Biology subjects

Simpson, M. J.

Publications and source records attributed to Simpson, M. J..

At least 19 recordsLinked to original sources

bioRxiv2026

Data requirements for accurate extinction-risk prediction in bistable populations

Understanding and predicting extinction risk is a central challenge in population biology. Mathematical models incorporating Allee thresholds are commonly used to understand population dynamics and to assess extinction risks. Inaccurate predictions can have serious consequences for conservation management. In this simulation study, we develop a likelihood-based inference and prediction workflow to estimate parameters, including the Allee threshold and population diffusivity parameters, using noisy count data generated using a well-defined discrete model. Although parameters are identifiable according to commonly used criteria, the accuracy of resulting predictions depends strongly on the quantity, quality, collection time and spatial resolution of the data. Our workflow demonstrates that seemingly reliable parameter estimates can lead to inaccurate predictions, highlighting the need for careful consideration of data quality and quantity to guide extinction-risk modelling and prediction. Open source software is provided on GitHub to replicate and extend all results considered.

ecology↗

bioRxiv2026

Distributed Clonal Deletion Prevents Autoimmune Disease Progression

Self-reactive B cells are generated during normal development and can acquire increased pathogenicity through activation-induced cytidine deaminase (AID)-mediated diversification following activation. Clonal deletion is thought to eliminate these cells, yet how deletion is distributed across developmental and activation stages to prevent autoimmune disease remains unclear. Here, we show that clonal deletion is enforced through temporally distinct mitochondrial apoptosis (MOMP) checkpoints that differentially regulate autoreactive B cell fate and disease progression. Using conditional Bcl-2 expression to inhibit MOMP either before or after B cell activation, we find that early inhibition permits the survival and maturation of autoreactive B cells after peripheral egress, expanding the pool of cells available for activation. These cells subsequently undergo AID-dependent diversification, producing class-switched IgG autoantibodies with expanded antigen breadth that target a wider range of self-antigens and drive lethal, female-biased autoimmune disease characterized by complement activation and kidney pathology. In contrast, inhibition of MOMP only after activation allows the accumulation of germinal center, switched memory, and plasma cells and promotes autoantibody production, but results in more restricted IgG autoreactivity, limited complement activation and limited tissue damage, and normal survival. Notably, early MOMP inhibition does not expand immature bone marrow B cells, indicating that a major clonal deletion checkpoint operates in the periphery rather than during initial B cell generation. Together, these findings support a Distributed Clonal Deletion Model in which early checkpoints restrict the entry of autoreactive B cells into diversification pathways, while later checkpoints limit the persistence of diversified autoreactive clones, thereby constraining autoimmune disease progression. One Sentence SummaryDistributed clonal deletion prevents autoimmune disease progression by restricting the breadth of autoreactive clones entering immune responses, with early MOMP checkpoints limiting diversification and later checkpoints constraining persistence.

immunology↗

bioRxiv2026

Mechanistic interpretation of biological tissue growth experiments with a computational model

The growth rates of biological tissues are influenced by the existing substrate geometry, mechanobiological processes and the interplay between them. Disentangling the contributions of geometry and mechanobiology experimentally is challenging, as mechanical properties are difficult to measure and tissue samples provide only static snapshot in time. However, the composition of a tissue preserves cues of the dynamic processes that shaped its architecture. In this work, we present a computational model of tissue growth that captures aspects of the interplay between geometry, mechanics, and stochastic biological processes, which we use to generate synthetic tissue compositions directly comparable with experimental samples. This framework enables quantitative analysis of tissue morphology, inference of underlying growth mechanisms, estimation of dynamic rates from single-time-point data, and investigation of how stochasticity contributes to emergent growth patterns. We demonstrate the applicability of the model to simulate the growth of different tissue types by applying this framework to two distinct tissue growth scenarios: (i) tissue grown within 3D-printed porous scaffolds, and (ii) bone formation in cortical pores.

bioengineering↗

bioRxiv2026

A mathematical model of curvature controlled tissue growth incorporating mechanical cell interactions

Many biological tissues grow at rates that depend strongly on the local geometry of the tissue interface, yet the cellular mechanisms underlying this dependence remain poorly understood. In this work, we develop a two-dimensional mathematical model of tissue growth to investigate how curvature-dependent growth emerges from the complex interactions between cell-scale crowding and mechanical interactions. The tissue interface is represented as a confluent layer of mechanically interacting cells that simultaneously produce new tissue. We formally derive a continuum limit of this discrete model, obtaining a system of partial differential equations governing the coupled evolution of cell density and tissue geometry. Although curvature is not explicitly represented in the discrete model, curvature-dependent tissue growth emerges naturally in the continuum limit through geometric crowding associated with tissue production. The continuum equations also establish a direct relationship between cellular mechanical properties and the effective diffusivity governing mechanical relaxation at the tissue scale. Numerical simulations illustrate that the competition between tissue production and mechanical relaxation can result in a variety of tissue evolutions, including tissue interface smoothing consistent with curvature-controlled in vitro growth in porous scaffolds and bone formation. Finally, the model predicts that the time required for scaffold pore closure scales with pore size but also depends on a pore shape factor, extending previous empirical observations of linear scaling with size. More broadly, this work provides an analytical framework linking cell-scale mechanics to tissue-scale growth and geometry.

