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Later Special Topics coursesMain focus: Crystallography & NMR
Majority of High resolution informationComputational methods of prediction
Molecular Dynamics; Simulated AnnealingOther methods not at Atomic Resolution
Circular Dichroism (CD) -- % secondary structureXAFS; EPR – details of local regions – near metals Mutagenesis – test importance, location
Cheaper, easier to start than other methodsResults can be misleading – many ways to skin a catUseful in combination w/ X-ray; NMR
Now concentrate on High Resolution General Methods
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ObjectivesLimited -- just enough to answer:
What sort of information is learned?How easy is it to get?How reliable is it?
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X-ray Crystallography
Greatest source of structural information
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X-ray Crystallography
“The X-ray study of proteins is sometimes regarded as an abstruse subject comprehensible only to specialists...""Diffraction without tears" (M.F. Perutz).What crystallographers and used car dealers have in common.
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An Inauspicious Start:"In 1934 J.D. Bernal and Dorothy Hodgkin... placed
a crystal of pepsin in an X-ray beam to see if it gave a diffraction pattern.
It was an unpromising experiment, because it had already been proven that protein crystals gave no diffraction pattern. This was only to be expected because the great German chemist XXXX had shown that proteins are colloids of random structure,…
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 2
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Getting less auspicious…and that the enzymatic activity of Northrop's
crystalline pepsin did not reside in the protein, which was but the inert carrier for its real, yet to be isolated, active principle (unmentionable references!).
Besides, even if the German chemists were wrong, and a diffraction pattern were obtained, it would clearly be impossible to deduce from it structures as large and complex as proteins.
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A Lesson for Graduate StudentsContrary to all reason,Contrary to all reason,…………or perhaps because they had not or perhaps because they had not
read the literatureread the literature, , Bernal and Hodgkin discovered Bernal and Hodgkin discovered
that pepsin crystals did give a that pepsin crystals did give a diffraction pattern. It was diffraction pattern. It was made up of sharp reflections made up of sharp reflections that extended to that extended to spacingsspacings of of the order of the order of interatomicinteratomicdistances, showing... that most distances, showing... that most of the 5,000 atoms occupy of the 5,000 atoms occupy definite placesdefinite places…”…” (M.F. (M.F. PerutzPerutz))
In many directions.By sample electrons.Intensities characteristic of structure
Scattering by electronsMap locations of electrons
Electron density mapWhat atomic structure is consistent w/ map & w/ scattering pattern?
X-rays
Sample
Diffraction
Now consider X-rays scattered in one direction, making a spot
(“reflection”)
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Components of a Reflection:
• Amplitude & phase depend on every• Component / atom• Just a little bit
• Many reflections, each w/ little information• Used to determine atom positions• Like complicated simultaneous equations
X-rays
Atom 1:
Atom 2:
Scattered XScattered X--raysrays (Atom 2 cf 1):•Same wavelength (monochromatic)•Greater intensity (more electrons in bigger atom)•Starting point (phase) depends on atom position
Phase: α or φ, relative to arbitrary standard
Phase difference
Sum of sine waves is a sine wave
Am
plitude
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Direct MethodsIntensities directly atom positionsSolving simultaneous equations
Non-linear;Probabilistic, not deterministicWon Karle & Hauptman Nobel prize
RequiresData: ~error-free & high resolution
Many data points per unknown atom positionDiscrete atoms
Small molecules solved in hours on computerProteins generally solved by Indirect Methods
Calculate electron density map – difficultBuild atomic model consistent w/ the map
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Electron Density CalculationDiffraction amplitudes = FT{Electron density}
FT: Fourier transformMore later, but proof beyond this courseA mathematical operation
Electron density by computing the inverse FTThus, structure determination involves:
Measuring diffraction amplitudesUsing a computer to calculate electron densityBuilding a model consistent w/ density
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 3
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Fourier Transforms (1)."Any" function can be approximated by a sum of sine waves.
