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Lecture 16

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1. Weconsideredamodelforgelelectrophoresisinrecitationandinclass.Inareal gel,doDNAsegmentspointedinonedirectiongofasterthantheothers?

A.YesB. No

E

perpendicularparallel

2.ConsideragainyouranalysisofgelelectrophoresisofDNAmolecules.Thinkaboutwhatfactorsareimportantincontrollingthemotionofthemoleculesthroughthegel.Isthedensity ofthegelimportant?Keepingall otherphysicalpropertiesthesame,whathappenstotheterminalvelocityifthedensityofthegelisdoubled?(Thinkabouttheinertialdragforceversustheviscousforce.)

A. Thereisnotenoughinfotosay.B. Theterminalvelocitywouldincreaseathigher

density.C. Itwoulddecrease.D. Itwouldstaythesame.

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1

Term

inalSpe

ed(cm/s)

PercentGlycerol

0

0.01

0.02

0.03

0.04

0.05

0.06

0% 20% 40% 60% 80% 100%

viscosity(Pas)

PercentGlycerol

Viscosity

3.Thetopfigureshowsyourdatafortheterminalvelocityofthelargesteelballinvariousmixturesofwaterandglycerol.Lowerfigureshowstheviscosityasafunctionofglycerolcontent.

Considerthishypothesis:Viscosityplaysamajorroleindeterminingtheterminalvelocity. Basedonthesedata,chooseone:

A. Yes,viscosityplaysamajorrole.

B. Viscosityclearlyisafactor,butmaybenotthemostimportantone.

C. Viscosityistotallyunimportant.

D. Thesedataarenotsufficienttotestthehypothesis.

Fq =

kCqQ1r12r1

E =Fqq= kCQ1

r12r1

4.AboveistheforceexertedbychargeQ1 ontestchargeq.OntherightistheelectricfieldduetochargeQ1.

Doesthiselectricfielddependonthetestchargeq?

A. YES

B. NO

3/6/18

1

1. Weconsideredamodelforgelelectrophoresisinrecitationandinclass.Inareal gel,doDNAsegmentspointedinonedirectiongofasterthantheothers?

A.YesB. No

E

perpendicularparallel

2.ConsideragainyouranalysisofgelelectrophoresisofDNAmolecules.Thinkaboutwhatfactorsareimportantincontrollingthemotionofthemoleculesthroughthegel.Isthedensity ofthegelimportant?Keepingall otherphysicalpropertiesthesame,whathappenstotheterminalvelocityifthedensityofthegelisdoubled?(Thinkabouttheinertialdragforceversustheviscousforce.)

A. Thereisnotenoughinfotosay.B. Theterminalvelocitywouldincreaseathigher

density.C. Itwoulddecrease.D. Itwouldstaythesame.

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1

Term

inalSpe

ed(cm/s)

PercentGlycerol

0

0.01

0.02

0.03

0.04

0.05

0.06

0% 20% 40% 60% 80% 100%

viscosity(Pas)

PercentGlycerol

Viscosity

3.Thetopfigureshowsyourdatafortheterminalvelocityofthelargesteelballinvariousmixturesofwaterandglycerol.Lowerfigureshowstheviscosityasafunctionofglycerolcontent.

Considerthishypothesis:Viscosityplaysamajorroleindeterminingtheterminalvelocity. Basedonthesedata,chooseone:

A. Yes,viscosityplaysamajorrole.

B. Viscosityclearlyisafactor,butmaybenotthemostimportantone.

C. Viscosityistotallyunimportant.

D. Thesedataarenotsufficienttotestthehypothesis.

Fq =

kCqQ1r12r1

E =Fqq

= kCQ1r12r1

4.AboveistheforceexertedbychargeQ1 ontestchargeq.OntherightistheelectricfieldduetochargeQ1.

Doesthiselectricfielddependonthetestchargeq?

A. YES

B. NO

Recoil• When an object at rest emits a part of itself,

in order to conserve momentum, it must go back in the opposite direction.

• When the heart ejects blood into the aorta, does it recoil?

(object goes backwards)

BEFORE:Everythingisatrest.TotalmassMT=mball +Meverything_else

AtrestmeansV=0,sototalmomentum=0.

𝑃 = 𝑀$𝑉$ = 0.

AFTER:Ballleavestheperson’shand,headingtowardspartitionwithspeedvball.

𝑃 = 𝑚)*++𝑣)*++ +𝑀-.-/012345_-+7-𝑉8*/1 = 0.

𝑉8*/1 = − ;<=>>?@A@BCDEFGH_@>I@

𝑣)*++

Fromconservationofmomentum,weseethevelocityofthecartisintheoppositedirectionfromthevelocityoftheball.

Question:Whathappensaftertheballbouncesoffthepartition?

RandomMotion Emergent PropertiesThe question: Can the properties of a system be explained in terms of the properties of its component parts (so, biology can be explained by chemistry, chemistry by physics)?

Emergence – some phenomena are undetectable when looked at “in the small”. They emerge only when looking at the system as a whole rather than its parts.

Spoiler alert: Diffusion is an emergent property!

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Biological Example of Emergence• Evolution

– If a single species of birds on an isolated island have a range of bill thicknesses, they may all surviveand interbreed well under normal circumstances.

– If the climate shifts so that the birds at the two extremes are more likely to survive than those in the middle – by only a little bit! – after a few decades the population may consist only of birds with only the smallest and largest bills.

