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Introduction
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September 1981 - August 1983
January 1986 - present
Publications
Publications (128)
Dual control denotes a class of control problems where the parameters governing the system are imperfectly known. The challenge is to find the optimal balance between probing, i.e. exciting the system to understand it more, and caution, i.e. selecting conservative controls based on current knowledge to achieve the control objective. Dynamic program...
We consider a one‐dimensional aggregation equation for a non‐negative density ρ(x,t) associated with a quartic potential W(x)=βx2+δx4 ( δ>0, β∈R). We show that for the case of symmetric initial data [ ρ(x,0)≡ρ(−x,0)], the solution of the aggregation equation can be expressed in terms of an explicit function of x, L(t), and Φ(t), where the functions...
Murine oligodendrocyte generation dynamics are considered distinct from those in the human, with implications for cross-species differences in neural homeostasis, injury response and ability to functionally adapt circuits through myelin plasticity. We identify that murine oligodendrocyte precursors do not vary their cell division times in vivo and...
In many biological systems, motile agents exhibit random motion with short-term directional persistence, together with crowding effects arising from spatial exclusion. We formulate and study a class of lattice-based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions, and use a mean-field approximatio...
In many biological systems, motile agents exhibit random motion with short-term directional persistence, together with crowding effects arising from spatial exclusion. We formulate and study a class of lattice-based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions, and use a mean-field approximatio...
We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allo...
In the version of this Comment originally published, equation (4) was incorrect; see the correction notice for details. This has now been corrected in the online versions of the Comment.
Measurement of the force between two atoms is performed routinely with the atomic force microscope. The shape of this interatomic force law is now found to directly regulate this capability: rapidly varying interatomic force laws, which are common in nature, can corrupt their own measurement.
Bullock et al. (Journal of Ecology 105:6-19, 2017) have suggested that the theory behind the Wald Analytical Long Distance (WALD) model for wind dispersal from a point source needs to be re-examined. This is on the basis that an inverse Gaussian probability density function (pdf) does not provide the best fit to seed shadows around individual sourc...
Motivated by in vitro time–lapse images of ovarian cancer spheroids inducing mesothelial cell clearance, the traditional agent–based model of cell migration, based on simple volume exclusion, was extended to include the possibility that a cell seeking to move into an occupied location may push the resident cell, and any cells neighbouring it, out o...
Crypt fission is an in vivo tissue deformation process that is involved in both intestinal homeostasis and colorectal tumourigenesis. Despite its importance, the mechanics underlying crypt fission are currently poorly understood. Recent experimental development of organoids, organ-like buds cultured from crypt stem cells in vitro, has shown promise...
Atomically-resolved imaging and force measurements using the atomic force microscope (AFM) are performed most commonly in a frequency-modulation (FM) mode. This has led to spectacular results, including direct observation of the atomic structure of complex molecules and quantification of chemical and frictional forces at the atomic scale. We addres...
The myelin sheath that insulates some axons in the central nervous system allows for faster signal conduction. Previously, axons were thought to be either unmyelinated or fully myelinated. Recent experimental work has discovered a new pattern of myelination (intermittent myelination) along axons in the mouse brain, in which long unmyelinated axon s...
In some disease systems, the process of waning immunity can be subtle, involving a complex relationship between the duration of immunity—acquired either through natural infection or vaccination—and subsequent boosting of immunity through asymptomatic re-exposure. We present and analyse a model of infectious disease transmission where primary and se...
Despite high vaccine coverage, pertussis has re-emerged as a public health concern in many countries. One hypothesis posed for re-emergence is the waning of immunity. In some disease systems, the process of waning immunity can be non-linear, involving a complex relationship between the duration of immunity and subsequent boosting of immunity throug...
The cellular mechanisms that regulate the topographic arrangement of myelin internodes along axons remain largely uncharacterized. Recent clonal analysis of oligodendrocyte morphologies in the mouse optic nerve revealed that adjacent oligodendrocytes frequently formed adjacent internodes on one or more axons in common, whereas oligodendrocytes in t...
Triply shared myelination.
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Calculating the overall probability of observing unique myelination given the data in Dumas et al. (2015).
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Sensitivity analysis.
