Measures for diffusion, ergodicity, & ageingRalf Metzler, U Potsdam & Wroc law U Science & Technology

— Lanzhou, 4th August 2018 —

– Typeset by FoilTEX– 1

Microscopical observations

on the particles contained in the pollen of plants; and on the generalexistence of active molecules in organic and inorganic bodies

Rocks of all ages, including thosein which organic remains havenever been found, yielded themolecules in abundance. Theirexistence was ascertained in eachof the constituent molecules ofgranite, a fragment of the Sphinxbeing one of the specimens ex-amined.

R Brown, Edinburgh new Philosophical Journal 358-371 (1828) 2

3

Brownian motion

P (r,∆t) =1

(4πK∆t)d/2

exp

(−

r2

4K∆t

)

Einstein-Smoluchowski relation:

K =kBT

mη=

(R/NA)T

mη∆t=30 sec

J Perrin, Comptes Rendus (Paris) 146 (1908) 967: NA = 70.5× 1022 4

5

6

Ivar Nordlund: 100+ years of SPT with time series analysis

Mercury droplet in aqueous solution

I Nordlund, Z Physik (1914): NA = 5.91× 1023 7

Eugen Kappler: ultimate diffusion measurements

Obiturary by L Reimer, Physikalische Blatter Feb 1978 pp 86 8

Eugen Kappler: ultimate diffusion measurements

E Kappler, Ann d Physik (1931): NA = 60.59× 1022 ± 1% 9

Brownian motion & Kappler’s diffusion measurements

Distribution @ fixed photographic plate

〈r2(t)〉 = 2dKt

P (r, t) = (4πKt)−d/2

exp(−r2/[4Kt])

E Kappler, Ann d Physik (1931): NA = 60.59× 1022 ± 1% 10

Single molecule imaging, state-of-the-art

10–6 10–5 10–4 10–3

Time (s)

10–6 10–5 10–4 10–3

Time (s)

GFP in water

GFP in cytoplasm

Diffraction limit

10–3

10–2

10–1

iM

SD

(µ

m2)

100

10–3

10–2

10–1

100

iM

SD

(µ

m2)

GFP dimer in water

GFP dimer in cytoplasm

Diffraction limit

100 µs

0.5 µs

C di Renzo et al, Nat Comm (2014)

Cou

rtes

yY

uva

lG

arin

i

S Shao, B Xue & Y Sun, Biophys J (2018) 11

When Brownian diffusion is not Gaussian

A B

-1 0

-1

0

1

2

log

<x

2>

/σ2

log t (s)

-60 -30 0 30 60

-3

-2

-1

0

log

Gs(x

,t)

x/σ

C

1

0.0 0.5 1.0

-1

0

1

log

Gs(r

,t)

r (μm)

C

-1 0 1 2-1

0

1

2

log

<Δ

r2>

/ξ2

log t (s)

B

1

A

ξ

2a

Colloidal beads diffusing on nanotubes Nanospheres diffusing in entangled actin

Wang et al, PNAS (2009); Nature Mat (2012) 12

Heterogeneous diffusion in population of nematodes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2468

101214(a)

diffusion coefficient (mm2 s–1)

probab

ility d

ensity

funct

ion

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.10.20.30.40.50.60.70.80.91.0

cumula

tive p

robabi

lity fu

nction

0

0.1

0.2

0.3

0.4

–3 –2 –1 0radians

relativ

e freq

uency

1 2 3

Soy

bea

ncy

stn

emat

od

e&

egg

Mal

eC

.el

egan

sn

emat

od

e

S Hapca, JW Crawford & IM Young, Roy Soc Interface (2009) 13

Non-Gaussian diffusion of Dictyostelium cells

0 100 200 300 400 500 600

0

100

200

300

400

500

600

x, m

y,m

1.21 t 20 sec

3 2 1 0 1 2 3

104

103

102

101

100

x, m

Nx

1.15 t 60 sec

5 0 5

104

103

102

101

100

x, m

Nx

1.09 t 200 sec

20 10 0 10 20

104

103

102

101

100

x, m

Nx

1.06 t 600 sec

30 20 10 0 10 20 30

104

103

102

101

100

x, m

Nx

AG Cherstvy, O Nagel, C Beta & RM (2018) 14

Non-Gaussian diffusion of Dictyostelium cells

4 2 0 2 4

0.0

0.2

0.4

0.6

0.8

z x s2 3

pz

e1.93

0 0.5 1 1.5 2

100

10 1

10 2

10 3

K,m2sec

nfit 5

e2.55

0 0.5 1 1.5 2

100

10 1

10 2

10 3

K,m2sec

nfit 15

Similarity function Kβ ' exp(−c1β + c2)

