Intrinsic Mean Square Intrinsic Mean Square Displacements in ProteinsDisplacements in Proteins

Henry R. Glyde

Department of Physics and Astronomy

University of Delaware,

Newark, Delaware 19716

JINS-ORNLOak Ridge, Tennessee 19 December 2013

Intrinsic Mean Square Displacements in Proteins Intrinsic Mean Square Displacements in Proteins

Collaborators:

Derya Vural University of Delaware

Liang Hong UT/ORNL Centre for Molecular Biophysics Oak Ridge National Laboratory

Oak Ridge, Tennessee

Jeremy Smith UT/ORNL Centre for Molecular Biophysics Oak Ridge National Laboratory Oak Ridge, Tennessee

1.1. MSDs widely measured with neutrons, quasi-elastic scattering MSDs widely measured with neutrons, quasi-elastic scattering

2.2. Observed MSD dominated by MSD of hydrogen.Observed MSD dominated by MSD of hydrogen.

3.3. MSD increases with increasing temperature.MSD increases with increasing temperature.

3.3. MSD shows a “Dynamical Transition” (DT) to large MSD shows a “Dynamical Transition” (DT) to large displacements at ~ 220 K in hydrated proteins.displacements at ~ 220 K in hydrated proteins.

4.4. Large MSD is generally associated with protein function.Large MSD is generally associated with protein function.

Mean Square Displacements in Proteins:Mean Square Displacements in Proteins:

Observed MSD in Lysozyme as a function of hydration h Observed MSD in Lysozyme as a function of hydration h

FWHM W= 1 μeV

To obtain intrinsic, long time MSD <rTo obtain intrinsic, long time MSD <r22> from simulations, the > from simulations, the same MSD as observed with neutrons.same MSD as observed with neutrons.

1.1. Observed MSD is instrument resolution dependent. Observed Observed MSD is instrument resolution dependent. Observed MSD increases with increased resolution.MSD increases with increased resolution.

2.2. Simulated MSD, increases with increasing simulation time.Simulated MSD, increases with increasing simulation time.

3. Differing MSD observed on different instruments.3. Differing MSD observed on different instruments.

Goal 1 of Talk:Goal 1 of Talk:

Mean Square Displacements in ProteinsMean Square Displacements in Proteins

Mean Square Displacement in ProteinsMean Square Displacement in Proteins

Observed Mean Square Displacements in ProteinsObserved Mean Square Displacements in Proteins

Define an Intrinsic MSD in ProteinsDefine an Intrinsic MSD in Proteins

Mean Square Displacements in ProteinsMean Square Displacements in Proteins

Simulations of LysozymeSimulations of Lysozyme

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme (h = 0.4), 1000 ns. Simulations of Lysozyme (h = 0.4), 1000 ns. ΔΔ(Q,t) out to 10 ns(Q,t) out to 10 ns

1.1. To obtained well defined MSDs from experiment. To obtained well defined MSDs from experiment.

2.2. To obtain time converged values from simulations.To obtain time converged values from simulations.

3.3. More profoundly and interestingly to obtain “equilibrium” More profoundly and interestingly to obtain “equilibrium” values of the MSD, the MSD that reflect the properties of the values of the MSD, the MSD that reflect the properties of the protein and the potential landscapes that are confining the Hprotein and the potential landscapes that are confining the H

and setting the possible motions: ---- to obtain the MSD that and setting the possible motions: ---- to obtain the MSD that would be predicted by statistical mechanics. would be predicted by statistical mechanics.

. Low T. Low T

. High T. High T

Why are we interested long-time Intrinsic MSDs?Why are we interested long-time Intrinsic MSDs?

Long time Intrinsic MSD in ProteinsLong time Intrinsic MSD in Proteins

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Fits of I(Q,t) and S(Q,Fits of I(Q,t) and S(Q,ωω) to Observed S(Q,) to Observed S(Q,ωω =0) =0)

To obtain intrinsic, wave vector, Q, independent MSD.To obtain intrinsic, wave vector, Q, independent MSD.

1.1. Observed MSD depends on Q value selected.Observed MSD depends on Q value selected.

2.2. Observed MSD decreases with increasing Q.Observed MSD decreases with increasing Q.

3. Does the Q dependence arise from? 3. Does the Q dependence arise from? 1.1. Gaussian approximation (neglecting higher Gaussian approximation (neglecting higher

cumulants in the ISF)cumulants in the ISF)2.2. Dynamical heterogeneity of H in the protein,Dynamical heterogeneity of H in the protein, in the ISFin the ISF

3. Or is there an “intrinsic” Q dependence in the 3. Or is there an “intrinsic” Q dependence in the MSD? The MSD is length scale dependent.MSD? The MSD is length scale dependent.

