Intrinsic Mean Square Displacements in Proteins Henry R. Glyde Department of Physics and Astronomy University of Delaware, Newark, Delaware JINS-ORNL Oak Ridge, Tennessee 19 December 2013

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.MSDs widely measured with neutrons, quasi-elastic scattering 2.Observed MSD dominated by MSD of hydrogen. 3.MSD increases with increasing temperature. 3.MSD shows a “Dynamical Transition” (DT) to large displacements at ~ 220 K in hydrated proteins. displacements at ~ 220 K in hydrated proteins. 4.Large MSD is generally associated with protein function. Mean Square Displacements in Proteins:

Observed MSD in Lysozyme as a function of hydration h FWHM W= 1 μeV

To obtain intrinsic, long time MSD from simulations, the same MSD as observed with neutrons. 1.Observed MSD is instrument resolution dependent. Observed MSD increases with increased resolution. 2.Simulated MSD, increases with increasing simulation time. 3. Differing MSD observed on different instruments. Goal 1 of Talk:

Mean Square Displacements in Proteins

Mean Square Displacement in Proteins

Observed Mean Square Displacements in Proteins

Define an Intrinsic MSD in Proteins

Mean Square Displacements in Proteins Simulations of Lysozyme

Intrinsic MSD in Proteins Simulations of Lysozyme (h = 0.4), 1000 ns. Δ(Q,t) out to 10 ns

1.To obtained well defined MSDs from experiment. 2.To obtain time converged values from simulations. 3.More profoundly and interestingly to obtain “equilibrium” values of the MSD, the MSD that reflect the properties of the protein and the potential landscapes that are confining the H and setting the possible motions: ---- to obtain the MSD that would be predicted by statistical mechanics. and setting the possible motions: ---- to obtain the MSD that would be predicted by statistical mechanics.. Low T. High T Why are we interested long-time Intrinsic MSDs? Why are we interested long-time Intrinsic MSDs?

Long time Intrinsic MSD in Proteins

Intrinsic MSD in Proteins Fits of I(Q,t) and S(Q,ω) to Observed S(Q,ω =0)

To obtain intrinsic, wave vector, Q, independent MSD. 1.Observed MSD depends on Q value selected. 2.Observed MSD decreases with increasing Q. 3. Does the Q dependence arise from? 1.Gaussian approximation (neglecting higher cumulants in the ISF) 2.Dynamical heterogeneity of H in the protein, in the ISF in the ISF 3. Or is there an “intrinsic” Q dependence in the MSD? The MSD is length scale dependent. Goal 2 of Talk:

The Q dependence of the MSD

1.Simulations of Lysozyme, calculations of I(Q,t). 2.Fits of model I(Q,t) to obtain, - also λ, β in stretched exponential. 3.Compare intrinsic MSD with resolution dependent MSD and with observed MSD. 4.Explore Dynamical Transition in the intrinsic MSD Outline of Talk:

1. Two proteins, arbitrary orientation, in water; h = Simulation 1: t = 100 ns, 19 temperatures Calculate I(Q,t) out to 1 ns 3.Simulation 2: t = 1000 ns, 5 temperatures Calculate I(Q,t) out to 10 ns. 4.Fit of model I(Q,t) to simulated I(Q,t) to obtain, also (λ, β in stretched exponential). Simulations of Lysozyme

Mean Square Displacements in Proteins Simulations of Proteins (Lysozyme)

Mean Square Displacements in Proteins

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: Fit model I(Q,t) to the simulated I inc (Q,t) Fit model I(Q,t) to the simulated I inc (Q,t) Obtain, (also λ, β in stretched exponential) from fit. Obtain, (also λ, β in stretched exponential) from fit. Application of Model I(Q,t)

I(Q,t) calculated out to 1 ns Mean Square Displacement in Proteins Simulations of Lysozyme, 100 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 100 ns. I(Q,t) out to 1 ns

I(Q,t) calculated out to 10 ns Mean Square Displacement in Proteins Simulations of Lysozyme, 1000 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 1000 ns. I(Q,t) out to 10 ns

Intrinsic MSD in Proteins Simulations of Lysozyme, 1000 ns. I(Q,t) out to 10 ns

Intrinsic MSD in Proteins Simulations of Lysozyme (h = 0.4), 1000 ns. Δ(Q,t) out to 10 ns

