Details of LS4107 (Autumn 2020)
| Level: 4 | Type: Theory | Credits: 4.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| LS4107 | Mathematical Biology | Dipjyoti Das |
| Preamble |
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| What to expect: Building upon basic mathematical skills (as introduced in Mathematics I & II), the course will develop the concepts of nonlinear dynamics and stochastic process, and will apply them in some detailed examples from biology and chemistry.
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| Syllabus |
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| Syllabus:
1. PART A: dynamical systems in biology: a. Introduction to dynamical systems: One- and two-dimensional flows, phase portraits, fixed points and linear satiability analysis, bifurcations, oscillatory systems. b. Population dynamics: Malthusian growth law, logistic model, model for species competition Lotka-Voltera model, Levins metapopulation theory. c. Biochemical reaction-kinetics: Law of mass action, enzyme kinetics, cooperativity, genetic switch. 2. PART B: Stochastic processes in biology: a. Basic probability theory: Discrete and continuous probability distributions, cumulative distributions, moments and correlations, generating functions (moment and cumulant series expansions), central limit theorem. b. Diffusion and generalization of random walk: Introduction to random walk/Brownian motion, Unbiased and biased diffusion equations, Master equation and Langevin formalism, Einstein's fluctuation dissipation relation, Auto-correlation functions. c. Stochastic biochemical reactions: Chemical master equation, introduction to polymer models, stochastic gene expression, Stochastic dynamics of biofilaments, Intracellular transport, Discussion on the role of intrinsic and extrinsic noise in phenotypic heterogeneity. d. Stochastic population biology: Stochastic birth-death processes in ecology, Symmetric/Asymmetric division of stem cells as branching process, Fission/Fusion of organelles. 3. PART C: Exposure to many-body problems and systems biology. Vicsek model and active matter. Collective behaviors in bird flocks and migrating cells, order-to-disorder phase transition. |
| Prerequisite |
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| Basic mathematics like algebra and basic calculus, familiarity with ordinary and partial differential equations.
Course credit options: 4th year MS students are eligible for taking this course as an elective. IP, MR, and RS students are also allowed provided they fulfill the prerequisite. |
| References |
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| References:
1. Mathematical Biology I, J D Murray (3rd Edition), Springer. 2. Nonlinear dynamics and Chaos, S. H. Strogatz, CRC Press. 3. Stochastic Processes in Physics and Chemistry (North-Holland) 3rd Edition, N.G. Van Kampen. 4. Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences (Springer) 3rd Edition, C. W. Gardiner. 5. A Guide to First-Passage Processes (Cambridge University Press), Sidney Redner. |
Course Credit Options
| Sl. No. | Programme | Semester No | Course Choice |
|---|---|---|---|
| 1 | IP | 1 | Elective |
| 2 | IP | 3 | Elective |
| 3 | IP | 5 | Elective |
| 4 | MR | 1 | Not Allowed |
| 5 | MR | 3 | Not Allowed |
| 6 | MS | 5 | Not Allowed |
| 7 | MS | 7 | Elective |
| 8 | MS | 9 | Elective |
| 9 | RS | 1 | Elective |
| 10 | RS | 2 | Elective |