arXiv subjects

Philip K. Maini

Publications and source records attributed to Philip K. Maini.

At least 55 records · Page 3Linked to original sources

arXiv2018

Patterns of non-normality in networked systems

Several mechanisms have been proposed to explain the spontaneous generation of self-organized patterns, hypothesised to play a role in the formation of many of the magnificent patterns observed in Nature. In several cases of interest, the system under scrutiny displays a homogeneous equilibrium, which is destabilized via a symmetry breaking instability which reflects the specificity of the problem being inspected. The Turing instability is among the most celebrated paradigms for pattern formation. In its original form, the diffusion constants of the two mobile species need to be quite different from each other for the instability to develop. Unfortunately, this condition limits the applicability of the theory. To overcome this impediment, and with the ambitious long term goal to eventually reconcile theory and experiments, we here propose an alternative mechanism for promoting the onset of patterns. To this end a multi-species reaction-diffusion system is studied on a discrete, network-like support: the instability is triggered by the non-normality of the embedding network. The non-normal character of the dynamics instigates a short time amplification of the imposed perturbation, thus making the system unstable for a choice of parameters that would yield stability under the conventional scenario. Importantly, non-normal networks are pervasively found, as we shall here briefly review.

nlin.PS↗

arXiv2018

Particle-based simulations of reaction-diffusion processes with Aboria

Mathematical models of transport and reactions in biological systems have been traditionally written in terms of partial differential equations (PDEs) that describe the time evolution of population-level variables. In recent years, the use of stochastic particle-based models, which keep track of the evolution of each organism in the system, has become widespread. These models provide a lot more detail than the population-based PDE models, for example by explicitly modelling particle-particle interactions, but bring with them many computational challenges. In this paper we overview Aboria, a powerful and flexible C++ library for the implementation of numerical methods for particle-based models. We demonstrate the use of Aboria with a commonly used model in mathematical biology, namely cell chemotaxis. Cells interact with each other and diffuse, biased by extracellular chemicals, that can be altered by the cells themselves. We use a hybrid approach where particle-based models of cells are coupled with a PDE for the concentration of the extracellular chemical.

cs.CE↗

arXiv2017

Inferring parameters of prey switching in a plankton 1 predator--2 prey system with a linear preference tradeoff

We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models' ability to fit data on freshwater plankton. We model the predator's switch from one prey to the other in two different ways: (1) smooth switching using a hyperbolic tangent function; and (2) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model predictions quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German--Swiss--Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate--algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.

math.DS↗

arXiv2017

Mathematical modelling of calcium signalling taking into account mechanical effects

Most of the calcium in the body is stored in bone. The rest is stored elsewhere, and calcium signalling is one of the most important mechanisms of information propagation in the body. Yet, many questions remain open. In this work, we initially consider the mathematical model proposed in Atri et al. \cite{ atri1993single}. Omitting diffusion, the model is a system of two nonlinear ordinary differential equations (ODEs) for the calcium concentration, and the fraction of $IP_3$ receptors that have not been inactivated by the calcium. We analyse in detail the system as the $IP_3$ concentration, the \textit{bifurcation parameter}, increases presenting some new insights. We analyse asymptotically the relaxation oscillations of the model by exploiting a separation of timescales. Furthermore, motivated by experimental evidence that cells release calcium when mechanically stimulated and that, in turn, calcium release affects the mechanical behaviour of cells, we propose an extension of the Atri model to a 3D nonlinear ODE mechanochemical model, where the additional equation, derived consistently from a full viscoelastic \emph{ansatz}, models the evolution of cell/tissue dilatation. Furthermore, in the calcium dynamics equation we introduce a new "stretch-activation" source term that induces calcium release and which involves a new bifurcation parameter, the "strength" of the source. Varying the two bifurcation parameters, we analyse in detail the interplay of the mechanical and the chemical effects, and we find that as the strength of the mechanical stimulus is increased, the $IP_3$ parameter range for which oscillations emerge decreases, until oscillations eventually vanish at a critical value. Finally, we analyse the model when the calcium dynamics are assumed faster than the dynamics of the other two variables.

math.DS↗

arXiv2016

A predator-2 prey fast-slow dynamical system for rapid predator evolution

We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast--3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast--slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.

