Protein Folding Explantions via Diffusion Bridge Score Matching

Technical Title: Sampling Transition Paths Between Molecular Conformations Using Diffusion Bridges and Score Matching.

Recent advances in Schrodinger Bridges [1,2] have enabled to learn stochastic mappings between 2 probability distributions (p(x) and q(x)) such that the stochastic map (which is modelled by a diffusion / SDE) is regularised by some prior process (whether it be computational or physical).

This project seeks to explore these methodologies in particular the simpler case studied in [3] where both the source and the target distributions are modeled as point masses (dirac delta functions / measures). We seek to apply the approach in [3] to sampling physically meaningful transition paths between two protein configurations as done in [4]. Ideally, we would aim to start working with simple/toy proteins and then move on to larger scale tasks where one of the protein configurations is a flat amino-acid and proteins produced by alpha fold, the end product would be to generate a video which gives a physically plausible folding process for alphafold [5] predictions.

An example Timeline of the project could be:

1. Get the codebase of [4] working and reproduce results on simple proteins.
2. Extend the work in [3] to the underdampened version. (Francisco can help with this)
3. Apply the extensions and adaptions of [3] to work on the simple proteins of [4] and compare to the method in [4].
4. Consider enhancements / extensions, would a full Schrödinger bridge work better here?
5. If time allowing, pick some of the most recently exciting discoveries from alphafold [5] that have a known potential and try and see if we can get it working.

Point 5. is an “if time allows” type of objective and I predict most the time will be spent in 3., successful completion of 3 could lead to a publication at a top venue whilst 5. could have a broader impact on the field.

Ideally a good background in the following could be very helpful for this project:

1. Timeseries models (Kalman filters, AR processes, Gaussian processes).
2. Introductory calculus (ODEs, Basic PDEs, limits).
3. Probability Theory (Limit Theorems, Change of Variables, basic concentration inequalities e.g., Markov/Chebyshev)
4. Variational Inference (MFVI, Amortised VI, Deep hierarchical latent variable models)

[1] https://arxiv.org/pdf/2106.01357.pdf
[2] https://arxiv.org/pdf/2106.02081.pdf
[3] https://arxiv.org/pdf/2207.02149.pdf
[4] https://arxiv.org/pdf/2111.07243.pdf
[5] https://alphafold.ebi.ac.uk/