Beyond Distance: Mapping Life with Mathematical Spaces

Author: Denis Avetisyan

A new mathematical framework moves beyond traditional distance-based models to provide a more flexible and nuanced representation of biological systems and their dynamic states.

This review proposes context-dependent, locally convex spaces to model biological processes, focusing on weak convergence and irreversible dynamics within admissible state spaces.

While physics leverages carefully constructed mathematical spaces to represent dynamical laws, biology lacks a unifying spatial framework capable of capturing its inherent complexity. In ‘Towards mathematical spaces for biological processes’, we introduce a novel mathematical formalism-built upon locally convex spaces indexed by context-to represent biological states and dynamics, addressing limitations of traditional metric-based approaches. This framework formalizes key biological features-including partial observability, irreversibility, and context-dependent stability-through admissible sets, observation maps, and proximity defined by families of seminorms. Can this approach facilitate quantitatively rigorous analyses of biological systems and ultimately reveal emergent principles governing life’s robustness and adaptability?

Beyond Static Snapshots: Embracing the Irreversible Nature of Life

Conventional biological models frequently operate under the assumption of equilibrium, portraying cells as static entities existing in defined states. However, this approach often overlooks a fundamental characteristic of life: its inherent irreversibility. Cellular processes are rarely, if ever, perfectly reversible; metabolic reactions proceed in a specific direction, signaling pathways activate and deactivate along a defined temporal course, and developmental programs unfold with a clear beginning and end. This unidirectional flow isn’t simply a matter of reaction rates; it’s deeply embedded in the physics and chemistry of biological systems, driven by entropy and the dissipation of energy. Consequently, a cell’s present state isn’t solely determined by its current conditions, but also by its past experiences and the trajectory it has already followed. Ignoring this historical dependence risks creating incomplete and potentially misleading representations of biological reality, hindering the development of truly predictive models.

Living systems are rarely defined by a singular, static condition; instead, biological reality resides in the continuous process of change. Cells don’t merely be cancerous or healthy, but rather progress towards those states through a series of molecular events influenced by both internal genetic programs and external environmental cues. This perspective acknowledges that a cell’s present state isn’t solely determined by its current molecular composition, but is fundamentally shaped by its past experiences – the sequence of signals it has received, the stresses it has endured, and the metabolic pathways it has previously navigated. Consequently, understanding biological function demands a shift from characterizing isolated states to deciphering the dynamic pathways and irreversible transitions that define life’s unfolding narrative, recognizing that history and context are integral components of any living system’s identity.

A comprehensive understanding of biological systems necessitates a shift from static descriptions to dynamic models that prioritize the process of change. Cells aren’t defined by what they are, but by how they become, and this ‘becoming’ is fundamentally directional, influenced by both internal cues and the external environment. Traditional approaches often fail to capture this inherent irreversibility; a cell’s history – the sequence of stimuli it has encountered – demonstrably shapes its present and future states. Therefore, a robust framework must explicitly incorporate these environmental influences and acknowledge that biological pathways aren’t simply reversible loops, but rather trajectories unfolding in time, guided by physical laws and responding to contextual signals. This perspective allows for predictions not just about a cell’s current condition, but about its potential future states and its response to novel challenges, ultimately offering a more nuanced and accurate representation of biological reality.

Mapping the Landscape: A Mathematical Foundation for Biological States

A Biological State Space is a mathematical construct used to represent the complete set of all possible conditions or configurations a biological system – such as a cell, tissue, or organism – can inhabit. This space isn’t limited to easily measurable parameters; it encompasses all relevant variables defining the system’s condition, whether those variables are gene expression levels, protein concentrations, or other biophysical properties. Formally, each point within this space represents a unique biological state, and the dimensionality of the space is determined by the number of variables considered. The use of such a space allows for a complete and quantifiable representation of system variability, providing a framework for analyzing transitions between states and predicting system behavior under different conditions.

Biological state spaces are formally constructed as Locally Convex Spaces (LCS). These spaces allow for the definition of continuous measures of distance or dissimilarity between biological states using [latex]\text{Seminorms}[/latex]. Unlike standard Euclidean distance, seminorms do not require the origin to be a fixed point, enabling the comparison of states based on specific features or characteristics without necessarily defining a ‘zero’ or baseline state. Multiple seminorms can be defined within a single LCS, each capturing a different aspect of state similarity. The combination of these seminorms defines the topology of the space, providing a rigorous mathematical foundation for quantifying and comparing the relative proximity of different biological states.

