The effectiveness of mathematics in biology

Pablo A. Marquet
Departamento de Ecología
Pontificia Universidad Católica de Chile

In 1960 Eugene P. Wigner wrote an influential essay on the “unreasonable effectiveness of mathematics in the natural sciences.” Although by natural sciences he meant physics, after 55 years it is possible to say that mathematics has had a more than reasonable effectiveness in biology too. In this talk, I will illustrate this point by first commenting on the role of mathematics in science, and particularly in biology, and then by referring to three areas where mathematics has contributed in an effective way to biology; these are the study of population genetics and in particular neutral theories in ecology, and the study of fluctuation and scaling in biodiversity.

How cancer shapes our bodies

Michael Hochberg CNRS, University of Montpellier
Santa Fe Institute

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection being a fundamental process leading to malignancy. Yet the integration of ecology and evolutionary biology to understanding and controlling neoplastic progression is still relatively uncommon. This is now beginning to change, and there is a growing interest in the interface between ecological and evolutionary theories and the idea of preventing or managing cancer as opposed to its eradication. I describe computational fronts where the ecological and evolutionary perspective is most developed, in particular, that cancer generates substantial levels of phenotypic diversity, which makes it a challenge to control even when discovered in early stages. Evolution has also apparently occurred at the species level, both in terms of influences on the prevalence of cancers across species with extremely different life-histories, and its prevalence across different tissues and organs within our bodies. I present theoretical results indicating that cancer is a constraint on different tissue families within our bodies, and that cancer poses a strong constraint on the evolution of life-span and body size.

Birth and death population networks and their diffusion approximation

Rolando Rebolledo
Centro de Análisis Estocástico
Facultades de Ingeniería y Matemáticas
Universidad Católica de Chile

The great diversity of relations existing in living entities represents an interesting mathematical challenge, to which different well-established theories look forward to providing suitable models. Population networks are among the most popular approaches and models, for instance, in Ecology.

Nevertheless, most of these applications, are built up based on systems of ordinary differential equations, the paradigm of a closed system dynamics. So that, to mimic at least partially, a more complex behavior, a number of researchers prefer to use birth and death stochastic processes to model the dynamics of a given population.

That kind of models allows, in effect, to start thinking the system of living entities in a richer relation with the environment, that the set of deterministic equations is unable to reflect. That is, the stochastic approach is consistent with an open system view on population dynamics.

However, like in the analysis of the evolution of a set of species in terms of the Neutral Theory of Biodiversity, it is important to deal with different time and space scales to obtain a biomass behavioral pattern, independent of specific times and places.

The conference will report a method seeking for adequate rescaling in a birth and death population network providing a limit consisting of diffusion processes on a graph.

A Simple Solver for Simulating Fluid-Fluid and Fluid-Solid Interactions

Jorge Balbás
Department of Mathematics
Cal State University
Northridge, CA 91330-8313

We propose a fully eulerian model for simulating the interaction of fluid-fluid and fluid-solid structures for incompressible fluids and a black-box type numerical scheme for computing its solution. The physical model consists of two Hamilton-Jacobi type PDEs: the fluid-fluid and fluid-solid interfaces are implicitly tracked by a level set equation, and the fluid motion is modeled by the vorticity formulation of Navier-Stokes equation, also of Hamilton-Jacobi type. This approach simplifies the task by allowing us to implement a single numerical scheme for solving both equations. We present the proposed scheme along with several numerical experiments simulating the coalescence of gas bubbles and the oscillations of an elastic membrane that demonstrate the simplicity and robustness of this approach.

Genome wide identification of transcription factors allele specific binding sites combining ChIP-seq data and transcription factor binding motifs

Alejandra Medina-Rivera
Laboratorio Internacional de Investigación sobre el Genoma Humano
UNAM