math.DS↗

arXiv2016

Optimisation of Simulations of Stochastic Processes by Removal of Opposing Reactions

Models invoking the chemical master equation are used in many areas of science, and, hence, their simulation is of interest to many researchers. The complexity of the problems at hand often requires considerable computational power, so a large number of algorithms have been developed to speed up simulations. However, a drawback of many of these algorithms is that their implementation is more complicated than, for instance, the Gillespie algorithm, which is widely used to simulate the chemical master equation, and can be implemented with a few lines of code. Here, we present an algorithm which does not modify the way in which the master equation is solved, but instead modifies the transition rates, and can thus be implemented with a few lines of code. It works for all models in which reversible reactions occur by replacing such reversible reactions with effective net reactions. Examples of such systems include reaction-diffusion systems, in which diffusion is modelled by a random walk. The random movement of particles between neighbouring sites is then replaced with a net random flux. Furthermore, as we modify the transition rates of the model, rather than its implementation on a computer, our method can be combined with existing algorithms that were designed to speed up simulations of the stochastic master equation. By focusing on some specific models, we show how our algorithm can significantly speed up model simulations while maintaining essential features of the original model.

physics.bio-ph↗

arXiv2016

The Critical Domain Size of Stochastic Population Models

Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population's ability to persist. Here we explore ways of including this variability. We study populations which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework, with distinct dispersal and sedentary stages. Individual based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up to the population level using the Central Limit Theorem, and the other a variation on both Galton-Watson branching processes and branching processes in random environments. These branching process models closely approximate the IBM and yield insight into the factors determining the critical domain size for a given population subject to stochasticity.

q-bio.PE↗

arXiv2015

Hybrid approaches for multiple-species stochastic reaction-diffusion models

Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. This way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

physics.comp-ph↗

arXiv2015

Multiscale modelling of intestinal crypt organization and carcinogenesis

Colorectal cancers are the third most common type of cancer. They originate from intestinal crypts, glands that descend from the intestinal lumen into the underlying connective tissue. Normal crypts are thought to exist in a dynamic equilibrium where the rate of cell production at the base of a crypt is matched by that of loss at the top. Understanding how genetic alterations accumulate and proceed to disrupt this dynamic equilibrium is fundamental to understanding the origins of colorectal cancer. Colorectal cancer emerges from the interaction of biological processes that span several spatial scales, from mutations that cause inappropriate intracellular responses to changes at the cell/tissue level, such as uncontrolled proliferation and altered motility and adhesion. Multiscale mathematical modelling can provide insight into the spatiotemporal organisation of such a complex, highly regulated and dynamic system. Moreover, the aforementioned challenges are inherent to the multiscale modelling of biological tissue more generally. In this review we describe the mathematical approaches that have been applied to investigate multiscale aspects of crypt behaviour, highlighting a number of model predictions that have since been validated experimentally. We also discuss some of the key mathematical and computational challenges associated with the multiscale modelling approach. We conclude by discussing recent efforts to derive coarse-grained descriptions of such models, which may offer one way of reducing the computational cost of simulation by leveraging well-established tools of mathematical analysis to address key problems in multiscale modelling.

q-bio.TO↗

arXiv2014

Reduction of chemical systems by delayed quasi-steady state assumptions

Mathematical analysis of mass action models of large complex chemical systems is typically only possible if the models are reduced. The most common reduction technique is based on quasi-steady state assumptions. To increase the accuracy of this technique we propose delayed quasi-steady state assumptions (D-QSSA) which yield systems of delay differential equations. We define the approximation based on D-QSSA, prove the corresponding error estimate, and show how it approximates the invariant manifold. Then we define a class of well mixed chemical systems and formulate assumptions enabling the application of D-QSSA. We also apply the D-QSSA to a model of Hes1 expression and to a cell-cycle model to illustrate the improved accuracy of the D-QSSA with respect to the standard quasi-steady state assumptions.

math.DS↗

arXiv2014

Mesoscopic and continuum modelling of angiogenesis

Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.

q-bio.TO↗

arXiv2013

A filter-flow perspective of hematogenous metastasis offers a non-genetic paradigm for personalized cancer therapy

