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.

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model.

The Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, and generalizations thereof, involves simple reaction-diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $\lambda $. For the Fisher-KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\sqrt {\lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed $c=2\sqrt {\lambda D}> 0$. This means that, for biologically relevant initial data, the Fisher-KPP model cannot be used to study invasion with $c \ne 2\sqrt {\lambda D}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, $-\infty < c < \infty $. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth.

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, [Formula: see text] that is coupled to the concentration of an immobile extracellular substrate, [Formula: see text]. Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to [Formula: see text], while a logistic growth term models cell proliferation. The extracellular substrate [Formula: see text] is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, [Formula: see text], as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, [Formula: see text]. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub .

A novel mathematical model of heterogeneous cell proliferation.

We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the nonnegativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations.

Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models.

HYPOTHESIS: Evaporation of surfactant droplets on leaves is complicated due to the complex physical and chemical properties of the leaf surfaces. However, for certain leaf surfaces for which the evaporation process appears to follow the standard constant-contact-radius or constant-contact-angle modes, it should be possible to mimic the droplet evaporation with both a well-chosen synthetic surface and a relatively simple mathematical model. EXPERIMENTS: Surfactant droplet evaporation experiments were performed on two commercial crop species, wheat and capsicum, along with two synthetic surfaces, up to a 90° incline. The time-dependence of the droplets' contact angles, height, volume and contact radius was measured throughout the evaporation experiments. Mathematical models were developed to simulate the experiments. FINDINGS: With one clear exception, for all combinations of surfaces, surfactant concentrations and angles, the experiments appear to follow the standard evaporation modes and are well described by the mathematical models (modified Popov and Young-Laplace-Popov). The exception is wheat with a high surfactant concentration, for which droplet evaporation appears nonstandard and deviates from the diffusion limited models, perhaps due to additional mechanisms such as the adsorption of surfactant, stomatal density or an elongated shape in the direction of the grooves in the wheat surface.

Invading and Receding Sharp-Fronted Travelling Waves.

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.

Examining Go-or-Grow Using Fluorescent Cell-Cycle Indicators and Cell-Cycle-Inhibiting Drugs.

The go-or-grow hypothesis states that adherent cells undergo reversible phenotype switching between migratory and proliferative states, with cells in the migratory state being more motile than cells in the proliferative state. Here, we examine go-or-grow in two-dimensional in vitro assays using melanoma cells with fluorescent cell-cycle indicators and cell-cycle-inhibiting drugs. We analyze the experimental data using single-cell tracking to calculate mean diffusivities and compare motility between cells in different cell-cycle phases and in cell-cycle arrest. Unequivocally, our analysis does not support the go-or-grow hypothesis. We present clear evidence that cell motility is independent of the cell-cycle phase and that nonproliferative arrested cells have the same motility as cycling cells.

Image analysis of shatter and pinning events on hard-to-wet leaf surfaces by drops containing surfactant.

BACKGROUND: A key challenge for developing computer models of spray retention by plants is to accurately predict how spray drops behave when impacting leaf surfaces. One poorly understood outcome occurs when drops bounce or shatter on impact but leave behind a proportion of the liquid on the surface (designated as pinning). This process is studied via impaction experiments with two hard-to-wet leaf surfaces (fat-hen: Chenopodium album and barnyard grass: Echinochloa crus-galli L. P. Beauv) and one hydrophobic artificial surface (Teflon) using three liquid formulations. RESULTS: Drops that impact upon Teflon underwent pinning shatter events via a well-known mechanism referred to as receding breakup. Drops impacting on leaf surfaces did not undergo receding breakup because the liquid rim was not in direct contact with the leaf surface when it broke into secondary droplets. However, pinning did occur on plant surfaces via a different mechanism, especially when using formulations containing a surfactant. CONCLUSION: Newly developed image analysis and methodology has allowed quantification of the volume fraction pinned to surfaces when drops shatter. The addition of surfactant can increase both the probability of pinning and the pinned volume when drops shatter on fat-hen or Teflon. However, the surfactants studied did not substantially improve the probability of pinning on barnyard grass. The difference in behaviour between the two leaf surfaces and the underlying mechanism is worth further study. © 2020 Society of Chemical Industry.

Simulating spray droplet impaction outcomes: comparison with experimental data.

BACKGROUND: A suite of plant retention spray models has been developed to simulate spray retention using virtual surfaces (either single leaves or whole plants) and their outputs compared with experimental data for the equivalent spray scenarios. RESULTS: The results for a single formulation (0.1% v/v lecithin mixture in water) and difficult to wet plant species Chenopodium album L (common lambsquarters) are presented. They include experimental observations with single leaves, as well as simulations of virtual impaction events, conducted to provide for the first time estimates of f (the proportion of theoretical impact drop diameter at shatter). With this factor prescribed, multi-plant simulations using a range of nozzle types and droplet sizes (volume mean diameter (VMD) range 241 to 530 µm) are compared with equivalent experimentally determined spray retention by real plants. The simulations demonstrated that impaction resulted predominantly in shatter with the production of daughter droplets, and that retention is mainly due to re-capture of these droplets. Overall the simulations show the same trends as experimental retention results from different nozzle applications, but at best predicted retention results were 68% to 79% of experimental percentage retention, depending on plant spacing. CONCLUSIONS: Retention is the result of some primary drop capture but predominantly by recapture of shatter droplets as the modelling illustrates. The value of f affects the droplet shatter outcome and can result in fewer, more energetic daughter droplets, or more droplets but with lower energies. However, this effect alone cannot explain the discrepancy between actual and simulated results. Possible operational influences are discussed. © 2020 Society of Chemical Industry.

