Otdfbt

For instance, take morphine. It is used as a baseline for measuring the potency of opioid agonists. Its structure looks like this:

But then, take heroin, around three times as potent, its structure is this:

6-Monoacetylmorphine is around 30% more potent than heroin itself:

If, however, the acetyl group is bonded to the top instead of the bottom, the resulting substance is less active than heroin.
U-47700, a research chemical, has a totally different structure but has around 7.5 times the potency of morphine in animal models:

Desomorphine is 8-10 times more potent as morphine:

Oxymorphone is 10 times as potent as morphine:

The common explanation that I've heard for these substances being potent opioid agonists is that they mimic the endorphins produced in the human brain. However, Beta-Endorphin is 18-33 times more potent than morphine, and its structure is thus:

That looks absolutely nothing like any of the substances mentioned above. How can molecules with structures so radically different from those of endorphins mimic the action of endorphins? And how come U-47700 has a potency similar to the more powerful morphine analogues but looks nothing like them?

But then, there's Fentanyl:

100 times more potent than morphine, making it a more powerful Opioid receptor agonist than the body's own endorphins, and its structure looks nothing like them or any of the other ones so far. Other analogues are even more potent. So what is it about a molecule's structure that makes it a potent opioid? The only structural similarity that I can spot is the presence of a tertiary amine group in the structure. And how does U-47700 have a potency similar to the more powerful morphine analogues?

Specific parts — moieties — of an agonist molecule bind to the receptor protein, causing the receptor to change shape, which in turn initiates a signaling pathway inside the cell.

Some agonists are better at causing the receptor to change to its "optimal shape" for relaying signal. These are called "full agonists".

Other agonists cause a partial change in receptor shape and reduced signal. These are called "partial agonists". The strength of the resulting signal is called "efficacy".

In addition to differences in efficacy between agonists, the parts of the agonist that bind to receptors can have a different chemical makeup and so create a different environment of electrostatic attractions and van der Waals forces between atoms in the agonist and in the atoms in the binding site in the receptor. Agonists that bind more strongly will stay bound to the receptor longer. These are said to have higher "affinity". Agonists that bind more weakly have lower affinity.

It is affinity and efficacy together that determine potency, or how well an agonist works.

In a biological environment, as well, these agonists swim in a pool with other agonist and antagonist molecules. For example, your agonist of interest might do a good job of changing a receptor's shape, but if it binds weakly, it will "pop off" the receptor quickly — perhaps too quickly to trigger the necessary signal. Another, different agonist molecule comes in and binds more tightly, but it doesn't create as strong a signal. Or perhaps a passing antagonist molecule floats around and pops onto the receptor, stopping any signal from that receptor and also keeping any agonist molecules from binding again for a while.

The population of agonists, antagonists, and receptors all change the number and kinds of binding events that take place, which determines whether you get the signal you want, as strong as you want it.

While the opioid molecules you are looking at have different numbers and varieties of atoms, it is the molecule's shape and electrochemistry that matter. Most specifically, it will be the part of the agonist molecule that actually binds to the receptor that matters most: how well it causes a conformal change in the receptor, and how strongly it binds.

These molecules could look very different from one another, but ultimately it is the part of the molecule that binds to the receptor, and how well it binds, that matters.

I don't know about the receptor-ligand kinetics of how different opioids bind to pain receptors, specifically, but hopefully there is some terminology in this answer that helps with further research.

I think you are confusing potency with receptor affinity.

The (unsourced) numbers you indicate appear to be DEA-style "potency", which probably refer to the effective dose (of some kind).

The fentanyl to morphine ratio on the κ-opioid receptor affinity, from human studies is only 1.35 to 1.14 (on average). In contrast the N-methyl fentanyl is estimated from animal studies to have a receptor affinity of around 18,000 (±3,000).

As pointed in the other answer, there numerous other issues besides receptor affinity that affect potency, like bioavailability by the various administration routes.

Receptor affinity alone is somewhat easier to deal with (e.g. in computational chemistry models), e.g. by "molecular docking":

Molecular docking, a type of virtual screening, places compounds or ligands into potential binding poses within the binding site of a biological target and assesses binding affinity with a scoring function. The scoring function evaluates the non-covalent interactions between the ligand and target to provide a binding energy (score) which can be used to rank compounds with increasing binding affinity

It's actually quite difficult to measure receptor affinity in practice; there's a large margin of error, better shown in this graph, from the source that that previous table is based on for human studies.

Also there are three opioid receptors to worry about. (The graph above is for μ.)

I think the reason why fentanyl (and its derivatives) is stronger than morphine in analgesic test is/are largely unknown. According to a 2014 paper

A large number of fentanyl analogs have been synthesized and structure activity relationships revealed, but the knowledge of their action is still incomplete and many questions remain unanswered. In general, the order of activity at receptor parallels that of binding affinity. However, the comparison of ED50 values, for μ- analgesics with their IC50 and Kd numbers at the corresponding binding site does not always correlate with experimental data. How to explain, why fentanyl (6), methylfentanyl (42), sufentanil (88), lofentanil (82) are more potent than it would be suggested from obtained binding values. Another example, the Kd of ohmefentanyl for the μ- site is approximately 5–10-times less than that of morphine, but it is nearly 7000-times more potent than morphine in analgesic tests. There is another contradiction, absence of correlation between lipid solubility and analgesic effect [195,196]. Another observation is that **in general fentanyls are compounds with the highest affinity for the μ-receptor and the most potency at that receptor. However not all fentanyl derivatives are highly μ- selective. They also produce actions through δ- and κ- receptors. Lofentanil (82) is the least and R30490 (90), are the most μ-selective compounds, among fentanyl series. Although their structures are very close, and affinities at the μ-receptor are similar, R30490 displays much lower δ- and κ-affinities than lofentanil.