Optimal search strategies for hidden targets

1. 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.

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

3. 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….] Intermittentsearch behavior, combining: • local scanning phases • relocating phases

4. 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 ?

5. Intermittent search : Animals searching for food

6. log-log Time spent in moving Time spent in searching Can we justify these observations by a simple model ? 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 correlationbetween these durations

7. Moving state 2 target Searching state 1 target Two state model of the intermittent behavior (1D model) • State 1: localscanning state • State 2: relocating state

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

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

10. L  O Efficiency of the search process ? • What is the search timem ? 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?

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

12. Results In the low density limit, where mlinearly depends on L !

13. then, m is minimum when 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)

14. 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 Limiting regimes: scaling laws

15. Histogram of t obtained from Bimodal distribution ! Regime S t= 0.1 s Regime M t= 25 s 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]

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

17. Intermittent search : Protein/DNA reactions

18. 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

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

20. 1D diffusion : experimental evidence [Bustamante] 1D diffusion is still to slow()

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

22. 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

23. 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)

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

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

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

27. 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 ?

28. 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 ?

29. 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

30. Two limiting modellings of the « slow » reactive phase In 2D,we have to define areaction radiusa « 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

31. 2D : Geometry and Basic equations Backward equations : System of integro-differential equations ( Ia is the indicatrix function of the target) The target (radius a) is centered in a spherical domain (radius b) with reflexing boundary conditions a b The searcher is initially uniformly distributed

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

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

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

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