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Cell invasion and cell plasticity are critical to human development but are also striking features of cancer metastasis. By distributing a multipotent cell type from a place of birth to distal locations, the vertebrate embryo builds organs. In comparison, metastatic tumor cells often acquire a de-differentiated phenotype and migrate away from a primary site to inhabit new microenvironments, disrupting normal organ function. Countless observations of both embryonic cell migration and tumor metastasis have demonstrated complex cell signaling and interactive behaviors that have long confounded scientist and clinician alike. James D. Murray realized the important role of mathematics in biology and developed a unique strategy to address complex biological questions such as these. His work offers a practical template for constructing clear, logical, direct and verifiable models that help to explain complex cell behaviors and direct new experiments. His pioneering work at the interface of development and cancer made significant contributions to glioblastoma cancer and embryonic pattern formation using often simple models with tremendous predictive potential. Here, we provide a brief overview of advances in cell invasion and cell plasticity using the embryonic neural crest and its ancestral relationship to aggressive cancers that put into current context the timeless aspects of his work. [ABSTRACT FROM AUTHOR]

In this work, we describe some effective teaching and research practices that can help to integrate mathematics and biology efficiently to enhance student learning at all levels. One of the successful approaches proposed is to employ mathematical modeling that can help transform pedagogical practices. In this regard, we introduce some modeling activities that have been shared with teachers through professional development programs and have been incorporated in the classrooms. We also present how engaging teachers in research experiences in mathematical modeling can help to transform their pedagogical practices and provide opportunities for students to consider pursuing areas at the interface of mathematics and biology. [ABSTRACT FROM AUTHOR]

Quantitative biology is a rapidly advancing field in the biological sciences, particularly given the rise of large datasets and computer processing capabilities that have continually expanded over the past 50 years. Thus, the question arises, How should K-12 biology teachers incorporate quantitative biology skills into their biology curriculum? The teaching of quantitative biology has not been readily integrated into undergraduate biology curricula that impact preservice teachers. This has potential to cascade effects downward into the quality of learning about quantitative biology that can be expected in K-12 contexts. In this paper, we present the perspectives of a mathematics educator, a science educator, and two biologists, and discuss how we have personally incorporated aspects of quantitative reasoning into our courses. We identify some common challenges relevant to expanding implementation of quantitative reasoning in undergraduate biology courses in order to serve the needs of preservice teachers—both in their disciplinary courses and methods courses. For example, time constraints, math pedagogical content knowledge, and personal views about the relevance of quantitative principles in biology teaching and learning can impact how and to what extent they become implemented in curricula. In addition, although national standards at the K-12 level do address quantitative reasoning, the emphasis and guidance provided are sparser than for other content standards. We predict that both K-12 standards and guidelines for undergraduate education will only increase in their emphasis on quantitative skills as computation, "big data," and statistical modeling are increasingly becoming requisite skills for biologists. [ABSTRACT FROM AUTHOR]

The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735–1741, 1977; Norton in Can Res 48:7067–7071, 1988; Hahnfeldt et al. in Can Res 59:4770–4775, 1999; Anderson et al. in Comput Math Methods Med 2:129–154, 2000. 10.1080/10273660008833042; Michor et al. in Nature 435:1267–1270, 2005. 10.1038/nature03669; Anderson et al. in Cell 127:905–915, 2006. 10.1016/j.cell.2006.09.042; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. 10.1371/journal.pcbi.1003800). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to. [ABSTRACT FROM AUTHOR]

The goals of this article and special issue are to highlight the value of mathematical biology approaches in industry, help foster future interactions, and suggest ways for mathematics Ph.D. students and postdocs to move into industry careers. We include a candid examination of the advantages and challenges of doing mathematics in the biopharma industry, a broad overview of the types of mathematics being applied, information about academic collaborations, and career advice for those looking to move from academia to industry (including graduating Ph.D. students). [ABSTRACT FROM AUTHOR]

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain "critical function" changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Gröbner bases. [ABSTRACT FROM AUTHOR]

