¾ Structural Methods

Structural Methods 10/18/2005

BCH 5205 (c) M.S.Chapman 1

10/18/2005 BCH 5505 Structural Methods 1

Structural Methods

BCH 5505 Structure & Function of Enzymes

© Michael S. Chapman, FSU, 1994-05


IntroductionEverything Superficial

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


ObjectivesLimited -- just enough to answer:

What sort of information is learned?How easy is it to get?How reliable is it?


X-ray Crystallography

Greatest source of structural information


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.


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,…




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.


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))


Diffraction -- scattering phenomenon

X-rays:Electromagnetic radn.Sinusoidal waves.Scattered:

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”)


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


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


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




Fourier Transforms (1)."Any" function can be approximated by a sum of sine waves.

Must be periodic (repeating)

Function can be reconstructed from

AmplitudesPhases (starting points)

RM Sweet © Academic Press10/18/2005 BCH 5505 Structural Methods 14

Fourier Transforms (2)Fourier coefficients = {Amplitudes, phases}

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


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


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


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?


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




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


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


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|


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|


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!


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




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:


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


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


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


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.


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




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


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


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

Torsion angles restrained > 2.8 ÅNo comments below 2.8 Å resolution


Poor starting models(Unpublishable) PoorPoor (MIR) mapmap Poor modelTypes of errors

Sometimes 2° structures connected incorrectly

Breaks in backboneConnections where should not be

Chemical sequence out of stepCarbonyls pointing in wrong directionTorsion angles incorrect

Or 3 Å with poor phases


Initial to Final ModelUse model to calculate improved mapIf atoms fit map exactly, no improvement

(FT(FT(f{x})) = f{x}To improve, must change

Move atoms consistent w/ map & chemistryMap reflects average of

Real structure – thro’ measured |F|Model – thro’ calculated phases

See how to improve model & repeat…Problem: model incorrect:

Map may or may not show how to change model10/18/2005 BCH 5505 Structural Methods 36

Map BiasMap may even look like incorrect prior model

Such is influence of phasesSeems to confirm erroneous structure

Very difficult to detectCarboxypeptidase; RuBisCO

Two ways to reduce bias1. Omit maps – the best:

Map Calculated in PiecesOmit local atoms from phase calculation

2. “2Fo-Fc” maps – the quickestSubtract ½ a map of the model, to “subtract” bias

Does not eliminate bias – care needed




Model Improvement – Refinement 1Real-space (rare, but easier to understand)

Computer method that moves atoms toImprove fit to the map by “least squares”

Min{ Uρ = Σx[ρcalc – ρobs]² }Improve agreement w/ known stereochemistry

Min{ Ug = Σn[lideal – lmodel]²+ …}Combine: Min{U = Σx[ρcalc – ρobs]²+Σn[lideal – lmodel]²+ …}

Problem: ρ depends on (poor) phasesReciprocal-space: Conventional refinement

Min{U = Σh[Fcalc – Fobs]²+Σn[lideal – lmodel]²+ …}Direct vs. diffraction amplitudesNot held up by poor phases


Refinement 2 – moving atomsGradient descent methods

Steepest Descent, Conjugate GradientMove atoms in directions that improve UCan get stuck in local minimum

Simulated annealingAtoms given kinetic energy

Repeatedly solve Newtonion equations: F = maMotions change according to interatomic forces

Can overcome potential barrierIf sufficient kinetic energy

Slowly cool (anneal) – settles in minimum

U

U


Refinement TargetsLeast squares:

Min{U = Σh[Fcalc – Fobs]²+Σn[lideal – lmodel]²+ …}Answer of least error

If errors are normally distributed

Easy to code

Maximum likelihoodP{model|observations}More complicatedSlightly better results


What can refinement do?Least squares

Bring atoms to correct positions if w/in ~ 0.5 Å.Correct torsion angles w/in 40°.

Simulated AnnealingWider convergence – change rotamers

Beyond Convergence RadiusManual rebuilding – computer modeling programs

Fitting structure into densityRefinement typically alternates automatic Refinement typically alternates automatic

procedures w/ procedures w/ ““manualmanual”” rebuilds to correct large rebuilds to correct large errorserrors


R-factors – Conventional quality indicatorR = Σh|Fcalc – Fobs|/ Σh|Fobs|

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.


R-factors – Measure Goodness of Fit

Sum of distances: Data to model“Model” is straight line

Similar to coefficient of regression




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


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


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


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


Model evaluation: a summary.First things:

Resolution; R-factors; Stereochemistry.“Global” indicators

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…




Crystallography and Reaction Mechanisms

ChallengeStructures change during reactionCrystallography gives time average

Over many hours

SolutionsSpecial Methods to take snapshotsIsolate stable intermediates


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.


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


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.


Biomolecular NMRAn Introduction


Biomolecular NMRPhysical basis

Solution NMRAnalysis of the dataAccuracy & reliabilityLogan / Cross: semester course every other Spring.




