How is geometry optimization of small molecules different from the protein folding problem?

Question

My layman's understanding is that finding the ground state geometry of small molecules is "hard". I don't have a good sense of how hard though. I suppose there are applications where methods like VSEPR are good enough, and other situation where some sort of ab initio calculations are required. But isn't the problem of protein folding essentially the same problem, albeit at a much larger scale? Yet we have methods like Alphafold which can produce good results in many cases. So what am I missing here? Why can't something like Alphafold give us ground state geometries of small small molecules if it can do something similar for much larger molecules?

Answer 1

tldr: For small rigid or semi-rigid molecules (e.g., 0-2 rotatable bonds) we can do a good job. For larger, more flexible molecules it's still hard.

One of the challenges with getting a good ground-state geometry is the accuracy of the methods to optimize the geometry.

AlphaFold and the various alternatives that have cropped up after its release, work in part from the vast amount of data in the Protein Data Bank (PDB) and similar databases. At the moment, there are almost 200,000 released structures. Moreover, because of the size of proteins, there's a lot of information that can be deduced from the secondary and tertiary structures.

There have also been numerous "blind tests" such as CASP, which started in 1994. The challenge is to predict the geometry / fold given the sequence. Such open challenges have definitely driven the field forward, since there's great prestige.

Add in that there has been a lot of work on bimolecular force field methods, such as AMBER, CHARMM, etc. Many groups around the world run molecular dynamics simulations with these methods and validate against experimental data, such as NMR.

One might expect that small molecules are easier – after all they are smaller!

Often that's true. For small molecules with zero "rotatable bonds" we have no conformers to predict and can often use quite accurate quantum mechanical methods to optimize the geometry. Bond lengths and angles are often very accurate when compared to experimental crystal structures, and the overall root mean square displacement (RMSD) of non-hydrogen atoms is often under 0.5Å. (Hydrogens are omitted because they usually are not available in crystal structures.)

For example, from our recent preprint we find the GFN2 semiempirical quantum method provides excellent agreement with crystal structures for zero or one rotatable bond (e.g., median RMSD ~0.2-0.3Å and mean RMSD ~0.5Å). That means for most of these molecules, atoms are only a small fraction of an Ångstrom out of place!

The challenge comes from a similar problem to protein structure prediction. There are lots of possible conformers for larger more flexible molecules. Many of these are not low-energy conformers and gas-phase optimizations neglect potential intermolecular stabilization in solution or a crystal structure. That's partly why the plot shows a slope for RMSD with increasing numbers of rotatable bonds. (Also, in large molecules, small errors in angles, etc. will add up.)

So why has AlphaFold done such a good job with protein structures, but small molecular conformers are still hard?

I can think of a few possible reasons:

1. More resources have been devoted to protein structure prediction. AlphaFold devoted a big team and lots of GPU/TPU time to attack it.
2. Proteins have fewer possible rotatable bonds. A recent paper by the Rarey group yielded a library of over 500 possible small molecule torsions with preferences.
3. Small molecule force fields are likely not as good as protein force fields, in my opinion. While faster approximate quantum methods like GFN2 (and some newer ML methods) exist, they're still fairly new and not as accurate as DFT on conformer energies.

There are many conformer sampling methods out there for small molecules. Unfortunately, very few support use with quantum methods.

Put that all together, and .. flexible small molecules are still tricky.

Answer 2

The combinatorial complexity of large structures means that simpler methods can never solve the problem

Guessing the structure for small molecules can require a variety of techniques that work reasonably well for small molecules. Even some very simple techniques can do a good job for molecules where the bonding is not complex. Simple molecular mechanics techniques, for example, ar often good enough to give moderately good 3D structures for molecules with dozens of atoms especially if the bonds don't involve complex electronic interactions (so chemical drawing tools often apply simple force-field models to estimate plausible 3D structures).

These tools use simple information on bond-lengths and spatial interactions (atoms far apart in bonding terms can't overlap in 3D space and van der Walls interactions can explain that and reduce spatial overlap). They sometimes give results that are not ridiculous.

It might seem that protein folding could be "solved" with similar approaches. After all the constraints on folding are often not long term electronic interactions but simple spatial overlaps (I'm simplifying a lot as there are some important interactions with specific amino acid residues a long way apart in from each other on the protein backbone but this doesn't affect the argument). Don't we just have to try a bunch of bond rotations in the backbone of the protein and residues and see whether we can find plausible structures where the spatial overlaps are minimised?

But, even ignoring those complex interactions, this is an essentially impossible task. While testing the overall structure by running through many bond-rotations and testing the spatial overlaps that result works when the number of rotations are small (perhaps dozens), the number required to test protein structures is far larger. The difficulty rises exponentially with the number of bonds that have to be tested. Imagine we can explore the configurations of a molecule by testing 10 different rotations of each rotatable bond (and doing some simple calculations for each). For a molecule with just one rotatable bond this takes 10 calculations. For one with 10 rotatable bonds this takes 10¹⁰ calculations, which is a lot more. For 100 rotatable bonds (very small for a protein) this grows to 10¹⁰⁰ calculations, which is not just a big number but an unimaginably big number vastly exceeding the number of particles in the known universe.

The combinatorial complexity grows exponentially with the number of bonds that need to be tested. This pretty rapidly exhausts the computer power of whatever computer you throw at the problem. Never mind that this is an extremely simplistic view of the complexity of the problem. We will never have the computer power to solve these problems.

What about Alphafold?

One might question that analysis by mentioning that Alphafold has made big progress on the protein folding problem so things can't be as bad as I suggest.

But this is to misunderstand how machine learning algorithms like Alphafold work (the saem applies to their similar achievement at Chess and Go both of which are intractable to brute force techniques largely because of their combinatorial complexity). Alphafold (and its equivalent for Chess and Go) does not attempt to calculate every possible configuration of the problem. It studies a very large number of known structures and tries to find common patterns that repeat across many structures. Machine learning is very good at spotting common motifs that are not very obvious to people in known structures. It then applies those to unknown protein sequences and tests whether they work. This is a far, far simpler approach than trying to work through all possible options and results in a higher, but still imperfect, guess as to the likely structure of the unknown protein.

What makes this work is that there are common features in many known structures that are very likely to repeat in others. And many of those are not easily spotted by people or other, existing, algorithms. But it is not a perfect method and does not solve the protein folding problem: it just does a much better job than other algorithms.

So the "success" of Alphafold does not undermine the argument that the fundamental barrier to using small molecule techniques to solver bigger problems is combinatorial complexity. The problem grows faster than increases our computing power as molecule size grows.