A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium

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 parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

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.

Implementing measurement error models with mechanistic mathematical models in a likelihood-based framework for estimation, identifiability analysis and prediction in the life sciences

Throughout the life sciences, we routinely seek to interpret measurements and observations using parametrized mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis and prediction, called profile-wise analysis. This frequentist approach to uncertainty quantification for mechanistic models leverages the profile likelihood for targeting parameters and understanding their influence on predictions. Case studies, motivated by simple caricature models routinely used in systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.

Systems Modeling to Quantify Safety Risks in Early Drug Development: Using Bifurcation Analysis and Agent-Based Modeling as Examples

Quantitative Systems Toxicology (QST) models, recapitulating pharmacokinetics and mechanism of action together with the organic response at multiple levels of biological organization, can provide predictions on the magnitude of injury and recovery dynamics to support study design and decision-making during drug development. Here, we highlight the application of QST models to predict toxicities of cancer treatments, such as cytopenia(s) and gastrointestinal adverse effects, where narrow therapeutic indexes need to be actively managed. The importance of bifurcation analysis is demonstrated in QST models of hematologic toxicity to understand how different regions of the parameter space generate different behaviors following cancer treatment, which results in asymptotically stable predictions, yet highly irregular for specific schedules, or oscillating predictions of blood cell levels. In addition, an agent-based model of the intestinal crypt was used to simulate how the spatial location of the injury within the crypt affects the villus disruption severity. We discuss the value of QST modeling approaches to support drug development and how they align with technological advances impacting trial design including patient selection, dose/regimen selection, and ultimately patient safety.

Modulating Oral Delivery and Gastrointestinal Kinetics of Recombinant Proteins via Engineered Fungi

A new modality in microbe-mediated drug delivery has recently emerged wherein genetically engineered microbes are used to locally deliver recombinant therapeutic proteins to the gastrointestinal tract. These engineered microbes are often referred to as live biotherapeutic products (LBPs). Despite advanced genetic engineering and recombinant protein expression approaches, little is known on how to control the spatiotemporal dynamics of LBPs and their secreted therapeutics within the gastrointestinal tract. To date, the fundamental pharmacokinetic analyses for microbe-mediated drug delivery systems have not been described. Here, we explore the pharmacokinetics of an engineered, model protein-secreting Saccharomyces cerevisiae, which serves as an ideal organism for the oral delivery of complex, post-translationally modified proteins. We establish three methods to modulate the pharmacokinetics of an engineered, recombinant protein-secreting fungi system: (i) altering oral dose of engineered fungi, (ii) co-administering antibiotics, and (iii) altering recombinant protein secretion titer. Our findings establish the fundamental pharmacokinetics which will be essential in controlling downstream therapeutic response for this new delivery modality.

Intestinal villus structure contributes to even shedding of epithelial cells

In the intestinal epithelium, proliferated epithelial cells ascend the crypts and villi and shed at the villus tips into the gut lumen. In this study, we theoretically investigate the roles of the villi on cell turnover. We present a stochastic model that focuses on the duration over which cells migrate the shortest paths between the crypt orifices and the villus tips, where shedding cells are randomly chosen from among those older than the shortest-path cell migration times. By extending the length of the shortest path to delay cell shedding, the finger-like shape of the villus would tightly regulate shedding-cell ages compared with flat surfaces and shorter projections; the villus allows epithelial cells to shed at around the same age, which limits them from shedding early or staying in the epithelium for long periods. Computational simulations of cell dynamics agreed well with the predictions. We also examine various mechanical conditions of cells and confirm that coordinated collective cell migration supports the predictions. These results suggest the important roles of the villi in homeostatic maintenance of the small intestine, and we discuss the applicability of our approach to other tissues with collective cell movement.

A mathematical model shows macrophages delay Staphylococcus aureus replication, but limitations in microbicidal capacity restrict bacterial clearance

S. aureus is a leading cause of bacterial infection. Macrophages, the first line of defence in the human immune response, phagocytose and kill S. aureus but the pathogen can evade these responses. Therefore, the exact role of macrophages is incompletely defined. We develop a mathematical model of macrophage - S. aureus dynamics, built on recent experimental data. We demonstrate that, while macrophages may not clear infection, they significantly delay its growth and potentially buy time for recruitment of further cells. We find that macrophage killing is a major obstacle to controlling infection and ingestion capacity also limits the response. We find bistability such that the infection can be limited at low doses. Our combination of experimental data, mathematical analysis and model fitting provide important insights in to the early stages of S. aureus infections, showing macrophages play an important role limiting bacterial replication but can be overwhelmed with large inocula.

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab^® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

Practical parameter identifiability for spatio-temporal models of cell invasion

We examine the practical identifiability of parameters in a spatio-temporal reaction-diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction-diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.