NMR Physics -- IntroductionNuclei spin

Some have magnetic moment

1H, 13C, 15N, 31PIn strong magnetic field

Spins alignParallel or anti-parallelAnti-|| slightly favored

Population differenceExcitation

Equilibrates || & anti-||Quantum ∆E = hγH0Low energyRadio-frequency RF pulse

Naturally return relaxation=> low energy state

Non-equal equilibrium~ 1st order kinetics

Emits RF energyE = hν => freq ∝ energy


Precession & frequenciesEach magnetic dipole processes about field vectorCharacteristic frequency: ω.

Emits radiation:ω (RF)

Frequencies from different groupsResonateBeat / Interfere

W/in free induction decay

Nucleus

External Field

Nucleus

Nucleus

Magneticdipole

ω


Chemical ShiftEnergies of spin states depend on:

Bonding environmentModulations in local “external field”

Frequency from FT of induction decay

Time frequency

Chemical ShiftModulations in frequencyRelative to group in “standard”environmentUnits: ppm, frequencyH in CH3 ≠ H in CH2

H in CH3 of Ala ≠ H in CH3 of Val etc..Most H of proteins would have different ω

If you could see them...1H spectroscopy common

others have similar principles


1-D NMRAncient history

Series of pulsesscanning freq., ω

After each pulsemeasure RF emission vs. time, same ω

Now: pulse w/ all ωEmitted mess: F.T. to resolve frequencies

Small proteinPeaks overlapCan not assign peaks to individual protons.


2D-NMR - Physics

2 pulses:Measure only after 2nd

But - Spins interact after 1stInteraction depends on time between pulses, or frequency (FT)Various “pulse sequences”

Dependent on ω1 & ω2 => 2DPeaks & Information spread out

Easier to resolve


2D-NMR - AnatomyDiagonal (ω1 = ω2) ≡ 1Dexpect RF emission ≡ excitation

Why off-diagonal peaks?ERF absorbed by one atom

(spin,E) transferred to neighborexcites spin of neighborneighbor relaxes =>

emission at ωRFcharacteristic of neighbor

(ωemission ≠ ωabsorption) ≡ off diagonal




Multidimensional NMRGoal: separate peaks for each 1H.If you don’t succeed at first…Add another dimension:

why stop @ 2 pulses? - “pulse sequences”Alternatively, different method to get different atomic interactions & different energy transfers


Common Protein NMR ExperimentsCOSY experiment: Correlation spectroscopy

transfer between 1H thro’ 2 or 3 bondsNOE or NOESY: Nuclear Overhauser Effect spectroscopy

non-bonded atoms, close in 3-D spaceTOCSY… new ones each day.Diagonal always the same 1D spectrum

Resolving (separating) the 1D in different waysStacking 2D spectra => 3D map


Sequential Resonance Assignment

Assume peaks separatedWhich corresponds to which 1H?One 1H excited, a few neighbors resonate

at same ω1 or ω2horizontal or vertical linejoin the dotsamino acid fingerprintsCOSY particularly useful

Peaks can not be connected if:Ambiguity from overlapped peaks“Missing peaks”

Switch to different 2-D experiment

Wutrich , Science © AAAS, 1989


Sequential resonance assignment - example

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).


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


Modeling NMR DataBuild models consistent with:

NOE distance constraintsStandard stereochemistry.

Start w/ random structureSearch for good modelMolecular dynamics

Family of structures




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


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


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!


NMR does more than solution structureBinding titrations affinities, footprints, ratesProtein dynamics rate constants

Relaxation rate depends on rates of protein motionMillisecond to picosecond regimes.

Also membrane protein structures

In membrane envirnoment


BibliographyPetsko & Ringe §§5.1-5.3(Branden & Tooze: pp 387-91)Wutrich, K. (1989) Science 243: 45-50

(Your homework!)For those w/ wetted appetites…

Methods in Enzymology 176 & 177Logan / Cross courses


Energy Calculations

Brief introduction




Quantum Calculation.Quantum theory; first principles.

Both nuclear and electronic structureCan predict changes in orbitals, covalent bonds, reactions...Applied to “small” molecules

Ab initio calculations:Approximations to quantum theory itself

Semi-empiricalSimpler functions w/o theoretical justification, but replicate essential properties

Too much calculation for proteins.


Molecular Mechanics Energy Calculation.Nuclei onlySemi-empirical Approximation.

Simplified functions quick computationLittle theoretical justification

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


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 …


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.


Macromolecular ApplicationsEnergy minimizationMolecular Dynamics


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




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


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


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

= −∇


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


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


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




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!


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.


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.


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.


Reading AssignmentsCrystallography:

Petsko & Ringe §§5.1-5.3(Branden & Tooze Chapter 18.)Methods in Enzymology: 114: Chapter 2.

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".

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¾ Structural Methods