A Model for Simulating Hypothetical Protein Crystallization Behaviors
ABSTRACT This paper describes the development of an approximate physical model that can be used for generating different types of protein crystallization behaviors or responses in varying physico-chemical environments. The modeling process involves three stages, two of which have been completed. The model developed in Stage I was over-simplified and was aimed at providing the right computational flavor of the process of vapor diffusion that is a generally used technique for macromolecular crystallization. After this model was thoroughly tested, we enhanced the simple model to include a more elaborate nucleation theory, that still is an approximation of the actual physical chemistry involved. This stage II model was evaluated and found to have face validity in that it is possible to simulate different types of protein response behaviors - for example, proteins that crystallize easily under a broad range of conditions, proteins that crystallize under a narrow range of conditions...
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Page 1
Vanathi
to
Gopalakrishnan
face
and
p
Bruce
to
G?
ulate
Buchanan
Intelligent SystemsLab oratory
Universityof Pittsburgh
Pittsburgh?PA ?????
fvanathi?buc hanang?cs?pitt?edu
JohnM?Rosenberg
DepartmentofBiologicalSciences
Univ ersityofPittsburgh
Pittsburgh?PA?????
jmr?jmr??xtal?pitt?edu
Abstract
Thispaperdescrib esthedevelopmentofanapproxi?
matephysicalmodelthat canbeusedforgenerating
di?erenttypesofproteincrystallizationbehaviorsor
responsesinvaryingphysico?chemicalen vironments?
Themodelingprocessinvolvesthreestages?twoof
whichhavebeencompleted?Themodeldevelopedin
StageIwasover?simpli?edandw asaimed at pro vid?
ingtherigh tcomputational?a voroftheprocessof
vapordi?usionthatisagenerally usedtechnique for
macromolecular
plemen
crystallizati
aluate
on?
ulation
Afterthis
del?
model
With
w as
this
thoroughlytested?weenhancedthesimplemodelto
includeamoreelaboraten ucleationtheory?that still
isanapproximation oftheactualphysicalchemistry
involved? ThisstageIImo delwas ev aluatedandfound
havevalidit yinthatitisossiblesim
di?eren
sim
ttypes
the
of protein
pro
resp
of
onse
v
beha
di?usion?
viors?for
whic
ex?
h
ample?proteinsthat crystallizeeasilyunderabroad
rangeofconditions?proteinsthat crystallizeundera
narrowrangeofconditions?andproteinsthattakea
verylongtimetonucleate andgrow?It tookapprox?
imately twoyears tocompletestages IandI I?Stage
IIIwouldbem
as
uch
b
more
eha
elab orate
as
and
ected
w
b
ould
y
inv
domain
olve
thejusti?cationofalltheequations inthe model rig?
orouslyfromaphysicalchemistrypoint ofview?This
couldtakeafair amountoftime?asactualexperimen?
tationwill needtobeperformed?
In troduction
In ordertoutilizeexistingtheories aboutthe process of
macromolecularcrystallizationfor practicalassistance
inexperimentdesign?itwasnecessarytodevelop?im?
tandevasimmo
goalinhand?wedevelopedappro ximatephysical mod?
els underlyingthesimulationintwostages? Inthe ?rst
stageofdevelopment?wefocusedoure?orts onbuild?
inganover?simpli?ed model that hadthe right?avor
ofulatingcessaporis
thedrivingprocess?butdidnot havesu?cientcom?
plexit ytomo delnaturallyoccurringphenomenasuc h
asnucleationandcrystal gro wth?Once weweresat?
is?edthatoursimplemodelhadbeenimplemented
correctly andwving expthe
experts?weproceededtothesecondstageof develop?
ingph ysicalmo dels to mirrornucleationandgro wth
ofmacromolecules?Thecomplexity of themo delde?
velop edinstageII providedamoreaccuraterepresen?
tationof crystallizationb ehavior of real?world macro?
molecules?Thoughsomechoicesmade inthe model
canbejusti?edbytheory?manyothersthoughrea?
sonable? cannotbestronglyjusti?edbasedonph ysical
theory?
OverviewoftheCrystallizationProcess
Crystallizationfromsolution?liketheformation of ice
fromwater?representsaphasec hange?andconstitutes
adiscontinuityinthebeha viorofasystem? Itoccurs
b ecauseanenergybarrierhasbeensurmoun ted and
the phasetransition leadstoamore favorable stateof
thesystemwithresp ectto energy?
Aswith any science?even though theoryand practi?
calexp erimentationare exp ectedto go hand?in?hand?
this is not alwa ys thecase in the domainof macro?
molecularcrystallization?Thegoal of exp erimentation
inthis?eldis to growa goodqualityX?raydi?ractible
crystal ofaprotein orDNA insolution? Asv erylittle
is actuallyknown about how thedi?erent experimen?
talvariables interactinsolution? the experimentdesign
isbasedmore ontrial?and?error andreliance onpast
exp erience insetting upandperformingexp eriments?
ratherthan theory?Manyscientistsbelievethatthe
theory?thoughweak? canandshouldbeused toguide
experimentation?especially ascrystallizationfrom so?
lutionisasequence of eventsthatoccurmore orless
consecutiv elybut arerarelycompletelyunconnected
?Boistelle?Astier??????
Ev enthougha completetheoryofthe crystalliza?
tionof macromoleculesis unknownatthepresent?ba?
sictheories ofphasechanges arewellundersto o d?In
broad brushterms? phase transitionb ehavior ofmacro?
moleculesissimilar to othermolecules? as isevidenced
inRosenb ergeret al? ?Rosenb erger et al??????? The
problemwith macromoleculesis thattherearemany
Page 2
imencrystallizationisap hanging
dropmethoprincipleandofthe
areedb
thebasisofguidancefromgeneraltheory andusev aria?
tionsinsolubilityparameterde?nitions tobring about
theidiosyncrasiesinindividual proteinbehaviorsbeing
simulated?
Acrystallizationset?up consistsofan initialsetof
chemicalsmixedtogetherinv aryingconcen trationsat
a particular temperatureandpH? alongwithprotein?
Therest ofwhathapp ensisbasicallyacontinuously
evolvingprocess?wherethesupersaturationoftheso?
lutionincreasesordecreases? triggeringeventssuchas
nucleationand growthof crystals?The levelof super?
saturationa?ectstherateat whichcrystals grow and
directlyimpactsthequalityofcrystal obtainedfrom
experimentation?Awidely adoptedmethod forexp er?
tsinthevordi?usion
d?Thepracticemethod
describelo w?
Principle ofV ap orDi?usion
A
exp
droplet
erimen
con
ts
taining
the
the
oratory
biological
?
macromol
tra
ecule?
bu?er?
?x?
precipitating
grid
agen
with
tand
cells
an
eac
y additiv
h
esin solu?
reser?
tion?
v
is
solutions?
equilibratedagainst
drop
areserv
from
oircon taining
top
aso?
v
lutionofprecipitatingagentata higherconcentration
thanthedroplet? Equilibration proceedsbydi?usion
ofwateror organicsolventuntilvaporpressurein the
droplet equalsthe oneinthereservoir?Ifequilibra?
tionoccursbywaterexchangefromthedroptothe
reservoir?itleadstoadropletvolumechange?Con?
sequen tly?theconcentrationsofalltheconstituen tsin
thedropwill change?Ducruix??editors??????? There
aremanytypes ofsolidphasesthatcanoccur whena
macromoleculechanges statefrom liquid to solid?The
most broadly de?nedsolidphasesaretheamorphous
andcrystallinephases?Thesecanbefurtherclassi?ed
based onsizeandshape?forexample? needlesand
platesaretwodi?erenttyp esof crystalline phases?
Figure?pro videsthesc hematicrepresentationof the
hangingdropv apordi?usionmethod asw elltheph ys?
icalrectangularplate?trayapparatusused top erform
in labThey isusually a
plasticbox ??containing
oir Thehangstheofacoer
?
Solubilityparametersrefertotheexp erimentalparam?
etersthata?ect thesolubilityofthe macromolecule? Sol?
ubilityofasolidphaseofproteinisde?nedasthecon?
centration oftheprotein inequilibriumwithsolidphase of
interest givenideal thermo dynamic conditions?
????
????
????
????
????
????
????
????
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????????
the
????????
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????????
A Tray
A well
(an experiment)
Several experiments in parallel
Figure ?? Hangingdropv ap ordi?usion method
slip?andcontainsthe
of
biological
our
macromolecule
Initial
solu?
for
tion? Itmustbenotedthattherealwa ysexistsasmall
leakfromthetray asthere isaseal?whichcannotbe
guaran teed tobeperfect?So?thev aporpressure in
the cellis alsoequilibratingwitham bientpressure
inadditiontothevaporpressurein thedropequili?
bratingwiththat ofthereservoir?Figure?isredrawn
fromMcPherson?McPhers on ?????? Itshowsastan?
dardcon?gurationforhanging drop proteincrystal?
lizationexp eriments?Aplastic platepro vides??w ells
for reservoirsofabout???to ????ml?Eachwellisan
exp eriment andcanbecoveredbyacoverslip witha
drop ofproteinsolution hanging fromitsunderside?
Thedrop of?to ???lequilibrateswiththe reservoir
solutionovertimethroughthev apor phase?causing
precipitant and proteinconcentrations toincrease in
the drop?andnucleation ortheinduction ofcrystal?
lization tooccur?Eachwell orexperiment isobserv ed
underamicroscopeperiodically toobservee?ects?
DescriptionModel
SimulatingtheV ap orDi?usionProcess
Our modelacceptsasinputvaluesforproteingiv enpa?
rameters?such asitsnameandisoelectricpoin t?and
simulationparameterssuch asthephase information
fortheproteinbeing simulated? Italso acceptssolu?
bilitytablede?nitionsthat describethesolubility of
this protein inindividualreagen tsandconditionssuch
aspH?Aset ofcontrolvariablevalues arealsoinput
that specifytheinitialvalues ofvariousconcen trations
ofchemicalsand theinitial pH?Wehavemodeledsol?
ubility ofahypotheticalproteinin?commonlyused
Page 3
rates?in concentrationofthevariouschemicalsand
proteinwithinthedrop?and updatesthesevalues at
discreteintervals?Themainhidden v ariablesin the
mo delarethesolubility andsaturationof each phase
ofprotein? Thesaturationofaparticularphaseofpro?
tein isdirectly prop ortionaltoproteinconcentration
in solutionandin v erselyprop ortionaltothesolubil?
ityofproteinphaseat thattime? If atanytime?the
saturation ofaparticularsolidphaseexceeds thein?
put saturation requiredfornucleation ?nuc
sat??asolid
massformswith size equalto ?????Ifsaturationofthat
phaseisgreaterthanthat requiredforgrowth?thesolid
massgrows inproportion toitssize?Otherwise ifthe
solutionisundersaturated ?thatis?saturation isbelow
??? then the solidmassshrinks?
We had to makeseveralsimplifyi ngassumptions in
ourmodel oftheprocess ofv apordi?usiondescribed
in anearliersection? Inthe?rststageof model de?
v elopment?wetried tomimic mainlythewaterv a?
p or exponen tialdecay?seeequation ??? withtwodriv?
ingcomp onen ts? namelytherateat whichthingsw ere
changing towardsequilibriumb etween the
?
reserv oirso?
lution and thedrop con tainingthemacromolecule ?the
equilibrationrate?andthe rateatwhich somev ap or
w
Grandpa
as leakingfrom
PEG
the
to
seal
depict
ofthe
PEGs
physical
of
apparatus
di?eren
?the
leakrate??We used discretetime units?whichcouldbe
varied?weexp erimentedwith hours and?ve?minute
interv als??Ateach timeunit? thev olumeof thesample
wouldbecalculated usingthew ater exponentialdecay
equation? Then?thesolubilit y ofeach solid phase of
proteinwouldbe calculatedasafunction of? solubili?
ties namely? solubility ofthat phasewith resp ect tosalt
concentration? precipitantconcen tration?andthe ionic
concentration?In the?rst stageof dev elopment?we ac?
tuallyusedgraphscontainingstraight?lineappro xima?
tions offunctions todescribesolubilityinformation of
solidphases ofah ypotheticalprotein? Thesesolubility
graphsw erelater describ edbysmo othmathematical
functionsand constitutethe ma jordomainknowledge
fordrivingthesim ulation?
