FROM CELL MEMBRANES TO ULTRACOLD GASES: CLASSICAL AND QUANTUM DIFFUSION IN INHOMOGENEOUS MEDIA Pietro Massignan ICFO - Institute of Photonic Sciences (Barcelona) In collaboration with: John Lapeyre (IDAEA & ICFO) • Miguel-Angel García-March (ICFO) • Aniello Lampo (ICFO) • Maciej Lewenstein (ICFO) • Carlo Manzo (ICFO) • Juan Torreno-Pina (ICFO) • María García-Parajo (ICFO) • Janek Wehr (Arizona U.) •
Outlook A. Classical • Anomalous diffusion in biological systems • Kinetic Ising Models B. Quantum • Quantum Kinetic Ising Models • Quantum Brownian Motion 2
Ordinary Random Walk Time-averaged mean squared displacement along a single trajectory: N − m 1 � 2 X � T-MSD( t lag = m ∆ t ) = x ( t i + m ∆ t ) − x ( t i ) N − m i =1 Ensemble-averaged mean squared displacement over J trajectories: J E-MSD( t lag = m ∆ t ) = 1 � 2 X � x j ( t i + m ∆ t ) − x j ( t i ) J j =1 An ordinary RW is • diffusive: MSD~t β , with β =1 • ergodic: T-MSD and E-MSD coincide at long times • stationary: MSD independent of the total observation time (at large times) 3
Diffusion in structured media A random walker may be strongly affected by: • repeated interactions • transient binding • clustering • crowding • multi-scale, or scale-free, disorder 4
Live cell membrane A random walker may be strongly affected by: • repeated interactions • transient binding • clustering • crowding • multi-scale, or scale-free, disorder 5
Trans-membrane receptors Single particle tracking of pathogen-recognizing receptors on live cellular membranes (M. García-Parajo’s group @ ICFO) 60 frames/s 20nm position accuracy Manzo, Torreno-Pina, PM, Lapeyre, Lewenstein & García-Parajo, Phys. Rev. X (2015) 6
Sampling of J=600 trajectories Time-averaged MSD Ensemble-averaged MSD Different scaling of T-MSD and E-MSD ➟ weak ergodicity breaking! Time&Ensemble averaged MSD: TE-MSD ∼ D TE ( T ) · t lag The TE-MSD depends on the total observation time T ➟ non-stationary! 7
Continuous-Time Random Walk -1- β with β≤ 1 (so that <t wait >= ∞ ) • CTRW: a fat-tailed distribution of waiting times ~t induces non-stationary (thus non-ergodic) subdiffusion β and TE-MSD~D TE (T)*t lag , with D TE ~T β -1 with E-MSD~t • Widely used model for transport in disordered media (initially developed for amorphous solids) Montroll & Weiss, 1965; Montroll & Scher, 1973 • However, are trapping events present in the ICFO experiment? only 5% of the trajectories contain events compatible with transient trapping and excluding these trajectories yields a very similar E-MSD exponent β • For CTRW, the long-time dynamics is R TH dominated by anomalous trapping events, so that the escape-time distribution at long times becomes independent of the trapping radius R TH 8
Strongly varying diffusivity Maps of receptor motion on the cell membrane highlight the • presence of patches with strongly varying diffusivity Many possible reasons: crossing regions of low/high • viscosity, transient binding, clustering, … Employ a likelihood-based Bayesian algorithm to detect • time-dependent changes of diffusivity within a single trajectory : 9
Theoretical model Ordinary Brownian motion with a diffusivity that varies randomly, • but it’s constant on time intervals with random duration (or patches with random sizes) power law at small D P D ( D ) ∼ D σ − 1 e − D/b Assume a distribution of diffusion coefficients • fast decay at large D mean transit time ~D - γ P τ ( τ | D ) ∼ D γ e − τ D γ /k and a conditional distribution of transit times each area has radius r~( τ D) -1/2 Three possible regimes: • (0) γ < σ : the long-time dynamics is compatible with regular Brownian motion ( β =1) (I) σ < γ < σ +1: the average transit-time diverges ➟ non-ergodic sub-diffusion ( β = σ / γ ) (II) γ > σ +1: both < τ > and <(r area ) 2 > diverge ➟ non-ergodic sub-diffusion ( β =1-1/ γ ) PM, Manzo, Torreno-Pina, García-Parajo, Lewenstein & Lapeyre, Phys. Rev. Lett. (2014) 10