The Biological State Space framework utilizes context-dependent proximity, recognizing that the similarity between biological states is not fixed but varies with environmental conditions. This means the ‘distance’ between two states is not an inherent property, but is defined relative to specific contexts. This approach is demonstrated by its ability to capture cell reclassification rates of 25% or greater in cells undergoing Epidermal Growth Factor Receptor Tyrosine Kinase Inhibitor (EGFR-TKI) treatment; these rates reflect changes in cellular identity triggered by the treatment, and are accurately modeled by the framework’s sensitivity to contextual shifts in state proximity.

Tracing the Path: Analyzing System Dynamics and Cellular Trajectories

Understanding cellular behavior requires analyzing Trajectories, which represent the time-ordered sequence of a cell’s states as it evolves within the Biological State Space. This space is a multi-dimensional representation of all possible cellular conditions, defined by measurable variables such as gene expression levels, protein concentrations, and metabolic activity. Rather than focusing on static snapshots, trajectory analysis tracks the dynamic changes in these variables over time, allowing researchers to characterize the progression of cells through different functional states. These trajectories are constructed from time-series data, often generated through single-cell RNA sequencing or other high-throughput omics technologies, and provide insights into developmental processes, disease progression, and cellular responses to stimuli. Analyzing these trajectories allows for the identification of key transitions, bifurcations, and attractors within the state space, providing a more comprehensive understanding of cellular dynamics than traditional static analyses.

Traditional definitions of stability in dynamical systems often focus on convergence to a fixed point; however, cellular stability is more accurately characterized by the tendency of a system to remain within a defined ‘Admissible Set’ of states. This set represents a range of physiological or functional states considered viable for the cell, rather than a single, static endpoint. A system exhibiting stability, therefore, doesn’t necessarily reach a specific point, but consistently returns to or remains within the boundaries of this admissible set following perturbation. This concept is crucial because biological systems are inherently noisy and dynamic; maintaining function relies on remaining within a functional range, not necessarily achieving a singular, perfectly defined state.

Stability analysis utilizes the criteria of Neighborhood Inclusion and Weak Convergence to assess cellular states. Neighborhood Inclusion determines stability by evaluating whether a trajectory remains within a defined proximity, or ‘neighborhood’, of an initial state; observed stabilization occurs within 24-48 hours based on this criterion. Weak Convergence, a mode of convergence allowing for some deviation, is then employed to analyze long-term trends in these trajectories, indicating the system’s overall behavior. Importantly, neighborhood inclusion and the resulting stabilization consistently precede transcriptomic convergence, suggesting this proximity-based assessment is an early indicator of broader systemic stabilization.

The Nuance of Resilience: Modeling Complexity and the Limits of Observation

Biological systems are frequently characterized by a phenomenon known as degeneracy, where a surprising diversity of underlying mechanisms and states can converge on the same observable outcome. This isn’t a flaw in design, but rather a robust feature that enhances adaptability and resilience. Consider, for example, that multiple different genetic mutations can sometimes result in identical phenotypic expressions, or that several distinct neural circuits might elicit the same behavioral response. This redundancy provides a buffer against perturbations; should one pathway fail, others can compensate, maintaining functionality. Degeneracy suggests that a one-to-one correspondence between genotype and phenotype, or between internal state and external output, is rarely accurate; instead, a complex landscape exists where numerous routes lead to the same result, creating a rich and nuanced biological reality.

The translation of a living system’s internal workings-its biological state space-into measurable data relies on what can be termed an ‘Observation Map’. This map isn’t a perfect reflection; it’s inherently reductive. Biological reality exists as a complex, high-dimensional space encompassing myriad internal variables. However, experiments can only access a limited subset of these variables – perhaps cell survival, protein levels, or specific behaviors. Consequently, the observation map collapses this rich internal state into a lower-dimensional representation, inevitably discarding information. This loss isn’t a flaw, but a fundamental constraint of measurement; multiple distinct internal states can manifest as the same observable output. Understanding this inherent limitation is crucial for interpreting experimental results, as observed patterns represent probabilities across a hidden state space, not a one-to-one correspondence with specific biological conditions.

Biological research frequently confronts the challenge of interpreting data generated from systems exhibiting inherent degeneracy – the capacity for diverse internal states to yield identical outputs. Recent investigations demonstrate that the level of this degeneracy is acutely sensitive to the granularity of observation; when researchers focus solely on survival – a coarse readout – degeneracy increases two to three times compared to studies employing multi-marker observation. This suggests that limiting the observable parameters can mask the true complexity of a biological system, leading to oversimplified conclusions. Recognizing this impact of observation granularity is crucial for nuanced data interpretation, allowing for a more accurate appreciation of the underlying biological reality and preventing the erroneous assumption of a single, definitive state.