Most genetic variants associated to a disease or trait in Genome Wide Association studies (GWAS) have been found outside protein coding sequences. a non-coding variant could be altering one or several regulatory mechanisms that could affect gene expression; however, understanding the mechanisms being altered by these variants poses an ongoing challenge.
Transcription factors (TFs) are DNA binding proteins that recognize and bind to sequence-specific motifs to regulate spatial-temporal gene expression. DNA sequence variants that disrupt such motifs will have differential binding effects and can result in either diminished or increased TF binding. This can occur directly through alteration of the TF binding motif, or indirectly by altering the motif a co-binding partner TF. Proposals have been made to detect allele specific binding by comparing ChIP-seq read coverage between two alleles, still, to accurately detect allele-specific binding events is necessary to account for read mapping biases.
Here we propose a two-step approach to identify allele specific binding sites for a given TF and the associated mechanism: 1) ChIP-seq read alignment using a custom alignment strategy that eliminates biases in allelic read mapping (WASP + ALEA), and 2) comparison of predicted binding sites confidence values between pairs of alleles (variation-scan).
We tested our approach in two human trio data sets for CTCF and identify a set of reliable allele specific binding events. Furthermore, to prove the tool can be used in other species we used mouse F1 data for three TFs (ONECUT1, HNF4a and CEBPa), finding similar results to the ones retrieved in human.
Sequence changes can influence binding by affecting the target TF motif, as well as co-bound factors binding. If allele coverage of one TF ChIP-seq reads differs between two alleles, but there is no impact on the TF binding motif this could be explained by the allele affecting co-factor binding. Using the mouse F1 dataset and previously identified co-factors that bind in clusters of regulatory modules in mouse, we were able to assess cofactor effect in allele-specific binding.

Identification of nonlinear parameters using recurrence plots

Juan Carlos Jauregui
Facultad de Ingeniería
Universidad Autónoma de Querétaro

The identification of nonlinear vibrations in mechanical systems is an unsolved problem. The structure of the measured data and wave forms have been studied for many years, and different techniques have been applied. Nevertheless, there is no a single technique for identifying the nonlinear parameters.

Parameter identification can be conducted either with a parametric approach or with a non-parametric approach. Among parametric approaches many researchers have work with the Hilbert transform, continuous and discrete wavelet transform, nonlinear modal analysis, phase space. Meanwhile, nonparametric approaches include fractal analysis, Hurt factor, and approximate entropy.

In this paper, the recurrence plots are applied for the identification of nonlinear parameters. Recurrence plots are constructed from the phase space. In order to calibrate the method, the recurrence plots were obtained from two theoretical models, a Van der Pol pendulum and a Duffing mass-lump model. Then, the recurrence plots were constructed from a mechanical gearbox.

Recurrence plots are an alternative solution for the identification of nonlinearities in mechanical systems.

Evaluation of white matter micro-structure through diffusion-weighted MRI

Luis Concha
Instituto de Neurobiología
UNAM, Campus Juriquilla

Emerging disease dynamics in a model coupling within-host and between-host systems

Zhilan Feng
Department of Mathematics
Purdue University

Epidemiological models and immunological models have been studied largely independently. However, the two processes (within- and between-host interactions) occur jointly and models that couple the two processes may generate new biological insights. Particularly, the threshold conditions for disease control may be dramatically different when compared with those generated from the epidemiological or immunological models separately. We developed and analyzed an ODE model, which links an SI epidemiological model and an immunological model for pathogen-cell dynamics. When the two subsystems are considered in isolation, the dynamics are standard and simple. That is, either the infection-free equilibrium is stable or a unique positive equilibrium is stable depending on the relevant reproduction number being less or greater than 1. However, when the two sub-systems are explicitly coupled, the full system exhibits more complex dynamics including backward bifurcations; that is, multiple positive equilibria exist with one of which being stable even if the reproduction number is less than 1. The biological implications of such bifurcations are illustrated using an example concerning the spread and control of toxoplasmosis.

Cographs in the Reconstruction of Evolutionary Histories

Maribel Hernández
Instituto de Matemáticas
UNAM, Unidad Juriquilla

A cograph is a $P_4$-free graph. Recently, it has been proved that graphs whose vertices represent genes and whose edges represent valid orthology relations between genes, must be cographs. However, due to divergence of genes during evolution, these orthology relations are not always predicted correctly. Therefore, there exists a need for methods to verify and correct those predictions. In this talk, we present applications of cographs for the proper reconstruction of evolutionary histories. Firstable, we present an orthology quality measure based on graphs that can be converted to cographs. Then we show how this method is applied to real data for the detection of ``noise'' which is defined as the set of induced subgraphs that do not allow a graph to be a cograph. Finally, we present an heuristic for the cograph editing problem, which is about converting an arbitrary graph into a cograph by applying edit operations, i.e. adding and removal of edges. This problem has been shown to be NP-complete and therefore the implementation of heuristics and/or approximation algorithms are necessary. These methods have been applied to the pangenomes of E.coli and have given us insights about the evolutionary events that have ocurred in the genomes of these bacteria. Additionally, we are currently using these methods for the reconstruction of the evolutionary histories of tRNAS in primates.