V
t
????V
init
?V
equilibration
??e
?
t
?
??
?
We use names such as Baby PEG? MamaPEG and
tmolecular
weights?Baby isthe low ermolecularweightPEG?PEG is
acommonly usedprecipitantand isthe acronymforPoly
Eth yleneGlycol?
time?or leakrate?usually?weeks??
The solubilitytables app eartobea goodway to
represent theb ehavior ofaparticularh ypothetical pro?
tein withresp ect todi?erent exp erimen tal con trolv ari?
ables?
Figure
They
??
havethe
y
poten
the
tial
crystalline
ofb eingav ery
of
?exible
protein
represen
with
tation
resp
for
Bab
describing
y
di?eren
concen
t functional
tration
b e?
ha viors ofaproteinin di?erentprecipitan ts andsalts?
In ourmodel?wegenerally use theErrorandGaussian
distributionfunctions
?
?Presset al??????torepresent
thebehaviorsofsolubilityofaparticularsolidphase of
proteinindi?erentsaltsandprecipitantsasafunction
oftheirvariousconcentrations?Thesolubilitytablepa?
rameters representthevalues suchasmid?pointsand
standarddeviations?requiredbythe individualdistri?
butionfunctions?Samplesolubilitygraphs areshown
inFigures?and?belo w?
0
5
10
15
20
25
30
020 4060 80100
Solubility (mg/ml)
Precipitant Concentration (% w/v)
Solubilitofphase
ect toPEG
5
6
7
8
9
10
11
12
13
14
15
02468 1012 14
Solubility (mg/ml)
pH
Figure??Solubilityoftheamorphousphaseof protein
withrespectto thepHofthe solution
Thesample solubilitygraphs depicted inFigures?
and?are represented asmathematical functionswith
parameters describ ed assolubility tables?F orexample?
the solubilitygraph depicted inFigure? isrepresented
asthe complement oftheerror function?Press etal?
??????We representthe corresponding parameters as
valuesforthemidpoint ofthexaxis?thestartandend
?
Allthemathematical function codeswere obtained
fromNumericalRecip es in C?
Page 4
deviationthatspeci?es theslop e at themid?pointofx
axis?
For purposes ofourmodeling?wehaveincorp orated
intothesolubility tablesforahyp otheticalprotein
calledAlphaGlobulase?numbersthat wereobtained
fromsolubility graphsprovidedbyourdomainexpert?
Thedi?cultlyinobtainingactual solubilitytable in?
formationarisesdueto thefactthat suchinformation
willhavetobegleanedfromactualexperimentation in
the laboratorybyaphysicalchemist?Solubilitygraphs
canbeextrap olatedfrompointsthatdepict solubility
ofthe solidphases ofagiv en proteinineachprecipi?
tantorsaltat di?erent concentrationsbyperforming
sev eralsimpleexperimentsinthelab oratorywithvery
smallamountsofproteinmaterial?Thus?in theory?it
ispossibletoobtain suchsolubilityinformation that
canbeusedasinput forourmodel?
Thein teractionb etw eenthedi?erentsolubility func?
tionsisspeci?edbya simpleequation ?seeequation ??
thatm ultiplies thesolubilities ofprotein in each ofpre?
cipitant? salt?andbu?er?pH??This isasimplifyingas?
sumptionthat theov erall solubilitycanberepresen ted
bymultiplyi ng thesolubility ofprotein in each of the
individualindependentvariables? Theoverallsolubil?
ity of proteininasolution is highlynon?linear due
totheinteractionsamongthe individualindependent
variables?Oursimplerepresen tation ofoverallsolu?
bility asmultiple ofindividualsolubilitiesworkswell
foroursimulation purposes ?intermsof expectedout?
putbehavior for giveninputs??F utureresearchmight
wellinvolv ev ariationsto this equationfordescribing
interactions inthis highlymultidim ensionalparameter
space?Solubilityofsolid phase i?Sol?i?? ofprotein is
calculatedas?
S ol?i??
?
q
Sol?i?
?PEG?
?Sol?i?
?Salt?
?S ol?i?
?Ionic?
???
wheretheconcen trations of precipitatingagent ?i?e?
?PEG
?
???saltandions a?ectthe solubility ofa partic?
ular phaseofprotein asgiv enby the solubilitytable
information anddomainkno wledgeofrelativeimpact
ofthe di?erentreagen tsforeachtype?Theconcentra?
tionofionsinsolutiondependson the saltconcentra?
tionand bu?er?that is?thepH??
?
PolyEth ylene Glycol?acommonly used precipitating
agent?availableinseveralmolecularweights?e?g??????
??????PEGconcen trationismeasured inunitsof?w?v
thatis?percentageweigh tp erv olume?
tocurrentvolumeofsample at timet?oftheirini?
tialconcentration ?seetheequations formoredetails??
Oncesolubility ofa solid phase iscalculated?the sat?
uration ofthatparticularsolid phaseofprotein in the
sampleiscalculated astheproteinconcentration in so?
lutiondividedbysolubility of this solidphase at this
time ?seeequation ???
Proteinconcentration isupdated as givenin equa?
tion??When thesaturation ofprotein inthe sample
reac hesthesaturationrequiredfornucleation tohap?
pen?anucleus wouldappearwith solidmass appro xi?
mately??ofexistingprotein concentrationin theso?
lution?Ifthesaturationlevelisbelow ??then thesolid
massstartsshrinking?Ifsaturationisabove??su?
persaturated??and thesaturationlevelis greaterthan
requiredforgro wth?then ucleusbeginstogrowand
accum ulatemassin thesolidphaseattheexpenseof
theliquidphase?TheoutputofstageImodel issimply
thedevelopment ofsolidmass?that is?thetransfer of
proteinfromliquidtothesolidphase?
ol
Equation?representsthecalculationofsaturation
ofaparticularsolidphaseofthegivenproteinat timet?