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Schematic of how an OL in our simulation model may myelinate the same axon twice given the internode and maximum primary process length constraints.
(PDF)
We develop an agent-based model of vasculogenesis, the de novo formation of blood vessels. Endothelial cells in the vessel network are viewed as linearly elastic spheres and are of two types: vessel elements are contained within the network; tip cells are located at endpoints. Tip cells move in response to forces due to interactions with neighbouri...
Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logi...
Oligodendrocytes are the myelin-producing cells of the central nervous system that are responsible for electrically insulating axons to speed the propagation of electrical impulses. A striking feature of oligodendrocyte development within white matter is that the cell bodies of many oligodendrocyte progenitor cells become organized into discrete li...
Incidence of whooping cough, an infection caused by Bordetella pertussis and Bordetella parapertussis, has been on the rise since the 1980s in many countries. Immunological interactions, such as immune boosting and cross-immunity between pathogens, have been hypothesised to be important drivers of epidemiological dynamics. We present a two-pathogen...
Incidence of whooping cough, an infection caused by Bordetella pertussis and Bordetella parapertussis, has been on the rise since the 1980s in many countries. Immunological interactions, such as immune boosting and cross-immunity between pathogens, have been hypothesised to be important drivers of epidemiological dynamics. We present a two-pathogen...
An enduring puzzle in evolutionary biology is to understand how individuals and populations adapt to fluctuating environments. Here we present an integro-differential model of adaptive dynamics in a phenotype-structured population whose fitness landscape evolves in time due to periodic environmental oscillations. The analytical tractability of our...
A single mathematical theme underpins disparate physical phenomena in classical, quantum and statistical mechanical contexts. This mathematical "correspondence principle", a kind of wave-particle duality with glorious realizations in classical and modern mathematical analysis, embodies fundamental geometrical and physical order, and yet in some sen...
Stochastic agent-based models are useful for modelling collective movement of biological cells. Lattice-based random walk models of interacting agents where each site can be occupied by at most one agent are called simple exclusion processes. An alternative motility mechanism to simple exclusion is formulated, in which agents are granted more freed...
The characteristic six-layered appearance of the neocortex arises from the correct positioning of pyramidal neurons during development and alterations in this process can cause intellectual disabilities and developmental delay. Malformations in cortical development arise when neurons either fail to migrate properly from the germinal zones or fail t...
We consider a model introduced by Baker et al. [Phys. Rev. E 88, 042113 (2013)] of a single lattice random walker moving on a domain of allowed sites, surrounded by blocked sites. The walker enlarges the allowed domain by eroding the boundary at its random encounters with blocked boundary sites: attempts to step onto blocked sites succeed with a gi...
We consider a discrete agent-based model on a one-dimensional lattice, where each agent occupies L sites and attempts movements over a distance of d lattice sites. Agents obey a strict simple exclusion rule. A discrete-time master equation is derived using a mean-field approximation and careful probability arguments. In the continuum limit, nonline...
We consider a class of lattice random walk models in which the random walker is initially confined to a finite connected set of allowed sites but has the opportunity to enlarge this set by colliding with its boundaries, each such collision having a given probability of breaking through. The model is motivated by an analogy to cell motility in tissu...
Mathematical models of swarms of moving agents with non-local
interactions have many applications and have been the subject of
considerable recent interest. For modest numbers of agents, cellular
automata or related algorithms can be used to study such systems, but in
the present work, instead of considering discrete agents, we discuss a
class of o...
The gubernaculum is postulated to grow like an embryonic limb bud during inguinoscrotal descent in rodents. Recently, modelling of limb bud growth suggests the undifferentiated, distal "progress zone" provides molecular morphogenic signals, rather than cell division, as previously thought. We aimed to develop a mathematical gubernacular growth mode...
Cell motility is a fundamental physiological process that regulates cellular fate in healthy and diseased systems. Cells cultured in 3D environments often exhibit biphasic dependence of migration speed with cell adhesion. Much is not understood about this very common behavior. A phenomenological model for 3D single-cell migration that exhibits biph...
Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the for...
We consider a discrete agent-based model on a one-dimensional lattice and a two-dimensional square lattice, where each agent is a dimer occupying two sites. Agents move by vacating one occupied site in favor of a nearest-neighbor site and obey either a strict simple exclusion rule or a weaker constraint that permits partial overlaps between dimers....