Irreversible thermodynamics of small biological systems

Ivan Santamaría
Unidad Multidisciplinaria de Docencia e Investigación
UNAM, Campus Juriquilla

In this talk we present and discuss a recent generalization of the Onsager-Machlup theory for the regression of fluctuations. The new formalism offers a powerful tool for analyzing the dynamics and thermodynamics of single small biological and technological systems, allowing to connect the fundamental physical and chemical laws with simple models that allow a better understanding and prediction of effects in biophysical systems. The theory is presented in a series of examples ranging from vesicle formation and pours dynamics to the folding-unfolding kinetics of RNA in stretching experiments.

Transmission Dynamics of Two Dengue Serotypes with vaccination scenarios

Mayra Núñez López
Universidad Autónoma Metropolitana
Unidad Cuajimalpa

We present a mathematical model that incorporates two Dengue serotypes, where we allow secondary infections from either of the two serotypes. The model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. We consider two viral strains and temporary cross-immunity with one vector mosquito population and estimate the basic reproduction number. The vaccine is introduced in two doses, and we perform numerical simulations for different scenarios of vaccination. Finally we present a spatial dynamics of dengue based on epidemiological data.

Stochastic dynamics of a two-pathogen epidemic system with partial vaccination

Marcos Capistrán
CIMAT AC
Guanajuato, México

The aim of this project is to study the dynamics of a seasonal two pathogen epidemic model where there is vaccination against one of them. We consider that both diseases are seasonal and there is cross-reaction between the pathogens. We would like to understand, using quantitative methods, how vaccination changes the dynamics of this system compared with the no vaccination regime.

A measure of regularity for polygonal mosaics in biological systems

José Luis Aragón
Centro de Física Aplicada y Tecnología Avanzada, UNAM

A measure of regularity in polygonal mosaics of different kinds in biological systems is proposed. It is based on the condition of "eutacticity", expressed in terms of eutactic stars, which is closely related to the regularity of polytopes. Thus it constitutes a natural measure of regularity. The proposed measure is tested with numerical and real data. Numerically is tested with a hexagonal lattice that is distorted progressively and with a non-periodic regular tiling. With real data, the distribution of oak trees in forests from three locations in the State of Queretaro, Mexico, and the spiral pattern of florets in a flowering plant are characterised.

Spot dynamics in a plant hair initiation model

Víctor F. Breña–Medina
Centro de Ciencias Matemáticas
UNAM Campus Morelia

Results on the dynamics of an initiation process in a root hair (RH) cell at a sub-cellular is presented. An earlier model proposed by Payne and Grierson captures key features of an interacting small G-protein family so-called Rho of Plants (ROPs). These proteins are in charge of promoting certain protuberances on root hair cells, which are crucial for nutrients uptake from the soil and anchorage, for instance. Here, consequences coming from cell growth and cell shape are taken into account. Auxins are a class of hormones that are known to take part on the morphogenesis of plants. As experimental observations show that a fast auxin flow is heterogeneously distributed along the cell at the ROPs diffusive scale, auxin catalysis is modelled as a spatially dependent coefficient controlling dominant cubic terms. Such a model consists of a generalised two-component Schnakenberg reaction-diffusion system, which is set up in a non-homogeneous domain. Previous results for a one-dimensional RH cell model suggests that ROP patch location is controlled by auxin gradient. Nonetheless, this approach is unable to disclose leaking auxin process from flanking non-RH cells outcome nor transversal geometrical aftereffects. In other words, upon considering a more realistic cell geometry, a two-dimensional RH cell gathers the essential ingredients that allows to analyse whether shape and form are relevant for patch location. A novel numerical bifurcation analysis, as well as time numerical simulations, and the theory of semi-strong interactions are performed in order to shed light on the understanding of RH formation dynamics.

Topological characterization of cell circuits in preBötC through the application of metrics of complex networks for analyzing pathological conditions

Elizabeth Santiago del Angel
Instituto de Matemáticas UNAM
Unidad Juriquilla

In this talk, a topological analysis of the correlations generated from the cell circuits located inpre-Bötzinger complex (preBötC) is carried out through the application of complex networks. Where the objective is to characterize the effect hypoxia and to identify topological patterns in both normal conditions and pathological conditions. Firstly, the modelling of cell circuits as complex networks and the description of the respective elements of a network are defined. Then, the centrality measures and connectivity are applied, and a topological classification based on the local and global communicability in the network is used. These metrics are applied to two blocks of data, where the first group of cell circuits is analyzed in two states (control-hypoxia), and a second group is studied in three states (control- isocitrate- hypoxia). The control, isocitrate and hypoxia states are associated to cell circuits in normal conditions, circuits influenced by hypoxia and circuits affected by the application of a drug named isocitrate; respectively. Finally, the results of the topological patterns found in each group are shown, and a type of topological class is defined for this class of cell networks obtained by the application of an approach of four classes based on connectivity.