biophysics↗

bioRxiv2025

Rapid Parameter Inference for Spatiotemporal Stochastic Biological Models using Neural Posterior Estimation

AO_SCPLOWBSTRACTC_SCPLOWCell migration is a key biological process underlying wound healing, tissue development, and cancer metastasis, yet calibrating mathematical models of migration to experimental data remains a major challenge. Scratch and barrier assays are widely used to study collective cell spreading, and agent-based random walk models provide a natural stochastic description of these experiments. However, parameter inference for such models is hampered by intractable likelihoods, forcing researchers to rely on Approximate Bayesian Computation, which introduces biases and tuning difficulties, or surrogate models that require potentially erroneous noise model specifications. Here, we overcome these limitations using neural posterior estimation, a simulation-based inference framework that learns the full posterior distribution directly from stochastic simulations without surrogate approximations or explicit noise model specifications. We deploy this framework on four progressively complex random walk models of barrier assay experiments describing in vitro cell migration: an isotropic baseline, a model with directional bias (chemotaxis), a model with cell proliferation, and a combined model incorporating both bias and proliferation. For each model, we demonstrate inference in two settings: using one-dimensional summary statistics (column counts), and using a convolutional neural network that enables inference directly from raw two-dimensional spatial data. Neural posterior estimation performs well across all four models, recovering biologically interpretable parameters (e.g. cell motility, directional bias, proliferation rates) from cases where classical surrogate-based methods are adequate through to the combined model where the interplay of multiple mechanisms renders surrogate approximations unreliable. We validate all posteriors using simulation-based calibration diagnostics and provide an open-source implementation of our pipeline to facilitate its adoption and extension to more complex, spatially-structured biological models.

systems biology↗

bioRxiv2025

Parameter-wise predictions and sensitivity analysis forrandom walk models in the life sciences

Sensitivity analysis characterises input-output relationships for mathematical models, and has been widely applied to deterministic models across many applications in the life sciences. In contrast, sensitivity analysis for stochastic models has received less attention, with most previous work focusing on well-mixed, non-spatial problems. For explicit spatio-temporal stochastic models, such as random walk models (RWMs), sensitivity analysis has received far less attention. Here we present a new type of sensitivity analysis, called parameter-wise prediction, for two types of biologically-motivated and computationally expensive RWMs. To overcome the limitations of directly analysing stochastic simulations, we employ continuum-limit partial differential equation (PDE) descriptions as surrogate models, and we link these efficient surrogate descriptions to the RWMs using a range of biophysically-motivated measurement error models. Our approach is likelihood-based, which means that we also consider likelihood-based parameter estimation and identifiability analysis along with parameter sensitivity. The new approach is presented for two important classes of lattice-based RWM including a classical model where crowding effects are neglected, and an exclusion process model that explicitly incorporates crowding. Our workflow illustrates how different process models can be combined with different measurement error models to reveal how each parameter impacts the outcome of the expensive stochastic simulation. Open-access software to replicate all results is available on GitHub.

biophysics↗

bioRxiv2025

Inference and prediction for random walkmodels in biology

Parameter inference is a critical step in the process of interpreting biological data using mathematical models. Inference provides a means of deriving quantitative, mechanistic insights from sparse, noisy data. While methods for parameter inference, parameter identifiability, and model prediction are well-developed for deterministic continuum models, working with biological applications often requires stochastic modelling approaches to capture inherent variability and randomness that can be prominent in biological measurements and data. Random walk models are especially useful for capturing spatiotemporal processes, such as ecological population dynamics, molecular transport phenomena, and collective behaviour associated with multicellular phenomena. This review focuses on parameter inference, identifiability analysis, and model prediction for a suite of biologically-inspired, stochastic agent-based models relevant to animal dispersal and populations of biological cells. With a particular emphasis on model prediction, we highlight roles for numerical optimisation and automatic differentiation. Open-source Julia code is provided to support scientific reproducibility. We encourage readers to use this code directly or adapt it to suit their interests and applications.

systems biology↗

bioRxiv2025

Data-informed model reduction for inference and prediction from non-identifiable models

Many mathematical models in the field of theoretical biology involve challenges relating to parameter identifiability. Non-identifiability implies that different combinations of parameter values lead to indistinguishable solutions of the mathematical model. This means that it is difficult, and sometimes impossible, to explain the mechanistic origin of observations using a non-identifiable mathematical model. A standard approach to deal with structurally non-identifiable models is to use reparameterisation, which typically focuses on the structure of the mathematical model without accounting for the impact of noisy, finite data. We present and explore a simple computational approach for model reduction, via likelihood reparameterisation, that can be applied to both structurally non-identifiable and practically non-identifiable problems. We construct simplified, identifiable mathematical models that enable model-based predictions for a range of continuum models based on different classes of commonly-used differential equations. Through a series of computational experiments, we illustrate how to deal with a range of noise models that relate the solution of the mathematical model with noisy observations. A key focus is to illustrate how computationally efficient model-based predictions can be made from reduced models.

