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In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the "ages" of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of Partial Differential Equations (PDEs) in the large-volume limit, as opposed to Ordinary Differential Equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems., Comment: 21 pages, 4 figures. Under review for publication

Profiling is a process that finds similarities between different RNA secondary structures by extracting signals from the Boltzmann sampling. The reproducibility of profiling can be identified by the standard deviation of number of features among Boltzmann samples. We found a strong relationship between the frequency of each helix class and its standard deviation of the frequency upon repeated Boltzmann sampling. We developed a perturbation technique to predict the stability of these featured helix classes without the need for repeated Boltzmann sampling, with accuracy between 84% and 94%, depending on the type of RNA. Our technique only requires 0.2% of the computation time compared to one profiling process., Comment: 14 pages

Radiation therapy has remained as one of the main cancer treatment modalities and a highly cost-effective single modality treatment of cancer care. Typical regimens for fractionated external beam radiotherapy comprise a constant dose administered on weekdays, and no radiation on weekends. However, every patient has a tumor with distinct properties depending on intra-tumor heterogeneity, aggressiveness, and interactive properties with other cells that may make it more resistant or sensitive to radiation treatment. Accordingly, the concept of personalized cancer treatment is emerging to specialize each patient treatment case to the unique properties of the tumor. In this paper, we examine adaptive radiation treatment strategies for heterogeneous tumors using a dynamical system model that consists of radiation-resistant and parental cell populations with unique interactive properties. We study different adaptive dosage strategies for PC3 and DU145 prostate cancer cell lines. We show that stronger doses of radiation given in longer time intervals, while keeping the overall dosage the same, reduce final tumor volume by more than half in PC3 cell lines, but by only five percent in DU145 cell lines. In addition, we tested an adaptive dosing schedule by administering a stronger dosage on Friday to compensate for the treatment-off period during the weekend, which was effective in decreasing the final tumor volume of both cell lines. This result creates interesting possibilities for new radiotherapy strategies at clinics that cannot provide treatment on weekends. Finally, we propose a dosage plan incorporating our findings., Comment: 15 pages, 9 figures

The purpose of this paper is to present a model of a phenomenon of plant stem morphogenesis observed by Cesar Gomez-Campo in 1970. We consider a simplified model of auxin dynamics in plant stems, showing that, after creation of the original primordium, it can represent random, distichous and spiral phyllotaxis (leaf arrangement) just by varying one parameter, the rate of diffusion. The same analysis extends to the $n$-jugate case where $n$ primordia are initiated at each plastochrone. Having validated the model, we consider how it can give rise to the Gomez-Campo phenomenon, showing how a stem with spiral phyllotaxis can produce branches of the same or opposite chirality. And finally, how the relationship can change from discordant to concordant over the course of a growing season., Comment: 27 pages, 17 figures

Single-cell RNA sequencing (scRNA-seq) enables dissecting cellular heterogeneity in tissues, resulting in numerous biological discoveries. Various computational methods have been devised to delineate cell types by clustering scRNA-seq data where the clusters are often annotated using prior knowledge of marker genes. In addition to identifying pure cell types, several methods have been developed to identify cells undergoing state transitions which often rely on prior clustering results. Present computational approaches predominantly investigate the local and first-order structures of scRNA-seq data using graph representations, while scRNA-seq data frequently displays complex high-dimensional structures. Here, we present a tool, scGeom for exploiting the multiscale and multidimensional structures in scRNA-seq data by inspecting the geometry via graph curvature and topology via persistent homology of both cell networks and gene networks. We demonstrate the utility of these structural features for reflecting biological properties and functions in several applications where we show that curvatures and topological signatures of cell and gene networks can help indicate transition cells and developmental potency of cells. We additionally illustrate that the structural characteristics can improve the classification of cell types.

This paper discusses, from a mathematician's point of view, the thesis formulated by Israel Gelfand, one of the greatest mathematicians of the 20th century, and one of the pioneers of mathematical biology: "There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology."

In applications of the optimal control theory to problems in medicine and biology, the dependency of the objective functional on the control itself is often a matter of controversy. In this paper, we explore the impact of the dependency using reasonably simple \emph{SIR} and \emph{SEIRS} epidemic models. To qualitatively compare the outcomes for different objective functionals, we apply the cost-effectiveness analysis. Our result shows that, at least for the comparatively inexpensive controls, the variation of the power at the controls in a biologically feasible range does not significantly affect the forms of the optimal controls and the corresponding optimal state solutions. Moreover, the costs and effectiveness are affected even less. At the same time, the dependency of the cost on the state variables can be very significant., Comment: 22 pages

