In our notes, in non-competitive inhibition, the substrate is still able to bind to the enzyme, but then the enzyme can't do anything with it. In Purves, it says that the presence of the non-competitive inhibitor in its own site on the enzyme distorts the shape of the enzyme and therefore makes it difficult for the substrate to bind to the enzme. Which one is it? Is it a combination of both? Yes, both. Purves does not offer the explanation I prefer, and vice-versa. I like to reserve the distortion case for allosteric inhibition, which is yet another situation (more complex).

Why does Km stay constant in non-competitive inihibition?  The non-competitive inhibitor does not bind to the substrate-binding site. Those enzyme molecules that are still active are responding to substrate normally, according to the intrinsic Km of the enzyme.

It seems to me that non-competitive inhibitors must bind irreversibly to the enzyme in order to exert their effects, and that substrate binding could be affected. What is wrong with this picture?
Irreversibility has nothing to do with it. If a particualr enzyme molecule is (reversibly) binding a non-competitive inhibitor (NCI) , then while it is doing so, it will be inactive and will not contribute to the formation of product, not contribute to the velocity of the reaction. So the velocity of the reaction will be less in that test tube for that moment because of the presence of the NCI on that molecule at that moment. The next moment, the NCI will have become disassociated from its enzyme target, but another NCI will have become associated with another enzyme molecule, thus taking it out of action. So statistically, summed over all enzyme molecules and all the time going by, there are effectively less active enzyme molecules in a test tube in the presence of this dynamically binding NCI. The presence of substarte has no effect on this state of affairs, as it binds to a different site on the enzyme whether the enzyme is active or not (by definition for a NCI).

What does the picture 6.19 in Purves have to do with non-competitive inhibition?
Non-competitive inhibition is defined as inhibition that is not affected by variations in the concentration of substrate. The example I gave in lecture was an inhibitor that prevented the k3 reaction (S -> P) without affecting k1 or k2. Such inhibition could be achieved by blocking an amino acid residue at the active site (but not at the substrate binding site of the active site) or by binding elsewhere, deforming the enzyme so as to inhibit the k3 reaction but not affecting its ability to biond substrate, as in the Purves example. I avoided this example, as it overlaps with a simple description of allosteric inhibition.

Non-competetive inhibition can't be drowned out by adding more S but can be by adding a lot of E, correct?
More S does not help restore acitivtyot an enzyme inhibited by a non-competitive inhibitor. Addng more E always helps (increases the rate of the reaction), no matter what the inhibiton, as long as the enzyme is not saturated with inhibitor.

Can I say a non-competitive inhibitor (NCI) binds to the enzyme not only at a different site, but with a much better binding ability than the substrate? So much better that the binding of NCI to enzyme is almost "permanent" compared to substrate binding. That's why the enzyme bound to NCI is "permanently" removed from the game (they are dead). No matter how much substrate we put in, the small amont of NCI (such as poison) is so potent that it binds the enzyme with extreme ability.
No. There is no relationship between the binding affinity of a non-competitive inhibitor (NCI) and that ofthe substrate. In most cases the binding of the NCI by the enzyme is not "permanent" but rather the enzyme is in equilibrium between the bound and free state. The lecture diagram showing some enzyme molecules with the NCI bound is meant to represent only a snapshot of the situation at a given moment in time,or an average of many such snapshots. In reality the NCI is continually binding to the enzyme and dissociating from it. But for the period of time the NCI is bound (which can be considerable), the enzyme cannot function.

You said that a non-competitive inhibitor does not change the active site of the enzyme, meaning that the substrate still binds but it can't do anything. In the picture link from Purves (figure 19.C), the non-competitive inhibitor changes the enzyme's conformation so that it can't bind the substrate. Does the substrate bind or not?
Purves says the non-competitive inhibitor affects the active site, but not the substrate binding site (p.127, last paragraph). The picture from the publisher's Web site differs from that in the current Purves (4th Ed.) text Fig. 6.19C. I believe the text paints a less confusing picture and have removed the linked picture. The active site would comprise the the substrate binding site plus those elements that facilitate the reaction, that help catayze the reaction, that produce a transition state. The catalytic activity is inhibited, but substrate binding per se is not. The important thing is that substrates do not compete with a noncompetitive inhibitor, and that non-competitive inhibitors do not resemble substrates.

Why can't non-competitive inhibitors be 'swamped out' by excess substrate?
Because they do no compete with substrate for binding. They are essentially oblivious to the substrate concentration.