Measurement of the power spectral density of (stochastic) Brownian fluctuations of micro- and nano-devices is used frequently to gain insight into their mechanistic properties. Noise is always present in these measurements and can directly influence any parameter estimation obtained through a least-squares analysis. Importantly, measurements of the...
A discrete agent-based model on a periodic lattice of arbitrary dimension is considered. Agents move to nearest-neighbor sites by a motility mechanism accounting for general interactions, which may include volume exclusion. The partial differential equation describing the average occupancy of the agent population is derived systematically. A diffus...
The thermal noise spectrum of nanomechanical devices is commonly used to characterize their mechanical properties and energy dissipation. This spectrum is measured from finite time series of Brownian motion of the device, which is windowed and Fourier transformed. Here, we present a theoretical and experimental investigation of the effect of such f...
Motivated by examples in developmental biology and ecology, we develop a model for convection-dominated invasion of a spatial region by initially motile agents which are able to settle permanently. The motion of the motile agents and their rate of settling are affected by the local concentration of settled agents. The model can be formulated as a n...
Sequential segmentation during embryogenesis involves the generation of a repeated pattern along the embryo, which is concurrently undergoing axial elongation by cell division. Most mathematical models of sequential segmentation involve inherent cellular oscillators, acting as a segmentation clock. The cellular oscillation is assumed to be governed...
In many developmental systems, spatial pattern arises from morphogen gradients, which provide positional information for cells to determine their fate. Typically, diffusion is thought to be the mechanism responsible for building a morphogen gradient. An alternative mechanism is investigated here. Using mathematical modeling, we demonstrate how a no...
Cell invasion involves a population of cells which are motile and proliferative. Tra-ditional discrete models of proliferation involve agents depositing daughter agents on nearest-neighbor lattice sites. Motivated by time-lapse images of cell invasion, we propose and analyze two new discrete proliferation models in the context of an exclusion proce...
Experimental observations of cell migration often describe the presence of mesoscale patterns within motile cell populations. These patterns can take the form of cells moving as aggregates or in chain-like formation. Here we present a discrete model capable of producing mesoscale patterns. These patterns are formed by biasing movements to favor a p...
Trajectory data from observations of a random-walk process are often used to characterize macroscopic transport coefficients and to make inferences about motility mechanisms. Continuum equations describing the average moments of the position of an agent in an exclusion process are derived and validated with simulation data. Unlike standard noninter...
Cell invasion is the basis of several fundamental biological systems including developmental morphogenesis and disease progression. Invasion processes involve combined cell motility and proliferation. Standard experimental approaches to characterize invasion systems focus on measuring population-level wavespeed data. However, continuum models which...
A motility mechanism based on a simple exclusion process, where the movement of discrete agents on a lattice is either unbiased (symmetric) or biased (asymmetric) is considered. Estimates of diffusivities from tracking data do not describe the population-level response of the system. This mismatch between the individual-level and population-level b...
The production of neurons to form the mammalian cortex, known as embryonic cortical neurogenesis, is a complex developmental process. Insight into the process of cell division during neurogenesis is provided by murine cortical cell lineage trees, recorded through experimental observation. Recurring patterns within cell lineage trees may be indicati...
The interaction of living cells with surfaces is important in applications of biomaterials, such as tissue engineering. Characterising and modelling the attachment, migration and proliferation of cells on materials used for tissue engineering provides valuable insight into their potential applications as well as a means of objective comparison. In...
Random walk phenomena abound in engineering contexts, from pedestrian traffic to cell motility in tissue engineering. We contrast two random walk models. The ghost model involves individuals who pass through each other unhindered. The folks model involves agents that interact by refusing to share the same location. Simple simulations reveal behavio...
Abstract The most complete form of academic timetabling problem is the population and course timetabling problem. In this problem, there may be multiple classes of each subject, and the decision on which students are to constitute each class is made in concert with the decision on the timetable for each class. In order to solve this problem, it is...
T cell development occurs in the thymus throughout life. Recent experimental findings show that the seeding of the thymus by multi-potent stem cells from the bone marrow is periodic rather than continuous, as previously assumed. However it is well known that the output rate of cells from the thymus is relatively constant. A quantitative model is us...
