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Lecture 4. Solving simple stoichiometric equations. A linear system of equations. The Gauß scheme. Multiplicative elements . A non-linear system. Matrix algebra deals essentially with linear linear systems. Solving a linear system.

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Lecture 4


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slide1

Lecture 4

Solving simple stoichiometric equations

A linear system of equations

TheGaußscheme

Multiplicativeelements.

A non-linear system

Matrix algebra dealsessentiallywithlinearlinear systems.

slide2

Solving a linear system

Thedivisionthrough a vectoror a matrixis not defined!

2 equations and fourunknowns

slide3

For a non-singularsquarematrixtheinverseisdefined as

Singularmatricesarethosewheresomerowsorcolumnscan be expressed by a linearcombination of others.

Suchcolumnsorrows do not containadditionalinformation.

Theyareredundant.

A matrixissingularifit’s determinant is zero.

r2=2r1

r3=2r1+r2

A linearcombination of vectors

Det A: determinant of A

A matrixissingularifatleast one of theparameters k is not zero.

slide4

Theinverse of a 2x2 matrix

Theinverse of a diagonal matrix

Determinant

Theinverse of a squarematrixonlyexistsifits determinant differsfrom zero.

Singularmatrices do not have an inverse

(A•B)-1 = B-1 •A-1 ≠ A-1 •B-1

Theinversecan be unequivocallycalculated by the Gauss-Jordan algorithm

slide6

The general solution of a linear system

Identitymatrix

OnlypossibleifAis not singular.

IfAissingularthe system has no solution.

Systems with a uniquesolution

Thenumber of independent equationsequalsthenumber of unknowns.

X: Not singular

TheaugmentedmatrixXaugis not singular and hasthe same rank as X.

Therank of a matrixis minimum number of rows/columns of thelargestnon-singularsubmatrix

slide10

Consistent

Rank(A) = rank(A:B) = n

Infinitenumber of solutions

Consistent

Rank(A) = rank(A:B) < n

No solution

Inconsistent

Rank(A) < rank(A:B)

Infinitenumber of solutions

Consistent

Rank(A) = rank(A:B) < n

Inconsistent

Rank(A) < rank(A:B)

No solution

Consistent

Rank(A) = rank(A:B) = n

Infinitenumber of solutions

slide11

We haveonlyfourequations but five unknowns.

The system isunderdetermined.

Themissingvalueisfound by dividingthevectorthroughitssmallestvalues to findthesmallestsolution for natural numbers.

slide12

Includinginformation on thevalences of elements

Equality of atomsinvolved

We have 16 unknows but withoutexperminetnalinformationonly 11 equations.

Such a system isunderdefined.

A system with n unknownsneedsatleast n independent and non-contradictoryequationsfor a uniquesolution.

If ni and aiareunknowns we have a non-linearsituation.

We eitherdetermine ni oraiormixedvariablessuchthat no multiplicationsoccur.

slide13

Thematrixissingularbecause a1, a7, and a10 do not containnewinformation

Matrix algebra helps to determinewhatinformationisneeded for an unequivocalinformation.

Fromtheknowledge of thesalts we get n1 to n5

slide14

We havesixvariables and sixequationsthatare not contradictory and containdifferentinformation.

Thematrixistherefore not singular.

slide15

Linearmodelsinbiology

The logistic model of population growth

K denotesthemaximumpossibledensity under resourcelimitation, thecarryingcapacity.

rdenotestheintrinsicpopulation growth rate. Ifr > 1 thepopulationgrowths, atr < 1 thepopulationshrinks.

t N

1 1

2 5

3 15

4 45

We needfourmeasurements

slide16

Population growth

We have an overshot.

In thenext time step thepopulationshoulddecreasebelowthecarryingcapacity.

Overshot

K

N

K/2

t

Fastestpopulation growth

slide17

Thetransitionmatrix

Assume a genewithfourdifferentalleles. Each allele canmutateintoanther allele.

Themutationprobabilitiescan be measured.

Initial allele frequencies

A→A

B→A

C→A

D→A

A→A

Whatarethefrequenciesinthenextgeneration?

A→B

A→C

A→D

Transitionmatrix

Probabilitymatrix

Sum

1

1

1

1

Σ = 1

Thefrequenciesat time t+1 do onlydepent on thefrequenciesat time t but not on earlierones.

Markovprocess

slide18

Doesthemutationprocessresultinstable allele frequencies?

Stable state vector

Eigenvector of A

Eigenvalue

Unit matrix

Eigenvector

Everyprobabilitymatrixhasatleast one eigenvalue = 1.

Thelargesteigenvaluedefinesthestable state vector

slide19

The insulin – glycogen system

At high bloodglucoselevels insulin stimulatesglycogensynthesis and inhibitsglycogenbreakdown.

ThechangeinglycogenconcentrationDN can be modelled by the sum of constantproductiong and concentration dependent breakdownfN.

Atequilibrium we have

Thesymmetric and squarematrixDthatcontainssquaredvaluesiscalledthedispersionmatrix

Thevector {-f,g} isthestationary state vector (thelargesteigenvector) of thedispersionmatrix and givestheequilibriumconditions (stationary point).

Theglycogenconcentrationatequilibrium:

Theequilbriumconcentrationdoes not depend on theinitialconcentrations

Thevalue -1 istheeigenvalue of this system.

slide21

Someproperties of eigenvectors

IfL isthe diagonal matrix of eigenvalues:

Theeigenvectors of symmetricmatricesareorthogonal

Eigenvectors do not changeafter a matrixismultiplied by a scalar k. Eigenvaluesarealsomultiplied by k.

Theproduct of alleigenvaluesequalsthe determinant of a matrix.

The determinant is zero ifatleast one of theeigenvaluesis zero.

In thiscasethematrixissingular.

If A istrianagularor diagonal theeigenvalues of A arethe diagonal entries of A.

slide22

Page Rank

Google sortsinternetpagesaccording to a ranking of websitesbased on theprobablitites to be directled to thispage.

Assume a surferclickswithprobability d to a certainwebsite A. Having N sitesintheworld (30 to 50 bilion) theprobability to reach A is d/N.

Assumefurther we havefoursite A, B, C, D, withlinks to A. AssumefurtherthefoursiteshavecA, cB, cC, and cDlinks and kA, kB, kC, and kDlinks to A.

Iftheprobability to be on one of thesesitesispA, pB, pC, and pD, theprobability to reach A fromany of thesitesistherefore

slide23

Thetotalprobability to reach A is

Google uses a fixedvalue of d=0.15.

Neededisthenumber of links per website.

In reality we have a linear system of 30-50 bilion equations!!!

ProbabilitymatrixP

Rankvectoru

Internet pagesarerankedaccording to probability to be reached

slide24

A

B

D

C

Larry Page (1973-

SergejBrin (1973-

slide25

Page Rank as an eigenvector problem

In reality theconstantisverysmall

Thefinalpagerankisgiven by thestationary state vector(thevector of thelargesteigenvalue).

home work and literature
Home work and literature
  • Refresh:
  • Linearequations
  • Inverse
  • Stochiometricequations
  • Prepare to thenextlecture:
  • Arithmetic, geometricseries
  • Limits of functions
  • Sums of series
  • Asymptotes

Literature:

Mathe-online

Asymptotes: www.nvcc.edu/home/.../MTH%20163%20Asymptotes%20Tutorial.pp

http://www.freemathhelp.com/asymptotes.html

Limits:

Pauls’sonlinemath

http://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx

Sums of series:

http://en.wikipedia.org/wiki/List_of_mathematical_series

http://en.wikipedia.org/wiki/Series_(mathematics)