– If the climate now stays shifted, after a few millennia, genetic drift can take the two populations apart so that they can no longer interbreed and would be identified as different species.

– The shifts are in fact visible over only a few generations.

Jonathan Weiner, The Beak of the Finch

BiologicalExampleofEmergence

Slimemoldaggregationfromchemotaxis

Inthissimulation,a“walker”startsat0andstepsleftandrightwithequalprobability.WewillletittakeNsteps.Ifwereleasealotofwalkersfromtheoriginatonce,ontheaverage,whatwillourdistributionofparticleslooklike?1. Therewillbepeakswithequal

numbersnear+N/2and–N/22. Theywillbemostlynear0nomatter

howmanystepsyoutake.3. Itwillpeakat0andgettingfarther

willdecreaseinprobability.4. Therewillbepeaksat+and– values

butnotat+N/2and–N/2;0willbelesslikely. Stp_RandomWalk1D.jar

0 1 2 3 4 5 6 7 8-1

Whiteboard:Denoteobjecttaking4stepstotherightasRRRR.(Left=L)

Assumeonly4stepsareallowed,anditisequallylikelytomoverightorleft.

Howmanydifferentwayscanitendupatx=4?x=2?x=0?x=-2?x=-4?

0

1

2

3

4

5

6

7

-15 -10 -5 0 5 10 15

P ~ exp −x2 /Dt⎡⎣ ⎤⎦

Thisiscalleda“normal”distribution,oraGaussianfunction.Itassumesrightandleftarerandomlydistributed.

0 1 2 3 4 5 6 7 8-1

0

0.05

0.1

0.15

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

0

0.2

0.4

-4 -2 0 2 4

N=4

N=16

N=64

x2 = 2Dt ~ N

1. Assumerandomwalk2. Applycountingstatistics3. Seehowdiffusiondependsontime.4. Fick’sLaw:

Findthatthespreaddoubleswhenthetimeincreasesbyafactorof4.

Diffusionemerges fromtherandomwalkmodel.

net flow: J = −D dndx

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Whathappenswhentherearealotofparticles?

Stp_RandomWalk1D.jar

1D 2D

Stp_RandomWalk2D.jar

Footholdideas:Randomwalkin1D

• Asaresultofrandommotion,aninitiallylocalizeddistributionwillspreadout,gettingwiderandwider.Thisphenomenoniscalleddiffusion

• Thewidthofthedistributionwillgrowlike

• Discalledthediffusionconstant andhasdimensionality[D]=L2/T

Δx( )2 = 2Dt

(Inthelab,r2=4Dtbecauseitisatwo-dimensionalsituation,not1D.)

Footholdprinciples:Randomness

• Matterismadeupofmoleculesinconstantmotionandinteraction.Thismotionmovesstuffaround.

• Ifthedistributionofachemicalisnon-uniform,therandomnessofmolecularmotionwilltendtoresultinmoleculesmovingfrommoredenseregionstoless.

• Thisisnot directedbutisanemergentphenomenonarisingfromthecombinationofrandommotionandnon-uniformconcentration.

Afinalquestiontoponder:

Attheheartofthismodelfordiffusionisthateachparticlemovesrandomly.Thatis,itisjustaslikelytomovetotherightasitistotheleft.

Ifthatistrue,whydoesdiffusioncausetheparticlestospreadout?

Forexample,whydoesanopenperfumebottleendupfillingtheroomwitharomaifitisjustaslikelyforaperfume“particle”tomovebacktowardsthebottleasitistomovetowardstherestoftheroom?

Sodiumionsareatdifferentdensitiesontheinsideandoutsideofacell.Assumeeachionmovesrandomlyasaresultofcollisionswithotheratomsandmolecules.

Asmallpatchofimpermeablemembrane(areaA)isshowninyellow.Therearemoreionsareontheleftthanontheright.

Whatdoyouexpectistrueabouttheionsontheleftsideofthemembrane?

A. MoregototherightB. MoregototheleftC. Equalamountgoesleft

andrightD. Thereisnotenough

informationtotell

Sodiumionsareatdifferentdensitiesontheinsideandoutsideofacell.Assumeeachionmovesrandomlyasaresultofcollisionswithotheratomsandmolecules.

Asmallpatchofimpermeablemembrane(areaA)isshowninyellow.Therearemoreionsontheleftthanontheright.

Whatdoyouexpectistrueabouttheionsontherightsideofthemembrane?

A. MoregototherightB. MoregototheleftC. EqualamountgoesleftD. Thereisnotenough

informationtotell

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Sodiumionsareatdifferentdensitiesontheinsideandoutsideofacell.Assumeeachionmovesrandomlyasaresultofcollisionswithotheratomsandmolecules.

Asmallpatchofsemi-permeablemembrane(areaA)isshowninyellow.Therearemoreionsareontheleftthanontheright.

Ifthemembraneallowsionstopassthroughwhatdoyouexpectwillbetrue?

A. TherewillbeanetflowofionstotherightB. TherewillbeanetflowofionstotheleftC. Therewillbenonetflow.Equalamounts

willgoleftandright.D. Thereisnotenoughinformationtotell

A. TherewillbeanetflowofionstotherightB. TherewillbeanetflowofionstotheleftC. Therewillbenonetflow.Equalamounts

willgoleftandright.D. Thereisnotenoughinformationtotell

net flow: J = −D dndx

ThisistheideabehindFick’sLaw– randommotionaloneleadstodiffusionwhenthereisaconcentrationgradient.

Emergence!