Individualbased simulations for cell biochemistry in crowded environments

1
Individual-based simulations for cell
biochemistry in crowded  environments
H. BERRY
Project-Team Alchemy, INRIA Saclay-Île de France
Research Centre, France
2
Outline

• Introduction
• Fluctuations in cell biochemistry beyond
  mass-action laws
• Michaelis-Menten reaction in crowded environments
• Aging and chaperone diffusion-aggregation in E.
  coli (preliminary project)
• Perspectives
• Hybrid discrete-continuous simulations

3

• Introduction
• Fluctuations in cell biochemistry beyond
  mass-action laws

4
The Laws of Mass-action

• Rooted in late19th century chemistry (vant Hoff,
  1884 Guldberg Waage, 1899)
• Used at every scale of biological modeling
• biochemical reactions (protein-ligand
  interactions, enzyme kinetics)
• cell biology /physiology (immune systems
  dynamics, endocrine regulation )
• population/ecosystems dynamics (e.g.
  Lotka-Volterra)
• BUT based on strict assumptions ( Mean-Field
  equations)
• 3D, dilute, perfectly mixed and spatially
  homogeneous media
• large copy numbers (law of large numbers)
• fluctuations / correlations assumed not
  significant

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The Case of Biological Cells

• Cellular media are
• not homogeneous (highly compartmented)
• not dilute solutions
• viscosity in the mitochondrion matrix ? 30 times
  that of a buffer (Scalettar et al., 1991)
• diffusion in cytoplasm ? 5-20 times lower than
  buffer (Verkman, 2002)

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

• Many regulatory proteins are present in low copy
  numbers
• 80 of the E. coli proteins have lt 100 copies
  (Guptasarma, 1995)
• down to only 4-20 copies (Yu et al., 2006)
• in yeasts, a large fraction of the proteins
  present as a few hundreds down to lt100
  copies per cell (Ghaemmaghami et al., 2003)
• Reactions in cells look stochastic, not
  deterministic

intrinsic noise
Elowitz et al., 2007
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Protein Diffusion in Cells is Anomalous

• Classical diffusion
• Single-cell measurements (FRAP, SPT, FCS)
• anomalous diffusion or subdiffusion
• Experimental measurements
• ? diffusion not a well-mixing process in cells

1.0
0.49
0.80
0.77
0.90
0.70
?
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Where does Anomalous Diffusion come from?

• Main source physical obstruction by immobile
  obstacles, ie crowding (Nicolau et al., 2007)

• Large-size obstacles actually abound in cells
• organelles (mitochondria, endosomes, Golgi),
• internal networks (endoplasmic reticulum,
  cytoskeleton)
• large macromolecular protein complexes (e.g.
  cytoskeletal)
• Other sources
• interactions with membrane lipid rafts or corals
• cytoskeleton picket fences

D. discoideum with cryoelectron tomography
Madalia et al., 2002
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Beyond Mass-action Laws

• How to model / simulate these effects?
• Langevin approach
• determining the correct noise parameters is a
  very hard task to carry on!!
• and must be re-done for every new case
• Gillespies exact SSA algorithm (Gillespie,
  1977)
• produce numerical realizations (samplings) for
  the time courses of the random variables
• E.g. plasticity and memory in neuron synapses

NA (t ) --NB (t ) --NC (t )
Delord, Berry, Guigon Genet (2007) PLoS
Computational Biology, 3 e124
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Simulating Crowded Environments

• Gillespie OK for questioning low copy numbers
• But
• Space must be homogeneous perfectly stirred
• Hardly accounts for space characteristics
  (depends on the scale)
• Spatial Gillespie (Stundzia . Lumsden, 1996)
• partition space into subvolumes within which
  mixing perfect
• diffusion from subvolumes ? reaction
• not for molecular crowding cause basically lacks
  excluded volume

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Individual-Based (Multi-Agent) Simulations

• Model explicitly each molecule ( 1 agent)
• Position molecules (/- obstacles) within space
  (lattice or not)
• Model diffusion as independent random walks of
  each agent
• Volumes (sites) occupied by obstacles are
  forbidden
• Spatial characteristics and obstacle positions
  are explicitly given

X
immobile obstacle
e.g. anomalous diffusion (Berry (2002) Biophys
J)
S
diffusing protein
X
X
S
X
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• Michaelis-Menten reaction in crowded environments

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Adding reaction Enzyme kinetics

• Mass-action laws
• Individual-based simulations

Simulating reaction
E
X
X
S

• E -S encounters yield C with proba a

E
C
S
S
X
E
S
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Adding reaction Enzyme kinetics

• Mass-action laws
• Individual-based simulations

Simulating reaction
X
X

• E -S encounters yield C with proba a

• C yields E S with proba b

C
E
S
X
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Adding reaction Enzyme kinetics

• Mass-action laws
• Individual-based simulations

Simulating reaction
X
X

• E -S encounters yield C with proba a

• C yields E S with proba b

C
E
C yields E P with proba g
X
P
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Fractal kinetics segregation
Berry (2002) Biophys J
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Fractal kinetics segregation
S
P
q 0
q 0.37 q m
q 0.61 q m
q 0.99 q m
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• Aging and chaperone diffusion-aggregation in E.
  coli (preliminary project)
• with A. Lindner F. Taddei (INSERM U571, Necker,
  Paris)

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Aging in E. coli experimental data

• Aging manifests as differences between cell poles
  (Stewart, Madden, Paul Taddei (2005) PLoS
  Biology)
• Older cells exhibit lower growth rates, decreased
  offspring production and increased death rate

young cells
old cells
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Aging in E. coli experimental data

• Related to protein aggregation of IbpA (Lindner
  et al. (2008), PNAS, in press)
• IbpA forms multimeric aggregates localized and
  progressively accumulating (within 3 generations)
  in the old cells.

