Solve the following problem. In your working, be meticulous and double check logic, assumptions and calculations. After arriving at final answers, perform a final double check to ensure consistency, correctness and absence of logical errors. As I cannot send you pictures, all figures will be described.

Enzyme E has five active sites, each of which is specific to one of the substrates SA, SB, or SC that are selectively transformed to products PA, PB, or PC, respectively. There is at least one active site for each substrate. Each active site is independent of the others.

Figure 1 description:

Left side (Diagram):
The diagram shows a large green irregular shape labeled "E" in the center, likely representing an enzyme. Around this central shape are five oval shapes of different colors, each labeled with a letter:


Yellow oval labeled "SA"
Blue oval labeled "S?"
Pink oval labeled "S?"
Gray oval labeled "SC"
Orange oval labeled "SB"

These ovals appear to be attached to or interacting with the central green shape, suggesting they may represent substrates or ligands binding to the enzyme.
Right side (Equations):
There are three similar chemical equations shown, each following the same general format but with different subscripts:

E + SA ⇌ ESA → E + PA
The forward reaction rate is labeled ka1, the reverse rate ka2, and the product formation rate ka3.
E + SB ⇌ ESB → E + PB
The forward reaction rate is labeled kb1, the reverse rate kb2, and the product formation rate kb3.
E + SC ⇌ ESC → E + PC
The forward reaction rate is labeled kc1, the reverse rate kc2, and the product formation rate kc3.

For E, it is known that:
1. The affinity for SC is higher than for SB.
2. The plot of 𝑣0𝑖 vs 𝑣0𝑖/[Si]0, known as Eadie–Hofstee plot, for SA, SB, SC, with rate given per active site (𝑣0𝑖), is shown below, but the scale and the legend are omitted.

This image shows a graph with the following characteristics:

The vertical axis is labeled "v0i", which likely represents initial reaction velocity.
The horizontal axis is labeled "v0i /[Si]0", which appears to be the ratio of initial velocity to initial substrate concentration.
The graph contains three separate lines, each starting from a different point on the vertical axis and extending towards the horizontal axis.
Each line consists of two parts:

A solid line segment in the middle
Dotted line segments at both ends (extending from the solid line towards the axes)


The lines have different slopes, with two lines having parallel steep slopes (one being above the other) and the bottom line having the shallowest slope.
All three lines appear to converge towards a single point on the horizontal axis, though they don't quite reach it in the visible part of the graph.
The graph does not include any numerical values on either axis, showing only qualitative relationships.

3. When E is saturated with SA, SB, SC, the catalytic turnover number (TON) for SC per site is 10200 min−1, and 2023 PA, PB, PC product molecules in total are synthesized per second. At the same time, no more than 5.94 ⋅ 106 molecules of PA and PB are detected to form per hour.
4. At equimolar concentrations of SA, SB, SC, which are at least 1000 times lower than corresponding 𝐾𝑀 values, the P , P , P concentrations become proportional to the catalytic efficiency (𝜀 = 𝑘𝑖3 ), and their
A B C
𝑖 𝐾𝑀,𝑖
ratio is 3 ∶ 2 ∶ 5, respectively.
5. Two ESi complexes have equal rate constants for dissociation to E and Si. The activation barrier for the reaction of ESC into the initial compounds is 1266 J mol−1 higher than for the reaction into the final products. Assume that the pre-exponential factors are equal for both reactions and 𝑇 = 25 ∘C.
6. For E + Si reaction: 𝑘𝑐1 = 1.57 ⋅ 107 M−1s−1 and 𝑘𝑎1 = 𝑘𝑏1.

Fill in the table (of all rate constants k1, k2 and k3 for all enzymatic sites, all KM values and the number of each sites Sa, Sb and Sc) on the answer sheet and provide your calculations. 17pt Hints:
• Use information from 1. and 2. to find the relation between 𝐾𝑀,𝐴, 𝐾𝑀,𝐵 and𝐾𝑀,𝐶 (<,>,=).
• Information from 3. and 5. allows you to complete the first column (num- ber of active sites for each substrate) and the last row (all the constants for substrate SC) of the table. Check that the sum of active sites is equal to 5.