Lagrangian particle calculations of distributive mixing: Limitations and applications

Discrete Manhattan and Chebyshev pair correlation functions in k dimensions

Pair correlation functions provide a summary statistic which quantifies the amount of spatial correlation between objects in a spatial domain. While pair correlation functions are commonly used to quantify continuous-space point processes, the on-lattice discrete case is less studied. Recent work has brought attention to the discrete case, wherein on-lattice pair correlation functions are formed by normalizing empirical pair distances against the probability distribution of random pair distances in a lattice with Manhattan and Chebyshev metrics. These distance distributions are typically derived on an ad hoc basis as required for specific applications. Here we present a generalized approach to deriving the probability distributions of pair distances in a lattice with discrete Manhattan and Chebyshev metrics, extending the Manhattan and Chebyshev pair correlation functions to lattices in k dimensions. We also quantify the variability of the Manhattan and Chebyshev pair correlation functions, which is important to understanding the reliability and confidence of the statistic.

Quantifying the dominant growth mechanisms of dimorphic yeast using a lattice-based model

A mathematical model is presented for the growth of yeast that incorporates both dimorphic behaviour and nutrient diffusion. The budding patterns observed in the standard and pseudohyphal growth modes are represented by a bias in the direction of cell proliferation. A set of spatial indices is developed to quantify the morphology and compare the relative importance of the directional bias to nutrient concentration and diffusivity on colony shape. It is found that there are three different growth modes: uniform growth, diffusion-limited growth (DLG) and an intermediate region in which the bias determines the morphology. The dimorphic transition due to nutrient limitation is investigated by relating the directional bias to the nutrient concentration, and this is shown to replicate the behaviour observed in vivo Comparisons are made with experimental data, from which it is found that the model captures many of the observed features. Both DLG and pseudohyphal growth are found to be capable of generating observed experimental morphologies.

Numerical Simulation of Chaotic Mixing in Single Screw Extruders with Different Baffle Heights

A kind of discontinuous baffle, which had the same length as the non-baffle zone distance, was inserted in the unwound channel of a single screw extruder to enhance mixing in the screw channel. The periodic unit of the flow channel was modeled as a dynamic system of complex duct flow. The finite volume method was used to solve the three-dimensional flow of purely viscous non-Newtonian fluid. Fluid particle tracking was performed by a fourth-order Runge–Kutta scheme. The effect of the baffle height on the mixing kinematics was investigated numerically. Poincaré sections were applied to reveal the different patterns and sizes of the KAM tubes. Distributive mixing was then visualized by the evolution of passive tracers initially located at different positions. The variance index and residence time distribution (RTD) were used to evaluate the statistical results. Among the four test cases, the results showed that the case with the baffle height equal to the screw depth had the largest chaotic mixing region, but the shortest mean residence time and the narrowest broadening of RTD under the same pressure gradient.

Quantifying two-dimensional filamentous and invasive growth spatial patterns in yeast colonies

The top-view, two-dimensional spatial patterning of non-uniform growth in a Saccharomyces cerevisiae yeast colony is considered. Experimental images are processed to obtain data sets that provide spatial information on the cell-area that is occupied by the colony. A method is developed that allows for the analysis of the spatial distribution with three metrics. The growth of the colony is quantified in both the radial direction from the centre of the colony and in the angular direction in a prescribed outer region of the colony. It is shown that during the period of 100–200 hours from the start of the growth of the colony there is an increasing amount of non-uniform growth. The statistical framework outlined in this work provides a platform for comparative quantitative assays of strain-specific mechanisms, with potential implementation in inferencing algorithms used for parameter-rate estimation.

In nutrient-depleted environments, it is commonly observed that strains of the yeast Saccharomyces cerevisiae forage by the mechanisms of filamentous and invasive growth. How do we quantify this spatial patterning of outward growth from a yeast colony? Previous studies have primarily relied on measuring the amount of growth, but do not take into account the spatial distribution of this highly non-uniform process. We fill this void by providing a statistical approach that enables the quantification of this important spatial information. This approach enables a more detailed mathematical analysis of the growth process and should allow the precise definition of mutant phenotypes, thus enabling a detailed analysis of the genetic control of morphogenesis.

Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions

We define a pair-correlation function that can be used to characterize spatiotemporal patterning in experimental images and snapshots from discrete simulations. Unlike previous pair-correlation functions, the pair-correlation functions developed here depend on the location and size of objects. The pair-correlation function can be used to indicate complete spatial randomness, aggregation, or segregation over a range of length scales, and quantifies spatial structures such as the shape, size, and distribution of clusters. Comparing pair-correlation data for various experimental and simulation images illustrates their potential use as a summary statistic for calibrating discrete models of various physical processes.

