Rapid mechanosensitive migration and dispersal of newly divided mesenchymal cells aid their recruitment into dermal condensates.

Embryonic mesenchymal cells are dispersed within an extracellular matrix but can coalesce to form condensates with key developmental roles. Cells within condensates undergo fate and morphological changes and induce cell fate changes in nearby epithelia to produce structures including hair follicles, feathers, or intestinal villi. Here, by imaging mouse and chicken embryonic skin, we find that mesenchymal cells undergo much of their dispersal in early interphase, in a stereotyped process of displacement driven by 3 hours of rapid and persistent migration followed by a long period of low motility. The cell division plane and the elevated migration speed and persistence of newly born mesenchymal cells are mechanosensitive, aligning with tissue tension, and are reliant on active WNT secretion. This behaviour disperses mesenchymal cells and allows daughters of recent divisions to travel long distances to enter dermal condensates, demonstrating an unanticipated effect of cell cycle subphase on core mesenchymal behaviour.

Pleiotropic constraints promote the evolution of cooperation in cellular groups.

The evolution of cooperation in cellular groups is threatened by lineages of cheaters that proliferate at the expense of the group. These cell lineages occur within microbial communities, and multicellular organisms in the form of tumours and cancer. In contrast to an earlier study, here we show how the evolution of pleiotropic genetic architectures-which link the expression of cooperative and private traits-can protect against cheater lineages and allow cooperation to evolve. We develop an age-structured model of cellular groups and show that cooperation breaks down more slowly within groups that tie expression to a private trait than in groups that do not. We then show that this results in group selection for pleiotropy, which strongly promotes cooperation by limiting the emergence of cheater lineages. These results predict that pleiotropy will rapidly evolve, so long as groups persist long enough for cheater lineages to threaten cooperation. Our results hold when pleiotropic links can be undermined by mutations, when pleiotropy is itself costly, and in mixed-genotype groups such as those that occur in microbes. Finally, we consider features of multicellular organisms-a germ line and delayed reproductive maturity-and show that pleiotropy is again predicted to be important for maintaining cooperation. The study of cancer in multicellular organisms provides the best evidence for pleiotropic constraints, where abberant cell proliferation is linked to apoptosis, senescence, and terminal differentiation. Alongside development from a single cell, we propose that the evolution of pleiotropic constraints has been critical for cooperation in many cellular groups.

Synchronized oscillations in growing cell populations are explained by demographic noise.

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery.

The invasion speed of cell migration models with realistic cell cycle time distributions.

Cell proliferation is typically incorporated into stochastic mathematical models of cell migration by assuming that cell divisions occur after an exponentially distributed waiting time. Experimental observations, however, show that this assumption is often far from the real cell cycle time distribution (CCTD). Recent studies have suggested an alternative approach to modelling cell proliferation based on a multi-stage representation of the CCTD. In this paper we investigate the connection between the CCTD and the speed of the collective invasion. We first state a result for a general CCTD, which allows the computation of the invasion speed using the Laplace transform of the CCTD. We use this to deduce the range of speeds for the general case. We then focus on the more realistic case of multi-stage models, using both a stochastic agent-based model and a set of reaction-diffusion equations for the cells' average density. By studying the corresponding travelling wave solutions, we obtain an analytical expression for the speed of invasion for a general N-stage model with identical transition rates, in which case the resulting cell cycle times are Erlang distributed. We show that, for a general N-stage model, the Erlang distribution and the exponential distribution lead to the minimum and maximum invasion speed, respectively. This result allows us to determine the range of possible invasion speeds in terms of the average proliferation time for any multi-stage model.

Correction to: A Multi-stage Representation of Cell Proliferation as a Markov Process.

Equations (9) and (10) were transcribed incorrectly.

Zebrafish adult pigment stem cells are multipotent and form pigment cells by a progressive fate restriction process: Clonal analysis identifies shared origin of all pigment cell types.

Skin pigment pattern formation is a paradigmatic example of pattern formation. In zebrafish, the adult body stripes are generated by coordinated rearrangement of three distinct pigment cell-types, black melanocytes, shiny iridophores and yellow xanthophores. A stem cell origin of melanocytes and iridophores has been proposed although the potency of those stem cells has remained unclear. Xanthophores, however, seemed to originate predominantly from proliferation of embryonic xanthophores. Now, data from Singh et al. shows that all three cell-types derive from shared stem cells, and that these cells generate peripheral neural cell-types too. Furthermore, clonal compositions are best explained by a progressive fate restriction model generating the individual cell-types. The numbers of adult pigment stem cells associated with the dorsal root ganglia remain low, but progenitor numbers increase significantly during larval development up to metamorphosis, likely via production of partially restricted progenitors on the spinal nerves.

