A mathematical model for nutrient-limited uniaxial growth of a compressible tissue.

We consider the uniaxial growth of a tissue or colony of cells, where a nutrient (or some other chemical) required for cell proliferation is supplied at one end, and is consumed by the cells. An example would be the growth of a cylindrical yeast colony in the experiments described by Vulin et al. (2014). We develop a reaction-diffusion model of this scenario which couples nutrient concentration and cell density on a growing domain. A novel element of our model is that the tissue is assumed to be compressible. We define replicative regions, where cells have sufficient nutrient to proliferate, and quiescent regions, where the nutrient level is insufficient for this to occur. We also define pathlines, which allow us to track individual cell paths within the tissue. We begin our investigation of the model by considering an incompressible tissue where cell density is constant before exploring the solution space of the full compressible model. In a large part of the parameter space, the incompressible and compressible models give qualitatively similar results for both the nutrient concentration and cell pathlines, with the key distinction being the variation in density in the compressible case. In particular, the replicative region is located at the base of the tissue, where nutrient is supplied, and nutrient concentration decreases monotonically with distance from the nutrient source. However, for a highly-compressible tissue with small nutrient consumption rate, we observe a counter-intuitive scenario where the nutrient concentration is not necessarily monotonically decreasing, and there can be two replicative regions. For parameter values given in the paper by Vulin et al. (2014), the incompressible model slightly overestimates the colony length compared to experimental observations; this suggests the colony may be somewhat compressible. Both incompressible and compressible models predict that, for these parameter values, cell proliferation is ultimately confined to a small region close to the colony base.

Understanding interactions between populations: Individual based modelling and quantification using pair correlation functions.

Understanding the underlying mechanisms that produce the huge variety of swarming and aggregation patterns in animals and cells is fundamental in ecology, developmental biology, and regenerative medicine, to name but a few examples. Depending upon the nature of the interactions between individuals (cells or animals), a variety of different large-scale spatial patterns can be observed in their distribution; examples include cell aggregates, stripes of different coloured skin cells, etc. For the case where all individuals are of the same type (i.e., all interactions are alike), a considerable literature already exists on how the collective organisation depends on the inter-individual interactions. Here, we focus on the less studied case where there are two different types of individuals present. Whilst a number of continuum models of this scenario exist, it can be difficult to compare these models to experimental data, since real cells and animals are discrete. In order to overcome this problem, we develop an agent-based model to simulate some archetypal mechanisms involving attraction and repulsion. However, with this approach (as with experiments), each realisation of the model is different, due to stochastic effects. In order to make useful comparisons between simulations and experimental data, we need to identify the robust features of the spatial distributions of the two species which persist over many realisations of the model (for example, the size of aggregates, degree of segregation or intermixing of the two species). In some cases, it is possible to do this by simple visual inspection. In others, the features of the pattern are not so clear to the unaided eye. In this paper, we introduce a pair correlation function (PCF), which allows us to analyse multi-species spatial distributions quantitatively. We show how the differing strengths of inter-individual attraction and repulsion between species give rise to different spatial patterns, and how the PCF can be used to quantify these differences, even when it might be impossible to recognise them visually.

Identifying the necrotic zone boundary in tumour spheroids with pair-correlation functions.

Automatic identification of the necrotic zone boundary is important in the assessment of treatments on in vitro tumour spheroids. This has been difficult especially when the difference in cell density between the necrotic and viable zones of a tumour spheroid is small. To help overcome this problem, we developed novel one-dimensional pair-correlation functions (PCFs) to provide quantitative estimates of the radial distance of the necrotic zone boundary from the centre of a tumour spheroid. We validate our approach on synthetic tumour spheroids in which the position of the necrotic zone boundary is known a priori It is then applied to nine real tumour spheroids imaged with light sheet-based fluorescence microscopy. PCF estimates of the necrotic zone boundary are compared with those of a human expert and an existing standard computational method.

On the derivation of approximations to cellular automata models and the assumption of independence.

