Christian A. Yates

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.

Ergodic directional switching in mobile insect groups

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the groups direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

Simplified Multitarget Tracking Using the PHD Filter for Microscopic Video Data

The probability hypothesis density (PHD) filter from the theory of random finite sets is a well-known method for multitarget tracking. We present the Gaussian mixture (GM) and improved sequential Monte Carlo implementations of the PHD filter for visual tracking. These implementations are shown to provide advantages over previous PHD filter implementations on visual data by removing complications such as clustering and data association and also having beneficial computational characteristics. The GM-PHD filter is deployed on microscopic visual data to extract trajectories of free-swimming bacteria in order to analyze their motion. Using this method, a significantly larger number of tracks are obtained than was previously possible. This permits calculation of reliable distributions for parameters of bacterial motion. The PHD filter output was tested by checking agreement with a careful manual analysis. A comparison between the PHD filter and alternative tracking methods was carried out using simulated data, demonstrating superior performance by the PHD filter in a range of realistic scenarios.

Ten Simple Rules for Effective Computational Research

In order to attempt to understand the complexity inherent in nature, mathematical, statistical and computational techniques are increasingly being employed in the life sciences. In particular, the use and development of software tools is becoming vital for investigating scientific hypotheses, and a wide range of scientists are finding software development playing a more central role in their day-to-day research. In fields such as biology and ecology, there has been a noticeable trend towards the use of quantitative methods for both making sense of ever-increasing amounts of data [1] and building or selecting models [2]. As Research Fellows of the “2020 Science” project (http://www.2020science.net), funded jointly by the EPSRC (Engineering and Physical Sciences Research Council) and Microsoft Research, we have firsthand experience of the challenges associated with carrying out multidisciplinary computation-based science [3]–[5]. In this paper we offer a jargon-free guide to best practice when developing and using software for scientific research. While many guides to software development exist, they are often aimed at computer scientists [6] or concentrate on large open-source projects [7]; the present guide is aimed specifically at the vast majority of scientific researchers: those without formal training in computer science. We present our ten simple rules with the aim of enabling scientists to be more effective in undertaking research and therefore maximise the impact of this research within the scientific community. While these rules are described individually, collectively they form a single vision for how to approach the practical side of computational science. Our rules are presented in roughly the chronological order in which they should be undertaken, beginning with things that, as a computational scientist, you should do before you even think about writing any code. For each rule, guides on getting started, links to relevant tutorials, and further reading are provided in the supplementary material (Text S1).

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion

Spatial reaction–diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction–diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as ‘compartment-based’ or ‘on-lattice’. These models are characterized by a discretization of the computational domain into a grid/lattice of ‘compartments’. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: ‘how are individual particles transported between the vastly different model descriptions?’ First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations.

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.

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.

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.

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.

Using approximate Bayesian computation to quantify cell–cell adhesion parameters in a cell migratory process

In this work, we implement approximate Bayesian computational methods to improve the design of a wound-healing assay used to quantify cell–cell interactions. This is important as cell–cell interactions, such as adhesion and repulsion, have been shown to play a role in cell migration. Initially, we demonstrate with a model of an unrealistic experiment that we are able to identify model parameters that describe agent motility and adhesion, given we choose appropriate summary statistics for our model data. Following this, we replace our model of an unrealistic experiment with a model representative of a practically realisable experiment. We demonstrate that, given the current (and commonly used) experimental set-up, our model parameters cannot be accurately identified using approximate Bayesian computation methods. We compare new experimental designs through simulation, and show more accurate identification of model parameters is possible by expanding the size of the domain upon which the experiment is performed, as opposed to increasing the number of experimental replicates. The results presented in this work, therefore, describe time and cost-saving alterations for a commonly performed experiment for identifying cell motility parameters. Moreover, this work will be of interest to those concerned with performing experiments that allow for the accurate identification of parameters governing cell migratory processes, especially cell migratory processes in which cell–cell adhesion or repulsion are known to play a significant role.Math heals: Using computational methods to improve design of wound healing assayCell motility is a central process in wound healing and relies on complex cell-cell interactions. A team of mathematicians led by Ruth Baker and Kit Yates at the University of Oxford utilised computer simulations to re-design wound-healing assays that efficiently identify cell motility parameters. New experimental designs through computer simulation can more accurately identify cell motility parameters by expanding the size of the domain upon which the experiment is performed, as opposed to increasing the number of experimental replicates. The results describe time and cost-saving alterations for an experimental method for evaluate complex cell-cell interactions.