anomalous diffusion in a finite world: time-scale dependent trajectory analysis

Anomalous diffusion in a finite world: time-scale

dependent trajectory analysis

Konrad Hinsen

Centre de Biophysique Moléculaire, Orléans, Franceand

Synchrotron SOLEIL, Saint Aubin, France

13 December 2016

Konrad Hinsen (CBM/SOLEIL) Anomalous diffusion in a finite world: time-scale dependent trajectory analysis13 December 2016 1 / 21

Anomalous diffusion

Defined by the asymptotic MSD for t →∞:

W(t) = 2Dαtα, 0 < α < 2

Study of different models that have this asymptotic behavior.

Asymptotic diffusion not accessible in experiment or simulation.


Diffusion in homogeneous molecular systems

Continuous process X(t)

W(t)→ kBTm t2 for t → 0 (ballistic motion)

No trapping, no rare events

Idea:

analyze finite-length trajectories at different time scales

use fractional Brownian Motion (fBM) as an analysis tool

use Bayesian inference to get error bars on the fBM parameters

K. Hinsen & G. Kneller, J. Chem. Phys. 145, 151101 (2016)


http://dx.doi.org/10.1063/1.4965881

Trajectory sampling at different time scales

1 Choose the number of samples L

2 Pick a lag time value ∆t

3 Analyze the discrete and finite series

X(∆t)i = X(i∆t), i = 1, . . . , L

4 Vary ∆t keeping L fixed


Fractional Brownian Motion

Probability density for a Gaussian L-step process X = (X0, . . . ,XL):

P(X) =1√

(2π)L|Σ|exp

(−1

2X ·Σ−1 · X

)

Fractional Brownian Motion:

Σi,j = Dα∆tα (iα + jα + |i− j|α)

Parameters: α,Dα


Bayesian inference

Goal: infer parameter distribution P(α,Dα) from observed L-stepprocesses X.Interpretation: P(α,Dα) represents the knowledge we have aboutthe parameters after exploiting the observations.

1 Assume a prior distribution P0(α,Dα)

2 Add information from independent observations X(j), j = 1, . . . ,N:

Pj(α,Dα) =P(X(j)|α,Dα)∫

dα∫dDαP(X(j)|α,Dα)

Pj−1(α,Dα),

3 Interpret the posterior distribution PN(α,Dα)


Bayesian inference for fBM

Normalization doesn’t matter for us, so let’s be sloppy!

P0(α,Dα) =

{1 if 0 < α < 2 and 0 < Dα∆tα < Dmax

0 otherwise

Dmax: upper limit introduced for normalizability

Pj(α,Dα) =1√

|Σ(α,Dα)|exp

(−1

2X(j) ·Σ−1(α,Dα) · X(j)

)Pj−1(α,Dα)

For numerical computations, use log P instead of P.


Pros and cons

Direct analysis of trajectories - no intermediate quantities

Estimation of the parameters and their uncertainties

We can know if our data is good enough to support conclusions

No check for model quality - if fBM is a bad match for the data,we can still get nice parameter distributions

Limited to L ≤ 500 with double-precision floats


Parameter distributions

Probability distributions of two variables are difficult to analyze

What we care most about is α

Dα turns out to be sharply localized around

(2Dα∆tα)est =⟨

(X(k)j+1 − X

(k)j )2

⟩j,k

(1)

I will show mostly 1D distributions P(α).


Analysis of fBM trajectories

α = 0.6, Dα = 1, L = 100, N = 500


Analysis of perfect fBM trajectories, α = 0.6, L = 10

0 10 20 30 40 50Number of trajectories

0.0

0.5

1.0

1.5

2.0

α


0.0

0.5

1.0

1.5

2.0

α

fBM trajectories, L= 10


Analysis of perfect fBM trajectories, α = 0.6, L = 100


0.0

0.2

0.4

0.6

0.8

1.0

α


0.0

0.2

0.4

0.6

0.8

1.0

α

fBM trajectories, L= 100

With a few hundred trajectories, a maximum-entropy estimate is OK.


Imperfect fBM trajectories

ck =< ∆X(t)∆X(t + kδt)) > /(Dαδtα)

0 2 4 6 8 10

k

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

c k

fBM α= 0. 6

fBM∗ α= 0. 6

cfBM0 = 1

cfBMk = (k + 1)α − 2kα + (k − 1)α for k > 0


Imperfect fBM trajectories: MSD

100 101 102 103

τ/δt

100

101

102

<W

(τ)>/(D·δtα

)fBM α= 0. 6

fBM∗ α= 0. 6


Analysis of imperfect fBM trajectories, α = 0.6


0.0

0.2

0.4

0.6

0.8

1.0

α


0.0

0.2

0.4

0.6

0.8

1.0

α

fBM with modified short-time behavior, L= 100, s= 10

Good convergence... to a wrong α.


Multiscale Bayesian inference as an analysis tool

Bayesian inference cannot say if fBM is a good model...

... but it always yields a number α ...

... that says something about the data.

Multiscale analysis: look at α as a function of time scale

For a scale-free model (fBM): no time scale dependence.


Multiscale analysis of fBM∗, α = 0.6, L = 100

0 20 40 60 80 100

∆t/δt

0.56

0.58

0.60

0.62

0.64

0.66αML


Lipid bilayer simulation

Coarse-grained MARTINI model

2033 POPC molecules, 57952 water beads

600 ns production run

NVT ensemble, T=320 K

Simulations run by Sławomir Stachura for his thesis

Lipid center-of-mass trajectories available from Zenodo:short-time, long-time

Preprocessing:

Use only in-plane coordinates x and y

Shift initial position to 0


http://www.theses.fr/2014PA066239

http://dx.doi.org/10.5281/zenodo.61742

http://dx.doi.org/10.5281/zenodo.61743

Convergence of Bayesian inference, L = 100


0.0

0.2

0.4

0.6

0.8

1.0

α

0 500 1000 1500 2000 2500 3000 3500 4000 4500Number of trajectories

0.0

0.2

0.4

0.6

0.8

1.0

α

lipid trajectories, ∆t= 18. 00 ps


Multiscale analysis, L = 100

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

αML

α= 0. 55

ballisticMSDWDFTMEE

10-2 10-1 100 101 102 103 104

∆t [ps]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

2Dα∆tα

[nm

2]

τ= 0. 59 ps

α= 0. 55

ballisticMSDWDFTMEE

fBM behavior for 10ps < ∆t < 10ns

Ballistic behavior for ∆t < 0.1ps


Outlook

Methodology:

Use the method on different data

Physics:

Understand the mechanisms that govern diffusion on each timescale


anomalous diffusion in a finite world: time-scale dependent trajectory analysis

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