This paper aims to evaluate the potential cost-effectiveness of healthcare interventions against human papillomavirus (HPV). For this, we consider a two-sex epidemic model for the transmission dynamics of HPV which includes screening, vaccination of adolescent boys and girls, and vaccination of sexually active adults. We first propose public health policies using constant control parameters and develop a cost-effectiveness analysis (CEA) to identify which intervention delivers the best effectiveness for the money invested. Secondly, we consider time-dependent control parameters and formulate an optimal control problem to obtain time-dependent versions of the interventions. As in the case of constant control parameters, we perform a CEA to investigate the cost-effectiveness of the time-dependent control interventions. Our findings suggest that females' vaccination, including adolescent girls and adult women, is the most cost-effective strategy. We also compare constant against the time-dependent healthcare interventions which are optimal in the sense that they minimize the objective functional of the optimal control problem. The results indicate that time-dependent controls are not always more cost-effective than constant controls., Comment: 22 pages, 5 figures

Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited success, so far, in treating solid cancers, it is increasingly successful, demonstrating to have a broader therapeutic potential. In this paper we develop a mathematical model to study the efficacy of engineered T cell receptor (TCR) T cell therapy targeting the E7 antigen in cervical cancer cell lines. We consider a dynamical system that follows the population of cancer cells, TCR T cells, and IL-2. We demonstrate that there exists a TCR T cell dosage window for a successful cancer elimination that can be expressed in terms of the initial tumor size. We obtain the TCR T cell dose for two cervical cancer cell lines: 4050 and CaSki. Finally, a combination therapy of TCR T cell and IL-2 treatment is studied. We show that certain treatment protocols can improve therapy responses in the 4050 cell line, but not in the CaSki cell line., Comment: 8 pages, 7 figures

We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities -- (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data -- inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomena can be modelled in an infinite number of different ways, through the adoption of an open/pluralistic approach. We explain the open approach using fish locomotion as a case study and illustrate some of the pitfalls -- universalism, creating models of models, etc. -- that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling. This article is dedicated to the memory of Edmund Crampin.

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratio R0 is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions proved. The model results suggest that as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium respectively. The analytical results, are supported by numerical simulations. Keywords: Time delay, Pneumococcal pneumonia, Vaccination, Stability, Hopf-bifurcation, Comment: 36 pages, 24 figures

During cell division paired (sister) chromosomes are observed to perform approximate saw-tooth oscillations across the cell mid-plane. Experimental data suggests that these oscillations are regulated through intersister tension. We propose a time dependent tension threshold model that exhibits three stable periodic solutions and the phase diagram can be generated semi-analytically. Incorporation of diffusive noise reproduces realistic oscillations, with realistic periods, amplitudes and reproducing the observation that either sister of the pair can switch direction first., Comment: 4 pages (including references and figure legends), 3 figures

In a previous paper (S. Ghosal and Z. Chen Bull. Math. Biol. 2010, vol. 72, pg. 2047) it was shown that the evolution of the solute concentration in capillary electrophoresis is described by a nonlinear wave equation that reduced to Burger's equation if the nonlinearity was weak. It was assumed that only strong electrolytes (fully dissociated) were present. In the present paper it is shown that the same governing equation also describes the situation where the electrolytic buffer consists of a single weak acid (or base). A simple approximate formula is derived for the dimensionless peak variance which is shown to agree well with published experimental data., Comment: 10 pages, 2 figures

The algebraic properties of flattenings and subflattenings provide direct methods for identifying edges in the true phylogeny -- and by extension the complete tree -- using pattern counts from a sequence alignment. The relatively small number of possible internal edges among a set of taxa (compared to the number of binary trees) makes these methods attractive, however more could be done to evaluate their effectiveness for inferring phylogenetic trees. This is the case particularly for subflattenings, and our work makes progress in this area. We introduce software for constructing and evaluating subflattenings for splits, utilising a number of methods to make computing subflattenings more tractable. We then present the results of simulations we have performed in order to compare the effectiveness of subflattenings to that of flattenings in terms of split score distributions, and susceptibility to possible biases. We find that subflattenings perform similarly to flattenings in terms of the distribution of split scores on the trees we examined, but may be less affected by bias arising from both split size/balance and long branch attraction. These insights are useful for developing effective algorithms to utilise these tools for the purpose of inferring phylogenetic trees., Comment: 21 pages, 13 figures

