biomath final

8/14/2019 Biomath Final

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Mathematics and BiologyMathematics and Biology

Biomathematics

eyond theeyond thesualsual

Brief Introduction toBrief Introduction to

MSP BiomathematicsInitiative


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Biomathematicsf Introduction tof Introduction to


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iomath as a fieldiomath as a fieldTheoretical Biology

omputational Biology

/Computational Biomodeling BiocomputingBioinformatics ,Creation of Algorithms Numerical Analysis and Simulation

ematical Biology

Biomathematics

iomodelinginding Solutions

nalysis

Experimental Biology

Biostatistics

Mathematical biophys

Systems Biology


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A LOOK INTO SYSTEMSBIOLOGY


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SIMULATIONIMULATION .OMP BIOOMP BIO


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EcologyEcologyPopulation dynamics

Malthus (1798):

Verhulst (1838, 1845):

(t)P0

P0P0

Biomath

/K 2


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EcologyEcologyInteraction of Species

Lotka-Volterra Predator-Prey Model:

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EcologyEcologyPopulation Model with Age Structure

Leslie Matrix Model

( )LESLIE MATRIX

Biomath


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EpidemiologyEpidemiologyEpidemics

SIR Compartmental Model:

Id t

d RIIS

d t

d IIS

d t

d S === ,,

+ I + R = 1

I

Biomath


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ENZYME KINETICSENZYME KINETICS

-ich a e lis M e n te nn zy m e K in e tics

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h


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Mathematical PhysiologyMathematical Physiology

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MATHs needed

Any branch of Mathematics can be; applied in Biology however

the field of biology should,dictate the maths needed not the.other way around

Biomathematics without biologistsBiomathematics without biologists.is too theoretical.is too theoretical


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Discrete MathDiscrete Math

Combinatorics (e.g. GenomeRearrangement)

Graph Theory and Network Analysis

Finite State Automata

Boolean Networks


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Knot Theory

AlgebraicStatistics(e.g.Phylogeny)

Fractals

o p o lo g y a n d M o d e rno p o lo g y a n d M o d e rnA lg e b ralg e b ra


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Optimization

EvolutionaryGame

Theory

DynamicProgrammin

g (e.g. DNASequenceAlignment)

p e ra tio n s R e se a rchp e ra tio n s R e se a rch


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Deterministic Models + Noise (e.g.diffusion)

Markov Chains

Bayesian Networks

Probabilistic Models (e.g. genetics)

;robability Stochasticrobability Stochasticrocess and Calculusrocess and Calculus


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Bifurcation and Chaos

Fib o n a cciS e q u e n c e

iffe re n ce E q u a tio niffe re n ce E q u a tio n


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Differential EquationsDifferential Equations

Every biomathematician should beEvery biomathematician should be

good in ODEs, PDEs and DDEs.good in ODEs, PDEs and DDEs.

Model and Analysis of DynamicalSystems

Control of Systems

MODEL PROCESSES AND CHANGE(cycles, switching, kinetics, rhythms,Input-Output, interaction,conduction, rate of growth/decayetc.)

/x dtx dt /y dty dt


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ODEsODEs

Solutions and Analysis of ODEs can bedone by:

1.Finding Explicit or Implicit Formula assolution

2.Finding series solution

3.Numerical analysis4.Representing solutions as graphs

5.Perturbation and stability analysis

6.Etc


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Perturbation & StabilityPerturbation & StabilityAnalysisAnalysis

- ( / /Fin d S te a d y sta te s E q u ilib riu m C ritica l R e st)Po in ts

- ,If S te a d y sta te is it sta b le o r u n sta b le ?Equilibrium Point

( )unstableEquilibrium Point

( )unstable

Equilibrium Point( )stable

Not in steady state


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Harvesting in CountvilleHarvesting in Countville

H a rv e stin g a S in g le Po p u la tio n w ith a co n sta n tyie ld Y 0 :

N

/dN dt

Y 0increasing

NFINITEECOVERYTIME

Y0= /rK 4

!Unstable

0


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Some Open ProblemsSome Open Problems(Chemostat Equations)(Chemostat Equations)

nn-species compete for-species compete for kkcomplementary resourcescomplementary resources

n=2 k=2 classic example n=3 k=2 competitive exclusion

n>3 k=2 open problem

n=4 k=3 co-existence may exist n=5 k=3 chaotic

n=12 k=5 coexists in oscillatory

form or chaotic form


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Romeo and JulietRomeo and Juliet

) , ( ) be Romeo s ardor for Juliet J t be Juliet s ardor for Rom

/ =dR dt aJ

/ =dJ dt bR


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Mathematics and BiologyMathematics and Biology

Bi th ti

eyond theeyond thesualsual

Brief Introduction toBrief Introduction to

MSP BiomathematicsInitiative

biomath final

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