Quantitative modeling of biological systems (Physiology 472/572)University of ArizonaFall 20099/01/09Instructors: Dr. Tim Secomb ([email protected]) Dr. Chris Bergevin ([email protected])Schedule: Tuesdays & Thursdays 9:30-10:45 (Modern Lang. 210)Website: http://www.physiology.arizona.edu/people/secomb/472572info09Lecture 3QualitativeQuantitative Analyticale.g. statistics e.g. mathematical modelse.g. numerical computationsDescribing Biological/Physiological Systems[words][numbers][equations]Functions of more than one variablefx, y- dependent variable- independent variables will be important for the various types of systems we will bedescribing analytically in the coursee.g. concentration of a solute in a solution (c) depends upon bothspatial location (x) and time (t)fx, y- reaction rate [mol/s]- concentration of inhibitor agents [mol]Biological Contextdon’t forgetabout units!can you think of an exampleof what the various variablesmight represent?solution todiffusion equation (or heat eqn.)Differentiation of functions of more than one variableDerivative (definition) for a function of a single variableFor multi-variable function, keep one variable constanty is effectively held constant herenote differencein notationDifferentiation of functions of more than one variableCan also take partial deriv.with respect to partial deriv.Simplified notationDifferentiation of functions of more than one variable: Examplesex.1 ex.2Differential EquationsODE (‘ordinary’):PDE (‘partial’): considers a function of more than one variableand how its various partial derivatives are relatedex.considers a function of one variable and how itchanges with respect to that variableex.ODE ReviewSolving an ODE: find c(t) for1. Separate variables and integrate2. Use an initial condition to find a unique solution. Suppose we havehttp://www.microscopy-uk.org.uk/dww/home/hombrown.htmBrownian Motionhttp://en.wikipedia.org/wiki/Image:Entropie.pngDiffusioneach particle undergoesBrownian motionDiffusion (1-D)- Adolf Fick (German physiologist, ~1855)[actually was the first to successfully put a contact lens on a person in 1888!]- Thomas Graham (Scottish chemist, ~1828-1833)[pioneered the concept of dialysis]“ A few years ago, Graham published an extensive investigationon the diffusion of salts in water, in which he more especiallycompared the diffusibility of different salts. It appears to me amatter of regret, however, that in such an exceedingly valuableand extensive investigation, the development of a fundamentallaw, for the operation of diffusion in a single element of space,was neglected, and I have therefore endeavoured to supply thisomission.”- A. Fick (1855)QualitativeQuantitative
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