Chronic TNFα-driven injury delays cell migration to villi in the intestinal epithelium

The intestinal epithelium is a single layer of cells which provides the first line of defence of the intestinal mucosa to bacterial infection. Cohesion of this physical barrier is supported by renewal of epithelial stem cells, residing in invaginations called crypts, and by crypt cell migration onto protrusions called villi; dysregulation of such mechanisms may render the gut susceptible to chronic inflammation. The impact that excessive or misplaced epithelial cell death may have on villus cell migration is currently unknown. We integrated cell-tracking methods with computational models to determine how epithelial homeostasis is affected by acute and chronic TNFα-driven epithelial cell death. Parameter inference reveals that acute inflammatory cell death has a transient effect on epithelial cell dynamics, whereas cell death caused by chronic elevated TNFα causes a delay in the accumulation of labelled cells onto the villus compared to the control. Such a delay may be reproduced by using a cell-based model to simulate the dynamics of each cell in a crypt-villus geometry, showing that a prolonged increase in cell death slows the migration of cells from the crypt to the villus. This investigation highlights which injuries (acute or chronic) may be regenerated and which cause disruption of healthy epithelial homeostasis.

Elevated apoptosis impairs epithelial cell turnover and shortens villi in TNF-driven intestinal inflammation

The intestinal epithelial monolayer, at the boundary between microbes and the host immune system, plays an important role in the development of inflammatory bowel disease (IBD), particularly as a target and producer of pro-inflammatory TNF. Chronic overexpression of TNF leads to IBD-like pathology over time, but the mechanisms driving early pathogenesis events are not clear. We studied the epithelial response to inflammation by combining mathematical models with in vivo experimental models resembling acute and chronic TNF-mediated injury. We found significant villus atrophy with increased epithelial cell death along the crypt-villus axis, most dramatically at the villus tips, in both acute and chronic inflammation. In the acute model, we observed overexpression of TNF receptor I in the villus tip rapidly after TNF injection and concurrent with elevated levels of intracellular TNF and rapid shedding at the tip. In the chronic model, sustained villus atrophy was accompanied by a reduction in absolute epithelial cell turnover. Mathematical modelling demonstrated that increased cell apoptosis on the villus body explains the reduction in epithelial cell turnover along the crypt-villus axis observed in chronic inflammation. Cell destruction in the villus was not accompanied by changes in proliferative cell number or division rate within the crypt. Epithelial morphology and immunological changes in the chronic setting suggest a repair response to cell damage although the villus length is not recovered. A better understanding of how this state is further destabilised and results in clinical pathology resembling IBD will help identify suitable pathways for therapeutic intervention.

A Bayesian Sequential Learning Framework to Parameterise Continuum Models of Melanoma Invasion into Human Skin

We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters: (i) the melanoma cell proliferation rate, [Formula: see text]; (ii) the melanoma cell diffusivity, D; and (iii) [Formula: see text], a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning framework involves a sequence of increasingly sophisticated experimental data from: (i) a spatially uniform cell proliferation assay; (ii) a two-dimensional circular barrier assay; and (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall, our approach to inference is simple-to-implement, computationally efficient, and well suited for many cell biology phenomena that can be described by low-dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.

Host-microbe interaction in the gastrointestinal tract

The gastrointestinal tract is a highly complex organ in which multiple dynamic physiological processes are tightly coordinated while interacting with a dense and extremely diverse microbial population. From establishment in early life, through to host-microbe symbiosis in adulthood, the gut microbiota plays a vital role in our development and health. The effect of the microbiota on gut development and physiology is highlighted by anatomical and functional changes in germ-free mice, affecting the gut epithelium, immune system and enteric nervous system. Microbial colonisation promotes competent innate and acquired mucosal immune systems, epithelial renewal, barrier integrity, and mucosal vascularisation and innervation. Interacting or shared signalling pathways across different physiological systems of the gut could explain how all these changes are coordinated during postnatal colonisation, or after the introduction of microbiota into germ-free models. The application of cell-based in-vitro experimental systems and mathematical modelling can shed light on the molecular and signalling pathways which regulate the development and maintenance of homeostasis in the gut and beyond.

Neutral dynamics and cell renewal of colonic crypts in homeostatic regime

The self renewal process in colonic crypts is the object of several studies. We present here a new compartment model with the following characteristics: (a) we distinguish different classes of cells: stem cells, six generations of transit amplifying cells and the differentiated cells; (b) in order to take into account the monoclonal character of crypts in homeostatic regimes we include symmetric divisions of the stem cells. We first consider the dynamic differential equations that describe the evolution of the mean values of the populations, but the small observed value of the total number of cells involved plus the huge dispersion of experimental data found in the literature leads us to study the stochastic discrete process. This analysis allows us to study fluctuations, the neutral drift that leads to monoclonality, and the effects of the fixation of mutant clones.