Modeling molecular and cellular biology: methods and … · Modeling molecular and cellular biology: methods and examples Lectures, Paris ... + regulation of cell activity) ... Motion

Modeling molecular and cellular biology: methods and examples

Lectures, Paris 2009D. Holcman

ENS

www.biologie.ens.fr/bcsmcbs/

Cellular organization and Trafficking

The major goal of the class is to present:1. Biophysical modeling2. mathematical methods3. Simulations tools to design rational simulations and

algorithmsFor studying cellular biophysics (neurons):

Find how cells function based on biophysical principles

Vesicular trafficking in a neuronal cell

Silvermann 2001

Biological MicrostructuresDefinition:Part of a cell, driven by molecular interactions underlyinga physiological unit.

• Synapses (transduce information between neurons)

• Outer segment of photoreceptors(a photon induces a hyperpolarization)

• Nucleus (guardian of the genetic material+ regulation of cell activity)

Cell-cell interaction and communications

-Dendrites

-Soma

-Axons

Hippocampal neurons

Questions: how to understand the interactions?How they work and what theory brings in?

Neurons connect to form networks

Synapse: connection between 2 neurons

Electrical activity of Neurons

Harris

Dendritic spine

-100 000 spines/neuron-Define the synaptic weight

What spines are good for?

Synapses

Synaptic terminal Quantum dots (QD) at a synapse

Why modeling microstructures?

1. Understand the function of microdomains and analyze the cell behavior in normal and pathological conditions.

2. Quantify:– small size of the structures.– low number of molecules (buffers, dyes introduced

experimentally may perturb the function).– Too many interactions: isolate pathways.– Predict of effect of drugs.

Role of modeling

• Derive physiology from statistical physicsproperties

• Compute properties at a biophysical level

Reward• Reconstruct the intact function• Study the effect of KO or adding drugs

1-What is memory at the cellular level?2-How neurons process information?3-How molecules such as DNAs move from one

position to another?4-How cells maintain and regulate proteins?5-How to estimate the mean time to travel from one

place to another?

Quantitative questions:6-How much ?How long?

Build model to answer quantitative questions.

Examples of questions

Syllabus1. Principle of diffusion: Brownian motion. Stochastic approach and partial differential equation. From stochastic

processes to Partial Differential Equations. Brownian motion, Ito calculus. Diffusion equations, Fokker-Planck equation. Motion of polymer. Modeling diffusion of shaped object, law of reflection, polymer dynamics using Rouse model.

2 Mean first passage time equations, conditional MFPT, distribution of exit points. The small hole theory, no potential, with an attracting and a repulsive potential. Small hole in a narrow domain (averaging methods). -The case of one and several holes.

3 Singular perturbation analysis. Mixed boundary value problems: Escape through a small hole. The small hole theory with a attracting (resp. repulsive) potential. Homogenisation theory with many small holes. Short and long time asymptotics, ray methods.

4 Minimization of MFPT, consequence hole distribution.-MFPT in random environments. Narrow escape time for a switching particle.-Homogenisation theory with many small holes.

5 Application to chemical reactions: Stochastic chemical reactions in a microdomain. Forward and backward chemical reactions .Theory of threshold. Clustering in confined environment, application to telomere organization in the nucleus. Application to morphogenesis: Patterning equations. Spreading of a Transcription factor, French flag, Brain territories and organization of the cortex.

6 Modeling a synapse and asymptotic estimation of the Dwell time. Synaptic transmission, synaptic weight. Synaptic cleft. Receptor trafficking, synaptic current, Dwell time of a receptor at the synapse. Calcium dynamics in dendritic spines. Neuron-Glia interactions.

7 Toward a quantitative cellular virology, cellular trafficking. Cellular trafficking: diffusion, active transport by molecular motors. Drift homogenisation. First arrival time of the first virus to a nuclear pore. Escape from endosome.

