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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Cell biochemistry in cytoplasms with large molecular crowding : anomalous diffusion and bacterial aging

Hugues BERRYProject-Team AlchemyINRIA Saclay, Francehttp://www-rocq.inria.fr/~hberry/

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

The problem of ritual bath in Ganges river

???

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

The interior of cells is as well crowded

E. coli, to-scale artist view Hoppert & Mayer, 1999

• [Macromolecules] > 300 g/L (4 M if 70 kDa !) Zimmerman & Trach, 1991

• Viscosity = 5-30 × a buffer Verkman, 2002

• Large effects on thermodynamics (association constants, protein folding…) Minton, 2000

• Large effects on diffusion

D. discoideum, cryoelectron tomography Madalia et al., 2002

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Protein Diffusion in Cells is Anomalous

• Classical diffusion:

• Single-cell measurements (FRAP, SPT, FCS):

• Anomalous Diffusion:

• Experimental values of vary, but < 1 ie subdiffusion Guigas et al., 2007

diffusion not a well-mixing process in cells

= 1

= 0.7

= 1 (norm

al)

= 0.7 (subdiff)

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Open problems

• Main source = physical obstruction (organelles, internal netwrks, large multimolec. complexes…) but importance of other sources (binding, lipid rafts, corals, picket fences) debated Nicolau et al., 2007

• Variable values of ? Reality? Interpretation?

• Influence on cell biochemistry / regulation networks?

1.00.49 0.800.77 0.900.52

membranes: Smith et al., 1999; Weiss et al., 2003

nucleus: Wachsmuth et al., 2000; Guigas et al.; 2007

cytoplasm: Wachsmuth et al., 2000; Weiss et al., 2004; Guigas et al., 2007

0.60

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

A rapid overview of the talk

Multi-agent based simulations to help :

• Interpret experimental data on anomalous diffusion in living cells

• Evaluate spatial effects of crowding on biochemical reactions• 2D enzyme reactions

• Protein aggregation and bacterial aging

• Toward hybrid modeling

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Variable exponents : data

• Experimental data in living cells :

• Interpretation (+ some simulations): α varies with obstacle density

0.49 0.800.77 0.900.52

membranes:nucleus:

cytoplasm:

0.60

= 1 (Ø obst.

)

(obst.)

e.g. Saxton, 1994; Weiss et al., 2004; Banks & Fradin, 2005

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Variable exponents : Theory

• A single anomalous value of e.g. 0.53 for d=3

• Three successive regimes : normal-anomalous-normal

• Crossover times (not ) modified by obstacle density

1 0.53 1

= 1 (Ø obst.

) 1

0.53obst.

Havlin & Ben-Avraham, 2002

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Multi-agent refined simulations

• lattice-free, very large space-domains (periodic BC)

space domain size = 5103× 5103 space domain size = 103×103×103

Berry & Chaté, in preparation

3 regimes but no straight lines at intermediate times

3 regimes but no straight lines at intermediate times

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Corresponding local exponents

• No power law (no horizontal segments) at intermediate times and intermediate obstacle densities

• Theory OK but crossovers hide α-regime

• No analytical expression available (for intermediate obst. density)

Berry & Chaté, in preparation

1

0.53

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Finite-size effects explain fake power-laws

• Pseudo-"Power law" regimes due to finite size (simulations)

• Could also be due to noise in experiments

• May explain experiments: α variations with obstacle density

space domain size = 200×200 space domain size = 200×200×200

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

An effect on cell biochemistry

• Mean-field kinetics:

• Simulations:• Agents: E, S, C, P molecules (diffusive) +Obstacles (immobile)

• Upon encounter:

2D Michaelis-Menten enzyme kinetics

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Crowding modifies the kinetics

Berry, 2002

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Crowding induces spatial segregation

=0 =0.37m

=0.61m =0.99m

S

P

Berry, 2002

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

A second effect on cell biochemistry

• Progressive loss of reproductive ability

• The most simple aging model

•Up to 40 % of aging due to protein aggregation

• Similar to human aging diseases?

