Mapping Meaning: How Embedding Geometry Reveals Protein Relationships

Author: Denis Avetisyan

New research shows that the geometric properties of language model embeddings can expose underlying structure in protein sequences, offering a powerful way to assess representation quality.

Dimensionality reduction applied to embeddings derived from a lung cancer dataset reveals inherent clusterings within the data, suggesting discernible patterns potentially indicative of distinct patient subtypes or disease characteristics.

Analyzing δ-hyperbolicity and ultrametricity in large language model embeddings provides insights into protein sequence relationships and improves classification and clustering tasks.

While large language models excel at various tasks, understanding the inherent structure of their learned representations remains a challenge. This is addressed in ‘Uncovering Hierarchical Structure in LLM Embeddings with $δ$-Hyperbolicity, Ultrametricity, and Neighbor Joining’, which investigates the geometric properties of LLM embeddings using metrics like δ-hyperbolicity and ultrametricity to reveal underlying hierarchical organization. The authors demonstrate that the degree to which these embeddings exhibit tree-like or hierarchical characteristics correlates with performance on downstream machine learning tasks, particularly in protein sequence analysis. Could these geometric insights provide a new framework for evaluating and improving the quality of LLM embeddings across diverse applications?

Beyond Sequences: Mapping the Protein Landscape

For decades, protein research has been largely defined by its focus on amino acid sequences – the linear order of building blocks that constitute these essential molecules. While sequence information provides a foundational understanding, it often fails to capture the complete picture of a protein’s behavior. Proteins don’t function in isolation; their three-dimensional structures and interactions with other molecules are paramount to their roles within a cell. Consequently, relying solely on sequence similarity can miss subtle yet critical relationships between proteins that share functional similarities but have diverged in their amino acid composition. This limitation poses a significant challenge to accurately predicting protein behavior, designing effective drugs, and fully unraveling the complexities of biological systems, highlighting the need for analytical methods that move beyond the confines of sequence-based comparisons.

The reliance on protein sequence alone presents significant challenges when attempting to model biological processes. Because protein function arises not simply from what amino acids are present, but also how those amino acids fold and interact, sequence similarity often fails to capture the full spectrum of relationships governing cellular behavior. This is especially true within complex systems, where proteins participate in intricate networks of interactions – pathways, regulatory loops, and signaling cascades – that are sensitive to even subtle changes in protein structure or dynamics. Consequently, predictions based solely on sequence can be misleading, overlooking crucial functional links and hindering a complete understanding of how proteins orchestrate life’s processes. A more holistic approach, accounting for these higher-order relationships, is therefore essential for accurately modeling and ultimately predicting protein behavior in vivo.

Current protein analysis frequently prioritizes linear amino acid sequences, a practice that often obscures the intricate ways proteins relate functionally and structurally. Researchers are increasingly focused on representing proteins not simply as strings of building blocks, but as points within a multi-dimensional ‘metric space’-a mathematical construct where distances reflect the degree of similarity. This innovative approach allows for the identification of relationships that sequence alone would miss; proteins with vastly different sequences can cluster closely if they share similar shapes, functions, or interaction patterns. By mapping proteins in this way, scientists can leverage geometric principles to predict protein behavior, discover novel connections within biological networks, and ultimately gain a more holistic understanding of cellular processes. The resulting spatial representations enable computational tools to identify distant evolutionary relatives, predict protein-protein interactions, and even design new proteins with desired characteristics, moving beyond the limitations of sequence-based comparisons.

Geometric Underpinnings: Defining Tree-Like Structure

Ultrametricity, in the context of protein embeddings, describes a hierarchical organization where distances satisfy the strong triangle inequality – effectively defining a perfect tree structure. In an ultrametric space, for any three points x, y, and z, the distance between x and y is always less than or equal to the maximum of the distances from x to z and from y to z. While theoretically ideal for representing evolutionary relationships or structural hierarchies, real-world protein embeddings generated from complex datasets rarely, if ever, perfectly satisfy this criterion due to noise, incomplete data, and the inherent complexity of protein families. Consequently, deviation from perfect ultrametricity is the norm, necessitating the use of more flexible metrics to assess the degree of tree-likeness present in these embeddings.

δ-hyperbolicity provides a relaxation of the strict requirements for ultrametricity when assessing the tree-like organization of embedding spaces. While perfect ultrametricity demands a complete hierarchical structure, δ-hyperbolicity quantifies the extent to which an embedding deviates from this ideal. A lower δ-hyperbolicity value indicates a stronger resemblance to a tree structure, acknowledging that real-world protein embeddings rarely exhibit perfect hierarchical organization. The metric is calculated using the Gromov product, which assesses the degree of non-positive curvature within the embedding space; thus, δ-hyperbolicity effectively measures the “thinness” of triangles formed by points in the embedding, with thinner triangles corresponding to more tree-like behavior.

δ-hyperbolicity is calculated using the Gromov Product, a metric that quantifies the extent of negative curvature present within the embedding space. Specifically, the Gromov Product of two points, $x$ and $y$ , is defined as $\frac{1}{2} (d(x,o)^2 + d(y,o)^2 - d(x,y)^2)$ , where $o$ is a reference point and $d(x,y)$ represents the distance between points $x$ and $y$ . A lower average Gromov Product across all point pairs indicates a more tree-like structure, as it reflects a greater prevalence of short paths and a reduced degree of non-positive curvature; conversely, higher values suggest a more complex, potentially graph-like, organization of the embedding space.

Analysis of ProtT5 embeddings generated from the PDB186 dataset revealed a δ-hyperbolicity value of 0.0418. This metric quantifies the deviation from an ideal tree structure, with lower values indicating a stronger resemblance to a true tree. In comparison, SeqVec embeddings, when evaluated on the same PDB186 dataset, exhibited a significantly higher δ-hyperbolicity of 3.2018. This substantial difference demonstrates that ProtT5 embeddings possess a more pronounced tree-like organization than those produced by SeqVec, suggesting a more efficient representation of hierarchical relationships within the protein data.

Ultrametricity, a measure of how closely a space resembles a perfect tree structure, was quantified for ProtT5 and SeqVec embeddings using the PDB186 dataset. ProtT5 demonstrated a significantly lower ultrametricity score of 0.1301 compared to SeqVec’s 16.6730. This substantial difference indicates that the geometric organization of ProtT5 embeddings more closely approximates an ideal tree structure than that of SeqVec, suggesting a more hierarchical and organized representation of protein sequences within the embedding space.

A t-SNE visualization of embeddings from the PDB 186 dataset reveals distinct clusters, best observed in color, representing underlying structural groupings.

Embedding Validation: Clustering and Biological Relevance

Protein embeddings generated by SeqVec, ESM-2, ProtT5, and TAPE were assessed for quality using unsupervised machine learning techniques. Specifically, k-means and Agglomerative Clustering algorithms were employed to determine the extent to which these embeddings could group proteins with similar characteristics without prior knowledge of their function or structure. The resulting cluster formations were then analyzed to evaluate the embedding’s ability to represent meaningful relationships within the protein sequence data, providing a quantitative measure of embedding quality prior to application in downstream tasks.

The PDB186 dataset was utilized to evaluate the capacity of generated protein embeddings to distinguish between interacting and non-interacting protein pairs. This benchmark dataset comprises protein sequences and associated data indicating DNA-protein binding interactions; successful differentiation by the embeddings suggests their effectiveness in representing biologically relevant relationships. Performance on PDB186 served as a proxy for assessing the embeddings’ ability to capture information crucial for understanding protein-DNA interactions, a fundamental process in cellular function.

Evaluation of the ProtT5 protein embeddings utilized Logistic Regression on the PDB186 dataset, a benchmark for DNA-protein binding prediction, resulting in a Receiver Operating Characteristic Area Under the Curve (ROC AUC) score of 0.7968. This metric quantifies the model’s ability to distinguish between interacting and non-interacting protein pairs, with higher values indicating better performance. The PDB186 dataset consists of experimentally validated DNA-protein interactions, providing a reliable basis for assessing the embedding’s capacity to capture biologically relevant information regarding protein binding.

Protein embeddings were evaluated for biological relevance by applying them to datasets containing protein sequences linked to breast and lung cancer activity. This assessment utilized Receiver Operating Characteristic (ROC) Area Under the Curve (AUC) as a performance metric. Results indicated that ESM2 embeddings achieved a ROC AUC of 0.8568 on the lung cancer dataset and 0.8385 on the breast cancer dataset, demonstrating the potential of these embeddings to differentiate between protein sequences associated with cancerous and non-cancerous activity.

Evaluation of ESM2 protein embeddings on datasets related to cancer activity yielded ROC AUC scores of 0.8568 for lung cancer and 0.8385 for breast cancer. These scores were obtained through predictive modeling using the ESM2 embeddings as input features, demonstrating the potential of these embeddings to differentiate between protein sequences associated with cancerous and non-cancerous conditions. The ROC AUC metric quantifies the model’s ability to distinguish between true positives and false positives, with higher values indicating improved performance.

Evaluation of protein embeddings generated by SeqVec, ESM-2, ProtT5, and TAPE revealed consistent clustering patterns, indicating that sequences with similar functional roles are represented closely in the embedding space. This was further supported by performance on biological datasets; specifically, a ROC AUC of 0.7968 was achieved using ProtT5 embeddings on the PDB186 DNA-protein binding prediction dataset. Additionally, ESM2 embeddings demonstrated a ROC AUC of 0.8568 on a lung cancer activity dataset and 0.8385 on a breast cancer activity dataset, suggesting these embeddings effectively capture information relevant to cancer-associated protein function and provide a quantifiable measure of embedding quality in a biologically relevant context.

A t-SNE visualization reveals distinct clusters within the Breast Cancer dataset embeddings, suggesting successful feature differentiation.

Beyond Prediction: A New Lens for Biological Understanding

The creation of accurate protein embeddings-numerical representations of proteins in a high-dimensional space-offers a powerful new approach to deciphering the complex web of life. These embeddings don’t simply catalog known interactions; rather, by mapping proteins based on shared characteristics and evolutionary relationships, they reveal previously unappreciated connections. Proteins appearing close together in this ‘embedding space’ are likely to participate in similar biological processes or even physically interact, allowing researchers to predict novel protein partnerships and functional associations with remarkable accuracy. This computational strategy bypasses the limitations of traditional experimental methods, offering a scalable and efficient way to explore the proteome and uncover hidden layers of biological organization, ultimately accelerating discoveries in areas like disease mechanisms and drug development.

Visualizing protein embeddings within a defined metric space offers a powerful new lens through which to examine the intricate organization of proteomes. This approach doesn’t merely catalog proteins, but maps their relationships based on shared features learned from vast datasets, revealing previously hidden patterns. Researchers are now able to observe how proteins cluster, indicating functional relatedness or evolutionary connections, and identify outliers that may represent novel mechanisms or adaptations. The resulting ‘maps’ of proteomes allow for the tracking of evolutionary trajectories, the identification of key proteins driving biological processes, and the detection of subtle shifts in protein function across different species or conditions, ultimately providing a deeper understanding of the fundamental principles governing life.

Protein embeddings are proving to be powerfully predictive when incorporated into machine learning frameworks. Researchers are demonstrating that these numerical representations, capturing complex relationships between proteins, significantly enhance the accuracy of predicting a protein’s three-dimensional structure – a longstanding challenge in biology. Beyond structure, these embeddings also facilitate the prediction of protein function, identifying likely enzymatic activities or cellular roles with increasing reliability. Crucially, the integration of protein embeddings extends to pharmaceutical applications, enabling the prediction of how proteins will respond to various drug candidates, potentially accelerating drug discovery and personalized medicine by identifying promising compounds and anticipating therapeutic efficacy. This ability to model complex biological interactions through machine learning holds substantial promise for unraveling the intricacies of life and developing targeted therapies.

The creation of robust protein embeddings represents a significant leap toward systems-level biological understanding, moving beyond the study of individual components to encompass the intricate web of interactions that define life. This foundational work promises to expedite therapeutic development by providing a powerful framework for identifying potential drug targets and predicting treatment efficacy. Researchers can now leverage these embeddings to model complex biological processes with unprecedented accuracy, simulating the effects of genetic mutations or environmental changes on cellular function. Consequently, the path from basic research to clinical application is substantially shortened, offering the potential for rapid innovation in areas such as personalized medicine and disease intervention. This approach doesn’t merely address symptoms; it seeks to understand and modulate the underlying mechanisms of illness, paving the way for genuinely preventative and curative strategies.

The pursuit of elegant embeddings, as this paper on δ\delta-hyperbolicity demonstrates, feels perpetually Sisyphean. Researchers meticulously analyze geometric properties, hoping to unlock better representations, yet the underlying chaos of biological sequences always reasserts itself. It’s a constant cycle of refinement, only to find that production data introduces unforeseen distortions. As Linus Torvalds once said, “Talk is cheap. Show me the code.” This obsession with theoretical purity-ultrametricity and ideal hierarchies-will inevitably collide with the messy reality of protein folding and evolutionary divergence. Everything new is just the old thing with worse docs, and this pursuit of mathematically ‘perfect’ embeddings will likely be no different.

What’s Next?

The pursuit of geometric signatures within the latent space of large language models, as demonstrated by analyses of δ\delta-hyperbolicity and ultrametricity, feels less like discovery and more like a refined articulation of inevitable compromise. These metrics offer a quantifiable lens through which to assess embedding quality, but the underlying assumption-that biological or semantic similarity neatly maps onto geometric properties-will undoubtedly encounter the messy realities of production data. Every metric optimized will one day be optimized back, tweaked to accommodate anomalies and edge cases until the original signal is lost in the noise.

Future work will likely focus on extending these geometric analyses beyond protein sequencing, applying them to other domains where LLMs generate embeddings. However, the real challenge isn’t scaling the technique, but acknowledging its inherent fragility. Architecture isn’t a diagram; it’s a compromise that survived deployment. The field should begin documenting failure modes – the types of data or tasks where these geometric properties break down – with the same rigor currently devoted to reporting positive correlations.

Ultimately, this research isn’t about finding the ‘correct’ embedding space, but about building tools to diagnose and resuscitate hope when the inevitable distortions occur. The search for an ideal representation is a ghost chase. The work lies in gracefully handling the imperfections-and the compromises-that will always remain.

Original article: https://arxiv.org/pdf/2512.20926.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Beyond Sequences: Mapping the Protein Landscape

Geometric Underpinnings: Defining Tree-Like Structure

Embedding Validation: Clustering and Biological Relevance

Beyond Prediction: A New Lens for Biological Understanding

What’s Next?

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