Saturationisdirectly prop ortionaltotheconcentration
ofproteinin solutionat thattime?andvaries inversely
withthesolubilityofthat phaseattime t?Solubil?
ity iscalculated asp erequation??wherewemultiply
theindividualsolubilitiesatdi?erentconcentrationsof
precipitantandsalt?andionicstrength?andtakethe
cuberoot?Wefoundthisassumptionabouthowsol?
ubilitiesin?uenceoneanothertoproduce thedesired
e?ectsin theoutputsfromthesimulator?Equations?
??depictthe c hangeinconcentrations ofthe protein?
saltand precipitant attime tas afunction ofthe ini?
tial concentration andratiochange inwatervolume of
the drop?
Saturation ofa solidphase ofprotein is calculated
as?
Satur ation?phase?
t
?
Pr oteinConc
t
S?phase?
t
???
ProteinConcentrationattimetisgivenby?
V
?
V
t
?
?
ProteinConc
?
?
?nucleii
X
i??
size?nucleus
i
?
?
???
Thesaltandprecipitantconcentrations areupdatedas
Page 5
In the ?rststage ofdev elopment ofthesimulation
model?thefocuswas onthedriving processofv apor
di?usion?and howtorepresen tit?In thenext stage?we
included actual even tssuchasnucleation andgrowthof
crystals?Nucleation isa stochastic even t?andsowein?
cluded someelement ofrandomnessin our model? The
main di?erenceisin thenucleationpro cess?wherewe
accountfor twodi?erenttyp es ofnucleationthat could
p ossibly happ ensimultaneously orseparately ?Thetwo
t yp esofnucleationthatoccur innature arereferred
to asheterogeneous andhomogeneousnucleation?We
will deferfurtherdescriptionof the stageIInucleation
mo del? un tilwehave describedthenature ofsolids and
thenucleation process in macromolecularcrystalliza?
tion?
DescriptionofSolidStates
There aretwogenerallyobserv ablesolidstates for
macromolecules?amorphousandcrystalline?Thedis?
tinguishingfeatureb etw eenthemis thelong?range or?
derof moleculesin thesolidmatter?Amorphoussolids
haveshort?range orderthat isap eriodic? while crys?
tallinesolids have long?rangeorderthatisp eriodic?
F or purposesof X?raydi?raction?p erio dicity isk ey
sincethedi?raction pattern ofhighlyp eriodicarrange?
men tsresult insp ots onthedi?raction screen? making
iteasier foranalysis?If onewereto di?ract an amor?
phousprecipitate?onew ouldgetahighly stringymess
on the di?ractionscreen?as opposed toa patternof
sp ots? makingit di?cult top erformany kindofanal?
ysis todetermine structure?
An importantquestion whendiscussing solidstates
ofmacromolecules is howtheydi?er fromotherc hem?
icals?b othinterms ofactual statesof existencein
the solid phase?aswell asindi?culty ofobtaining
thesolidstates themselv es?Although conventional
solution?gro wnorganicandinorganiccrystals mayap?
pearsuper?ciallysimilar tomacromolecular crystals
grown fromsolution?there aremanyimp ortantdi?er?
ences in physical andmechanical properties?The dif?
ferences inprop erties? inturn?a?ect theway inwhich
crystals aregro wnandusedin the laboratory? Equally
signi?cantfromastructuralanalysis p ointof viewis
thatproteincrystalsseemtoreac hamore or lesster?
minalsizeandseldomexceedit?Proteins alsotend to
denatureeasily?
Amorphous phaseisusually the kinetically morefa?
vored solid state?sinceaggregation ofmoleculeso ccurs
vin termsofandmoney obtainsu?cien
Nucleationand wthof
toincorp oratethe followingthermo dynamic?aand b?
andkineticfeatures ?c???a? Ingeneral? thereexistsa
rangeofconditionsinwhich thecrystalline phaseis
morestable than theamorphousphase? ?b? Thesecon?
ditions arenear theboundary
?
b etween clear?i?e?? no
observ ableresultorc hange inliquid phase? andamor?
phous? and?c? Ifthermo dynamics favorthecrystalline
phase? thekinetics couldfavortheamorphousphase?
Thekineticsarebasedonthe ratesofc hangeof the
various parametersusedinexperimen tation?
It ism uchharder toobtainthesolidstates ofmacro?
molecules asopposed tosimpleinorganicororganic
molecules? becauseof thevastn umb er ofparameters
thatcana?ectnucleation andgro wth?thee?orts in?
olvedtime tot
quan tities ofproteinandpurifyit?andthefact thata
proteincandenature easily?Thenextsubsection de?
scrib esthe phenomenonofn ucleation?
Gro
MacromolecularCrystals
Nucleation
and
isav erycomplex
in
phenomenon
v
and
are
ev en
physical chemistsdo not understandeverydetailofthe
mannerinwhichthiseventoccurs formacromolecules?
Somesimpleliquid?solidtransitionssuchasthatofwa?
tertoiceisundersto o d?buteven in thatcase?the pro?
cessofliquidcondensationfromthevaporphaseisthe
phenomenonthatismostw ell?understood?Aleapof
faithismade withrespectto thephasetransitionfrom
liquid tothesolidphase?Whileasampleofmatterin
itsliquidandvaporstatescanbe describedin terms
of itsvolume?pressure?temp eratureandamountof
substancepresentinit?descriptionofasampleinthe
solidstaterequiresastudy of itsshape andsymmetry?
re?ectingthearrangementofmolecules ina structured
fashion or lattice?
Inthe caseofliquid?v aporphaseboundary?anypoint
onthisboundaryrepresen tsa stateofdynamic equi?
libriuminwhichvaporizationandcondensationcon?
tinueatmatching rates?A tkins?????? Moleculesare
leavingthesurfaceoftheliquid ataparticular rate?
molecules alreadytheap orphasereturn?
?
Thebiggestfactorincrystal gro wthistime?Th us?even
ifathermo dynamicallystablestateis favored farfrom the
boundary?itw ouldtakeam uchlongertime toobtainsuch
acrystal thanonethat isstable kinetically?duetotheslow
rate ofc hangethatdrivestheprocessofcrystalgrowth?
Page 6
butthey are doingsoata ratethatexactly matches
therateatwhichmoleculesalreadyin theliquid are
settling onto thesurfaceofthesolidandcontributing
tothesolidphase?