A Multi-stage Representation of Cell Proliferation as a Markov Process.

The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation-vital to the accurate modelling of many biological processes-whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.

Extending the Multi-level Method for the Simulation of Stochastic Biological Systems.

The multi-level method for discrete-state systems, first introduced by Anderson and Higham (SIAM Multiscale Model Simul 10(1):146-179, 2012), is a highly efficient simulation technique that can be used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. In this paper, we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multi-level method. The second part provides the means for a deft implementation of the technique and concludes with a discussion of a number of open problems.

Discrete and continuous models for tissue growth and shrinkage.

The incorporation of domain growth into stochastic models of biological processes is of increasing interest to mathematical modellers and biologists alike. In many situations, especially in developmental biology, the growth of the underlying tissue domain plays an important role in the redistribution of particles (be they cells or molecules) which may move and react atop the domain. Although such processes have largely been modelled using deterministic, continuum models there is an increasing appetite for individual-based stochastic models which can capture the fine details of the biological movement processes which are being elucidated by modern experimental techniques, and also incorporate the inherent stochasticity of such systems. In this work we study a simple stochastic model of domain growth. From a basic version of this model, Hywood et al. (2013) were able to derive a Fokker-Plank equation (FPE) (in this case an advection-diffusion partial differential equation on a growing domain) which describes the evolution of the probability density of some tracer particles on the domain. We extend their work so that a variety of different domain growth mechanisms can be incorporated and demonstrate a good agreement between the mean tracer density and the solution of the FPE in each case. In addition we incorporate domain shrinkage (via element death) into our individual-level model and demonstrate that we are able to derive coefficients for the FPE in this case as well. For situations in which the drift and diffusion coefficients are not readily available we introduce a numerical coefficient estimation approach and demonstrate the accuracy of this approach by comparing it with situations in which an analytical solution is obtainable.

Novel methods for analysing bacterial tracks reveal persistence in Rhodobacter sphaeroides.

Tracking bacteria using video microscopy is a powerful experimental approach to probe their motile behaviour. The trajectories obtained contain much information relating to the complex patterns of bacterial motility. However, methods for the quantitative analysis of such data are limited. Most swimming bacteria move in approximately straight lines, interspersed with random reorientation phases. It is therefore necessary to segment observed tracks into swimming and reorientation phases to extract useful statistics. We present novel robust analysis tools to discern these two phases in tracks. Our methods comprise a simple and effective protocol for removing spurious tracks from tracking datasets, followed by analysis based on a two-state hidden Markov model, taking advantage of the availability of mutant strains that exhibit swimming-only or reorientating-only motion to generate an empirical prior distribution. Using simulated tracks with varying levels of added noise, we validate our methods and compare them with an existing heuristic method. To our knowledge this is the first example of a systematic assessment of analysis methods in this field. The new methods are substantially more robust to noise and introduce less systematic bias than the heuristic method. We apply our methods to tracks obtained from the bacterial species Rhodobacter sphaeroides and Escherichia coli. Our results demonstrate that R. sphaeroides exhibits persistence over the course of a tumbling event, which is a novel result with important implications in the study of this and similar species.

Minimal reaction schemes for pattern formation.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Recycling random numbers in the stochastic simulation algorithm.

The stochastic simulation algorithm (SSA) was introduced by Gillespie and in a different form by Kurtz. Since its original formulation there have been several attempts at improving the efficiency and hence the speed of the algorithm. We briefly discuss some of these methods before outlining our own simple improvement, the recycling direct method (RDM), and demonstrating that it is capable of increasing the speed of most stochastic simulations. The RDM involves the statistically acceptable recycling of random numbers in order to reduce the computational cost associated with their generation and is compatible with several of the pre-existing improvements on the original SSA. Our improvement is also sufficiently simple (one additional line of code) that we hope will be adopted by both trained mathematical modelers and experimentalists wishing to simulate their model systems.

Stochastic drift in discrete waves of nonlocally interacting particles.