Chimeric antigen receptor (CAR) T-cell based immunotherapy has shown its potential in treating blood cancers, and its application to solid tumors is currently being extensively investigated. For glioma brain tumors, various CAR T-cell targets include IL13Ra2, EGFRvIII, HER2, EphA2, GD2, B7-H3, and chlorotoxin. In this work, we are interested in developing a mathematical model of IL13Ra2 targeting CAR T-cells for treating glioma. We focus on extending the work of Kuznetsov et al. (1994) by considering binding of multiple CAR T-cells to a single glioma cell, and the dynamics of these multi-cellular conjugates. Our model more accurately describes experimentally observed CAR T-cell killing assay data than a model which does not consider cell binding. Moreover, we derive conditions in the CAR T-cell expansion rate that determines treatment success or failure. Finally, we show that our model captures distinct CAR T-cell killing dynamics at low, medium, and high antigen receptor densities in patient-derived brain tumor cells., Comment: 12 pages, 9 figures

RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simulations and the theoretical analysis of the model. In order to perform the simulations, we have implemented the mathematical model into a computational platform capable of running simulations and presenting the results in a graphical format in real time. Among other things, we show that the establishment of virus populations may display four distinct regimes from its introduction into new hosts until achieving equilibrium or undergoing extinction. Also, we were able to simulate different fitness distributions representing distinct environments within a host which could either be favorable or hostile to the viral success. We addressed the most used mechanisms for explaining the extinction of RNA virus populations called lethal mutagenesis and mutational meltdown. We were able to demonstrate a correspondence between these two mechanisms implying the existence of a unifying principle leading to the extinction of RNA viruses., Comment: 34 pages, 10 figures, 4 tables, 1 appendix

The males of the specie of frogs Engystomops pustulosus produce simple and com- plex calls to lure females, as a way of Intersexual selection. Complex calls lead males to a greater reproductive success than simple calls do. However, the complex calls are also more attractive to the main predator of these amphibians, the bat Trachops cirrhosus. Therefore, as M. Ryan suggests, the complexity of the calls let the frogs keep a trade off between reproductive success and predation. In this paper, we first propose to model the proportion of simple to complex calls as a symmetric game of two strategies. We also propose a model with three strategies (simple callers, complex callers and quiet males), where we assess the effect of a male that keeps quiet and intercepts females, which would play a role of Intrasexual selection. We analyze the stable points of the replicator equations of the models that we propose. Under the assumption that the decision of the males takes into account this trade off between reproductive success and predation, our model reproduces the observed behavior reported in the literature with minimal assumption on the parameters. From the three strategies model, we verify that the quiet strategy could only coexists with the simple and complex strategies as long as the rate at which quiet males intercept females is high. We conclude that the reproductive strategy of the male frog Engystomops pustulosus is rational., Comment: 17 pages, 6 figures

Understanding whether a population will survive and flourish or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction-diffusion equations, where migration is usually represented as a linear diffusion term, and birth-death is represented with a bistable source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalise the constant diffusivity, $D$, to a nonlinear diffusivity function $D(C)$, where $C>0$ is the density. While it has been formally established that the choice of $D(C)$ affects long-term survival or extinction of a bistable population, working solely in a classical continuum framework makes it difficult to understand precisely how the choice of $D(C)$ affects survival or extinction. Here, we address this question by working with a simple discrete simulation model that is easy to interpret. The continuum limit of the discrete model is a nonlinear reaction-diffusion equation, where the flux involves a nonlinear diffusion term and the source term is given by the strong Allee effect bistable model. We study population extinction/survival using this very intuitive discrete framework together with numerical solutions of the reaction-diffusion continuum limit equation. This approach provides clear insight into how the choice of $D(C)$ either encourages or suppresses population extinction relative to the classical linear diffusion model., Comment: 40 pages, 11 figures, 1 supplementary material document

Age at infection is often an important factor in epidemic dynamics. In this paper a disease transmission model of SIS type with age dependent infection on a heterogeneous network is discussed. The model allows the infectious rate and the recovery rate to vary and depend on the age of the infected individual at the time of infection. We address the threshold property of the basic reproduction number and present the global dynamical properties of the disease-free and endemic equilibria in the model. Finally, some numerical simulations are carried out to illustrate the main results. The combined effects of the network structure and the age dependent factor on the disease dynamics are displayed., Comment: 29 papges, 5 figures

We comment on Alan Turing's celebrated paper "The Chemical Basis of Morphogenesis" published in 1952 in the "Philosophical Transactions of the Royal Society of London". It is a typical example of a pioneering and inspired work in the domain of mathematical modelling., Comment: 33 pages, 19 figures

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis) very few have been proposed for the regeneration of the lymphatic network. Moreover, lymphangiogenesis is markedly distinct from angiogenesis, occurring at different times and in a different manner. Here a model of five ordinary differential equations is presented to describe the formation of lymphatic capillaries following a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system is then solved numerically and the results are compared with the available biological literature. Finally, a parameter sensitivity analysis of the model is taken as a starting point for suggesting new therapeutic approaches targeting the enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper understanding of the phenomenon in question, clarifying the main factors involved. In particular, the balance between TGF-$\beta$ and VEGF levels, rather than their absolute values, is identified as crucial to effective lymphangiogenesis. In addition, the results indicate lowering the macrophage-mediated activation of TGF-$\beta$ and increasing the basal lymphatic endothelial cell growth rate, \emph{inter alia}, as potential treatments. It is hoped the findings of this paper may be considered in the development of future experiments investigating novel lymphangiogenic therapies.

Calibration of a SIR (Susceptibles-Infected-Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of calibrated model parameters and, ultimately, of model predictions.

The emerging global infectious COVID-19 coronavirus disease by novel Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) presents critical threats to global public health and the economy since it was identified in late December 2019 in China. The virus has gone through various pathways of evolution. For understanding the evolution and transmission of SARS-CoV-2, genotyping of virus isolates is of great importance. We present an accurate method for effectively genotyping SARS-CoV-2 viruses using complete genomes. The method employs the multiple sequence alignments of the genome isolates with the SARS-CoV-2 reference genome. The SNP genotypes are then measured by Jaccard distances to track the relationship of virus isolates. The genotyping analysis of SARS-CoV-2 isolates from the globe reveals that specific multiple mutations are the predominated mutation type during the current epidemic. Our method serves a promising tool for monitoring and tracking the epidemic of pathogenic viruses in their gradual and local genetic variations. The genotyping analysis shows that the genes encoding the S proteins and RNA polymerase, RNA primase, and nucleoprotein, undergo frequent mutations. These mutations are critical for vaccine development in disease control.

The mathematical analysis of a three-tiered food-web describing anaerobic chlorophenol mineralisation has suggested the emergence of interesting dynamical behaviour through its specific ecological interactions, which include competition, syntrophy and product inhibition. Previous numerical analyses have revealed the possibility for a Hopf bifurcation occurring through the interior equilibrium and the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired stable positive steady-state, where full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and prove the occurrence of a supercritical Hopf bifurcation resulting from variations in these operating parameters. Further, using numerical estimation, we also show that variations in the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations. Finally, we are able to show apply persistence theory for a range of parameter sets to demonstrate unique persistence in the cases where chlorophenol and hydrogen are extraneously added to the system, mirroring recent applied studies highlighting the role of hydrogen in maintaining stable anaerobic microbial communities., Comment: 31 pages, 9 figures

In December 2019, a novel coronavirus (SARS-CoV-2) has been identified to cause acute respiratory disease in humans. An outbreak of this disease has been reported in mainland China with the city of Wuhan as the recognized epicenter. The disease has also been exported to other countries, including the Philippines, but the level of spread is still under control (as of 08 February 2020). To describe and predict the dynamics of the disease, several preliminary mathematical models are formulated by various international study groups. Here, the insights that can be drawn from these models are discussed, especially as inputs for designing strategies to control the epidemics. Proposed model-based strategies on how to prevent the spread of the disease in local setting, such as during large social gatherings, are also presented. The model shows that the exposure time is a significant factor in spreading the disease. With a basic reproduction number equal to 2, and 14-day infectious period, an infected person staying more than 9 hours in the event could infect other people. Assuming the exposure time is 18 hours, the model recommends that attendees of the social gathering should have a protection with more than 70 percent effectiveness., Comment: 35 pages

Recent whole-brain calcium imaging recordings of the nematode C. elegans have demonstrated that neural activity is dominated by dynamics on a low-dimensional manifold that can be clustered according to behavioral states. Despite progress in modeling the dynamics with linear or locally linear models, it remains unclear how a single network of neurons can produce the observed features. In particular, there are multiple clusters, or fixed points, observed in the data which cannot be characterized by a single linear model. We propose a nonlinear control model which is global and parameterized by only four free parameters that match the features displayed by the low-dimensional C. elegans neural activity. In addition to reproducing the average probability distribution of the data, long and short time-scale changes in transition statistics can be characterized via changes in a single parameter. Some of these macro-scale transitions have experimental correlates to single neuro-modulators that seem to act as biological controls, allowing this model to generate testable hypotheses about the effect of these neuro-modulators on the global dynamics. The theory provides an elegant characterization of the neuron population dynamics in C. elegans. Moreover, the mathematical structure of the nonlinear control framework provides a paradigm that can be generalized to more complex systems with an arbitrary number of behavioral states.

