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Optimal search strategies for hidden targets
O. Bénichou, M. Coppey, C. Loverdo, M. Moreau,
P.H. Suet, R. Voituriez
Laboratoire de Physique Théorique de la Matière
Condensée, Université Pierre et Marie Curie,
Paris.
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Search processes

• Examples
• Microscopic scale - diffusion limited
  reactions
• - protein
  searching for its specific site on a
  DNA strand
• Macroscopic scale - rescuers searching for lost
  victims in avalanches
• - animals searching for
  food

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Outline

• Intermittent strategies a widely observed
  search behaviour (i) Ecology search
  behaviour of animals (ii) Molecular biology
  search for a specific sequence on DNA
• Intermittent strategies a generic search
  mechanism ?

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Intermittent search Animals searching for food
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Intermittent search strategies in behavioural
ecology
Bell, OBrien
Observations many animals adopt a  saltatory 
behavior

• Displacement phases alternate with
   stationary  phases
• The durations of these two phases vary widely
  according to species
• There is a correlation between these durations

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Two state model of the intermittent behavior (1D
model)

• State 1 local scanning state
• State 2 relocating state

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Modelling of the searcher
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Modelling of the targets

• real situation targets are hidden at unknown
  sites, randomly distributed, with a small density
  1/L

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Efficiency of the search process ?

• What is the search time m ?

where t(x,i) is the mean first passage time at
the target, for a searcher initially at the
position x in the state i.

• Is there an optimal strategy with respect to f1
  and f2?

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Basic equations
By using the backward Chapman-Kolmogorov
differential equations, we obtain
Boundary conditions
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Results
In the low density limit,
where
m linearly depends on L !
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Minimization of the search time m (f1, f2)

• no global minimum for m (f1, f2)

• but f1 is bounded by f1max (analysing the
  information received by sensory organs requires a
  minimum time)

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Limiting regimes scaling laws
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Comparison with experimental data (1)
Experimental data for f1 and f2 are available
for fishes, birds and lizards (18 different
species) OBrien, Kramer and Mc Laughin
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Comparison with experimental data Log-Log plots
of f1 and f2
Bénichou et al, Phys. Rev. Lett. 94, 198101 (2005)
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Intermittent search Protein/DNA reactions
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Intermittent search strategies in molecular
biology
Protein searching for a target site on DNA
Transcription factor, Restriction enzyme
Typical reaction time (Smoluchowski)
1000s
??
1s
Experiments give
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First idea 1D diffusion
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1D diffusion experimental evidence
Bustamante
1D diffusion is still to slow ( )
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Berg et al (1981)
What is the mean first passage time at the target
? Is there an optimal strategy with respect to l ?

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First passage density
is the probability density that the protein
leaves the DNA at time ti, without reaching the
target site
is the probability density that the protein comes
back onto the DNA after a 3D excursion of
duration ti
is the probability density that the protein finds
the target at time tn, without leaving the DNA
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Mean first passage time (MFPT)
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Results
Optimal strategy
if
Optimal strategy in the large size limit
Mirny and Slutsky
(2004)
Coppey et al, Biophys. J. 87, 1640 (2004)
Further models Lombholt et al (2005), Zhou et
al (2005), Sokolov et al (2005), Grosberg et al
(2006)
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Experimental confirmation
Stanford et al. (2000)
1
c
a
b
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C
AB
Preference
Distance between targets
The model gives
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Outline

• Intermittent strategies a widely observed
  search behaviour
• Intermittent strategies a generic search
  mechanism ?

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The target (radius a) is centered in a spherical
domain (radius b) with reflexing boundary
conditions
a
b
The searcher is initially uniformly distributed
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Decoupling approximation
Auxiliary functions
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Conclusion
Intermittent search strategies are widely
observed because they are efficient
Losing time in relocating phases can speed up a
search process