From Patterson Maps to Atomic Coordinates: Training a Deep Neural Network to Solve the Phase Problem for a Simplified Case (2003.13767v1)

Abstract: This work demonstrates that, for a simple case of 10 randomly positioned atoms, a neural network can be trained to infer atomic coordinates from Patterson maps. The network was trained entirely on synthetic data. For the training set, the network outputs were 3D maps of randomly positioned atoms. From each output map, a Patterson map was generated and used as input to the network. The network generalized to cases not in the test set, inferring atom positions from Patterson maps. A key finding in this work is that the Patterson maps presented to the network input during training must uniquely describe the atomic coordinates they are paired with on the network output or the network will not train and it will not generalize. The network cannot train on conflicting data. Avoiding conflicts is handled in 3 ways: 1. Patterson maps are invariant to translation. To remove this degree of freedom, output maps are centered on the average of their atom positions. 2. Patterson maps are invariant to centrosymmetric inversion. This conflict is removed by presenting the network output with both the atoms used to make the Patterson Map and their centrosymmetry-related counterparts simultaneously. 3. The Patterson map does not uniquely describe a set of coordinates because the origin for each vector in the Patterson map is ambiguous. By adding empty space around the atoms in the output map, this ambiguity is removed. Forcing output atoms to be closer than half the output box edge dimension means the origin of each peak in the Patterson map must be the origin to which it is closest.

• The paper introduces a deep learning approach using a convolutional neural network to solve the phase problem by translating Patterson maps into atomic coordinates for a simplified case of 10 atoms.
• The method achieved an average positional accuracy of 0.283 pixels for inferred atomic positions in the simplified synthetic dataset.
• While promising for small atom sets, further research is needed to scale this deep learning approach to larger, more complex molecular structures found in real experimental data.

Solving the Phase Problem in X-ray Crystallography Using Deep Learning

The paper "From Patterson Maps to Atomic Coordinates: Training a Deep Neural Network to Solve the Phase Problem for a Simplified Case" presents a novel approach for addressing the phase problem inherent in X-ray crystallography. By leveraging deep learning techniques, specifically convolutional neural networks (CNNs), the author aims to translate Patterson maps into atomic coordinates, showcasing the potential of artificial intelligence in structural determination.

Overview and Methodology

The study focuses on a simplified scenario involving 10 randomly positioned atoms. It employs a synthetic dataset to train a neural network for inferring the atomic arrangement from Patterson maps, which are derived from the diffraction magnitudes squared and represent vectors between atom pairs. The process can be seen as a deconvolution problem similar to other imaging tasks addressed by neural networks, such as super-resolution.

Key to the training protocol is ensuring the Patterson map uniquely describes the corresponding atomic coordinates. To achieve this, three main strategies are employed:

1. Translation Invariance: The atomic coordinates output by the network are centered to address the invariance of Patterson maps to translation.
2. Centrosymmetry Ambiguity: By including both original and centrosymmetric atoms in the training output, the network is presented a non-redundant mapping of coordinates.
3. Vector Origin Ambiguity: Adding empty space around the atoms confines them such that Patterson vectors can be faithfully interpreted with their closest origin.

The network architecture utilizes 12 3D convolutional layers within the Keras framework and is trained over numerous epochs, adjusting hyperparameters to optimize performance. The CNN's ability to generalize is evidenced by comparable losses in training and validation datasets, indicating efficacy beyond mere memorization.

Results and Implications

The network's ability to generalize is further confirmed through comparative analysis of inferred and true density maps—achieving an average positional accuracy of 0.283 pixels. Additionally, the necessity of simultaneously training on both original and centrosymmetric atoms is underscored, as omitting this step resulted in significant performance degradation. Similarly, the need for empty space surrounding atoms in training data was validated, with increased box size correlating with diminished training efficacy.

Conceptually, this work posits that a neural network could substitute traditional direct methods in crystallography by effectively solving the probabilistic equations associated with phase determination. However, while the results demonstrate promise for small, random atom sets, extrapolation to larger, non-random molecular structures remains uncertain.

Future Directions

The findings open several avenues for future research. Scaling the network for larger atom assemblages and testing on actual experimental data stands as a primary objective. Here, the transition from synthetic to empirical scenarios will test the robustness of the outlined strategies, such as the inclusion of empty space and the handling of centrosymmetry. Moreover, exploring whether the network functions as a memory device might reveal insights into its generalization capabilities, necessitating tests that isolate parts of the configuration space.

In conclusion, this study lays foundational work toward a deep learning-based resolution to the phase problem in X-ray crystallography, with potential implications for the structural analysis of more complex molecular systems. While the current scope is limited, ongoing advancements in neural network methodologies and computational resources may extend the practicality of this approach, heralding significant strides in crystallographic techniques and beyond.