During a wound-healing cell migration assay experiment, cells are observed to detach and undergo mitosis before reattaching as a pair of cells on the substrate. During experiments with mice 3T3 fibroblasts, cell detachment can be confined to the wavefront region or it can occur over the whole wave profile. A multi-species continuum approach to mode...
Interpretive and predictive tools are needed to assist in the understanding of cell invasion processes. Cell invasion involves cell motility and proliferation, and is central to many biological processes including developmental morphogenesis and tumor invasion. Experimental data can be collected across a wide range of scales, from the population sc...
A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhib...
This article deals with the theoretical size distribution (of number of sub-taxa) of a fossil taxon arising from a simple null model of macroevolution.
New species arise through speciations occurring independently and at random at a fixed probability rate, while extinctions either occur independently and at random (background extinctions) or catacl...
Recently, a suite of cell migration assays were conducted to investigate the migration of neural crest (NC) cells along the gut during the development of the enteric nervous system (ENS). The NC cells colonise the gastro-intestinal tract as a rostro-caudal wave. Local behaviour was shown to be controlled by position relative to the leading edge of...
We introduce and review a number of topics drawn from the theories of random processes and random systems. In particular we
address the following subjects: random walks in continuous spaces and on lattices; continuum limits of random walks and stable
distributions; master equations, generalized master equations and continuous-time random walks; sel...
We discuss a class of models for the evolution of tree networks in which new nodes are recruited into the network at random times, and nodes already in the network may die at random times. Stochastic mechanisms for growth and death of the network that are either sensitive or insensitive to the coordination number or degree of nodes are studied usin...
Although cell migration is an essential process in development, how cells reach their final destination is not well understood. Secreted molecules are known to have a migratory effect, but it remains unclear whether such molecules act as directional guidance cues or as motility regulators. There is potential to use signalling molecules in new medic...
We derive expressions for the density of the time to ruin given that ruin occurs in a Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are distributed as Erlang(n, β). We provide numerical illustrations of finite time ruin prob-abilities, as well as illustrating features of the density fu...
This article deals with the theoretical size distribution of gene and protein families in complete genomes. A simple evolutionary model for the development of such families in which genes in a family are formed or selected against independently and at random, and in which new families are formed by the random splitting of existing families, is used...
We discuss a class of models for the evolution of networks in which new nodes are recruited into the network at random times, and links between existing nodes that are not yet directly connected may also form at random times. The class contains both models that produce "small-world" networks and less tightly linked models. We produce both trees, ap...
Examination timetabling is a well-studied combinatorial optimization problem. We present a new hybrid algorithm for examination
timetabling, consisting of three phases: a constraint programming phase to develop an initial solution, a simulated annealing
phase to improve the quality of solution, and a hill climbing phase for further improvement. The...
We present a simple explanation for the occurrence of power-law tails in statistical distributions by showing that if stochastic processes with exponential growth in expectation are killed (or observed) randomly, the distribution of the killed or observed state exhibits power-law behavior in one or both tails. This simple mechanism can explain powe...
A possible explanation for the frequent occurrence of power-law distributions in biology and elsewhere comes from an analysis of the interplay between random time evolution and random observation or killing time. If the system population or its topological parameters grow exponentially with time, and observations on the system correspond to stoppin...
We present a stochastic model for the size of a taxon in paleobiology, in which we allow for the evolution of new taxon members, and both individual and catastrophic extinction events. The model uses ideas from the theory of birth and death processes. Some general properties of the model are developed, and a fuller discussion is given for specific...
We present a model for the distribution of family names that explains the power-law decay of the probability distribution for the number of people with a given family name. The model includes a description of the process of generation or importation of new names, and a description of the growth of the number of individuals with a name, and correspo...
This article deals with the theoretical size (number of species) distribution of live genera, arising from a simple model of macroevolution in which speciations and extinctions are assumed to occur independently and at random, and in which new genera are formed by the random splitting of existing genera. Mathematically, the distribution is that of...
Examination timetabling is a well-studied combinatorial optimization problem. We present a new hybrid algorithm for examination timetabling, consisting of three phases: a constraint programming phase to develop an initial solution, a simulated annealing phase to improve the quality of solution, and a hill climbing phase for further improvement.
