Module

Topics

Module- I: Calculus

Lecture 1:  Introduction

Keywords:  Mathematics as a language, Need of learning mathematics, Applications of mathematics in BIology

Lecture 2:  Graphs and functions - I

Keywords: Linear function, Quadratic function, Exponential function

Lecture 3: Graphs and functions - II

Keywords: Periodic functions, Combination of simple functions, Examples from Biology

Lecture 4: Functions and derivatives

Keywords: Logarithmic function, Slope of curves, Idea of derivative

Lecture 5: Calculation of  derivatives

Keywords: Derivatives of simple functions, Derivative of exponential function, Derivative of sum of two functions

Lecture 6: Differentiation and its application in Biology - I

Keywords: Product rule in differentiation, Derivatives of Sine and Cosine functions, Plotting derivatives, Differential calculus to understand actin polymerization

Lecture 7 : Differentiation and its application in Biology - II

Keywords: Enthalpy and Entropy of a chemical reaction, Growth curve, Idea of curvature

Lecture 8 : Differentiation and its application in Biology - III

Keywords: Curvature, Free energy, Energy of spring-like protein, Maxima and Minima of a function

Lecture 9: Differentiation and its application in Biology - IV

Keywords: Force and energy, DNA unzipping, Plotting mathematical functions

Lecture 10: Integration -I

Keywords: Indefinite integrals, integration of simple functions, Integral as “anti-derivative”

Lecture 11: Integration - II

Keywords: Definite integrals, Integral as area under a curve, Integration by parts, Finding derivative and integral given a set of data points

Module: II: Differential Equations

Lecture: 12: Differential equations-I

Keywords: Simple differential equations, First order differential equations, Examples: Polymerizing and depolymerizing filaments, Cell growth

Lecture : 13: Differential equations - II

Keywords: Concentration gradient, Second order differential equations. Motion of an object under external force : Newton’s equations

Module III: Vectors

Lecture 14: Vectors - I

Keywords: Physical quantities like position and force as  vectors, Attracting and repelling charges, Vector addition

Lecture 15: Vectors - II

Keywords: Calculation of forces in a system of charges, Calculation of magnitude and direction of a vector, Unit vectors, Calculation of resultant force

Lecture 16: Vectors - III

Keywords: Dot product and cross product, Polar coordinate system, Gradient of a scalar

Module IV: Applications of calculus and vector algebra in biology

Lecture 17:  Nernst equation

Keywords: Potential difference across a membrane, Flow of ions due to diffusion, Flow of  ions due to electrostatic interactions

Lecture 18: Diffusion-I : Diffusion equation

Keywords: Continuity equation, Diffusion equation, Mean-square position

Lecture 19: Diffusion - II: Mean-square displacement

Keywords: Mean-square displacement, Derivation of mean-square displacement, Mean-square distance scaling with time, Diffusion timescale

Lecture 20: Diffusion-III : Einstein’s relation

Keywords: Mean displacement, Diffusion coefficient, Einstein's relation, Diffusion under external field

Module V: Probability and statistics in Biology

Lecture 21 : Statistics : Mean and variance

Keywords: Introduction to statistics, Mean/Average, Variance, Standard deviation

Lecture 22: Statistics: Distribution function

Keywords: Introduction to distribution functions, Normal distribution, Examples from biology: End-to-end vector distribution of DNA, Concentration distribution

Lecture 23 : Understanding Normal distribution

Keywords: Gaussian function, Peak as average of normal distribution, Width of a Gaussian and standard deviation

Lecture 24: Fitting a function to experimental data

Keywords: Linear fit, Least-square fit, Errors

Lecture 25 :  Size of a flexible protein: Simplest model

Keywords: Flexible protein chain,  End-to-end distance, End-to-end distance scaling with polymer length, Random walk, Normal distribution, Exponential distribution

Lecture 26: Uniform and Poisson distributions; Knudson’s analysis

Keywords: Uniform distribution, Poisson distribution, Knudson’s analysis of retinoblastoma patients, Poisson statistics and tumor

Module VI : Fourier series and Fourier transform

Lecture 27: Fourier Series-I

Keywords: Introduction to Fourier series, Fourier coefficients, Calculation of Fourier series for simple functions, Sum of periodic functions

Lecture 28: Fourier Series-II

Keywords: Fourier coefficients with more examples, Calculation of Fourier series for square-wave-like function, Learning Fourier series by plotting functions

Lecture 29: Fourier transform

Keywords: Introduction to Fourier transform, Fourier space, Inverse Fourier transform, Application of Fourier transform: X-ray crystallography,  structure studies of proteins, Z-transform

Module VII: Mathematical models in biology

Lecture 30: Master equation: Polymerization dynamics, Molecular motor motion

Keywords: Simple model for polymerization depolymerization dynamics, Simple model for molecular motor motion, Biased walk, Growth velocity of polymerizing filaments, Master equation, Solving master equation

Lecture 31: Evolution: Simplest model

Keywords: Wright-Fisher model, Simplest model in population genetics/evolution, Binomial distribution, Evolution

Module VIII: Tutorials

Lecture 32: Tutorial - I

Keywords: Microtubule dynamics, Dynamic instability,  application of functions and derivatives, Enzyme kinetics

Lecture 33: Tutorial-II

Keywords: Vectors, Pulling chromosome, Diffusion coefficient, Integral of a Gaussian function

Module IX : Statistical thermodynamics of biological systems

Lecture 34: Temperature, Energy and Entropy

Keywords: Definition of temperature, Definition of internal energy, Definition of entropy, Calculation of entropy, entropy of a flexible protein

Lecture 35: Partition function, Free energy

Keywords: Definitions, Calculation of partition function, Calculation of Free energy, Thermal Equilibrium, Bending of  DNA

Lecture 36: Bending fluctuations of DNA and spring-like proteins

Keywords: Worm-like chain model, Partition function, Gibbs free energy

Lecture 37: Force-extension and looping of DNA

Keywords: Force extension relation of single stranded DNA, Persistence length, Looping of DNA,

Lecture 38: Thermodynamics of protein organization along DNA

Keywords: Proteins binding on DNA, Calculation of energy, entropy and free energy, Thought-experiment on DNA melting

Lecture 39: Learning mathematics with the help of a computer

Keywords: Plotting functions using computer, gnuplot demonstration, numerical calculations, Interpolation