Anomalous diffusion in cells: experimental data and modelling

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1 Anomalous diffusion in cells: experimental data and modelling Hugues BERRY INRIA Lyon, France CIMPA School Mathematical models in biology and medicine H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, Mauritius, Dec Dec hugues.berry@inria.fr

2 H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

3 The ongoing revolution of super-resolution fluorescence microscopy Go below the diffraction limit (10-40 nm) to localize singlemolecules and their trajectories in living cells Das et al PNAS 2015 Rac1 (membrane) Izzedin et al elife 2014 c-myc (nucleus) Ibac et al PLoS One 2015 EGFR (membrane) H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

4 New experimental methods generate new questions Now that we can see individual molecule movements: Do they have remarkable properties / specificities beyond (partial) stochasticity? Nair D et al. J Neurosci AMPAR For instance, a hallmark of the classical Brownian motion / diffusion is the linear scaling with time of the mean-squared displacement (MSD): x 2 (t) =2Dt H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

5 In cells, the MSD is frequently sublinear with time: Cell$ Probe$ Methods$ scale$ α$ Ref$ E.#coli#(C)$ mrna$par,cles$ spt/videomic$ 1s$ $10 3 s$ 0.71$ Golding@Cox$PRL$2006$ S.#pombe#(C)# lipid$granules$ spt/videomic$ 10ms$ $1s$ 0.80$ Jeon$et$al$PRL$2011$ Min@6$(C)$ Insulin$granules$ spt/confocal$ 1@100s$ 0.76$ Tabei$et$al$PNAS#2013$ mammal$(n)$ P@TEFb$ sptpalm$ 10@100ms$ 0.60$ Izeddin$et$al$eLife$2015$ rodent$(m)$ IgE$receptor$ FRAP,$spt$ 1@100s$ 0.65$ Feder$et$al$Biophys#J$1996$ oligoden.$(m)$ MOG$(prot.)$ FCS$ 0.01@100ms$ 0.59$ Gielen$et$al$C#R#Biol#2005$ HeLa$(M)$ Gal@Transferase$ FCS$ 0.3@100$ms$ 0.75$ Weiss$et$al$Biophys#J$2003$ HEK$(M)$ K TIRFM$ 0.1@100s$ 0.80$ Weigel$et$al$PNAS$2011$ Cos@7$(M)$ Gag$(HIV@1)$ sptpalm$ 20@500$ms$ 0.60$ Manley$et$al$Nat#Meth'08$ see e.g. Höfling and Franosch Rep Prog Phys 2013 anomalous diffusion / subdiffusion H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

6 New experimental methods generate new questions Now that we can see individual molecule movements: Do they have remarkable properties / specificities beyond (partial) stochasticity? YES: subdiffusion Nair D et al. J Neurosci AMPAR What are the underlying microscopic mechanisms that are responsible for subdiffusion? When coupled with reaction, doe subdiffusion change the spatio-temporal dynamics of the biochemical reactions? H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

7 Outline 1. The Reference: Brownian Motion 2. Continuous-Time Random Walks (CTRW) as Models for subdiffusion 3. From CTRW to fractional diffusion and to renewal equation H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

8 1. THE REFERENCE : BROWNIAN MOTION H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

9 Microscopic origins of Brownian motion A random walk is the result H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

10 Continuous-space model for Brownian motion Lattice-free (overdamped) Langevin dynamics where i are i.i.d. N (0, 1) H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

11 Properties of Brownian motion R(t) = 0 in 1d: x(t + t) =x(t)+ p 2D t R 1 (t) R 2 (t) R 3 (t) so that hx(t) x(0)i = p 2Dt h i =0 The$mean$distance$is$zero$ H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

12 MSD as a hallmark of Brownian motion time (t) Ensemble Average MSD : x 2 (t) =1/N P N i=1 [x i(t) x i (0)] 2 space (x) in 1d: x(t + t) =x(t)+ p 2D t 10 2 so that D(x(t) x(0)) 2E = 2Dt 2 = 2DtVar( ) = 2Dt H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

13 2. CONTINUOUS-TIME RANDOM WALKS (CTRW) AS MODELS FOR SUBDIFFUSION H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

14 The main models for subdiffusion I. Fractional Brownian motion Brownian motion with long time correlations Motion in a viscoelastic fluid II. Brownian motion over a fractal space Randomly located immobile obstacles close to the percolation threshold : molecular crowding III. Continuous-time random walks Power-law distributed residence time: trapping and interactions (no clear microscopic mechanism for power-law) H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

15 Continuous Time Random Walks (CTRW) Metzler & Klafter, Phys Rep 2000 Introduced by Montroll & Weiss in 1965 (J. Math. Phys. 6:167) Both the time between two consecutive jumps and the jump distance are random variable: The waiting time (residence) between two jumps is a random variable τ with distribution ( ) The jump distance is a random variable ξ with distribution ( ) H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

