Quantifying Intermittent Transport in Cell Cytoplasm

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

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

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

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).

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

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?

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)

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.

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

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

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?

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

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

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 )

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

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.

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.

Mean binding radius (2) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Exit radius distribution 0.040 0.035 0.030 0.025 0.020 Dotted line: Theoretical exit radius distribution Solid line: Numerical distribution (Brownian trajectories) 0.015 0.010 initial radius r0=100 0.005 0 0 10 20 30 40 50 60 70 80 90 100 Radius For Θ << 1 r(r 0 ) r 0 (1 + Θ2 12 )

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 2 0 12. 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)

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 2 0 12. 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 0.010 0.008 0.006 0.004 0.002 Solid line: Numerical Distribution (intermittent Brownian trajectories) Dotted line: theoretical distribution obtained with Langevin description 0 1 3 5 7 9 11 13 15 17 19 Radius Quantifying (µm) Intermittent Transport in Cell Cytoplasm

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.

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)

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.

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

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)).

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.

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

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

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??

Introduction Asymptotics for structured targets The lab http://www.biologie.ens.fr/bcsmcbs/ lagache@biologie.ens.fr

Negative drift Introduction Asymptotics for structured targets Noise due to reflecting external membrane Steady state distribution 0.0040 0.0035 0.0030 0.0025 0.0020 Dashed line: Theoretical Langevin distribution Solid line: Intermittent Brownian simulations 0.0015 0.0010 0.0005 0.0000 0 2 4 6 8 10 12 14 16 18 20 Radius (µm)

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

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),

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

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

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