We use the modified Bigeleisen–Mayer equation to compute kinetic isotope effect values for non-enzymatic phosphoryl transfer reactions from classical and path integral molecular dynamics umbrella sampling. The modified form of the Bigeleisen–Mayer equation consists of a ratio of imaginary mode vibrational frequencies and a contribution arising from the isotopic substitution’s effect on the activation free energy, which can be computed from path integral simulation. In the present study, we describe a practical method for estimating the frequency ratio correction directly from umbrella sampling in a manner that does not require normal mode analysis of many geometry optimized structures. Instead, the method relates the frequency ratio to the change in the mass weighted coordinate representation of the minimum free energy path at the transition state induced by isotopic substitution. The method is applied to the calculation of 16/18O and 32/34S primary kinetic isotope effect values for six non-enzymatic phosphoryl transfer reactions. We demonstrate that the results are consistent with the analysis of geometry optimized transition state ensembles using the traditional Bigeleisen–Mayer equation. The method thus presents a new practical tool to enable facile calculation of kinetic isotope effect values for complex chemical reactions in the condensed phase.
REFERENCES
1.
M.
Garcia-Viloca
, J.
Gao
, M.
Karplus
, and D. G.
Truhlar
, “How enzymes work: Analysis by modern rate theory and computer simulations
,” Science
303
, 186
–195
(2004
).2.
A.
Ganguly
, B. P.
Weissman
, T. J.
Giese
, N.-S.
Li
, S.
Hoshika
, S.
Rao
, S. A.
Benner
, J. A.
Piccirilli
, and D. M.
York
, “Confluence of theory and experiment reveals the catalytic mechanism of the Varkud satellite ribozyme
,” Nat. Chem.
12
, 193
–201
(2020
).3.
C. S.
Gaines
, J. A.
Piccirilli
, and D. M.
York
, “The L-platform/L-scaffold framework: A blueprint for RNA-cleaving nucleic acid enzyme design
,” RNA
26
, 111
–125
(2020
).4.
A. C.
Hengge
, “Isotope effects in the study of phosphoryl and sulfuryl transfer reactions
,” Acc. Chem. Res.
35
, 105
–112
(2002
).5.
P.
Paneth
, “Chlorine kinetic isotope effects on enzymatic dehalogenations
,” Acc. Chem. Res.
36
, 120
–126
(2003
).6.
W. W.
Cleland
, “The use of isotope effects to determine enzyme mechanisms
,” Arch. Biochem. Biophys.
433
, 2
–12
(2005
).7.
B.
Weissman
, Ş.
Ekesan
, H.-C.
Lin
, S.
Gardezi
, N.-S.
Li
, T. J.
Giese
, E.
McCarthy
, M. E.
Harris
, D. M.
York
, and J. A.
Piccirilli
, “Dissociative transition state in hepatitis delta virus ribozyme catalysis
,” J. Am. Chem. Soc.
145
, 2830
–2839
(2023
).8.
J.
Bigeleisen
and M.
Wolfsberg
, “Theoretical and experimental aspects of isotope effects in chemical kinetics
,” Adv. Chem. Phys.
1
, 15
–76
(1958
).9.
W.-P.
Hu
and D. G.
Truhlar
, “Deuterium kinetic isotope effects and their temperature dependence in the gas-phase SN2 reactions X− + CH3Y → CH3 + Y− (X, Y = Cl, Br, I)
,” J. Am. Chem. Soc.
117
, 10726
–10734
(1995
).10.
A.
Fernández-Ramos
, J. A.
Miller
, S. J.
Klippenstein
, and D. G.
Truhlar
, “Modeling the kinetics of biomolecular reactions
,” Chem. Rev.
106
, 4518
–4584
(2006
).11.
T.
Yu
, J.
Zheng
, and D. G.
Truhlar
, “Multi-structural variational transition state theory. Kinetics of the 1,4-hydrogen shift isomerization of the pentyl radical with torsional anharmonicity
,” Chem. Sci.
2
, 2199
(2011
).12.
T. J.
Giese
, Ş.
Ekesan
, and D. M.
York
, “Extension of the variational free energy profile and multistate Bennett acceptance ratio methods for high-dimensional potential of mean force profile analysis
,” J. Phys. Chem. A
125
, 4216
–4232
(2021
).13.