Modeling and Control of Insects

B.E. Ainseba
Bordeaux University
Institute of Mathematics IMB UMR CNRS 5251

In this talk we will give an overview of the mathematical modeling of important insect dynamics that cause many economical and/or epidemiogical damages. The obtained models are nonlinear partial differential systems with nonlocal terms corresponding to birth and transition rates from one stage of the insect to the next one. Many parameters included in these models cannot be achieved experimentally and depending on the collected data we will discuss the identification of these missing data. Control theory technics can be used to reduce the size of the insect/pest population. Some open questions in connection with controllability of age-structured models will be discussed.

Orthology, Paralogy, and Cographs: How to use paralogs in phylogenomics

Peter Stadler
Bioinformatik, Inst.f.Informatik
Univ.Leipzig, Germany

Recent advances in mathematical phylogenetics set the stage to actively making use of paralogous genes in phylogenomics application rather than spending efforts to exclude them as a nuisances. Orthology and (conversely) paralogy of genes can be estimated with reasonable accuracy from sequence similarity data. The orthology and paralogy relations mathematically are cographs. The empirical estimates thus can be corrected computationally to conform to the required mathematical structures. The resulting co-graphs in turn are equivalent to homeomorphic images of the gene trees together with an event labeling that assigns to each interior node whether it corresponds to a speciation or a gene duplication. Finally, one can show that the underlying species tree must display a certain subset of the triples defined by the event-labeled gene trees. Thus the orthology relation, or more precisely, the paralogs, convey independent information that can be used to successfully reconstruct the species tree itself. These observations pose new challenges: for this approach to be of practical use, good and efficient heuristics for the co-graph editing problem are required. A particularly promising approach starts from the modular decomposition, which exists for all graphs, and stepwisely converts the modular decomposition tree into a co-tree.

The transmission of dengue in Mexico

José Ramos
Instituto Nacional de Salud Pública
Cuernavaca, México

The dengue in Mexico, as is in the majority of the Americas regions, is the most prevalent vector-borne disease, reporting thousands of cases every year. So far, strategies based on vector control with the aim of interrupting the epidemic episodes do not show proven efficacy mainly because structural deficiencies in the epidemiologic surveillance systems, but also because the basis of the endemic transmission is not well understood. Heterogeneity in epidemiological dynamics is a major issue and models, such as the Ross-MacDonald, have received a great deal of criticism. It seems that endemic dengue transmission relies on three major variables: seroprevalence, serotype circulation and age dependent mobility of the subjects; these combines in a complex way and the resulting forces drives transmission. Here, I will present some results from a cohort study developed in two endemic communities in Morelos, Mexico, compatible with the scenario described above.

The DNA fractal structure

Pedro Miramontes
Facultad de Ciencias UNAM

Early in the nineties, several researchers started to study DNA sequences in the framework of time-series analysis. Some of them reported self-similar mosaic arrangements typical of fractal geometry. Since then, it has been clear that behind the fractal-like structure of DNA there is a rich arrangement of correlations at all scales. In this presentation, we show that the fractal structure is intrinsic to DNA. We show several examples and propose to study the organization of DNA as an outcome of a dynamical system.

Exocytosis through a fusion pore without vesicle collapse

Guillermo Ramírez-Santiago
Instituto de Matemáticas
UNAM, Campus Juriquilla Qro.

The hypothesis of chemical transmission proposes that neurotransmitter is discharged in molecular packets from synaptic vesicles. The current view of the exocytosis of transmitter molecules is that it starts with the formation of a pore that builds up upon the full fusion of the vesicle’s membrane with the cell’s plasma membrane. This pore connects the intra-vesicular and the extracellular spaces, and after the transmitter is released, the vesicle collapses. A second scenario is that under certain circumstances, a rapid closure of the pore, before the full vesicle fusion occurs, produces only a partial release of the transmitter. Here it is shown that there exists a third scenario in which the whole release of the transmitter occurs through pores that remain open for tens of milliseconds without vesicle collapse. This scenario was discovered from amperometric measurements of serotonin release from electrodense vesicles in the axon of leech Retzius neurons and mathematical modelling. By modelling the transmitter release with a diffusion equation subjected to boundary conditions defined by the experiment, it is shown that those pores with a fast half rise time constant remained opened and allowed the full quantum release without vesicle collapse, whereas pores with a slow rise time constant closed rapidly, thus leading to a transmitter partial release. Therefore, it is concluded that a full transmitter release also occurs through the fusion pore in the absence of vesicle collapse.

