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.

Search processes How long does it take to the searcher to find the target ? 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 The search time is generally a limiting factor which has to be optimized !

What is the quickest way to find a randomly hidden object ? Everyday life example : A B Small keys lost between two distant points A and B of a pavement, with no other information about their location Systematic exploration [« Search theory » : Stone….] Intermittent search behavior, combining: local scanning phases relocating phases

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 ?

Animals searching for food Intermittent search : Animals searching for food

Intermittent search strategies in behavioural ecology [Bell, O’Brien] 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 Time spent in searching Time spent in moving log-log Can we justify these observations by a simple model ?

Searching state 1 Moving state 2 Two state model of the intermittent behavior (1D model) Searching state 1 target Moving state 2 State 1: local scanning state State 2: relocating state

Modelling of the searcher Phase 2 of duration T2 : Ballistic moving phase (constant velocity v) Prob(T2 > t) = exp(-f2t) T2 = 1/f2 = t2 Phase 1 of duration T1 : Diffusive moving phase (diffusion coefficient D) Prob(T1 > t) = exp(-f1t) T1 = 1/f1 = t1

Modelling of the targets real situation: targets are hidden at unknown sites, randomly distributed, with a small density 1/L modelling: the targets distribution is assumed to be regular L searcher target L  O Single target problem on a circle, with a searcher initially randomly distributed on the circle

Efficiency of the search process ? What is the search time m ? L  O 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?

Basic equations By using the backward Chapman-Kolmogorov differential equations, we obtain Boundary conditions:

Results In the low density limit, where m linearly depends on L !

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) then, m is minimum when

Limiting regimes: scaling laws If In this regime S, the searcher spends more time searching than moving If In this regime M, the searcher spends more time moving than searching Note that t a priori depends on the nature of the animal

Comparison with experimental data (1) Experimental data for f1 and f2 are available for fishes, birds and lizards (18 different species) [O’Brien, Kramer and Mc Laughin] Histogram of t obtained from Bimodal distribution ! Regime M t = 25 s Regime S t = 0.1 s

Comparison with experimental data: Log-Log plots of f1 and f2 Bénichou et al, Phys. Rev. Lett. 94, 198101 (2005)

Protein/DNA reactions Intermittent search : Protein/DNA reactions

?? Intermittent search strategies in molecular biology Protein searching for a target site on DNA : Transcription factor, Restriction enzyme… Typical reaction time (Smoluchowski) ~ 1000s ?? Experiments give ~ 1s

Non specific 1D diffusion on DNA First idea : 1D diffusion protein target DNA domain Non specific 1D diffusion on DNA

1D diffusion is still to slow ( ) 1D diffusion : experimental evidence [Bustamante] 1D diffusion is still to slow ( )

 Random relocation after each 3D excursion A model of intermittent search processes [Berg et al (1981)] 3D time distribution: 1D time distribution: target 3D 1D protein Strong assumption:  Random relocation after each 3D excursion No correlations of 3D excursions What is the mean first passage time at the target ? Is there an optimal strategy with respect to l ?

Basic equations (1) 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

Basic equations (2) Laplace transform : where is the Laplace tranform of the first passage time density of the mere 1D diffusion Mean first passage time (MFPT)

proportional to L+M in the large size limit MFPT Results proportional to L+M in the large size limit MFPT 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)]

Experimental confirmation [Stanford et al. (2000)] A BC 1 a b c 2 AB C Preference = Distance between targets The model gives To be compared with [Halford 2005, Desbiolles 2005]

Outline Intermittent strategies : a widely observed search behaviour Intermittent strategies : a generic search mechanism ?

Intermittence : a generic search mechanism ? Intermittence is involved at very different scales (animals, protein …) In the two previous 1D examples : combination of two regimes A « slow », but reactive motion : scanning phases for animals, diffusion along DNA A « fast », but  non reactive  motion : relocating phases for animals, excursions of proteins Is the efficiency of intermittence specific to the 1D case ? 2D model, involving intermittent strategies, which optimizes the encounter rate ?

Optimizing the encounter rate : Lévy strategies ? [Viswanathan et al., Nature (1999)] One state model I Non destructive search [many visits to the same target]: optimizes the encounter rate II Destructive search [target vanishes after first encounter] : is optimal : straight ballistic motion! What if searching and moving are incompatible ?

An alternative to Lévy strategies : intermittent strategies We consider a bidimensional two-state searcher, alternating « slow » reactive phases (state 1) « fast » relocating phases (state 2) Durations of each phase i are exponential, with mean ti = absence of temporal memory Fast  relocating phases are ballistic flights of constant velocity v and random direction = absence of orientational memory The searcher enjoys « minimum » memory skills

Two limiting modellings of the « slow » reactive phase In 2D, we have to define a reaction radius a « Dynamic » mode of detection The searcher diffuses in the reactive phase, reaction being infinitely efficient « Static » mode of detection The searcher is immobile in the reactive phase, reaction occuring at rate k

2D : Geometry and Basic equations The target (radius a) is centered in a spherical domain (radius b) with reflexing boundary conditions a b The searcher is initially uniformly distributed Backward equations : ( Ia is the indicatrix function of the target) System of integro-differential equations

Decoupling approximation Auxiliary functions : Assume : [Many reorientations before finding the target. Exact for d=1] Back to Linear ODE With Then

Results (1) : comparison with numerical simulations ln Dynamic mode Plain line = approximate theory Symbols = numerical simulations <t> Static mode Plain line = approximate theory Symbols = numerical simulations Good agreement for a wide range of parameters

Global optimal stategy Results (2) : optimal strategies (small density limit) ? Dynamic mode Global minimum Static mode Global minimum In both cases : Global optimal stategy O. Bénichou et al, to appear in Phys. Rev. E, Rapid Communication

“Losing time” in relocating phases can speed up a search process Conclusion Intermittent search strategies are widely observed because they are efficient “Losing time” in relocating phases can speed up a search process