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Models for the calculation of peptide vibrational spectra [Elektronische Ressource] / von Roman D. Gorbunov

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Models for the Calculation ofPeptide Vibrational SpectraDISSERTATIONzur Erlangung des Doktorgradesder Naturwissenschaftenvorgelegt beim FachbereichChemische und Pharmazeutische Wissenschaftender Johann Wolfgang Goethe-Universit¨atin Frankfurt am MainvonRoman D. Gorbunovaus DniprodzerzhinskFrankfurt am Main2007(DF1)1vom Fachbereich Chemische und Pharmazeutische Wissenschaften derJohann Wolfgang Goethe-Universit¨at als Dissertation angenommen.Dekan: Prof. Dr. Harald Schwalbe1. Gutachter: Prof. Dr. Gerhard Stock2. Gutachter: Prof. Dr. Josef WachtveitlDatum der Disputation: .............................................2Contents1 Introduction 112 Ab initio models of amide I vibrations 172.1 Exciton Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Terms Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Choice of Theory Level and Basis Set . . . . . . . . . . . . . . . . . 192.2 Parameterization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Construction of Amide I Local Modes . . . . . . . . . . . . . . . . . 242.2.3 Step for the Numerical Differentiation. . . . . . . . . . . . . . . . . 322.2.4 Hessian Matrix Reconstruction Method . . . . . . . . . . . . . . . . 362.2.5 Usage of the NMA-Based Local Modes . . . . . . . . . . . . . . . . 382.2.6 Carbonyl Coordinate Displacement Method . . . .

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Models for the Calculation of
Peptide Vibrational Spectra
DISSERTATION
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich
Chemische und Pharmazeutische Wissenschaften
der Johann Wolfgang Goethe-Universit¨at
in Frankfurt am Main
von
Roman D. Gorbunov
aus Dniprodzerzhinsk
Frankfurt am Main
2007
(DF1)
1vom Fachbereich Chemische und Pharmazeutische Wissenschaften der
Johann Wolfgang Goethe-Universit¨at als Dissertation angenommen.
Dekan: Prof. Dr. Harald Schwalbe
1. Gutachter: Prof. Dr. Gerhard Stock
2. Gutachter: Prof. Dr. Josef Wachtveitl
Datum der Disputation: .............................................
2Contents
1 Introduction 11
2 Ab initio models of amide I vibrations 17
2.1 Exciton Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Terms Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Choice of Theory Level and Basis Set . . . . . . . . . . . . . . . . . 19
2.2 Parameterization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Construction of Amide I Local Modes . . . . . . . . . . . . . . . . . 24
2.2.3 Step for the Numerical Differentiation. . . . . . . . . . . . . . . . . 32
2.2.4 Hessian Matrix Reconstruction Method . . . . . . . . . . . . . . . . 36
2.2.5 Usage of the NMA-Based Local Modes . . . . . . . . . . . . . . . . 38
2.2.6 Carbonyl Coordinate Displacement Method . . . . . . . . . . . . . 41
2.2.7 Comparison of the Parameterization Schemes . . . . . . . . . . . . 42
2.3 Amide I Anharmonicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1 Anharmonicities in NMA . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 Anharmonicities in the GD: Conformational Dependency . . . . . . 51
2.3.3 Explanation of the Disagreement Between FD and HMR Methods . 54
2.3.4 Effect of the Anharmonicities on the Transition Frequencies . . . . 56
2.4 Building Block Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.2 First-Neighbor Couplings . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.3 Second-Neighbor Coupling . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.4 Site Energies of the Terminal Residues . . . . . . . . . . . . . . . . 62
2.4.5 Site Energies of the Inner Peptide Unit . . . . . . . . . . . . . . . . 63
2.4.6 Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3CONTENTS CONTENTS
2.5 Transferability of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5.1 Effect of Side Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5.2 Effect of End Groups (Different Protonation States) . . . . . . . . . 66
2.5.3 Comparison of GD and AAA Maps . . . . . . . . . . . . . . . . . . 66
3 Calculation of Infrared Spectra 69
3.1 Trialanine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.1 Semiclassical Line Shape Theory . . . . . . . . . . . . . . . . . . . 70
3.1.2 Solvent-Induced Frequency Shift . . . . . . . . . . . . . . . . . . . . 73
3.1.3 Distributions of Vibrational Frequencies . . . . . . . . . . . . . . . 74
3.1.4 Calculation of the Absorption Spectrum . . . . . . . . . . . . . . . 77
3.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Photoswitchable Bicyclic Azobenzene Octapeptide . . . . . . . . . . . . . . 84