8 Vesicle trafficking, neurite outgrowth, DNA movement in the nucleus in normal and pathological conditions. (signaling in virus trafficking, growth cone dynamics, DNA breaks, and many others).

9 Summary: Toward a quantitative approach in cellular biology.

Class organization-WEB site announcement: class 2009http://www.biologie.ens.fr/bcsmcbs/

Requirements: partial differential equations, probability,Laplace's method, some notions of cellular biology.

Evaluation: small projects

Additional readings and classes: -Brownian motion, probability-Partial differential equations and differential geometry-Dynamical systems-Cellular biology-Program a computer.

Further background

• Pasteur institute: viruses.

• Statistical physics: ENS+Paris VI,XI: classes

• Curie Institute and Gustave Roussy: cellular biology and cancer research.

Some References (on line)-Z. Schuss A, Singer, D. Holcman, PNAS 2007.

-D. Holcman, Z. Schuss. 2004. Diffusion of receptors on a postsynaptic membrane:exit through a small opening, J. of Statistical Physics, 117, 5/6 191-230.

-A. Singer Z, Schuss, D. Holcman, Narrow Escape I, II and III, in J. of Statistical Physics, 2006, Vol 122, N. 3, p 437 -563.

-D. Holcman, Z. Schuss. 2005. A theory of stochastic chemical reactions in confined microstructures, Journal of Chemical Physics 122, 114710, 2005.

-D. Holcman A. Marchevska Z. Schuss, The survival probability of diffusion with trapping in cellular biology Phys.Rev E Stat Nonlin Soft Matter Phys. 2005,72: 031910.

Books:

-Molecular Biology of the Cell, B. Alberts, et al, 4rd ed., 2002.

-Ion channels of excitable membrane, 3rd ed, B. Hille 2001

- Schuss, Z., 1980, Theory and Applications of Stochastic Differential, New York, Wiley. (Book)

-Roger and Williams, stochastic processes, Brownian motion

-P. Garabedian, Partial Differential Equations

ReferencesBasics:• -Molecular Biology of the Cell, B. Alberts, et al,4rd ed., 2002.

-D. Holcman, Computational challenges in synaptic transmission, AMS Contemporary Mathematics : Proceedings of the Conference on Imaging Microstructures : Mathematical and Computational Challenges, Edited by H. Ammari and H. Kang. 2009

• -Schuss, Z., 1980, Theory and Applications of Stochastic Differential, New York, Wiley.

Advanced:• -D. Holcman, Z. Schuss. 2004. Diffusion of receptors on a postsynaptic membrane:exit through a small

opening, J. of Statistical Physics, 117, 5/6 191-230. -A. Singer Z, Schuss, D. Holcman, Narrow Escape I, II and III, in J. of Statistical Physics, 2006, Vol 122, N. 3, p 437 - 563. -D. Holcman A. Singer, Z. Schuss, Narrow escape and leakage of Brownian particles. PRE 78:051111 (2008).

• -D. Holcman, Z. Schuss. 2005. A theory of stochastic chemical reactions in confined microstructures, Journal of Chemical Physics 122, 114710, 2005. -Reingruber D. Holcman, Narrow escape for a switching state Brownian particle

• -A. Taflia D. Holcman, The Optimal PSD of synaptic transmission-J. Reingruber and D. Holcman, Diffusion in narrow domains and application to phototransduction, Phys Rev E 2009 ;79(3 Pt 1):030904.

• -T. Lagache E. Dauty D. Holcman, Physical principles and models describing intracellular virus particle dynamics, Current Opinion in Microbiology, 12,4 (2009).

• - D. Coombs and R. Straube M Ward Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points (SIAM J. Appl. Math., Vol. 70, No. 1, (2009), pp. 302-332.)

• - S. Pillay, A. Peirce, and T. Kolokolnikov, M. Ward, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains (SIAM Multiscale Modeling and Simulation, (March 2009), 28 pages.)