1.Resolution broadened MSD 2.Other simulated MSD for Lysozyme. 3.Observed MSD in Lysozyme. 3.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 T D ) “time window” effect modifies the DT (increases T D ) Compare long-time, Intrinsic MSD with:

Observed Mean Square Displacements in Proteins

Intrinsic MSD and Resolution Broadened MSD Compared: identical I(Q,t) for Lysozyme

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

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

Observed MSD in Lysozyme, h = 0.4 g water/g protein FWHM W= 3.5 μeV

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

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

1.Observed MSD depends on instrument resolution FWHM. 2.Simulated MSD increase with increasing simulation time. 3. Can extract the long time, intrinsic MSD 3.Concept is to fit a model which contains long time, intrinsic as a parameter to data or finite time I(Q,t). 4.Obtain, also (λ, β in stretched exponential) as fitting parameters. Summary

1.An intrinsic can be defined and determined from simulations. The always greater than R. In lysozyme R ~ 2 R for W = 1 μeV. In lysozyme R ~ 2 R for W = 1 μeV. 2.The intrinsic MSD relative to R depends on the decay times in the protein relative to the cut off time τ ~ ħ/W set by the instrument resolution. Rapid decay times means close to R. 3. Intrinsic MSD also depends on the function used to represent C(t). A stretched exponential rather than simple exponential, means a larger intrinsic. 3.The intrinsic MSD,, shows a Dynamical Transition. A finite resolution displaces T D to a higher temperature. A finite resolution displaces T D to a higher temperature. Conclusions:

Obtain a wave vector, Q, independent MSD. Exploring beyond the Gaussian Approximation Exploring beyond the Gaussian Approximation (higher cumulants) Does the Q dependence arise from limits of the analysis? Does the Q dependence arise from limits of the analysis? i.Gaussian approximation ii.Dynamical diversity of H in the protein iii.An intrinsic Q dependence? Goal 2 of Talk:

The Q dependence of the MSD

The Full I i (Q,t) and the Gaussian approximation I iG (Q,t)

Mean Square Displacements in Proteins

The Full I i (Q,t) and the Gaussian approximation I iG (Q,t)

Compare Full I i (Q,t) and Gaussian approximation I iG (Q,t)

Compare the Full I i (Q,t) and the Gaussian approximation I iG (Q,t)

Intrinsic MSD obtained by fitting to Full I i (Q,t) (red) and to Gaussian approximation (blue)

1.The higher order cumulants (e.g. 4 th order) contribute little to the ISF I i (Q,t). the ISF I i (Q,t). 2.The fitted intrinsic MSD changes little when higher order cumulants are omitted. They remain Q dependent. Thus Q dependence does not arise from making the Gaussian approximation in the cumulant expansion of I i (Q,t). 3. Rather it appears to arise from the dynamical diversity in the Gaussian term of I i (Q,t). Conclusions:

ASN39 ALA32 LYS33 VAL109 I(Q,t) of Individual H in Lysozyme

I(Q,t) of Individual H in Lysozyme: (1) Full ISF I i (Q,t) (1) Full ISF I i (Q,t) (2) Gaussian Approx. I iG (Q,t) (2) Gaussian Approx. I iG (Q,t)

I(Q,t) of selected H: (1) Full ISF I i (Q,t) and (2) Gaussian Approx. I iG (Q,t) (1) Full ISF I i (Q,t) and (2) Gaussian Approx. I iG (Q,t)

I(Q,t) of selected H: MSD and relaxation parameter λ MSD and relaxation parameter λ

I(Q,t) of selected H, VAL109: MSD and relaxation parameter λ (1) Full ISF I i (Q,t) and (2) Gaussian Approx. I iG (Q,t) (1) Full ISF I i (Q,t) and (2) Gaussian Approx. I iG (Q,t)

Magnitude of Terms Beyond the Gaussian approximation.

The Kurtosis, γ, and the Magnitude of the Fourth Cumulant.

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

1.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 I i (Q,t). The fitted intrinsic MSD changes little the ISF I i (Q,t). The fitted intrinsic MSD changes little when higher order cumulants are omitted. The Gaussian approximation to I i (Q,t) is valid. when higher order cumulants are omitted. The Gaussian approximation to I i (Q,t) is valid. 2. Rather the Q dependence appears to arise largely from the dynamical heterogeneity in the Gaussian term of I i (Q,t). 3. There appears to be some intrinsic Q dependence of the MSD in the absence of dynamical heterogeneity. Conclusions: