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Kate Discussion Engel Page 1 Handout 11/3/08 Chemical Potential & Progress Variable Recall that for a reaction aA + bB cC + dD Free energy= G= Chemical potential= i= A similar equation to the above G can be written in terms of chemical potential for each reactant/product: i= These can be used to write the free energy for the reaction: G= For a reaction, we care what the value of G is, not G . G describes the reaction conditions and will dictate the direction of spontaneous reaction. So when G 0, G= Q is the reactant quotient and describes the present conditions that are not at equilibrium: Q= Now we can define a progress variable, , which measures the extent of the reaction, i.e. how far the reaction has proceeded. What are the units? What does =0 mean? How about =1? Discussion Kate Engel Page 2 11/3/08 reflects that as a reaction progresses, products are made and reactants are reduced. The numbers of moles of products and reactants are coupled to each other: na=na0- a dna=- ad nb=nb0- b dnb=- bd nc=nc0+ c dnc=+ cd nd=nd0+ d dnd=+ dd We can write the change in free energy in relation to change in product/reactant number: dG= Substituting in the above equations, we see that dG= At equilibrium, dG=0, so this makes c c + d d = a a + b b The bottom line of these derivations is that at equilibrium, reactants balance products. pKa The acid dissociation constant Ka= It is an equilibrium constant for the dissociation of an acid to its conjugate base. To obtain the value pKa, simply take the log10 of the Ka value. pH is derived similarly what is it? The Ka for water (written as Kw) is 1.0 x 10-14. What is the standard free energy of its dissociation? What is the pH? Finally, there is the Henderson-Hasselbach Equation: What does the Henderson-Hasselbach tell us? Discussion Kate Engel van't Hoff Equation lnKeq=(- H /RT)+ S /R Page 3 11/3/08 Provided that H and S are of independent temperature, we can measure Keq at varying temperatures and plot Keq vs. 1/T. What can we derive from such a graph? Protein Folding U represents the unfolded state; F represents the folded state. What are the equilibrium constants for folding and unfolding? How are they related? Protein folding is a balance between energy and entropy. Energetic contributions include: These factors do not contribute much to free energy. Why not? Entropy is a strong driving force for protein folding, mainly by the hydrophobic effect. What does this mean? Considering entropy more in depth now, the entropy of the unfolded protein itself is greater than the folded protein. Why? Discussion Kate Engel Page 4 11/3/08 Using the probabilistic definition of entropy, and assuming each unfolded protein state is equivalent in energy (i.e. no energetic multiplicity), then the entropy of the unfolded state per mole, assuming standard state conditions is is: S unfolded=RlnNc Where NC is the number of possible conformations. To determine this value, we must consider the torsional angles and rotamer conformations. We said that for each residue there are 3 unique torsional angle conformations, and 2 unique rotamer conformations. That means that there are a total of 2 x 3 = 6 unique conformations for each residue. How is Nc calculated, then, for the protein? What are the values of S unfolded and S folded? An increase in water entropy drives protein folding. When a hydrophobic residue is surrounded by water, the rotation of the surrounding water molecules is restricted. Our simple model for this change in entropy is S water, unfolded=RNresln2 S water, folded=RNresln6, making S (water)=RNres(ln6-ln2). Note:Nres here refers to only the hydrophobic residues. The S (water) term outweighs the S (protein) and therefore causes protein folding to be favorable! Melting Temperature What is the melting temperature? How can you measure the melting temperature experimentally?

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