• - A. Cheviakov and R. Straube M. Ward, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere (SIAM Multiscale Modeling and Simulation, (March 2009), 32 pages.)






5 Application to chemical reactions: Stochastic chemical reactions in a microdomain. Forward and backward chemical reactions .Theory of threshold. Clustering in confined environment, application to telomere organization in the nucleus. Application to morphogenesis: Patterning equations. Spreading of a Transciption factor, French flag, Brain territories and organization of the cortex.

6 Modeling a synapse and asymptotic estimation of the Dwell time. Synaptic transmission, synaptic weight. Synaptic cleft. Receptor trafficking, synaptic current, Dwell time of a receptor at the synapse. Calcium dynamics in a dendritic spine. Neuron-Glia interactions.




Langevin dynamics: description of particle movement

• electrostatic interaction V’(x)• Ito calculus, stochastic integral.• Stochastic equations, dynamical systemsSolution=trajectory: find recurrence

[ ] '( ) 2

[ ] '( ) 2

x x V x w

x V x w

γ εγγ

γ εγ

•• • •

• •

+ + =

→ ∞

+ =Particle X(x,t) follows

-Fokker-Planck equation

• Diffusion equation: Brownian motion

• Probability to be in A at time t:

• Solution, interpretation, asymptotic computations, boundary conditions, First eigenvalue and the time to escape a domain

( , ) Pr ( ) | (0) 0( )

( ) for

p x t X t x dV X

p x tD p x t x

t

= ∈ + =∂ , = ∆ , ∈ Ω,

∂

Pr ( ) | (0) 0 ( , )A

X t A X p x t dx∈ = =

General theory: Fokker-Planck equations

( ) 0 for 0 ap x t y t x y, | = > , ∈∂Ω, ∈Ω

[ ]( )( ) '( ) ( ) for

( 0 ) ( ) for

p x t yD p x t y x p x t y x y

tp x y x y x y

φ

δ

∂ , | = ∆ , | + ∇ , | , ∈Ω,∂

, | = − , ∈Ω

[ ]( )( ) '( ) ( ) ( ) 0 for 0 aD p x t y x p x t y x t x yφ ν− ∇ , | + , | ⋅ = > , ∈∂Ω − ∂Ω , ∈Ω.

'( ) 2 ,x x D w Dεφγ

• •= − + =

Pr ( ) | (0) ( , | )X t x dx X y p x t y dx∈ + = = satisfies

Derivation of the Boundary conditions from the Wiener path Integral

absorption

reflection

Motion of an ellipsoid, law of reflection

• Brownian Motion of an Ellipsoid

-10 -8 -6 -4 -2 0 2 4 6 8 10

-2

-1

0

-25 -20 -15 -10 -5 0 5

-2

0

2

•Reflection on a surface, partial absorption.

Polymers to a small hole

Rouse model of polymer diffusion:individual beads connected by springs.

N = 2 molecules N = 8 molecules

Polymers

( )211 2 1 2 0 1 22( , )U k l= − − −x x x x x x

Langevin Equation:

1 1( , , ) 2i i i i i iU D− ++ ∇ =x x x x w

Potential function for simple harmonic spring:

1 1 1 1( , , ) ( , ) ( , )i i i i i i iU U U− + − += +x x x x x x x



2 Mean first passage time equations, conditional MFPT, distribution of exit points. Exit from a well. The small hole theory, no potential, with an attracting and a repulsive potential. Small hole in a narrow domain (averaging methods). -The case of one and several holes.








Escape from a well• Off rate: Arhenius and Kramer theory of chemical

reaction rate. Escape from a domain

• Off rate= rate to escape the domain.• Asymptotic expansion when is small.

[ ] '( ) 2 bassin of attraction

inf 0, ( ) ( )?

x V x win

t X t

E

γ εγτ

τ

• •+ = Ω =

= > ∈ ∂Ω

1/ ( )E τε

How long it takes to escape through a narrow pore ?

[1 (1)]4V

oaD

τ = +

2 Dimensions 3 Dimensions

MFPT = Mean First Passage Time =

The Narrow Escape Problem for Diffusion in Cellular Microdomains. Holcman-Schuss. J.Stat.Phys. 2004Singer-Schuss-Holcman, PNAS, JSP 2006

1ln (1)

AO

Dτ

π ε= +

Small Hole Theory

12

sr

επ

= <<

How long it takes for a Brownian Molecule to reach a small hole?

Small aspect ratio assumption

Ward-Keller 1993, Berezchkoskiiet al. 2002, R. Pinsky 2003Holcman-Schuss. J.Stat.Phys. 2004Singer-Schuss-Holcman JSP, PRE 2006-2008

Other MFPT • Entering in an annulus:

• Entering in a cone of angle

• Exit through a cusp, between 2 circles:

• Extension to Riemannian manifolds

α

1log (1)gE O

Dτ

α ε| Ω | = + ,

1

1(1)

( 1)E O

d Dτ

ε−

| Ω | = + −

22 2 2 2 4 222 1 2 22

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

2 1 4R

E R R R O Rτ β ε βε β β

= − + + + − + , .−

1

2

1RR

β = <

averaged with respect to a uniform initial distribution

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

Exocytosis/Activation rate

rv

a

arv

xn

txpnp

D

xtxp

xpxp

ppDt

p

xxdxxtxdxxtxp

Ω∂∈=∂∂−

∂∂=⋅

Ω∂∈==

Ω∂Ω∂=Ω∂∇⋅∇−∆=∂

∂=+∈=

for ,0),(nJ

for ,0),(

)()0,(

in ,)(

)0(|)(Pr)|,(

0

00

φ

φ

δδ

δ

δ

δδδ

δ

FPE:

The average time the vesicle spent at point x prior to exocytosis:

Solution of

With above boundary conditions and the MFPT is

0

( ) ( , ) ,u x p x t dtδ δ

∞

=

0 ( ), in v r ap D p pδ δ φ− = ∆ − ∇ ⋅ ∇ ∂Ω = ∂Ω ∂Ω

( )u x dxδ δτΩ

=

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Chemical reactions

• Chemical reactions in solutions:

• Limited by diffusion

Solve equations

f

free

b

k

M S MS

k

+ ,

=forward binding rate=depends on global properties

=geometry, distribution of substract, diffusion constantfk

backward binding rate=depends on local interactionsbk =

0

[ ] [ ][ ]

[ ] [ ]

mean number

f b

dMk SM k M S

dtM SM M

M

= −

+ ==

[ ][ ] [ ]

[ ][ ][ ] [ ]

f b

f b

MD M k M S k SM

tSM

k M S k SMt

∂ = ∆ − +∂

∂ = −∂

4a-Chemical reactions: rate

• On and Off rate: Arhenius and Kramer theory of chemical reaction rate. Escape from a domain

• Off rate= rate to escape the domain.• Asymptotic expansion when is small.• On rate=flux to the sites.

[ ] '( ) 2 bassin of attraction

inf 0, ( ) ( ) ?

x V x win

t X t

E

γ εγτ

τ

• •+ = Ω =

= > ∈ ∂Ω

1/ ( )E τε

Backward rate

Definition:Inverse of the MFPT to escape the potential well of

binding site due to thermal noise. It is the Arrhenius law (Kramer theory 40’s, Matkowsky-Schuss 75)

bk

,

for large

aEkT

b

A C

k Ce

Cω ω γ

πγ

−=

=

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

A

C

V=potential

''( ), ''( )A CV A V Cω ω= = −

Forward rateReciprocal of the mean first passage time for a

Brownian molecule to find a binding site + eventual potential activation barrier.

