diffrac’oncristalografiaproteinas.qb.fcen.uba.ar/docs/clasex - difracción.pdfx-ray microscopy:...
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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