UNSATURATED REGION -
SOLID PHASE DISSOLVES
diagram
SUPERSATURATED
REGION
Labile Region - Stable nucleii
spontaneously form and grow
Metastable Region -
Stable nucleii grow
but do not initiate
PROTEIN CONCENTRATION
PRECIPITANT CONCENTRATION
Figure??Twoimportant dimensionsofphasediagram
forproteincrystallization?from?Feigelson??????
Figure? ?Feigelson????? represen tsatraditional
phasediagramthatre?ects themostcommontech?
niquefordrivingamacromoleculefromsolution?called
saltingout?Theheavy solidline represen tsthe equili?
bration linecorrespondingtoasaturatedsolution?At
concentrations ofsoluteb elow thesolubilitylimitfor
values alongthehorizon talaxis?the system issaid to
beunsaturated? Above thatlimit? the region iscalled
supersaturation? Aswe can see fromthe ?gure?the
supersaturation requiredfornucleiito formis greater
than thatrequiredforgro wth?Because sev eralph ysi?
caland chemicalparameters mayin?uence solubility?
the phaseforaproteinornucleicacidistypi?
callymulti?dimensional? Ourmodeltakesasinputthis
complexphasediagramandattempts toprovidethe
response of thatproteinundervarying experimental
conditions?Eachsolubilitytablerepresen tsap ortion
oftherealphase diagramof theprotein? Each solubil?
itygraph hasonev ariablesuch as precipitant concen?
trationonthehorizontalaxisandsolubility?basically
proteinconcentration inmg?ml?valuesonthevertical
axis?
F ornucleationtohappen?itisnecessaryforasuf?
fects themean ofthesize distribution anditsvariance?
The moleculesarecon tinuously bangingagainst one
anotherwith tremendousforce?andtheb om bardment
causes dislodgemen t ofmolecules from an aggregate?
When the solution is above saturationor is supersat?
urated?
the
ab
then
ove
at
acritical
some
size?
t
aggregation
time?
is
will
fav
b
ored?
e
Thus?anucleus can formand growwithoutb ound?
The criticalsize? orthe meanofthe sizedistribution
ofaggregates insolution?becomes low er assaturation
lev el increases?Hence?weobservethee?ect of spon?
taneousn ucleationhapp ening ataparticularsatura?
tion level? asthecriticalmasshasdecreased toapoint
whereaggregation ofmolecules is favored?
In homogeneousn ucleation?weseearandomcolli?
sion ofmolecules formingaggregates? Itisobserv ed
thatinhomogeneousn ucleation?we oftengetashower
ofmicro?crystals?The processofnucleationishighly
non?linearascanbe inferredfrom theobserv ed e?ects?
Suddenly? therewillappearashower ofmicro?crystals
and theirn umb erisprop ortional tothep ercentageof
protein
just
inthe
ed
solution?
?see
This e?ect
???
canbeexplainedby
usingthe following analogy?Ifwe thinkofcritical mass
asahigh?jump bar thatisbeinglowered constantly as
thelevel ofsaturationincreases?andthe ?ick ering ag?
gregates as manychildrenhopping tov ariouslev elsof
bar?poin inthebarlow
enoughthata certainp ercentageof c hildrencouldeas?
ilyhop over?This analogycanbeusedto explainthe
o ccurrenceofa suddensho w er ofmicrocrystals?andis
re?ectedinourmo del?In heterogeneousn ucleation?
thereis a substratethat binds molecules?Duetothe
existence ofasubstrate? heterogeneousnucleationis
foundtooccurmoreofteninrealexperimen tation?as
theaggregatesformeddueto bindingtendtoexceed
critical mass earlierthanthoseformedasaresultof
randomcollisionsalone?
Wehavemodelednucleationinthesimulatortore?
?ect thisprocess?byimposingacritical mass forany
n ucleus?Thiscritical massforanyphase decreasesas
thesaturation increases?therebysim ulatingthee?ect
describ equation
criticalmass?phase??
?
saturation?phase?
???
Rosenb ergeretal??Rosenb ergeretal??????have
performedextensivestudieswiththemacromolecule
lysozymefromhenegg?whitethatiseasy toobtain?
Page 7
bergeret al???????
Thehigher thesupersaturation?thesmaller the
criticalnucleusand? th us? thehighertheprobabil?
ityfornucleation?Casey?Wilson??????
Weha vebasedourmo del ofnucleation onaphe?
nomenologicalexplanation? Both? homogeneousand
heterogeneousnucleation are modeled assto c hastic
pro cessesthathappenasafunctionofc hanceoncethe
saturationrequired fornucleationhasbeenattained?
Oneofthe input parametersto thesimulator mod?
eledinstageIIisthenum berofheterogeneoussub?
stratespresentforeachofthephases thatcouldpos?
siblybindtoproteinmoleculesto formheterogeneous
nucleii?Usually?thisnumberissmall? e?g?????In any
sim ulation run?it isthereforepossible toobtaineither
orb othhomogeneousand heterogeneousn ucleii that
form and growasoneormoreofthesolid phases?Ho?
mogeneousnucleationismodeledso astoresultinthe
formationofashowerofmicrocrystals?based onthe
calculation ofprobability ofoccurrence ofahomoge?
neouscrystalperunitvolume ofthe samplemultiplied
bythe actualvolume attimet ofsimulation?
In thecase ofhomogeneousn ucleation?theprobabil?
ity of?ndingan ucleusofsizeequaltothecriticalmass
is dependent onthevolume ofthesample? The prob?
abilitydensityfunction forcalculatingtheprobability
ofnucleiiofparticularsize formingper unitvolume
ofsampleis mo deledasaGaussian ?midpt? width??
wheremidptand widthdescribetheaspectsof sizeof
nucleus? whichv ariesdirectly asa functionofsatura?
tion?Aftercalculating thecritical mass requiredfor
nucleating that particularphase?we thencalculate the
probability of?ndinganucleus ofsize?critical massby
referring to theprobabilitydensit y function ?equation
??? In thiscase ofhomogeneousn ucleation?equation
???wem ultiplytheprobabilityp erunitvolume bythe
Volumeat timet? soas to yieldthedesirede?ect ?of
relating volumeof sampletonucleationprobability??
Thus?theprobabilityofhomogeneousn ucleationp er
unit v olume ofsampleforeach solidphaseisgiv en
by?Mandel??????
