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

3. Conformational Entropy 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. 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 Where W is the number of different conformations populated in the molecule, R is the gas constant

4. Conformational Entropy 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 -1 per 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?

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 p i 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! g+ g- anti E g+ = 0.75 kcal/mol E anti = 0.00 kcal/mol E g- = 0.75 kcal/mol

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

8. Entropies from Boltzmann Probabilites RotamerRelative Energy (kcal/mol) Probability of being Populated p i ln(p i )Entropy -Rp i ln(p i ) kcal/mol/K Entropic Energy Contribution at 300K gauche+0.750.18-0.3090.000610.18 gauche-0.750.18-0.3090.000610.18 anti-0.00.64-0.2860.000570.17 Total----1.00-0.9040.001790.54 where R is the gas constant (0.001987 kcal/mol/K) and p i is the probability of a residue being in rotamer i. 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. Entropies from Vibrational Modes Where S i is the entropy associated with vibrational mode i. 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? Where i 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 1 2 3 4 chemwiki.ucdavis.edu

10. Computational Identification of Vibrational Modes www.sciencetweets.eu 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 In practice: 1)Minimize the molecule (protein) to ensure that it is at the bottom of the potential energy well 2)Compute the vibrational frequencies for 3N-6 vibrational modes 3)Convert into entropies

11. 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?