• Molecular link between aggregation and aging?

pole1

0

pole

1

0

cellgrowth

1

0

2

0

0

1

start celldivision

old pole cell

new pole cell

Stewart et al., 2005

ibpA aggregates (yellow) accumulate in the old pole cells

Lindner et al., 2008

[old with agg.] - [new w/o agg.] ≈ - 1 s-1

Av. growth rate differences (1 generation):

[o∨n with agg.] - [o∨n w/o agg.] ≈ - 0.4 s-1

E. coli aging and protein aggregation

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Simulation of IbpA aggregation

Cell model:

Protein parameters (monomer):• radius:

– rIbpA 1.5 nm (van Montfort et al.,2001); rGFP 2 nm (Reka et al., 2002)

rIbpA-YFP = r1 = 3.0 nm

• diffusion coefficient:– DGFP (28 kDa) = 7.7 µm2/s; DGFP-MBP (72 kDa) = 2.5 µm2/s (Elowitz et al.,1999)

DIbpA-YFP (39 kDa) = D1 = 4.4 µm2/s

Zimmerman, 2006

high crowding within nucleoids

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Simulation of IbpA aggregation

• 3-d off-lattice agent-based simulations

• n-mer aggregate:• radius:

– globular proteins

• diffusion coefficient:

• Benchmark : aggregate localization at 1st appearance : experimental data

r1 = 3 nm

D1= 4.4 µm2/s

r2 = 3.78 nm

D2= 3.49 µm2/s

r3 = 4.33 nm

D3= 3.05 µm2/s

pag

Pro

babi

lity

position along large axisLindner et al., 2008

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Location of aggregates at first appearance

Berry & Lindner, in preparation

nucl. nucl.

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Comparison with experiments

experimentally determined simulations (600 obst/µm2)

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Comparison with experiments

• Qualitative agreement for intermediate obstacle densities and ≈25-mers aggregates

• To consider: cell size mixing, membrane interactions, folding-dependent aggregation, cell shape, burst expression

• Indicate that ibpA aggregate formation (and localization) = purely passive (crowding in nucleoids)

experimentally determined simulations (600 obst/µm2)

Berry & Lindner, in preparation

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Hybrid Discrete-Continuous Simulations

• Discrete compartment = area of interest (eg nucleoids)

• Or automatic switching discrete continuous on the basis of a threshold in the copy number of molecules in the compartments

continuous nodes

discrete compartment

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Proof of concept : 1d diffusion

C D C

l

• Parameters:• Diffusion coefficient:

• Compartment junctions:

update: update:

Chopard & Droz, 1998

Adjust boundary conditionsHybrid continuous/discrete interface

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Conclusions : Influences of crowding

• Anomalous Thermodynamics :• association constants

• protein folding, aggregation…

• Anomalous Diffusion :• transient subdiffusion

• affects species spatial distributions / fluctuations within the cell

• modifies reaction kinetics Berry, 2002; Saxton, 2002; Nicolaus & Burrage, 2008…

• increases nearby target finding probabilities of transcription factors Golding & Cox, 2006; Guigas & Weiss, 2008

• No analytic form for transient anomalous subdiffusion : Multi-agent based modeling (/hybrid techniques) especially well suited

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Acknowledgement

• Funding :• INRIA (Nat'l. Inst. Comp. Sci. & Control)

• INSERM (Nat'l. Inst. Med. Sci. & Health)

• Collaborations:

Hugues ChatéAtomic Energy Commission (CEA), Saclay, France

Ariel Lindner

François Taddei INSERM U571

Medical School Cochin,

Paris, France

Annick LesneUniv. P & M Curie-Paris 6, Paris & IHES, Bures-sur-Yvette, France

Olivier MichelUniv. Creteil-Paris 12

Creteil, France

• ISC (Complex Systems Inst., Paris)

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Quantification of segregation

• Segregation occurs even for mild obstacle densities

= 1: perfectly mixed>1 : segregation

Berry, 2002

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2nd Paris Workshop on MAS in Biology, June 23rd, 2009

Example: Synapses (dendrites)

Lau & Zuckin, 2007

Ca2+ diffusion in dendrites is anomalous Santamaria et al,, 2006