The evolution equations in real space and time corresponding to a class of anomalous diffusion processes are examined. As special cases, evolution equations corresponding to stable processes are derived using the theory of generalized functions, recovering some known results differently interpreted, and an evolution law for stable processes of orde...
Calculates the critical exponent of percolation conductivity t for two- and three-dimensional networks by using a finite-size scaling technique. In two dimensions the authors obtain t=1.264+or-0.054, in excellent agreement with the recent transfer-matrix calculation of Derrida and Vannimenus (t=1.28+or-0.03, 1982). In three dimensions the finite-si...
The authors give the exact solution in one dimension of the continuum analogue of a model of Stanley et al. (1983) which interpolates between Markovian (Polya) and self-avoiding random walks on lattices, and make some heuristic comments on the nature of phase transitions in the model in higher dimensions.
Based on numerical evidence, the authors conjecture a connection between the bond percolation threshold of Bravais lattices in three or more dimensions and the value at the origin of a lattice Green function related to the probability of return to the origin for a Polya random walk.
A case is argued for the retention within our undergraduate and graduate applied mathematics programmes of classical analysis, as exemplified by the text of Whittaker and Watson (4th ed., 1927). The case is based around research experiences in applied probability, mathematical physics and chemical physics.
We discuss the concept of the surface energy of a solid. Our point of
view is that the surface enrgy, as classically defined, is but one
attribute of a more general distance-dependent interaction between
bodies. We argue that the mechanics of adhesion and fracture of solids
can be correctly modeled by the incorporation of classical surface
energy i...
The observed minimum in the normal stress beneath the highest portion of a ‘sandpile’ (a pile of granular material) is a counter-intuitive result that has long remained unexplained. In this paper, we suggest that spatial size differentiation, where the larger particles are separated from smaller particles within the pile, creates an increase in the...
The displacement of a high-viscosity non-Newtonian fluid by a low-viscosity Newtonian fluid in a Hele-Shaw cell is capable of producing ramified viscous-fingering patterns exhibiting fractal characteristics. Recently, it was established that interfacial tension has little influence on the formation of these fractal patterns. However, the precise me...
We examine in detail a model for transport in heterogeneous solids and porous media which contain N distinct families of transport paths (with N>=2), recently proposed by the authors [Phys. Rev. Lett. 70, 2581 (1993)]. The model is relevant to transport in metals, polycrystals, porous catalysts, coalbed methane reservoirs, and geological systems wi...
We present solutions for the effective stress induced by gas flow through a porous solid into a borehole resulting from sudden pressure reduction. Tensile effective stress that exceeds the strength of the solid will lead to borehole failure. This has applications to the intentional creation of cavities, relevant to the efficient recovery of coalbed...
We propose a new model for transport in disordered solids and rocks which contain N distinct families of transport paths (N>=2), and study the behavior of its effective transport properties. The model is relevant to transport in metals, polycrystals, porous catalysts, coalbed methane reservoirs, and geological systems with fractures and pores.
We present numerical solutions and analytical approximate solutions to problems of gas flow in porous media arising in the modelling of outbursts in coal mines and the efficient recovery of methane from coal seams.
We have constructed a discrete element computational code which solves the equations of motion that describe particle-particle interactions. As a first example we have applied the code to a two dimensional triangular pile of discrete particles. We take this to be a two dimensional analogue of a conical pile of particulate material. With this simple...
We examine some results and techniques of analytic number theory which have application, or potential application, in mathematical physics. We consider inversion formulae for lattice sums, various transformations of infinite series and products, functional equations and scaling relations, with selected applications in electrostatics and statistical...
As a primitive model for structural breakdown in elastic media, we analyze the failure of random resistor-fuse networks with various distributions of properties. We show that variations in breakdown voltage have a more significant effect than variations in resistance values. This is analogous to the fluid-displacement problem [D.Y.C. Chan, B. D. Hu...
Nan-xian Chen [Phys. Rev. Lett. 64, 1193 (1990)] has generalized a formula of classical algebraic number theory to continuous variables and noted some useful consequences of the generalization. We present an alternative view of this analysis, based on the Mellin transformation and Riemann's zeta function.