Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions From normal to anomalous deterministic diffusion Part 3: Anomalous diffusion Rainer Klages Queen Mary University of London, School of Mathematical Sciences Sperlonga, 20-24 September 2010 From normal to anomalous diffusion 3 Rainer Klages 1
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Outline yesterday: From normal to anomalous deterministic diffusion: 2 normal diffusion in particle billiards and anomalous diffusion in intermittent maps note: work by T.Akimoto From normal to anomalous diffusion 3 Rainer Klages 2
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Outline yesterday: From normal to anomalous deterministic diffusion: 2 normal diffusion in particle billiards and anomalous diffusion in intermittent maps note: work by T.Akimoto today: Anomalous diffusion: 3 generalized diffusion and Langevin equations, biological cell migration and fluctuation relations From normal to anomalous diffusion 3 Rainer Klages 2
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Reminder: Intermittent map and CTRW theory 3 M subdiffusion coefficient calculated 2 from CTRW theory key: solve Montroll-Weiss equation 1 in Fourier-Laplace space, 0 w ( s ) ̺ ( k , s ) = 1 − ˜ 1 -1 0 1 2 ˆ x ˜ s λ ( k ) ˜ w ( s ) x 0 1 − ˆ -1 -2 From normal to anomalous diffusion 3 Rainer Klages 3
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Time-fractional equation for subdiffusion For the lifted PM map M ( x ) = x + ax z mod 1, the MW equation in long-time and large-space asymptotic form reads p ℓ 2 a γ s γ ˆ ̺ − s γ − 1 = − 2 Γ( 1 − γ ) γ γ k 2 ˆ ̺ , γ := 1 / ( z − 1 ) ˜ ˜ From normal to anomalous diffusion 3 Rainer Klages 4
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Time-fractional equation for subdiffusion For the lifted PM map M ( x ) = x + ax z mod 1, the MW equation in long-time and large-space asymptotic form reads p ℓ 2 a γ s γ ˆ ̺ − s γ − 1 = − 2 Γ( 1 − γ ) γ γ k 2 ˆ ̺ , γ := 1 / ( z − 1 ) ˜ ˜ LHS is the Laplace transform of the Caputo fractional derivative � ∂̺ ∂ γ ̺ γ = 1 ∂ t � t ∂ t γ := 0 dt ′ ( t − t 1 ′ ) − γ ∂̺ 0 < γ < 1 ∂ t ′ Γ( 1 − γ ) From normal to anomalous diffusion 3 Rainer Klages 4
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Time-fractional equation for subdiffusion For the lifted PM map M ( x ) = x + ax z mod 1, the MW equation in long-time and large-space asymptotic form reads p ℓ 2 a γ s γ ˆ ̺ − s γ − 1 = − 2 Γ( 1 − γ ) γ γ k 2 ˆ ̺ , γ := 1 / ( z − 1 ) ˜ ˜ LHS is the Laplace transform of the Caputo fractional derivative � ∂̺ ∂ γ ̺ γ = 1 ∂ t � t ∂ t γ := 0 dt ′ ( t − t 1 ′ ) − γ ∂̺ 0 < γ < 1 ∂ t ′ Γ( 1 − γ ) transforming the Montroll-Weiss eq. back to real space yields the time-fractional (sub)diffusion equation ∂ γ ̺ ( x , t ) ∂ 2 ̺ ( x , t ) = K Γ( 1 + α ) ∂ t γ ∂ x 2 2 From normal to anomalous diffusion 3 Rainer Klages 4
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Interlude: What is a fractional derivative? d 1 / 2 letter from Leibniz to L ’Hôpital (1695): dx 1 / 2 =? one way to proceed: we know that for integer m , n d m n ! Γ( n + 1 ) dx m x n = ( n − m )! x n − m = Γ( n − m + 1 ) x n − m ; assume that this also holds for m = 1 / 2 , n = 1 d 1 / 2 2 dx 1 / 2 x = √ π x 1 / 2 ⇒ From normal to anomalous diffusion 3 Rainer Klages 5
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Interlude: What is a fractional derivative? d 1 / 2 letter from Leibniz to L ’Hôpital (1695): dx 1 / 2 =? one way to proceed: we know that for integer m , n d m n ! Γ( n + 1 ) dx m x n = ( n − m )! x n − m = Γ( n − m + 1 ) x n − m ; assume that this also holds for m = 1 / 2 , n = 1 d 1 / 2 2 dx 1 / 2 x = √ π x 1 / 2 ⇒ fractional derivatives are defined via power law memory kernels, which yield power laws in Fourier (Laplace) space: d γ dx γ F ( x ) ↔ ( ik ) γ ˜ F ( k ) ∃ well-developed mathematical theory of fractional calculus ; see Sokolov, Klafter, Blumen, Phys. Today 2002 for a short intro From normal to anomalous diffusion 3 Rainer Klages 5