Cellular automata are discrete agent-based models, generally used in cell-based applications. There is much interest in obtaining continuum models that describe the mean behaviour of the agents in these models. Previously, continuum models have been derived for agents undergoing motility and proliferation processes, however, these models only hold under restricted conditions. In order to narrow down the reason for these restrictions, we explore three possible sources of error in deriving the model. These sources are the choice of limiting arguments, the use of a discrete-time model as opposed to a continuous-time model and the assumption of independence between the state of sites. We present a rigorous analysis in order to gain a greater understanding of the significance of these three issues. By finding a limiting regime that accurately approximates the conservation equation for the cellular automata, we are able to conclude that the inaccuracy between our approximation and the cellular automata is completely based on the assumption of independence.

Distinguishing between mechanisms of cell aggregation using pair-correlation functions.

Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.

Ecological genomics of marine Roseobacters.

Bacterioplankton of the marine Roseobacter clade have genomes that reflect a dynamic environment and diverse interactions with marine plankton. Comparative genome sequence analysis of three cultured representatives suggests that cellular requirements for nitrogen are largely provided by regenerated ammonium and organic compounds (polyamines, allophanate, and urea), while typical sources of carbon include amino acids, glyoxylate, and aromatic metabolites. An unexpectedly large number of genes are predicted to encode proteins involved in the production, degradation, and efflux of toxins and metabolites. A mechanism likely involved in cell-to-cell DNA or protein transfer was also discovered: vir-related genes encoding a type IV secretion system typical of bacterial pathogens. These suggest a potential for interacting with neighboring cells and impacting the routing of organic matter into the microbial loop. Genes shared among the three roseobacters and also common in nine draft Roseobacter genomes include those for carbon monoxide oxidation, dimethylsulfoniopropionate demethylation, and aromatic compound degradation. Genes shared with other cultured marine bacteria include those for utilizing sodium gradients, transport and metabolism of sulfate, and osmoregulation.

Forced solitary waves and fronts past submerged obstacles.

Herein, an efficient numerical method is presented to describe the flow of a liquid in an open channel with various types of bottom configurations. The method is developed for steady two-dimensional potential free surface flows. The resulting nonlinear problem is solved numerically by boundary integral equation methods. In addition weakly nonlinear solutions are derived. New solutions which complement those of Dias and Vanden-Broeck [J. Fluid Mech. 59, 93-102 (2004)] are presented. Furthermore some solutions for channel flows past dips in the bottom are discussed.

Application of digital image analysis and flow cytometry to enumerate marine viruses stained with SYBR gold.

A novel nucleic acid stain, SYBR Gold, was used to stain marine viral particles in various types of samples. Viral particles stained with SYBR Gold yielded bright and stable fluorescent signals that could be detected by a cooled charge-coupled device camera or by flow cytometry. The fluorescent signal strength of SYBR Gold-stained viruses was about twice that of SYBR Green I-stained viruses. Digital images of SYBR Gold-stained viral particles were processed to enumerate the concentration of viral particles by using digital image analysis software. Estimates of viral concentration based on digitized images were 1.3 times higher than those based on direct counting by epifluorescence microscopy. Direct epifluorescence counts of SYBR Gold-stained viral particles were in turn about 1.34 times higher than those estimated by the transmission electron microscope method. Bacteriophage lysates stained with SYBR Gold formed a distinct population in flow cytometric signatures. Flow cytometric analysis revealed at least four viral subpopulations for a Lake Erie sample and two subpopulations for a Georgia coastal sample. Flow cytometry-based viral counts for various types of samples averaged 1.1 times higher than direct epifluorescence microscopic counts. The potential application of digital image analysis and flow cytometry for rapid and accurate measurement of viral abundance in aquatic environments is discussed.

In situ hybridization of Prochlorococcus and Synechococcus (marine cyanobacteria) spp. with RRNA-targeted peptide nucleic acid probes.

