Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry.

Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

A Mathematical Modelling Study of Chemotactic Dynamics in Cell Cultures: The Impact of Spatio-temporal Heterogeneity.

As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in microdevices, we develop a framework for constructing approximate analytical solutions for the location, speed and cellular densities for cell chemotaxis waves in heterogeneous fields of chemoattractant from the underlying partial differential equation models. In particular, such chemotactic waves are not in general translationally invariant travelling waves, but possess a spatial variation that evolves in time, and may even oscillate back and forth in time, according to the details of the chemotactic gradients. The analytical framework exploits the observation that unbiased cellular diffusive flux is typically small compared to chemotactic fluxes and is first developed and validated for a range of exemplar scenarios. The framework is subsequently applied to more complex models considering the chemoattractant dynamics under more general settings, potentially including those of relevance for representing pathophysiology scenarios in microdevice studies. In particular, even though solutions cannot be constructed in all cases, a wide variety of scenarios can be considered analytically, firstly providing global insight into the important mechanisms and features of cell motility in complex spatiotemporal fields of chemoattractant. Such analytical solutions also provide a means of rapid evaluation of model predictions, with the prospect of application in computationally demanding investigations relating theoretical models and experimental observation, such as Bayesian parameter estimation.

Squirmer hydrodynamics near a periodic surface topography.

The behaviour of microscopic swimmers has previously been explored near large-scale confining geometries and in the presence of very small-scale surface roughness. Here, we consider an intermediate case of how a simple microswimmer, the tangential spherical squirmer, behaves adjacent to singly and doubly periodic sinusoidal surface topographies that spatially oscillate with an amplitude that is an order of magnitude less than the swimmer size and wavelengths that are also within an order of magnitude of this scale. The nearest neighbour regularised Stokeslet method is used for numerical explorations after validating its accuracy for a spherical tangential squirmer that swims stably near a flat surface. The same squirmer is then introduced to different surface topographies. The key governing factor in the resulting swimming behaviour is the size of the squirmer relative to the surface topography wavelength. For instance, directional guidance is not observed when the squirmer is much larger, or much smaller, than the surface topography wavelength. In contrast, once the squirmer size is on the scale of the topography wavelength, limited guidance is possible, often with local capture in the topography troughs. However, complex dynamics can also emerge, especially when the initial configuration is not close to alignment along topography troughs or above topography crests. In contrast to sensitivity in alignment and topography wavelength, reductions in the amplitude of the surface topography or variations in the shape of the periodic surface topography do not have extensive impacts on the squirmer behaviour. Our findings more generally highlight that the numerical framework provides an essential basis to elucidate how swimmers may be guided by surface topography.

The developmental basis of fingerprint pattern formation and variation.

Fingerprints are complex and individually unique patterns in the skin. Established prenatally, the molecular and cellular mechanisms that guide fingerprint ridge formation and their intricate arrangements are unknown. Here we show that fingerprint ridges are epithelial structures that undergo a truncated hair follicle developmental program and fail to recruit a mesenchymal condensate. Their spatial pattern is established by a Turing reaction-diffusion system, based on signaling between EDAR, WNT, and antagonistic BMP pathways. These signals resolve epithelial growth into bands of focalized proliferation under a precociously differentiated suprabasal layer. Ridge formation occurs as a set of waves spreading from variable initiation sites defined by the local signaling environments and anatomical intricacies of the digit, with the propagation and meeting of these waves determining the type of pattern that forms. Relying on a dynamic patterning system triggered at spatially distinct sites generates the characteristic types and unending variation of human fingerprint patterns.

Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems.

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction-diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

De-risking clinical trial failure through mechanistic simulation.