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Deterministic vs. stochastic density initial value problem for fractional diffusion equation can be solved exactly; compare with simulation results for P = ̺ n ( x ) : 0 10 2 r 1 -1 10 0 0 0.5 1 x Log P -2 10 -3 10 -20 0 20 x Gaussian and non-Gaussian envelopes (blue) reflect intermittency fine structure due to density on the unit interval r = ̺ n ( x ) ( n ≫ 1 ) (see inset) From normal to anomalous diffusion 3 Rainer Klages 6
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Escape rate theory for anomalous diffusion? recall the escape rate theory of Lecture 1 expressing the (normal) diffusion coefficient in terms of chaos quantities: � L � 2 D = lim [ λ ( R L ) − h KS ( R L )] L →∞ π Q: Can this also be worked out for the subdiffusive PM map? From normal to anomalous diffusion 3 Rainer Klages 7
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Escape rate theory for anomalous diffusion? recall the escape rate theory of Lecture 1 expressing the (normal) diffusion coefficient in terms of chaos quantities: � L � 2 D = lim [ λ ( R L ) − h KS ( R L )] L →∞ π Q: Can this also be worked out for the subdiffusive PM map? solve the previous fractional subdiffusion equation for 1 absorbing boundaries: can be done solve the Frobenius-Perron equation of the subdiffusive 2 PM map: ?? ( ∃ methods by Tasaki, Gaspard (2004)) even if step 2 possible and modes can be matched: ∃ an 3 anomalous escape rate formula ??? two big open questions... From normal to anomalous diffusion 3 Rainer Klages 7
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Motivation: biological cell migration Brownian motion 3 colloidal particles of radius 0 . 53 µ m; positions every 30 seconds, joined by straight lines (Perrin, 1913) From normal to anomalous diffusion 3 Rainer Klages 8
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Motivation: biological cell migration Brownian motion single biological cell crawling on 3 colloidal particles of radius a substrate (Dieterich, R.K. et 0 . 53 µ m; positions every 30 al., PNAS, 2008) seconds, joined by straight Brownian motion? lines (Perrin, 1913) From normal to anomalous diffusion 3 Rainer Klages 8
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Our cell types and how they migrate MDCK-F (Madin-Darby canine kidney) cells two types: wildtype ( NHE + ) and NHE-deficient ( NHE − ) movie: NHE+: t=210min, dt=3min From normal to anomalous diffusion 3 Rainer Klages 9
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Our cell types and how they migrate MDCK-F (Madin-Darby canine kidney) cells two types: wildtype ( NHE + ) and NHE-deficient ( NHE − ) movie: NHE+: t=210min, dt=3min note: the microscopic origin of cell migration is a highly complex process involving a huge number of proteins and signaling mechanisms in the cytoskeleton , which is a complicated biopolymer gel – we do not consider this here! From normal to anomalous diffusion 3 Rainer Klages 9
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Measuring cell migration From normal to anomalous diffusion 3 Rainer Klages 10
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Theoretical modeling: the Langevin equation Newton’s law for a particle of mass m and velocity v immersed in a fluid m ˙ v = F d ( t ) + F r ( t ) with total force of surrounding particles decomposed into viscous damping F d ( t ) and random kicks F r ( t ) From normal to anomalous diffusion 3 Rainer Klages 11
Outline Weakly chaotic map Anomalous cell migration Fluctuation relations Conclusions Theoretical modeling: the Langevin equation Newton’s law for a particle of mass m and velocity v immersed in a fluid m ˙ v = F d ( t ) + F r ( t ) with total force of surrounding particles decomposed into viscous damping F d ( t ) and random kicks F r ( t ) suppose F d ( t ) / m = − κ v and F r ( t ) / m = √ ζ ξ ( t ) as Gaussian white noise of strength √ ζ : v + κ v = √ ζ ξ ( t ) ˙ Langevin equation (1908) ‘Newton’s law of stochastic physics’: apply to cell migration? note: Brownian particles passively driven, whereas cells move actively by themselves! From normal to anomalous diffusion 3 Rainer Klages 11
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