P?homounit??
?
?
p
??
e
?
?homounit ?homounit
?
?
?
??
?
???
wherehomounit?criticalmass of solid phaseatthis
particular time?
Pr ob?homonuc??Prob?homounit??V
t
???
wheretheprobabilityp erunit v olumefor thatphase
is calculatedfrom aGaussian probability distribution
withvariable widthandmid?points ?thatdep endon
thesaturation atany giventime??seeequation ??
In thecase of heterogeneousnucleation?we start
withauser?de?nedn umber of?xedheterogeneousn u?
cleii?F oreachnucleii?we havea di?erentprobability
densityfunction?describ ed asGaussian?midpt?width??
wheremidptand widthv arydep ending onsatura?
tion?F or eachheterogeneousnucleusthat exists?we
useequation ?? todetermine theprobabilityfor it to
nucleate?Theprobability ofheterogeneousnucleation
happ eningisdescrib edb y?
P?het??
?
?
p
??
e
?
?het?het
?
?
?
??
?
????
where het?critical mass ofthesolidphase atthis
particulartime?
?
?
?varianceinthesizeofcriticalmass fornucleation
ofthisheterogeneousn ucleus inparticularsolidphase?
het
?
?meancriticalmass orsizerequired forn ucle?
ation ofthisheterogeneousnucleus inparticular solid
phase?
Growthrate ofanucleus hasbeenfound inpractice
to increaserapidlywith higher supersaturationlev els
thereby yielding crystalsofpo orqualityor precipita?
tion?Thegrowth?rateis modeledasadirectfunc?
tion ofsaturation ?seeequation ????Rosenbergeretal?
?Rosenbergeretal??????observethatsmaller crystals
growslowerthanlargerones?Therefore?eachnucleus
gro wsor shrinksin prop ortiontoitssize? Figure?
depictsour modelingof thegrowthphenomenon?
growth
rate?nucleus?? saturation?phase?????
Insummary?wecouldhaveseveralnucleiiineach
solidphaseofaprotein?andtheycouldbeeitherho?
mogeneousorheterogeneous?Homogeneousnucleation
canproducesev eralsmalln ucleii of thesamesize ?
thesewould resem blethe showerofmicrocrystalsthat
areoftenfoundinnature andthelaboratory?
TheCharacterizationofCrystalQuality
X?raydi?raction providesdataforanumberof kinds of
quan titativeanalyses oftherelative qualityofcrystals
Page 8
a familyof planesinthe crystal?Thelimit at which
di?ractionintensities disappearforaparticular crys?
tal de?nesitsresolution?and thisin turndetermines
theprecisionofthestructuralmodelderivedfromthat
analysis?
Anotherfeature ofthedi?ractionpattern thattends
to re?ecttheinherentorderofacrystalisitsmosaicity
?that ismanifestedbythewidth? orspread?ofthe
intensities? Mosaicit y?inasense? re?ectsameasureof
thevarious kindsofphysical defectsinacrystal?The
conceptof mosaicity isre?ected inour modelas the
B?factor?which isan umb erthatre?ectsthe internal
orderof thecrystal?This internal orderisusuallyb est
attime ofnucleation?andgraduallyworsens as the
crystal grows?
Theresolution limitof di?raction?di?im?ofacrys?
tal isusuallyan umb erb etween??? and???Angstroms?
and lowernumb ersimplyverygo odcrystalquality?for
example?a crystalwith di?im ???Angstromsdi?racts
verywell?andis saidto beacrystal of excellent qual?
ity?? The?nalqualityofagro wn crystal isindicated
by itsdi?imv alue andis calculatedusingequation ???
In thesimulatormo delwe re?ect theinternalorder
within
where
asolid
?
phase as
limit
theB?factor?
of
which
?
essen
a
tially
con?
indicatesan umbervaluethatislowest atthe timeof
nucleation?andsteadilyincreases asgro wth proceeds?
If
analysis??
a crystalgrows too rapidly? theB?factorb ecomes
av ery largen umber? leadingtoa larger?nal di?im
v alue?indicating poor crystalqualitywithresp ectto
X?ray di?raction? Growth rate ofa crystalismo deled
asafunction ofsaturation? Ifthe supersaturation level
isv eryhigh? the growth rate isalso high? Thecalcula?
tionfor?nal di?imis based ontwo mainfactors that
determinethe quality ofa gro wncrystal?namely ? its
internal order?represen tedastheB?factor?? andits
size ?or theillumi natedv olumeforX?ra ydi?raction
analysis??Theequation forcalculating the?nal crys?
tal qualityordi?imisshownb elo w?
d?f?
r
B
l nV
????
dresolutiondi?raction?f real
stant?B? B?factor?in ternal orderre?ector?andV?
size ofcrystal ?illuminatedvolumefor X?raydi?raction
1. Obtain initial state + other givens
2. Set up domain knowledge into tables
3. Initialize data structures
2. For each phase do:
2. Print final outcome(s)
Phase Calculations
For #time units simulation is to be run do:
Time++
Phase++
3. Update current state
(eqs. 4, 5, 6)
1. Calculate volume at time t (eq. 1)
1. Calculate final outcome (eq. 12)
A
B
Figure ?? Flowc hartdescribing the mo del ofv ap or dif?
fusion?Main Flow
Flowc hartsfor Simulation Mo del in
StageII
The?owc hartsthatdepict thepro cessing inv olved in
the stageII simulationmodel is shown in?gures?? ??
Thebo xes labeledas A?B? C?D?E andFrepresen t
thepartsthat areretained fromthe stageImodel?
SensitivityAnalysis
Thissimulator canbeused toproducetwo di?erent
types ofoutputs?asinglerealn umb errepresen ting the
qualityofitsb estnucleus orb estresult ataparticu?
lartime ofobservation?andseveral graphicaloutputs
thatshow thechangesinthevariousc hemicalconcen?
trations and thenucleationandgro wth of solid mass or
nucleiiin eachsolidphase oftheprotein? Thegraphi?
cal outputsw ereusedfor subjective evaluation of the
mo delbydomain experts?Wehavetried toplotsev?
eral di?erentvariables inonegraph? inorder to depict
the process?We ?rstc hoose oneprecipitating agent?
namelyBabyPEG orlowmolecularweight PEG?and
show thegraphical output froma sim ulatorrunthat
has noheterogeneousn ucleii?Onlythe totalsolid mass
that isthe sum ofsev eralmicro crystalsthatformaho?
mogeneousnucleus? issho wn asthesolid mass for each
Page 9
Number of nucleii
in phase i > 0?