Drug development typically comprises a combination of pre-clinical experimentation, clinical trials, and statistical data-driven analyses. Therapeutic failure in late-stage clinical development costs the pharmaceutical industry billions of USD per year. Clinical trial simulation represents a key derisking strategy and combining them with mechanistic models allows one to test hypotheses for mechanisms of failure and to improve trial designs. This is illustrated with a T-cell activation model, used to simulate the clinical trials of IMA901, a short-peptide cancer vaccine. Simulation results were consistent with observed outcomes and predicted that responses are limited by peptide off-rates, peptide competition for dendritic cell (DC) binding, and DC migration times. These insights were used to hypothesise alternate trial designs predicted to improve efficacy outcomes. This framework illustrates how mechanistic models can complement clinical, experimental, and data-driven studies to understand, test, and improve trial designs, and how results may differ between humans and mice.

A Mathematical Model of Aqueous Humor Production and Composition.

Purpose: We develop a mathematical model that predicts aqueous humor (AH) production rate by the ciliary processes and aqueous composition in the posterior chamber (PC), with the aim of estimating how the aqueous production rate depends on the controlling parameters and how it can be manipulated. Methods: We propose a compartmental mathematical model that considers the stromal region, ciliary epithelium, and PC. All domains contain an aqueous solution with different chemical species. We impose the concentration of all species on the stromal side and exploit the various ion channels present on the cell membrane to compute the water flux produced by osmosis, the solute concentrations in the AH and the transepithelial potential difference. Results: With a feasible set of parameters, the model predictions of water flux from the stroma to the PC and of the solute concentrations in the AH are in good agreement with measurements. Key parameters which impact the aqueous production rate are identified. A relevant role is predicted to be played by cell membrane permeability to \(\text{K}^+\) and \(\text{Cl}^-\), by the level of transport due to the Na+-H+ exchanger and to the co-transporter of Na+/K+/2Cl-; and by carbonic anhydrase. Conclusions: The mathematical model predicts the formation and composition of AH, based on the structure of the ciliary epithelium. The model provides insight into the physical processes underlying the functioning of drugs that are adopted to regulate the aqueous production. It also suggests ion channels and cell membrane properties that may be targeted to manipulate the aqueous production rate.

Analysis of cellular kinetic models suggest that physiologically based model parameters may be inherently, practically unidentifiable.

Physiologically-based pharmacokinetic and cellular kinetic models are used extensively to predict concentration profiles of drugs or adoptively transferred cells in patients and laboratory animals. Models are fit to data by the numerical optimisation of appropriate parameter values. When quantities such as the area under the curve are all that is desired, only a close qualitative fit to data is required. When the biological interpretation of the model that produced the fit is important, an assessment of uncertainties is often also warranted. Often, a goal of fitting PBPK models to data is to estimate parameter values, which can then be used to assess characteristics of the fit system or applied to inform new modelling efforts and extrapolation, to inform a prediction under new conditions. However, the parameters that yield a particular model output may not necessarily be unique, in which case the parameters are said to be unidentifiable. We show that the parameters in three published physiologically-based pharmacokinetic models are practically (deterministically) unidentifiable and that it is challenging to assess the associated parameter uncertainty with simple curve fitting techniques. This result could affect many physiologically-based pharmacokinetic models, and we advocate more widespread use of thorough techniques and analyses to address these issues, such as established Markov Chain Monte Carlo and Bayesian methodologies. Greater handling and reporting of uncertainty and identifiability of fit parameters would directly and positively impact interpretation and translation for physiologically-based model applications, enhancing their capacity to inform new model development efforts and extrapolation in support of future clinical decision-making.

Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems.

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction-diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction-diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction-diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, [Formula: see text], is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with [Formula: see text], and conjecture that this is a general impact of time delay on reaction-diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction-diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

Fock-space approach to stochastic susceptible-infected-recovered models.

We investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equations for the subpopulation sizes of S, I, and R individuals, the stochastic SIR model is driven by a master equation governing the transition probabilities among the system's states defined by SIR occupation numbers. In the Fock-space approach the master equation is recast in the form of a real-valued Schrödinger-type equation with a second quantization Hamiltonian-like operator describing the infection and recovery processes. We find exact analytic expressions for the Hamiltonian eigenvalues for any population size N. We present small- and large-N results for the average numbers of SIR individuals and basic reproduction number. For small N we also obtain the probability distributions of SIR states, epidemic sizes and durations, which cannot be found from deterministic SIR models. Our Fock-space approach to stochastic SIR models introduces a powerful set of tools to calculate central quantities of epidemic processes, especially for relatively small populations where statistical fluctuations not captured by conventional deterministic SIR models play a crucial role.

