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Lecture 2 Some basic functions and their application in biology. C 0 /2. C 0 /2. C 1 /2. C 1 /2. C 2 /2. C 2 /2. The exponential function is a general description of a random process , where the probability of a certain event is independent of time.

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**Lecture 2**Somebasicfunctions and theirapplicationinbiology C0/2 C0/2 C1/2 C1/2 C2/2 C2/2 Theexponentialfunctionis a general description of a random process, wheretheprobability of a certaineventis independent of time**C14 has a half time of 5568 years.**How long would it take until of one mol C14 only one atom remains? One mol contains 6.0210*1023atoms**The age of fossilized organic matter can be determined by**the C14-method of radioactive decay. The half-live of C14 is 5568 years. The equilibrium content of C14 in living plants is about 10-6ppm (parts per million). How old is a fossilized plant with a C14 content of 1.5*10-7ppm?**Thelogarithmicfunction**Log-seriesrelativeabundancedistribution Species – area relation**Selfsimilarobjects**The Sierpinski triangle • Start with a triangle, • shrink to 1/2 size, • make three copies • arrange the three copies in quadrants 2,3, and 4 • goto (2).**Scalingfactor = 1 / unit of measurement**= magnification a iscalledthenormalizationconstant**Whatisthevalue of a?**IfourobjectisclassicalEuclidean d = 0 Rulerlength l Bothequationsmatchif b = cs Now we consider an area. Theareascales to thesquare of therulerlength Bothequationsmatchif b = cs2**.**Euclideandimension E=0 Euclideandimension E=1 Euclideandimension E=2 0 < d < 1 Euclideandimension E=3 E+dtakesalwaysvaluesbetweentheactualEuclideandimension and thenexthigher one. Itiscommonlytermedthefractaldimensionof an object. An importantclass of fractalobjectsareselfsimilarobjects. We describethem by powerfunctions.**What are the relation between radius, volumen and surface in**such a branching pattern? A branchingpattern X = 0.75**Calculatethetotalleafarea of thisfern**We have to measuretwoleafsatdifferentscale to getthescalingexponent of thearea - lengthrelationship Lettheaveragelength of thesmallestleaflets be 1 cm and itsarea 3 cm2. Atthenexthigherscaleleafletlengthmight be 10 cm and therespectivearea 35 cm2. Thewholefernis 1 m long.**How should population density scale to body weight?**Populationdensityisproportional to availablespace and to available energy.**MD is proportional to total population biomass**What is if z is about 0.75? Energy equivalence(Damuth’s) rule (for poikilotherms: equalbiomasshypothesis)**The mean number of bee species per km2 in Poland [312685**km2]is 110, the total number of Polish bees is 463. Estimate the number of bees in the district of Kujaw-Pommern [17970 km2]. Observed: unknown The mean number of bird species in Poland is about 430, the total European [10500000 km2] species number is about 800. How many species do you expect for France [543965 km2]? Would it make sense to estimate the species number of Luxembourg [2586 km2]? What about Kujaw-Pommern [17970 km2]? Observed: 530 Observed: 250 Observed: 262**The inverse hyperbola**Michaelis-Menten equation Monod function**Haemoglobin or myoglobin bind oxygen according to the**partial pressure of O2 Denoting y for [MbO], p(O2) for the partial pressure of oxygen and using [MbO] + [Mb] = const we get Hill equation of oxygen binding**Home work and literature**• Refresh: • Fractal geometry • Selfsimilarity • Branchingprocesses • Logarithmictransformations • Species – arearelationships • Radioactivedecay • Prepare to thenextlecture: • Vectors • Vector operations (sum, S-product, scalarproduct) • Scalarproduct of orthogonalvectors • Distancemetrics (Euclidean, Manhattan, Minkowski) • Cartesian system, orthogonalvectors • Matrix • Types of matrices • Basic matrix operations (sum, S-product, dotproduct) Literature: Mathe-online Fractal geometry: http://classes.yale.edu/fractals/ Fractals: http://en.wikipedia.org/wiki/Fractal

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