831 research outputs found
Global dynamics of a novel delayed logistic equation arising from cell biology
The delayed logistic equation (also known as Hutchinson's equation or
Wright's equation) was originally introduced to explain oscillatory phenomena
in ecological dynamics. While it motivated the development of a large number of
mathematical tools in the study of nonlinear delay differential equations, it
also received criticism from modellers because of the lack of a mechanistic
biological derivation and interpretation. Here we propose a new delayed
logistic equation, which has clear biological underpinning coming from cell
population modelling. This nonlinear differential equation includes terms with
discrete and distributed delays. The global dynamics is completely described,
and it is proven that all feasible nontrivial solutions converge to the
positive equilibrium. The main tools of the proof rely on persistence theory,
comparison principles and an -perturbation technique. Using local
invariant manifolds, a unique heteroclinic orbit is constructed that connects
the unstable zero and the stable positive equilibrium, and we show that these
three complete orbits constitute the global attractor of the system. Despite
global attractivity, the dynamics is not trivial as we can observe long-lasting
transient oscillatory patterns of various shapes. We also discuss the
biological implications of these findings and their relations to other logistic
type models of growth with delays
An automatic adaptive method to combine summary statistics in approximate Bayesian computation
To infer the parameters of mechanistic models with intractable likelihoods,
techniques such as approximate Bayesian computation (ABC) are increasingly
being adopted. One of the main disadvantages of ABC in practical situations,
however, is that parameter inference must generally rely on summary statistics
of the data. This is particularly the case for problems involving
high-dimensional data, such as biological imaging experiments. However, some
summary statistics contain more information about parameters of interest than
others, and it is not always clear how to weight their contributions within the
ABC framework. We address this problem by developing an automatic, adaptive
algorithm that chooses weights for each summary statistic. Our algorithm aims
to maximize the distance between the prior and the approximate posterior by
automatically adapting the weights within the ABC distance function.
Computationally, we use a nearest neighbour estimator of the distance between
distributions. We justify the algorithm theoretically based on properties of
the nearest neighbour distance estimator. To demonstrate the effectiveness of
our algorithm, we apply it to a variety of test problems, including several
stochastic models of biochemical reaction networks, and a spatial model of
diffusion, and compare our results with existing algorithms
Travelling gradients in interacting morphogen systems
Morphogen gradients are well known to play several important roles in development; however the mechanisms underlying the formation and maintenance of these gradients are often not well understood. In this work, we investigate whether the presence of a secondary morphogen can increase the robustness of the primary morphogen gradient to perturbation, thereby providing a more stable mechanism for development. We base our model around the interactions of Fibroblast Growth Factor 8 and retinoic acid, which have been shown to act as morphogens in many developmental systems. In particular, we investigate the formation of opposing gradients of these morphogens along the antero-posterior axis of vertebrate embryos, thereby controlling temporal and spatial aspects of axis segmentation and neuronal differentiation
The impact of temporal sampling resolution on parameter inference for biological transport models
Imaging data has become widely available to study biological systems at
various scales, for example the motile behaviour of bacteria or the transport
of mRNA, and it has the potential to transform our understanding of key
transport mechanisms. Often these imaging studies require us to compare
biological species or mutants, and to do this we need to quantitatively
characterise their behaviour. Mathematical models offer a quantitative
description of a system that enables us to perform this comparison, but to
relate these mechanistic mathematical models to imaging data, we need to
estimate the parameters of the models. In this work, we study the impact of
collecting data at different temporal resolutions on parameter inference for
biological transport models by performing exact inference for simple velocity
jump process models in a Bayesian framework. This issue is prominent in a host
of studies because the majority of imaging technologies place constraints on
the frequency with which images can be collected, and the discrete nature of
observations can introduce errors into parameter estimates. In this work, we
avoid such errors by formulating the velocity jump process model within a
hidden states framework. This allows us to obtain estimates of the
reorientation rate and noise amplitude for noisy observations of a simple
velocity jump process. We demonstrate the sensitivity of these estimates to
temporal variations in the sampling resolution and extent of measurement noise.
We use our methodology to provide experimental guidelines for researchers
aiming to characterise motile behaviour that can be described by a velocity
jump process. In particular, we consider how experimental constraints resulting
in a trade-off between temporal sampling resolution and observation noise may
affect parameter estimates.Comment: Published in PLOS Computational Biolog
A mathematical investigation of a clock and wavefront model for somitogenesis
Abstract Somites are transient blocks of cells that form sequentially along the antero-posterior axis of vertebrate embryos. They give rise to the vertebrae, ribs and other associated features of the trunk. In this work we develop and analyse a mathematical formulation of a version of the Clock and Wavefront model for somite formation, where the clock controls when the boundaries of the somites form and the wavefront determines where they form. Our analysis indicates that this interaction between a segmentation clock and a wavefront can explain the periodic pattern of somites observed in normal embryos. We can also show that a simplification of the model provides a mechanism for predicting the anomalies resulting from perturbation of the wavefront
Waves and patterning in developmental biology: vertebrate segmentation and feather bud formation as case studies
In this article we will discuss the integration of developmental patterning mechanisms with waves of competency that control the ability of a homogeneous field of cells to react to pattern forming cues and generate spatially heterogeneous patterns. We base our discussion around two well known patterning events that take place in the early embryo: somitogenesis and feather bud formation. We outline mathematical models to describe each patterning mechanism, present the results of numerical simulations and discuss the validity of each model in relation to our example patterning processes
Formation of vertebral precursors: Past models and future predictions
Disruption of normal vertebral development results from abnormal formation and segmentation of the vertebral precursors, called somites. Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. We review the most popular models of somite formation: Cooke and Zeeman's clock and wavefront model, Meinhardt's reaction-diffusion model and the cell cycle model of Stern and co-workers, and discuss the consistency of each in the light of recent experimental findings concerning FGF-8 signalling in the presomitic mesoderm (PSM). We present an extension of the cell cycle model to take account of this new experimental evidence, which shows the existence of a determination front whose position in the PSM is controlled by FGF-8 signalling, and which controls the ability of cells to become competent to segment. We conclude that it is, at this stage, perhaps erroneous to favour one of these models over the others
Formation of vertebral precursors: Past models and future predictions
Disruption of normal vertebral development results from abnormal formation and segmentation of the vertebral precursors, called somites. Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. We review the most popular models of somite formation: Cooke and Zeeman's clock and wavefront model, Meinhardt's reaction-diffusion model and the cell cycle model of Stern and co-workers, and discuss the consistency of each in the light of recent experimental findings concerning FGF-8 signalling in the presomitic mesoderm (PSM). We present an extension of the cell cycle model to take account of this new experimental evidence, which shows the existence of a determination front whose position in the PSM is controlled by FGF-8 signalling, and which controls the ability of cells to become competent to segment. We conclude that it is, at this stage, perhaps erroneous to favour one of these models over the others
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