Goal 2 of Talk:Goal 2 of Talk:

The Q dependence of the MSD The Q dependence of the MSD

The Q dependence of the MSD The Q dependence of the MSD

1.1. Simulations of Lysozyme, calculations of I(Q,t).Simulations of Lysozyme, calculations of I(Q,t).

2.2. Fits of model I(Q,t) to obtain <rFits of model I(Q,t) to obtain <r22> , > , - also - also λλ, , ββ in stretched exponential. in stretched exponential.

3.3. Compare intrinsic MSD with resolution dependent MSD and Compare intrinsic MSD with resolution dependent MSD and with observed MSD. with observed MSD.

4.4. Explore Dynamical Transition in the intrinsic MSDExplore Dynamical Transition in the intrinsic MSD

Outline of Talk:Outline of Talk:

1. Two proteins, arbitrary orientation, in water; h = 0.4.1. Two proteins, arbitrary orientation, in water; h = 0.4.

2.2. Simulation 1: t = 100 ns, 19 temperaturesSimulation 1: t = 100 ns, 19 temperaturesCalculate I(Q,t) out to 1 nsCalculate I(Q,t) out to 1 ns

3.3. Simulation 2: t = 1000 ns, 5 temperatures Simulation 2: t = 1000 ns, 5 temperatures Calculate I(Q,t) out to 10 ns.Calculate I(Q,t) out to 10 ns.

4.4. Fit of model I(Q,t) to simulated I(Q,t) to obtain <rFit of model I(Q,t) to simulated I(Q,t) to obtain <r22>, >, also (also (λλ, , ββ in stretched exponential). in stretched exponential).

Simulations of LysozymeSimulations of Lysozyme

Mean Square Displacements in ProteinsMean Square Displacements in Proteins

Simulations of Proteins (Lysozyme)Simulations of Proteins (Lysozyme)

Mean Square Displacements in ProteinsMean Square Displacements in Proteins

1. Experiment, measure:1. Experiment, measure:

Use model I(Q,t) to calculate S(Q,0) and fit to experiment.Use model I(Q,t) to calculate S(Q,0) and fit to experiment.

2. Simulation, calculate:2. Simulation, calculate:

Fit model I(Q,t) to the simulated IFit model I(Q,t) to the simulated Iincinc(Q,t) (Q,t)

Obtain <rObtain <r22>, (also >, (also λλ, , ββ in stretched exponential) from fit. in stretched exponential) from fit.

Application of Model I(Q,t)Application of Model I(Q,t)

I(Q,t) calculated out to 1 ns

Mean Square Displacement in ProteinsMean Square Displacement in ProteinsSimulations of Lysozyme, 100 nsSimulations of Lysozyme, 100 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 nsSimulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 nsSimulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 nsSimulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 nsSimulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

I(Q,t) calculated out to 10 ns

Mean Square Displacement in ProteinsMean Square Displacement in ProteinsSimulations of Lysozyme, 1000 nsSimulations of Lysozyme, 1000 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 1000 ns. I(Q,t) out to 10 nsSimulations of Lysozyme, 1000 ns. I(Q,t) out to 10 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme, 1000 ns. I(Q,t) out to 10 nsSimulations of Lysozyme, 1000 ns. I(Q,t) out to 10 ns

Intrinsic MSD in ProteinsIntrinsic MSD in Proteins

Simulations of Lysozyme (h = 0.4), 1000 ns. Simulations of Lysozyme (h = 0.4), 1000 ns. ΔΔ(Q,t) out to 10 ns(Q,t) out to 10 ns

1.1. Resolution broadened MSD Resolution broadened MSD

2.2. Other simulated MSD for Lysozyme.Other simulated MSD for Lysozyme.

3.3. Observed MSD in Lysozyme.Observed MSD in Lysozyme.

3.3. Intrinsic MSD shows a “Dynamical Transition” (DT). Intrinsic MSD shows a “Dynamical Transition” (DT). Thus DT an intrinsic property, not a “time window” effect.Thus DT an intrinsic property, not a “time window” effect. “ “time window” effect modifies the DT (increases Ttime window” effect modifies the DT (increases TDD))

