Free Energy Profiles

Part 1: Free Energy Profiles

1. Given a set of rate constants for an enzymatic reaction and the knowledge of transition state theory you should be able
   to construct a free energy profile/diagram for the reaction. The following enzyme (E) converts substrate (S) to product (P)
   via one chemical intermediate (I). It has four transition states and rate constants listed below.
   k1 = 107 M-1s-1
   k-1 = 1.1x104 s-1
   k2 = 2 s-1
   k-2 = 12 s-1
   k3 = 1.3x102 s-1
   k-3 = 33 s-1
   k4 = 1.9x103 s-1
   k-4 = 3.7x108 M-1s-1
   A) Write out the kinetic mechanism for the enzyme (e.g., , etc.), with the corresponding rate constants (e.g.,
   k1, k-1, etc.). Draw the free energy profile, label the steps (e.g., E+S, E•S, etc.), and indicate which barriers correspond to
   the rate constants above.
   B) Using the rate constants, calculate barrier heights (DGǂ, kcal/mol) in the forward and reverse directions. Assume that the
   concentration of substrate and product are at their intracellular levels, which are 40 µM and 2 µM, respectively. Hint: you
   need this information to compute the pseudo-first order rate constants, k1 and k-4. Assume the enzyme works inside human
   body, which is maintained at 37 ºC.
   You may need the following constants: h = 6.6256x10-27 erg sec
   kB = 1.38x10-16 erg K-1
   R = 1.987 cal K-1mol-1
   (“erg” = 10-7 J and cal = 4.184 J)
   See if you need to re-adjust your free energy diagram in part A, based on barrier heights you calculated here.
   C) Show the calculation you did to get the pseudo-first order rate constant for k1 and k-4
   D) Show your calculation to determine the DGǂ (kcal/mol) for the step involving k1
   E) Using information you obtained in parts B-D, complete the table below. Include units for each parameter.
   Step Rate constant
   (forward or reverse)
   Pseudo-first order rate
   constant (k1 and k-4)
   DGǂ (kcal/mol) for
   forward direction
   DGǂ (kcal/mol) for
   reverse direction
   1
   2
   3
   4
   F) What is the rate limiting step in your profile? Explain your answer.
   2
2. Komives et al. constructed free energy profiles for triosephosphate isomerase (TIM) reaction using the above approach
   (see the paper assigned for Problem Set 1). Experimentally measured kinetic parameters are summarized in Table I and
   Kinetics portion of the Results. Assume reaction kinetics were measured at 37 ºC.
   A) Write out the kinetic mechanism for TIM, going from DHAP to GAP, with the corresponding rate constants (e.g., k1, k-1,
   etc.).
   B) Using the information in Table I and assumptions described in the Kinetics portion of the Results, complete tables below
   for the reaction in part A. Include units for each parameter.
   Wild-type TIM:
   Step Rate constant
   (forward or reverse)
   Pseudo-first order rate
   constant (k1 and k-3)
   DGǂ (kcal/mol) for
   forward direction
   DGǂ (kcal/mol) for
   reverse direction
   1
   2
   3
   H95Q TIM:
   Step Rate constant
   (forward or reverse)
   Pseudo-first order rate
   constant (k1 and k-3)
   DGǂ (kcal/mol) for
   forward direction
   DGǂ (kcal/mol) for
   reverse direction
   1
   2
   3
   C) Show the calculation you did to get k-1 and k3 for both the wild-type and H95Q mutant TIM. Don’t forget the units.
   D) Show your calculation to determine the DGǂ (kcal/mol) for the step involving k2 in both the wild-type and H95Q mutant
   TIM.
   E) What is the rate-limiting step in wild-type and H95Q TIM? Is the same step rate-limiting in both enzymes or a different
   step?
   F) Using your calculated parameters from part B, draw out (no screenshots from the paper!) free energy profiles for the wildtype and H95Q TIM below. Label each step (e.g., E+DHAP, E•DHAP, etc.). Which step in the kinetic mechanism is affected
   most by the H95Q mutation? Does the mutation speed that step up, slow it down or has no effect? Explain why that step is
   sped up, slowed down, or unaffected by the mutation, using the measured kinetic parameters (Table I in the paper) and
   chemical mechanism you wrote out for H95Q enzyme in Problem Set 1.
   Part 2: Transition States
   Alkaline phosphatase (AP) catalyzes the hydrolysis of a variety of phosphate ester compounds, via a covalent
   phosphorylated-enzyme reaction intermediate (shown below), which is then hydrolyzed to release inorganic phosphate:
   3
   The first step in AP-catalyzed reaction is a nucleophilic substitution (SN), with enzymatic Serine serving as a nucleophile
   and phosphate ester substituent (–OR) as a leaving group. Roston et al. use AP as a working system to explore effects of
   changing the leaving group on the structure of the transition state in an enzyme-catalyzed reaction (see posted J Am Chem
   Soc 2016 paper). Please answer the following questions about the paper and AP:
3. Draw out possible transition states (TS) for AP-catalyzed reaction shown above. What do the authors mean by “tight” vs
   “loose” TS? How does the charge on phosphate oxygens differ between these different TSs? Which path (tight or loose) do
   the substrates tested in the paper follow?
4. What makes a “good” or “bad” leaving group in a nucleophilic substitution reaction? Draw out an example of a good and
   a bad leaving group tested in the paper.
5. Does the electronic structure of the TS change or not across different leaving groups tested in the paper? Draw out
   representative TSs for a phosphate monoester with a good and a bad leaving group. Show relevant distances between
   atoms. What are the consequences of a changing TS structure (if it changes) to how the reacting atoms interact with the
   enzyme at the TS?
6. Classic models predict linear free-energy relationship (FER) between log of reaction rate (V/K in the paper) as a function
   of the leaving group pKa in a substitution reaction. The authors and experimentalists before them, however, observe curved
   FERs for AP-catalyzed substitution of phosphate esters. Explain why that is.
7. How do the authors explain the observation that AP is more catalytically proficient with phosphate diesters that have a
   leaving group pKa ~9 vs. those with a leaving group pKa above or below ~9 (see the maximum value in Fig. 4C plot)?
8. The authors propose several experimental tests of the predictions made by their computational study. List at least two
   such prediction/experimental test pairs. Whenever applicable, state if that experiment has already been done and what the
   results were.
9. In your opinion, what is the main lesson the paper is trying to get across to the scientific community?

Sample Solution