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Space and timeaveraging

Data collection for X-ray crystallography is performed on protein crystals thatcontain many millions of molecules and data collection takes on the order ofminutes to hours. Over this time period all molecules in the crystal are agitatedby thermal energy and explore their conformational energy landscape. Since X-ray diffraction data is averaged both across a large number of molecules andover a long period of time this corresponds to a very thorough sampling of theprotein’s energy landscape. The electron density maps we obtain from X-raycrystallography, are blurred by those protein motions and this “ blurring”contains a tremendous amount of information about protein conformationalenergy landscapes. The challenge is to extract that information.

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B= 8 π2 rmsd2

B (Å2) rms displacement (Å)

1 0.115 0.2510 0.3650 0.8100 1.14

-1σ 1σ

The simplest way of modeling this blurring of atomic positions is toapproximate them as gaussian distributions by assigning a so-called B-factor toeach atom. The width (I.e. the rmsd or sigma) of this distributions is related tothe B-factor according to the equation shown above.Modeling conformational variability with a single B-factor makes theassumption a) that the distribution is gaussian, b) that the atomic motions areindependent of one another and c) that the width of the distribution is same in alldirections. Based on what we learned in class so far, these assumptions shouldstrike you as rather unreasonable.For example, we know that the mean free path within a protein is on the order of0.1-0.2 Angstrom. So an rmsd displacement of 0.8 Ang. (I.e. a B-factor of 50),which is quite common in proteins, clearly requiresthe concerted motions of multiple sidechains. I.e. it requires an atom orsidechain to “ leave the cage” of sourounding residues. Therefore thedistribution would probably have several bumps in it and not be smooth. Also, ifatoms are chemically bonded to a neighbor it is extremely unlikely that theywould actually be able to move independently, as this would require chemicalbonds to strech by an Angstrom or two – which is clearly unreasonable.

Despite all these shortcomings, the B-factor is the most common way to modelconformational dynamics and variability in protein crystallography. The mainreason is that using a B-factor is computationally easy and requires very fewparameters.Most X-ray crystal structures are determined from data sets that barely containenough data points to determine the structure in the first place. As a result, thereis generally not enough data available to use more appropriate model for theatomic motions.

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U11 U12 U13U21 U22 U23U31 U32 U33

][ 3 Eigenvectors3 Eigenvalues

From Eigenvalues:shape of the ellipse

i.e. σ of a normal distribution

From Eigenvectors:orientation of ellipse

in space

50% is at ~ 1.2 σ

The number of reflections (I.e. data points) in a crystallographic data set isproportional to 1/r3 (where r is the resolution) of that data set. In other word avery high resolution data set extending to 0.8 Ang. contains 33 = 27 times asmany data points as a 2.4 Angstrom resolution data set. With all this dataavailable it is then possible to use more sophisticated models to describe theensemble of conformations adopted by the protein molecules in the crystal.

One popular way of doing modeling these displacements, is to describe thedisplacement not as an isotropic gaussian distribution, I.e. a distribution, whichis equally wide in all directions, but to use a model in which the width of thedistribution varies in different directions. While not perfect, this modelapproximates the true distribution of conformations more accurately.In particular the use of these so-called anisotropic B-factors allows theidentification of the preferred directions of displacement. In other words, itallows one to determine in which direction the energy landscape of an atom isflat (large sigma in this direction) and in which directions the energy landscapeis steep (small sigma).

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From: Ujj et al. Bipohys. J. 1998

Photoactive Yellow Protein

Photoactive yellow protein is one of the few proteins, in which crystals are sowell ordered that they diffract to very high resolution, so that anisotropic B-factor parameters can be determine with great confidence.

The intial light activation reaction that allows PYP to sense light, is the photon-driven cis-trans isomerization of the chromophore (center). In this isomerizationreaction occurs by a 180 degree flip of the thioester linkage that connects thechromophore to the protein backbone. In this reaction the thioester sulfur atomand oxygen atom essentially trade place.

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By analyzing the anisotropic B-factor parameters of atoms in the PYP active site,it was possible to determine that the motions of the chromophore andsurrounding residues anticipate the trajectory of the isomerization reaction evenbefore the protein is excited by light. In other words, the analysis shows that theenergy landscape is quite flat along the trajectory of the isomerization reaction(I.e. thermal motions lead to large displacements along this direction) so thatatomic motions are primed to favor the light sensing reaction.

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x

ρ

ReducedNoise Level

Better models

Better phases

Lower noise maps

Better Data

Higher “Optical” Resolution

Better electron density maps

Very high-resolution data also means that closely spaced structural states can beresolved. As a result it is often possible to identify distinct structural substates.

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ρrelative Occupancy and

relative Energy are related by the

Maxwell- Boltzmann Equation

exp

[ΔG

/RT]

e−ΔGR T[Conf. A]

[Conf. B]=

By knowing the relative occupancy, with which two conformational states areobserved in the crystal allows the calculation of the difference in free energybetween these two conformations. In other words there is a direct link betweenthe conformational energy landscape of a protein and the electron densityobserved in a crytallographic experiment.

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Here is an example of an alpha helix (found in PYP) that adopts two distinctconformations (one adopted by 75% and the other adopted by 25% of themolecules in the crystal) . The wire cages shown in blue and brown are electrondensity maps calculated to selectively show the two conformations (omit maps).The maps clearly show that the conformational variablity of this helix can beefficiently described by a model of two separate distinct minima in theconformational energy landscape, rather than by a single broad energy minimum.

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3.3o

rotation aroundpsi of Phe 76

Here is a picture of the two conformations of that helix without the electrondensity maps. The two conformations are related by a simple rotation around asingle bond axis.

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Green structure from P65 crystal form (Van Aalten et al. Protein Sci. ,2000)

Top View Side View

When PYP is crystalized in different space groups, this the major conformationof that helix in the new space group (show in green) corresponds to the minorconformation in the first space group. Apparently the different crystallizationconditions or the differing contacts in the two crystal forms have altered theconformation of this helix by shifting the relative population of the twoconformers.

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From: Pellequer, JL, Brudler, R,Getzoff ED, Curr. Biol 1999

Co-FactorFixL HemePYP PCAHERG None

PYP alt. helix conf.

Interestingly the conformational variability of the helix observed in the PYPstructure is mirrored by the conformational variability of that helix betweendifferent member of the same protein fold. PYP belongs to the family ofPAS/GAF/LOV domains. These domains are involved in sensing environmentalsignals. Many of these domains have a bound cofactor and the way thesedomains accommodate cofactors of various different sizes in an otherwisehighly conserved structure is by adjusting the very same helix that showsconformational variability in our PYP structure.

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So far I have only talked about the use of X-ray crystallography in looking atequilibrium dynamics of a protein.In other words we looked at how over time the thermal energy causes thestructure of the protein to fluctuate on its conformational energy land scape.

Another way of looking at protein dynamics with protein crystallography is bythe perturbation relaxation approach.I this method one perturbs the protein by some physical stimulus, which ineffect change the conformational energy landscape of that protein. One thenwatches how the distribution of conformational states in the protein populationchanges to reflect that new energy landscape. Since protein motions aregenerally very fast, most proteins will relax from a perturbation long before onehas had time to deliver the stimulus or before one has been able to collect a dataset that allows structure determination. What one needs then is an experimentalsystem, in which one can triggers a protein conformational change on thetimescale of nanoseconds or faster and in which one can collect acrystallographic data set with the same speed.

There are only a handful of such systems out there. And fortunately, my favoritemolecule PYP is one of them. The picture above is a diagram of theexperimental setup I used during my Ph.D. to determine the structure of atransiently populated state in PYP’s response to light activation and watched thisintermediate decay over time.

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The collection of X-ray diffraction data on the millisecond time scale requiresthe use of a special X-ray crystallographicTechnique called the “ Laue Method” . Use of this technique is technicallychallenging and requires crystals of exceptional quality. On the other hand, thediffraction patterns one obtains from Laue experiments are simply beautiful.

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Here is a snapshot of PYP during its photocycle, where red areas show regionsfrom which atoms moved and blue areas show regions too which atoms moved.

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Here is a close up view of PYP’s active site, in which the protein dark adapted(I.e. unperturbed) structure of the proteinIs shown in yellow (the chromophore is on the right) and the structural changefollowing the perturbation is shown in white.Also shown are three different kinds of electron density maps that were used todetermine the structure of the “ perturbed” state. It takes a few microseconds forthe protein from the time the stimulus (in this case a photon) is delivered and thetime the protein has responded with the structural change between from theyellow to the white structure.

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Relaxation- Return toRelaxation- Return toEquilibriumEquilibrium

t

z axisx,y plane

0

1

2

t

0

1

2

8 8

E-t/T2

t

1-e-t/T1

t

LongitudinalTransverse

Transverse always faster!

NMR spectroscopy is another method that most of us associate with thedetermination of static structures, butNMR spectroscopy is also an excellent tool to study protein dynamics, inparticular equilibrium dynamics.

Unlike X-ray crystallography, which gave information about the relativepopulation of different conformational states and the direction, along whichdisplacements involve large or small energetic penalties, NMR spectroscopyprovides information about the rate, with which structures fluctuate. In otherwords X-ray crystallography provides mostly information about the directionand amplitude of atomic motions while NMR tells us the rate with which thosemotions occur.

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One popular method used by NMR spectroscopists to look at dynamics is theanalysis of the “ dynamic exchange broadening effect” . A change in a protein’sconformation will change the local molecular environment of the atoms close tothe site of the conformational change and as a result the frequency, at which thenuclei of those atoms absorb electromagnetic radiation will also change. If thedifference in the frequencies corresponding to the two conformations is on thesame time scale as the molecular motion that induces this change in frequencies,one will observe the broadening effect.

