Exploring Linear Transformations in AlphaFold 3 Pairformer: A Simplified Demo

Introduction

This work is about understanding the pairformer module - triangle attention of Alpha Fold 3.

This module is used to enrich the token pair representation Z through a series of attention steps that are simulated in this work for understanding and education.

Demonstration programs have been developed to reproduce some key features of the real process while making simplifying assumptions.

This is the 3d iteration to develop correct demo programs. A key part of this development process is done by GPT4 and details are here.? However, GPT4 only provided the correct demo program after several iterations, while the code and explanations in intermediate iterations were wrong.

Program linear-OUTGOINg_c.py

The source code is available in github

Program Description

Goal: mimic the Alpha Fold 3 Pairformer Process

Analyzing the Diagram and Text Description

• Element-wise multiplication: For each pair (i,j), the computation involves an element-wise multiplication of features from row i of a with row j of b across all k and summing these products.
• Gating mechanism: The summed result is then gated by the matrix g, using the corresponding position (i,j).

Revised Program Analysis

Here's a review based on the corrected understanding:

1. Initialization: a, b, and g Initialization: Correctly initializes 3D tensors which aligns with needing distinct transformations for each interaction scenario.
2. Triangle Update Calculation: Iterates over all positions (i,j). For each position, it computes a weighted sum of element-wise products of vectors taken from a[i,k,:]? and b[j,k,:] for all k. The summation captures the aggregate influence from all tokens k on the interaction between tokens i and j. Applies gating with g[i,j,:] to modulate this interaction.
3. Correctness of Element-wise Multiplication: a[i,k,:] times b[j, k, :] is the correct formula based on the diagram. This considers each feature of i and j for every k, reflecting how different tokens (indexed by k) contribute to the interaction between i and j.

Questions

(i) Does this program match the diagram and this text?

• Yes, the program correctly implements the logic as depicted and described by using row i of a and row j of b, summing across all tokens k, and gating the result.

(ii) Does the program implement the Triangle Update outgoing edges algorithm?

• Yes, it does. The program accurately models the computation of outgoing edges by considering how each token k influences the interaction between tokens i and j, and then controlling this interaction through a gating mechanism.

the summation formula at point 3 of the preceding paragraph exactly matches the left side diagram

(iii) Explanation on how it's implemented:

• By iterating over each possible pair (i,j) in the output matrix and calculating the contribution of all tokens k to the interaction between i and j as influenced by the transformations in a and b, and finally gating the result.

Conclusion

The provided code after the clarification effectively represents the intended mechanics of the Triangle Update for outgoing edges in the Pairformer architecture of AlphaFold 3. This includes the aggregation of influences across different contexts k and their modulation through the gating matrix g, matching the computational needs for modeling complex biological interactions like protein folding.

Program linear-INCOMING_c.py

The source code is available in github

Program Description

Goal: mimic the Alpha Fold 3 Pairformer Process

Analyzing the Diagram and determine if the program matches it

This Python program simulates a model similar to the PairFormer mechanism used in AlphaFold for updating protein representations based on "incoming" triangle updates. Here's a detailed description of each part of the program:

1. Random Seed Fixation np.random.seed(42): Ensures reproducibility by fixing the random seed, making the random numbers predictable.
2. Initialization of 3D Tensors: initialize_projections: Initializes three 3D tensors a, b, and g, each with dimensions [Ntokens, Ntokens, Cz], representing transformation matrices for the incoming edges update. Here, Ntokens represents the number of tokens (or nodes in the context of graph theory), and Cz represents the number of channels or features per token.
3. Triangle Update Function for Incoming Edges: apply_triangle_update_incoming: Applies the triangle update logic specific to incoming edges. For each pair of indices (i, j), it calculates a new value for each element in the tensor Z by: Iterating over all tokens k and performing element-wise multiplication of the i-th column of tensor a and the j-th column of tensor b, summing these products. The summed product is then gated by multiplying with the corresponding elements from tensor g at (i, j). This operation models the interaction between different features across tokens, factoring in the influence based on adjacency or direct connections in the modeled biological or chemical structure.
4. Training Loop: simple_training_loop_incoming: Simulates training over a specified number of epochs. In each epoch: The apply_triangle_update_incoming function is called to update the feature tensor Z based on the current state of the transformation matrices a, b, and gating matrix g. The matrices a, b, and g are slightly adjusted after each epoch to mimic learning or adaptation, using a simple decrement based on random values scaled by 0.01.
5. Example Usage: Sets up a basic scenario with Ntokens = 5 and Cz = 3, representing a small feature tensor Z. Initializes the tensor and runs the incoming triangle update simulation over a predefined number of epochs, outputting the projections (a, b, g) and the updated tensor Z.
6. Output: After training, the program prints out the final states of the projections a, b, g and the updated feature tensor Z, showing how each token's representation has evolved.