Many of the chemical reactions that take place within a living cell are irreversible. Due to evolutionary pressures, the number of allowable reactions within these systems are highly constrained and thus the resulting metabolic networks display considerable asymmetry. In this paper, we explore possible evolutionary factors pertaining to the reduced symmetry observed in these networks, and demonstrate the important role environmental variability plays in shaping their structural organization. Interpreting the returnability index as an equilibrium constant for a reaction network in equilibrium with a hypothetical reference system, enables us to quantify the extent to which a metabolic network is in disequilibrium. Further, by introducing a new directed centrality measure via an extension of the subgraph centrality metric to directed networks, we are able to characterise individual metabolites by their participation within metabolic pathways. To demonstrate these ideas, we study 116 metabolic networks of bacteria. In particular, we find that the equilibrium constant for the metabolic networks decreases significantly in-line with variability in bacterial habitats, supporting the view that environmental variability promotes disequilibrium within these biochemical reaction systems.

Electrophoretic separation of a mixture of chemical species is a fundamental technique of great usefulness in biology, health care and forensics. In capillary electrophoresis the sample migrates in a microcapillary in the presence of a background electrolyte. When the ionic concentration of the sample is sufficiently high, the signal is known to exhibit features reminiscent of nonlinear waves including sharp concentration `shocks'.In this paper we consider a simplified model consisting of a single sample ion and a background electrolyte consisting of a single co-ion and a counterion in the absence of any processes that might change the ionization states of the constituents. If the ionic diffusivities are assumed to be the same for all constituents the concentration of sample ion is shown to obey a one dimensional advection diffusion equation with a concentration dependent advection velocity.If the analyte concentration is sufficiently low in a suitable non-dimensional sense, Burgers' equation is recovered, and thus, the time dependent problem is exactly solvable with arbitrary initial conditions. In the case of small diffusivity either a leading edge or trailing edge shock is formed depending on the electrophoretic mobility of the sample ion relative to the background ions. Analytical formulas are presented for the shape, width and migration velocity of the sample peak and it is shown that axial dispersion at long times may be characterized by an effective diffusivity that is exactly calculated. These results are consistent with known observations from physical and numerical simulation experiments., Comment: 20 pages, 4 figures

We derive the amino acid assignment to one codon representation (typical 64-dimensional irreducible representation) of the basic classical Lie superalgebra osp(5|2) from biochemical arguments. We motivate the approach of mathematical symmetries to the classification of the building constituents of the biosphere by analogy of its success in particle physics and chemistry. The model enables to calculate polarity and molecular volume of amino acids to a good approximation., Comment: 19 pages, 4 pictures, Part of this work has been submitted internally at Computer Science Corporation for the LEADING EDGE FORUM CSC_PAPERS 2011

The brain is one of the most studied and highly complex systems in the biological world. It is the information center behind all vertebrate and most invertebrate life, and thus has become a major focus in current research. While many of these studies have concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural aspects of the brain should elucidate some of its functional properties. In this paper we analyze the brain of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian matrix of the 279-neuron "giant component" of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot (in eigenfunction coordinates) both 2 and 3 dimensional visualizations of the neural network. Further analysis examines the small-world properties of the system, including average path length and clustering coefficient. We then test for localization of eigenfunctions, using graph energy and spacial variance. To better understand these results, all of these calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been "rewired" to have small-world properties. This analysis is one of many stepping-stones in the research of neural networks. While many of the structures and functions within the brain are known, understanding how the two interact is also important. A firmer grasp on the structural properties of the neural network is a key step in this process, Comment: 24 Pages, 6 Figures

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption., Comment: 19 pages, 2 figures

This tutorial introduces participants to the design and implementation of an agent-based model using NetLogo through one of two different projects: modelling T cell movement within a lymph node or modelling the progress of a viral infection in an in vitro cell culture monolayer. Each project is broken into a series of incremental steps of increasing complexity. Each step is described in detail and the code to type in is initially provided. However, each project has room to grow in complexity and biological realism so participants are encouraged to expand their project beyond the scope of the tutorial or to develop a project of their own.

Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models.~The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erd\'ely-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a {\it ggBm-like} construction of the random variable., Comment: 8 pages

This study focuses on analyzing a deterministic SIR model governing the dynamics of the hosts and vectors on an urban network. Our analysis scrutinizes the typical existence--stability of the equilibria as well as the sensitivity of the basic reproductive number. The latter is a groundbreaking finding that has a strong applicative relevance, particularly for addressing the problem of the distribution of job opportunities toward diminishing the disease persistence on the entire network., Comment: 7 pages

We derived an ordinary differential equation model to capture the disease dynamics during blood-stage malaria. The model was directly derived from an earlier age-structured partial differential equation model. The original model was simplified due to experimental constraints. Here we calibrated the simplified model with experimental data using a multiple objective genetic algorithm. Through the calibration process, we quantified the removal of healthy red blood cells and the the preferential infection of reticulocytes during \textit{Plamodium cynomolgi} infection of \textit{Macaca mulatta}. The calibration of our model also revealed the existence of host erythropoietic response prior to blood stage infection., Comment: 17 pages, 14 figures

The central nervous system is composed of many individual units -- from cells to areas -- that are connected with one another in a complex pattern of functional interactions that supports perception, action, and cognition. One natural and parsimonious representation of such a system is a graph in which nodes (units) are connected by edges (interactions). While applicable across spatiotemporal scales, species, and cohorts, the traditional graph approach is unable to address the complexity of time-varying connectivity patterns that may be critically important for an understanding of emotional and cognitive state, task-switching, adaptation and development, or aging and disease progression. Here we survey a set of tools from applied mathematics that offer measures to characterize dynamic graphs. Along with this survey, we offer suggestions for visualization and a publicly-available MATLAB toolbox to facilitate the application of these metrics to existing or yet-to-be acquired neuroimaging data. We illustrate the toolbox by applying it to a previously published data set of time-varying functional graphs, but note that the tools can also be applied to time-varying structural graphs or to other sorts of relational data entirely. Our aim is to provide the neuroimaging community with a useful set of tools, and an intuition regarding how to use them, for addressing emerging questions that hinge on accurate and creative analyses of dynamic graphs., Comment: 21 pages, 5 figures. Toolbox not yet publicly available

Falciparum malaria is a major parasitic disease causing widespread morbidity and mortality globally. Artemisinin derivatives---the most effective and widely-used antimalarials that have helped reduce the burden of malaria by 60% in some areas over the past decade---have recently been found to induce growth retardation of blood-stage Plasmodium falciparum when applied at clinically relevant concentrations. To date, no model has been designed to quantify the growth retardation effect and to predict the influence of this property on in vivo parasite killing. Here we introduce a mechanistic model of parasite growth from the ring to trophozoite stage of the parasite's life cycle, and by modelling the level of staining with an RNA-binding dye, we demonstrate that the model is able to reproduce fluorescence distribution data from in vitro experiments using the laboratory 3D7 strain. We quantify the dependence of growth retardation on drug concentration and demonstrate the model's utility as a platform to propose experimentally-testable mechanisms of growth retardation. Furthermore we illustrate that a drug-induced delay in growth may significantly influence in vivo parasite dynamics, demonstrating the importance of considering growth retardation in the design of optimal artemisinin-based dosing regimens., Comment: 44 pages, 20 figures (including 8 supplementary figures), 8 tables (including 6 supplementary tables)

Artemisinin resistance constitutes a major threat to the continued success of control programs for malaria. With alternative antimalarial drugs not yet available, improving our understanding of how artemisinin-based drugs act and how resistance manifests is essential to enable optimisation of dosing regimens in order to prolong the lifespan of current first-line treatment options. Here, through introduction of a novel model of the dynamics of the parasites' response to drug, we explore how artemisinin-based therapies may be adjusted to maintain efficacy and how artemisinin resistance may manifest and be overcome. We introduce a dynamic mathematical model, extending on the traditional pharmacokinetic-pharmacodynamic framework, to capture the time-dependent development of a stress response in parasites. We fit the model to in vitro data and establish that the parasites' stress response explains the recently identified complex interplay between drug concentration, exposure time and parasite viability. Our model demonstrates that the previously reported hypersensitivity of early ring stage parasites of the 3D7 strain to dihydroartemisinin (DHA) is primarily due to the rapid development of stress, rather than any change in the maximum achievable killing rate. Of direct clinical relevance, we demonstrate that the complex temporal features of artemisinin action observed in vitro have a significant impact on predictions of in vivo parasite clearance using PK-PD models. Given the important role that such models play in the design and evaluation of clinical trials for alternative drug dosing regimens, our model contributes an enhanced predictive platform for the continued efforts to minimise the burden of malaria., Comment: 24 Pages, 9 figures, 2 tables

Forests are complex systems, and it is necessary to include this characteristic in every forest definition, in order to consider the restriction that this imposes in terms of prediction and control. This lost of predictability and controllability should be incorporated in every Environmental Impact Assessment or management program. We present two case-studies located in Mexico and one in the US to illustrate three relevant indicators of complexity. First, we introduce an informational framework to measure the Zoquiapan forest systemic complexity. Then, we analyze complexity changes among different types of forest and management systems, related with spatial distributions, using data from a floristic study in the Montes Azules National Park. Finally, we analyze time series of $CO_{2}$ fluctuations taken from AMERIFLUX data bases. Our results show firstly that it is possible to measure the systemic complexity of different forests, characteized by a criticality state (1/f noise) which has been proposed as a finger print of complexity. And secondly, that this characteristic can be used as a proxy of their state of conservation, where the lowest complexity values are found in perturbed areas showing the relevance of the concept and its measurement for forest conservation and management., Comment: 16 pages, 7 figures

