C7790 Introduction to Molecular Modelling -1C7790 Introduction to Molecular Modelling TSM Modelling Molecular Structures Petr KulhΓ‘nek kulhanek@chemi.muni.cz National Centre for Biomolecular Research, Faculty of Science Masaryk University, Kamenice 5, CZ-62500 Brno PS/2020 Distant Form of Teaching: Rev1 Lesson 21 Kinetic Isotope Effect (KIE) C7790 Introduction to Molecular Modelling -2- Context C7790 Introduction to Molecular Modelling -3- Revision 𝐸 = 𝐸(π‘Ÿπ‘œ) + 𝐸 𝑉𝑅𝑇 r0 a) we need to find a potential energy minimum b) we can further evaluate vibrations from the PES curvature at the minimum Remember: ➒ PES cannot describe mass effect of nuclei; it only describes electronic effects. To characterize a quantum state: (too difficult to calculate, thus it is usually neglected) C7790 Introduction to Molecular Modelling -4Revision: H2, D2, T2 DEr H2 r0 DEr D2 r0 DEr T2 r0 𝜐 = 1 2πœ‹ 𝐾 πœ‡ different: bigger mass -> smaller frequency -> lower energy < <|DEr||DEr| |DEr| r0 r0 r0= = 𝐸 𝑉 = 𝑣 + 1 2 β„Žπœ Ev re re re~ ~ observable equilibrium bond lengths impact of anharmonicity and QM character of vibrations !! not in scale !! the same for all systems due to the same PES C7790 Introduction to Molecular Modelling -5Kinetic Isotope Effect C7790 Introduction to Molecular Modelling -6Kinetic Isotope Effect The kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes: 𝐾𝐼𝐸 = π‘˜π‘™π‘–π‘”β„Žπ‘‘ π‘˜β„Žπ‘’π‘Žπ‘£π‘¦ https://en.wikipedia.org/wiki/Kinetic_isotope_effect The major contributing factor is the change in ZPVE (zero-point vibrational energy). However, other factors can also be involved including: ➒ breaking symmetry (entropic factor) ➒ changes in rotation and/or translational energies ➒ tunneling Example: C7790 Introduction to Molecular Modelling -7ZPVE as Major Contributor to KIE g g potentialenergy reaction coordinate }imaginary frequencies, these vibrations do not contribute into ZPVE PES is the same for all isotope substitution!!! Remember: This is 1D projection of E(R), which is a function of 3N variables (N-number of atoms). primary KIE secondary KIE TS P R THE SAME PES (but different projections) vibrational states C7790 Introduction to Molecular Modelling -8ZPVE as Major Contributor to KIE g g potentialenergy reaction coordinate }imaginary frequencies, these vibrations do not contribute into ZPVE effect of isotope substitution PES is the same for all isotope substitution!!! Remember: This is 1D projection of E(R), which is a function of 3N variables (N-number of atoms). light isotope heavy isotope primary KIE secondary KIE TS P R C7790 Introduction to Molecular Modelling -9Activation Energy Δ𝐺≠ β‰ˆ Δ𝐸≠ = 𝐸 𝑇𝑆 βˆ’ 𝐸 𝑅 Assume the following approximation: 𝐸 𝑅 = 𝐸 𝑹 𝑅 + ෍ π‘˜=1 3π‘βˆ’6 𝐸 𝑉 𝑅 ,π‘˜πΈ 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + ෍ π‘˜=1 3π‘βˆ’6 𝐸 𝑉(𝑇𝑆),π‘˜ 3N-6 degrees of freedom (N - number of atoms) = number of normal vibrational modes potential energy zero-point vibrational energy C7790 Introduction to Molecular Modelling -10Activation Energy, cont. Δ𝐺≠ β‰ˆ Δ𝐸≠ = 𝐸 𝑇𝑆 βˆ’ 𝐸 𝑅 Assume the following approximation: 𝐸 𝑅 = 𝐸 𝑹 𝑅 + ෍ π‘˜=1 3π‘βˆ’6 𝐸 𝑉 𝑅 ,π‘˜πΈ 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + ෍ π‘˜=1 3π‘βˆ’6 𝐸 𝑉(𝑇𝑆),π‘˜ 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 3N-6 degrees of freedom (N - number of atoms) = number of normal vibrational modes imaginary vibration, it does not contribute to the total energy consider two similar vibrational modes in R and TS, which mainly include atoms involved in creating/breaking bonds C7790 Introduction to Molecular Modelling -11Activation Energy, cont. Δ𝐺≠ β‰ˆ Δ𝐸≠ = 𝐸 𝑇𝑆 βˆ’ 𝐸 𝑅 Assume the following approximation: 3N-6 degrees of freedom (N - number of atoms) = number of normal vibrational modes 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 