Comparison with experiments Simulation of WT dynamics σ =1.16 γ =1.38 regime (I) ☟ non-ergodic subdiffusion with β = σ / γ =0.84 Instead, mutated receptors with impaired function ( Δ rep ) may be best simulated with γ < σ , i.e., they perform usual ergodic and diffusive Brownian motion Transport properties strongly linked to molecular function Manzo, Torreno-Pina, PM, Lapeyre, Lewenstein & García-Parajo, Phys. Rev. X (2015) 11
⬇ ⬆ ⬆ ⬆ ⬇ Include receptor/membrane interactions? [see Poster P.03 “Random walks in Random Environments” by M.-A. García-March] X Model the membrane as an Ising lattice: H = − J σ i = ↑ , ↓ σ i σ i +1 i Link diffusivity to local membrane state (e.g. faster diffusion on ⬆ background) Allow walker to locally modify the membrane state 12
Kinetic Ising Model for the membrane ˙ X [ w ( σ 0 → σ ) P ( σ 0 , t ) − w ( σ → σ 0 ) P ( σ , t )] Kinetic Ising Model: • P ( σ , t ) = w: transition rate σ 0 α /2: single spin-flip rate w ( σ i → − σ i ) = α h 1 − γ i Interacting membrane: 2 σ i ( σ i − 1 + σ i +1 ) • γ >0: IFM 2 γ <0: AFM Detailed balance at equilibrium: • P eq ( σ 1 , . . . , σ i , . . . , σ n ) w ( σ i → − σ i ) = P eq ( σ 1 , . . . , − σ i , . . . , σ n ) w ( − σ i → σ i ) q i ( t ) = h σ i ( t ) i P eq ( σ 1 , . . . , σ i , . . . , σ n ) ∝ exp[ β J σ i ( σ i − 1 + σ i +1 )] • r i,k ( t ) = h σ i ( t ) σ k ( t ) i 8 9 P ( σ , t ) = 1 N ) P ( σ 0 , t ) = 1 < = X X X (1 + σ 1 σ 0 1 ) . . . (1 + σ N σ 0 : 1 + σ i q i ( t ) + σ i σ k r i,k ( t ) + . . . • 2 N 2 N ; { σ 0 } i i 6 = k α − 1 ˙ Recursive system of differential equations: e.g., • q i ( t ) = − q i ( t ) + γ / 2[ q i − 1 ( t ) + q i +1 ( t )] Exact time-dependent solutions may be found for: infinite lattice, single spin fixed, (time- • delayed) spin correlations, spin systems in a magnetic field, … R. Glauber, Time-dependent statistics of the Ising model , J. Math. Phys. (1963) 13
Kinetic Ising Model for membrane+walker s: spins (=membrane) ˙ Master equation: • P ( σ , x, t ) = [ γ s L s + γ w L w ] P ( σ , x, t ) w: walker x: position of the walker The walker may act as a localized potential for the spins, or it may change the tunneling • between the neighboring spins If either the spins or the walker dynamics is fast, we may use an adiabatic approximation • � L s + γ w P ( σ , x, t ) ∼ P w ( x, t | σ ) P (eq) L = γ s L w E.g., and ( σ | x ) • γ s � γ w ! s γ s polaronic behavior (rapid formation of a dressing cloud, dragged along by the walker) • In the opposite limit instead, the walker spreads very fast, and will act as a diffuse potential • (mean-field) for the membrane 14
(Quantum) Kinetic Ising Models q Ansatz: • P ( σ , t ) = P eq ( σ ) φ ( σ , t ) ˙ X The KIM master equation may be written as H σσ 0 φ ( σ 0 , t ) φ ( σ , t ) = − • s σ 0 P eq ( σ 0 ) X w ( σ → σ 00 ) δ σσ 0 − w ( σ 0 → σ ) H σσ 0 = P eq ( σ ) σ 00 If the KIM satisfies a detailed-balanced condition, then H is a (real) symmetric matrix, so that • the KIM-ME may be seen as a Schrödinger equation in imaginary time, which converges exponentially to the thermal equilibrium solution. Diagonalization of H is then equivalent to the complete solution of the KIM-ME Classical spins may be promoted to non-commuting Pauli matrices to obtain quantum models • 1 X e − β H ( σ ) / 2 | σ i p Z N | Ψ i = Quantum states built from thermal states of classical Hamiltonians, e.g., • σ fulfill the area law in any dimension, even at criticality , and can be represented efficiently as matrix product states (MPS), or projected entangled pair states (PEPS) Augusiak, Cucchietti, Haake & Lewenstein, New J. Phys. (2010) 15
Quantum Brownian Motion A small system interacting with a large thermal bath: • H = H S + H B + H I x k : position of the k th oscillator X Simplest interaction: • H I = − κ k x k x κ k : coupling constant x: position of the system k Z t ρ S ( t ) = − 1 ME for the reduced density matrix: ˙ d s Tr B [ H I ( t ) , [ H I ( s ) , ρ ( s )]] • ~ 2 0 Born approx.: • ρ ( t ) ' ρ S ( t ) ⌦ ρ B (0) Markov approx.: the bath evolves on timescales much faster than the system, and as such • effectively retains no memory of the system’s dynamics κ 2 The spectral density contains all details of the bath-system coupling: X • k J ( ω ) ≡ δ ( ω − ω k ) 2 m k ω k k 16
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