Narrow Escape Time General introduction

D. Holcman

ENS Paris

Lecture notes

Overview

-General motivation: Search of a target:

1. Nucleus

2. Synaptic transmission

3. Motion of receptor on two-dimensional

membrane

4. Amplification process in phototransduction

-Mathematical formulation

-Coarse graining in simulations

Cells

Diffusion in dendritic spines

Diffusion in dendritic spines

Synaptic transmission

Synapses Pre-

Post-

What determines the synaptic response

What determines the synaptic response

( ) ( ) ( , ) ( , , ) , linear approximation

Conductivity of type k-receptor

post k k

k

k

I t t N x t p k x t dx V

-Depends on molecular interactions

-can change by diffusion

What determines the synaptic response

( ) ( ) ( , ) ( , , ) , linear approximation

Conductivity of type k-receptor

( , ) # k-receptors at position

( , , ) Pr{Receptor k at opens at time t}

post k k

k

k

k

I t t N x t p k x t dx V

N x t x

p k x t x

-Depends on molecular interactions

-can change by diffusion

-Depends on the cleft geometry and

intrinsic biophysical properties.

Conclusion: system at the limit between continuous and discreet

receptor

Glutamate trajectory

Binding to a receptor

Exit trajectory Released vesicle

Rs~ 100s of nm

PSD

h=cleft height

~ 10s of nm

Modeling the synaptic cleft

Density of extra-PSD receptors:

Few 10s.

Density of PSD receptors transporter

RPSD

dr

dg

3000 neurotransmitters

Stochastic simulations of the

synaptic current

D. Fresche et al PloS One 2011

Trafficking on cellular membrane

Organization of receptor at the PSD

M. Kennedy Science 2000

Few numbers of receptors shape the synaptic response

PSD

Receptor trafficking from SPT

-Chen, Science 2000

-D. Borgoff, Science. 2002

-Newpher, Neuron 2008

Local neuronal membrane

organization

Kusumi, 2005

Q: How the dendrite heterogeneity affects receptor trafficking and synaptic transmission ?

Application effect of crowding on

diffusion

• J. Eisinger, J. Flores and W.P. Petersen, Biophys. J. 49, 987 (1986).

• H.X. Zhou R. and Zwanzig, J. Chem. Phys. 94, 6147 (1991).

• A. Kusumi, Y. Sako and M. Yamamoto, Biophys. J. 65, 2021 (1993).

• M.J. Saxton, Biophys. J. 69, 389 (1995).

• P.A. Netz and T. Dorfmüller, J. Chem. Phys. 107, 9221 (1997).

• M.J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).

• Berezhkovskii, A.M., V.Yu. Zitserman, and S.Y. Shvartsman, J. Chem. Phys. (15), p.7146 (2003).

• D Holcman, Z Schuss Control of flux by narrow passages and hidden targets in cellular biology, Reports on

Progress in Physics 76 (7), 074601 2013.

O. Medalia, Science 2002

Visualization of actin network, membranes, and

cytoplasmic macromolecular complexes.

Phototransduction

Rod and Cone: Functional subunits

Outer

Segment

Inner

Segment

Synaptic

Terminal

Phototransduction cascade and rod inner structure

Reingruber Holcman (BioPhys. J.2008, Phy Rev E2009)

Impact of geometry on biochemical reaction rates

Compartmentalized

Rod Outer Segment

Encounter Rate

Mean First Passage Time Framework

Toad Mouse

Finding a target in the cell

Nucleus

Nucleus

How the DNA is organized in the nucleus?

Polymer Looping induces gene activation

• The lac operon: The DNA bends

to bring the regulatory site for the

lac-repressor close to an operator to

block transcription.

Question: What the looping rate

depends on? – distance between the encounter sites

– mechanical properties of the polymer.

Looping for a polymer

R1

RN

e

Example of the boundary shape for

a rigid Polymer in dim 2

1( ) ( ) exp( ( ))i i it t b i t R R

1

( ) exp( ( ))k

k i

i

t b i t

R

2i id DdB

2

2

1

1

1 1

{( ) such that exp( ) }

..

N

N N k

k

N

B T i

T S S

e e

A. Amitai, SIAM MMS 2012

Shape of the boundary for N=4

2

1 2 3 4exp( ) exp( ) exp( ) exp( )i i i i e

A. Amitai, SIAM MMS 2012

The search for a target on the

nuclear periphery, a simulation

approach

A. Amitai et al. J. Chem. Phys. (2012) .

Narrow escape theory

Narrow escape theory

1 2(0) (0)(0) 1[1 ln( ) (1)], (0)

4 2

V Ha O H

aD a

2 Dimensions 3 Dimensions

MFPT = Mean First Passage Time = τ

depends on the initial position

1ln (1)

AO

D

e

Narrow escape and Dire Strait theory

12

ORD R

e

e

Three dimensional funnel

1

1 1

(1 )| |

2sin

1

a

D

1( )( ) (1 (1)) 0 near

(1 )

xr x a o x

3 21

(1 (1)) for2

cc

c

Ro a R

a R D

Two dimensional funnel

Other MFPT • Entering in an annulus:

• Entering in a cone of angle

• Exit through a cusp, between 2 circles:

1log (1)

gE O

D

e

1

1(1)

( 1)E O

d D

e

22 2 2 2 4 222 1 2 22

1 1 1 1( ) log log 2 2 log ( )

2 1 4

RE R R R O R e

e

1

2

1R

R

averaged with respect to a uniform initial distribution

Singer-Schuss-Holcman J.Stat.Phys. 2006

Dynkin’s equations

1 on

0 on

0 on

| |1

| |

a

r

a

r

D u

u

u

n

e

Expand the solution u for small e

Dynkin’s equations u(x)=MFPT conditioned to start at x

The residence time of a receptor in a microdomain

In agreement with measurement (Borgoff, Science 2002)

Conclusion (2004): a confined domain restricts Brownian

diffusion, but not long enough at the time scale of synaptic

transmission

(dim 2)

2

| | 1ln

500 ,

1/100

experiment=0.02 m /

D

R nm

D s

e

e

T=50 seconds

Holcman-Schuss, JSP 2004

1

1(1)

( 1)E O

d D

e

Obstacle

Singer et al, JSP 2006 T=1000 seconds

12

ORD R

e

e

T=300 seconds

Coarse-Graining

Conclusion

Green’s function Matching asymptotics

-Holcman-Schuss. J.Stat.Phys. 2004, -Singer-Schuss-Holcman, JSP 2006, -Ward-Keller SIAP 1993

-Singer-Schuss-Holcman PRE2008, -Singer-Schuss-Holcman SIAP 2008 -Ward Henshaw KellerSIAP 1993

-Singer-Schuss-Holcman, PNAS, 2007, -Holcman-Schuss J. Phys. A: Math.2008 -Pillay &al 2010-2011

-J. Newby-Bressloff, RMP, 2013 -Cheviakov Ward MMS2010

-Holcman-Schuss, MMS, 2012 -Holcman-Schuss, PRE, 2011 -R. Voituriez Benichou, PRL 2008

- Holcman-Hoze-Schuss, PRE, 2011 -Reingruber JCP, 2008 -Holcman Schuss SIAM REV 2013 -

D. Holcman Z. Schuss Phys. Prog Rep 2013

-Asymptotic expression classes: Narrow Escape Time

-Explore the parameter space

-Coarse-graining in numerical simulations

References: