From reductionism to realism: Holistic mathematical modelling for complex biological systems

TL;DR

The paper argues that biological systems resist concise reductionist modelling and calls for a holistic mathematical framework that integrates multi-scale biology with empirical data. It outlines three pillars—rich representational formalisms (e.g., multilayer and annotated networks), simulation-based modelling (ABMs and digital twins), and inverse-problem data-driven dynamics—to bridge mechanistic insight with predictive capability. Key contributions include a principled stance on combining mechanistic and data-driven approaches, a roadmap for scalable inference and model reduction, and links to foundational theories to guide modelling. This framework aims to catalyze predictive, interpretable models of brain and other biological systems, leveraging HPC and generative AI to move toward a quantitative theory of life.

Abstract

At its core, the physics paradigm adopts a reductionist approach, aiming to understand fundamental phenomena by decomposing them into simpler, elementary processes. While this strategy has been tremendously successful in physics, it has often fallen short in addressing fundamental questions in the biological sciences. This arises from the inherent complexity of biological systems, characterised by heterogeneity, polyfunctionality and interactions across spatiotemporal scales. Nevertheless, the traditional framework of complex systems modelling falls short, as its emphasis on broad theoretical principles has often failed to produce predictive, empirically-grounded insights. To advance towards actionable mathematical models in biology, we argue, using neuroscience as a case study, that it is necessary to move beyond reductionist approaches and instead embrace the complexity of biological systems - leveraging the growing availability of high-resolution data and advances in high-performance computing. We advocate for a holistic mathematical modelling paradigm that harnesses rich representational structures such as annotated and multilayer networks, employs agent-based models and simulation-based approaches, and focuses on the inverse problem of inferring system dynamics from observations. We emphasise that this approach is fully compatible with the search for fundamental biophysical principles, and highlight the potential it holds to drive progress in mathematical biology over the next two decades.

Figures (5)

• Figure 1: The spectrum of interaction structures in mathematical models. Models of physical system exist on a spectrum from the regular to the random. For regular structures, continuum assumptions are often valid leading to a low number of degrees of freedom. In the random case, behaviour is typically described by a statistical distribution rather than individual elements. Between these extrema are complex systems, where interactions are heterogeneous leading to high-dimensional dynamics.
• Figure 2: Richer representations for complex systems. Networks have become the dominant paradigm for representing interactions in complex systems. However, a range of extensions exist that have the expressiveness to represent more complex interactions as well as their dynamics. Examples include hypergraphs, simplicial complexes, as well as temporal, multilayer and annotated networks.
• Figure 3: Spatiotemporal scales in Alzheimer's disease. Complex biological processes, such as the development of neurodegenerative diseases such as Alzheimer's disease (AD), evolve over a range of interacting spatiotemporal scales. Focussing on AD, protein misfolding ultimately leads to the spread of toxic tau and amyloid-$\beta$ proteins through axonal fibres, which is partially cleared through cell regulation. Ultimately, this leads to cell death and pathological neural dynamics. This manifests in tau and amyloid pathology and, finally, cognitive decline which can be combatted with therapy. A MM of this process must take into account the full range of spatiotemporal scales in order to build a mechanistic understanding of AD.
• Figure 4: Simulation-based modelling. A. Agent-based models are powerful tools for describing the behaviour of heterogeneous populations of agentic units. Examples include the collective dynamics of swarms and flocks, where each agent can be modelled with its own dynamical properties. B. Digital twins are simulation-based models of physical systems implemented in a computer. One application of biomedicine is to use personalised recordings of neural activity or heartbeat data to calibrate individual in-silico models. Ultimately, this will lead to a virtual version of the biological system where interventions can be tested via simulation.
• Figure 5: Latent representations for observed dynamical systems. A. High-dimensional data from complex biological systems is often clustered onto a low-dimensional manifold. Using data science techniques such as variational auto-encoders, we can discover the low-dimensional latent dynamics of the process. B. Following the theory of the Koopman operator, nonlinear dynamics can be 'lifted' into high-dimensional spaces where the dynamics are approximately linear. This projection can be achieved with deep learning techniques such as neural networks.