Notes and Corrections
Chapter 2
pp. 43-44. Box 2A: infππ should be
Rπ
−π .
pp. 57. Equation (2.2.25) should be
∂ p(x,t)
∂ [A(x)p(x,t)] 1 ∂ 2 B(x)2 p(x,t)
,
=−
+
∂t
∂x
2
∂ x2
pp. 59. Equation above (2.2.31) should be
dX(t) = −κX(t)dt + dW (t),
pp. 65. First line below equation (2.3.8): the boundary condition should be absorbing, that is, τ(y) = 0 for y ∈ ∂ Ω .
pp. 68. Third equation. Remove factor of r2 in the denominator of right-hand expression
Chapter 3
pp. 114. Equation for Γ (z) below (3.2.8). Summation in second line should be with
respect to n rather than m.
Chapter 4
pp. 165. Last equation should be
1
λ 2 − (1 + r)λ + r = 0,
r=
π
.
ε
pp. 170. Equation below (4.1.10) should be
v+
∂ P+
∂ P−
− v−
= 0,
∂x
∂x
pp. 170. line 9 from bottom: P(x) = P(0)e−V x/D should be P(x) = P(0)eV x/D .
pp. 195. In equation (4.4.12) v f − should be v f + .
Chapter 6
pp. 276 Line 3: “using po (n, 0) = δn,0 ”
pp. 279 Fig. 6.3: rate of protein production should be r rather than k.
pp. 281 Caption of Fig. 6.4: ratios k± /k should be k± /γ.
pp. 281 3rd line below Fig. 6.4: “rate of degradation k” should be “rate of degradation γ”
pp. 292,293. In equations (6.4.3) and (6.4.6) change ∑ j=0,1,2 to ∑k=0,1,2
pp. 318: Last paragraph: ”nucleitide” should be ”nucleotide”
pp. 319 Just above equation (6.6.4):The cumulative distribution function is 1 −
S(n0 ,t) [rather than 1 − S(n0 , 0)]
Chapter 7
pp. 387.
We use a slight abuse of notation by referring to the matrix Anm = A(n, m, ; x) in
the Chapman-Kolmogorov equation as a transition matrix. Strictly speaking, the
transpose of A is the generator of the underlying Markov chain.
pp. 388. The quasi-steady-state reduction and boundary conditions.
• Suppose that the discrete states n = 1, . . . , N in the CK equation (7.4.5),
∂p
∂ [p(x, n,t)] 1 N
= −vn
+ ∑ A(n, n0 ; x)p(x, n0 ,t),
∂t
∂x
ε n0 =1
2
have non-zero velocities with vn > 0 for 1 ≤ n ≤ m and vn < 0 for m < n ≤ N.
Equation (7.4.5) is then an N-th order quasilinear hyperbolic PDE. In a bounded
domain x ∈ [0, L], it is necessary to impose a total of N boundary conditions. If both
ends are absorbing boundaries then we require
pn (0,t) = 0,
1≤n≤m
pn (L,t) = 0,
m < n ≤ N.
and
If the left-hand boundary is reflecting then we require
N
∑ vn pn (0,t) = 0.
n=1
However, this has to be supplemented by an additional m − 1 boundary conditions.
• On the other hand, the FP equation obtained by performing a QSS reduction is
a second-order parabolic PDE and one typically has to specify one boundary condition at either end. It follows that for N > 2, there is a mismatch in the number
of boundary conditions between the CK equation and the FP equation. This implies
that the QSS reduction may break down in a small neighborhood of either boundary,
as reflected by the existence of boundary layers. One way to eliminate the existence
of boundary layers is to ensure that the boundary conditions of the CK equation are
compatible with the QSS reduction. For example at x = 0,
pn (0,t) = ρn u(0,t) + O(ε),
where ρn are the components of the stationary density of the discrete Markov process. The absorbing boundary conditions of the CK equation are compatible with
u(0,t) = 0, and the reflecting boundary condition of the CK equation is compatible
with −Vu(0,t) + εD∂x u(0,t) = 0. One way to ensure compatibility of the remaining
m − 1 conditions at x = 0 is to set
pn (0,t) ρn
= ,
p1 (0,t) ρ1
n = 2, . . . , m.
• If there also exists a non-ballistic diffusing state n = 0 with v0 and diffusion coefficient D0 , then the order of the CK equation becomes N + 2. The reflecting boundary
condition of the CK equation at x = 0 becomes
N
∑ vn pn (0,t) − D0
n=1
∂ p0 (x,t) = 0,
∂ x x=0
whereas the absorbing boundary condition is supplemented by p0 (0,t) = 0. Again
there is the possibility of boundary layers.
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• The nature of boundary conditions in the case of x-dependent velocities is more
complicated because the partitioning of left and right moving states may differ at
the boundaries x = 0 and x = L. One has to look more closely at the characteristics
of the underlying quasilinear PDE.
pp. 389. In equation at the bottom of the page, the diffusion coefficient should be
(v −V )2 (v +V )2
.
+
D=ε
γβ+2
γβ−2
pp. 399 Section 7.5.2: We have absorbed a factor of 2 into ε.
pp. 401 Line 7: J0 = 0.25 + O(ε 2 /L2 ) rather than J0 = 0.5 + O(ε 2 /L2 )
Chapter 9
pp. 511.. Equation (9.2.6b) should be
∂v
= ∇2 v + κ(−u + bv + uv2 +Cuv)
∂t
pp. 528. In equation below (9.3.12) there is a missing delta function
hη`, j (t)η`0 , j0 (t 0 )i = D`, j;`0 , j0 δ (t − t 0 ).
pp. 550. In the Langevin equation (9.5.13) there is a spurious factor of U`
dU` (t)
= V` (U) + Ω −1/2 η` (t),
dt
Chapter 11
pp. 619. First paragraph: ”In this appendix...” should be ”In this chapter...”
pp.619. First line of section 11.1: ”In appendix A” should be ”In Chap. 10”
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