Depends mostly on the geometry.Smoluchowski computation: Flux in a sphere Zwanzig computation (PNAS, 1990)Berezhkovskii (J. Chem. Phys. 2004)Small hole

Smoluchowski computation

• The computation is obtained as the probability flux through a sphere of radius R in dimension 3:

• For one molecule

• For a concentration

• LIMIT=INFINIT SPACE: not the case in biology

4

radius of the binding siteD=diffusion constant

fk DR

R

π=

=

04fk DRcπ=0c

Chemical reaction: Microdomains. rate Markovian Approach

In -mobile agonist molecules -receptors located at the surface

The probability distribution of the first passage time is approximately exponential with rate

(area of the absorbing boundary (the channels) is small relative to the

surface area of the reflective boundary)

f

free

b

k

M S MS

k

+ ,3RΩ ⊂

M

11

1λτ

= ,

S

Chemical reactions with few components in a microdomains

• Find mean and variance of bounded molecules.

• Use Master equations to findIn a closed domain:

( ) Pr ( ) kp t SM t k= =

( ) ( )( )

0

0

Pr ( ) Pr ( ) 1 1 Pr ( ) 1 1

Pr ( ) 1 ( )

b f

b f

SM t t k SM t k k t k SM t k k t S k

SM t k kk t S k k t

+ ∆ = = = + ∆ + + = − ∆ − +

+ = − ∆ − − ∆

( ) ( ) ( )0 1 0 1( ) ( ) ( ) 1 ( ) 1 ( )b f k b k f kkp t kk S k k p t k k p t k S k p t•

+ − = − − − + + + − +

For k >0,

Standard equations used in queuing theory: References: Saaty 1981, Karlin-Taylor 1975

Chemical reactions in detached excised patch, cGMP-channels

Fluctuation due to binding and unbinding+ channel noise

Simulation, based on the small hole approximation in a Markov model.

0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

# agonist molecules

Var

ianc

e sq

uare











What underlies neuronal activity?Ionic Channels

Origin of a depolarizationsynapse

What determines synaptic transmission

( ) ( ) ( , ) ( , , )

Conductivity of type k-receptor

( , ) # k-receptors at position

( , , ) PrReceptor at opens at time t

post k kk

k

k

I t t N x t p k x t dx

N x t x

p k x t x

γ

γ

=

=

=

=

Depends on molecular interactions

can change by diffusion

Depends on diffusion in the cleft and intrinsic properties: to be estimated

receptor

Glutamate trajectoryBinding to a receptor

Exit trajectoryReleased vesicle

Rs~ 100s of nm

PSD

h=cleft height~ 10s of nm

Modeling of the synaptic cleft

Density of extra-PSD receptors:Few 10s.

Density of PSD receptorstransporter

RPSD

dr

dg3000 neurotransmitters

Parameters of synaptic transmission• Synaptic cleft geometry (Rs, h,dg)

• PSD properties ( RPSD), #, distribution, types of Receptor.

• Position and # of release vesicles dr

• Number and Intrinsic properties of the channels

• Uptake from (# and distribution of) transporters

– Holmes, BJ 1995, Barbour JNS, 2001: Reaction-diffusion equations– Bartol-Sejnowski 2003, Rusakov BJ, 2001: Brownian simulations– Jahr, Xie PNAS, 1997: modeling +physiology

Goal Find a quantitative framework

kγ

Approaches

• Estimate analytically the number of open receptors (Mixed boundary value problem in degenerated microdomains)

• Design rational Simulations:– Brownian simulations (need calibration)– Solve the associated Reaction-Diffusion

equations

Movement of AMPAR during LTD/LTP

Collindgridge, 2004

Receptor trafficking

D. Choquet A. Triller

QD-GluR2 (Hippocampus)

D. Choquet, personal communication

Dynamics of synaptic receptor: Markovian approach

Master equations:1)exchange of q receptors with the rest of the dendrite 2) k binding/unbinding with scaffolds