A model of interspecific host competition in a system with one parasite (butterfly—Maculinea) and multiple potential hosts (ants—Myrmica) is presented. Results indicate that host interspecific competition increases the occurrence of multiple host behaviour in Maculinea natural populations but decreases the ability of the parasite populations to adapt to the most abundant host species. These qualitative predictions were compared with data on host specificity, with good agreement. Analysis of the data also indicates that Maculinea teleius and Maculinea arion respond differently to changes in relative host abundances. Maculinea teleius shows a larger fraction of sites where it displays multiple host behaviour and a larger fraction of sites where the niches of the hosts overlap. In some instances, Maculinea teleius is adapted to Myrmica hosts that are present in lower frequencies. Maculinea arion is locally more host-specific and occurs at sites where host interspecific competition is unlikely and is more frequently adapted to the most abundant host species. [ABSTRACT FROM AUTHOR]

There are a growing number of studies that model immunological processes in the artery wall that lead to the development of atherosclerotic plaques. However, few of these models use parameters that are obtained from experimental data even though data-driven models are vital if mathematical models are to become clinically relevant. We present the development and analysis of a quantitative mathematical model for the coupled inflammatory, lipid and macrophage dynamics in early atherosclerotic plaques. Our modeling approach is similar to the biologists' experimental approach where the bigger picture of atherosclerosis is put together from many smaller observations and findings from in vitro experiments. We first develop a series of three simpler submodels which are least-squares fitted to various in vitro experimental results from the literature. Subsequently, we use these three submodels to construct a quantitative model of the development of early atherosclerotic plaques. We perform a local sensitivity analysis of the model with respect to its parameters that identifies critical parameters and processes. Further, we present a systematic analysis of the long-term outcome of the model which produces a characterization of the stability of model plaques based on the rates of recruitment of low-density lipoproteins, high-density lipoproteins and macrophages. The analysis of the model suggests that further experimental work quantifying the different fates of macrophages as a function of cholesterol load and the balance between free cholesterol and cholesterol ester inside macrophages may give valuable insight into long-term atherosclerotic plaque outcomes. This model is an important step toward models applicable in a clinical setting. [ABSTRACT FROM AUTHOR]

A spatio-temporal evolution of chemicals appearing in a reversible enzyme reaction and modelled by a four-component reaction-diffusion system with the reaction terms obtained by the law of mass action is considered. The large time behaviour of the system is studied by means of entropy methods. [ABSTRACT FROM AUTHOR]

The present paper analyzes an ODE model of a certain class of (open) enzymatic reactions. This type of model is used, for instance, to describe the interactions between messenger RNAs and microRNAs. It is shown that solutions defined by positive initial conditions are well defined and bounded on $$[0, \infty )$$ and that the positive octant of $${\mathbb {R}}^3$$ is a positively invariant set. We prove further that in this positive octant there exists a unique equilibrium point, which is asymptotically stable and a global attractor for any initial state with positive components; a controllability property is emphasized. We also investigate the qualitative behavior of the QSSA system in the phase plane $${\mathbb {R}}^2$$ . For this planar system we obtain similar results regarding global stability by using Lyapunov theory, invariant regions and controllability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions (MRFs) with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy (MPE) model is discussed in great detail. Mathematically, unlike the previous persistent entropy (Chintakunta et al. in Pattern Recognit 48(2):391-401, 2015; Merelli et al. in Entropy 17(10):6872-6892, 2015; Rucco et al. in: Proceedings of ECCS 2014, Springer, pp 117-128, 2016), a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our MPE can be used in conformational entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of an MRF built from angular distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the intermediate range. Moreover, by comparing with traditional entropies from various grid sizes, bond angle-based methods and a persistent homology-based support vector machine method (Cang et al. in Mol Based Math Biol 3:140-162, 2015), we find that our MPE method gives the best results in terms of average true positive rate in a classic protein structure classification test. More interestingly, all-alpha and all-beta protein classes can be clearly separated from each other with zero error only in our model. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the 'regularity' of protein structures. Basically, a protein structure is deemed as regular if it has a consistent and orderly configuration. Our PSI model is tested on a database of 110 proteins; we find that structures with larger portions of loops and intrinsically disorder regions are always associated with larger PSI, meaning an irregular configuration, while proteins with larger portions of secondary structures, i.e., alpha-helix or beta-sheet, have smaller PSI. Essentially, PSI can be used to describe the 'regularity' information in any systems. [ABSTRACT FROM AUTHOR]

Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we study the probabilities of splits on gene trees under the multispecies coalescent model, and how their features might inform species tree inference. After investigating the behavior of split consensus methods, we investigate split invariants-that is, polynomial relationships between split probabilities. These invariants are then used to show that, even though a split is an unrooted notion, split probabilities retain enough information to identify the rooted species tree topology for trees of 5 or more taxa, with one possible 6-taxon exception. [ABSTRACT FROM AUTHOR]

