C8863 Free Energy Calculations -1C8863 Free Energy Calculations Petr Kulhánek kulhanek@chemi.muni.cz National Centre for Biomolecular Research, Faculty of Science Masaryk University, Kamenice 5, CZ-62500 Brno JS/2022 Present Form of Teaching: Rev1 Lesson 4 Equilibrium - Practicals C8863 Free Energy Calculations -2- Overview C8863 Free Energy Calculations -3- Revisions ▪ At the given temperature and definition of the standard state, the equilibrium constant is determined only by the standard reaction Gibbs energy: ▪ The equilibrium constant K is proportional to activities of all compounds in the equilibrium. ▪ For ideal (diluted) solutions, activities can be approximated by molar concentrations: Δ𝐺𝑟 0 = −𝑅𝑇 ln 𝐾 𝐾 = ෑ 𝑖=1 𝑁 𝑎 𝑟,𝑖 𝜐 𝑖 𝐾 ≈ ෑ 𝑖=1 𝑁 [𝑋𝑖] 𝑟 𝜐 𝑖 Sign convention for stochiometric coefficients ni products (end state) - positive value reactants (initial state) - negative value C8863 Free Energy Calculations -4- Equilibrium multiple chemical processes C8863 Free Energy Calculations -5Complex Chemical Mixtures Composition of the chemical system with multiple reactions is determined by a system of equations. These equations include ➢ each equilibrium process ➢ balance of all reacting compounds A + 2B AB2 2A + C A2C 𝐾1 = 𝐴𝐵2 𝐴 𝐵 2 𝐾2 = 𝐴2 𝐶 𝐴 2 𝐶 Example: C8863 Free Energy Calculations -6Complex Chemical Mixtures Composition of the chemical system with multiple reactions is determined by a system of equations. These equations include ➢ each equilibrium process ➢ balance of all reacting compounds A + 2B AB2 2A + C A2C 𝐾1 = 𝐴𝐵2 𝐴 𝐵 2 𝐾2 = 𝐴2 𝐶 𝐴 2 𝐶 Example: 𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2 𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶 Unknowns: → 5 equations 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶 balance equilibria initial amount C8863 Free Energy Calculations -7Numerical Solution I A + 2B AB2 2A + C A2C 𝐾1 = 𝐴𝐵2 𝐴 𝐵 2 𝐾2 = 𝐴2 𝐶 𝐴 2 𝐶 Example: 𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2 𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶 Unknowns: → 5 equations 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶 balance equilibria initial amount Only two components are independent: • five components • three balances C8863 Free Energy Calculations -8𝐴2 𝐶 = 1 2 𝑐0,𝐴 − 1 2 𝐴 − 1 2 𝐴𝐵2 Numerical Solution I, cont. 𝐾1 = 𝐴𝐵2 𝐴 𝐵 2 𝐾2 = 𝐴2 𝐶 𝐴 2 𝐶 𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶 0 = log 𝐴𝐵2 − log 𝐴 − 2 log 𝐵 − log(𝐾1) 0 = log 𝐴2 𝐶 − 2log 𝐴 − log 𝐶 − log(𝐾2) Find [A] and [B] such that the last two equations are satisfied: 𝐴𝐵2 = 1 2 𝑐0,𝐵 − 1 2 𝐵 𝐶 = 𝑐0,𝐶 − 𝐴2 𝐶 1. Determine dependent parameters: 2. Solve system of independent equations: Octave, Matlab: lsqnonlin 𝑓 𝑿 = 𝟎 C8863 Free Energy Calculations -9Numerical Solution II Find concentration of all components such that all equations are satisfied: 1. Solve system of equations: 𝐾1 = 𝐴𝐵2 𝐴 𝐵 2 𝐾2 = 𝐴2 𝐶 𝐴 2 𝐶 0 = log 𝐴𝐵2 − log 𝐴 − 2 log 𝐵 − log(𝐾1) 0 = log 𝐴2 𝐶 − 2log 𝐴 − log 𝐶 − log(𝐾2) 𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶 𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶 0 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 − 𝑐0,𝐴 0 = 𝐵 + 2 𝐴𝐵2 − 𝑐0,𝐵 0 = 𝐶 + 𝐴2 𝐶 − 𝑐0,𝐶 𝑓 𝑿 = 𝟎 this might be numerically less stable C8863 Free Energy Calculations -10- Problems C8863 Free Energy Calculations -11Host with two binding sites + H + G HG + + HG + G HG2 𝐾1 𝐾2 𝐾1 𝐾2 𝐾 = 𝐾1 𝐾2 Note: binding sites are chemically equivalent host (H) guest (G) C8863 Free Energy Calculations -12Host with two binding sites, tasks 1. Are 𝐾1 and 𝐾2 equal? 2. Determine the composition of the reaction mixture for 𝑐0,𝐻 = 1 𝑚𝑀 titrated by guest up to 6 molar equivalents for: • 𝐾1 = 102 • 𝐾1 = 105 3. Determine Job Plots for 𝑐0,𝐻 = 1 𝑚𝑀 and • 𝐾1 = 101 • 𝐾1 = 102 • 𝐾1 = 103 • 𝐾1 = 104 C8863 Free Energy Calculations -13Host Dimerization KD + KD 2H H2 • What is 𝐾 𝐷 for dimerization process of the host? Selected 1H NMR signal (fast exchange) undergoes the following change during the sample dilution. TBA C8863 Free Energy Calculations -14- References • Hibbert, D. B.; Thordarson, P. The Death of the Job Plot, Transparency, Open Science and Online Tools, Uncertainty Estimation Methods and Other Developments in Supramolecular Chemistry Data Analysis. Chemical Communications 2016, 52 (87), 12792–12805. https://doi.org/10.1039/C6CC03888C. • Gilson, M. K.; Irikura, K. K. Symmetry Numbers for Rigid, Flexible, and Fluxional Molecules: Theory and Applications. J. Phys. Chem. B 2010, 114 (49), 16304–16317. https://doi.org/10.1021/jp110434s. • Duboué-Dijon, E.; Hénin, J. Building Intuition for Binding Free Energy Calculations: Bound State Definition, Restraints, and Symmetry. J. Chem. Phys. 2021, 154 (20), 204101. https://doi.org/10.1063/5.0046853. https://arxiv.org/abs/2102.06089