Research into mechanisms of hematogenous metastasis has largely become genetic in focus, attempting to understand the molecular basis of `seed-soil' relationships. Preceeding this biological mechanism is the physical process of dissemination of circulating tumour cells (CTCs). We utilize a `filter-flow' paradigm to show that assumptions about CTC dynamics strongly affect metastatic efficiency: without data on CTC dynamics, any attempt to predict metastatic spread in individual patients is impossible.

q-bio.TO↗

arXiv2013

Mathematical modeling of the metastatic process

Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competition between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.

q-bio.TO↗

arXiv2013

A Markov chain model of evolution in asexually reproducing populations: insight and analytical tractability in the evolutionary process

The evolutionary process has been modelled in many ways using both stochastic and deterministic models. We develop an algebraic model of evolution in a population of asexually reproducing organisms in which we represent a stochastic walk in phenotype space, constrained to the edges of an underlying graph representing the genotype, with a time-homogeneous Markov Chain. We show its equivalence to a more standard, explicit stochastic model and show the algebraic model's superiority in computational efficiency. Because of this increase in efficiency, we offer the ability to simulate the evolution of much larger populations in more realistic genotype spaces. Further, we show how the algebraic properties of the Markov Chain model can give insight into the evolutionary process and allow for analysis using familiar linear algebraic methods.

q-bio.PE↗

arXiv2011

Advection, diffusion and delivery over a network

Many biological, geophysical and technological systems involve the transport of resource over a network. In this paper we present an algorithm for calculating the exact concentration of resource at any point in space or time, given that the resource in the network is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. We consider the implications of advection, diffusion and delivery for simple models of glucose delivery through a vascular network, and conclude that in certain circumstances, increasing the volume of blood and the number of glucose transporters can actually decrease the total rate of glucose delivery. We also consider the case of empirically determined fungal networks, and analyze the distribution of resource that emerges as such networks grow over time. Fungal growth involves the expansion of fluid filled vessels, which necessarily involves the movement of fluid. In three empirically determined fungal networks we found that the minimum currents consistent with the observed growth would effectively transport resource throughout the network over the time-scale of growth. This suggests that in foraging fungi, the active transport mechanisms observed in the growing tips may not be required for long range transport.

q-bio.TO↗

arXiv2010

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 non-monotonic behavior of several order/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 group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

q-bio.PE↗

arXiv2010

Growth-induced mass flows in fungal networks

Cord-forming fungi form extensive networks that continuously adapt to maintain an efficient transport system. As osmotically driven water uptake is often distal from the tips, and aqueous fluids are incompressible, we propose that growth induces mass flows across the mycelium, whether or not there are intrahyphal concentration gradients. We imaged the temporal evolution of networks formed by Phanerochaete velutina, and at each stage calculated the unique set of currents that account for the observed changes in cord volume, while minimising the work required to overcome viscous drag. Predicted speeds were in reasonable agreement with experimental data, and the pressure gradients needed to produce these flows are small. Furthermore, cords that were predicted to carry fast-moving or large currents were significantly more likely to increase in size than cords with slow-moving or small currents. The incompressibility of the fluids within fungi means there is a rapid global response to local fluid movements. Hence velocity of fluid flow is a local signal that conveys quasi-global information about the role of a cord within the mycelium. We suggest that fluid incompressibility and the coupling of growth and mass flow are critical physical features that enable the development of efficient, adaptive, biological transport networks.

q-bio.TO↗

arXiv2006

An analysis of B cell selection mechanisms in germinal centers

Affinity maturation of antibodies during immune responses is achieved by multiple rounds of somatic hypermutation and subsequent preferential selection of those B cells that express B cell receptors with improved binding characteristics for the antigen. The mechanism underlying B cell selection has not yet been defined. By employing an agent-based model, we show that for physiologically reasonable parameter values affinity maturation can neither be driven by competition for binding sites nor antigen -- even in the presence of competing secreted antibodies. Within the tested mechanisms, only clonal competition for T cell help or a refractory time for the interaction of centrocytes with follicular dendritic cells are found to enable affinity maturation while generating the experimentally observed germinal center characteristics and tolerating large variations in the initial antigen density.

q-bio.CB↗