AG Cherstvy, O Nagel, C Beta & RM (2018) 15

Fickian, non-Gaussian diffusion with diffusing diffusivity

B Wang, J Kuo, SC Bae & S Granick, Nat Mat (2012): 〈x2(t)〉 = 2K1t, yet P (x, t)

non-Gaussian. Superstatistical approach P (x, t) =∫∞

0G(x, t)p(D)dD

MV Chubinsky & G Slater, PRL (2014); R Jain & KL Sebastian, JPC B (2016): diffusing

diffusivity

Our minimal model for diffusing diffusivity:

x(t) =√

2D(t)ξ(t)

D(t) = y2(t)

y(t) = −τ−1y + ση(t)

0.001

0.01

0.1

1

-4 -2 0 2 4

P(x,t)

x

Sim t=0.5Sim t=0.2Sim t=0.1Eq (27) t=0.5Eq (27) t=0.2Eq (27) t=0.1

0.001

0.01

0.1

1

-20 -15 -10 -5 0 5 10 15 20

P(x,t)

x

Sim t=100Sim t=10Sim t=1IFT t=100IFT t=10IFT t=1

0

4000

8000

12000

16000

20000

0 4000 8000 12000 16000 20000

<x2 (t)>

t 2

3

4

5

6

7

8

9

0.01 0.1 1 10 100 1000 10000

kurto

sist

TheorySimulation

AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017); generalised γ(D): V Sposini, AV Chechkin, G Pagnini, F Seno & RM, NJP (2018) 16

Passive motion of submicron tracers in cells is viscoelastic

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bead

sin

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likem

icellarsolu

tion−→

Lip

idgran

ules

inlivin

gyeast

cells↓

JH Jeon, . . . L Oddershede & RM, PRL (2011); JH Jeon, N Leijnse, L Oddershede & RM, NJP (2013) 17

Superdiffusion in supercrowded Acanthamoeba castellani

inhibits myosin II motors

depolymerises microtubules

inhibits actin polymerisation

y motors cause superdiffusion

Centrosome

MicrotubulusNucleus

GranuleAc n filaments

Vesicle

Diges ve

vacuole

Contrac le

vacuoleAcanthapodium

A B

JF Reverey, J-H Jeon, H Bao, M Leippe, RM & C Selhuber-Unkel, Sci Rep (2015) 18

Non-Gaussian diffusion in viscoelastic systems

So far consensus: submicron tracer motion in cytoplasm is FBM-like, i.e., Gaussian

RNA-protein particles in E.coli & S.cerevisiae perform exponential anomalous diffusion:

5 4 3 2 1 0 1 2 3 4 510

3

10 2

10 1

100

!x"/#

"

P(!

x"/#

")

Laplace

Gaussian

1.5 1.0 0.5 0.0 0.5 1.0 1.510

2

10 1

100

101

!x"

(µm)

P(!

x")

(µm

1)

"=1 s

"=10 s

"=2 min

"=17 min

a

c

E. coli

5 4 3 2 1 0 1 2 3 4 510

3

10 2

10 1

100

!x"/#

"

P(!

x"/#

")

Laplace

Gaussian

1.5 1.0 0.5 0.0 0.5 1.0 1.510

2

10 1

100

101

!x"

(µm)

P(!

x")

(µm

1)

"=0.015 s

"=0.15 s

"=1.5 s

"=15 s

b

d

S. cerevisiae

E. coli S. cerevisiae

0 1 2 3 4 5 6 7 810

3

10 2

10 1

100

D/<D>

P(D

/<D

>)

E. coli 1 15 s

E. coli 1 15 min

S. cerevisiae 0.015 0.75 s

Exponential

10-710-610-510-410-310-210-1100

-10 -5 0 5 10

P(x,t)

x

Modelling based on grey GLE: J Slezak, RM & M Magdziarz, NJP (2018)

TJ Lampo, S Stylianidou, MP Backlund, PA Wiggins & AJ Spakowitz, BPJ (2017); N&V: RM, BPJ (2017) 19

