I think biology needs to become more mathematical. Not in the sense that every biologist should become a mathematician, but in the sense that understanding the mathematics behind a problem should be as important as knowing which experimental technique or computational tool to use. We are now able to measure biological systems at a scale that would have been difficult to imagine a few decades ago. Single-cell sequencing, spatial transcriptomics and multi-omics technologies produce enormous amounts of data. Yet collecting more measurements does not necessarily mean that we understand the systems producing them.

What concerns me is how easily data analysis can become a process of applying established methods to increasingly complex datasets. We learn how to normalise expression matrices, reduce dimensionality, cluster cells, infer trajectories and identify differentially expressed genes. These techniques are useful, but they are not substitutes for understanding the problem mathematically. A method can produce a convincing result while relying on assumptions that are poorly suited to the biological question.

I believe the next generation of computational biologists will need to do more than apply existing statistical methods. They will need enough mathematical understanding to recognise when a problem has been framed incorrectly, to identify what current methods cannot answer and to develop new approaches when necessary. Biology has many questions for which the correct mathematical description is not yet obvious. That is precisely why the intersection between mathematics and biology is so interesting to me.

One example is the problem of cellular identity. We often describe a cell using its gene expression profile at a particular moment. We can compare thousands of genes between young and old cells, identify differences and construct a list of pathways associated with ageing. But a cell is not a static collection of expression values. It is a system of interacting components that continuously responds to internal and external signals.

Two cells could have similar expression profiles and still behave differently when exposed to the same disturbance. One might recover quickly while the other enters a persistent state of dysfunction. If we only measure their current molecular states, we may not understand this difference. To explain it, we need to understand how the systems evolve.

This is where dynamical systems become useful. A dynamical system describes how the state of a system changes over time according to a set of rules. In biology, the state might be represented by the expression levels of thousands of genes, the activity of regulatory proteins or a combination of molecular measurements. The interactions between these variables determine how the system moves through its possible states.

Instead of asking only which genes distinguish a young cell from an old one, we could ask how the cell responds to a perturbation, how quickly it recovers and whether it returns to its previous state. These questions are much closer to understanding the mechanisms that maintain cellular function.

Cells as attractors

One concept I find particularly useful is the attractor. In a dynamical system, an attractor is a state or pattern of behaviour towards which the system tends to evolve. A stable cellular identity can be understood as an attractor created by the interactions within a gene regulatory network.

The familiar Waddington landscape provides an intuitive way to visualise this idea. Imagine a ball resting in a valley. The ball represents the cell, and the landscape represents the dynamics of the regulatory system. A small disturbance pushes the ball up the side of the valley, but the ball eventually rolls back down. A larger disturbance might push it over a ridge and into another valley, corresponding to a transition into a different cellular state.

This is more than a metaphor. In a 2018 study published in PLOS ONE, Chong and colleagues constructed a mathematical model of an ageing-related gene regulatory network involving p53 and cell-cycle control. Their model produced different attractors associated with homeostasis, cell-cycle arrest and senescence or apoptosis. In silico perturbation of the network also produced a cancer-associated attractor.

The study was theoretical, so its landscape should not be interpreted as a direct measurement of the physical state of a living cell. Nevertheless, it illustrates how mathematical models can connect regulatory interactions to the stability of cellular states.

What interests me most is how this framework could help explain ageing. Perhaps the problem is not simply that old cells accumulate molecular damage. Perhaps some cells also become less able to maintain their normal regulatory state when disturbances occur.

In the landscape picture, this could correspond to a shallower valley, a changed boundary between states or a reduction in the range of perturbations from which the system can recover. A cell might take longer to return to its original state or settle into a different state altogether.

I would not assume that every ageing cell follows this pattern. Cellular ageing is heterogeneous, and different tissues may behave differently. The depth of an attractor is also a model-dependent concept, not something that can be inferred directly from a single gene expression measurement. But the hypothesis is interesting because it gives us testable questions about resilience rather than simply describing molecular differences associated with age.

From snapshots to trajectories

RNA velocity is one of the attempts to move beyond static measurements. In their 2018 Nature Biotechnology paper, La Manno and colleagues introduced a method for estimating the future state of a cell from the relative abundance of unspliced and spliced RNA. The basic idea is that these measurements contain information about the dynamics of transcription and RNA processing.

The later scVelo framework developed a more general model of RNA kinetics and made it possible to estimate velocity across a wider range of genes and cellular states.

I find this shift from measuring states to estimating movement particularly important. A conventional single-cell experiment tells us what RNA is present in each sampled cell. RNA velocity attempts to infer the direction in which expression is changing. Instead of seeing only a collection of points in a high-dimensional space, we begin to estimate how those points might move.

However, an arrow on a UMAP plot is not direct evidence of a cell's future. RNA velocity depends on assumptions about RNA kinetics, and the inferred direction can be sensitive to the model and the data. Pseudotime and velocity are also different concepts. Pseudotime orders cells along an inferred progression, whereas velocity estimates local changes in expression. Neither automatically reconstructs the complete dynamics of a biological system.

This is an example of why mathematical understanding matters. It is easy to generate a trajectory using a computational package. It is much harder to determine what that trajectory means, which assumptions support it and whether the result can be validated experimentally.

I am particularly interested in what happens when these methods are connected to gene regulatory networks. A trajectory may indicate that a cell is moving towards a different state, but it does not necessarily explain why. Which transcription factors are driving the change? Which feedback loops stabilise the original state? Which regulatory interactions determine whether the transition is reversible?

In RNA-ODE, published in the Journal of Molecular Biology in 2022, researchers used ordinary differential equations to model gene expression dynamics and connect RNA velocity to trajectory inference and gene regulatory network reconstruction. More recently, the STICCC method used gene expression data and regulatory interactions to infer reversible and irreversible cellular state transitions.

These approaches illustrate an important direction for the field. Rather than treating expression, velocity and regulatory networks as separate analyses, we can try to connect them within a common mathematical framework.

But even a model that accurately predicts a trajectory may not have identified the correct biological mechanism. Different regulatory networks can sometimes produce similar observable behaviour. Perturbation experiments are therefore essential if we want to move from predicting cellular changes to explaining their causes.

The geometry of cell states

Another mathematical idea that deserves more attention in biology is topology.

Single-cell datasets are often represented as collections of points in a high-dimensional space. Each point corresponds to a cell, and the distance between points reflects some measure of similarity between their molecular profiles. Dimensionality reduction then produces a two-dimensional representation that we can visualise.

These plots are useful, but their appearance can depend on the choice of method and parameters. A cluster on a UMAP plot is not necessarily a distinct biological state, and the apparent distance between two clusters should not automatically be interpreted as a meaningful biological distance.

Topology offers a different way of studying these data. Rather than focusing only on exact distances, topological data analysis examines the structure of the data across different scales. Are there distinct groups? Does the data form a branching structure? Are there loops or other patterns that persist as the scale of analysis changes?

One important technique is persistent homology, which measures how topological features appear and disappear as we vary the scale at which the data are examined. Another is the Mapper algorithm, which can construct a graph summarising the organisation of high-dimensional data.

A 2025 review of topological data analysis in single-cell biology describes how these methods can help identify rare cell populations, transitional states and branching differentiation trajectories. The approach is appealing because biological processes do not necessarily follow a single linear path. Cells can branch into different fates, converge towards similar states or move through cyclic programmes.

Topology could therefore help us ask whether the organisation of cellular state space changes with age or disease. For example, does the structure of a cell population become more fragmented? Do alternative routes between states disappear? Are there changes in the connectivity of transitional states?

These are questions I would like to see explored more systematically. However, topology does not directly reveal the forces governing cellular movement. A topological analysis can describe the organisation of sampled data, but it cannot by itself establish whether a cell will return to its original state after a perturbation. The conclusions also depend on sampling, preprocessing and the way the underlying space is constructed.

I see topology as complementary to dynamical systems. Topology helps describe the structure of the space in which cells exist, while dynamical models attempt to explain how cells move through that space.

Information and ageing

Information theory adds another dimension to this problem. Originally developed to understand communication, it provides mathematical tools for measuring uncertainty and how much information one variable contains about another.

At the molecular level, cells constantly transmit and process information. Transcription factors regulate target genes, signalling pathways communicate changes in the environment, and feedback mechanisms help maintain stable patterns of gene expression. If ageing changes these relationships, the cell may become less effective at coordinating its own activity.

