On equivalence between maximal and maximum antichains over range-restricted vectors with integral coordinates and the number of such maximal antichains

Consider a strict partially ordered set $\mathcal{P}$ consisting of all $d$-dimensional vectors with integral coordinates restricted in a certain range. We found that any maximal antichain is also maximum, and the maximum size has a simple expression in terms of the range. Properties of the number of maximal antichains given the range are explored. We present our proof, the application on other areas, and some open questions.Comment: Submitted partial result to Journal Orde

Sharper bounds and structural results for minimally nonlinear 0-1 matrices

The extremal function ex(n, P) is the maximum possible number of ones in any 0-1 matrix with n rows and n columns that avoids P. A 0-1 matrix P is called minimally nonlinear if ex(n, P) = ω(n) but ex(n, P′) = O(n) for every P′ that is contained in P but not equal to P. Bounds on the number of ones and the number of columns in a minimally non-linear 0-1 matrix with k rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with k rows from 5k − 3 to 4k − 4. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with k rows from 4k − 2 to 4k − 4. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly d columns between them, then within these columns there are at most 2d − 1 rows above and 2d − 1 rows below with ones

Sharper bounds and structural results for minimally nonlinear 0-1 matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the maximum number of ones and the maximum number of columns in a minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the bound on the maximum number of ones in a minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally non-linear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally non-linear matrix and that there are not more than six ones in any other row of a minimally non-linear matrix. Furthermore, we prove that if a minimally non-linear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones

Population genetics in microchannels

Spatial constraints such as rigid barriers affect the dynamics of cell populations, potentially altering the course of natural evolution. In this paper, we study the population genetics of Escherichia coli proliferating in microchannels with open ends. Our experiments reveal that competition among two fluorescently labeled E. coli strains growing in a microchannel generates a self-organized stripe pattern aligned with the axial direction of the channel. To account for this observation, we employ a lattice population model in which reproducing cells push entire lanes of cells towards the open ends of the channel. By combining mathematical theory, numerical simulations, and experiments, we find that the fixation dynamics is extremely fast along the axial direction, with a logarithmic dependence on the number of cells per lane. In contrast, competition among lanes is a much slower process. We also demonstrate that random mutations appearing in the middle of the channel and close to its walls are much more likely to reach fixation than mutations occurring elsewhere.Comment: 21 pages, 14 figure

Paper-based tuberculosis diagnostic devices with colorimetric gold nanoparticles

A colorimetric sensing strategy employing gold nanoparticles and a paper assay platform has been developed for tuberculosis diagnosis. Unmodified gold nanoparticles and single-stranded detection oligonucleotides are used to achieve rapid diagnosis without complicated and time-consuming thiolated or other surface-modified probe preparation processes. To eliminate the use of sophisticated equipment for data analysis, the color variance for multiple detection results was simultaneously collected and concentrated on cellulose paper with the data readout transmitted for cloud computing via a smartphone. The results show that the 2.6 nM tuberculosis mycobacterium target sequences extracted from patients can easily be detected, and the turnaround time after the human DNA is extracted from clinical samples was approximately 1 h

Voltage-gated ion channels mediate the electrotaxis of glioblastoma cells in a hybrid PMMA/PDMS microdevice

Transformed astrocytes in the most aggressive form cause glioblastoma, the most common cancer in the central nervous system with high mortality. The physiological electric field by neuronal local field potentials and tissue polarity may guide the infiltration of glioblastoma cells through the electrotaxis process. However, microenvironments with multiplex gradients are difficult to create. In this work, we have developed a hybrid microfluidic platform to study glioblastoma electrotaxis in controlled microenvironments with high throughput quantitative analysis by machine learning-powered single cell tracking software. By equalizing the hydrostatic pressure difference between inlets and outlets of the microchannel, uniform single cells can be seeded reliably inside the microdevice. The electrotaxis of two glioblastoma models, T98G and U-251MG, requires an optimal laminin-containing extracellular matrix and exhibits opposite directional and electro-alignment tendencies. Calcium signaling is a key contributor in glioblastoma pathophysiology but its role in glioblastoma electrotaxis is still an open question. Anodal T98G electrotaxis and cathodal U-251MG electrotaxis require the presence of extracellular calcium cations. U-251MG electrotaxis is dependent on the P/Q-type voltage-gated calcium channel (VGCC) and T98G is dependent on the R-type VGCC. U-251MG electrotaxis and T98G electrotaxis are also mediated by A-type (rapidly inactivating) voltage-gated potassium channels and acid-sensing sodium channels. The involvement of multiple ion channels suggests that the glioblastoma electrotaxis is complex and patient-specific ion channel expression can be critical to develop personalized therapeutics to fight against cancer metastasis. The hybrid microfluidic design and machine learning-powered single cell analysis provide a simple and flexible platform for quantitative investigation of complicated biological systems

Graphlet and Orbit Computation on Heterogeneous Graphs

Many applications, ranging from natural to social sciences, rely on graphlet analysis for the intuitive and meaningful characterization of networks employing micro-level structures as building blocks. However, it has not been thoroughly explored in heterogeneous graphs, which comprise various types of nodes and edges. Finding graphlets and orbits for heterogeneous graphs is difficult because of the heterogeneity and abundance of semantic information. We consider heterogeneous graphs, which can be treated as colored graphs. By applying the canonical label technique, we determine the graph isomorphism problem with multiple states on nodes and edges. With minimal parameters, we build all non-isomorphic graphs and associated orbits. We provide a Python package that can be used to generate orbits for colored directed graphs and determine the frequency of orbit occurrence. Finally, we provide four examples to illustrate the use of the Python package.Comment: 13 pages, 7 figure

Glioblastoma adhesion in a quick-fit hybrid microdevice

Translational research requires reliable biomedical microdevices (BMMD) to mimic physiological conditions and answer biological questions. In this work, we introduce a reversibly sealed quick-fit hybrid BMMD that is operator-friendly and bubble-free, requires low reagent and cell consumption, enables robust and high throughput performance for biomedical experiments. Specifically, we fabricate a quick-fit poly(methyl methacrylate) and poly(dimethyl siloxane) (PMMA/PDMS) prototype to illustrate its utilities by probing the adhesion of glioblastoma cells (T98G and U251MG) to primary endothelial cells. In static condition, we confirm that angiopoietin-Tie2 signaling increases the adhesion of glioblastoma cells to endothelial cells. Next, to mimic the physiological hemodynamic flow and investigate the effect of physiological electric field, the endothelial cells are pre-conditioned with concurrent shear flow (with fixed 1 Pa shear stress) and direct current electric field (dcEF) in the quick-fit PMMA/PDMS BMMD. With shear flow alone, endothelial cells exhibit classical parallel alignment; while under a concurrent dcEF, the cells align perpendicularly to the electric current when the dcEF is greater than 154 V m(-1). Moreover, with fixed shear stress of 1 Pa, T98G glioblastoma cells demonstrate increased adhesion to endothelial cells conditioned in dcEF of 154 V m(-1), while U251MG glioblastoma cells show no difference. The quick-fit hybrid BMMD provides a simple and flexible platform to create multiplex systems, making it possible to investigate complicated biological conditions for translational research