Single lipid motion in bilayer membrane MD simulations

Liquid disordered Liquid ordered Gel phase

J-H Jeon, H Martinez-Seara Monne, M Javanainen & RM, PRL (2012) 20

Sample trajectories for the lipid & cholesterol motion

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Liquid disordered Liquid ordered Gel phase

� � �Liquid ordered/gel phases: extended anomalous diffusion

J-H Jeon, H Martinez-Seara Monne, M Javanainen & RM, PRL (2012) 21

Tempered FBM & FLE motion: sub- to normal diffusion

Consider tempered fGn:

〈ξ(t)ξ(t+ τ)〉 =

C

Γ(2H − 1)τ

2H−2e−τ/τ?

C

Γ(2H − 1)τ

2H−2

(1 +

τ

τ?

)−µ

10-2 100 102 104 106

0.2

1

5

20

t

<x2HtL>�t

10-2 10-1 100 101 102

10-2

10-1

100

101

t, ns

<x2HtL>

,nm

2

D Molina-Garcia, T Sandev, G Pagnini, AV Chechkin & RM (2018) 22

Reproducible TA MSD & antipersistent correlations

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Rattling dynamics: exptl first passage PDF y FLE motion

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D Molina-Garcia, T Sandev, G Pagnini, AV Chechkin & RM (2018) 23

Protein crowded membranes reduce effective mobility

1:50

1:75

1:100

1:150

1:200

1:INF

0

2

4

·10−7

0.10.5

1.01.01.4

2.0 0.61.2

1.52.0

2.4

3.3Diffusioncoeffi

cientof

lipids(cm

2 /s)

1:50

1:75

1:100

1:150

1:200

1:INF

0

20

40

60

10−9

0.41.52.74.610.9

40.6

2.411.513.0

28.3

40.2

59.8

Diffusioncoeffi

cientof

proteins

(cm

2 /s)

Protein crowding effects anomalous lipid diffusion

1 ns 10 ns 100 ns 1 µs 10 µs 100 µs

0.6

0.8

1

Time

α

1 ns 10 ns 100 ns 1 µs 10 µs 100 µs

0.8

0.9

1

1.1

Time

α

Left: DPPC (protein-aggregating) case. Right: DLPC protein non-aggregating case.

Blu

e:D

LP

C.

Red

:D

PP

C

:::: ::::

M Javanainen, H Hammaren, L Monticelli, JH Jeon, RM & I Vattulainen, Faraday Disc (2013) 24

Crowding in membranes increases dynamic heterogeneity

0

0.5

1

α

0

0.5

1

0.8

0.9

1

100

101

102

103

104

105

lag time ∆ (ns)

10-2

100

102

104

100

101

102

103

104

105

lag time ∆ (ns)

10-2

100

102

104

100

101

102

103

104

lag time ∆ (ns)

10-2

100

102

104

δ2(∆

)

0.7

0.8

0.9

1

0.6

0.8

1

0

0.5

1

Single NaK channel DLPC (non-aggregating) DPPC (aggregating)

↓B

lue:

lipid

s.R

ed:

protein(s)

0 100 200 300 400 500

lag time ∆ (ns)

-0.2

0

0.2

0.4

Cδt(∆

)/C

δt(0

)

DPPC (crowded membrane)

FLE theoryLipids & proteins behave quite differently

Increment correlation no longer simple FBM −→

J-H Jeon, M Javanainen, H Martinez-Seara, RM & I Vattulainen, PRX (2016) 25

Crowding in membranes: non-Gaussian lipid/protein diffusion

!""# $"%&'()*

+,-$*.

10-2

100

102

104

10-1

100

101

-log(1

-Π)

10-2

10-1

100

101

102

103

10410

-1

100

101

-log(1

-Π)

10-2

10-1

100

101

102

103

104

r2 (nm

2)

10-1

100

101

-log(1

-Π)

~r2

~r1.6

~r1.6

~r1.4~r

1.6

~r1.6

~r1.6

~r1.7

Dilute membrane: P (r, t) Gauss

Crowded membrane (δ ≈ 1.3 . . . 1.7):

P (r, t) ∝ exp

(−[ r

ctα/2

]δ)10-2 10-1 100 101 102

r2 (nm2)

10-1

100

101

-log(1

-

0 1 2 3 4 5r (nm)

-6-4-2024

log P r(r,

)

10-3 10-2 10-1 100 101

r2 (nm2)10-1

100

101

-log(1

-0 0.5 1 1.5 2

r (nm)-8-6-4-2024

log P r(r,

)

10-3 10-2 10-1 100 101 102

r2 (nm2)10-1

100

101

-log(1

-Π)

0 2 4 6r (nm)

-5

0

5

log P r(r,∆

)

2=1.691=1.56

δ2=1.58 δ2=1.28

δ1=1.39

~r2

~r2

δ2=1.58

δ1=1.93~r2

δ2=1.28δ2=1.53

δ1=1.67

1=1.83

1=1.60

2=1.26 2=1.16

J-H Jeon, M Javanainen, H Martinez-Seara, RM & I Vattulainen, PRX (2016) 26

Crowding in membranes increases dynamic heterogeneity

0 5 10 15 20 25

0.1

0.2K

α(t

)

0 5 10 15 20 25time t (µs)

0

0.1

0.2

Kα(t

)

0 0.05 0.1 0.15 0.2 0.25K

α

0

5

10

15

20

25

P(K

α)

DPPC lipids (dilute)

0 20 40 60 800

0.05

0.1

Kα(t

)

0 20 40 60 80time t (µs)

0

0.05

0.1

Kα(t

)

0 0.05 0.1 0.15 0.2K

α

0

5

10

15P

(Kα)

Diffusivity(t) for two lipids

Lipid diffusivity, dilute membrane

Lipid diffusivity, crowded membrane

J-H Jeon, M Javanainen, H Seara Monne, RM & I Vattulainen, PRX (2016) 27

Confinement in argon system shows geometric origin

a b c

d e

f g h

J-H Jeon, M Javanainen, H Seara Monne, RM & I Vattulainen, PRX (2016) 28

Geometry-induced violation of Saffman-Delbruck relation

P12K1L P21QJP P32F1C P41NQG P52W5J P61LDF P73TDP

0

0.2

0.4

0.6

0.8Dilute

D/D

0

CG2DLJ

1 3 100.030.10.3

0 5 10 15 200

0.2

0.4 Crowded

r/r0

D/D

0

D∝R−1 D∝ln R−11 3 100.01

0.030.10.3

Dilute system: Saffman-Delbruck law

D(R) ' log(1/R)

Crowded membrane & 2DLJ discs:

D(R) ' 1/R

M Javanainen, H Seara Monne, RM & I Vattulainen, JPC Lett (2017) 29

CTRW-like motion of Ka channels in plasma membraneB CA B

○

ψ(τ) ' τ−1−α scale free

δ2(∆) apparently random

δ2(∆) 6= 〈r2(∆)〉 WEB

T

AV Weigel, B Simon, MM Tamkun & D Krapf, PNAS (2011) 30

Time averaged MSD & weak ergodicity breaking (WEB)

Time averaged MSD ' ∆ is pseudo-Brownian and ageing (〈x2(t)〉 ' Kαtα):⟨

δ2(∆)⟩∼

1

N

N∑i

δ2i (∆) ∼

2dKα

Γ(1 + α)

∆

T 1−α ∴ Kα ≡〈δr2〉2τα

Amplitude distribution δ2 of trajectories (ξ ≡ δ2/〈δ2〉):

φα(ξ) ∼Γ1/α(1 + α)

αξ1+1/αL

+α

(Γ1/α(1 + α)

ξ1/α

)

φ1/2(ξ) =2

πexp

(−ξ2

π

); φ1(ξ) = δ(ξ − 1)

Confinement does not effect a plateau (〈x2(t)〉 ' const(T )):⟨δ2(∆)

⟩∼(⟨

x2⟩B−〈x〉2B

) 2 sin(πα)

(1− α)απ

(∆

T

)1−α

;1

(Kαλ1)1/α� ∆� T

0 1 2 3 4 50

0.5

1

0 1 2 3 4 50

0.5

1

ξ

α = 0.5

α = 0.75

Y He, S Burov, RM & E Barkai, PRL (2008); S Burov, RM, & E Barkai, PNAS (2010) 31

Granule subdiffusion in harmonic optical tweezer potential

!" ! ! !

!

!"

#

!"

#

!"

#

!"

#

!"

!"

!"#

$%

!"

!"#

!! "#$ !