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow.

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analog of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterized by a blowup of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent α=1/3 just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.

Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy.

The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c≪1.

Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation.

We present a suite of experimental data showing that cell proliferation assays, prepared using standard methods thought to produce asynchronous cell populations, persistently exhibit inherent synchronization. Our experiments use fluorescent cell cycle indicators to reveal the normally hidden cell synchronization, by highlighting oscillatory subpopulations within the total cell population. These oscillatory subpopulations would never be observed without these cell cycle indicators. On the other hand, our experimental data show that the total cell population appears to grow exponentially, as in an asynchronous population. We reconcile these seemingly inconsistent observations by employing a multi-stage mathematical model of cell proliferation that can replicate the oscillatory subpopulations. Our study has important implications for understanding and improving experimental reproducibility. In particular, inherent synchronization may affect the experimental reproducibility of studies aiming to investigate cell cycle-dependent mechanisms, including changes in migration and drug response.

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics.

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.

Extended logistic growth model for heterogeneous populations.

Cell proliferation is the most important cellular-level mechanism responsible for regulating cell population dynamics in living tissues. Modern experimental procedures show that the proliferation rates of individual cells can vary significantly within the same cell line. However, in the mathematical biology literature, cell proliferation is typically modelled using a classical logistic equation which neglects variations in the proliferation rate. In this work, we consider a discrete mathematical model of cell migration and cell proliferation, modulated by volume exclusion (crowding) effects, with variable rates of proliferation across the total population. We refer to this variability as heterogeneity. Constructing the continuum limit of the discrete model leads to a generalisation of the classical logistic growth model. Comparing numerical solutions of the model to averaged data from discrete simulations shows that the new model captures the key features of the discrete process. Applying the extended logistic model to simulate a proliferation assay using rates from recent experimental literature shows that neglecting the role of heterogeneity can, at times, lead to misleading results.

Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data.

Collective cell spreading takes place in spatially continuous environments, yet it is often modelled using discrete lattice-based approaches. Here, we use data from a series of cell proliferation assays, with a prostate cancer cell line, to calibrate a spatially continuous individual based model (IBM) of collective cell migration and proliferation. The IBM explicitly accounts for crowding effects by modifying the rate of movement, direction of movement, and the rate of proliferation by accounting for pair-wise interactions. Taking a Bayesian approach we estimate the free parameters in the IBM using rejection sampling on three separate, independent experimental data sets. Since the posterior distributions for each experiment are similar, we perform simulations with parameters sampled from a new posterior distribution generated by combining the three data sets. To explore the predictive power of the calibrated IBM, we forecast the evolution of a fourth experimental data set. Overall, we show how to calibrate a lattice-free IBM to experimental data, and our work highlights the importance of interactions between individuals. Despite great care taken to distribute cells as uniformly as possible experimentally, we find evidence of significant spatial clustering over short distances, suggesting that standard mean-field models could be inappropriate.

A computational modelling framework to quantify the effects of passaging cell lines.

In vitro cell culture is routinely used to grow and supply a sufficiently large number of cells for various types of cell biology experiments. Previous experimental studies report that cell characteristics evolve as the passage number increases, and various cell lines can behave differently at high passage numbers. To provide insight into the putative mechanisms that might give rise to these differences, we perform in silico experiments using a random walk model to mimic the in vitro cell culture process. Our results show that it is possible for the average proliferation rate to either increase or decrease as the passaging process takes place, and this is due to a competition between the initial heterogeneity and the degree to which passaging damages the cells. We also simulate a suite of scratch assays with cells from near-homogeneous and heterogeneous cell lines, at both high and low passage numbers. Although it is common in the literature to report experimental results without disclosing the passage number, our results show that we obtain significantly different closure rates when performing in silico scratch assays using cells with different passage numbers. Therefore, we suggest that the passage number should always be reported to ensure that the experiment is as reproducible as possible. Furthermore, our modelling also suggests some avenues for further experimental examination that could be used to validate or refine our simulation results.

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell.

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

A Bayesian Computational Approach to Explore the Optimal Duration of a Cell Proliferation Assay.

Cell proliferation assays are routinely used to explore how a low-density monolayer of cells grows with time. For a typical cell line with a doubling time of 12 h (or longer), a standard cell proliferation assay conducted over 24 h provides excellent information about the low-density exponential growth rate, but limited information about crowding effects that occur at higher densities. To explore how we can best detect and quantify crowding effects, we present a suite of in silico proliferation assays where cells proliferate according to a generalised logistic growth model. Using approximate Bayesian computation we show that data from a standard cell proliferation assay cannot reliably distinguish between classical logistic growth and more general non-logistic growth models. We then explore, and quantify, the trade-off between increasing the duration of the experiment and the associated decrease in uncertainty in the crowding mechanism.

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices.

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.

Logistic Proliferation of Cells in Scratch Assays is Delayed.

Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction-diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation; however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay; at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.