Category Archives: Diffusion

Diffusion Models Can’t Give You a Likelihood… So How Do We Score Inverse-Folded Sequences?

I’ve been working with inverse folding models for sequence design for a while now, and one question kept coming up. How would you score a sequence’s likelihood under these models? It turns out the answer depends heavily on whether the model is autoregressive or diffusion-based.

Autoregressive models, like GPT-style language models, make this straightforward. You can calculate the probability of a sequence directly, by breaking it into one prediction per position and multiplying them together. Diffusion models define a probability distribution over outputs too, but getting the likelihood of a particular output means accounting for many unobserved intermediate states, and that turns out to be much harder. This is why diffusion models typically rely on the Evidence Lower Bound, or ELBO, a computable estimate, rather than the exact likelihood itself.

The easy case: autoregressive models

An autoregressive model breaks the probability of a sequence into a product of conditional probabilities:

p(x1:n)=∏i=1np(xi|x<i)p(x_{1:n}) = \prod_{i=1}^{n} p(x_i \mid x_{<i})
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A timeline of sampling methods of diffusion models

When approaching the methods used in de-novo protein design, one is quickly confronted with a plethora of overlapping formulations of what looks superficially like “the same thing”. One paper trains an ϵ\boldsymbol{\epsilon}-prediction network with a simple MSE loss; another trains a score network with a stochastic-differential-equation justification; a third trains a clean-data predictor under yet another schedule. Each formulation carries its own notation, its own variance schedule, and its own sampler. Qualitatively, this zoo of formulations is doing the same thing: it starts from some unstructured noise and iteratively refines it to eventually produce a protein structure similar (but different!) to other proteins we have experimentally determined in the past. What is not immediately obvious to a newcomer is that all of these formulations are historical descendants of a small number of foundational ideas, and that essentially every architectural and algorithmic decision in a modern protein-design diffusion model has a specific paper of origin and a specific motivation for being there.

This post is my attempt to put these formulations onto a single timeline. I trace the trajectory of the field through four foundational works: DDPM (Ho et al., 2020), DDIM (Song et al., 2021a), the score-based SDE unification (Song et al., 2021b), and EDM (Karras et al., 2022), explaining at each step what specific problem with the previous formulation the next paper was attacking and how the new formulation generalises or simplifies the old one. The goal is coherent motivation rather than exhaustive coverage; the reader interested in implementation details is referred to the original papers and the references at the end.

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Analyzing AlphaFold 3’s Diffusion Trajectory

A useful way to understand AlphaFold 3’s sampling behavior is to look not only at the final predicted structure, but at what happens along the reverse diffusion trajectory itself. If we track quantities such as the physical energy of samples, noise scale, and update magnitude over time, a very clear pattern emerges: structures remain physically imperfect for most of sampling, and only take proper global shape in the final low-noise steps.

This behavior is a result of the diffusion procedure implemented in Algorithm 18, Sample Diffusion, which follows an EDM-style sampler with churn. Rather than simply marching monotonically from noise to structure, the sampler repeatedly perturbs the current coordinates, denoises them, and then takes a Euler-like update step. Because of the churn mechanism, AlphaFold 3 deliberately injects additional noise during part of the trajectory, which encourages exploration but also delays local geometric convergence. This mechanism is shown in step 4 -7 of the Sample Diffusion Algorithm from Alphafold3 Supplementary Information.

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SigmaDock: untwisting molecular docking with fragment-based SE(3) diffusion

Alvaro Prat, Leo Zhang, Charlotte Deane, Yee Whye Teh, & Garrett M. Morris
International Conference On Learning Representations (ICLR 2026)

Molecular docking sits at the heart of structure-based drug discovery. If we can reliably predict how a small molecule binds in a protein pocket, we can prioritize compounds faster, reason about interactions more clearly, and build better pipelines for hit discovery and lead optimization. But in practice, docking is still a difficult problem: classical methods are often robust but imperfect, while recent deep learning approaches have sometimes looked promising on headline metrics without consistently producing chemically plausible poses.

SigmaDock was built to address exactly that gap. Instead of treating docking as a problem of directly diffusing on torsion angles or unconstrained atomic coordinates, SigmaDock represents ligands as collections of rigid fragments and learns how to reassemble them inside the binding pocket using diffusion on SE(3)\text{SE}(3). In plain English: rather than trying to “wiggle” every flexible degree of freedom in a tangled way, SigmaDock breaks the ligand into chemically meaningful rigid pieces and learns where those pieces should go, and how they should reorient, to recover a valid bound pose.

Figure 1: Illustration of SigmaDock using PDB 1V4S and ligand MRK. We create an initial conformation of a query ligand where we define our mm rigid body fragments (colour coded). The corresponding forward diffusion process operates in SE(3)m\text{SE}(3)^m via independent roto-translations.
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Controlling the Diffusion Denoising Process: A Molecular Show

This blog post is supporting my poster at Young Modellers Forum and makes things way easier to see and understand. Underneath each GIF, is the explanation of what you should look for as things denoise throughout the diffusion trajectory. Click the GIFs for higher quality viewing!

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