Many aquatic organisms detect the presence of moving objects in their environment, such as predators, by sensing the hydrodynamic disturbances the movements produce. The resultant water flow is readily detectable by stationary organisms, but free-swimming organisms are carried with the surrounding water and may not detect the bulk surrounding flow, which limits the available information about the source. We have developed a theory that clarifies what information is contained in disturbances generated by an attacking predator that is available to a free-swimming organism and might be extracted from local flow deformations alone. The theory shows that, depending on how well the deformations can be measured in space and time, an organism can reduce the range of possible locations, speeds, sizes, and arrival times of the predator. We apply the theory to planktonic copepods that have mechanosensory hairs along a pair of antennules. The study reveals the presence of 'blind spots,' potential ambiguities in resolving from which of two sides a predator attacks, and whether it generates a bow wave or suction. Our findings lead to specific testable hypotheses concerning optimal escape strategies, which are helpful for interpreting the behavior of evasive prey and designing free-swimming robots with sensory capabilities. [ABSTRACT FROM AUTHOR]

We use the reinfection SIRI epidemiological model to analyze the impact of education programs and vaccine scares on individuals decisions to vaccinate or not. The presence of the reinfection provokes the novelty of the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model studied by Bauch and Earn (PNAS 101:13391-13394, 2004). The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient they need to implement a mechanism to suddenly increase the vaccination coverage level. [ABSTRACT FROM AUTHOR]

We consider a two-patch model for a single species with dispersal and time delay. For some explicit range of dispersal rates, we show that there exists a critical value τ for the time delay τ such that the unique positive equilibrium of the system is locally asymptotically stable for τ∈[0, τ) and unstable for τ> τ. [ABSTRACT FROM AUTHOR]

We consider a general class of mathematical models for stochastic gene expression where the transcription rate is allowed to depend on a promoter state variable that can take an arbitrary (finite) number of values. We provide the solution of the master equations in the stationary limit, based on a factorization of the stochastic transition matrix that separates timescales and relative interaction strengths, and we express its entries in terms of parameters that have a natural physical and/or biological interpretation. The solution illustrates the capacity of multiple states promoters to generate multimodal distributions of gene products, without the need for feedback. Furthermore, using the example of a three states promoter operating at low, high, and intermediate expression levels, we show that using multiple states operons will typically lead to a significant reduction of noise in the system. The underlying mechanism is that a three-states promoter can change its level of expression from low to high by passing through an intermediate state with a much smaller increase of fluctuations than by means of a direct transition. [ABSTRACT FROM AUTHOR]

Recently, we have shown that calculating the minimum-temporal-hybridization number for a set ${\mathcal{P}}$ of rooted binary phylogenetic trees is NP-hard and have characterized this minimum number when ${\mathcal{P}}$ consists of exactly two trees. In this paper, we give the first characterization of the problem for ${\mathcal{P}}$ being arbitrarily large. The characterization is in terms of cherries and the existence of a particular type of sequence. Furthermore, in an online appendix to the paper, we show that this new characterization can be used to show that computing the minimum-temporal hybridization number for two trees is fixed-parameter tractable. [ABSTRACT FROM AUTHOR]

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such as the properties of burst action potential firing and their associated Ca transients. These previous studies were based on experimental recordings taken from the soma of GnRH neurons. However, some research groups have shown that the dendrites of GnRH neurons play very important roles. In particular, it is now known that the site of action potential initiation in these neurons is often in the dendrite, over 100 μm from the soma. This raises an important question. Since some of the mechanisms for controlling the burst length and interburst interval are located in the soma, how can electrical bursting be controlled when initiated at a site located some distance from these controlling mechanisms? In order to answer this question, we construct a spatio-temporal mathematical model that includes both the soma and the dendrite. Our model shows that the diffusion coefficient for the spread of electrical potentials in the dendrite is large enough to coordinate burst firing of action potentials when the initiation site is located at some distance from the soma. [ABSTRACT FROM AUTHOR]