Conventional symbols: F; φ or α.FT is mathematical operation
Yielding Fourier coefficientsFrom Function
ReversibleFT{FT{f(x)}} = f(x)Or f(x) FT {F,φ}If we wish to be more specific, might say
{F,φ} is the Fourier Transform of f(x)f(x) is the Inverse (Fourier) Transform of {F,φ} Inverse transform abbreviated FT-1.Choice of which “forward” is arbitrary
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Fourier Transforms – Relevance to DiffractionThe scattered x-rays have amplitudes given by Fourier coefficients of electron density.
Can measure amplitudesIf we could also measure phases
Could compute electron density by inverse Fourier transformFit a model to the density
Phases are extremely difficult to measure, hence
The The ““phase problemphase problem””
The biggest challenge of macromolecular structure determination
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Calculation of PhasesFrom a known structure
Known electron densityPhases from Fourier transformation
Catch-22:Why would we need to calculate phases if we know the structure?
Not all daft… consider information in a map½ from amplitudes; ½ from phasesSuppose phases from approximate structureAmplitudes measured for real structureMap “average” of real & approx structures
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Bootstrapping Phases (– whence the name?)If both {F}, {φ} from approx structure
Map = approx structure – no new informationIf {F} from real structure
Map is average of approx & real structuresConfusingMight give some indication of how they differ
Can build an approx model that is closer to realityUse improved model improved phasesModel Model Phases Phases Map Map Model Model Phases Phases ……
End of structure determination: “Refinement”Where does the 1st model come from?
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Importance of Model in BootstrappingTransforming {F,φ} map {F,φ} w/o a model intermediate generates no new information
FT is mathematically reversibleDensity from a fitted model differs
Forced consistency w/ stereochemistryE.g. blobs (atoms) 1.5 Å apartSide chains in order of 1° sequence
Imposition of a priori stereochemical knowledge is the source of phase improvement
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 4
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Initial Phases by Molecular ReplacementCalculated from a related structure
E.g. use creatine kinase structure to solve arginine kinase
ChallengesOrienting & positioning the model within the crystal before it can be visualized
“Rotation” and “Translation” functionsMap may look more like the related structure
Difficult to tell where the real structure differsRequires structure of reasonably close relative
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Isomorphous ReplacementReplace a few protein atoms with ones that disproportionately affect diffraction
“Heavy” atoms – Pb, U, Pt, Hg…Impact ∝ Z√N; Z = atomic #; N = number
Each Hg ≈ 170 carbons20% of a 150 aa protein
Need four Hg’s for 20% impact on 300 aa proteinUsing difference diffraction: Derivative – native
Solve positions of (only) heavy atomsMethods like those used to solve small structures
Phases calculated from heavy atoms used to crudely approximate protein phases
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Structure Factors as VectorsFourier coefficients of diffraction known as
“Structure Factors”.Magnitude and phasePhase can be considered direction of structure factor “vector”.
Argand notation in complex space.Convenient for addingstructure factors(Note diagrams refer toone of many reflections)
ℜe
im
|F|
φ
|F2|
|F+F 2|
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Measure magnitudes: native (P) & derivative (PH)(Can not measure phases for P & PH)
Calculate magnitude & phase for heavy atomsPH should be sum of P + H
Vector sumProviding protein “isomorphous”
Unchanged by Heavy atomsTriangulate to determine 2 possible phases2nd derivative resolves ambiguityErrors huge, so often > 2 derivatives
Multiple Isomorphous Replacement (MIR)
ℜe
im
|F H|
|FPH
|
|FPH|
|FP|
|FP|
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Limitations of Isomorphous Replacement Phasing
Isomorphism: attaching a few heavy atoms without changing the protein structureLarge errors
Phasing depends on small difference between 2 diffraction patternsEach has much errorDifference has huge errorPhasing error rarely less than 60 degrees!
