AIS Mathematical Biology (2018) - Speakers and Syllabus

The intersection of the mathematical sciences with topics at the forefront of current biology is a rich source of mathematical problems. Quantifying biological phenomena through simple and more complex models, as well as achieving a deeper, more systematic understanding of these models, is a challenge that draws upon multiple skills. Interested mathematicians and scientists must draw upon very disparate aspects of mathematics. These include nonlinear dynamical systems, partial differential equations (such as the Navier-Stokes equations for fluids or the equations that govern biological pattern formation), information theory,stochastics and probability, network models and discrete mathematics relevant to computation.

This AIS is among the first of its kind in India to introduce a broad audience consisting of applied and pure mathematicians, physicists, engineers and some biologists to this active field. It provides a selection of topics that are among the forefront of those being studied internationally. It is aimed at those pursuing (or intending to pursue) research in any related area of mathematics (interpreted in a broad sense). A basic facility with di↵erential equations and some elementary probability will be assumed, but otherwise there are no prerequisites.

 Speaker No. of lectures Syllabus I (KJ) Prof. Kavita Jain, JNCASR, Bengaluru 6 Mathematical Modelling in Population Biology:-Lecture-1  will introduce some basic concepts and models in population genetics. Ref: Ewens, Mathematical Population Genetics [book] Lecture-2 Mutation rates are also subject to the action of evolutionary forces. I will discuss somerecent experiments and stochastic models to understand the evolution of mutation rates. Ref:Lynch et al., Nat Rev Genet 17 (2016), 704 [review]Jain and James, J. Theo Biol. 433 (2017), 85 Lectures- 3 & 4  will introduce some basic results in quantitative genetics and discuss how one canunderstand them starting from population genetics theory. Ref: Jain and Stephan, Mol Biol Evol 34 (2017), 3169 [review] Tutorials-1 & 2: Will be devoted to exploring in more detail specific problems discussed in the lectures II (MB)Prof. Malay Banerjee, IIT Kanpur 6 A survey of Ecological Models:-Lecture -1 Introduction to single species population models This lecture is mainly focused on the ordinary differential equation models of singlespecies population growth. Some basic local stability theory will be discussed to understand the dynamics of the models under consideration. Lecture-2 Introduction to two species population models There are several types of population models of two interacting species - competitive,cooperative, prey-predator etc. Various types of models will be discussed in the context of ecology and their mathematical analysis with the help of linear stability theory. Tutorial-1Hands on training for the modelling and analysis of single and two species models. Lecture-3 Stability and bifurcation analysis for multi-species population models Mathematical models of interacting species with three and higher trophic levels canbe divided into two parts - food chains and food webs. Main motivation of this talk will be to illustrate these types of models and how to perform preliminary bifurcation analysis tounderstand the change in system behavior depending upon parameter(s). Lecture-4 Spatio-temporal model - Turing instability Spatio-temporal models of interacting populations are capable to capture the effect of heterogeneous distribution of various species over their habitats on the resultingdynamics. This heterogeneous distribution can affect the stability behavior, in particular the temporal perturbation around homogeneous steady-state can leads to stabilizationhowever heterogeneous perturbation can destabilize the system and is responsible for spatial pattern formation. Main objective of this lecture is to introduce to spatio-temporal models of interacting populations andderivation of Turing instability condition for suitable model(s). Tutorial-2Analysis of multi-species model and derivation of Turing instability condition by hand. III (LN) Prof. Leelavati Narlikar, NCL, Pune 6 Sequence Analysis :- Lecture-1 Introduction to Algorithms and Probability Theory Analyzing algorthms, big O notation Dynamic programming Probability distributions, Bayes theorem Inference: MLE and MAP estimate Lecture-2 Modeling biological sequences I Sequences in Biology, how can we understand function from them? Stochastic processes, Markov chains HMMs explained with the Casino problem Dynamic programming applied here Lecture-3 Modeling biological sequences II Gene finding Motif finding Classifying regulatory regions Lecture-4: Meta Genomics Introduction to the area Characterizing microbial diversity Discussion of open problems Tutorial 1Implement Viterbi and Baum-Welch, or do on paper (their wish, based on their programming capabilities)Tutorial 2Handle real data from Galaxy (will need internet connection) IV (GIM) Prof. Gautam I Menon, IMSc, Chennai 6 The Mathematics of Infectious Diseases:- Lecture-1 Introduction to the Modelling of Infectious Diseases This lecture provides the basic phenomena studied in infectious disease modelling, introduces the SI, SIR and more general compartmental models, and motivates thecalculation of the basic reproductive ratio as the fundamental parameter characterising epidemic diseases Lecture-2 Host heterogeneities and multi-host, multi-pathogen models Populations are heterogeneous and the generalisation of simple compartmental models to study such heterogeneities will be discussed. Several diseases can infect multiplehosts and be transmitted, while others require an obligate intermediate host (e.g. mosquitos) to complete the transmission cycle. A number of models will be described and their mathematical study. Tutorial-3 Implementing SIR and related models in simple computer programs Lecture-4 Temporal Forcing and Stochastic Models Many diseases undergo forcing from some external factor, such as climate, rainfall or temperature. The basics of the modelling of forced systems will be described, as also thegeneralisation of the compartmental deterministic models to the stochastic case Lecture-5 Spatial and Network models Spatial and network generalizations of the classic disease models, as well as their generalizations to agent-based models will be described. The basics of the Gillespie method will also be described. Tutorial-6 Implementing agent-based models in simple computer modelsRef: Modeling Infectious Diseases in Humans and Animals, by Matt J. Keeling & Pejman Rohani (book) V (MT) Prof. Mukund Thattai, NCBS, Bengaluru 6 Models in Systems Biology:-Will  cover experiments and models of a variety of biological networks, including gene regulatory networks, signalling networks, and intracellular traffic networks. Lecture-1 Gene regulatory networks. Models of transcriptional regulationLecture-2 Dynamical systems models Multi-stable and oscillatory systems. Memory in biological networksTutorial-1 MATLAB tutorial on bifurcations in gene regulatory networksLecture-3 Signalling networks Noise in biological systems. Physical limits of signallingLecture-4 Intracellular traffic networks Origin and maintenance of organelles in eukaryotic cells. Tutorial-2 Detailed reading of classic quantitative experiments on biologicanetworks. VI (MI) Prof. Mandar Inamdar, IIT Bombay 6 Mathematical Models for Continuous Media and Biological Applications :- The broad scope of these lectures is to give the students basic understanding and implementable skill to model various phenomena in migration and mechanics of collective cell migration. To that end the lectures will be planned as follows. Each lecture/tutorial described below will be of 90 minutes duration. Lecture-1 Introduction to continuum mechanicsTopics: Meaning of continuum for mechanobiological systems Basic introduction to various mechanical phenomena observed during morphogenesis with experimental case studies and their quantification. Kinematics of continuum systems, stress tensor, and conservation equations in Lagrangian and Eulerian framework. Thermodynamics and constitutive modelling for passive and active tissues. Lecture-2 Application of continuum modelling for biological tissues Topics: A few experimental case studies from epithelial mechanics and collective cell migration, with quantification of data in the light of continuum mechanics:kinematics from image segmentation, PIV analysis, traction force microscopy etc. General modeling using ideas from solid mechanics, fluid mechanics, poroelasticty and plasticity, with the addition of active components, such as actomyosin signalling, motility, and cell-division, to incorporate unique features of living matter. Simple 1D models such as Fisher model for population dynamics and diffusion,and active Couette flows in tissues. Formulation of 2D flows in epithelial sheets using the active mechanics theory. Lecture-3 Numerical techniques to solve continuum equationsTopics: A brief exposure to the need and philosophy of numerical techniques. Basic introduction to the Finite Element Method with special emphasis of finite element software FENiCs. Demonstration of solving simple to cutting edge problems described in Lecture-2 using this approach. Lecture-4/Tutorial: Solve mid-level to simple problems in the three topics discussed above. Topics: The tutorial will include hands-on implementation of the FEA software FENiCs from scratch, including installation. Lecture-5 Discrete modelling of tissues and understanding the emergence of continuum level properties Topics: Formulations that detail how to coarse-grain cell-level kinematics, to obtain tissue level continuum kinematics. Describe cell level models such as the cell-centre model, cellular Potts model and vertex model Theory of using the triangulation technique implemented in Tissue-Miner software, to coarse grain model output from the vertex model to get insights into the continuum level behaviour described in the first four lectures. Lecture-6/Tutorial Hands-on implementation of mesoscale modelling involving coarse-graining of the vertex model to continuum level description Topic: Installation and hands-on implementation of cell-based modeling software CHASTE and coarse-graining software Tissue-Miner on problems involving discussed in the previous lectures, but now first modelled using the vertex model, and then coarse-grained using the triangulation approach. VII (PG) Prof. Pranay Goel, IISER Pune 6 Electrophysiological Models :- Lecture-1 Introduction to electrical phenomena in the nervous system . will introduce the basic biology of the nervous system; and the elementary ideas from circuit theory and their extensions that will be needed in later lectures. This will set thestage for the subsequent development of the Hodgkin-Huxley equations of action potential propagation in nerves. Lecture-2 Modeling excitable media: The Hodgkin-Huxley equations of action potentials.We will develop the celebrated Hodgkin-Huxley model for electrical activity in nerves. We will then describe two broad classes of properties - Types I and II - that action potentials can be grouped into. Tutorial-1 Introduction to XPP, and modelling excitable media. We will introduce XPP, a differential equations solutions package with a graphical interface. We will explore the Hodgkin-Huxley equations, and identify properties of Type I and II neurons using numerical experiments. Lecture-3: Reduced models and phase-plane analysis of excitability. We will describe two-dimensional models that successfully reproduce the essential properties of Type I and II neurons. We will carry out a phase-plane analysis to understandthese differences as essentially arising out of the saddle-node and Hopf bifurcations, respectively, in models. Lecture-4 Modeling neural networks.  will investigate synapses, and the connectivity of neurons in networks. We will introduce questions of emergent behaviours in neural networks, and briefly touch upon some methods that been used to investigate these phenomena. Tutorial 2 Neural networks.This tutorial will not only explore the phase planes described in the lectures but also begin to connect neurons into (small) networks via synapses. We will explore some consequences of network connectivity for the synchronization of action potentials. VIII (SS) Prof. Sitabhra Sinha, IMSc, Chennai 6 Networks Lecture-1 Networks: An unifying framework for biology across scalesLecture-2 From Euler to Google: Using information about network structure to solve complex problemsLecture-3 Micro, Meso and Macro: Looking at different scales in a networkLecture-4 Are complex systems inherently unstable ? Investigating dynamics on networksTutorial-1 Measuring network metrics: Path length, Clustering and Centrality Tutorial-2 Identifying communities: Modularity, Graph partitioning and Synchronization & other collective dynamics in modular networks