3.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.2 frequencies Distributions . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.3 Frequencies Correlation Functions . . . . . . . . . . . . . . . . . . . 90
3.2.4 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Conclusions 104
References 109
Acknowledgments 118
Zusammenfassung 120
Lebenslauf 124
Publikationen 125
4List of Figures
1.1 “Glycine dipeptide” (GD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+1.2 Scheme and atom labeling of trialanine cation A . . . . . . . . . . . . . . . 153
2.1 Comparison of the amide I vibrational coupling k and force constant k12 1
◦of GD as obtained for φ = −60 at three levels of theory: Hartree Fock,
density functional theory, and Møller-Plesset perturbation theory. . . . . . 20
2.2 AmidenormalmodefrequenciesoftheNMAandGDmoleculesasfunctions
of the basis set. GD molecule is in the parallel sheet conformation β (-P
119,113). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Scheme of the “glycine dipeptide analog” (GD) molecule, introducing the
two local coordinate systems, which are employed to define NMA-based
amide I local modes of the system. . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Theforceandcouplingconstantasfunctionofthestepusedinthenumerical
differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Dependence of the functions ξ (Δq) on the accuracy of the minimum defi-j
nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Examples of the functions ξ (Δq). . . . . . . . . . . . . . . . . . . . . . . . 35j
2.7 Difference between the exact potential energy and the fitting polynomial of
different order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Force constant calculated by numerical differentiation of the ab initio po-
tential energy and the fitting cubic polynomial. . . . . . . . . . . . . . . . 37
2.9 Couplingconstants,differencebetweentheforceconstants,andaverageforce
constant calculated by FD and HMR methods. . . . . . . . . . . . . . . . 43
2.10 The coupling and difference between the force constants calculated by the
HMR method in combinations with the NMA-based local modes and CCD
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5LISTOFFIGURES LISTOFFIGURES
2.11 The coupling constant, difference between the force constant, and the aver-
ageforceconstantcalculatedforthefullyoptimizedandrestrictedgeometries. 46
2.12 Transition frequencies of the first two excited states as well ground state
energy as functions of size of the basis set. . . . . . . . . . . . . . . . . . . 48
2.13 Difference between the first two transition frequencies ((E −E )−(E −E )) 492 1 1 0
2.14 (E −E )−(E −E ) as a function of the normal mode coordinate range2 1 1 0
used during the fitting of the amide I potential energy. Different curves
correspond to different order of polynomial used during fitting. . . . . . . . 50
2.15 Valuesofthelocalcoordinatewhichcorrespondtofullyoptimizedgeometry
of GD as well as mesh used for calculation of cubic anharmonicities in GD. 52
2.16 Average frequencies, frequency splitting and site energies obtained with the
usage of full optimized, restricted and restricted tuned geometries. . . . . . 55
2.17 Transitionfrequenciesshiftedbytheanharmonicitiesareshownasfunctions
of the not shifted values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.18 Blockedglycinepeptidesunderconsideration: AN-methylacetamide(NMA),
B “glycine dipeptide” (GD), and C “glycine tripeptide” (GT). . . . . . . . 58
2.19 Amide I local-mode frequencies ε (panels B and C) and associated vibra-n
tional couplings β (panels A) of glycine tripeptide. Compared are resultsnm
obtained directly from DFT calculations (“Reference”) and from various
approximate schemes (“Model”), see text. . . . . . . . . . . . . . . . . . . 60
2.20 TransferabilityoftheamideIvibrationalconstantsforAc-Gly-NHCH (GD)3
to peptides with a hydrophilic side chain Ac-Asp-NHCH (Asp) and a hy-3
drophobic side chain Ac-Phy-NHCH (Phy), respectively. Shown are (in3
−1cm ) the intersite coupling β, the site energies ε and ε , as well as the1 2
frequency gap Δω =ω −ω . . . . . . . . . . . . . . . . . . . . . . . . . . 65+ −
2.21 TransferabilityoftheamideIvibrationalconstantsforAc-Gly-NHCH (GD)3
+− + −totrialanineinitszwitterionic(A ),cationic(A ),andanionic(A )state.3 3 3
−1Shown are (in cm ) the intersite coupling β, the site energies ε and ε , as1 2
well as the frequency gap Δω =ω −ω . . . . . . . . . . . . . . . . . . . . 67+ −
2.22 (φ,ψ)-maps of the mean ω¯ (top) and the splitting Δω (bottom) of the two
amideIfrequencies,asobtainedforisolatedglycinedipeptide(left),isolated
trialanine (middle), and trialanine in D O (right). . . . . . . . . . . . . . . 682
6LISTOFFIGURES LISTOFFIGURES
3.1 Distribution of the amide I normal-mode frequencies obtained for the three
conformational states of trialanine (a) in the gas phase and (b) in solution.
Panel (c) shows the corresponding absorption bands calculated within the
cumulant approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 DistributionofthefrequencysplittingΔωwithout(left)andwith(right)the
inclusion the solvent contribution, as obtained for the three conformational
states of glycine dipeptide (top) and trialanine (bottom). . . . . . . . . . . 76
3.3 Correlation functionshM (t)i of the total transition dipole moment, shownk s
for both amide I normal modes (k = 1,2) and the conformations s =α , β,R
and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78II
3.4 AmideIabsorptionbandsσ (ω)oftrialanineobtainedfortheconformationss
s =α , β, and P . Compared are results calculated directly from semiclas-R II
sical line shape theory (via Eq. (3.5), thick black lines), by invoking only
the adiabatic approximation (via Eq. (3.8, thin red lines), and by invoking
adiabatic and cumulant approximations (via Eq. (3.8), blue dashed lines). . 79
3.5 Comparison of experimental (Ref. [1], green dashed line) and calculated
amide I absorption spectra of trialanine. The latter were obtained directly
from semiclassical line shape theory (via Eq. (3.5), thick black line) and by
invoking adiabatic and cumulant approximations (via Eq. (3.8), thin red line). 81
3.6 StructureandaminoacidlabelingofthebicyclicazobenzenepeptidebcAMPB. 84
3.7 Frequency distributions corresponding to the beginning and end of the time
evolution of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.8 Dependency of the frequency distribution on time. . . . . . . . . . . . . . . 88
3.9 Dependencies of the fitting coefficients on frequency. . . . . . . . . . . . . . 88
3.10 Correlation between coefficients of the fitting. . . . . . . . . . . . . . . . . 89
3.11 Time dependent changes of the frequency distribution. . . . . . . . . . . . 90
3.12 Time dependent changes of the frequency distribution. The time depen-
dency is calculated with the usage of the fitting functions. . . . . . . . . . 91
3.13 Averaged[overtime-segmentsandMDtrajectories]normalmodefrequencies
as functions of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.14 Averaged[overtime-segmentsandMDtrajectories]siteenergiesasfunctions
of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.15 Contributions to the first tree site energies as functions of time. . . . . . . 96
7LISTOFFIGURES LISTOFFIGURES
3.16 Ab initio maps of the ε combined with the distributions of the dihedraln
angles φ and ψ for the beginning and end of the time evolution. . . . . . . 97
3.17 Ab initio maps of the ε combined with the distributions of the dihedralc
angles φ and ψ for the beginning and end of the time evolution. . . . . . . 98
3.18 Fittingparameterofthecorrelationfunctionsofthe8normalmodefrequen-
cies as functions of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.19 Fitting parameter of the correlation functions of the normal modes frequen-
cies obtained by the averaging over time (red curves) as well as by linear fit
of the time dependencies (blue curves). . . . . . . . . . . . . . . . . . . . . 101
3.20 Time dependent spectra of photoswitchable peptide. . . . . . . . . . . . . . 102
3.21 Changes of the vibrational spectrum of photoswitchable peptide. . . . . . . 102
8List of Tables
2.1 Basis-setdependencyofthediagonalforceconstantsk1andk andthevibra-2
tional coupling k as obtained from DFT calculations on glycine dipeptide.12
Data are shown for the following conformations and Ramachandran angles
(φ,ψ): Parallel β-sheet β (-119,113), antiparallel β-sheet β (-139,135),P AP
right-handedα-helixα (-57,-47), andleft-handedα-helicesα (57,47)andR L1
α (90,-90). Therootmeansquaredeviation(RMSD)iscomputedwithre-L2
specttothe6-311+G(2df,2p)data,usingbesidesthefivelistedstructures12
additionalconformationswith(φ,ψ) = (−n·90,−m·90),withn,m = 1,2,3.
˚Units are mdyn/Au. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9LISTOFTABLES LISTOFTABLES
10