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MAD PhasingMultiwavelength Anomalous DiffractionRecent method – possible with synchrotron x-raysSimilar to MIRSome atoms scatter x-rays differently at different x-ray wavelengthsOne derivative (at most)Collect at several wavelengthsTriangulate wavelengths as beforeSolve structure of derivativeAdvantage: no native – don’t have to worry about isomorphism – better phases & maps
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 5
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Embarrassing question 1: Why X-rays?
Above: sum hardly depends on exact position of atom 2Right: exact position affects phase of wave 2 greatlyKey: wavelength similar to distances measuredC—C = 1.5 Å: choose similar λ.
X-rays
Atom 1:
Atom 2:
X-rays
Atom 1:
Atom 2:
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Embarrassing question 2: Why Crystals?Probability of a photon being scattered by each atom is very lowSingle molecule in beam
Experiment takes 1011 yearsNeed many molecules
Orientations different – get average structureSize etc., but a mess!Require identical orientations
Crystal!containing ~ 1016 molecules
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Question 3: So what does resolution mean?Detail that can be seen.
May vary in different parts of map
Technical definition:Based on Fourier transformLow periodicity terms add detail
Periodicity aka “d-spacing”Resolution ≡ smallest d-spacing used
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Resolution: Why it is limited1. Higher resolution diffraction weak or absent
Diffraction is Scattering from identically positioned atoms in different moleculesMolecules in approximately same position strong low resolution termsNot exactly same position:
Disorder – more than one conformationMotion
Weak high resolution terms2. Experimenter may limit
Exact map requires infinite # Fourier termsTime: may not use all measurable terms
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Resolution: Complication 1 – the phasesTo use a Fourier term, we must know its phaseRandom phase error blurred map
Like missing high resolution termsResolution is therefore a measure of the maximum detail that could be seen
With perfect phasesThus, we talk of good and bad 3 Å maps
Not all structures at a resolution are equalIn particular, MIR map may be very poor
Lots of errors in initial structure.
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Resolution & PrecisionCrystallographers claim better precision than resolution – How so?At low resolution, single blob of density may contain several atoms
Atoms could go anywhere???Chemical constraints
How many atoms expected a blobHow they are spaced
(C—C = 1.5 Å)Limits ways that they can all be fitted into blob
Precision of atoms better than resolution of blob
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 6
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Quality of maps at different resolutions
2 ÅCarbonyls clearside-chains clear
3 ÅNo carbonylsPath unambiguousPeptide ambiguous
4 ÅSide chain density poorAdditional backbone connections
α-helix β-sheet
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Importance of Chemical knowledgeChemical sequences
Early 2 Å structures, before chem. Sequences50% side chains recognized correctlyNow never solve structures before sequencesAllow path determination at 3.25 Å
StereochemistryRigid fragments
Aromatic rings; peptide planes…Fitting groups of atoms to blobs of density
Bond lengths, anglesRestrain where atoms can be placed
Allows ¼ Å precision w/ 2.5 Å data
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Information in a restrained structureA structure should conform to restraints applied
Says nothing about the restraintExample: Require C—C = 1.5 Å
C—C will be seen at ≈ 1.5 Å whether or not they really are
Bond lengths, angles restrained @ resolutions > 1.5 ÅNo comments below 1.5 Å resolution
Discrepancy between observed data and thatcalculated from modelLike standard deviation, but not the same
R < 20% indicates good structureRemaining discrepancies
Disorder not modeled – protein, solventExperimental error in dataRemaining errors in model
R > ~26% may indicate problems ifRefinement has been attemptedMay be OK - Depends on resolution etc.
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R-factors – Measure Goodness of Fit
Sum of distances: Data to model“Model” is straight line
Similar to coefficient of regression
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 8
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Improving R (Goodness of Fit)
1) Improve the model (change
the line)
2) Make model more flexible:a) Add parameters:
y = ax + c y = ax²+ bx + cb) Adding H2O, Bs etc.
c) Relaxing stereochemistry
3) Discard dataEasier to fit, but worse model
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Problem with R-factorsMeasures how well model fit to data
Not quality of modelRefinement minimizes difference between data and model
R-factor measures the same discrepancyImproved by giving freedom to model to fit data
Stereochemical flexibilityLimited # data points
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Not Just low R-factorHow many data points for each parameter?
Data points depend on inverse cube resolutionCan refine fewer parameters at low resolution
Were the stereochemical restraints too flexible?Rmsd bond lengths ~ 0.01 Å, angles 2.5°…
Tables of such parametersAre φ,ψ allowed – Ramachandran plotBest tests are of unrestrained geometries
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Cross-validated “free”-R-factorsSet aside ~ 10% data – not used in refinementOnly used to assess quality of model
Calculate Rfree against only this dataNot refined, so independent of stereochemical restraints, # data etc..Indicator of model quality.(1 to 5% Higher than conventional R-factor)Rfree < 30% means structure approx. correct
Quality may vary – need “Local” IndicatorsHow reliable is a particular amino acid?No panacea. Several methods – each has problems:
Fit to map:Inspection or Real-space R = Σx|ρcalc – ρobs|/ Σh|ρobs|Does phase calculation biased map?
Thermal (B) factors indicate flexibilityWhen a single conformation is inappropriateWhen refinement can’t find the single conformation
Users of structure - need 2B sophisticated consumer.Crystallographers tend 2B over-confident of results
like most scientists10/18/2005 BCH 5505 Structural Methods 48
Realistically, what can you learn?4.5 Å Domain structure: α, β, α/β etc..3.5 Å Trace Backbone:3.2 Å Most Side Chain locations:
Important players by considering alsoSequence conservation; expected pKOther data: mutants, chemical modification…
3.0 Å Precision ~ 0.5 Å:Enough to define H-bonds?
Error on distance measurement = √(0.52 + 0.52) = 0.7Can’t be confident of individual interactions
2.5 Å Water molecules:2.0 Å Precision ~ 0.2 Å: H-bonds etc..1.2 Å Precision ~ 0.1 Å: Geometric distortions…
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 9
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Crystallography and Reaction Mechanisms
ChallengeStructures change during reactionCrystallography gives time average
Over many hours
SolutionsSpecial Methods to take snapshotsIsolate stable intermediates
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Snap-shots w/ Laue Crystallography““LaueLaue”” diffraction collected in a few millisecondsdiffraction collected in a few milliseconds
Intense Intense polychromaticpolychromatic xx--rays at synchrotron rays at synchrotron Slow reaction to millisecond timescaleSlow reaction to millisecond timescale
Naturally slow; Mutants; Cool sampleNaturally slow; Mutants; Cool sample……Synchronized: all molecules at same stepSynchronized: all molecules at same step
““CagedCaged”” substratessubstratesActive form released by photoActive form released by photo--induced reactioninduced reaction
Stimulated by laser flashStimulated by laser flashTechnique demandingTechnique demandingFew reactions are amenable, but some important examplesFew reactions are amenable, but some important examples
Dissociation of Dissociation of carbmonoxycarbmonoxy myoglobinmyoglobin::See helices move etc..See helices move etc..
Reaction of glycogen Reaction of glycogen phosphorylasephosphorylase b:b:PhotoreleasedPhotoreleased phosphate: 3,5phosphate: 3,5--dinitrophenyl phosphate.dinitrophenyl phosphate.
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More Common “Difference” MethodsComplexes that are stable for days
transition state analogs, inhibitors, productsConventional data collection -- days
Diffuse inhibitors etc into crystalsProtein structure almost unchanged
Use native phases Quick structure determination.“Difference” map
Fourier coefficients = Protein+inhibitor -Protein_aloneShows positions of inhibitors & changes to protein
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Obstacles to crystal structures.Or Why we don't have more structures...
Crystals!Sample purity, quantity.
Heavy atom derivatives.Tend to denature.
Poor MIR phases.Getting w/in convergence radius of refinement.Uncovering model errors.
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Biomolecular NMRAn Introduction
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Biomolecular NMRPhysical basis
Solution NMRAnalysis of the dataAccuracy & reliabilityLogan / Cross: semester course every other Spring.
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 10
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NMR Physics -- IntroductionNuclei spin
Some have magnetic moment
1H, 13C, 15N, 31PIn strong magnetic field
Spins alignParallel or anti-parallelAnti-|| slightly favored
Davulcu, O., Clark, S., Chapman, M. S. & Skalicky, J. J. Main chain 1H, 13C, and 15N resonance assignments of the 42-kDa enzyme arginine kinase. J Biomol NMR 32, 178 (2005).
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Where is structural information?(We don’t need NMR for covalent connectivity.)
3º structure from NOE interactionsthrough spacedipole-dipole interaction
NOE ∝ 1/r6
Intensity => distancein principle, not practice
spin diffusion:direct interaction, or through intermediary?
also depends on correlation time, τc, varies in proteinuse r only as constraint:
If see peak, then atoms closer than 5 ÅOther types of NMR give torsion angles directly
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Modeling NMR DataBuild models consistent with:
NOE distance constraintsStandard stereochemistry.
Start w/ random structureSearch for good modelMolecular dynamics
Family of structures
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 12
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Limits & Future of NMR~ 20 kD due to peak overlap
but increasing daily, some near 100 kDnew pulse sequencesisotopic labeling:
only labeled atoms (or neighbors) excitedfewer peaksnow easier: express in enriched / depleted media
Macromol. NMR is developing: 1st protein 1985New techniquesQuality controlHealthy discussions
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Quality of NMR structuresPrecision:
Variation between models that satisfy constraintsRMSD: root mean square deviation - < 1 Å, ideally for backbone
E minimization / stereochemical constraints reduce rmsdAlso: ratio of distance constraints to atoms
want 10 to 18 per amino acidCatastophic errors: Very rare:
1 mis-assignment => many mis-assignments if not detectedcheck that complete, unambiguous assignments
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NMR vs. Crystallography
NMR Crystallogr.
Precision (positions)
Best: 0.8-1.5Å (1989)
0.1 to 0.8 Å
(distances) much better worse
Disorder, flexibility & motion
Structures, frequencies
Little information
Limitations Size Crystals!
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NMR does more than solution structureBinding titrations affinities, footprints, ratesProtein dynamics rate constants
Relaxation rate depends on rates of protein motionMillisecond to picosecond regimes.
e.g. bond potential:Not a parabolaApproximation for limited distortions only
Emin in right placero & kb are “force field parameters”, constants: Constants determined by fitting to…
Accurate ab initio calculationsPrecise experimental data
Structures, infra-red spectroscopy…
( )2
0∑ −= rrkE bb
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Empirical Energy FunctionsTotal energy is a sum of many components.
Bond length, bond angle: kθ[θ - θ0]²Torsion angles, H-bonds, electrostatics…
Kluges:Van der Waals: E1,2 = (A1,2/r12 – B1,2/r6)
Should correct torsion angles, but poor approx.Omit vdW for next nearest neighborsInclude explicit torsion angle term: |kφ| - kφcos nφ
No physical basis – ad hoc
Contention:Electrostatics: what is the dielectric?How to represent unseen solvent interaction …
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Energy calculation: Can's and Can'tsCan be used to calculate the energies of alternative conformations.Can't predict the fold of a protein: too many alternatives to calculate.Can predict, for example, the conformation of a bound substrate; rotamer of a site-directed mutant... "simple" things.As the size of molecules increase, it quickly takes too long to calculate the energy for all possible conformations.
Energy Minimization:Start with known experimental structure
assume it is an approximation to real structure
Move atoms to reduce the potential energy.