T. D.
Poulsen
, M.
Garcia-Viloca
, J.
Gao
, and D. G.
Truhlar
, “Free energy surface, reaction paths, and kinetic isotope effect of short-chain acyl-CoA dehydrogenase
,” J. Phys. Chem. B
107
, 9567
–9578
(2003
).14.
T. J.
Giese
, J.
Zeng
, Ş.
Ekesan
, and D. M.
York
, “Combined QM/MM, machine learning path integral approach to compute free energy profiles and kinetic isotope effects in RNA cleavage reactions
,” J. Chem. Theory Comput.
18
, 4304
–4317
(2022
).15.
J.
Gao
, “Chapter fourteen—Enzymatic kinetic isotope effects from path-integral free energy perturbation theory
,” Methods Enzymol.
577
, 359
–388
(2016
).16.
A.
Vardi-Kilshtain
, N.
Nitoker
, and D. T.
Major
, “Nuclear quantum effects and kinetic isotope effects in enzyme reactions
,” Arch. Biochem. Biophys.
582
, 18
–27
(2015
).17.
J. K.
Hwang
, Z. T.
Chu
, A.
Yadav
, and A.
Warshel
, “Simulations of quantum mechanical corrections for rate constants of hydride-transfer reactions in enzymes and solutions
,” J. Phys. Chem.
95
, 8445
–8448
(1991
).18.
J. K.
Hwang
and A.
Warshel
, “A quantized classical path approach for calculations of quantum mechanical rate constants
,” J. Phys. Chem.
97
, 10053
–10058
(1993
).19.
J.-K.
Hwang
and A.
Warshel
, “How important are quantum mechanical nuclear motions in enzyme catalysis?
,” J. Am. Chem. Soc.
118
, 11745
–11751
(1996
).20.
K.-Y.
Wong
, J. P.
Richard
, and J.
Gao
, “Theoretical analysis of kinetic isotope effects on proton transfer reactions between substituted α-methoxystyrenes and substituted acetic acids
,” J. Am. Chem. Soc.
131
, 13963
–13971
(2009
).21.
K.-Y.
Wong
, Y.
Xu
, and L.
Xu
, “Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman’s path integral
,” Biochim. Biophys. Acta
1854
, 1782
–1794
(2015
).22.
K.-Y.
Wong
, Y.
Xu
, and D. M.
York
, “Ab initio path-integral calculations of kinetic and equilibrium isotope effects on base-catalyzed RNA transphosphorylation models
,” J. Comput. Chem.
35
, 1302
–1316
(2014
).23.
H.
Gu
, S.
Zhang
, K.-Y.
Wong
, B. K.
Radak
, T.
Dissanayake
, D. L.
Kellerman
, Q.
Dai
, M.
Miyagi
, V. E.
Anderson
, D. M.
York
, J. A.
Piccirilli
, and M. E.
Harris
, “Experimental and computational analysis of the transition state for ribonuclease A-catalyzed RNA 2′-O-transphosphorylation
,” Proc. Natl. Acad. Sci. U. S. A.
110
, 13002
–13007
(2013
).24.
À.
González-Lafont
and J. M.
Lluch
, “Kinetic isotope effects in chemical and biochemical reactions: Physical basis and theoretical methods of calculation
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
6
, 584
–603
(2016
).25.
O.
Redlich
, “Eine allgemeine beziehung zwischen den schwingungsfrequenzen isotoper molekeln
,” Z. Phys. Chem.
28B
, 371
–382
(1935
).26.
W. R.
Angus
, C. R.
Bailey
, C. K.
Ingold
, A. H.
Leckie
, C. G.
Raisin
, J. W.
Thompson
, and C. L.
Wilson
, “Infra-red spectrum of hexadeuterobenzene and the structure of benzene
,” Nature
136
, 680
(1935
).27.
W. R.
Angus
, A. H.
Leckie
, C. R.
Bailey
, C. G.
Raisin
, J. L.
Gleave
, C. L.
Wilson
, and C. K.
Ingold
, “Raman spectra of deuterobenzenes and the structure of benzene
,” Nature
135
, 1033
–1034
(1935
).28.
M.
Ceriotti
and T. E.
Markland
, “Efficient methods and practical guidelines for simulating isotope effects
,” J. Chem. Phys.
138
, 014112
(2013
).29.
D. T.
Major
and J.
Gao
, “An integrated path integral and free-energy perturbation-umbrella sampling method for computing kinetic isotope effects of chemical reactions in solution and in enzymes
,” J. Chem. Theory Comput.
3
, 949
–960
(2007
).30.
J.
Gao
, “Computation of kinetic isotope effects for enzymatic reactions
,” Sci. China. Chem.
54
, 1841
–1850
(2011
).31.
Y.
Fan
, A.
Cembran
, S.
Ma
, and J.
Gao
, “Connecting protein conformational dynamics with catalytic function as illustrated in dihydrofolate reductase
,” Biochemistry
52
, 2036
–2049
(2013
).32.
K.-Y.
Wong
, H.
Gu
, S.
Zhang
, J. A.
Piccirilli
, M. E.
Harris
, and D. M.
York
, “Characterization of the reaction path and transition states for RNA transphosphorylation models from theory and experiment
,” Angew. Chem., Int. Ed.
51
, 647
–651
(2012
).33.
P.
Paneth
and A.
Dybala-Defratyka
, “Isotope effects as analytical probes: Applications of computational theory
,” in Computational Techniques for Analytical Chemistry and Bioanalysis
(The Royal Society of Chemistry
, 2020
).34.
J.
Bigeleisen
and M. G.
Mayer
, “Calculation of equilibrium constants for isotopic exchange reactions
,” J. Chem. Phys.
15
, 261
–267
(1947
).35.
V.
López-Canut
, J.
Ruiz-Pernía
, I.
Tuñón
, S.
Ferrer
, and V.
Moliner
, “Theoretical modeling on the reaction mechanism of p-nitrophenylmethylphosphate alkaline hydrolysis and its kinetic isotope effects
,” J. Chem. Theory Comput.
5
, 439
–442
(2009
).36.
D. G.
Truhlar
, J.
Gao
, M.
Garcia-Viloca
, C.
Alhambra
, J.
Corchado
, M.
Luz Sanchez
, and T. D.
Poulsen
, “Ensemble-averaged variational transition state theory with optimized multidimensional tunneling for enzyme kinetics and other condensed-phase reactions
,” Int. J. Quantum Chem.
100
, 1136
–1152
(2004
).37.
C.
Alhambra
, J.
Corchado
, M. L.
Sánchez
, M.
Garcia-Viloca
, J.
Gao
, and D. G.
Truhlar
, “Canonical variational theory for enzyme kinetics with the protein mean force and multidimensional quantum mechanical tunneling dynamics. Theory and application to liver alcohol dehydrogenase
,” J. Phys. Chem. B
105
, 11326
–11340
(2001
).38.
D. G.
Truhlar
, J.
Gao
, C.
Alhambra
, M.
Garcia-Viloca
, J.
Corchado
, M. L.
Sánchez
, and J.
Villà
, “The incorporation of quantum effects in enzyme kinetics modeling
,” Acc. Chem. Res.
35
, 341
–349
(2002
).39.
N.
Kanaan
, S.
Ferrer
, S.
Martí
, M.
Garcia-Viloca
, A.
Kohen
, and V.
Moliner
, “Temperature dependence of the kinetic isotope effects in thymidylate synthase. A theoretical study
,” J. Am. Chem. Soc.
133
, 6692
–6702
(2011
).40.
J.
Pu
, J.
Gao
, and D. G.
Truhlar
, “Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions
,” Chem. Rev.
106
, 3140
–3169
(2006
).41.
A.
Kuppermann
and D. G.
Truhlar
, “Exact tunneling calculations
,” J. Am. Chem. Soc.
93
, 1840
–1851
(1971
).42.
B. C.
Garrett
and D. G.
Truhlar
, “Semiclassical tunneling calculations
,” J. Phys. Chem.
83
, 2921
–2926
(1979
).43.
M.
Garcia-Viloca
, C.
Alhambra
, D. G.
Truhlar
, and J.
Gao
, “Hydride transfer catalyzed by xylose isomerase: Mechanism and quantum effects
,” J. Comput. Chem.
24
, 177
–190
(2003
).44.
J.
Pang
, J.
Pu
, J.
Gao
, D. G.
Truhlar
, and R. K.
Allemann
, “Hydride transfer reaction catalyzed by hyperthermophilic dihydrofolate reductase is dominated by quantum mechanical tunneling and is promoted by both inter- and intramonomeric correlated motions
,” J. Am. Chem. Soc.
128
, 8015
–8023
(2006
).45.
Q.
Liu
, J. A.
Tossell
, and Y.
Liu
, “On the proper use of the Bigeleisen–Mayer equation and corrections to it in the calculation of isotopic fractionation equilibrium constants
,” Geochim. Cosmochim. Acta
74
, 6965
–6983
(2010
).46.
I. H.
Williams
, “Kinetic isotope effects from QM/MM subset Hessians: ‘Cut-off’ analysis for SN2 methyl transfer in solution
,” J. Chem. Theory Comput.
8
, 542
–553
(2012
).47.
J.
Cao
and G. A.
Voth
, “The formulation of quantum statistical mechanics based on the Feynman path centroid density. IV. Algorithms for centroid molecular dynamics
,” J. Chem. Phys.
101
, 6168
–6183
(1994
).48.
M.
Ceriotti
, J.
More
, and D. E.
Manolopoulos
, “i-PI: A Python interface for ab initio path integral molecular dynamics simulations
,” Comput. Phys. Commun.
185
, 1019
–1026
(2014
).49.
M.
Ceriotti
and D. E.
Manolopoulos
, “Efficient first-principles calculation of the quantum kinetic energy and momentum distribution of nuclei
,” Phys. Rev. Lett.
109
, 100604
(2012
).50.
M.
Ceriotti
, D. E.
Manolopoulos
, and M.
Parrinello
, “Accelerating the convergence of path integral dynamics with a generalized Langevin equation
,” J. Chem. Phys.
134
, 084104
(2011
).51.
K.-Y.
Wong
, “Simulating biochemical physics with computers
,” Ph.D. thesis, University of Minnesota
, 2008
.52.
K.-Y.
Wong
, “Review of Feynman’s path integral in quantum statistics: From the molecular Schrödinger equation to Kleinert’s variational perturbation theory
,” Commun. Comput. Phys.
15
, 853
–894
(2014
).53.
M.
Rostkowski
, H. K. V.
Schürner
, A.
Sowińska
, L.
Vasquez
, M.
Przydacz
, M.
Elsner
, and A.
Dybala-Defratyka
, “Isotope effects on the vaporization of organic compounds from an aqueous solution–insight from experiment and computations
,” J. Phys. Chem. B
125
, 13868
–13885
(2021
).54.
J.
Kästner
and W.
Thiel
, “Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: ‘Umbrella integration
,’” J. Chem. Phys.
123
, 144104
(2005
).55.
B. W.
Zhang
, J.
Xia
, Z.
Tan
, and R. M.
Levy
, “A stochastic solution to the unbinned WHAM equations
,” J. Phys. Chem. Lett.
6
, 3834
–3840
(2015
).56.
M. R.
Shirts
and J. D.
Chodera
, “Statistically optimal analysis of samples from multiple equilibrium states
,” J. Chem. Phys.
129
, 124105
(2008
).57.
T.-S.
Lee
, B. K.
Radak
, A.
Pabis
, and D. M.
York
, “A new maximum likelihood approach for free energy profile construction from molecular simulations
,” J. Chem. Theory Comput.
9
, 153
–164
(2013
).58.
T.-S.
Lee
, B. K.
Radak
, M.
Huang
, K.-Y.
Wong
, and D. M.
York
, “Roadmaps through free energy landscapes calculated using the multidimensional vFEP approach
,” J. Chem. Theory Comput.
10
, 24
–34
(2014
).59.
E.
Vanden-Eijnden
and M.
Venturoli
, “Revisiting the finite temperature string method for the calculation of reaction tubes and free energies
,” J. Chem. Phys.
130
, 194103
(2009
).60.
W.
E
, W.
Ren
, and E.
Vanden-Eijnden
, “Finite temperature string method for the study of rare events
,” J. Phys. Chem. B
109
, 6688
–6693
(2005
).61.
L.
Maragliano
, A.
Fischer
, E.
Vanden-Eijnden
, and G.
Ciccotti
, “String method in collective variables: Minimum free energy paths and isocommittor surfaces
,” J. Chem. Phys.
125
, 024106
(2006
).62.
Y.
Zhou
, P.
Ojeda-May
, M.
Nagaraju
, B.
Kim
, and J.
Pu
, “Mapping free energy pathways for ATP hydrolysis in the E. coli ABC transporter HlyB by the string method
,” Molecules
23
, 2652
(2018
).63.
C. A.
Gonzalez
, T. C.
Allison
, and F.
Louis
, “General expression for the effective mass in the one-dimensional treatment of tunneling corrections
,” J. Phys. Chem. A
105
, 11034
–11040
(2001
).64.
D. G.
Truhlar
and B. C.
Garrett
, “Reduced mass in the one-dimensional treatment of tunneling
,” J. Phys. Chem. A
107
, 4006
–4007
(2003
).65.
M.
Wolfsberg
and M. J.
Stern
, “Validity of some approximation procedures used in the theoretical calculation of isotope effects
,” Pure Appl. Chem.
8
, 225
–242
(1964
).66.
M. J.
Stern
and M.
Wolfsberg
, “Simplified procedure for the theoretical calculation of isotope effects involving large molecules
,” J. Chem. Phys.
45
, 4105
–4124
(1966
).67.
L.
Melander
and W. H.
Saunders
, Jr., Reaction Rates of Isotopic Molecules
(John Wiley and Sons
, New York
, 1980
).68.
E. K.
Thornton
and E. R.
Thornton
, Isotope Effects in Chemical Reactions
(Van Nostrand Reinhold
, New York
, 1970
), Chap. 4, pp. 213
–285
.69.
G. D.
Ruggiero
, S. J.
Guy
, S.
Martí
, V.
Moliner
, and I. H.
Williams
, “Vibrational analysis of the chorismate rearrangement: Relaxed force constants, isotope effects and activation entropies calculated for reaction in vacuum, water and the active site of chorismate mutase
,” J. Phys. Org. Chem.
17
, 592
–601
(2004
).70.
H.
Li
and J. H.
Jensen
, “Partial Hessian vibrational analysis: The localization of the molecular vibrational energy and entropy
,” Theor. Chim. Acta
107
, 211
–219
(2002
).71.
Y.
Tao
, W.
Zou
, S.
Nanayakkara
, M.
Freindorf
, and E.
Kraka
, “A revised formulation of the generalized subsystem vibrational analysis (GSVA)
,” Theor. Chem. Acc.
140
, 31
(2021
).72.
K.
Swiderek
, A.
Dybala-Defratyka
, and D. R.
Rohr
, “A new scheme to calculate isotope effects
,” J. Mol. Model.
17
, 2175
–2182
(2011
).73.
T. J.
Giese
, B. A.
Gregersen
, Y.
Liu
, K.
Nam
, E.
Mayaan
, A.
Moser
, K.
Range
, O. N.
Faza
, C. S.
Lopez
, A. R.
de Lera
, G.
Schaftenaar
, X.
Lopez
, T.-S.
Lee
, G.
Karypis
, and D. M.
York
, “QCRNA 1.0: A database of quantum calculations of RNA catalysis
,” J. Mol. Graphics Modell.
25
, 423
–433
(2006
).74.
J.
Zeng
, T. J.
Giese
, Ş.
Ekesan
, and D. M.
York
, “Development of range-corrected deep learning potentials for fast, accurate quantum mechanical/molecular mechanical simulations of chemical reactions in solution
,” J. Chem. Theory Comput.
17
, 6993
–7009
(2021
).75.
Q.
Cui
, M.
Elstner
, E.
Kaxiras
, T.
Frauenheim
, and M.
Karplus
, “A QM/MM implementation of the self-consistent charge density functional tight binding (SCC-DFTB) method
,” J. Phys. Chem. B
105
, 569
–585
(2001
).76.
M.
Elstner
, T.
Frauenheim
, E.
Kaxiras
, G.
Seifert
, and S.
Suhai
, “A self-consistent charge density-functional based tight-binding scheme for large biomolecules
,” Phys. Status Solidi B
217
, 357
–376
(2000
).77.
T. A.
Niehaus
, M.
Elstner
, T.
Frauenheim
, and S.
Suhai
, “Application of an approximate density-functional method to sulfur containing compounds
,” J. Mol. Struct.: THEOCHEM
541
, 185
–194
(2001
).78.
