BIOMODELAT MATEMATIC (MATHEMATICAL BIOMODELING)
This course introduces the basic elements to address the problem of how to formulate very simple mathematical models capable of addressing a number of relevant problems related to cell and population growth, stability and instability and the logic of living systems. Most models are based on one-dimensional differential equations. These include cancer, cell growth and differentiation, epidemic spreading, population dynamics, gene and neural networks (at a very introductory level). The course is taught in catalan.
CONTINGUTS / CONTENTS1. Linear Models. Exponential growth. Estimating cell division time. Mortality and exponential decay. Constant growth + linear decay:constitutive gene expression and other examples. Growth in cell cultures: early stages. 2. Linear diffusion between two compartments. Fick's law and relaxation to equilibrium. Chemical equilibrium. 3. Transitions between multiple states: chemical reaction chains and protein folding. Introduction to first-order linear differential equations. 3. Bimolecular reactions: introduction to nonlinear models. DNA hybridisation and Cot curves. Hybridisation curves and the structure of prokaryotic versus eukaryotic genomes. Receptor-ligand kinetics and active transport. Imatinib and tyrosine-kinase interactions in cancer. Facilitated diffusion and active transport. 4. Population explosions and singularities. Second-order growth equations and critical times. Hyperbolic dynamics. 5. Logistic growth dynamics. Derivation of the logistic equation and its exact solution. Other models of growth under constraints: Gompertz equation and embryonic development. The importance of spatial dynamics. 6. Linear stability and equilibrium (fixed) points. The concept of attractor. Bifurcations and bifurcation diagrams.Potential function: definition, use and meaning. 7. Symmetry breaking and historical dynamics. Frozen accidents and multi stability. Path dependence andevolution. Bottlenecks and biological diversity. Human diversity and cancer heterogeneity. 8. Epidemic spreading. Historical perspective and case studies. Evidence for transitions in epidemic thresholds.The SIS model: assumptions and implications. Eradication thresholds and vaccination strategies. 9. Competition between populations. The constant population constraint (CPC) model. Competitive exclusion. Applications to microbial populations and cancer dynamics. 10. Discrete dynamical systems. Boolean models and Boolean logic. Biological computation. McCulloch-Pitts logic neurons andthreshold neuron models. Computation in small neural systems. Genetic circuits and the lambda switch.
VIDEOS/LECTURESMIT lecture: Differential equations of growth https://www.youtube.com/watch?v=IDo4uPyqQbQhttps://www.youtube.com/watch?v=Lp3T4DtAyOo Transport across cell membranes and chemical equilibrium: https://www.youtube.com/watch?v=RPAZvs4hvGAhttps://www.youtube.com/watch?v=knv4fNNoEG8&list=PLbKSbFnKYVY25Xu3-g8SZzyvmhle-oYqAhttps://www.youtube.com/watch?v=lmwuP_iuFCM MIT Differential equations https://www.youtube.com/watch?v=tVzaX9u6YAE Facilitated diffusion: the concept: https://www.youtube.com/watch?v=vKGN_Zhz8AY TED Talk. Global population growth by Hans Rosling: https://www.youtube.com/watch?v=fTznEIZRkLg Regulated/unregulated cancer cell growth https://www.youtube.com/watch?v=IeUANxFVXKc&list=PLl4eoRGS3f1v1wbJuSBBl6GeBGJwUKxEV Stable/unstable equilibrium points https://www.youtube.com/watch?v=JrlZSfRM-IY https://www.youtube.com/watch?v=YNIm2Op7UPg&index=2&list=PLj_l4pOO0YKhJHLQVbjpPlbNyYtT8aAQz Population bottleneck and genetic drifthttps://www.youtube.com/watch?v=VM7Y1D8NMo8https://www.youtube.com/watch?v=mjQ_yN5znyk Pandemics and future epidemics TED Talks https://www.youtube.com/watch?v=UG8YbNbdacohttps://www.youtube.com/watch?v=6Af6b_wyiwI Growth, competition and space in Physarum plasmodium: https://www.youtube.com/watch?v=BZUQQmcR5-g Neural and genetic Boolean models https://www.youtube.com/watch?v=q0pm3BrIUFo&list=PLUl4u3cNGP63gFHB6xb-kVBiQHYe_4hSi&index=12https://www.youtube.com/watch?v=gc3O2sKIsX4&list=PLUl4u3cNGP63OI3pSKo8Ha_DFBMxm23xOhttps://www.youtube.com/watch?v=vA9D727AGHI