Computational Biology

Research notes spanning three related areas: protein folding (Levinthal’s paradox, energy landscape theory, inverse folding with Potts models, dynamics-informed drug design), the 3D point set alignment algorithms that underpin structural comparison (Kabsch, Horn, Arun, Umeyama), and computational metabolic pathway design (graph-grammar chemical space expansion with ILP flow queries).

Protein Folding

Year	Paper	Key Idea
1969	How to Fold Graciously	Levinthal’s paradox: random search cannot explain protein folding
1995	Funnels, Pathways, and Energy Landscapes	Energy landscape theory with folding funnels resolves Levinthal’s paradox
2025	DynamicFlow	Flow matching co-generating ligands and flexible protein pockets
2025	InvMSAFold	Potts model parameter prediction for fast diverse inverse folding

Point Set Alignment

Year	Paper	Key Idea
1976	Kabsch Algorithm	Closed-form optimal rotation via cross-covariance eigendecomposition
1987	Arun et al.: SVD-Based Point Fitting	SVD-based rotation and translation for 3D point sets
1987	Horn’s Quaternion Method	Absolute orientation via unit quaternion eigenvector solution
1988	Horn et al.: Orthonormal Matrices	Polar decomposition alternative to quaternion-based alignment
1991	Umeyama’s Method	Corrected SVD alignment that always produces proper rotations

Metabolic Pathway Design

Year	Paper	Key Idea
2026	Graph Grammar and ILP for Carbon Fixation Pathways	MØD chemical space expansion plus ILP flow queries discover novel autocatalytic carbon fixation cycles

Computational Biology

Graph-grammar expansion of a carbon fixation reaction network with ILP flow queries selecting short autocatalytic cycles

Graph Grammar and ILP for Carbon Fixation Pathways

A graph-grammar cheminformatics workflow expands the carbon fixation reaction network, then uses integer linear programming flow queries to surface short autocatalytic pathways producing Acetyl-CoA and Malate with efficiencies approaching the CETCH cycle.

Computational Biology

Three-panel diagram showing input point sets, SVD factorization of the cross-covariance matrix, and the aligned result

Arun et al.: SVD-Based Least-Squares Fitting of 3D Points

Presents a concise SVD-based algorithm for finding the optimal rotation and translation between two 3D point sets, with analysis of the degenerate reflection case that Umeyama later corrected.

Computational Biology

Diagram showing the polar decomposition of the cross-covariance matrix M into orthonormal factor U and positive semidefinite square root

Horn et al.: Absolute Orientation Using Orthonormal Matrices

The matrix-based companion to Horn’s 1987 quaternion method, deriving the optimal rotation as the orthonormal factor in the polar decomposition of the cross-covariance matrix via eigendecomposition of a 3x3 symmetric matrix.

Computational Biology

Side-by-side comparison showing naive SVD producing a reflected alignment versus Umeyama's corrected proper rotation

Umeyama's Method: Corrected SVD for Point Alignment

Corrects a flaw in prior SVD-based alignment methods (Arun et al., Horn et al.) that could produce reflections instead of rotations under noisy data, and provides a complete closed-form solution for similarity transformations in arbitrary dimensions.

Computational Biology

3D scatter plot showing left and right point sets with rotation axis and quaternion rotation arc

Horn's Method: Absolute Orientation via Unit Quaternions

Derives the optimal rotation between two 3D point sets as the eigenvector of a 4x4 symmetric matrix built from cross-covariance sums, using unit quaternions to enforce the orthogonality constraint.

Computational Biology

3D scatter plot showing source points, target points, and Kabsch-aligned points overlapping the targets

Kabsch Algorithm: Optimal Rotation for Point Set Alignment

A foundational 1976 short communication presenting a direct, non-iterative method for finding the best rotation matrix between two point sets via eigendecomposition of a cross-covariance matrix.

Computational Biology

DynamicFlow illustration showing the transformation from apo pocket to holo pocket with ligand molecule generation

DynamicFlow: Integrating Protein Dynamics into Drug Design

This paper introduces DynamicFlow, a full-atom stochastic flow matching model that simultaneously generates ligand molecules and transforms protein pockets from apo to holo states. It also contributes a new dataset of MD-simulated apo-holo pairs derived from MISATO.

Computational Biology

InvMSAFold generates diverse protein sequences from structure using a Potts model

InvMSAFold: Generative Inverse Folding with Potts Models

InvMSAFold replaces autoregressive decoding with a Potts model parameter generator, enabling diverse protein sequence sampling orders of magnitude faster than ESM-IF1.

Computational Biology

Four types of protein folding energy landscapes from left to right: smooth funnel, rugged funnel with kinetic traps, moat funnel, and champagne glass funnel

Funnels, Pathways, and Energy Landscapes of Protein Folding

This paper resolves Levinthal’s paradox by replacing the single-pathway view with a statistical energy landscape approach. It introduces the concepts of the folding funnel, the glass transition in proteins, and the ‘stability gap’ as a design principle for foldable sequences.

Computational Biology

Protein folding energy landscape funnel showing high-energy unfolded states converging to the native state

How to Fold Graciously: Levinthal's Paradox (1969)

Levinthal’s 1969 perspective paper defined the protein folding paradox by demonstrating the impossibility of random search, establishing the need for kinetic pathways that guide folding faster than thermodynamic equilibration allows.

Protein Folding#

Point Set Alignment#

Metabolic Pathway Design#

Protein Folding

Point Set Alignment

Metabolic Pathway Design