Modelling articular cartilage: the relative motion of two adjacent poroviscoelastic layers.

In skeletal joints two layers of adjacent cartilage are often in relative motion. The individual cartilage layers are often modelled as a poroviscoelastic material. To model the relative motion, noting the separation of scales between the pore level and the macroscale, a homogenization based on multiple scale asymptotic analysis has been used in this study to derive a macroscale model for the relative translation of two poroviscoelastic layers separated by a very thin layer of fluid. In particular the fluid layer thickness is essentially zero at the macroscale so that the two poroviscoelastic layers are effectively in contact and their interaction is captured in the derived model via a set of interfacial conditions, including a generalization of the Beavers-Joseph condition at the interface between a viscous fluid and a porous medium. In the simplifying context of a uniform geometry, constant fixed charge density, a Newtonian interstitial fluid and a viscoelastic scaffold, modelled via finite deformation theory, we present preliminary simulations that may be used to highlight predictions for how oscillatory relative movement of cartilage under load influences the peak force the cartilage experiences and the extent of the associated deformations. In addition to highlighting such cartilage mechanics, the systematic derivation of the macroscale models will enable the study of how nanoscale cartilage physics, such as the swelling pressure induced by fixed charges, manifests in cartilage mechanics at much higher lengthscales.

A method for the inference of cytokine interaction networks.

Cell-cell communication is mediated by many soluble mediators, including over 40 cytokines. Cytokines, e.g. TNF, IL1ß, IL5, IL6, IL12 and IL23, represent important therapeutic targets in immune-mediated inflammatory diseases (IMIDs), such as inflammatory bowel disease (IBD), psoriasis, asthma, rheumatoid and juvenile arthritis. The identification of cytokines that are causative drivers of, and not just associated with, inflammation is fundamental for selecting therapeutic targets that should be studied in clinical trials. As in vitro models of cytokine interactions provide a simplified framework to study complex in vivo interactions, and can easily be perturbed experimentally, they are key for identifying such targets. We present a method to extract a minimal, weighted cytokine interaction network, given in vitro data on the effects of the blockage of single cytokine receptors on the secretion rate of other cytokines. Existing biological network inference methods typically consider the correlation structure of the underlying dataset, but this can make them poorly suited for highly connected, non-linear cytokine interaction data. Our method uses ordinary differential equation systems to represent cytokine interactions, and efficiently computes the configuration with the lowest Akaike information criterion value for all possible network configurations. It enables us to study indirect cytokine interactions and quantify inhibition effects. The extracted network can also be used to predict the combined effects of inhibiting various cytokines simultaneously. The model equations can easily be adjusted to incorporate more complicated dynamics and accommodate temporal data. We validate our method using synthetic datasets and apply our method to an experimental dataset on the regulation of IL23, a cytokine with therapeutic relevance in psoriasis and IBD. We validate several model predictions against experimental data that were not used for model fitting. In summary, we present a novel method specifically designed to efficiently infer cytokine interaction networks from cytokine perturbation data in the context of IMIDs.

Modern perspectives on near-equilibrium analysis of Turing systems.

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Introduction to 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Modelling Motility: The Mathematics of Spermatozoa.

In one of the first examples of how mechanics can inform axonemal mechanism, Machin's study in the 1950s highlighted that observations of sperm motility cannot be explained by molecular motors in the cell membrane, but would instead require motors distributed along the flagellum. Ever since, mechanics and hydrodynamics have been recognised as important in explaining the dynamics, regulation, and guidance of sperm. More recently, the digitisation of sperm videomicroscopy, coupled with numerous modelling and methodological advances, has been bringing forth a new era of scientific discovery in this field. In this review, we survey these advances before highlighting the opportunities that have been generated for both recent research and the development of further open questions, in terms of the detailed characterisation of the sperm flagellum beat and its mechanics, together with the associated impact on cell behaviour. In particular, diverse examples are explored within this theme, ranging from how collective behaviours emerge from individual cell responses, including how these responses are impacted by the local microenvironment, to the integration of separate advances in the fields of flagellar analysis and flagellar mechanics.

