Concentration Control Coefficient Explorer

Same two-enzyme pathway as before (Source → E1 → S → E2 → Product), but now we track the steady-state pool size of the intermediate metabolite, S = u1/(u1+u2), instead of the flux.

current point    tangent line (slope = dS/du₁, or CE1S on the ln–ln plot)
[S]
dS/du₁ (slope)
CE1S
CE2S
C1 + C2
■ negative 0 positive ■
Because this toy pathway has only two enzymes, the summation theorem (ΣCEiS = 0) forces CE2S to always equal −CE1S exactly. In a real pathway with many enzymes, the total across all of them still sums to zero, but any two individual enzymes don't have to be exact opposites — some might be small, some large, some positive, some negative.

Try it yourself

1. Baseline and sign asymmetry. Set u₁=u₂=1. Note the signs of CE1S and CE2S. Both enzymes have positive flux control coefficients (from the first explorer) — so why does increasing E1 raise [S] while increasing E2 lowers it, even though both increase the flux J?

2. Scarce producer. Set u₁=0.1, u₂=1 (E1 is the weak link). What happens to [S] and to CE1S? Interpret what a coefficient close to +1 means for how tightly [S] tracks E1's activity.

3. Crippled consumer. Set u₂=0.1, u₁=1 (E2 activity cut to 10%). [S] rises sharply. The chapter text notes that cutting an enzyme's activity to 0.1× can require a substrate to rise substantially to keep flux constant. Compare the CE2S value you see here to that idea — do the signs and rough magnitudes agree, even though the exact numbers differ (this toy model lets both S and J change together, while the text's example holds J fixed)?

4. Summation theorem for n=2. Try several different u₁/u₂ combinations and watch C1+C2 in the stat panel. Does it ever move off zero? Using S = u₁/(u₁+u₂), show algebraically why CE1S + CE2S must equal 0.

5. Flux control vs. concentration control. Recall from the flux explorer that CE2J is always positive (E2 always helps drive flux). Here, CE2S is always negative. What does it mean, biologically, for one enzyme to simultaneously be a positive controller of pathway flux and a negative controller of its own substrate's concentration?