• Presence of the aggregates in cells
  statistically explains 40 of aging

1
new
old
2
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Main experimental questions

• At the temporal resolution (2.5 min), a given
  cell usually contains a single aggregate
• Spatial distribution of the aggregate
• segregation at one pole or the cell midplane
• Passive or active?
• Interaction with nucleoids (obstacles)?
• Quantitative experimental results
• P(0 agg ? 1) 0.71, while P (1 agg ? 2) very
  low
• P(0 agg ? 1) significantly lower if mother cell
  had a focus and passed it to its other offspring,
  w.r.t when mother free of focus.

young pole
old pole
PURIFICATION?
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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
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Simulation of IbpA aggregation

• Morphology of the aggregate not taken into
  account remains a globular protein (sphere of
  constant density)
• n-mer aggregate
• radius
• globular proteins rn r1 n1/3
• effective radius
• diffusion coefficient
• Dn ? 1/rn ? Dn D1 n-1/3

pag
pag
r3 4.33 nm D3 3.05 µm2/s
r2 3.78 nm D2 3.49 µm2/s
r1 3 nm D1 4.4 µm2/s
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Capture time in homogeneous spheres
R
r1
Adam Delbrück, 1968
R
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Example 1

• Initial conditions 100 IbpA (monomers)
  homogenous distribution pag 1

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Average over 2000 runs kinetics
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Effect of the aggregation probability
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Average over 2000 runs distribution

• Final position of the final survivors is
  homogeneous (uniform)

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Putative mechanisms to be tested

• Nucleoids densely packed highly crowded
  regions
• add static obstacles in the nucleoids (ongoing
  simulations).
• Random transcription bursts (40 proteins/burst)
  Pedraza Paulsson, 2008
• periodic or stochastic (Poisson) timing
• Co-aggregate of the IbpA with other (unfolded)
  proteins?
• Interactions with the cell membrane?
• Explicitly model cell growth and division.
• Should allow confronting the simulation results
  with the observed variations in the presence
  probability of the aggregation focus upon cell
  division.

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• Perspectives
• Hybrid discrete-continuous simulations
• with O. Michel (IBISC, CNRS FRE 2873, Evry)
• and A. Lesne (LPTMC, Univ. P M. Curie, Paris)

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Hybrid Discrete-Continuous Simulations
discrete compartment
continuous nodes

• Discrete compartment area of interest (e.g.
  high molecular crowding)
• Or automatic switching discrete ? continuous on
  the basis of a threshold in the copy number of
  molecules in the compartments

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Example Synapses
Lau Zuckin, 2007
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1d Diffusion
l
l
C
C
C
D
C
C
C
update
update

• Restrictions
• CCCC DDDD
• CDC DDD

Chopard Droz, 1998
???????????
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Acknowledgements

• B. Delord, S. Genet, E. Guigon INSERM U742,
  UPMC, Paris
• A. Lindner, F. Taddei INSERM U571, Necker, Paris
• O. Michel, J.L. Giavitto IBISC, CNRS FRE 2873,
  Evry
• A. Lesne LPTMC, CNRS UMR7600 UPMC, Paris IHES,
  Bures
• D. Coore Univ. West Indies, Mona, Jamaica

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A posteriori evaluations

• Record g(t), the total number of enzyme-substrate
  collisions that have effectively given rise to
  complex formation after time t.

?
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Quantifying Segregation
1 perfectly mixed gt1 segregation

• Segregation exists even for mild obstacle
  densities

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Comparison to Theoretical Values
Segregation
No segregation

• h depends on ds (spectral dimension), but this
  dependence changes when segregation occurs
  (Argyrakis et al., 1993)
• ds varies with q (empirical values only,
  Argyrakis Kopelman,1984)
• ds(0) ? 1.80 ds(qc) ? 4/3

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Ex. 2 Taking nucleoids into account

• Initial conditions 502 IbpA (2 bursts)
  homogenous distribution restricted to the
  nucleoids pag 1

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The Problem of Computational Costs
Turner et al., 2004
SMA - 3D
SMA - 2D
Gillespie

• direct Monte Carlo approaches suffer from
  the drawback of requiring large amounts of
  computer resources for problems of realistic
  dimensions, if the system is built up molecule by
  molecule. We argue that the best way forward is
  along a middle path, involving multiscale
  simulation methods that deal with heterogeneity
  and nondeterminism at the scales at which these
  are appropriate but can retain the powerful
  approach of differential equations over all other
  scales. (Nicolau et al., 2007)