By K. Schulten
Lectures in Theoretical Biophysics via ok. Schulten
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We will see below that for three-dimensional Brownian motion not all particles, even after arbitrary long time, will encounter a reactive boundary of finite size. 39), is ∂t N (t|x0 , t0 ) = − x0 2π D (t − t0 ) (t − t0 ) exp − x20 4 D (t − t0 ) . 21) which reads for the case considered here, ∂t N (t|x0 , t0 ) = −D ∂x p(x, t|x0 , t0 ) x=0 . 42) Evaluation of this expression yields the same result as Eq. 41). 21) can be. 3 Fluorescence Microphotolysis Fluorescence microphotolysis is a method to measure the diffusion of molecular components (lipids or proteins) in biological membranes.
110). For the exponent one can write − (r0 − a)2 4 D (t − t0 ) = (r0 − a) α + D (t − t0 ) α2 − (r0 − a + 2 D (t − t0 ) α)2 . 111) = z 2 (t ) We introduce the functions x(t ), y(t ), and z(t ) for notational convenience. 112) π D (t − t0 ) = = √ 2 πDα √ 2 πDα D (r0 − a) 4 D (t − t0 ) 3/2 − D (r0 − a) 4 D (t − t0 ) = dx(t )/dt Preliminary version 3/2 − D (r0 − a) 4 D (t − t0 ) 3/2 D2 (t − t0 ) α + 2 D (t − t0 ) (r0 − a) 2 (t − t0 ) 4 D (t − t0 ) + 3/2 Dα 4 D (t − t0 ) . 7: The left plot shows the fraction of particles that react at boundary r = a.
145) + i 1 2 [B·BT ]ij ∂i ∂j f[x] p(x, t|x0 , t0 ) . i,j Partial integration assuming a volume Ω with a surface ∂Ω allows one to change the order of the partial differential operators. For example, the first sum becomes Ai ∂i f [x] p(x, t|x0 , t0 ) = − dx Ω dx f [x] Ω i + i dx Ω = − ∂i Ai p(x, t|x0 , t0 ) ∂i Ai f [x] p(x, t|x0 , t0 ) i dx f [x] Ω ∂i Ai p(x, t|x0 , t0 ) i da · A f [x] p(x, t|x0 , t0 ) . 146) ∂Ω Assuming a p(x, t|x0 , t0 ) of finite spatial extent, such that it vanishes on the boundary ∂Ω, we can neglect the surface term.