# Direct Phasing of Protein Crystals with Continuous Iterative Projection Algorithms and Refined Envelope Reconstruction

## TL;DR

This paper introduces new algorithms and methods to improve the accuracy of solving protein crystal structures with limited solvent content.

## Contribution

The study introduces continuous iterative projection algorithms and a refined envelope reconstruction method to enhance direct phasing of protein crystals.

## Key findings

- The refined envelope scheme increased success rates of continuous algorithms by 45.7% and classical algorithms by 60.5%.
- Integrating a genetic algorithm improved success rates by 2.5-fold and reduced phase errors by 6.83°.
- The new methods enabled atomic models with backbone RMSD typically below 0.5 Å compared to PDB structures.

## Abstract

Direct methods provide a model-free approach to solving the crystallographic phase problem and deliver unbiased atomic structures. However, conventional iterative projection algorithms such as Hybrid Input–Output (HIO) face two critical challenges: discontinuous density modification at the protein-solvent boundary and inaccurate molecular envelope reconstruction that fails to account for trapped solvent, particularly in crystals with solvent content approaching the lower limits of direct phasing applicability. We introduced four continuous iterative projection algorithms, including our improved continuous version, which implements smooth density modification at protein-solvent interfaces. To address envelope inaccuracy, we developed a two-step refined reconstruction scheme using sequential large-radius and small-radius Gaussian filters to identify trapped solvent molecules within surface cavities and internal channels. This scheme enhances the performance of both continuous and classical algorithms, including HIO, the difference map, and our improved versions. Benchmarking on 28 protein structures (solvent contents 55–78%, resolutions 1.46–3.2 Å, reported R-factor less than 0.22) showed that the refined envelope scheme increased average success rates of continuous algorithms by 45.7% and classical algorithms by 60.5%. The performance of continuous algorithms and improved classical algorithms proved comparable to the well-established HIO algorithm, forming a top-tier group that exceeded other classical algorithms. Integrating a genetic algorithm co-evolution strategy further enhanced average success rates by approximately 2.5-fold and accelerated convergence through population-wide information sharing. Although the success rate correlates with solvent content, our strategy improved success probability at any given solvent level, extending the practical boundaries of direct methods. The high success rate enabled averaging of multiple independent solutions, which reduced mean phase error by approximately 6.83° and yielded atomic models with backbone root-mean-square deviation (RMSD) typically below 0.5 Å relative to structures reported in the Protein Data Bank (PDB). This work introduces novel algorithms, a refined envelope reconstruction methodology, and an effective optimization strategy with genetic algorithm evolution. The complete framework enhances the capability and reliability of direct methods for phasing protein crystals with limited solvent content and provides a toolkit for addressing challenging cases in structural biology.