Predicted limited redistribution of T cells to secondary lymphoid tissue correlates with increased risk of haematological malignancies in asplenic patients.

The spleen, a secondary lymphoid tissue (SLT), has an important role in generation of adaptive immune responses. Although splenectomy remains a common procedure, recent studies reported poor prognosis and increased risk of haematological malignancies in asplenic patients. The high baseline trafficking of T lymphocytes to splenic tissue suggests splenectomy may lead to loss of blood-borne malignant immunosurveillance that is not compensated for by the remaining SLT. To date, no quantitative analysis of the impact of splenectomy on the human T cell trafficking dynamics and tissue localisation has been reported. We developed a quantitative computational model that describes organ distribution and trafficking of human lymphocytes to explore the likely impact of splenectomy on immune cell distributions. In silico splenectomy resulted in an average reduction of T cell numbers in SLT by 35% (95%CI 0.12-0.97) and a comparatively lower, 9% (95%CI 0.17-1.43), mean decrease of T cell concentration in SLT. These results suggest that the surveillance capacity of the remaining SLT insufficiently compensates for the absence of the spleen. This may, in part, explain haematological malignancy risk in asplenic patients and raises the question of whether splenectomy has a clinically meaningful impact on patient responses to immunotherapy.

Control and controllability of microswimmers by a shearing flow.

With the continuing rapid development of artificial microrobots and active particles, questions of microswimmer guidance and control are becoming ever more relevant and prevalent. In both the applications and theoretical study of such microscale swimmers, control is often mediated by an engineered property of the swimmer, such as in the case of magnetically propelled microrobots. In this work, we will consider a modality of control that is applicable in more generality, effecting guidance via modulation of a background fluid flow. Here, considering a model swimmer in a commonplace flow and simple geometry, we analyse and subsequently establish the efficacy of flow-mediated microswimmer positional control, later touching upon a question of optimal control. Moving beyond idealized notions of controllability and towards considerations of practical utility, we then evaluate the robustness of this control modality to sources of variation that may be present in applications, examining in particular the effects of measurement inaccuracy and rotational noise. This exploration gives rise to a number of cautionary observations, which, overall, demonstrate the need for the careful assessment of both policy and behavioural robustness when designing control schemes for use in practice.

Isolating Patterns in Open Reaction-Diffusion Systems.

Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Patterns of bacterial motility in microfluidics-confining environments.

Understanding the motility behavior of bacteria in confining microenvironments, in which they search for available physical space and move in response to stimuli, is important for environmental, food industry, and biomedical applications. We studied the motility of five bacterial species with various sizes and flagellar architectures (Vibrio natriegens, Magnetococcus marinus, Pseudomonas putida, Vibrio fischeri, and Escherichia coli) in microfluidic environments presenting various levels of confinement and geometrical complexity, in the absence of external flow and concentration gradients. When the confinement is moderate, such as in quasi-open spaces with only one limiting wall, and in wide channels, the motility behavior of bacteria with complex flagellar architectures approximately follows the hydrodynamics-based predictions developed for simple monotrichous bacteria. Specifically, V. natriegens and V. fischeri moved parallel to the wall and P. putida and E. coli presented a stable movement parallel to the wall but with incidental wall escape events, while M. marinus exhibited frequent flipping between wall accumulator and wall escaper regimes. Conversely, in tighter confining environments, the motility is governed by the steric interactions between bacteria and the surrounding walls. In mesoscale regions, where the impacts of hydrodynamics and steric interactions overlap, these mechanisms can either push bacteria in the same directions in linear channels, leading to smooth bacterial movement, or they could be oppositional (e.g., in mesoscale-sized meandered channels), leading to chaotic movement and subsequent bacterial trapping. The study provides a methodological template for the design of microfluidic devices for single-cell genomic screening, bacterial entrapment for diagnostics, or biocomputation.