1q q→ −1q q→ + 1k k→ +1k k→ −

-2 2 0 -2 2 0in k,q k+1,q k-1,qk,q1

in k,q-1 k,q+11

receptor binds/unbindsNothing happens

receptor

q-k= -[k k+k (q-k)(S -k)+<J >+ ]p + k (k+1)p (t)+k (q-k+1)(S -k+1)pp

q+1-k + <J >p + p

enters/exits











dsDNA repair

• Double strand DNA cut induced by radiation can result in alteration of genetic information (deletion)

How long does it take for dsDNA break to glue?(model effect of constraint environment: known in the case of Radiodurans.

Chromatin was present in all compartments, and it adopted a distinctive toroidal shape (A and B). DNA toroids are present in all compartments of stationary-state bacteria, the nucleoid in one or two compartmentsof growing cells is dispersed (Fig. 1, C and D). After irradiation, the toroidal DNA structure underwent a transition into an open S-like morphology, followed by progressive spreading of DNA between two compartments through a membrane orifice

DNA-Rouse model of DNA dynamics in a microdomain

-40 -35 -30 -25 -20 -15 -10 -5 0-2

-1

0

1

2

L

0 5 10 15 20 25 30 35-2

0

2

Retraction

L

0 5 10 15 20 25 30 35-2

0

2

l0

h

Effect of restricted microdomains

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

h = 0.05 x L0

time (in seconds)

Frac

tion

of s

trand

s st

ill f

ree

L = 0.1 x L0L = 0.3 x L0L = 0.5 x L0L = 0.7 x L0L = 0.9 x L0

0 1 2 3 4 5 6

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

fraction of strands still free vs time for L = 0.1 x L0

time (in seconds)

Frac

tion

of s

trand

s st

ill fr

ee

actual valuesexponential fit

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

Pro

babi

lity

of R

ecom

bini

ng

h = 0.05 x L0h = 0.1 x L0h = 0.2 x L0

Ziskind-Shaul-Holcman 2008

Effect of initial position in 1D (2-Brownian particles)

0 0.2 0.4 0.6 0.8 0.990

0.2

0.4

0.6

0.8

1

1.2

1.4

x1

Pro

babi

lity

P(x2=0.99)

Holcman-Kupka-JPA 2008

1 2 2 1

2 ( )( ) log 1

8M

Z aP x x m P Z x x

Lω

π− − , = ℑ , = + −

32 4P PP = − ,′

3 2

31 22 8 1 854 5 244115106

4

dp

p pω

∞/

/= = ∗ . = . .

−

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

x1

Pro

babi

lity

P(x2=0.5)

Goal1. Modeling virus trafficking in cells.2. Estimate efficiency of infection3. The role of parameters:

– Geometry of the cell– Cytoplasmic Organization – Cell signaling (Rab, Calcium,…)

Toward a computerized study of cellular trafficking

in vitro efficiency(genes/cell)

gene size(kbp)

1-100

3 - 8

particle size30-100 nm

104-106

> 150

low efficiency

++++

++++

+

variable

++

+

++

New types of biodegradable polymers are developed to optimize drug and gene delivery.

Plasmid Cytoplasmic Trafficking

The endosomal step for AAV

Ding Gene Therapy (2005) 12, 873–880.2005

Cytoplasmic phase

Escape phase

Endosomal trafficking

Endocytozed phase

How to model endosomal steps ?

Hemagglutinin dynamics

Capsid organization

Hemagglutinin molecule

Hemagglutinin in action

How to model virus and DNA delivery?

Arhel N et.al Nat Methods. 2006 (10):817-24. Brandenburg B, Zhuang X. Nat Rev Microbiol. 2007 Mar;5(3):197-208. Review.

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Modeling molecular and cellular biology: methods and … · Modeling molecular and cellular biology: methods and examples Lectures, Paris ... + regulation of cell activity) ... Motion