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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

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

conformational entropy1

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

conformational entropy2

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

conformational entropy3

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

conformational entropy as a function of state populations

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.

deriving probabilites or populations from energies

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

probabilites

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?

entropies from boltzmann probabilites

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

entropies from vibrational modes

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

computational identification of vibrational modes

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

protein folding enthalpy versus entropy

Protein Folding: Enthalpy versus Entropy

Probing the protein folding mechanism by simulation of dynamics and nonlinear infrared spectroscopy.

Doctoral Thesis / Dissertation, 2010, 157 Pages

how much entropy is present in amino acid side chains2

How Much Entropy is Present in Amino Acid Side Chains?

How much energy is -2.2 cal/K/mol at 300K?