W.O. Kermack and A.G. McKendrick introduced in their fundamental paper, A Contribution to the Mathematical Theory of Epidemics, published in 1927, a deterministic model that captured the qualitative dynamic behavior of single infectious disease outbreaks. A Kermack–McKendrick discrete-time general framework, motivated by the emergence of a multitude of models used to forecast the dynamics of epidemics, is introduced in this manuscript. Results that allow us to measure quantitatively the role of classical and general distributions on disease dynamics are presented. The case of the geometric distribution is used to evaluate the impact of waiting-time distributions on epidemiological processes or public health interventions. In short, the geometric distribution is used to set up the baseline or null epidemiological model used to test the relevance of realistic stage-period distribution on the dynamics of single epidemic outbreaks. A final size relationship involving the control reproduction number, a function of transmission parameters and the means of distributions used to model disease or intervention control measures, is computed. Model results and simulations highlight the inconsistencies in forecasting that emerge from the use of specific parametric distributions. Examples, using the geometric, Poisson and binomial distributions, are used to highlight the impact of the choices made in quantifying the risk posed by single outbreaks and the relative importance of various control measures. [ABSTRACT FROM AUTHOR]

Around the world, infectious disease epidemics continue to threaten people’s health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called “social distancing.” Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic. [ABSTRACT FROM AUTHOR]

Pattern formation occurs in a wide range of biological systems. This pattern formation can occur in mathematical models because of diffusion-driven instability or due to the interaction between reaction, diffusion, and chemotaxis. In this paper, we investigate the spatial pattern formation of attack clusters in a system for Mountain Pine Beetle. The pattern formation (aggregation) of the Mountain Pine Beetle in order to attack susceptible trees is crucial for their survival and reproduction. We use a reaction-diffusion equation with chemotaxis to model the interaction between Mountain Pine Beetle, Mountain Pine Beetle pheromones, and susceptible trees. Mathematical analysis is utilized to discover the spacing in-between beetle attacks on the susceptible landscape. The model predictions are verified by analysing aerial detection survey data of Mountain Pine Beetle Attack from the Sawtooth National Recreation Area. We find that the distance between Mountain Pine Beetle attack clusters predicted by our model closely corresponds to the observed attack data in the Sawtooth National Recreation Area. These results clarify the spatial mechanisms controlling the transition from incipient to epidemic populations and may lead to control measures which protect forests from Mountain Pine Beetle outbreak. [ABSTRACT FROM AUTHOR]

We provide a simple mathematical model of the bioremediation of contaminated wastewater leaching into the subsoil below a septic tank percolation system. The model comprises a description of the percolation system’s flows, together with equations describing the growth of biomass and the uptake of an organic contaminant concentration. By first rendering the model dimensionless, it can be partially solved, to provide simple insights into the processes which control the efficacy of the system. In particular, we provide quantitative insight into the effect of a near surface biomat on subsoil permeability; this can lead to trench ponding, and thus propagation of effluent further down the trench. Using the computed vadose zone flow field, the model can be simply extended to include reactive transport of other contaminants of interest. [ABSTRACT FROM AUTHOR]

We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious individuals, our method enables a data-based model prediction of the number of unobserved exposed individuals over very long times. [ABSTRACT FROM AUTHOR]

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hyperplasia. Necrotic areas are typically surrounded by hypercellular regions known as 'pseudopalisades' originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes. [ABSTRACT FROM AUTHOR]

A modelling approach is used for studying the effects of population vaccination on the epidemic dynamics of a set of n cities interconnected by a complex transportation network. The model is based on a sophisticated mover-stayer formulation of inter-city population migration, upon which is included the classical SIS dynamics of disease transmission which operates within each city. Our analysis studies the stability properties of the Disease-Free Equilibrium (DFE) of the full n-city system in terms of the reproductive number R. Should vaccination reduce R below unity, the disease will be eradicated in all n-cities. We determine the precise conditions for which this occurs, and show that disease eradication by vaccination depend on the transportation structure of the migration network in a very direct manner. Several concrete examples are presented and discussed, and some counter-intuitive results found. [ABSTRACT FROM AUTHOR]

The Venus flytrap has long been regarded as one of the most amazing examples of movement in the plant kingdom. The trapping ability of the flytrap consists of three unique features. First, trap closure represents one of the fastest movements in the plant kingdom. Second, a decision-making stage allows the plant to 'decide' whether to completely close or open the trap, based on stimuli provided from the trapped object. Finally, the Venus flytrap contains a 'memory function' that requires two mechanical stimuli within about 30 seconds to initiate trap closure. The movement involved in trap closure consists of nonlinear dynamics that have not been well understood. By understanding the movement, through nonlinear dynamics analysis, it will be possible to better understand this biological process. A mathematical model describing the movement of the Venus flytrap was first proposed by the authors in Yang et al., Plant Signal. Behav. 5(8), 968-978 (). In the current work, the earlier research has been advanced and an in-depth nonlinear and control analysis of the dynamic process has been provided. [ABSTRACT FROM AUTHOR]

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present. [ABSTRACT FROM AUTHOR]

In this paper, we describe a project-based mathematical lab implemented in our Applied Mathematics in Biology course. The Leaky Bucket Lab allows students to parameterize and test Torricelli's law and develop and compare their own alternative models to describe the dynamics of water draining from perforated containers. In the context of this lab students build facility in a variety of applied biomathematical tools and gain confidence in applying these tools in data-driven environments. We survey analytic approaches developed by students to illustrate the creativity this encourages as well as prepare other instructors to scaffold the student learning experience. Pedagogical results based on classroom videography support the notion that the Biology-Applied Math Instructional Model, the teaching framework encompassing the lab, is effective in encouraging and maintaining high-level cognition among students. Research-based pedagogical approaches that support the lab are discussed. [ABSTRACT FROM AUTHOR]

Staphylococcus aureus uses quorum sensing (QS) to enhance its pathogenicity. An intriguing aspect of this is that different strains are capable of inactivating the QS systems of opposing strains. In Part 1 of this study, we presented a model of this phenomenon in a well-mixed environment; here, we incorporate spatial structure. Two competitive strains occupying adjacent habitats with freely diffusing QS signal molecules (QSSMs) are considered. We investigate the effect of the QSSM diffusion coefficient and the relative size of the two populations on the ability of one population to dominate the other. Regarding population size, a larger population is generally at an advantage (initial conditions permitting), while the implications of different diffusivities are more complex and depend upon the sizes of the populations. [ABSTRACT FROM AUTHOR]

Cell biologists have developed methods to label membrane proteins with gold nanoparticles and then extract spatial point patterns of the gold particles from transmission electron microscopy images using image processing software. Previously, the resulting patterns were analyzed using the Hopkins statistic, which distinguishes nonclustered from modestly and highly clustered distributions, but is not designed to quantify the number or sizes of the clusters. Clusters were defined by the partitional clustering approach which required the choice of a distance. Two points from a pattern were put in the same cluster if they were closer than this distance. In this study, we present a new methodology based on hierarchical clustering to quantify clustering. An intrinsic distance is computed, which is the distance that produces the maximum number of clusters in the biological data, eliminating the need to choose a distance. To quantify the extent of clustering, we compare the clustering distance between the experimental data being analyzed with that from simulated random data. Results are then expressed as a dimensionless number, the clustering ratio that facilitates the comparison of clustering between experiments. Replacing the chosen cluster distance by the intrinsic clustering distance emphasizes densely packed clusters that are likely more important to downstream signaling events. We test our new clustering analysis approach against electron microscopy images from an experiment in which mast cells were exposed for 1 or 2 minutes to increasing concentrations of antigen that crosslink IgE bound to its high affinity receptor, Fc ϵRI, then fixed and the Fc ϵRI β subunit labeled with 5 nm gold particles. The clustering ratio analysis confirms the increase in clustering with increasing antigen dose predicted from visual analysis and from the Hopkins statistic. Access to a robust and sensitive tool to both observe and quantify clustering is a key step toward understanding the detailed fine scale structure of the membrane, and ultimately to determining the role of spatial organization in the regulation of transmembrane signaling. [ABSTRACT FROM AUTHOR]

Posttranslational modification of proteins is key in transmission of signals in cells. Many signaling pathways contain several layers of modification cycles that mediate and change the signal through the pathway. Here, we study a simple signaling cascade consisting of n layers of modification cycles such that the modified protein of one layer acts as modifier in the next layer. Assuming mass-action kinetics and taking the formation of intermediate complexes into account, we show that the steady states are solutions to a polynomial in one variable and in fact that there is exactly one steady state for any given total amounts of substrates and enzymes. We demonstrate that many steady-state concentrations are related through rational functions that can be found recursively. For example, stimulus-response curves arise as inverse functions to explicit rational functions. We show that the stimulus-response curves of the modified substrates are shifted to the left as we move down the cascade. Further, our approach allows us to study enzyme competition, sequestration, and how the steady state changes in response to changes in the total amount of substrates. Our approach is essentially algebraic and follows recent trends in the study of posttranslational modification systems. [ABSTRACT FROM AUTHOR]

Necessity to understand the role of additional food as a tool in biological control programs is being increasingly felt, particularly due to its eco-friendly nature. A thorough mathematical analysis in this direction revealed the vital role of quality and quantity of the additional food in the controllability of the predator-prey systems. In this article controllability of the additional food-provided predator-prey system is studied from perspectives of pest eradication and biological conservation. Time optimal paths have been constructed to drive the state of the system to a desired terminal state by choosing quantity of the additional food as control variable. The theory developed in this article has been illustrated by solving problems related to pest eradication and biological conservation. [ABSTRACT FROM AUTHOR]

The Levin-Stewart model of bacteriophage predation of bacteria in a chemostat is modified for a flow reactor in which bacteria are motile, phage diffuse, and advection brings fresh nutrient and removes medium, cells and phage. A fixed latent period for phage results in a system of delayed reaction-diffusion equations with non-local nonlinearities. Basic reproductive numbers are obtained for bacteria and for phage which predict survival of each in the bio-reactor. These are expressed in terms of physical and biological parameters. Persistence and extinction results are obtained for both bacteria and phage. Numerical simulations are in general agreement with those for the chemostat model. [ABSTRACT FROM AUTHOR]

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number ${\mathcal{R}_{0}}$: There is a unique positive solution giving the size of the epidemic if and only if ${\mathcal{R}_{0}}$ exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number ${\mathcal{R}_{0}}$. For other mixing patterns, the relation may be reversed. [ABSTRACT FROM AUTHOR]

Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail. [ABSTRACT FROM AUTHOR]

Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived. [ABSTRACT FROM AUTHOR]

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, ). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. [ABSTRACT FROM AUTHOR]

numerical method is implemented for computing unsteady blood flow through a branching capillary network. The evolution of the discharge hematocrit along each capillary segment is computed by integrating in time a one-dimensional convection equation using a finite-difference method. The convection velocity is determined by the local and instantaneous effective capillary blood viscosity, while the tube to discharge hematocrit ratio is deduced from available correlations. Boundary conditions for the discharge hematocrit at divergent bifurcations arise from the partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q≥1. When q=1, the cells are partitioned in proportion to the flow rate; as q tends to infinity, the cells are channeled into the branch with the highest flow rate. Simulations are performed for a tree-like, perfectly symmetric or randomly perturbed capillary network with m generations. When the tree involves more than a few generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous self-sustained oscillations in the absence of external forcing. A phase diagram in the m- q plane is presented to establish conditions for unsteady flow, and the effect of various geometrical and physical parameters is examined. For a given network tree order, m, oscillations can be induced for a sufficiently high value of q by increasing the apparent intrinsic viscosity, decreasing the ratio of the vessel diameter from one generation to the next, or by decreasing the diameter of the terminal vessels. With other parameters fixed, oscillations are inhibited by increasing m. The results of the continuum model are in excellent agreement with the predictions of a discrete model where the motion of individual cells is followed from inlet to outlet. [ABSTRACT FROM AUTHOR]

Multiplex DNA profiles are used extensively for biomedical and forensic purposes. However, while DNA profile data generation is automated, human analysis of those data is not, and the need for speed combined with accuracy demands a computer-automated approach to sample interpretation and quality assessment. In this paper, we describe an integrated mathematical approach to modeling the data and extracting the relevant information, while rejecting noise and sample artifacts. We conclude with examples showing the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-population continuous integro-differential model of cell differentiation, using a non-local term to describe the influence of the local environment on differentiation. We investigate three different versions of the model, with differentiation being cell autonomous, regulated via a community effect, or weakly dependent on the local cellular environment. We consider the spatial patterns that such different modes of differentiation produce, and investigate the formation of both stripes and spots by the model. We show that pattern formation only occurs when differentiation is regulated by a strong community effect. In this case, permanent spatial patterns only occur under a precise relationship between the parameters characterising cell dynamics, although transient patterns can persist for biologically relevant timescales when this condition is relaxed. In all cases, the long-lived patterns consist only of stripes, not spots. [ABSTRACT FROM AUTHOR]

