Implementation of linear minimum area enclosing traingle algorithm

This article has been made available through the Brunel Open Access Publishing Fund.An algorithm which computes the minimum area triangle enclosing a convex polygon in linear time already exists in the literature. The paper describing the algorithm also proves that the provided solution is optimal and a lower complexity sequential algorithm cannot exist. However, only a high-level description of the algorithm was provided, making the implementation difficult to reproduce. The present note aims to contribute to the field by providing a detailed description of the algorithm which is easy to implement and reproduce, and a benchmark comprising 10,000 variable sized, randomly generated convex polygons for illustrating the linearity of the algorithm

Spatial-temporal modelling and analysis of bacterial colonies with phase variable genes

2015 Copyright is held by the owner/author(s). This article defines a novel spatial-temporal modelling and analysis methodology applied to a systems biology case study, namely phase variation patterning in bacterial colony growth. We employ coloured stochastic Petri nets to construct the model and run stochastic simulations to record the development of the circular colonies over time and space. The simulation output is visualised in 2D, and sector-like patterns are automatically detected and analysed. Space is modelled using 2.5 dimensions considering both a rectangular and circular geometry, and the effects of imposing different geometries on space are measured. We close by outlining an interpretation of the Petri net model in terms of finite difference approximations of partial differential equations (PDEs). One result is the derivation of the “best” nine-point diffusion model. Our multidimensional modelling and analysis approach is a precursor to potential future work on more complex multiscale modelling.EPSRC Research Grant EP I036168/1; German BMBF Research Grant 0315449H

A Novel Method to Verify Multilevel Computational Models of Biological Systems Using Multiscale Spatio-Temporal Meta Model Checking

Insights gained from multilevel computational models of biological systems can be translated into real-life applications only if the model correctness has been verified first. One of the most frequently employed in silico techniques for computational model verification is model checking. Traditional model checking approaches only consider the evolution of numeric values, such as concentrations, over time and are appropriate for computational models of small scale systems (e.g. intracellular networks). However for gaining a systems level understanding of how biological organisms function it is essential to consider more complex large scale biological systems (e.g. organs). Verifying computational models of such systems requires capturing both how numeric values and properties of (emergent) spatial structures (e.g. area of multicellular population) change over time and across multiple levels of organization, which are not considered by existing model checking approaches. To address this limitation we have developed a novel approximate probabilistic multiscale spatio-temporal meta model checking methodology for verifying multilevel computational models relative to specifications describing the desired/expected system behaviour. The methodology is generic and supports computational models encoded using various high-level modelling formalisms because it is defined relative to time series data and not the models used to generate it. In addition, the methodology can be automatically adapted to case study specific types of spatial structures and properties using the spatio-temporal meta model checking concept. To automate the computational model verification process we have implemented the model checking approach in the software tool Mule (http://mule.modelchecking.org). Its applicability is illustrated against four systems biology computational models previously published in the literature encoding the rat cardiovascular system dynamics, the uterine contractions of labour, the Xenopus laevis cell cycle and the acute inflammation of the gut and lung. Our methodology and software will enable computational biologists to efficiently develop reliable multilevel computational models of biological systems

Workflow for creating multiscale spatio-temporal model checking methodology instances.

<p>The workflow comprises two levels, the upper generic (meta) level, and the lower specific (instance) level. The upper level comprises the multiscale spatio-temporal meta model checking methodology. Conversely the lower level consists of the specific collections of spatial entity types and measures employed to create multiscale spatio-temporal model checking methodology instances. For each considered pair (e.g. m) of spatial entity types and spatial measures collections a corresponding multiscale model checking methodology instance is created. The resulting methodology instances (e.g. m) can then be employed for various case studies (e.g. n) to decide if computational models (e.g. m,n) are correct relative to corresponding formal specifications (e.g. m,n) or not. Rounded rectangles and arrows having the same border/line colour correspond to the same collections of spatial entity types and spatial measures.</p

Model simulation and analysis execution times for the rat cardiovascular system dynamics, the uterine contractions of labour, the <i>Xenopus laevis</i> cell cycle, and the acute inflammation of the gut and lung case studies.

<p>Model simulation and analysis execution times for the rat cardiovascular system dynamics, the uterine contractions of labour, the <i>Xenopus laevis</i> cell cycle, and the acute inflammation of the gut and lung case studies.</p

The state space of the system i.e. all possible states which can be reached from the initial state <i>S</i>₀.

<p><i>Cell</i> and <i>A</i>_<i>extracellular</i> are the spatial state variables representing the position of the cell, respectively distribution of nutrient <i>A</i> in the environment. <i>A</i>_<i>intracellular</i> and <i>Energy</i> represent the intracellular availability of nutrient <i>A</i>, respectively energy. The percentage associated with the arrows connecting each pair of states represents the probability of transitioning from one state to the other.</p

Initial state of the system.

<p><i>Cell</i> and <i>A</i>_<i>extracellular</i> are the spatial state variables representing the position of the cell, respectively distribution of nutrient <i>A</i> in the environment. <i>A</i>_<i>intracellular</i> and <i>Energy</i> represent the intracellular availability of nutrient <i>A</i>, respectively energy.</p

The multiscale spatio-temporal analysis workflow.

<p>An MSSpDES model of the system under consideration is constructed and simulated to generate time series data. This time series data is split up into subsets (1) such that each subset corresponds to a single subsystem and scale. The time series data subsets are passed to a uniscale spatio-temporal analysis module (2) which automatically detects, analyses and annotates spatial entities with their corresponding scale and subsystem. The results of the uniscale spatio-temporal analysis are then merged (3) such that spatial entities corresponding to the same time point are grouped together. If more simulations are required, a new time series dataset is generated, for which steps (1)–(3) are repeated.</p

Natural language descriptions of the formal specifications employed for the rat cardiovascular system dynamics, the uterine contractions of labour, the <i>Xenopus laevis</i> cell cycle, and the acute inflammation of the gut and lung case studies.

<p>Natural language descriptions of the formal specifications employed for the rat cardiovascular system dynamics, the uterine contractions of labour, the <i>Xenopus laevis</i> cell cycle, and the acute inflammation of the gut and lung case studies.</p