Quantitative Physiology I / Molecular and Cellular Systems; BMEN E4001x

HW1: Chemical Kinetics & Equilibria

Due September 25, 2024, 11:00PM US Eastern Time

1) Enzyme kinetics (10 points total)

The reaction rate for an enzyme you are investigating increases with increasing concentration of substrate (S) at low concentrations of S. However, this rate decreases as concentration of S increases at higher concentrations. You formulate the following mechanism for this response:

.   The Enzyme (E) binds reversibly to Substrate (S), leading to production of Product (P) and regeneration of E. This is the basic enzymatic reaction we covered in class.

.   S can also reversibly dimerize with another molecule of S. This dimer acts as a competitive inhibitor of E.

Note: dimerization of Sand binding to E happen in separate steps. For here, don’t include a trimolecular interaction of S, a secondS, and E binding together in one step.

Your goal for the next week is to test this model. The first step is to generate a quantitative prediction of reaction rate vs. Total Enzyme (E0) and S.

1.1) Draw a diagram of the reaction system. Introduce and indicate fundamental rate constants.

1.2) List the set of equations that describe this system. This includes differential equations  describing the time evolution of E, S, P, and all intermediates. Also provide a conservation equation that includes all forms of the enzyme.

1.3) List the simplified equations that result from applying the Equilibrium approximation to this system. At this step, do not solve the system of equations.

1.4) List the simplified equations that result from applying the Quasi-State approximation to this system. At this step, do not solve the system of equations.

1.5) Derive an expression for reaction rate (V) as a function of the concentrations of substrate (S), total enzyme concentration E0, and rate constants.

Moment for reflection (not graded): Does your result make intuitive sense and capture the experimental observation? What is the reaction rate as S → 0? How about at large values of S? Compare to other inhibition mechanisms.

2&3) Enzyme inhibition (10 points)

2) The antibiotic sulfanilamide is structurally similar to para-aminobenzoic acid (PABA), a  substrate that’s used in the production of folic acid. Since folic acid is vital for bacterial  growth, sulfanilamide has been previously used to competitively inhibit the enzyme that would otherwise bind PABA and thus stem bacterial growth.

You investigate sulfanilamide as an antibiotic against two bacterial strains, A and B, which have different versions of the enzyme used to produce folic acid from PABA, and different tolerances for sulfanilamide:

.   Substrate, S, is present at 15 mM.

.   Sulfanilamide inhibits the enzyme via competitive inhibition.

.   At [I] = 200 mM, strain A bacteria are killed.

.   Without sulfanilamide, both strains of bacteria produce folic acid at 5 mM / min.

.   Km,A = 15 mM, Km,B  = 30 mM, Ki,A = 100 mM, and Ki,B  = 60 mM.

2.1) Determine the maximum reaction rate of strain A compared to strain B. That is, calculate Vmax,A∕Vmax,B.

2.2) For strain A, determine the reaction rate below which cells are killed.

2.3) What concentration of Inhibitor is needed to kill strain B? Assume strain B dies at the same reaction rate as for strain A, which you calculated in 2.2.

3)  Data of an enzyme inhibition experiment are presented in the table below, consisting of the

rate of reaction (in mM /s) as a function of substrate concentration [S] (in mM) and inhibitor concentration [I] (also in mM). Assuming that enzyme is always present at a concentration of 100 nM:

3.1)  Estimate Vmax, Km, and Ki ; remember, Vmax  and Km  are based on just the interactions between substrate and enzyme, and are independent of the effect of inhibitor.

3.2)  Identify the mode of inhibition (competitive, noncompetitive, or uncompetitive), and explain this choice.

Please note, problem 2 is separate from 3. These are different experiments.

4) Hemoglobin (10 points)

In class, it was calculated that the capacity of blood to contain dissolved oxygen is not enough to supply the body’s oxygen needs. Specifically:

.   A 70kg body at rest needs 1.2e-2 mol/min O2

.   [O2]dissolved=sO2 *PO2 . For O2  into aqueous solutions, sO2=1.4e-6 M/mmHg

.   In the lungs, PO2=100 mmHg, in tissues, PO2=40 mmHg.

.   Cardiac output is 5 L/min.

Thus, the rate of delivery of O2 by dissolved gas is: [PO2,lungs * sO2- PO2,tissues * sO2]*cardiac utput =

[(100 mmHg)*(1.4e-6M/mmHg)-(40 mmHg)*(1.4e-6M/mmHg)*(5 L/min) = 4.2e-4 mol/min Hemoglobin (Hb) helps to meet this demand:

.   Hb is present at 2.3 mM

.   Binding constant of Hb with O2:  KM = 25.85 mmHg

.   Each Hb molecule can bind 4 O2 molecules

.   The behavior. of Hb can be estimated with good success by the Hill equation, with n = 2.45

4.1) With these assumptions, what is the oxygen delivery capacity (how fast oxygen can be transported from lungs to tissue) of blood (in mol/min) with this hemoglobin?

4.2) Suppose we could control the Hill coefficient for Hemoglobin. What is the oxygen delivery capacity for n=1? for n=4?

4.3) What is the relative increase in delivery capacity afforded by cooperativity between the Hb subunits? That is, how much does capacity increase when n=2.45 relative to n=1? Explain    the results for n=4.

4.4) Under conditions of increased oxygen demand, the PO2  of tissues drops to 20 mmHg. What is the relative increase in oxygen delivery capacity at n=2.45 versus n=1?

4.5) As a result of an unexpected, global change, oxygen in the atmosphere drops, resulting in a change of PO2  in the lungs from 100 mmHg to 80 mmHg. Identify a single adaptation in the parameters used in this problem (such as KM  or n) that would restore oxygen delivery to the level you identified in part 4.1. Provide a quantitative change, such as KM  increases/decreases to ??? mmHg.

Moment for reflection (not graded): Cooperative binding is often posed as a way for amplifying oxygen transport. Can you define the conditions for which this is true?