Perform
Growth Calculations
Perform
Nucleation Check
current state, current phase i
YESNO
1. Calculate solubility[i] (eq. 2)
2. Calculate saturation[i] (eq. 3)
C
Figure??PhaseCalculations
current state, current phase i
Is saturation[i] <= nuc-sat?RETURN
5. If nucleation occurs, set appropriate parameters for nucleii, and update
number of nucleii in phase I accordingly
YES
NO
Return the number of nucleii in phase i
D
1. Calculate critical mass required for nucleation (eq. 7)
2. Calculate probability of homogeneous nucleation occurring (eq. 9)
for nucleii (i.e. size, #centers, intrinsic B-factor)
3. If nucleation occurs (i.e. prob > threshold), set appropriate parameters
4. Calculate probability of heterogeneous nucleation occuring (eq. 10 )
Figure??Nucleation Check
current state, current phase i
Is saturation[i] > growth-sat?
For each nucleus j in phase i,
1. Size of nucleus j increases
proportional to growth-rate
2. B-factor of nucleus j decreases
proportional to growth-rate
For each nucleus j in phase i,
1. Size of nucleus j decreases
proportional to growth-rate
(= shrink-rate)
2. B-factor of nucleus j varies
accordingly
RETURN
Is saturation[i] < 1.0?
YESNO
YES
NO
Grow
Mode
Shrink
Mode
Obtain growth-rate for saturation[i] (eq. 11)
E
F
Figure??GrowthCalculations
0
10
20
30
40
50
60
70
80
0204060 80100120140
Concentrations and Water Volume
# Days
’Protein-Conc’
’PEG-Conc’
’NaCl-Conc’
’Water-vol’
Figure??Control parameterswith ???w?vBab y
PEG? ??mg?ml protein????MNaCl ?
solidphase?
ThegraphinFigure?illustratesthe c hangein the
control v aluesof four of the parametersthat in?uence
the process? The exponen tialdeca y ofthew aterv ol?
umeinthehangingdropisshown?Salt concentration
isavery smallvalueandincreaseswith time?Thereare
several factorsoftheexponentialdecaythatcanvary
the outputfromthe model?Inthis example?wehave
c hosena fairlylargesize drop?thatis ???l?Normally ?
a sizeb etw een? and ???l ischosen? How ev er?for
thepurposes oftesting our mo del? thisw asacceptable?
Thev olume atwhichequilibrationo ccursis usually
set at half thev olumeoftheoriginal drop?Th us?we
de?nedV
equilibration
tobe??l?Thevolume ofw ater
thatleak ed fromthe sealwas setto beequalto half
V
equilibr ation
?Thus?V
leak
issetto????l?The rateat
whichw atervap or leaks isrepresentedby thesecond
exp onent??
?
? in equation? andis normally setto?
weeks?Therate atwhich equilibrationoccursis rep?
Page 10
0
10
20
30
40
50
60
70
80
90
100
0 2040 60 80 100120140
Concentration and Solid Mass
# Days
’Protein-Conc’
’PEG-Conc’
’AmorMass’
’XtalMass’
Figure??? Appearanceofsolidphasesin ???w?vBaby
PEG???mg?ml protein????MNaCl
0
10
20
30
40
50
60
70
80
90
100
020406080100 120140
Concentrations and Solid Mass
# Days
’Protein-Conc’
’PEG-Conc’
’AmorMass’
’XtalMass’
Figure???App earance ofsolidphasesin???w?vBaby
PEG??? mg?mlprotein????MNaCl
resentedbythe?rstexp onent??
?
?inequation?and
issettobe?week forAlphaGlobulase?Byvarying
thetwoexponentsandthevolumesinthew aterde?
cayequation?we cansimulate di?erent ratesatwhich
thecrystallization process takesplace? The simulation
w asp erformedusingdiscretetime units?each ofwhich
represented? min utes?
Figure ?? depictsthe app earanceof?rst the amor?
phoussolidaround?? da ys aftersettingup the ex?
p eriment?and its subsequentgrowth?Thecrystalline
phasenucleates at appro ximately??da ysaftersetup?
Thecrystallinesolidphasegro wsattheexpenseofthe
amorphoussolidphase? Th us?wecanseetheamor?
phoussolidmassshrinkingasthecrystallinesolidmass
increasesinsize steadilystartingjustpriortoday????
Figure??depictstheoutputs fromanotherex?
perimentinitializedwithexactlythesameparame?
tersinthe?rstexperiment? witha smallchange?
Theprecipitantconcentrationisincreased by??w?v
?weigh t?v olume??We can seethedi?erence intwo
graphs???and??? fairly obviously? The satura?
tion lev elsrequiredfornucleation andgrowthofthe
crystallinephase are achiev edearlierintime?
aphaseofthegiv
e?ectsofsubtlefactorsthatcrystallization
sucasgromab noticeableexcept
der certainconditions?Th itbnecessaryto
b toelopdsthatcanp
formelofsensitivanalysisrequiredb
used
of
salts
complexit
and?commonl
y
y
ould
usedprecipitan
part
ts?Apart
w
from
W
the
can
table
also
de?nitions?
no
the
necessit
simulator
y
requires
?rst
?pa?
to
rameter
dev
de?nitions
and
for
the
describing
mo
thephase
ed
informa?
this
tion
b
for
e
eachsolidphase
to
ofthe
rigorous
protein?Th us?
c
we
need
istry
??
exp
parameters
erimen
todescrib
and
e?solidphases?namely
amorphousandcrystalline?Additionalinputsinclude
controlparametersandothergivens?Eachtable re?
quiresapproximately ?realnumbersforde?ningthe
parametersofthecorrespondingmathematical func?
tion?Thus?the totaln umberofinputs tothesimulator
isapproximately???real numbers?