Non-linear optimizationMoving on one cycle affects other interactions
Improving torsion angle may close contact…Need many cycles
Mathematically, what do we know about Emin? The derivative is 0.We know when we are there!
∑ −atomsi
totalE
ixδδ
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 14
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Energy Minimization – a ProblemAccuracy no better than 0.5 Å
When checked vs. experimental structureCan be better if combined with X-ray or NMR experimental data, but often need computation because experiment too difficult…So, energy functions are not good enough alone
Why?Force fields are approximateUncertainty of dielectricSolvent effects – solvent often missing from model
Not protein in vacuoUse bulk, statistical or ensemble approximations
“Missing” forces
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Another limitation of Energy Minimization.Process can only reduce energyFinds local minimumCan not pass through unfavorable state to find a better one
Can’t change rotamerSmall molecules
Systematically optimize all conformersProteins – not possible – too many
Molecular DynamicsStudy MotionAlso for energy minimization
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Molecular Dynamics.Mostly used to study molecular motion(Also can be used as a tool in energy minimization.
See “Simulated Annealing” later)Add kinetic energy to atomSolve Newton’s equation of motion
∇ is directional gradientKinetic energy can be converted to potential
Like swinging pendulumOvercome a potential barrier
Find new rotamers
δδ
2
2
xi xi
tE
mi
= −∇
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Kinetic EnergyStatic structure – motion must be addedKinetic energy associated w/ temperatureOverall motion chosen to corresponds to stated TInitial atomic components have random speed, direction
Many possible starting pointsOften repeat ensemble of structures
After initial cycle, new motions from Newton’s equations
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Evolution, Sampling SpaceAs atoms move, forces changeRepeat the calculation about every 0.2 psWhen to stop?
Believe have sampled much conformational space
Perhaps nanoseconds of simulationSome approach milliseconds
When run out of computer time
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Slow cooling & Simulated AnnealingMolecular Dynamics Techniques for Energy MinimizationStart w/ High kinetic energy (3-5,000 K)Let molecule explore different conformersSlowly reduce temperature
Molecule less able to switch rotamersHopefully, most settle in best rotamer
Structural Methods 10/18/2005
BCH 5205 (c) M.S.Chapman 15
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Uses/Abuses of Molecular Dynamics.☺ X-ray and NMR structure refinement.☺ Understanding general principles of molecular motion.
Small conformational changes, substrate binding etc.. (Langevin dynamics; HRV14).
[Wild] predictions of large conformation change.Some basics still controversial (solvent, dielectric, correlation distance?)Few (no?) examples of a dynamics prediction of a large change later verified by experiment... Usually the experiment is very difficult!Doesn't stop the predictions!
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QM/MM = Quantum Mechanical / Molecular Mechanical
Can not yet model changes during an enzyme reaction.
Molecular mechanics can not describe changes to bonding / electronic structureQuantum mechanics too slow for proteins
Combine in QM/MMQuantum theory applied to handful of atomsNeighborhood: assume only nuclei important & treat w/ molecular mechanicsChallenges: Interface; Semi-empirical approxs.Attempted by a few research groups.
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CD & other spectroscopy.Circular dichroism can be used to estimate the proportions of α,β ±10%.
Not the resolution required for analysis of mechanism.
Absorption, EPR, XAFS... spectroscopies are used in special cases.
Will discuss these techniques as we encounter them.
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Mutation/Chemical Modification.Monitor function of modified proteins.Compared to crystallography/NMR.
Faster, Cheaper, Less equipmentProblem: rarely enhance function.
Many ways to reduce activity:Need to rule out a general conformational changeVery careful controls etc..
When structure unknown, ~30% mutants interpreted correctly – when structure becomes knownBest used in combination w/ known structure.
Energy Minimization/Dynamics:Brooks et al. & Karplus, "CHARMM: A Program for Macromolecular Energy, Minimization & Dynamics calculations", J. Comput. Chem. 4: 187-217 (1983).Brunger: "X-Plor Reference manual".