It turns out that the time-scale, at which these effects occur happens to beobserved for proteins falls into the micro to millisecond time-scale on which alot of the interesting protein motions take place.

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fs ps ns µs ms s hr

Protein dynamics by NMR

hetero nuclear relax. HD exchangeSpin-latice relax.

Ex. broadening.

The effects of protein dynamics manifest themselves not only in terms of linebroadening, but also in the rate with which the NMR signal decays. These socalled relaxation phenomena can provide insights into protein dynamics ondifferent time-scales. Recent years have seen a boom in the number of NMRexperiments that monitor these relaxation phenomena and NMR spectroscopycan now provide information about dynamics across nearly the entire range ofmolecular motions relevant for protein function. However, while NMRspectroscopy provides us with a lot of information about the rate, with which aprotein’s structure fluctuates it tells us little about the distance or direction of thatstructural change.

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NtrC + P-NtrCKern Lab

Site of phosphorylation

NtrC is a response regulator protein in a bacterial histidine kinase signaltransduction pathway. NtrC’s biological activity is regulated by thephosphorylation of a specific residue in the sensor domain (shown above).This phosphorylation induces a large protein conformational change in one halfof the molecule that involves the partial unfolding of helices and atomicmovements by many Angstroms. This structural change then propagates to amultimerization domain (not shown) that effects DNA binding, which in turnregulates transcription of genes involved in nitrogen fixation.

Here we are only concerned with the structural change in the sensory domain(shown above).Structure determination of that sensor domain in the unphosphorylated andphosphorylated form showed one curious fact. The site of the phosphorylationis accessible to the kinase only in the phosphorylated form while it is tuckedaway in the protein interior in the un-phosphorylated form. This then presentsus with somewhat of a chicken-and-egg problem. In order to make thephosphorylation site accessible to the enzyme that will phosphorylate it, the sitehas to be phosphorylated already.

This lead to the idea that NtrC always exists in a dynamic equilibrium betweenthe active and inactive conformation, evenWhen it is not phosphorylated. In other words, one would expect that the un-phosphorylated protein transiently adopts the activated conformation for briefperiods of time, and that the kinase uses this transient state as the substrate.

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Flexibility onps-ns timescale

Flexibility onµs-ms timescale

Backbone rmsdΔ15N chemical shiftNtrC vs. P-NtrC

NtrC Mutant P-NtrC

NMR dynamics data shows that those regions of the protein that are involved inthe conformational change betweenNtrC’s two states show pronounced dynamics on the microsecond tomillisecond time scale.

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Asp 88

Using exchange broadening experiments, Kern and colleagues were able toshow that a series of mutations resulted in a shift between the structurescorresponding to the phosphorylated and un-phosphorylated state or the protein,even in the absence of any phosphorylation. The researchers were further able toshow that those same mutations affected DNA binding, where the mutations thatshifted the equilibrium to the “ phosphorylated” conformation also showedincreased affinity for DNA.

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Figure 2 Variability in structural ensembles of ubiquitin. a, Backbonetrace of 15 representative conformations obtained from a clusteringprocedure. The structures are coloured from the N terminus (red) tothe C terminus (blue) and are traced within an atomic density map 46representing the 20% amplitude isosurface of the density of atoms inthe polypeptide main chain. The r.m.s.d. values of backbone (C )atoms ( b) and side-chain atoms ( c) in ubiquitin ensembles weredetermined by dynamic-ensemble refinement, NMR X-ray diffractionand molecular dynamics (MD) simulations

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Liquid vs. Solid

Δlindemann= rmsddaverage

Liquid Δlindemann > 0.15

Solid Δlindemann < 0.15

The lindemann delta value is used as a molecular scale measure to distinguishbetween a liquid and a solid.The lindemann delta value is simply the rmsd of atomic positions divided by themean free path between atoms.

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Backbone Sidechain

Overall 0.14 0.29

Internal core (r=6Å) 0.12 0.25

Exterior of protein 0.17 >0.30

Δlindemann values from DER

Sidechains, even in the interior of the protein behave like liquids

Dynamic ensemble refinement, which combines NMR spectroscopy withMolecular dynamics simulations shows that a protein’s backbone typicallybehaves very much like a solid, while the side chains, even those burried in theprotein interior, show clear liquid-like behavior.

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DER X-ray Rapper NMR

How is liquid-like behavior possible when we know that the interior of proteinsis very tightly packed.There is a simple case study of the packing of two residues in the hydrophobiccore of the protein ubiquitin.Dynamic Ensemble refinement shows that these sidechains exchange rapidlybetween four differentConfigurations which are quite different from one another, but where each ofthese configurations shows the tight packing we come know from protein crystalstructures.