Giventhecomplexity ofthe mo del? itisp ossible to
noticethee?ects ofcritical variables such asprotein
concen tration thatin?uence theoverallsolubility and
saturation ofsolidenprotein?while
in?uence
hwth?rateynoteun?
us?ecomes
e able dev automated methoer?
theleveyamodel
this ?Thiswbeoffutureork?
ewsee thetobe able
elop studydeldescribinpaper
eforattemptingp erform physicalhem?
tation testing?
V alidationofthePredictiveModelfor
CrystalGrowth
Themodeldevelopedinthispaperhasbeenevalu?
atedsubjectivelywiththesatisfactionofthedomain
expertsastheprimarycriterion?Inordertoestablish
facevalidityofthemodel?wetunedthevariouspa?
rametersofthesimulatortoproduce?classesofhypo?
theticalmacromolecules?Theseinclude?proteinsthat
crystallizeeasily?proteins thatprecipitateeasilybut
haveanarrowrangeofcrystallizationconditions?and
proteinsthattakealongtimetocrystallize?Crystallo?
grapherswerepresentedwiththeobservede?ectsfrom
themodelforeachtypeofproteinandwereaskedques?
tionsaboutobserveddi?erences inbehavior?Wede?ne
overallbeha viorastheresponseofaproteininseveral
commonlyused chemicalandphysicalenvironments?
asdescribedbytheobservedoutcomesovertime?
Weweresurprisedandhappyto?ndthattuning the
modeltoproducedi?erentresponse behaviorsturned
outtobeafairlyeasytaskonceaninitialmodel
hadbeendevelopedtosimulateAlphaGlobulase?a
Page 11
Objective?Toproduceamodelthatsim ulatesthe ?a?
vorofaproteinwithslowgrowth?rate?thatis nothing
will bevisibleforanyexperimen twithsomelevelsof
protein concen trationforabout?monthsoftime?
Method?Bysimply studyingthemodel?itbecomes
apparentthat thekeyparameterstovaryaretheex?
p onents? ?
?
and?
?
of equation ??astheydirectly af?
fect therate atwhich things move tow ardequilibrium?
Hence?wew ereable to produceresponseb ehaviorthat
accuratelymodeled thisdesiredob jective?Togivean
example?wew ereabletoproducethefollowingt ype
ofresponseforapproximately ???? experiments that
representedafactorialsearch oftheparameterspace
?withmediumgran ularity??For lo w erlevelsofprotein
concentrationbelow??mg?ml?therewerenoobserv?
ableresults ?clearresult?forov er?months? whileifwe
weretoincreasetheproteinconcentration?thenweare
likelyto seeaspread ofcrystalsappearing sometime
after?weeksorso?
Summary
Wehave relied onsubjectiveev aluationof the mo del
due totheunav ailabilityof data?It isinteresting
to notethat somecommonly adoptedheuristicsfor
searchingbasedon partialresultsfromcrystallization
trialsyield successfulcrystals forman yof theh ypo?
theticalproteins generatedby the model?Gopalakr?
ishnan??????
T able?showsa summaryofthedesirablee?ects
fromthemodel?Atickmark?
p
? indicatesfeatures
that existinourmodel?The otherfeatures arepossibly
requiredinthefuture?
FutureWork
Thecurrent modelwill needtobetestedwith solubility
dataobtained from real experimentations inthelabo?
ratoryorfromreporteddataintheliterature? Once
thisdataiscollected?thevalidity ofthemodelcan
thenbetested againstexperimen tsperformedinthe
lab oratory?A thoroughquantitativeevaluation ofthe
modelis notpossible atthis time?as itwillrequire
thecollectionofseveralthousand experimen tal results
orobserv ations?whichwouldbeverytedious?Ho w?
ever? sincethemodelhasbeenconstructedusingex?
pertknowledge?itcancertainlybeused asatutoring
to olfor novicecrystallographers inthefuture? Utiliz?
ingsuchatutoringsystemwouldprovidetremendous
?Abilityfordi?eren tphases
to existsimultaneously
p
?Abilit y toincorp orateNeeds
solubilitytableinformationWork
?Resolutionlimitbest atnucleation
p
??Finalcrystalqualityisa
function ofdi?imandsize
p
??Rate ofchangeofwatervolume
indropa?ectsthe entire process
p
Table??Summaryof desirede?ectsinmodel
savingsintermsoftime?money?e?ortandresources
spentontraining crystallographers?If themodeland
thephysicalchemistrybehinditcanpro videfairlyac?
curatepredictionsregardingtheexperiments thatare
morelikelytobeyieldcrystals?given initialsolubil?
ity information?thenit canbeusedas anintelligent
decision?makingaidforexperimentdesign?
References
Atkins?P?W??????Theelementsofphysicalchem?
istry?Oxford UniversityPress?
Boistelle?R??andAstier? J??????Crystallization
mechanisms insolution? In Geig?e?R??Ducruix?A??
Fontecilla?Camps?J? C??Feigelson? R?S?? Kern? R??
andMcPherson?A?? eds??CrystalGrowthofBiological
Macromolecules?NorthHolland?Proceedingsofthe
SecondInternationalConferenceonProteinCrystal
Growth?Bischenberg?Strasbourg?France?AFEBS
AdvancedLectureCourse?
Casey?G?A??and Wilson?W?????? JournalofCrys?
talGrowth???????
Ducruix?A??and?editors??R?G??????Crystalliza?
tion ofNucleicAcids andPr oteins?a Practic alAp?
proach?Oxford Press?
Feigelson?R?S??????Therelevanceofsmallmolecule
crystalgrowththeoriesandtechniquestothegrowth
ofbiologicalmacromolecules?Journalof Crystal
Growth ??????
Gopalakrishnan? V? ?????ParallelExp eriment Plan?
ning?Macr omole cularCrystallizationCaseStudy?
Ph?D?Dissertation?UniversityofPittsburgh?
Mandel?J??????TheStatisticalAnalysisof Experi?
mentalData?DoverPublications?Inc? NY?
Page 12
Press?
Rosenberger?F??Vekilov?P?? Muschol?M??and
B?R?Thomas? ?????Nucleation andcrystallization of
globularproteins?whatweknowandwhat ismissing?
Journalof Crystal Growth ?????
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