Lamination and mixing in three fundamental flow sequences driven by electromagnetic body forces

This article pursues the idea that the degree of striations, called lamination, could be engineered to complement stretching and to design new sequential mixers. It explores lamination and mixing in three new mixing sequences experimentally driven by electromagnetic body forces. To generate these three mixing sequences, Lorentz body forces are dynamically controlled to vary the flow geometry produced by a pair of local jets. The first two sequences are inspired from the "tendril and whorl" and "blinking vortex" flows. The third novel sequence is called the "cat's eyes flip." These three mixing sequences exponentially stretch and laminate material lines representing the interface between two domains to be mixed. Moreover, the mixing coefficient (defined as 1-σ(2)/σ(0)(2) where σ(2)/σ(0)(2) is the rescaled variance) and its rate grow exponentially before saturation. This saturation of the mixing process is related to the departure of the mixing rate from an exponential growth when the striations' thicknesses reach the diffusive length scale of the measurements or species and dyes. Incidentally, in our experiments, for the same energy or forcing input, the cat's eyes flip sequence has higher lamination, stretching, and mixing rates than the tendril and whorl and the blinking vortex sequences. These features show that bakerlike in situ mixers can be conceived by dynamically controlling a pair of local jets and by integrating lamination during stirring stages with persistent geometries. Combined with novel insights provided by the quantification of the lamination, this paper should offer perspectives for the development of new sequential mixers, possibly on all scales.

Generalized index for spatial data sets as a measure of complete spatial randomness

Spatial data sets, generated from a wide range of physical systems can be analyzed by counting the number of objects in a set of bins. Previous work has been limited to equal-sized bins, which are inappropriate for some domains (e.g., circular). We consider a nonequal size bin configuration whereby overlapping or nonoverlapping bins cover the domain. A generalized index, defined in terms of a variance between bin counts, is developed to indicate whether or not a spatial data set, generated from exclusion or nonexclusion processes, is at the complete spatial randomness (CSR) state. Limiting values of the index are determined. Using examples, we investigate trends in the generalized index as a function of density and compare the results with those using equal size bins. The smallest bin size must be much larger than the mean size of the objects. We can determine whether a spatial data set is at the CSR state or not by comparing the values of a generalized index for different bin configurations-the values will be approximately the same if the data is at the CSR state, while the values will differ if the data set is not at the CSR state. In general, the generalized index is lower than the limiting value of the index, since objects do not have access to the entire region due to blocking by other objects. These methods are applied to two applications: (i) spatial data sets generated from a cellular automata model of cell aggregation in the enteric nervous system and (ii) a known plant data distribution.

Analysis of chaotic mixing in plugs moving in meandering microchannels

Droplets moving in meandering microchannels can serve as a passive and robust strategy to produce chaotic mixing of species in droplet-based microfluidics. In this paper, a simplified theoretical model is proposed for plug-shaped droplets moving in meandering microchannels at Stokes flow. With this model to provide the velocity field, particle tracking, which requires a large computation time, is performed directly and easily without interpolation. With this convenience, a broad survey of the parameter space is carried out to investigate chaotic mixing in plugs, including the channel curvature, the Peclet number, the viscosity ratio, and the plug length. The results show that in order to achieve rapid mixing in plugs in meandering microchannels, a large curvature, a small Peclet number, a moderate viscosity ratio, and a moderate plug length are preferred.

Spatial analysis of multi-species exclusion processes: application to neural crest cell migration in the embryonic gut

Hindbrain (vagal) neural crest cells become relatively uniformly distributed along the embryonic intestine during the rostral to caudal colonization wave which forms the enteric nervous system (ENS). When vagal neural crest cells are labeled before migration in avian embryos by in ovo electroporation, the distribution of labeled neural crest cells in the ENS varies vastly. In some cases, the labeled neural crest cells appear evenly distributed and interspersed with unlabeled neural crest cells along the entire intestine. However, in most specimens, labeled cells occur in relatively discrete patches of varying position, area, and cell number. To determine reasons for these differences, we use a discrete cellular automata (CA) model incorporating the underlying cellular processes of neural crest cell movement and proliferation on a growing domain, representing the elongation of the intestine during development. We use multi-species CA agents corresponding to labeled and unlabeled neural crest cells. The spatial distributions of the CA agents are quantified in terms of an index. This investigation suggests that (i) the percentage of the initial neural crest cell population that is labeled and (ii) the ratio of cell proliferation to motility are the two key parameters producing the extreme differences in spatial distributions observed in avian embryos.

Quantifying evenly distributed states in exclusion and nonexclusion processes

Spatial-point data sets, generated from a wide range of physical systems and mathematical models, can be analyzed by counting the number of objects in equally sized bins. We find that the bin counts are related to the Pólya distribution. New measures are developed which indicate whether or not a spatial data set, generated from an exclusion process, is at its most evenly distributed state, the complete spatial randomness (CSR) state. To this end, we define an index in terms of the variance between the bin counts. Limiting values of the index are determined when objects have access to the entire domain and when there are subregions of the domain that are inaccessible to objects. Using three case studies (Lagrangian fluid particles in chaotic laminar flows, cellular automata agents in discrete models, and biological cells within colonies), we calculate the indexes and verify that our theoretical CSR limit accurately predicts the state of the system. These measures should prove useful in many biological applications.