Time Series Approaches to Understanding Protein Dynamics

CHEONG Siew Ann

Division of Physics and Applied PhysicsSchool of Physical and Mathematical Sciences

Email: cheongsa@ntu.edu.sgHomepage: http://www1.spms.ntu.edu.sg/~cheongsa/

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Macroscopic Thermal Physics

• Macroscopic order parameters differentiate– Solid (S)– Liquid (L)– Gas (G)

S

G

L

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Microscopic Statistical Physics

• S, L, G time series distinguishable• S, L, G phase within single time

series distinguishable

δr fluctuates about 0,δr2 = αT time-independent

diffusive trajectories,δr2 increases with time

ballistic trajectories,infrequent collisions

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From Micro to Macro

• Group statistically similar time series• Discover presence of different phases

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Dynamics of a Small Protein

• ACE-(ALA)5-NME– 5 alanine repeated units– Capped by acetyl and methylamide groups– 62 atoms in all

• Simulation in water– MU Yuguang, School of Biological Sciences, NTU

• Fold into α-helix twice during simulation

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Cross Correlations Between Time Series

• Pearson• Spearman• Digital

( )

Dow Jones US economic sector indices

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

j

jj

i

iiij

xtxxtxCσσ

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Vector Pearson Correlations

• Velocity time series– Means– Covariances

• Basis-independent square deviations

• Scaled deviations

• Vector correlations

( ) ( )tt ji vv ,

jiijC ξξrr

⋅=

ji vv ,

ji ΣΣ ,

( )[ ] ( )[ ] ( )[ ] ( )[ ]jjjT

jjiiiT

ii tttt vvvvvvvv −Σ−−Σ− −− 11 ,

( )[ ]( )[ ]jjjj

iiii

t

t

vv

vv

−Σ=

−Σ=−

−

2/1

2/1 ,

ξ

ξr

r

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Sliding Windows

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Correlation Matrix(1250, 1500)

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Hierarchical Clustering

• Complete linkage algorithm• Pairwise distance

• Determine as function of time– Number of clusters– Composition of clusters– Thresholds of clusters

( )ijij Cd −= 12

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Complete Linkage Dendrogram(1250, 1500)

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Reordered Correlation Matrix(1250, 1500)

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Effective Variables(2250, 2500)

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Effective Interactions(0, 500)

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Effective Interactions(1250, 1500)

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Effective Dynamics

TA TB

Ta

TA

TB

Ta

Tb

unfold?

unfold?

unfold?

fold?

fold? fold?

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Segmentation vs Clustering

• Time Series Clustering– Discover effective mesoscopic variables in given time window– Discover slow time evolution of effective variables by sliding time

window• Time Series Segmentation

– Discover number/type of macroscopic phases– Discover lifetimes of macroscopic phases– Discover time scales of transitions between macroscopic phases

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Nonstationarity in Time Series

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Modeling Nonstationary Time Series

• Assume non-stationary time series– (v(1), v(2), …, v(t), …, v(N))– M stationary segments– In segment m, data points drawn from (μm, Σm) Gaussian

distribution• Recursive segmentation

– One time series → two segments– Each segment → two subsegments– Iterate + optimize– Terminate

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Jensen-Shannon Divergence

• Single-segment likelihood for (v(1), v(2), …, v(N))

• Two-segment likelihood for (v(1), …, v(t), v(t+1), …, v(N))

• ML estimates• Jensen-Shannon divergence

( )[ ] ( )[ ]{ }∏=

− −Σ−−Σ

=N

i

T iiL1

11 exp

21 μμπ

vv

( ) ( )[ ] ( )[ ]{ } ( )[ ] ( )[ ]{ }∏∏+=

−

=

− −Σ−−Σ

−Σ−−Σ

=N

tiRR

TR

R

t

iLL

TL

L

iiiitL1

1

1

12 exp

21exp

21 μμ

πμμ

πvvvv

RLRL ΣΣΣ ˆ,ˆ,ˆ,ˆ,ˆ,ˆ μμμ( ) ( ) 0ln

1

2 ≥=ΔL

tLt

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Recursive Segmentation

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Start of Segmentation

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End of Segmentation

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Final Segmentations

few segments

many segments

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Fluctuations Ellipsoid and Color Map

• Velocity fluctuations in v(t) characterized by covariance matrix Σ

• Eigenvalues λ1, λ2, λ3, semi-axes of fluctuations ellipsoid• Color map for segments

– Gray = spherical– Reddish/purplish = prolate– Greenish = oblate

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

max,1

3

max,1

2

max,1

1 ,,,,λλ

λλ

λλBGR

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Temporal Distributions of Eccentricity

prolate → oblate + oblate → prolate

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Temporal Distributions of Eccentricity

global transformations: not all H’s are involved!

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Conclusions

• Dynamics of small protein from microscopic time series• Time series clustering

– Two synchronized clusters• Interaction clusters

– Effective dynamics from thresholds• Identify folding & unfolding events

– Nucleation from midpoint of protein• Time series segmentation

– Precisely identified global vs local events– Changes in fluctuations ellipsoid

• Potential to understand mechanisms– Nucleation from middle of protein

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Acknowledgments

• Time Series Clustering– Mikhail FILIPPOV (PhD)

• Time Series Segmentation– Jeremy HADIDJOJO (PAP/4)

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Thank You!