In this paper, we investigate a generalized model of N particles undergoing second-order nonlocal interactions on a lattice. Our results have applications across many research areas, including the modeling of migration, information dynamics, and Muller's ratchet-the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as (lnN)^{-2} and (lnN)^{-3}, respectively-much slower than the usual N^{-1/2} factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

Swapping in lattice-based cell migration models.

Cell migration is frequently modeled using on-lattice agent-based models (ABMs) that employ the excluded volume interaction. However, cells are also capable of exhibiting more complex cell-cell interactions, such as adhesion, repulsion, pulling, pushing, and swapping. Although the first four of these have already been incorporated into mathematical models for cell migration, swapping has not been well studied in this context. In this paper, we develop an ABM for cell movement in which an active agent can "swap" its position with another agent in its neighborhood with a given swapping probability. We consider a two-species system for which we derive the corresponding macroscopic model and compare it with the average behavior of the ABM. We see good agreement between the ABM and the macroscopic density. We also analyze the movement of agents at an individual level in the single-species as well as two-species scenarios to quantify the effects of swapping on an agent's motility.

Mathematical methods for scaling from within-host to population-scale in infectious disease systems.

Mathematical modellers model infectious disease dynamics at different scales. Within-host models represent the spread of pathogens inside an individual, whilst between-host models track transmission between individuals. However, pathogen dynamics at one scale affect those at another. This has led to the development of multiscale models that connect within-host and between-host dynamics. In this article, we systematically review the literature on multiscale infectious disease modelling according to PRISMA guidelines, dividing previously published models into five categories governing their methodological approaches (Garira (2017)), explaining their benefits and limitations. We provide a primer on developing multiscale models of infectious diseases.

Modelling cell migration and adhesion during development.

Cell-cell adhesion is essential for biological development: cells migrate to their target sites, where cell-cell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395-427, 2009) that incorporates both cell-cell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.

Inherent noise can facilitate coherence in collective swarm motion.

Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the self-propelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

Equivalence framework for an age-structured multistage representation of the cell cycle.

We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multistage Markov process for approximating cell cycle time distributions. We further demonstrate that the resulting mean behavior is equivalent, over large timescales, to the classical McKendrick-von Foerster integropartial differential equation. We conclude by extending this framework to a spatial context, facilitating the modeling of traveling wave phenomena and cell-mediated pattern formation. More generally, this methodology may be extended to myriad reaction-diffusion processes for which the age of individuals is relevant to the dynamics.

Misinformation can prevent the suppression of epidemics.

The effectiveness of non-pharmaceutical interventions, such as mask-wearing and social distancing, as control measures for pandemic disease relies upon a conscientious and well-informed public who are aware of and prepared to follow advice. Unfortunately, public health messages can be undermined by competing misinformation and conspiracy theories, spread virally through communities that are already distrustful of expert opinion. In this article, we propose and analyse a simple model of the interaction between disease spread and awareness dynamics in a heterogeneous population composed of both trusting individuals who seek better quality information and will take precautionary measures, and distrusting individuals who reject better quality information and have overall riskier behaviour. We show that, as the density of the distrusting population increases, the model passes through a phase transition to a state in which major outbreaks cannot be suppressed. Our work highlights the urgent need for effective interventions to increase trust and inform the public.

Critical weaknesses in shielding strategies for COVID-19.

The COVID-19 pandemic, caused by the coronavirus SARS-CoV-2, has led to a wide range of non-pharmaceutical interventions being implemented around the world to curb transmission. However, the economic and social costs of some of these measures, especially lockdowns, has been high. An alternative and widely discussed public health strategy for the COVID-19 pandemic would have been to 'shield' those most vulnerable to COVID-19 (minimising their contacts with others), while allowing infection to spread among lower risk individuals with the aim of reaching herd immunity. Here we retrospectively explore the effectiveness of this strategy using a stochastic SEIR framework, showing that even under the unrealistic assumption of perfect shielding, hospitals would have been rapidly overwhelmed with many avoidable deaths among lower risk individuals. Crucially, even a small (20%) reduction in the effectiveness of shielding would have likely led to a large increase (>150%) in the number of deaths compared to perfect shielding. Our findings demonstrate that shielding the vulnerable while allowing infections to spread among the wider population would not have been a viable public health strategy for COVID-19 and is unlikely to be effective for future pandemics.