Compare long-time, Intrinsic MSD with:Compare long-time, Intrinsic MSD with:

Observed Mean Square Displacements in ProteinsObserved Mean Square Displacements in Proteins

Intrinsic MSD and Resolution Broadened MSD Intrinsic MSD and Resolution Broadened MSD

Compared: identical I(Q,t) for LysozymeCompared: identical I(Q,t) for Lysozyme

Intrinsic and Resolution Broadened MSD Compared Intrinsic and Resolution Broadened MSD Compared Simulations of Lysozyme, Roh et al. for h = 0.43 Simulations of Lysozyme, Roh et al. for h = 0.43

Present Intrinsic MSD (Lysozyme (h = 0.4) ComparedPresent Intrinsic MSD (Lysozyme (h = 0.4) Comparedwith Experiment (Lysozyme, variable h, W = 1 microeV) with Experiment (Lysozyme, variable h, W = 1 microeV)

Observed MSD in Lysozyme, h = 0.4 g water/g proteinObserved MSD in Lysozyme, h = 0.4 g water/g protein

FWHM W= 3.5 μeV

Intrinsic MSD shows a Dynamical Transition Intrinsic MSD shows a Dynamical Transition Resolution Broadening moves TResolution Broadening moves TDD to higher T to higher T

Intrinsic MSD shows a Dynamical Transition Intrinsic MSD shows a Dynamical Transition Resolution Broadening moves TResolution Broadening moves TDD to higher T to higher T

1.1.Observed MSD depends on instrument resolution FWHM.Observed MSD depends on instrument resolution FWHM.

2.2.Simulated MSD increase with increasing simulation time.Simulated MSD increase with increasing simulation time.

3. Can extract the long time, intrinsic MSD3. Can extract the long time, intrinsic MSD

3.3. Concept is to fit a model which contains long time, intrinsic Concept is to fit a model which contains long time, intrinsic as a parameter to data or finite time I(Q,t).as a parameter to data or finite time I(Q,t).

4.4.Obtain <rObtain <r22> , also (> , also (λλ, , ββ in stretched exponential) as fitting in stretched exponential) as fitting parameters.parameters.

SummarySummary

1.1. An intrinsic <rAn intrinsic <r22> can be defined and determined from > can be defined and determined from simulations. The <rsimulations. The <r22> always greater than <r> always greater than <r22>> R R. .

In lysozyme <rIn lysozyme <r22>> R R ~ 2 <r~ 2 <r22>> R R for W = 1 for W = 1 μμeV.eV.

1.1. The intrinsic MSD <rThe intrinsic MSD <r22> relative to <r> relative to <r22>> R R depends on the decay depends on the decay

times in the protein relative to the cut off time times in the protein relative to the cut off time ττ ~ ħ/W set by the ~ ħ/W set by the instrument resolution. Rapid decay times means <rinstrument resolution. Rapid decay times means <r22>> close to close to

<r<r22>> R R ..

2.2. Intrinsic MSD also depends on the function used to represent Intrinsic MSD also depends on the function used to represent C(t). A stretched exponential rather than simple exponential, C(t). A stretched exponential rather than simple exponential, means a larger intrinsic <rmeans a larger intrinsic <r22>> . .

1.1. The intrinsic MSD, <rThe intrinsic MSD, <r22>, shows a Dynamical Transition. >, shows a Dynamical Transition. A finite resolution displaces TA finite resolution displaces TD D to a higher temperature. to a higher temperature.

Conclusions:Conclusions:

Obtain a wave vector, Q, independent MSD.Obtain a wave vector, Q, independent MSD.

Exploring beyond the Gaussian Approximation Exploring beyond the Gaussian Approximation (higher cumulants)(higher cumulants)

Does the Q dependence arise from limits of the analysis? Does the Q dependence arise from limits of the analysis? i.i. Gaussian approximationGaussian approximationii.ii. Dynamical diversity of H in the proteinDynamical diversity of H in the proteiniii.iii. An intrinsic Q dependence?An intrinsic Q dependence?

