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Graphs and functions, Derivative of a function, Techniques of differentiation Differentiation and its application in Biology, Finding maxima, minima, Plotting functions, Integrals, Techniques of Integration

Scalars and vectors. Force, Concentration gradient, Polar coordinates

Differential equations, Nernst Equation, Diffusion Equation, Mean-square displacement, Einstein’s relation

Probability and Statistics: Mean and variance, Distribution functions: Normal Distribution, Uniform distribution, Poisson distributions, Knudson’s analysis, Wright-Fisher model, Fitting a function to experimental data

Fourier Series, Fourier transform, Z-transform, Discussion of the use of Fourier transformation in X-ray crystallography, and other areas in biology.

Modeling biological problems: Statistical thermodynamics,
Flexible proteins--size and conformations, Polymerization dynamics, Molecular motor motion, Bending and looping of DNA, Protein organization along DNA

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

  • Mathematics for Biological Scientists, M. Aitken, B. Broadhursts, S. Haldky, Garland Science (2009)
  • Introduction to Mathematics for Life Scientists, E. Batschelet, Springer Verlag, 3rd edition (2003)
  • Calculus for Life Sciences, R. De Sapio, W. H. Freeman and Co. (1976)
  • Physical Biology of the Cell, R Phillips, J Kondev, J. Theriot, Garland Science (2009)
  • Random Walks in Biology, H. C. Berg, Princeton university press (1993)

  1. Biological Physics, Philip Nelson, W. H. Freeman, 1st edition (2007)
  2. Mechanics of Motor Proteins and the Cytoskeleton, J. Howard, Sinauer Associates; New edition (2001)
  3. Calculus and Analytic Geometry, Georege Thomas, Ross Finney Addison Wesley, 9 edition (1995)


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