A simple method for whole-cell hybridization using fluorescently labeled rRNA-targeted peptide nucleic acid (PNA) probes was developed for use in marine cyanobacterial picoplankton. In contrast to established protocols, this method is capable of detecting rRNA in Prochlorococcus, the most abundant unicellular marine cyanobacterium. Because the method avoids the use of alcohol fixation, the chlorophyll content of Prochlorococcus cells is preserved, facilitating the identification of these cells in natural samples. PNA probe-conferred fluorescence was measured flow cytometrically and was always significantly higher than that of the negative control probe, with positive/negative ratio varying between 4 and 10, depending on strain and culture growth conditions. Prochlorococcus cells from open ocean samples were detectable with this method. RNase treatment reduced probe-conferred fluorescence to background levels, demonstrating that this signal was in fact related to the presence of rRNA. In another marine cyanobacterium, Synechococcus, in which both PNA and oligonucleotide probes can be used in whole-cell hybridizations, the magnitude of fluorescence from the former was fivefold higher than that from the latter, although the positive/negative ratio was comparable for both probes. In Synechococcus cells growing at a range of growth rates (and thus having different rRNA concentrations per cell), the PNA- and oligonucleotide-derived signals were highly correlated (r = 0.99). The chemical nature of PNA, the sensitivity of PNA-RNA binding to single-base-pair mismatches, and the preservation of cellular integrity by this method suggest that it may be useful for phylogenetic probing of whole cells in the natural environment.

Application of neural networks to flow cytometry data analysis and real-time cell classification.

Conventional analysis of flow cytometric data requires that population identification be performed graphically after a sample has been run using two-parameter scatter plots. As more parameters are measured, the number of possible two-parameter plots increases geometrically, making data analysis increasingly cumbersome. Artificial Neural Systems (ANS), also known as neural networks, are a powerful and convenient method for overcoming this data bottleneck. ANS "learn" to make classifications using all of the measured parameters simultaneously. Mathematical models and programming expertise are not required. ANS are inherently parallel so that high processing speed can be achieved. Because ANS are nonlinear, curved class boundaries and other nonlinearities can emerge naturally. Here, we present biomedical and oceanographic data to demonstrate the useful properties of neural networks for processing and analyzing flow cytometry data. We show that ANS are equally useful for human leukocytes and marine plankton data. They can easily accommodate nonlinear variations in data, detect subtle changes in measurements, interpolate and classify cells they were not trained on, and analyze multiparameter cell data in real time. Real-time classification of a mixture of six cyanobacteria strains was achieved with an average accuracy of 98%.

Cell Cycle Regulation in Marine Synechococcus sp. Strains.

The cell cycle behavior of four marine strains of the unicellular cyanobacterium Synechococcus sp. was analyzed by examining the DNA frequency distributions of exponentially growing and dark-blocked populations and by considering the patterns of change in these distributions during growth under a diel light-dark cycle. The two modes of cell cycle regulation previously identified in a freshwater and coastal marine Synechococcus isolate, respectively, were represented among the three open-ocean strains we examined. The first of these modes of regulation is consistent with the slow-growth case of the widely accepted prokaryotic cell cycle paradigm. The second appears to involve asynchronous initiation of chromosome replication, the presence of multiple chromosome copies at low growth rates, and variability in chromosome copy number among cells in the population. These characteristics suggest the involvement of a large probabilistic component in cell cycle regulation which could make the application of cell cycle-based estimators of in situ growth rate to Synechococcus populations problematic.

Combination of 16S rRNA-targeted oligonucleotide probes with flow cytometry for analyzing mixed microbial populations.

Fluorescent oligonucleotide hybridization probes were used to label bacterial cells for analysis by flow cytometry. The probes, complementary to short sequence elements within the 16S rRNA common to phylogenetically coherent assemblages of microorganisms, were labeled with tetramethylrhodamine and hybridized to suspensions of fixed cells. Flow cytometry was used to resolve individual target and nontarget bacteria (1 to 5 microns) via probe-conferred fluorescence. Target cells were quantified in an excess of nontarget cells. The intensity of fluorescence was increased additively by the combined use of two or three fluorescent probes complementary to different regions of the same 16S rRNA.

Relationship between DNA cycle and growth rate in Synechococcus sp. strain PCC 6301.

Flow cytometry was used to examine cell cycle regulation in Synechococcus sp. strain PCC 6301 under a variety of growth conditions. The DNA frequency distributions of exponentially growing and dark-blocked populations confirmed that this cyanobacterium contains multiple chromosome copies even at very slow growth rates. Furthermore, the presence of major peaks corresponding to other than 2" chromosome copies strongly suggests that DNA replication is initiated asynchronously. Although this suggestion is at odds with the standard formulation of the procaryotic cell cycle model, it is similar to recent observations of asynchrony in Escherichia coli replication mutants.