16 Continuous Time Random Walks (CTRW) Metzler & Klafter, Phys Rep 2000 Define p(x, t) =p(x, 0) time t (master equation): Noting: apple 1 p(x, t) Z t 0 as the probability that the RW is at position x at (t 0 )dt 0 + Z t 0 ˆf(s)= Laplace transform (in time) of f(t) Z R (t t 0 ) (x x 0 )p(x x 0,t t 0 )dt 0 dx 0 so that ˆ p(k, s) = 1 s ˆ(s) p(k, 0) 1 ˆ(s) (k) H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

17 The case of finite mean residence time For illustration, assume Poissonian residence ( ) = and Gaussian lengths ( ) =(2 2 ) 1/2 e m e /m then and 1 ˆ(s) = ms +1 1 ms + O(s2 ) for s! 0 (k) =e k k2 + O(k 4 ) for k! 0 2 p(k, 0) ˆ p(k, s) s + Dk 2 with D = 2m and O(sk2 )! 0 ) sˆ p(k, s) p(k, 0) = Dk 2 ˆ p(k, s) now, inverse transformation t p(x, t) =D@ xx p(x, t) and using dhx 2 i(s) kk ˆ p(0,s) : x 2 (t) =2Dt yields Brownian motion / diffusion H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

18 The case of infinite mean residence time Now, assume power-law residence where Z 1 O ( )d!1if 0 < < 1 ( ) =A m (1+ ) then ˆ(s) 1 (ms) + O(s 2 ) for s! 0 TauberianTh. and ˆ p(k, s) = 1 ˆ(s) p(k, 0) s 1 ˆ(s) (k) p(k, 0) ) ˆ p(k, s) s(1 + K s k 2 ) with K = 2 2m and O(s k 2 )! 0 so that x 2 (t) = 2K (1 + ) t yields subdiffusion H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

19 3. FROM CTRW TO FRACTIONAL DIFFUSION AND TO RENEWAL EQUATION H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

20 The diffusion equation with CTRW ˆ p(k, s) = p(k, 0) s(1 + K s k 2 ) ) sˆ p(k, s) p(k, 0) = K s 1 k 2 ˆ p(k, s) inverse Fourrier gives s p(x, s) p(x, 0) = K s xx p(x, s) then inverse Laplace t p(x, t) =K 0 Dt xx p(x, t) with the Riemann-Liouville operator for fractional differentiation so CTRW gives subdiffusion and a fractional diffusion equation H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

21 CTRWs as a Mean-Field / Continuous limit to anomalous diffusion 1 α = α = t = t = p(x,t) 0.4 p(x,t) space x Anomalous subdiffusion space x Brownian diffusion H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

22 However, the Riemann-Liouville operator makes things complicated First, it is non-local in time i.e. non-markovian: 0Dt xx p(x, t) = 1 Z t xx p(x, t 0 ) (t t 0 ) 1 dt0 As a result, Reaction-Diffusion independence is lost, e.g.: Brownian diffusion: H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

23 Ex.: (Henry et al. PRE t A(x, t) =D e applet 0D t 1 e xx A(x, t) applea(x, t) n(x,t) n(x,t) x x H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

24 Even worse, the macroscopic description depends on microscopic details The correct macroscopic equation depends on e.g. whether reaction occurs only upon a jump or can occur between jumps or Henry et al. PRE 2006 H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

25 Modeling CTRW as an age-renewal equation with age-resetting jumps with V. Calvez, T. Lepoutre & A Mateos Define the age a of the walker as the time spent at the same position since the last jump Denote u(t, x, a) the concentration of walkers that are at position x at time t and that have been there for the last a time steps Upon a jump the age is reset to zero This defines a Markovian age-structured PDE model 8 t u(t, x, a)+@ a u(t, x, a)+ (a)u(t, x, a) =0 u(t, x, 0) = R 1 R 0 R >: (a) (x x0 )u(t, x 0,a)dxda u(0,x,a)=u 0 (x, a). H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

26 Modeling CTRW as an age-renewal equation with age-resetting jumps Relation with residence time distribution: leads to Brownian diffusion whereas, e.g. (a) = yields subdiffusion 1+a with V. Calvez, T. Lepoutre & A Mateos Reaction-Diffusion is still separable, ex. 8 t u(t, x, a)+@ a u(t, x, a)+ (a)u(t, x, a)+ku(t, x, a) =0 u(t, x, 0) = R 1 R 0 R >: (a) (x x0 )u(t, x 0,a)dxda u(0,x,a)=u 0 (x, a). However microscopic details are also an issue, ex. 8 >< >: A + A * apple t u(t, x, a)+@ a u(t, x, a)+ (a)u(t, x, a)+ku(t, x, a) R R u(t, x, a0 )da 0 =0 u(t, x, 0) = R 1 R 0 R (a) (x x0 )u(t, x 0,a)dxda u(0,x,a)=u 0 (x, a). H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec

27 Thanks Funding: for$computer$ressources$ Collaborations: Beagle Group INRIA Lyon H. Berry G. Beslon B. Caré A.S. Coquel P. Gabriel A. Lo Van H. Soula M. Vangkeosay INRIA Lyon V. Calvez T. Lepoutre A. Mateos School Maths. Univ Manchester UK S. Fedotov H. BERRY - Anomalous diffusion in cells CIMPA School Mauritius, 08 Dec