Stability analysis for atmospheric dynamics

Gerardo Hernández Dueñas
Instituto de Matemáticas
UNAM, Campus Juriquilla

The aim of this talk is to identify stability and instability criteria for idealized precipitating hydrodynamics. A linear stability analysis is presented for fluid dynamics with water vapor and precipitation, where the rain falls relative to the fluid at speed $V_T$. Previous considerations include non-precipitating atmospheres ($V_T=0$), or fluids where any rain formed is immediately removed from the parcel of moist air (infinitely fast $V_T$). The aim here is to bridge two extreme cases of saturated precipitating atmospheres by considering the full range of $V_T$ values, and including hydrodynamics. We will show results that have some resemblance to the notion of conditional instability: separate sufficient conditions exist for stability versus instability, with an intermediate range of environmental states where stability or instability is not definitive. This is joint work with Samuel N. Stechmann and Leslie M. Smith.

PDE models for bacterial aggregation with non-linear cross diffusion: asymptotics, numerical simulations, and stability of fronts

Ramón Plaza
Instituto de Investigación en Matemáticas Aplicadas y Sistemas
UNAM

Motivated by biological applications (e.g. spatial ecology, bacterial aggregation), several reaction-diffusion models consider density-dependent diffusion coefficients which, in addition, are degenerate in one (or more) equilibrium points of the reaction. These degenerate diffusions describe, for example, the avoidance of crowded areas by individuals of certain biological populations. In this talk I present a generic bacterial aggregation reaction-diffusion-chemotaxis model with nonlinear cross-diffusion. High resolution numerical simulations using Graphic Processing Units (GPUs) are presented, showing that the chemotactic term enhances the velocity of propagation of the colony envelope for dense-branching morphologies. In addition, the chemotaxis seems to stabilize the formation of branches in the soft-agar, low-nutrient regime. An asymptotic estimation predicts the growth velocity of the colony envelope as a function of both the nutrient concentration and the chemotactic sensitivity. For fixed nutrient concentrations, the growth velocity is an increasing function of the chemotactic sensitivity. Furthermore, I will discuss new results and techniques in the study of stability of traveling fronts for scalar reaction-diffusion equations with degenerate diffusion. I will explain the two main ideas to control, on one hand, the essential spectrum and, on the other hand, the point spectrum of the linearized operator around the wave. Both techniques are designed to overcome the degeneracy of the diffusion at the equilibrium points of the reaction. This is joint work with J. Francisco Leyva and Carlos Málaga (UNAM).

Predictive logical modelling of T-helper cell (re)programming

Denis THIEFFRY
Institut de Biologie de l'École Normale Supérieure
Paris, France

T helper (Th) lymphocytes play a key role in the regulation of immune responses. Upon presentation of antigens, naive Th cells differentiate into various subsets, which secrete distinct sets of cytokines. This process requires the integration of multiple signals triggering the T cell receptor, co-stimulatory molecules, and cytokine receptors. Diverse signal combinations lead to the differentiation of naïve Th cells into diverse subsets. Recent experimental studies demonstrated extensive diversity and plasticity of Th lymphocytes.
To study the mechanisms underlying this diversity and plasticity, we have built a comprehensive logical model of Th differentiation, which integrates 20 signalling pathways, a dozen transcription factors, along with cytokines produced by Th cells. To analyse the dynamical properties of this model, we have used an automated model reduction method along with an algorithm enabling an efficient determination of stable states. We further defined temporal logic statements and used a symbolic model checker (NuSMV-ARCTL) to verify the response of virtual Th cells depending on triggering conditions.
We have assessed the consistency of our model by comparing its behaviour with published experimental observations on Th differentiation and plasticity. Our model qualitatively reproduces the polarisation of naïve Th cells into various canonical subsets (activated, quiescent, or anergic Th1, Th2, Th17, iTreg, Th9, Th22 and Tfh). Furthermore, our model predicts various hybrid Th cells expressing combinations of markers associated with different canonical subsets. Finally, we used our model to analyse the plasticity of canonical subsets depending on signalling inputs, and to delineate specific Th cell reprogramming strategies.

On the controllability of coupled parabolic equations

Luz de Teresa
IM-UNAM, C.U.

We present some results on the controllability of coupled parabolic equations. We work with equations that are linearized versions of systems that arise in the study of chemical reactions and in a wide variety of mathematical biology and physical situations.