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Conformational Entropy

Conformational Entropy. Entropy is an essential component in Δ G and must be considered in order to model many chemical processes, including protein folding, and protein – ligand binding

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Conformational Entropy

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  1. Conformational Entropy Entropy is an essential component in ΔG and must be considered in order to model many chemical processes, including protein folding, and protein – ligand binding Conformational Entropy – relates to changes in entropy that arise from changes in molecular shape or dynamics ΔG = ΔH – TΔS

  2. Conformational Entropy Enthalpy (DH) is favorable – due to the formation of hydrogen bonds, salt-bridges, dipolar interactions, van der Waals contacts and other dispersive interactions Entropy (DS) is unfavorable – due to a reduction in the number of degrees of freedom of the molecule – that is, entropy favors disorder The entropy of heterogeneous random coil or denatured proteins is significantly higher than that of the folded native state tertiary structure

  3. Conformational Entropy In proteins, backbone dihedral angles and side chain rotamers are commonly used as conformational descriptors. These characteristics are used to define the degrees of freedom available to the molecule. Discretize = To convert a continuous space into an equivalent discrete space for the purposes of easier calculation To calculate conformational entropy, the possible conformations may first be discretized into a finite number of states, usually characterized by unique combinations of certain structural parameters, such as rotamers, each of which has been assigned an energy level. Where W is the number of different conformations populated in the molecule, R is the gas constant

  4. Where W is the number of different conformations populated in the molecule, R is the gas constant For a single C-C bond (sp3-sp3) there are 3 possible rotamers (gauche+, gauche+, anti-). If we assume that each is equally populated, that is, each bond is 33% g+, 33% g-, and 33% anti Then W = 3 And S = – Rln3 = –2.2 cal.K-1.mol-1per rotatable bond How much energy is this at 300K? 0.66 kcal/mol – can you derive this? But, what if the rotamers are not populated equally? Conformational Entropy

  5. Conformational Entropy as a Function of State Populations The conformational entropy associated with a particular conformation is then dependent on the probability associated with the occupancy of that state. Conformational entropies can be defined by assuming a Boltzmann distribution of populations for all possible rotameric states [1]: where R is the gas constant and pi is the probability of a residue being in rotamer i. 1. Pickett SD, Sternberg MJ. (1993). Empirical scale of side-chain conformational entropy in protein folding. J Mol Biol 231(3):825-39.

  6. Deriving Probabilites or Populations from Energies But how do we derive the probabilities (or populations) that a particular state will be occupied? Boltzmann to the rescue! Eg+ = 0.75 kcal/mol Eanti = 0.00 kcal/mol Eg- = 0.75 kcal/mol g- g+ anti

  7. For rotamer 1 (Eg+): Probabilites For rotamer 3 (Eg-): For rotamer 2 (Eanti): And the sum: For the three rotamers: Eg+ = 0.75 kcal/mol, Eanti = 0.0 kcal/mol, Eg- = 0.75 kcal/mol Now the populations (or probabilities, pi) can be computed easily for each rotamer as: And panti = 0.64, can you derive this?

  8. where R is the gas constant (0.001987 kcal/mol/K) and pi is the probability of a residue being in rotamer i. Entropies from Boltzmann Probabilites Conclusion? A single rotatable bond has about 0.5 kcal/mol of entropic energy Thus, if a single bond becomes rigid upon binding to a receptor, it will cost about 0.5 kcal/mol

  9. In addition to bonds being prevented from rotating, several other physical properties change upon ligand binding. In general the protein also becomes more rigid. Put another way, it’s vibrational modes change. How can we capture this Vibrational Entropy? Entropies from Vibrational Modes Where Si is the entropy associated with vibrational modei. n1 n2 Where ni is the vibrational frequency of mode i, h = Planck’s constant k = Boltzmann’s constant Thus, we need to identify all of the vibrational modes in the protein n3 n4 chemwiki.ucdavis.edu

  10. In general non-linear molecules have 3N-6 normal modes, where N is the number of atoms. This is the same as the number of internal coordinates ;-) Assume all vibrational motions are harmonic – that is they are simple oscillations around an equilibrium position This is a good approximation for force fields since the bonds and angles are modeled using Hooke’s Law Computational Identification of Vibrational Modes www.sciencetweets.eu In practice: Minimize the molecule (protein) to ensure that it is at the bottom of the potential energy well Compute the vibrational frequencies for 3N-6 vibrational modes Convert into entropies

  11. How Much Entropy is Present in Amino Acid Side Chains?

  12. How Much Entropy is Present in Amino Acid Side Chains?

  13. Protein Folding: Enthalpy versus Entropy Probing the protein folding mechanism by simulation of dynamics and nonlinear infrared spectroscopy. Doctoral Thesis / Dissertation, 2010, 157 Pages

  14. How Much Entropy is Present in Amino Acid Side Chains? How much energy is -2.2 cal/K/mol at 300K?

  15. How Much Entropy is Present in Amino Acid Side Chains?

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