Goal 2 of Talk:Goal 2 of Talk:

The Q dependence of the MSD The Q dependence of the MSD

The Full IThe Full Iii(Q,t) and the Gaussian (Q,t) and the Gaussian

approximation Iapproximation IiGiG(Q,t)(Q,t)

Mean Square Displacements in ProteinsMean Square Displacements in Proteins

The Full IThe Full Iii(Q,t) and the Gaussian (Q,t) and the Gaussian

approximation Iapproximation IiGiG(Q,t)(Q,t)

Compare Full ICompare Full Iii(Q,t) and Gaussian (Q,t) and Gaussian

approximation Iapproximation IiGiG(Q,t)(Q,t)

Compare the Full ICompare the Full I ii(Q,t) and the Gaussian (Q,t) and the Gaussian

approximation Iapproximation IiGiG(Q,t)(Q,t)

Intrinsic MSD obtained by fitting to Full IIntrinsic MSD obtained by fitting to Full I ii(Q,t) (Q,t)

(red) and to Gaussian approximation (blue) (red) and to Gaussian approximation (blue)

1.1. The higher order cumulants (e.g. 4 th order) contribute little toThe higher order cumulants (e.g. 4 th order) contribute little to the ISF Ithe ISF Iii(Q,t). (Q,t).

2.2. The fitted intrinsic MSD <rThe fitted intrinsic MSD <r22> changes little when higher order > changes little when higher order cumulants are omitted. They remain Q dependent. Thus Q cumulants are omitted. They remain Q dependent. Thus Q dependence does not arise from making the Gaussian dependence does not arise from making the Gaussian approximation in the cumulant expansion of Iapproximation in the cumulant expansion of Iii(Q,t).(Q,t).

3.3. Rather it appears to arise from the dynamical diversity in the Rather it appears to arise from the dynamical diversity in the Gaussian term of IGaussian term of Iii(Q,t) . (Q,t) .

Conclusions:Conclusions:

ASN39ASN39 ALA32 ALA32 LYS33 LYS33 VAL109VAL109

I(Q,t) of Individual H in Lysozyme I(Q,t) of Individual H in Lysozyme

I(Q,t) of Individual H in Lysozyme: I(Q,t) of Individual H in Lysozyme:

(1) Full ISF I(1) Full ISF Iii(Q,t)(Q,t)

(2) Gaussian Approx. I(2) Gaussian Approx. I iGiG(Q,t)(Q,t)

I(Q,t) of selected H: I(Q,t) of selected H:

(1) Full ISF I(1) Full ISF Iii(Q,t) and (2) Gaussian Approx. I(Q,t) and (2) Gaussian Approx. I iGiG(Q,t)(Q,t)

I(Q,t) of selected H: I(Q,t) of selected H:

MSD <rMSD <r22> and relaxation parameter > and relaxation parameter λλ

I(Q,t) of selected H, VAL109: I(Q,t) of selected H, VAL109:

MSD <rMSD <r22> and relaxation parameter > and relaxation parameter λλ (1) Full ISF I(1) Full ISF Iii(Q,t) and (2) Gaussian Approx. I(Q,t) and (2) Gaussian Approx. I iGiG(Q,t)(Q,t)

Magnitude of Terms Beyond the Gaussian Magnitude of Terms Beyond the Gaussian

approximation.approximation.

The Kurtosis, The Kurtosis, γγ, and the Magnitude of the , and the Magnitude of the

Fourth Cumulant. Fourth Cumulant.

Beyond the Gaussian approximationBeyond the Gaussian approximationThe Kurtosis, (4 th order cumulant) The Kurtosis, (4 th order cumulant)

1.1. The higher order cumulants (e.g. 4 th order) arising from The higher order cumulants (e.g. 4 th order) arising from non-Gaussian motional distributions can be neglected in non-Gaussian motional distributions can be neglected in the ISF Ithe ISF Iii(Q,t). The fitted intrinsic MSD <r(Q,t). The fitted intrinsic MSD <r22> changes little> changes little

when higher order cumulants are omitted. The Gaussian when higher order cumulants are omitted. The Gaussian approximation to Iapproximation to Iii(Q,t) is valid.(Q,t) is valid.

2.2. Rather the Q dependence appears to arise largely from the Rather the Q dependence appears to arise largely from the dynamical heterogeneity in the Gaussian term of Idynamical heterogeneity in the Gaussian term of Iii(Q,t) . (Q,t) .

3. There appears to be some intrinsic Q dependence of the MSD in 3. There appears to be some intrinsic Q dependence of the MSD in the absence of dynamical heterogeneity. the absence of dynamical heterogeneity.

Conclusions:Conclusions: