Quantifying Intermittent Transport in Cell Cytoplasm

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1 Quantifying Intermittent Transport in Cell Cytoplasm Ecole Normale Supérieure, Mathematics and Biology Department. Paris, France. May 19 th 2009

2 Cellular Transport Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Extra and Intracellular communication Intermittent transport: diffusion and active motion alternation Active motion along microtubules (MTs) via molecular motors

3 Intermittent Search Mechanism Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Alternation between diffusion and directed motion to a target mrna granules to synaptic targets along a dendrite. DNA-viruses to nuclear pores

4 Early steps of viral infection Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations 1-2: extracellular diffusion and membrane exploring 3: Entry 3-4: Intermittent transport: diffusion and directed motion along MTs 4: Nuclear delivery of DNA Figure: G. Seisengerger et al., Science 294, 1929 (2001).

5 Scheme Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations

6 Motivations Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Deriving drift accounting for intermittent transport Langevin description of trajectories Application to viral infection analysis: possible degradation in cytoplasm Mean Time τ e and Probability P e a virus enters a nuclear pore?

7 Langevin Description Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Left-Hand side: Intermittent Dynamics ẋ = 2Dẇ Free Particle, ẋ = V Bound Particle. Right-Hand side: Langevin Dynamics ẋ = b(x) + 2Dẇ +killing field k(x)

8 Fokker-Planck Equation Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Survival probability:p(x, y, t) = Pr{X (t) x + dx X (0) = y } Forward Fokker-Planck Equation t p = D p (p b (x)) k (x) p boundary conditions: p = 0 on Ω a (nuclear pores) and p n = 0 on Ω Ω a.

9 Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Probality P e and mean time τ e to a nuclear pore P e and τ e P e = 1 τ e = 0 0 Ω k(x) p(x, t)dxdt k(x)t p(x, t)dxdt Ω p(x, t)dxdt 0 where p(x, t) = Ω p(x, y, t)p i(y)dy P e Ω

10 Asymptotic Results Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Nuclear pores ( Ω a )= small holes Ωa Ω = ɛ 1 Asymptotic Results in ɛ P e = ln( 1 ɛ) Dπ Φ(x) 1 Ω R Ω e D ds x, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x τ e = ln ( 1 ɛ ) Dπ ln ( 1 ɛ ) Dπ RΩ Φ(x) e D dx, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x for b = Φ

11 Asymptotic Results Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Nuclear pores ( Ω a )= small holes Ωa Ω = ɛ 1 Asymptotic Results in ɛ P e = ln( 1 ɛ) Dπ Φ(x) 1 Ω R Ω e D ds x, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x τ e = ln ( 1 ɛ ) Dπ ln ( 1 ɛ ) Dπ RΩ Φ(x) e D dx, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x for b = Φ PROBLEM: b?

12 Principle Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case MFPTs from x 0 to x f are equal. In the small diffusion limit: x f x 0 b(x 0 ) = τ(x 0 ) + t m

13 Cell representation Introduction Two-dimensional radial cell with N uniformly distributed microtubules: Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Neurite cross section with N thin cylindrical MTs

14 Two-dimensional representation Principle Cell Representation Two-dimensional radial case Cylindrical neurite case In the small diffusion limit r 0 r f b(r 0 ) = r 0 ( r(r 0 ) d m ) = τ(r 0 ) + t m b(r 0 )

15 MFPT to a microtubule Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Reflecting boundary!" r virus Dynkin s system!" a binding site #!" a brownian motion R D u(r, θ) = 1 in Ω u(r, 0) = u(r, Θ) = 0, u (R, θ) r = 0. Absorbing boundary For Θ << 1 τ(r 0 ) = 1 Θ Θ 0 u(r 0, θ)dθ r 2 0 Θ 2 12D

16 Mean binding radius (1) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Heat equation D p(r, θ, t) = p (r, θ, t) in Ω t p(r, 0, t) = p(r, Θ), t = 0, p (R, θ, t) r = 0.

17 Mean binding radius (1) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Heat equation Indeed, r(r 0 ) = 1 Θ D p(r, θ, t) = p (r, θ, t) in Ω t p(r, 0, t) = p(r, Θ), t = 0, p (R, θ, t) r = 0. Θ R 0 0 rɛ(r r 0, θ 0 )dθ 0 with ɛ(r r 0, θ 0 ) = 0 j(r, t r 0, θ 0 )dt = D p 0 n (r, t r 0, θ 0 )dt.

18 Mean binding radius (2) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Exit radius distribution Dotted line: Theoretical exit radius distribution Solid line: Numerical distribution (Brownian trajectories) initial radius r0= Radius For Θ << 1 r(r 0 ) r 0 (1 + Θ2 12 )

19 Results Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Effective drift amplitude b(r 0 ) = r 0 ( r(r 0 ) d m ) = d Θ m r τ(r 0 ) + t m t m + r0 2 Θ2 12D ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2)

20 Results Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Effective drift amplitude b(r 0 ) = r 0 ( r(r 0 ) d m ) = d Θ m r τ(r 0 ) + t m t m + r0 2 Θ2 12D ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2) Steady State Distribution Solid line: Numerical Distribution (intermittent Brownian trajectories) Dotted line: theoretical distribution obtained with Langevin description Radius Quantifying (µm) Intermittent Transport in Cell Cytoplasm

21 Cylindrical neurite case Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Cross section of a neurite In the small diffusion limit b = d m t m + τ with τ 1 λ 1 = Ω ln( 1 ɛ) 2πN the MFPT to a microtubule.

22 Results(1) with the two-dimensional potential ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2)

23 Results(1) with the two-dimensional potential ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2) Probability and mean time to a nuclear pore P e τ e ( d m d m + K K k (d m + K) where K = 2k 0 δt m ln ( 1 ɛ 1 Kδ (d ) mδ + Dt m ) 12Dt m d m (d m + K) Θ2 ( 1 + δ (d ) mδ + Dt m ) 12Dt m (d m + K) Θ2. ) ( ) and α = 1 + R+δ 1 d m 24.

24 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min.

25 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min. coherent with the reported total entry time of 15min. (G. Seisengerger et al., Science 294, 1929 (2001)).

26 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min. coherent with the reported total entry time of 15min. (G. Seisengerger et al., Science 294, 1929 (2001)). without drift: τ e 15min.

27 General framework to analyze intermittent search processes Application to viral entry modelling

28 Introduction Asymptotics for structured targets Other steps of viral infection (endosome escape... ) Asymptotics for structured targets (many nuclear pores on a spherical nuclear pore... )

29 Asymptotics for structured targets Asymptotics for structured targets (pure diffusion b = 0) n disks (nuclear pores) of radius η located on a microdomain (capacitance C S : for a spherical nucleus of radius δ, C S = 4πδ) Old Asymptotics τ e = 1 + ( Ω 4Dnη ( R ) Ω k(x)dx 4Dnη Problem: lim n,nɛ 2 1 τ e = 0 ) New Asymptotics τ e = 1 + ( Ω D C ( R ) Ω k(x)dx D C where 1 C 1 C S + 1 4nη ) New asymptotics with a drift??

30 Introduction Asymptotics for structured targets The lab

31 Negative drift Introduction Asymptotics for structured targets Noise due to reflecting external membrane Steady state distribution Dashed line: Theoretical Langevin distribution Solid line: Intermittent Brownian simulations Radius (µm)

32 Limit radius Introduction Asymptotics for structured targets In cell of radius 50µm, positive drift for d m 1µm

33 Escape through a small hole (1) Asymptotics for structured targets How long it takes for a brownian particle confined to a domain Ω to escape through a small opening Ω a (ɛ = Ωa Ω << 1)? Mean escape time τ = Ω πd ln τ = Ω 4ɛD ( ) 1 ɛ (2-dimensional case), (3-dimensional case),

34 Escape through a small hole (2) Dynkin s system Asymptotics for structured targets u(x) = 1 D in Ω Neumann Function N (x, ξ) u(x) = 0 on Ω a u n (x) = 0 on Ω r = Ω Ω a. N (x, ξ) = δ(x ξ) for x, ξ Ω N 1 (x, ξ) = for x Ω, ξ Ω. n Ω

35 Escape through a small hole (3) Asymptotics for structured targets Ω and N (x, ξ) u(x) N (x, ξ)u(x)dx = Ω N (x, ξ) u(x) N (x, ξ)u(x)dx = u(ξ) 1 D thus u(ξ) 1 D Ω + N (x, ξ) u Ω a n (x)dx 1 u(x)dx Ω Ω Ω N (x, ξ)dx N (x, ξ)dx = N (x, ξ) u Ω a n (x)dx + 1 Ω Ω u(x)dx

36 Escape through a small hole (4) Asymptotics for structured targets For ξ Ω a, C 0 the constant leading order in ɛ of u(x) and g(s) = g 0 u ɛ the local expansion of on the boundary: 2 s 2 1 N (x, ξ)dx = N (s)g(s)ds + C 0 D Ω Ω a N (s) = 1 4π s + regular function, 1 D Ω N (x, ξ)dx is bounded and g 0 = (compatibility condition). Thus: Ω 2πɛD n u(x) C 0 = Ω 4ɛD

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