, H/D exchange reaction is

19

currently widely used for the investigation of the conformational dynamic of biological

20

molecules both in the solution4,5 6 7,8 9 and in the gas phase10,11 12 13 14.

21 22

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:


2

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1 2 3

AX + BX* ↔ AX* + BX

(1)

-ln(1-F)=R[(a+b)ab]t

(2)

can be described by a formula:

4

Here F is the fraction (progress variable) of isotopic exchange, R- the rate of exchange, [AX] +

5

[AX*] = a, [BX] + [BX*] = b15-18. In a case of complex mechanism19,20 the exact equations

6

describing the exchange can be complicated but the equation (2) remains valid to certain extent.

7

It is obvious that if a molecule containing several labile atoms is subjected to the isotope

8

exchange reaction molecules with different degree of substitution will be present in a system.

9

The regular “bulk” chemistry cannot distinguish such molecules, but use of mass spectrometry

10

can effectively separate them and reveal a real form of isotopic distribution.

11

Normally the isotopic distribution has a Gaussian form, though sometimes it unusually

12

broadens21-23 or even become bimodal24,25 what means that target molecule can exists in different

13

conformations26,27. The shape of the isotopic distribution in the equilibrium obey the binomial

14

distribution and there are rapid methods to calculate it28,29. Recent progress in the quantum

15

chemistry and molecular dynamics made it possible to accurately describe the isotope exchange

16

process even in the case of peptide30-33. Khakinejad et al. demonstrated that gas phase

17

Hydrogen/Deuterium of a peptide can be accurately simulated using molecular dynamics and

18

hydrogen accessibility scoring-number of effective collisions model33. Such simulations are

19

complicated and require a consideration of several hundreds of possible structures of peptide.


3


Page 4 of 23

1

At the same time, to our knowledge no simple theory was developed to describe the evolution

2

of the isotopic distribution when an isotopically pure molecule is subjected to the isotope

3

exchange reaction.

4

Here we present the analytical solution of the general system of rate equations describing the

5

isotopic exchange of an arbitrary macromolecule. For simplicity, we will consider a case of

6

Hydrogen/Deuterium exchange reaction.

7

THEORY OF THE EXCHANGE KINETIC

8

Let’s consider a molecule M that contains N equivalent labile atoms. Molecule is placed

9

in media filled with D2O and H2O. Molecules of media continuously interact with the target

10

molecule. If the concentration of M is relatively low, than exchange reaction can be considered as

11

a pseudo-1st-order reactions with kD and kH – constants of forward and back exchange reaction.

12

Let’s denote Mi – the concentration of molecule in which i atoms of hydrogen are replaced for

13

deuterium. Molecules of media continually interact with M, but only those collisions in which

14

D2O interacts with labile hydrogen and H2O interacts with labile deuterium will lead to the

15

exchange. The portion of such collisions are i/N and (N-i)/N correspondingly (see Fig.1). So, the

16

exchange reaction can be described as following system of rate equations: dM 0 1 = −kD M 0 + k H M1 dt N ..........................................

17

dM i  N −i i  i +1 N − (i − 1) =  − kD − kH M i + kH M i +1 + k D M i −1 , dt N N N N  ..........................................

(3)

dM N 1 = k D M N −1 − k H M N dt N


4

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1


with the initial conditions:

M 0 t =0 = M 00 ; M i t =0 = 0, 0 < i ≤ N .

2

(4)

3

This system cannot be solved directly, but using the software Mathematica 7.0, that can

4

process analytical formulas, it is possible to find an analytical solution for any N. Analyzing

5

solutions found for several values of N (see Appendix 1 for example) we proposed that the

6

general solution should be: i

1 ( − k − k )t    1 ( − k − k )t  (− 1)  − 1 + e N d h  kdi  e N h d kd + kh  N!     M i (t ) = N i!( N − i )! (kd + kh )

N −i

i

7

8 9

10

11 12

13

14 15

.

(5)

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): dM i  N −i i  i +1 N − (i − 1) =  − kD − k H M i + k H M i +1 + k D M i −1 dt N N N N 

(6)

This can be done using Mathematica 7.0 (see Supplementary Information). The equation (5) can be rewritten as:

M i (t ) =

N!  1− e i!( N − i)! 

1 ( − k d − k h )t N

   kd   kd     k + k  1 − k + k d   h d  h  i

i

1 ( − k h − k d )t   eN 1 − kh + kd  

    

N −i

(7)

We can see, that equation (7) has the form of binomial distribution, which can be approximated with a Gaussian distribution:


5


− 1 Mi ≈ e σ 2π

(i − µ )2 2σ 2

;

1 ( − k d − k h )t  kd  N  ; 1 e − µ=N  kd + kh  

1

Page 6 of 23

(8)

1 1 ( − k d − k h )t   ( − k d − k h )t     N N      1 k d 1 − e k e − d     1−   . σ =N    kd + kh kd + kh      

2

We can see, that the evolution of the isotopic envelope is described by the time

3

dependence of expected value (average number of exchanges) and variance (width of the isotopic

4

envelope). The full code for obtaining (guessing and proving) the solution using Mathematica 7.0

5

is given in the Supplementary Information.

6

There are 2 important limiting cases:

7

1) t → ∞ .

8 9

This is the case of the equilibration of forward and backward exchange reactions. The equilibrium is described with following parameters:

− 1 Mi ≈ e σ 2π

10

11

(i − µ )2 2σ 2

;

σ =N

kd  kd  kd kh 1 −  = N ; kd + kh  kd + kh  (kd + kh )2

µ=N

kd . kd + kh

(9)

2) kd >> kh


6

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1


For this case, the solution has the following form:

− 1 Mi ≈ e σ 2π

4

2σ 2

;

k − dt   µ = N 1 − e N ;   kd k − t − dt  N   σ = N 1 − e e N  

2

3

( i − µ )2

(10)

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

Tmax σ =

5

N ln 2 , kd

(11)

6

and then shrinks to the zero width.

7

Using the equations (10) it is possible to determine activation energy of the exchange

8

reaction if the temperature dependence of the reaction degree is given. The second equation in

9

(10) can be rewritten as:

kd = −

10

N  µ ln1 −  t  N

11

Substituting the expression for k using Arrhenius equation:

12

kd = k d e

13

0

−

Ea kT

,

(12)

(13)

we obtain:


7


−

1

2

 N   N  Ea  µ  µ   = ln − 0 ln1 −   = ln 0  + ln − ln1 −   . kT N  N     kd t   kd t 

Page 8 of 23

(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

3

(15)

4

The developed theory has following limitations: labile atoms on the surface of molecule are not

5

always equivalent, the molecule can change its conformation, so additional atoms will become

6

available for the exchange. Special approaches based on the molecular dynamics and hydrogen

7

accessibility scoring-number of effective collisions model32,33 should be used in order to

8

accurately describe the isotope exchange of the large molecule taking into account conformer

9

dynamics, effects of charge etc.

10

SIMULATION AND EXPERIMENT

11

We have simulated the evolution dynamics of the deuterium distribution for a molecule that

12

contains N=100 equivalent labile hydrogens and for kD=1, kH=0. The results are presented in

13

Fig.2. It can be seen, that the isotopic distribution adopts a Gaussian shape, shifts and then

14

shrinks.

15

Using equation (14) it is possible to estimate the activation energy if the temperature

16

dependence of the isotope exchange reaction degree is obtained. Recently we have proposed a

17

simple approach for performing of the H/D exchange directly in the ESI source34-39. If one infuse

18

D2O into the ESI source, then molecular ions interact with D2O in the heated desolvating

19

capillary and exchange labile hydrogens for deuterium40,41. Variation of the temperature of the


8

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1

desolvating capillary allows varying the degree of the exchange reaction. Therefore, using (14) it

2

is possible to estimate the activation energy. All our experiments were performed on an LTQ FT

3

Ultra (Thermo Electron Corp., Bremen, Germany) mass-spectrometer equipped with a 7T

4

superconducting magnet. Ions were generated by an IonMax Electrospray ion source (Thermo

5

Electron Corp., Bremen, Germany) in positive and negative ESI mode. The temperature of the

6

desolvating capillary was varied from 50 oC to 450 oC. The length of the capillary was 105 mm

7

and its inner diameter was 0.5 mm. The infusion rate of the sample was 1 µl /min and the needle

8

voltage was 4000 V. For in-ESI source isotope exchange, the atmosphere was saturated with the

9

D2O (or H218O) vapors in the region between ESI tip and the inlet of desolvating capillary by

10

placing 400 µL of D2O (or H218O) on a copper plate positioned approximately 7mm below the

11

ESI needle (see Fig.3).

12

In the Fig.4 the kinetic of the following exchange reactions is presented: 16

16

O/18O exchange42

13

for [IO3]- ion,

O/18O exchange for [I3O13]-, H/D exchange reaction43,44 of ubiquitin6+,

14

angiotensin+, dopamine+, resorcinol+, oligonucleotide45 A206- and oligosaccharide dextran+46. It

15

can be seen, that the isotopic distribution adopts a Gaussian shape for all samples, except dextran

16

and resorcinol for both H/D and

17

atoms are as follows: 3 for [IO3]-, 13 for [I3O13]-, 150 for [ubiquitin+6H]6+, 155 for

18

[ubiquitin+11H]11+, 15 for [angiotensin +H]+, 5 for [dopamine +H]+, 55 for [A20-6H] 6-, 14 for

19

[dextran5+Na]+ and 6 for [resorcinol+H]+. We can see that the deuterium distribution in the case

20

of dextran becomes bimodal. Previously we have explained this effect by the assumption that

21

ions of oligosaccharides can be formed in 2 conformations47,48. One of the conformations is

22

folded and the exchange goes slow, the other one is unfolded and the exchange goes fast.

23

Molecule of resorcinol has 2 –OH groups, nevertheless, protonated ion demonstrates 6

18

O/16O exchange reactions. Total numbers of exchangeable


9


Page 10 of 23

1

exchanges. This can be explained by keto-enol tautomerism, that facilitate the exchange in

2

adjacent –CH positions. Previously such effects were observed for several organic molecules46,49.

3

The mechanism of the H/D exchange involves the formation of the hydrogen bond between

4

deuterium of the D2O and lone electron pair of the heteroatom (nitrogen or oxygen) of the

5

molecule. Schematically the reaction is represented in the Scheme 150:

6

X

H

+ Y

D

D

Y

X

H

X

D

+ Y

H

7

Scheme 1. The H/D reaction in the complex formed by hydrogen bond

8

This is why the exchange occurs almost instantly in O-H, Cl-H, Br-H, I-H and doesn’t occur in

9

Si-H, P-H and C-H bonds. In neutral N-H groups the exchange is fast but protonation of nitrogen

10

makes electron pair shared with a substituent and the rate of exchange considerably decrease.

11

Oxygen exchange occurs via nucleophilic substitution. For the exchange to take place, the

12

central atom must be able to increase its coordination number and must carry positive charge. For

13

example, the in the HClO4 the Cl atom cannot increase its coordination number, but iodine in

14

[IO4]- easily increases it from 4 to 650.

15

We have summarized the results of the observation of the kinetic of isotope exchange reactions

16

for many compounds in the Fig.5. In the Fig.5A is represented the temperature dependence of the

17

isotopic exchange reaction degree. We can see that for many compounds the maximum

18

percentage of exchange is about only 50%. This can be explained by the slow rate of the

19

exchange in certain groups (for example in –NH2) and by the conformational effects that protect

20

certain hydrogens from the contact with D2O. The effect of back exchange is negligible because

21

the atmosphere in the ESI source is saturated with D2O. Increasing the reaction time by


10

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1

increasing the length of the desolvating capillary can help to increase the degree of the

2

exchange51,52.

3

The determination of the activation energy is shown in the Fig.5B. We can see that for [IO3]-

4

and [I3O13]- ions the charts obtained using (15) have the form of almost perfect lines. It means

5

that the developed theory is valid and that oxygen atoms in those ions are equivalent. For other

6

ions the labile atoms cannot be considered equivalent and that can explain the deviation of the

7

curves in Fig.5B from straight lines, moreover, for large ions (proteins, oligosaccharides,

8

oligonucleotides) we must take into account possible conformational changes that can occur

9

during the ionization and transport21,22,33,53,54. In addition, the important question is the variation

10

of the gas dynamics in the desolvating capillary with the temperature. Changes in the gas velocity

11

leads to the variation of t, and changes in pressure leads to the variation of k d0 in (14). But

12

previously performed computer simulation52,55,56 of the gas dynamics in the inlet capillary and

13

the fact that logarithm is slow changing function allows to neglect this effect.

14

The activation energy was estimated for several ions and was found to be in a range 0.04 eV -

15

0.3 eV. We can see that the value of the activation energy differ for compounds belonging to

16

different classes. Indeed, the activation energies for H/D exchange reaction for dopamine and

17

resorcinol were found to be 0.058 eV and 0.048 eV correspondingly; angiotensin, ubiquitin and

18

oligonucleotide demonstrate activation energy in the range 0.072 eV -0.085 eV. Oligosaccharide

19

has activation energy ~0.17 eV. Based on this result, it will be possible using in-ESI source H/D

20

exchange to understand the nature of functional groups for unknown compounds and relate those

21

compounds to most probable class.

22


11


Page 12 of 23

1

CONCLUSION

2

We have obtained the analytical solution for the system of rate equations describing the isotope

3

exchange process for molecule. We treat the molecule as a sphere in which all labile atoms are

4

equivalent and placed on the surface. The exchange occurs only in those collisions of molecules

5

of media with target molecule in which participate different isotopes. For example, the exchange

6

doesn’t occur when D2O interacts with already deuterated part of the molecule. The exact

7

solution of the system (3) was obtained using Mathematica 7.0 software and this solution has the

8

form of the time-dependent Gaussian distribution. For the case when forward exchange

9

considerably overlaps the back exchange it is possible to estimate the activation energy of the

10

reaction by obtaining a temperature dependence of the reaction degree. The activation energy was

11

estimated for several ions and was found to be in a range 0.04 eV - 0.3 eV. The value of

12

activation energy can indicate the presence of certain functional groups in the molecule. The

13

deviation of the relationship (15) from the straight line can indicate the presence in the molecule

14

nonequivalent labile atoms. This can have potential analytical applications for determining of

15

types of functional groups in complex mixtures, such as petroleum, humic substances, bio-oil etc.

16 17

Appendix 1.

18

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


12

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5

4

1 ( − k d − k h )t   1 ( − k h − k d )t  15 (− k h − k d )t    5 e   kd  e 5  5 1 k + k − − + e k + k d h d h        ; M (t ) =     ; M 0 (t ) =  1 (kd + kh )5 (kd + kh )5 2

1

3

1 ( − k d − k h )t  2  1 ( − k h − k d )t    kd  e 5  10 − 1 + e 5 k + k d h        ; M 2 (t ) = 5 (kd + kh )

 − 10 − 1 + e  M 3 (t ) =

3

 3  kd  e     (kd + kh )5

1 ( − k d − k h )t 5

4

1 ( − k h − k d )t 5

(10)

2

 kd + kh   ; 1

5

1 1 ( − k d − k h )t  4  1 ( − k h − k d )t ( − k d − k h )t  5    5 5 5       kd 5 − 1 + e 1 k e k + k − − + e d d h         ; M (t ) =   . M 4 (t ) =  5 (kd + kh )5 (kd + kh )5

2 3 4

AUTHOR INFORMATION

5

Corresponding Author

6

* Eugene Nikolaev [email protected]

7

ASSOCIATED CONTENT

8

The Supporting Information is available free of charge on the ACS Publications website at DOI:

9

10.1021/acs.analchem.xxxxxxx.

10

AUTHOR CONTRIBUTIONS

11

The manuscript was written through contributions of all authors. All authors have given approval

12

to the final version of the manuscript.

13

FUNDING SOURCES


13


1

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

2

COMPETING FINANCIAL INTEREST

3

Authors declare no competing financial interests.

Page 14 of 23

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FIGURES

26


14

Page 15 of 23


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Fig. TOC graphic.

1 2 3 4


15


Page 16 of 23

1 2 3 4

Fig.1 The 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 – total number of labile atoms.

5 6 7 8 9 10 11


16

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1 2 3

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

4 5 6


17


Page 18 of 23

1 2

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

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


18

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1 2 3 4

Fig.4. The 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]+.

5 6 7 8 9 10 11 12 13 14 15 16 17


19


Page 20 of 23

1 2 3 4

Fig.5 Determination of activation energy. A – dependence of the isotope exchange reaction degree on the temperature; B – estimation of the activation energy.

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D Exchange Kinetic of Macromolecule

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