$%&'()*+ !"#$

%&

〈r2(t)〉 ' tα 〈r2(t)〉 ' const

α ≈ 0.80..0.86

J-H Jeon, V Tejedor, S Burov, E Barkai, C Selhuber-Unkel, K Berg-Sørensen, L Oddershede & RM, PRL (2011) 32

Ageing effects in single trajectory time averages

atageing period measurement t

Ageing mean squared displacement (Λ(z) = (1 + z)α − zα)⟨δ2(∆)

⟩a

=Λα(ta/T )

Γ(1 + α)

g(∆)

T 1−α < 〈x2(t)〉a '

{tα, ta � t

tα−1a t, ta � t

Probability to make at least one step during [ta, ta + T ]: population splitting

mα(T/ta) ' (T/ta)1−α

, T � ta

10−1

100

10110

−5

10−3

10−1

mα=1

∆

δ2

10−1

100

10110

−5

10−3

10−1

mα=0.063

∆

δ2

J Schulz, E Barkai & RM, PRL (2013), PRX (2014) 33

Self-similar internal protein dynamics: 13 decades of ageing

10−5

−2−12 −14

−5 −4 −3 −2 −1 0

−12 −10

−10

−8

−6

−6

−2

2

6−4

−2

0

2

0

0

PGK domains

PGK residues

K-RAS segments

ePepN, dom. 1−2

ePepN, dom. 2−3

ePepN, dom. 2−4

ePepN, dom. 4-rest

Ref. 3

Ref. 4

∝t0.9

2 4 6 8 10 12 14 16

2

4

6

8

log

10(

c/

10−

12 s

)τ

log10(t/10−12 s)

log

10[S

(f)/

(nm

2 1

0−

12 H

z−1 )

]

log10(f/10 12 Hz)

10

12 Experiments

Experiments

MD data

100 101 102 103 104 105 106

10−4100 ps10 ns500 ns

17 µs

100 ps

100 ps

10 ns

500 ns

17 µs

10 ns

500 ns

17 µs

∝ f−1

∝ f−3.6

Supplementary Equation 9

10−3

10−2a

10−2

10−1

100

1/e

C(

; t

)

1.5

(ps)

10−1 100 101 102 103 104 105 107106

0.2

0.1

b

c d

∆

2 (

; t)

δ∆

∆

∆

∆

∆

(ps)∆

d(t)

local conformational

change

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

5051

52

53

54

55

56

57

58

5960

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141142143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167168

169

170171

172

173

174

175

176

177

178 179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236 237

238

239

240

241

242

243

© 2016 Macmillan Publishers Limited. All rights reserved

X Hu, L Hong, MD Smith, T Neusius, X Cheng & JC Smith, Nature Phys (2016); N&V RM Nature Phys (2016) 34

Power spectral density of a single Brownian trajectory

-30-20-10

0 10 20 30 40

2 4 6 8 10 12

ln(f-2)

ln S(f

)

ln f 0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

P(A(γ S

))

A(γS)

)

)

Power spectrum /w conserved slope Amplitude distribution for various dT

heory

Exp

erimen

t

D Krapf, E Marinari, RM, G Oshanin, A Squarcini & X Xu, NJP (2018); Perspective: ND Schnellbacher & US Schwarz, NJP (2018) 35

Power spectral density of a single FBM trajectory

✶ ✶! ✶!! ✶!!! ✶!✹

✶!✺

✁✶✂!

✶✂✶

✶✂✄

✶✂☎

✶✂✆

✝❙✭❍✱✁✮

✞❂✸✴✟

✞❂✷✴✸

✞❂✠✴✷

✞❂✠✴✸

✞❂✠✴✟

✞❂✼✴✽

γ =

(〈S2

T (f)〉 − 〈ST (f)〉2)1/2

〈ST (f)〉

D Krapf, N Lukat, E Marinari, RM, G Oshanin, C Selhuber-Unkel, A Squarcini, L Stadler, M Weiss & X Xu (2018) 36

Power spectral density of a single FBM trajectory

Agaroseα = 0.8

Telomeres α = 0.5

Vacuolesα = 1.3

Amoebaα = 2

0 2 4 60.0

0.5

1.0

P(A

)

A0 2 4 6

0.0

0.4

0.8

P(A

)

A0 2 4 6

0.0

0.5

1.0

1.5

2.0

P(B

)

B

S(2

) (f =

0)

time (s)