In The Information Theory of Aging, published in Nature Aging in 2023, Lu, Tian and Sinclair proposed that the progressive loss of epigenetic information contributes to ageing. Their perspective distinguishes the relatively stable DNA sequence from the epigenetic information that helps maintain cellular identity and regulate gene expression. It also discusses the possibility that epigenetic reprogramming can restore aspects of youthful cellular function.

I find this framework compelling, although I would distinguish the broad hypothesis from what has been conclusively established. Ageing involves many interacting processes, and it remains an open question how much of the decline in cellular function can be explained by loss of epigenetic information.

More directly relevant to regulatory dynamics is the paper Decay in transcriptional information flow is a hallmark of cellular aging. The researchers used information theory to examine communication between transcription factors and their target genes in single-cell RNA sequencing datasets from ten tissues. They reported a decline in mutual information between regulators and their targets with age. Their analysis linked this decline to changes in regulatory coordination, network structure and stabilising feedback.

Mutual information measures how much knowing one variable reduces uncertainty about another. In this context, it can help quantify how informative transcription-factor activity is about the expression of its target genes.

This is a more precise idea than saying that an ageing cell simply contains less information. The genome does not necessarily lose its sequence information. Instead, the relationship between regulatory signals and gene expression may become less coordinated. The cell may retain its molecular components while becoming less effective at using them to maintain normal function.

I think this distinction could be important for understanding resilience. If regulatory interactions become less coordinated, a cell might struggle to respond appropriately to perturbations even when many of the relevant genes are still present.

Information theory could help quantify that loss of coordination. Dynamical systems could help describe how it affects the cell's behaviour over time. Together, these approaches might explain why two cells with similar molecular profiles respond differently to the same stimulus.

Of course, mutual information does not establish causation. A reduction in the measured relationship between transcription factors and their targets could arise for several reasons, including changes in cell composition, noise, unmeasured variables or the assumptions used to infer regulatory relationships. The important next step is to determine which mechanisms produce the observed changes and whether modifying them can restore function.

Beyond established methods

What I find most exciting is the possibility of combining these mathematical perspectives rather than treating them as independent analytical techniques.

Imagine studying the same cell population over time while measuring gene expression, DNA methylation and responses to controlled perturbations. We could use dynamical models to estimate how cells move through molecular state space. We could use topology to characterise the organisation of that space. Information theory could help quantify changes in regulatory communication, while gene regulatory networks could provide hypotheses about the mechanisms responsible for the observed dynamics.

The resulting model might allow us to ask whether an aged cell is simply in a different state or whether its capacity to recover from perturbations has changed. It might help distinguish a state that is temporarily disrupted from one that has become difficult to reverse. It could also generate predictions about which regulatory interactions are worth targeting experimentally.

This would require more than combining several existing software packages. It would require careful mathematical thinking about what the state variables represent, which dynamics are identifiable from the data and how uncertainty should be handled. It would also require experiments designed to distinguish competing explanations.

For example, if a model predicts that a particular regulatory network has become less resilient with age, we could perturb the system and measure its recovery over time. We could compare the response of young and old cells, test whether the predicted regulatory interactions are causal and determine whether the original state can be restored. These experiments would connect mathematical predictions to observable biological behaviour.

I do not think that every biological problem needs a complex dynamical model, or that mathematical sophistication automatically leads to better science. Sometimes a simple statistical analysis is exactly what is required. But we should understand why a method is appropriate rather than use it simply because it is established or convenient.

That is why I believe mathematical training should become a more central part of modern biology. Statistics, probability and linear algebra are essential, but so are differential equations, nonlinear dynamics, information theory, graph theory and topology. The point is not to use all of these tools in every project. It is to understand enough of them to recognise which ideas are relevant and to create new approaches when existing methods are inadequate.

Biology is full of problems that do not fit neatly into standard analytical frameworks. Cellular identity, ageing, development and disease involve systems with feedback, memory, noise, changing interactions and behaviour that emerges from many components acting together. There is no reason to expect one established technique to capture all of this complexity.

For me, the future of computational biology is not simply about producing larger datasets or more sophisticated visualisations. It is about developing mathematical models that explain biological behaviour and make predictions that can be tested. We should be asking not only what a cell looks like at a particular moment, but what determines its next state, how stable that state is and whether it can recover when something goes wrong.

If we can begin to answer these questions, mathematics becomes more than a set of tools for analysing biological data. It becomes a way of formulating new biological questions in the first place.