H.
Wang
, L.
Zhang
, J.
Han
, and W.
E
, “DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics
,” Comput. Phys. Commun.
228
, 178
–184
(2018
).79.
D. A.
Case
, K.
Belfon
, I. Y.
Ben-Shalom
, S. R.
Brozell
, D. S.
Cerutti
, T. E.
Cheatham
III, V. W. D.
Cruzeiro
, T. A.
Darden
, R. E.
Duke
, G.
Giambasu
, M. K.
Gilson
, H.
Gohlke
, A. W.
Goetz
, R.
Harris
, S.
Izadi
, S. A.
Izmailov
, K.
Kasavajhala
, K.
Kovalenko
, R.
Krasny
, T.
Kurtzman
, T.
Lee
, S.
Le-Grand
, P.
Li
, C.
Lin
, J.
Liu
, T.
Luchko
, R.
Luo
, V.
Man
, K.
Merz
, Y.
Miao
, O.
Mikhailovskii
, G.
Monard
, H.
Nguyen
, A.
Onufriev
, F.
Pan
, S.
Pantano
, R.
Qi
, D. R.
Roe
, A.
Roitberg
, C.
Sagui
, S.
Schott-Verdugo
, J.
Shen
, C. L.
Simmerling
, N.
Skrynnikov
, J.
Smith
, J.
Swails
, R. C.
Walker
, J.
Wang
, R. M.
Wilson
, R. M.
Wolf
, X.
Wu
, Y.
Xiong
, Y.
Xue
, D. M.
York
, and P. A.
Kollman
, AMBER 20, University of California
, San Francisco, CA
, 2020
.80.
R.
Zwanzig
, “Nonlinear generalized Langevin equations
,” J. Stat. Phys.
9
, 215
–220
(1973
).81.
K.
Nam
, J.
Gao
, and D. M.
York
, “An efficient linear-scaling Ewald method for long-range electrostatic interactions in combined QM/MM calculations
,” J. Chem. Theory Comput.
1
, 2
–13
(2005
).82.
T.
Kubař
, K.
Welke
, and G.
Groenhof
, “New QM/MM implementation of the DFTB3 method in the gromacs package
,” J. Comput. Chem.
36
, 1978
–1989
(2015
).83.
T.
Darden
, D.
York
, and L.
Pedersen
, “Particle mesh Ewald: An N · log(N) method for Ewald sums in large systems
,” J. Chem. Phys.
98
, 10089
–10092
(1993
).84.
U.
Essmann
, L.
Perera
, M. L.
Berkowitz
, T.
Darden
, H.
Lee
, and L. G.
Pedersen
, “A smooth particle mesh Ewald method
,” J. Chem. Phys.
103
, 8577
–8593
(1995
).85.
T. J.
Giese
and D. M.
York
, “A GPU-accelerated parameter interpolation thermodynamic integration free energy method
,” J. Chem. Theory Comput.
14
, 1564
–1582
(2018
).86.
H. J. C.
Berendsen
, J. P. M.
Postma
, W. F.
van Gunsteren
, A.
Dinola
, and J. R.
Haak
, “Molecular dynamics with coupling to an external bath
,” J. Chem. Phys.
81
, 3684
–3690
(1984
).87.
T. J.
Giese
, J.
Zeng
, and D. M.
York
, “Multireference generalization of the weighted thermodynamic perturbation method
,” J. Phys. Chem. A
126
, 8519
–8533
(2022
).88.
J.
Kästner
, J. M.
Carr
, T. W.
Keal
, W.
Thiel
, A.
Wander
, and P.
Sherwood
, “DL-FIND: An open-source geometry optimizer for atomistic simulations
,” J. Phys. Chem. A
113
, 11856
–11865
(2009
).89.
J.
Baker
, “An algorithm for the location of transition states
,” J. Comput. Chem.
7
, 385
–395
(1986
).90.
P. Y.
Ayala
and H. B.
Schlegel
, “A combined method for determining reaction paths minima and transition state geometries
,” J. Chem. Phys.
107
, 375
–384
(1997
).91.
C.
Choi
and R.
Elber
, “Reaction path study of helix formation in tetrapeptides: Effect of side chains
,” J. Chem. Phys.
94
, 751
–760
(1991
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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