Diffrac'on

IIEscueladeCristalogra5aenBiologíaEstructural

BuenosAires27.08.2018-07.09.2018

X-rayscaCeringbyelectrons

e-E

hCps://en.wikipedia.org/wiki/Lorentz_force

Lorentz’sforce:F=qE+qv×B

F(v=0)

INCOMINGX-RAYBEAM

SCATTEREDX-RAYBEAM

Radia'onscaCeredbyelectronsinanatomisweakerathigherangles

Atomicformfactors

X-rayatomicformfactorsofoxygen(blue)O,chlorine(green),Cl−(magenta),and

K+(red);smallerchargedistribu'onshaveawiderformfactor.

hCps://en.wikipedia.org/wiki/Atomic_form_factor

X-raymicroscopy:X-raysscaCeredbynon-periodical

systems

Projec'onthesingleyeastcell

PNAS.2005Oct25;102(43):15343-6.Epub2005Oct11.BiologicalimagingbysoDx-raydiffracIonmicroscopy.

DavidSayre’sgroup

ScaCeringofX-raysbyasingleyeastcell

IfonlywecouldcollectX-raysscaCeredbyasinglemolecule!

TIGRESSgamma-raydetector

UNFORTUNATELY1.X-raydetectorsarenotthatsensi've

2.OrienIngamoleculeinthebeamisnotthatsimple3.X-raysourcesarenotthatpowerful

Brightidea(?):let’sputmoleculesinacrystal

1.X-raydetectorsaresensi'vetoscaCeringbybillionsof

molecules

2.OrienIngacrystalinthebeamissimpler

3.X-raysourcesarepowerful

enough

Beijingsynchrotron

Unfortunately….

AllisleooftheunderlyingunitcellscaCeringpaCern

isacollec'onofspots

INCOMINGX-RAYBEAM

In-phasescaCeringislostalmosteverywhere

…usingcrystalstomeasureX-rayscaCeringisaloss-of-informa'on

disaster

ScaCeringofX-raysbyasingleunitcell

Whatwewouldliketomeasure:

ScaCeringofX-raysbyacrystal

Whatwecanmeasure:

Theconvolu'ontheoremexplainswhy

g:electrondensityofthecrystalf:electrondensityoftheunitcellh:crystallaqce

G:diffrac'onpaCernofthecrystalF:scaCeringpaCernoftheunitcellH:reciprocalcrystallaqce

F

x

H

=

G=F×H

TheFouriertransformoftheconvolu'onoftwofunc'onsistheproductoftheFouriertransformsofthesamefunc'ons.

ThecrystalX-raydiffrac+onpa/ernisallisleo

oftheunderlyingunitcelltransform

IsthispaCerngoodorisitbad? Somepeoplelikecrystaldiffrac'onbeCer

thantheunitcellscaCeringpaCern

Ques'on:ifwegotosmallerλ,arethehigher-resolu'onspots

stronger?

λ=1.0Å

Otherrelevantquan''esthatdependonλ:beamintensity,airabsorp'on;crystalabsorp'on

λ>>1.0Å

Circulardetector

ScaCeringfrommolecules(andcrystals)doesNOTdecreasemonotonicallywithresolu'on

ρHSpherically averaged atomice lect ron dens i'es decaymonotonically with increasingr ad iu s and so do the i rs c aC e r i n g f a c t o r s w i t hincreas ing length of thescaCeringvector.

Sphericallyaveragedmolecularelectrondensi'es(e.g.P(r),the

transformofSAXSscaCeringprofiles)andsphericallyaveragedunitcellcrystalelectronsdensi'esdoNOT

decaymonotonicallywiththeradius–nordotheirtransforms.Moleculesandcrystalpackingcanhavehollowsin

theirinterior!

SAXSandP(r)ofMOS1transferase

Solu'onconforma'onsofearlyintermediatesinMos1transposi'onNucleicAcidsResearch41(3)2012

InterferenceofscaCeringfromtwoelectrons

DirecIonofradiaIon:k=n/λwith|n|=n=1i.e.nisaunitvectorpoin'nginthedirec'onofthebeam

DirecIonofscaSeredwaves:k’=n’/λwith|n’|=n’=1i.e.n’isaunitvectorpoin'nginthedirec'onwewanttomeasurethescaCeredwaves

Electron1

Electron2

Twoelectronsareseparatedinspacebyavectord

Radia'onimpingesontheelectonsalongk.Wecollecttheradia'ontheyscaCeralongk’.

InterferenceofscaCeringfromtwoelectrons

ComputethedifferenceinopIcalpathsx1+x2betweenthewavesscaSeredbythefirstandsecondelectron

Electron1

Electron2

x2=d�k’x1=-d�k

x2=dcosθ’=d�k’

x1=dcos(π-θ)=-dcos(θ)=-d�k

x1+x2=d�(k’-k)

IfwewantthescaCeredwavestobeinphase,weneed:

d�(k’-k)=n

becauseitgives:d�(n’-n)=nλ

InterferenceofscaCeringfromelectronsequivalentunderlaqcetransla'on

(Laueconstruc'on)IfelectronsarerelatedbyalaqcevectorD:

D=ma+nb+ocwitha,b,cthevectorsdefiningthelaqce(i.e.theunitcell)

andm,n,oareintegers thescaCeredwaveswillbeinphase,if:

D�(k’-k)=N,Nisaninteger

Inalldirec'onsk’=K+kthatobeythisruleD�K=N,therewillbeconstrucIveinterferencebetweenoneelectronandanyotherelectronwhichisequivalentbylaZcetranslaIon.

D=ma+nb+oc

k’

k withK=k’-kD�K=N

KvectorsthatobeyD�K=nformalaqcecalledreciprocallaZce

Let’swriteK=ha*+kb*+lc* h,k,lareintegers

Itwouldbegoodifa�a*andb�b*andc�c*wereequalto1andallthecrossscalarproductsa�b*,…,b�c*were0

Howdowefindthelaqcea*,b*,c*thattheseKvectorsbelongto?Weneedtosolvetheequa'on:

D�K=Ni.e.(ma+nb+oc)�(ha*+kb*+lc*)=N

mha�a*+mka�b*+…..+olc�c*=N

Solet’schoosec*:•  alongthevectorproducta×b,sothatc*�a=c*�b=0•  andchoosealsoc�c*=1andsimilarlyfora*andb*

IfX-raysalongkhitacrystaloflaqcea,b,cthereciprocallaqceaxesa*,b*,c*predictdiffrac'onraysalongk’=K+k

Soifwesubs'tutein:D�K=N

i.e.(ma+nb+oc)�(ha*+kb*+lc*)=Nmha�a*+mka�b*+…..+olc�c*=N

Weget:

mh+0+0+…+nk+…+ol=N

Let’schoose:

a*=a�(b×c)

b×c

b*=b�(c×a)

c×a

c*=c�(a×b)

a×b

SoalltheKvectorsthatbelongtothisa*,b*,c*laqcewillobeyD�K=N

andwecanpredictthattherewillbediffractedraysinalldirec'onsk’=K+k