0.1 1 10 1001E-6

1E-4

0.01

1

0.01 0.1 1 100.01

1

100

1E4

1E61E8

0.1 1 10

1E-4

0.01

1

100

0.01 0.1 1 10 100

1E-3

1

1000

1E6 agarose

telomeres vacuoles amoeba

0.01 0.1 1 10

0.01

1

PS

D (µ

m2 /H

z)

f (Hz)f (Hz)f (Hz)f (Hz)

PS

D (µ

m2 /H

z)

PS

D (µ

m2 /H

z)

PS

D (µ

m2 /H

z)1E-4

Agarose Telomeres Vacuoles

D Krapf, N Lukat, E Marinari, RM, G Oshanin, C Selhuber-Unkel, A Squarcini, L Stadler, M Weiss & X Xu (2018) 37

First-past-the-post: few-encounter limit & geometry control

r0

Θ

1

θ0

0 0.2 0.4 0.6 0.8 1ω

0.0

0.5

1.0

1.5

2.0

P(

ω)

(b)10

-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

10-4

10-2

100

102

℘(t

)

t

(a)

10-3

10-2

10-1

�

T Mattos, C Mejıa-Monasterio, RM & G Oshanin, PRE (2012); A Godec & RM, PRX (2016); Sci Rep (2016) 38

Geometry control, defocusing, & finite target reactivity

10-4 10-2 100 10210-4

10-2

100

102 (a)

10-3 10-1 101 103 10510-6

10-4

10-2

100(b)

10-7 10-5 10-3 10-1 101 103

10-4

10-2

100

102

104 (a) r/R = 0.02r/R = 0.1r/R = 0.5r/R = 1

10-7 10-5 10-3 10-1 101 103 10510-6

10-4

10-2

100

102 (b) r/R = 0.02r/R = 0.1r/R = 0.5r/R = 1

10-3 10-1 101 103reaction time (a.u.)

10-4

10-2

100

102

104

reactiv

ity (a.

u.)

10-1410-1210-1010-810-610-4

10-7 10-5 10-3 10-1 101 103 105reaction time (a.u.)

0

0.2

0.4

0.6

0.8

1

distan

ce to

targe

t (a.u.

)

10-14

10-12

10-10

10-8

10-6

10-4

10-2

D Grebenkov, RM & G Oshanin, NJP (2017); PCCP (2018); E-print (2018) 39

Maximum likelihood implementation of diffusing diffusivity

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

Pro

b(M

odel|D

ata

)

∆t

BMBM+Noise

FBMFBM+Noise

DD

S Thapa, AG Cherstvy & RM (2018); model see AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017) 40

Application to mucin data

0

0.2

0.4

0.6

0.8

1

5 10 15 20

Pro

b(M

odel

|Dat

a)

Trajectory Number

BMBM+Noise

FBMFBM+Noise

DD

S Thapa, AG Cherstvy & RM (2018); model see AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017) 41

Stochasticity of fixational eye movementsM

SD

Discp

lacemen

tA

CF

CJJ Herrmann, RM & R Engbert, Sci Rep (2017) 42

Time averages & ageing in financial market time series

dX(t) = µX(t)dt+ σX(t)dW (t)

δ2d(∆) =

T−∆∫td

[X(t+ ∆)−X(t)]2 dt

T − td −∆

∼∆

T − tdX

20

(eσ2T − eσ

2td

)

log[⟨δ2d(∆, td)

⟩/⟨δ2(∆)

⟩]∼ td/T

AG Cherstvy, D Vinod, E Aghion, AV Chechkin & RM, NJP (2017) 43

Overview articles

: Single particle manipulation & tracking:C Nørregaard, RM, CM Ritter, K Berg-Sørensen & LB Oddershede,Chem Rev 117, 4342 (2017)

:: Anomalous diffusion models, WEB & ageing:RM, JH Jeon, AG Cherstvy & E Barkai, Phys Chem Chem Phys 16,24128 (2014)

::: Ageing renewal theory:JHP Schulz, E Barkai & RM, Phys Rev X 4, 011028 (2014)

:::: Anomalous diffusion in membranes:RM, JH Jeon & AG Cherstvy, Biochimica et Biophysica Acta - Biomem-branes 1858, 2451 (2016)

; Polymer translocation:V Palyulin, T Ala-Nissila & RM, Soft Matter 10, 9016 (2014)

44