Problems in diffusion and absorption: How fast can you hit a target with a random walk?

Transcription

1 Problems in diffusion and absorption: How fast can you hit a target with a random walk? Andrew J. Bernoff Harvey Mudd College In collaboration with Alan Lindsay (Notre Dame) Thanks to Alan Lindsay, Michael Ward (UBC), Justin Tzou (UBC) and Theodore Kolokolnikov for introducing me to these problems Supported in part by Simons Foundation Grant #

2 Random Signaling Problems

3 Pollination What what probability will a pollen grain released from the stamen of one flower find the pistil of another flower? On average, how long does this take? How does the shape of the stamen affect this?

4 Foraging Suppose an animal is randomly foraging for food. Will the anteater find the ants? On average, how long does this take?

5 Molecular Signaling T Cell Antigens When an antigen (a toxin or a protein that promotes an immune response) binds to a receptor on a T-cell it can trigger the creation of antibodies. What is the probability of this binding occurring? On average, how long does this take? How does the distribution of the receptors affect this?

6 Modeling Diffusion & Capture pollen on a stamen

7 Brownian Motion Brownian motion is the random walk a molecule takes due to collisions with other molecules. Typically it can be describe by a distribution of velocities. Let V0 be the average speed L0 be the mean free path

8 Random Walks Due to the Central Limit Theorem after many steps the density, (~x, t), of the random walk converges to a Gaussian distribution. In N-dimensions, the distribution approaches (~x, t) = ~x 2 R N 1 p 4 Dt N e ~x 2 /(4Dt) p Dt (~x, t) The diffusion constant, D, scales as ~x D / (V 0 ) 2 /L 0 [Length] 2 x [Time] We call (~x, t) the diffusion kernel.

9 The Diffusion Equation The density of molecules, (~x, t), satisfies the diffusion equation in some domain. = 0 2 t = Dr 2 in 2 1 r ˆn = 0 1

10 Flux & Neumann BC s The flux of molecules is given by the Fourier Law ~F = Dru [velocity]x[density] At an impermeable boundary, we have a no-flux BC: ~F ˆn =0 ) ru ˆn = 0 ˆn ˆn ˆn ˆn ˆn At the particle level, particles are reflected. Angle of incidence =Angle of reflection

11 Capture & Dirichlet BC s At a boundary that captures every molecule that impacts it the density will drop to zero u = 0 At the particle level, particles are captured.

12 Partial Capture & Robin BC s Consider a boundary that reflects some, but not all, of the particles that strike it. ru ˆn + appleu = 0 apple 1 Mostly Reflecting apple 1 Mostly Absorbing ˆn ˆn ˆn ˆn ˆn At the particle level, some particles are captured while some are reflected.

13 Metrics for Capture

14 Probability of Capture Consider a domain with a target boundary and an escape boundary. The probability, f(x), that a molecule is captured satisfies Laplace s equation: r 2 =0in =1 for capture =0 for escape A 1D Escape Problem =1 x/l x L This is sometimes called harmonic measure.

15 Distribution of First Passage Time (FPT) Consider a domain with a target boundary. The density of mass capture per unit time, p(t), can be computed by integrating the flux over the target boundary. I p(t) = Dr ˆndS 2 [Mass] [Time] t = Dr 2 in = 0 2 ˆn 2 1 r ˆn = 0 1

16 The Importance of Dimension 1D x 0 R 2D r=a r=r 3D r=a r=r

17 Capture in 1D Consider a d-function release at x=r. u(x, t) u t = Du xx x>0, t > 0 u(x, 0) = (x R) x>0 u(0,t)=0 t>0 x 0 R By method of images: u(x, t) = 1 2 p Dt (X R) [e 2 /4Dt e (X+R)2 /4Dt ] FPT: p(t) = Du x (0,t)= Re R 2 /4Dt 2 p Dt 3/2 Probability of capture: p(t) Capture Time t t 3/2 (R) =1

18 Capture in 2D Consider a d-function release at r=r and a circular trap of radius a. 1 ut = D urr + ur r r=r r=a u(r, 0) = R) r > a u(a, t) = 0 t > 0 a2 R a2 t FPT: p(t) = d q(, ) = a = d Z 2 J0 (s)y0 ( s) 2 1 J0 ( s)y0 (s) q(, ) = se s ds s=0 [J0 (s)]2 + [Y0 (s)]2 Obtained with a Laplace Transform Probability of capture: Capture Time q(, 3) 2 [ln ]2 t 1 (r 2 R r > 0, t > 0 (R) = 1

19 r=r Capture in 3D Consider a d-function release at r=r and a spherical trap of radius a. r=a u t = D apple u rr + 2 r u r r>0, t > 0 u(r, 0) = 1 (r R) r>a 4 R2 u(a, t) =0 t>0 FPT: q(, 3) p(t) = a2 d q(, ) Capture Time q(, )= 1 = R a 1 e ( 1)2 4 2 p 3/2 = a2 t d (r) Probability of capture: (r) = a r 3/2 t r

20 Homogenization & Effective Boundary Conditions

21 The Basic Question Can you replace a complicated boundary condition or geometry with a much simpler one? Asymptotics Numerics Irregular shape cell Many pores Spherical Trap

22 The steady-state ansatz Suppose we are looking at a small target in a large domain. 2 When can we assume that the density is steady near the target? 1 L t = Dr 2 in L r 2 =0in = 0 2 r ˆn = 0 1 p Dt = 0 2! 0 for ~x L

23 A simple example Sphere with a partial absorbing boundary Smaller sphere with a totally absorbing boundary r 2 u =0 r>a u r (a) appleu(a) =0 lim u(r) =C 1 r!1 u(r) =C /applea a r r 2 u =0 u(b) =0 r>b lim r!1 u(r) =C 1 u(r) =C 1 1 b r Problems are equivalent when b = a 1+1/applea

24 A Pore on a Plane Consider a circular pore of radius a on an infinite plane. There is an exact solution for u(r, z). r 2 u =0 z>0 u z (r, 0) = 0 for r>a u(r, 0) = 0 for r<a lim u(r, z) =C 1 z!1 On the surface, the flux is: ( 0 r>a u z (r, 0) = pa2 C 1 r<a r 2 2 The total flux into the pore is: F pore =4aC 1 1 Flux ~ Perimeter!! a z 2 r=a r u z (r, 0) a r Sneddon, Elements of PDE, (1957).

25 Homogenization: Many pores on a planes For N pores of radius a: F pore =4aNC a2 Berg & Purcell 1977 NX i=1 NX j=1 j6=i 1 d ij Bernoff & Lindsay 2016 When the distance between pores, d ij a. This tells us how to replace many pores with one.

26 Homogenization: Pores on a Sphere This is a classic problem considered by Berg & Purcell (1977) in the dilute pore limit. F total =4aDNC Na/ R Knowing the flux allows you to replace the sphere with either a smaller sphere or a Robin BC. The next order correction in the asymptotics has been computed by Lindsay & Ward (2016).

27 Random Walk Numerics random walk simulation for an absorbing sphere

28 3D Random Walk Numerics Consider a d-function release at r=r and a spherical trap of radius a. r=a Numerical Method: % Captured r=r u t = D apple u rr + 2 r u r r>0, t > 0 u(r, 0) = 1 (r R) r>a 4 R2 u(a, t) =0 t>0 N particles Take Gaussian distributed steps of Variance = DDt Stop when particle is inside target. a =1,R=3,D =1 1/3 0 0 Exact Numerics Time N = 1000, t =.001 Challenges: Slow convergence. Long-time asymptotics often impossible to verify. Good for capture %, but difficult to obtain capture rate.

29 Challenge Problems

30 A Challenge Problem Suppose a set of particles are released at the center of a sphere of radius R. The sphere has N pores of radius a on its surface. Compute the distribution of exit times for the sphere. Notes: You can do this fairly effectively with a random walk particle code.

31 The Homogenized Challenge Problem Suppose a set of particles are released at the center of a sphere of radius R. The exit of the particles is modeled by a Robin boundary condition. Compute the distribution of exit times for the sphere. apple u t = D u rr + 2 r u r R>r>0, t > 0 u(r, 0) = 1 (r) R>r>0, 4 r2 u r (R, t)+appleu(r, t) =0 t>0 Notes: You may wish to start out with the 1D version. Make sure you consider apple,dsmall and large.

32 References Berg, Howard C. (1993). Random walks in biology. Princeton University Press. Berg, Howard C. & Edward M. Purcell. Physics of chemoreception. Biophysical journal 20.2 (1977) 193. Bialek, William & Sima Setayeshgar. Physical limits to biochemical signaling Proceedings of the National Academy of Sciences of the United States of America (2005): Cheviakov, Alexei F., Michael J. Ward, & Ronny Straube. "An asymptotic analysis of the mean first passage time for narrow escape problems: Part II: The sphere." Multiscale Modeling & Simulation 8.3 (2010): Berezhkovskii, A. M., Dagdug, L., Lizunov, V. A., Zimmerberg, J., & Bezrukov, S. M. (2012). Communication: Clusters of absorbing disks on a reflecting wall: Competition for diffusing particles. Journal of Chemical Physics, 136(21), Lindsay, Alan E. & Michael J. Ward. First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps. Preprint (2016).