imaginary vibration, it does not contribute to the total energy consider two similar vibrational modes in R and TS, which mainly includes atoms involved in creating/breaking bonds 𝐸 𝑹 𝑇𝑆 βˆ’ 𝐸 𝑹 𝑅 > 𝐸 𝑇𝑆 βˆ’ 𝐸 𝑅 First consequence: REMEMBER: neglecting ZPVE has a greater impact on calculated activation energy than on reaction energy because of one imaginary vibrational mode in TS C7790 Introduction to Molecular Modelling -12Activation Energy, cont. Δ𝐺≠ β‰ˆ Δ𝐸≠ = 𝐸 𝑇𝑆 βˆ’ 𝐸 𝑅 Assume the following approximation: 3N-6 degrees of freedom (N - number of atoms) = number of normal vibrational modes 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 imaginary vibration, it does not contribute to the total energy consider two similar vibrational modes in R and TS, which mainly include atoms involved in creating/breaking bonds Second consequence: The isotope substitution dominantly influences only on ONE state, which is the reactant. As a result, the activation barrier is changed. C7790 Introduction to Molecular Modelling -13Types of KIE 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 primary KIEsecondary KIE Main conclusion: heavier isotopes makes the reaction slower* 𝐾𝐼𝐸 = π‘˜π‘™π‘–π‘”β„Žπ‘‘ π‘˜β„Žπ‘’π‘Žπ‘£π‘¦ > 1 *) there are certain exceptions, see for the inverse kinetic isotope effect C7790 Introduction to Molecular Modelling -14Types of KIE 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 primary KIEsecondary KIE Main conclusion: heavier isotopes makes the reaction slower* 𝐾𝐼𝐸 = π‘˜π‘™π‘–π‘”β„Žπ‘‘ π‘˜β„Žπ‘’π‘Žπ‘£π‘¦ > 1 *) there are certain exceptions, see for the inverse kinetic isotope effect R TS effect of isotope substitution light isotope heavy isotope Ξ”πΊπ‘™π‘–π‘”β„Žπ‘‘ β‰  Ξ”πΊβ„Žπ‘’π‘Žπ‘£π‘¦ β‰  states freeenergy C7790 Introduction to Molecular Modelling -15Types of KIE 𝐸 𝑅 = 𝐸 𝑹 𝑅 + 𝐸 𝑉 𝑅 ,1 + 𝐸 𝑉 𝑅 ,2 + β‹― + 𝐸 𝑉 𝑅 ,3π‘βˆ’7 + 𝐸 𝑉 𝑅 ,3π‘βˆ’6 𝐸 𝑇𝑆 = 𝐸 𝑹 𝑇𝑆 + 𝐸 𝑉 𝑇𝑆 ,1 + 𝐸 𝑉 𝑇𝑆 ,2 + β‹― + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’7 + 𝐸 𝑉 𝑇𝑆 ,3π‘βˆ’6 primary KIEsecondary KIE isotope substitution takes place in the reaction center (substituted atom is involved in forming/breaking bonds) isotope substitution takes place nearby the reaction center since ALL atoms are involved in ALL normal mode vibrations, the isotope effect is different for both states Typical use of KIE: ➒ KIE is employed in experimental validation of reaction mechanisms < C7790 Introduction to Molecular Modelling -16- Summary ➒ KIE is an experimental method important for studying reaction mechanisms. ➒ KIE is a consequence of quantum behavior of molecular vibrations, which due to Heisenberg principle of uncertainty cannot posses zero energy. ➒ In general, neglecting ZPVE in calculations of activation energies can introduce nonnegligible error. ➒ ZPVE cannot be modelled by methods employing classical physical laws (molecular dynamics, Monte-Carlo simulations). It is necessary to use special techniques such as the path integral molecular dynamics, which can describe these quantum effects even when employing non-QM potentials such as molecular mechanics. ➒ Very accurate QM calculations are required to predict/quantify KIE. In many cases, calculations will not be accurate enough to predict KIE. ➒ Instead, the modelling can provide supplementary data such as suggesting suitable atoms for isotope substitutions based on analysis of molecular vibrations. C7790 Introduction to Molecular Modelling -17- Homework C7790 Introduction to Molecular Modelling -18- Homework ➒ Find some experimental study employing KIE. ➒ Can be KIE used for studying mechanisms of enzymatic reactions? ➒ What is the largest KIE observed so far. What is the reason for such high value? ➒ What is the change of the activation free energy responsible for KIE from two previous points?