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.
•New stochastic model of heterogeneous cell migration and proliferation.•Continuum limit related to a novel generalisation of classical logistic growth.•Parameterise model with experimentally-derived ...heterogeneous proliferation rates.•New quantitative framework to explore the implication of neglecting heterogeneity.•Perturbation solutions provide analytical insight into the role of heterogeneity.
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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.
Functional analyses of the 4.4 Ma hominin Ardipithecus ramidus postcrania revealed a previously unknown and unpredicted locomotor pattern combining arboreal clambering and a form of terrestrial ...bipedality. To date, all of the fossil evidence of Ar. ramidus locomotion has been collected from the Aramis area of the Middle Awash Research Project in Ethiopia. Here, we present the results of an analysis of additional early Pliocene Ar. ramidus fossils from the Gona Project study area, Ethiopia, that includes a fragmentary but informative partial skeleton (GWM67/P2) and additional isolated manual remains. While we reinforce the original functional interpretations of Ar. ramidus of having a mixed locomotor adaptation of terrestrial bipedality and arboreal clambering, we broaden our understanding of the nature of its locomotor pattern by documenting better the function of the hip, ankle, and foot. The newly recovered fossils document a greater adaptation to bipedality in the Ar. ramidus ankle and hallux than previously recognized. In addition, a newly discovered scaphoid bone with a fusing os centrale provides further evidence about the nature of hominin hand evolution.
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.
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,
κ
, 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
κ
so that commonly reported experimental estimates of
c
can be used to provide estimates of the unknown parameter
κ
. 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
c
<
2
are normally disregarded. This means that our analysis of the Fisher–Stefan model has both practical value and an inherent mathematical value.
Summary
Background
Recessive forms of congenital ichthyosis encompass a group of rare inherited disorders of keratinization leading to dry, scaly skin. So far, 13 genes have been implicated, but ...there is a paucity of data on genotype–phenotype correlation in some populations.
Objectives
We compiled an English cohort of 146 individuals with recessive ichthyosis and assessed genotype–phenotype correlation.
Methods
Deep phenotyping was undertaken by history‐taking and clinical examination. DNA was screened for mutations using a next‐generation sequencing ichthyosis gene panel and Sanger sequencing.
Results
Cases were recruited from 13 National Health Service sites in England, with 65% of patients aged < 16 years at enrolment. Pathogenic biallelic mutations were found in 83% of cases, with the candidate gene spread as follows: TGM1 29%, NIPAL4 12%, ABCA12 12%, ALOX12B 9%, ALOXE3 7%, SLC27A4 5%, CERS3 3%, CYP4F22 3%, PNPLA1 2%, SDR9C7 1%. Clinically, a new sign, an anteriorly overfolded ear at birth, was noted in 43% of patients with ALOX12B mutations. The need for intensive care stay (P = 0·004), and hand deformities (P < 0·001), were associated with ABCA12 mutations. Self‐improving collodion ichthyosis occurred in 8% of the cases (mostly TGM1 and ALOX12B mutations) but could not be predicted precisely from neonatal phenotype or genotype.
Conclusions
These data refine genotype–phenotype correlation for recessive forms of ichthyosis in England, demonstrating the spectrum of disease features and comorbidities, as well as the gene pathologies therein. Collectively, the data from these patients provide a valuable resource for further clinical assessment, improving clinical care and the possibility of future stratified management.
What's already known about this topic?
Recessive forms of ichthyosis are rare but often difficult to diagnose.
Mutations in 13 genes are known to cause recessive forms of ichthyosis: ABCA12, ALOX12B, ALOXE3, CERS3, CYP4F22, LIPN, NIPAL4, PNPLA1, SDR9C7, SLC27A4, SULT2B1, ST14 and TGM1.
Some phenotypic features may associate with certain gene mutations, but paradigms for genotype–phenotype correlation need refining.
What does this study add?
The genotypic spectrum of recessive ichthyosis in England (based on 146 cases) comprises TGM1 (29%), NIPAL4 (12%), ABCA12 (12%), ALOX12B (9%), ALOXE3 (7%), SLC27A4 (5%), CERS3 (3%), CYP4F22 (3%), PNPLA1 (2%) and SDR9C7 (1%).
New or particular phenotypic clues were defined for mutations in ALOX12B, ABCA12, CYP4F22, NIPAL4, SDR9C7 and TGM1, either in neonates or in later life, which allow for greater diagnostic precision.
In around 17% of cases, the molecular basis of recessive ichthyosis remains unknown.
Linked Editorial: Steele and O’Toole. Br J Dermatol 2020; 182:521–522.
Plain language summary available online
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,
u
^
(
x
^
,
t
^
)
that is coupled to the concentration of an immobile extracellular substrate,
s
^
(
x
^
,
t
^
)
. Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to
s
^
, while a logistic growth term models cell proliferation. The extracellular substrate
s
^
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,
c
=
c
min
, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed,
c
>
c
min
. 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
.
Abstract
Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first ...passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well-known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed-form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally-occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. MATLAB software to implement all calculations is available at
https://github.com/ProfMJSimpson/Exit_time
.
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.
Accurate characterization of sexual dimorphism is crucial in evolutionary biology because of its significance in understanding present and past adaptations involving reproductive and resource use ...strategies of species. However, inferring dimorphism in fossil assemblages is difficult, particularly with relatively low dimorphism. Commonly used methods of estimating dimorphism levels in fossils include the mean method, the binomial dimorphism index, and the coefficient of variation method. These methods have been reported to overestimate low levels of dimorphism, which is problematic when investigating issues such as canine size dimorphism in primates and its relation to reproductive strategies. Here, we introduce the posterior density peak (pdPeak) method that utilizes the Bayesian inference to provide posterior probability densities of dimorphism levels and within-sex variance. The highest posterior density point is termed the pdPeak. We investigated performance of the pdPeak method and made comparisons with the above-mentioned conventional methods via 1) computer-generated samples simulating a range of conditions and 2) application to canine crown-diameter datasets of extant known-sex anthropoids. Results showed that the pdPeak method is capable of unbiased estimates in a broader range of dimorphism levels than the other methods and uniquely provides reliable interval estimates. Although attention is required to its underestimation tendency when some of the distributional assumptions are violated, we demonstrate that the pdPeak method enables a more accurate dimorphism estimate at lower dimorphism levels than previously possible, which is important to illuminating human evolution.
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