We present a mathematical model, based on ordinary differential equations, for the evolution of solid tumors and their response to treatment. Specifically the effects of a cytotoxic agent and a monoclonal antibody are included as control term in the equations. The variables considered here are: the number of cancerous cells sensitive to chemotherapy, the number of cancerous cells resistant to chemotherapy, the degree of angiogenesis and the average intensity of VEGF. The rules that govern the quantities mentioned above are based on a geometrical argument: we approximate the tumor mass as a sphere and thus derive basic formulae for the normoxic cells and for VEGF production. The monoclonal antibody acts on VEGF and thus has in uence to the global degree of angiogenesis. Numerical estimates on some of the parameters are performed in order to match the main landmark in tumor progression and reaction to treatment in the specific case of colorectal cancer., Comment: 21 pages, 7 figures

Replicative senescence, induced by telomere shortening, exhibits considerable asynchrony and heterogeneity, the origins of which remain unclear. Here, we formally study how telomere shortening mechanisms impact on senescence kinetics and define two regimes of senescence, depending on the initial telomere length variance. We provide analytical solutions to the model, highlighting a non-linear relationship between senescence onset and initial telomere length distribution. This study reveals the complexity of the collective behavior of telomeres as they shorten, leading to senescence heterogeneity.

We propose a multiscale model of the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We assume that the epithelial mesenchymal-like transition between them is driven primarily by the epidermal growth factors. We moreover take into account the transidifferentiation program of the cancer stem cells and the cancer associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix. The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose "mathematical-treatment" scenarios to control the aggressiveness of the tumour.

In theoretical studies, the most commonly used measure of ecological stability is resilience: ecosystems asymptotic rate of return to equilibrium after a pulse-perturbation $-$or shock. A complementary notion of growing popularity is reactivity: the strongest initial response to shocks. On the other hand, empirical stability is often quantified as the inverse of temporal variability, directly estimated on data, and reflecting ecosystems response to persistent and erratic environmental disturbances. It is unclear whether and how this empirical measure is related to resilience and reactivity. Here, we establish a connection by introducing two variability-based stability measures belonging to the theoretical realm of resilience and reactivity. We call them intrinsic, stochastic and deterministic invariability; respectively defined as the inverse of the strongest stationary response to white-noise and to single-frequency perturbations. We prove that they predict ecosystems worst response to broad classes of disturbances, including realistic models of environmental fluctuations. We show that they are intermediate measures between resilience and reactivity and that, although defined with respect to persistent perturbations, they can be related to the whole transient regime following a shock, making them more integrative notions than reactivity and resilience. We argue that invariability measures constitute a stepping stone, and discuss the challenges ahead to further unify theoretical and empirical approaches to stability., Comment: 35 pages, 7 figures, 2 tables

Directional growth caused by gravitropism and corresponding bending of plant cells has been explored since 19th century, however, many aspects of mechanisms underlying the perception of gravity at the molecular level are still not well known. Perception of gravity in root and shoot gravitropisms is usually attributed to gravisensitive cells, called statocytes, which exploit sedimentation of macroscopic and heavy organelles, amyloplasts, to sense the direction of gravity. Gravity stimulus is then transduced into distal elongation zone, which is several mm far from statocytes, where it causes stretching. It is suggested that gravity stimulus is conveyed by gradients in auxin flux. We propose a theoretical model that may explain how concentration gradients and/or stretching may indirectly affect the global orientation of cortical microtubules, attached to the cell membrane and induce their dynamic reorientation perpendicular to the gradients. In turn, oriented microtubules arrays direct the growth and orientation of cellulose microfibrils, forming part of the cell external skeleton and determine the shape of the cell. Reorientation of microtubules is also observed in reaction to light in phototropism and mechanical bending, thus suggesting universality of the proposed mechanism., Comment: published

Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 50's with the KiSS (after Kierstead, Slobodkin and Skellam) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size $L$ surrounded by a deadly environment with infinite mortality -- i.e. an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence. Here, we introduce an individual-based analogue of the KiSS model to investigate the effects of discreteness and demographic stochasticity. In particular, we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasi-stationary distribution of the number of individuals for patch sizes above the critical threshold., Comment: 9 pages, 7 figures

Colonies of ants are systems of interacting living organisms in which interactions between individuals and their environment can produce a reliable performance of a complex tasks without the need for centralised control. Particularly remarkable is the process of formation of refined paths between the nest and food sources that is essential for successful foraging. We have designed a simple stochastic off-lattice model of ant foraging in the absence of direct communication. The motion of ants is governed by two components - a random change in direction of motion that improves ability to explore the environment (facilitating food discovery), and a non-random global indirect interaction component based on pheromone signalling. Using numerical simulations we have studied the model behaviour in different parameter regimes and tested the ability of our model ants to adapt to changes in the external environment. The simulated behaviour of ants in the model recapitulated the experimentally observed behaviours of real ants. Furthermore we observe interesting synchronization in ant movement patterns which vary with the diffusion properties of the pheromones.