Herein, we rigorously develop novel 3-dimensional algebraic models called Genetic Hotels of the Standard Genetic Code (SGC). We start by considering the primeval RNA genetic code which consists of the 16 codons of type RNY (purine-any base-pyrimidine). Using simple algebraic operations, we show how the RNA code could have evolved toward the current SGC via two different intermediate evolutionary stages called Extended RNA code type I and II. By rotations or translations of the subset RNY, we arrive at the SGC via the former (type I) or via the latter (type II), respectively. Biologically, the Extended RNA code type I, consists of all codons of the type RNY plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The Extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. Since the dimensions of remarkable subsets of the Genetic Hotels are not necessarily integer numbers, we also introduce the concept of algebraic fractal dimension. A general decoding function which maps each codon to its corresponding amino acid or the stop signals is also derived. The Phenotypic Hotel of amino acids is also illustrated. The proposed evolutionary paths are discussed in terms of the existing theories of the evolution of the SGC. The adoption of 3-dimensional models of the Genetic and Phenotypic Hotels will facilitate the understanding of the biological properties of the SGC. [ABSTRACT FROM AUTHOR]

gent-based modeling and simulation is a useful method to study biological phenomena in a wide range of fields, from molecular biology to ecology. Since there is currently no agreed-upon standard way to specify such models, it is not always easy to use published models. Also, since model descriptions are not usually given in mathematical terms, it is difficult to bring mathematical analysis tools to bear, so that models are typically studied through simulation. In order to address this issue, Grimm et al. proposed a protocol for model specification, the so-called ODD protocol, which provides a standard way to describe models. This paper proposes an addition to the ODD protocol which allows the description of an agent-based model as a dynamical system, which provides access to computational and theoretical tools for its analysis. The mathematical framework is that of algebraic models, that is, time-discrete dynamical systems with algebraic structure. It is shown by way of several examples how this mathematical specification can help with model analysis. This mathematical framework can also accommodate other model types such as Boolean networks and the more general logical models, as well as Petri nets. [ABSTRACT FROM AUTHOR]

We give a formal study of the relationships between the transition cost parameters and the generalized maximum parsimonious reconstructions of unknown (ancestral) binary character states $\{\mathtt{0},\mathtt{1}\}$ over a phylogenetic tree. As a main result, we show there are two thresholds $\lambda^{\mathtt{1}}_{n}$ and $\lambda^{\mathtt{0}}_{n}$, generally confounded, associated to each node n of the phylogenetic tree and such that there exists a maximum parsimonious reconstruction associating state $\mathtt{1}$ to n (resp. state $\mathtt{0}$ to n) if the ratio ' $\mathtt{1}\mathtt{0}$-cost'/' $\mathtt{0}\mathtt{1}$-cost' is smaller than $\lambda^{\mathtt{1}}_{n}$ (resp. greater than $\lambda^{\mathtt{0}}_{n}$). We propose a dynamic programming algorithm computing these thresholds in a quadratic time with the size of tree. We briefly illustrate some possible applications of this work over a biological dataset. In particular, the thresholds provide a natural way to quantify the degree of support for states reconstructed as well as to determine what kind of evolutionary assumptions in terms of costs are necessary to a given reconstruction. [ABSTRACT FROM AUTHOR]

Bursting electrical activity is ubiquitous in excitable cells such as neurons and many endocrine cells. The technique of fast/slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the design principles for bursting in endocrine cells are just quantitative variations of those for bursting in neurons. [ABSTRACT FROM AUTHOR]

In this paper, we present a mathematical model describing the effect of polar lipids, excreted by glands in the eyelid and present on the surface of the tear film, on the evolution of a pre-corneal tear film. We aim to explain the interesting experimentally observed phenomenon that the tear film continues to move upward even after the upper eyelid has become stationary. The polar lipid is an insoluble surface species that locally alters the surface tension of the tear film. In the lubrication limit, the model reduces to two coupled non-linear partial differential equations for the film thickness and the concentration of lipid. We solve the system numerically and observe that increasing the concentration of the lipid increases the flow of liquid up the eye. We further exploit the size of the parameters in the problem to explain the initial evolution of the system. [ABSTRACT FROM AUTHOR]

It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this 'rogue taxon' effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a 'rogue taxon' such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we computationally construct polyhedral cones that give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation. [ABSTRACT FROM AUTHOR]

model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (Evolutionary Biology, vol. 14, pp. 61-204, ) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models. A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface-a generalization of the hyperplane found by Zhivotovsky et al. (Proc. Natl. Acad. Sci. USA 91, 1079-1083, ) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance. Other results include the nonexistence of 'viability analogous, Hardy-Weinberg' modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the presence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains. [ABSTRACT FROM AUTHOR]