synthetic biology↗

bioRxiv2023

Making predictions using poorly identified mathematical models

Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub. Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of some parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

systems biology↗

bioRxiv2023

How osteons form: A quantitative hypothesis-testing analysis of cortical pore filling and wall asymmetry

Osteon morphology provides valuable information about the interplay between different processes involved in bone remodelling. The correct quantitative interpretation of these morphological features is challenging due to the complexity of interactions between osteoblast behaviour, and the evolving geometry of cortical pores during pore closing. We present a combined experimental and mathematical modelling study to provide insights into bone formation mechanisms during cortical bone remodelling based on histological cross-sections of quiescent human osteons and hypothesis-testing analyses. We introduce wall thickness asymmetry as a measure of the local asymmetry of bone formation within an osteon and examine the frequency distribution of wall thickness asymmetry in cortical osteons from human iliac crest bone samples from women 16-78 years old. Our measurements show that most osteons possess some degree of asymmetry, and that the average degree of osteon asymmetry in cortical bone evolves with age. We then propose a comprehensive mathematical model of cortical pore filling that includes osteoblast secretory activity, osteoblast elimination, osteoblast embedment as osteocytes, and osteoblast crowding and redistribution along the bone surface. The mathematical model is first calibrated to symmetric osteon data, and then used to test three mechanisms of asymmetric wall formation against osteon data: (i) delays in the onset of infilling around the cement line; (ii) heterogeneous osteoblastogenesis around the bone perimeter; and (iii) heterogeneous osteoblast secretory rate around the bone perimeter. Our results suggest that wall thickness asymmetry due to off-centred Haversian pores within osteons, and that nonuniform lamellar thicknesses within osteons are important morphological features that can indicate the prevalence of specific asymmetry-generating mechanisms. This has significant implications for the study of disruptions of bone formation as it could indicate what biological bone formation processes may become disrupted with age or disease.

biophysics↗

bioRxiv2023

Modelling count data with partial differential equation models in biology

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

systems biology↗

bioRxiv2022

Formation and growth characteristics of co-culture tumour spheroids

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret coculture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspans seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspans seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.

biophysics↗

bioRxiv2022

A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing profile-wise prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.

systems biology↗

bioRxiv2022

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

ecology↗

bioRxiv2022

Computationally efficient framework for diagnosing, understanding, and predicting biphasic population growth

Throughout the life sciences, biological populations undergo multiple phases of growth, often referred to as biphasic growth for the commonly-encountered situation involving two phases. Biphasic population growth occurs over a massive range of spatial and temporal scales, ranging from microscopic growth of tumours over several days, to decades-long re-growth of corals in coral reefs that can extend for hundreds of kilometres. Different mathematical models and statistical methods are used to diagnose, understand, and predict biphasic growth. Common approaches can lead to inaccurate predictions of future growth that may result in inappropriate management and intervention strategies being implemented. Here we develop a very general computationally efficient framework, based on profile likelihood analysis, for diagnosing, understanding, and predicting biphasic population growth. The two key components of the framework are: (i) an efficient method to form approximate confidence intervals for the change point of the growth dynamics and model parameters; and, (ii) parameter-wise profile predictions that systematically reveal the influence of individual model parameters on predictions. To illustrate our framework we explore real-world case studies across the life sciences.

systems biology↗

bioRxiv2022

Computationally efficient mechanism discovery for cell invasion with uncertainty quantification

Parameter estimation for mathematical models of biological processes is often difficult and depends significantly on the quality and quantity of available data. We introduce an efficient frame-work using Gaussian processes to discover mechanisms underlying delay, migration, and proliferation in a cell invasion experiment. Gaussian processes are leveraged with bootstrapping to provide uncertainty quantification for the mechanisms that drive the invasion process. Our frame-work is efficient, parallelisable, and can be applied to other biological problems. We illustrate our methods using a canonical scratch assay experiment, demonstrating how simply we can explore different functional forms and develop and test hypotheses about underlying mechanisms, such as whether delay is present. All code and data to reproduce this work are available at https://github.com/DanielVandH/EquationLearning.jl. 1 Author summaryIn this work we introduce uncertainty quantification into equation learning methods, such as physics-informed and biologically-informed neural networks. Our framework is computationally efficient and applicable to problems with unknown nonlinear mechanisms that we wish to learn from experiments where only sparse noisy data are available. We demonstrate our methods on a canonical scratch assay experiment from cell biology and show the underlying mechanisms can be learned, providing confidence intervals for functional forms and for solutions to partial differential equation models believed to describe the experiment.

cell biology↗

bioRxiv2022

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability

Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.

cancer biology↗

bioRxiv2022

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

biophysics↗