This program essentially provides a framework to model how information might flow and be processed in a network where nodes (tokens) receive contributions from their neighbors, influenced by specific transformation rules encoded in the tensors a, b, and gated by g. It captures an abstract aspect of the computational mechanisms likely employed in PairFormer modules within the AlphaFold architecture.

Clarifying Question

Does the program implement the Triangle Update incoming edges algorithm?

• Yes, the program now implements the Triangle Update for incoming edges correctly. It does this by selecting columns from a and b for each token interaction, effectively flipping the dimensions used in the outgoing update scenario. This matches the operational flow illustrated in the diagram where column elements are used for generating updates based on interactions across different tokens

.

the summation formula update_value += a[k, i, :] * b[k, j, :]? exactly matches the right? side diagram

Triangle Attention

To complete the pairformer review, one needs to speak about the triangle attention which is explained in some detail in this paper. If you are familiar with the attention algorithm, and you have followed through the preceding paragraphs you will also understand how it works. The scheme is the similar as described before, and they set up a triangle attention algorithm for starting nodes, matching the scheme of outgoing edges, and for ending nodes, matching the scheme for incoming edges. I believe it would be redundant to develop additional programs to cover this topic. If you are not familiar with attention, there are excellent references:? my favorite is this one, provided by Google.?

Triangle Attention - Starting Node

Triangle Attention - Ending Node

Summary

I created demonstration programs that replicate certain functionalities of the Pairformer module within the Alpha Fold 3 architecture, specifically focusing on "triangle attention." The document explains both outgoing and incoming triangle updates through Python simulation scripts, aiming to mimic the complex protein folding interactions modeled by Alpha Fold 3.

Two separate Python scripts have been made available in github : one for outgoing edges and another for incoming edges. These scripts are designed to demonstrate how different tokens (atoms or amino acids) influence each other in the protein folding process modeled by Alpha Fold 3.

Core Logic of Triangle Updates:

• For outgoing edges, the program processes the interactions where a token pair influences other tokens based on predefined transformations.
• For incoming edges, the operations are mirrored to reflect how each token pair is influenced by other tokens.

Technical Details:

• Both scripts use three-dimensional tensors a, b, and g to simulate transformation matrices and gating mechanisms that dictate the interaction dynamics among tokens.
• The programs iterate through these tensors to apply the triangle update logic based on element-wise multiplication and aggregation operations, subsequently gated by the matrix g.

Goals: The primary goal is to simplify and explain how transformations and interactions within the Pairformer component of Alpha Fold 3 contribute to its ability to predict protein structures.

Evaluation and Questions: The document also poses questions about whether the programs correctly implement the intended algorithms and matches the conceptual diagrams provided in the original scientific discussions of Alpha Fold 3.

Using LLMs

I have used GPT4 as a programming and Q&A aid throughout this project. It took several iterations to get the bot to write a demo program that realistically matches the algorithms implemented in Alpha Fold. It was even more difficult to get correct answers on how the program works and whether it matches the real algorithm.?

From a programming perspective, you can easily be fooled into an incorrect algorithm if you are not familiar, in this case with tensor operations.

This should be nothing new for LLM users, and it proves once more that the user must be very critical before accepting the system responses as valid ones.Chain-of-thoughts helped, but even so there were many responses that while appearing plausible did not match the ground truth.