Identifying the entire set of complexes is essential not only to understand complex formations, but also to map the high level organisation of the cell. Computational prediction of protein complexes faces several challenges including the lack of sufficient protein interactions, presence of noise in protein interaction datasets and difficulty in predicting small and sparse complexes. These challenges are covered in most reviews of complex prediction methods. However, an important challenge that needs to be addressed is the prediction of membrane complexes. These are often ignored because existing protein interaction detection techniques do not detect interactions between membrane proteins. But, recently there have been several new experimental techniques including MY2H that are capable of detecting membrane protein interactions. In the light of this new data, we discuss here new challenges and the kind of open problems that need to be solved to effectively detect membrane complexes., Comment: 7 pages

According to the World Health Organization, dengue fever is the most important mosquito-borne disease of humans, and it is currently estimated that there may be 50 - 100 million yearly dengue infections worldwide. For the purpose to provide new techniques to public health policies in course, we introduce a predictive non-linear population dynamics model to describe the population size of four stages of the development of \emph{Aedes aegypti}, having the coefficients set to be dependent on the rainfall index data. In spite of the population dynamics of the \emph{Ae. aegypti} be mainly ruled by the rainfall regime, most models are dedicated exclusively to effects of temperature and only few models are dedicated to influence of rainfall. Vector control actions are also implemented in many periods of the year in order to compare relative efficiency of public health policies. The analysis of equilibrium and stability was performed. Field rainfall time series data from the City of Lavras (Minas Gerais, Brazil) was used for the model evaluation. The model was validated in a comparison with experimental mosquito abundance data acquired by field health agents. We evaluated and validated an entomological conjecture that claims that control actions should be performed during the dry season, instead of the common procedure adopted by vector control programs, in which those are mainly applied in the rainy season., Comment: 21 pages, 8 figures. Submitted to Mathematical Biosciences Journal

Bacterial resistance to antibiotic treatment is a huge concern: introduction of any new antibiotic is shortly followed by the emergence of resistant bacterial isolates in the clinic. This issue is compounded by a severe lack of new antibiotics reaching the market. The significant rise in clinical resistance to antibiotics is especially problematic in nosocomial infections, where already vulnerable patients may fail to respond to treatment, causing even greater health concern. A recent focus has been on the development of anti-virulence drugs as a second line of defence in the treatment of antibiotic-resistant infections. This treatment, which weakens bacteria by reducing their virulence rather than killing them, should allow infections to be cleared through the body's natural defence mechanisms. In this way there should be little to no selective pressure exerted on the organism and, as such, a predominantly resistant population would be unlikely to emerge. However, much controversy surrounds this approach with many believing it would not be powerful enough to clear existing infections, restricting its potential application to prophylaxis. We have developed a mathematical model that provides a theoretical framework to reveal the circumstances under which anti-virulence drugs may or may not be successful. We demonstrate that by harnessing and combining the advantages of antibiotics with those provided by anti-virulence drugs, given infection-specific parameters, it is possible to identify treatment strategies that would efficiently clear bacterial infections, while preventing the emergence of resistant subpopulations. Our findings strongly support the continuation of research into anti-virulence drugs and demonstrate that their applicability may reach beyond infection prevention., Comment: Pre-review manuscript. Submitted to Journal of Theoretical Biology, July 21st 2014

T11 Target structure (T11TS), a membrane glycoprotein isolated from sheep erythrocytes, reverses the immune suppressed state of brain tumor induced animals by boosting the functional status of the immune cells. This study aims at aiding in the design of more efficacious brain tumor therapies with T11 target structure. We propose a mathematical model for brain tumor (glioma) and the immune system interactions, which aims in designing efficacious brain tumor therapy. The model encompasses considerations of the interactive dynamics of macrophages, cytotoxic T lymphocytes, glioma cells, TGF-$\beta$, IFN-$\gamma$ and the T11TS. The system undergoes sensitivity analysis, that determines which state variables are sensitive to the given parameters and the parameters are estimated from the published data. Computer simulations were used for model verification and validation, which highlight the importance of T11 target structure in brain tumor therapy.