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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