Toward a Predictive Biology: Unifying Theory with Single-Cell Data

The concept of a ‘Fiber Bundle’ represents a crucial advancement in understanding cellular states by formally establishing a connection between the context space – encompassing external signals and environmental factors – and the state space, which defines the internal condition of a cell. This mathematical framework treats cellular states not as isolated points, but as cross-sections of a higher-dimensional bundle, where each ‘fiber’ represents a possible internal state for a given external context. By rigorously linking these spaces, the Fiber Bundle allows researchers to predict how changes in the cellular environment will influence internal states and vice versa, moving beyond simple correlations to a more mechanistic understanding of cellular behavior. This approach provides a powerful tool for analyzing complex single-cell data and interpreting how cells respond and adapt to diverse stimuli, ultimately offering insights into phenomena like drug resistance and phenotypic heterogeneity.

Investigating cellular behavior requires robust data, and single-cell analysis offers an increasingly detailed view of complex biological systems. Data derived from models like the PC9 cell line, representing EGFR-mutant Non-Small Cell Lung Cancer (NSCLC), is particularly valuable for validating computational models of cellular resilience. This approach allows researchers to move beyond population averages and examine the heterogeneity within cell populations, revealing how individual cells respond to stress and potentially develop drug tolerance. By comparing model predictions with observations from these single-cell experiments, scientists can refine their understanding of the underlying mechanisms governing cell fate and identify potential targets for therapeutic intervention, ultimately improving the accuracy and predictive power of these models.

Early stabilization within cellular states suggests a remarkable capacity for resilience and adaptation, as evidenced by recent analyses of single-cell data. Researchers found that cells rapidly converge towards stable phenotypes, and, crucially, observed a rare fraction – approximately 0.3% – exhibiting tolerance to perturbation. This frequency aligns closely with previously reported baseline levels of Drug-Tolerant Persisters (DTPs) identified in the PC9 cell line, a model for EGFR-mutant Non-Small Cell Lung Cancer. While powerful, this analytical approach isn’t without limitations; a compression artifact rate of 5-10% was noted, indicating a degree of data redundancy that must be considered when interpreting the results, yet the observed convergence towards stability provides valuable insight into the mechanisms underlying cellular robustness.

The pursuit of mathematically rigorous frameworks for biological processes necessitates a careful consideration of underlying assumptions. This paper’s exploration of locally convex spaces, particularly their capacity to model context-dependent biological states, echoes a fundamental philosophical inquiry. As René Descartes famously stated, “Doubt is not a pleasant condition, but it is necessary for a clear understanding.” Similarly, the authors challenge conventional metric-based approaches, embracing a more nuanced system to represent irreversible dynamics and observational maps. This deliberate questioning and refinement of existing paradigms underscores the elegance of a well-defined system, where form and function harmonize to reveal deeper truths about the natural world.

The Horizon Beckons

The pursuit of a suitable mathematical language for biology perpetually feels like chiseling at granite. Traditional metric spaces, while serviceable, often impose a rigidity that living systems simply do not possess. This work, by embracing the more pliable structure of locally convex spaces, offers a glimpse of a potentially more harmonious representation-one where equivalence isn’t merely sameness, but a resonant similarity sculpted by context. Yet, the devil, predictably, resides in the details. Translating the abstract elegance of these spaces into predictive models requires grappling with the inherent noisiness of biological observation, and the difficulty of defining truly ‘admissible’ sets without falling into overly restrictive assumptions.

The challenge now isn’t merely to map biological processes onto these spaces, but to allow the spaces themselves to sing – to reveal hidden dynamics and emergent properties. Weak convergence, so central to the mathematical framework, demands careful interpretation; a convergence in this abstract realm does not automatically guarantee a biologically meaningful outcome. The true test will lie in applying these tools to systems where irreversibility is paramount – where the arrow of time is not a mathematical convenience, but a fundamental aspect of life itself.

One anticipates that future work will focus on developing computational methods for navigating these high-dimensional, context-dependent spaces. Perhaps, a fruitful avenue lies in exploring connections with information theory-measuring the ‘distance’ between biological states not by Euclidean distance, but by the information required to distinguish between them. Ultimately, the goal isn’t to simply describe life, but to understand the principles that allow it to flourish, even-and perhaps especially-in the face of imperfection.

Original article: https://arxiv.org/pdf/2601.15854.pdf

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