Chapter 12
Nucleation and growth of molecular clusters
In Sect. 4.1, we considered the polymerization of actin and microtubular filaments.
This is just one example of a self-organizing process that involves the nucleation
and growth of molecular clusters. Other biological examples include the formation
of lipid bilayers, spherical and cylindrical vesicles, and protein aggregates such as
amyloid fibrils. Classical nucleation theory was originally developed within the context of physical processes such as liquid condensation in vapors [37, 11]. The main
assumptions of the classical approach are that (i) the system is below the temperature of the gas/liquid phase transition and (ii) the density of liquid droplets (clusters
or nuclei of vapor molecules) is dilute. One can then subdivide the system into an
ensemble of nuclei that are separated from the pure vapor by distinct interfacial
boundaries with associated macroscopic surface energies. Moreover, interactions
between nuclei can be neglected. The system is said to be in a weakly supersaturated metastable state, in the sense that there is an energy barrier between the given
state and a lower energy state where there is an irreversible conversion of monomers
to supercritical clusters that grow ever larger, resulting in the separation of the liquid and vapor phases. Another classical example, which is closely analogous to the
aggregation of biomolecules within a cell, is the crystallization of a colloid1 . In this
section we review some of the mathematical models that are used to describe the
dynamics of the nucleation and growth of clusters of solute molecules in a dilute
solution, and then apply the theory to various biological examples such as the formation of micelles and vesicles, and protein aggregation.
12.1 Becker-Doring model of aggregation-fragmentation
A classical approach to modeling aggregation and fragmentation processes is to
consider a deterministic (mean-field) model in which cluster size is a discrete
1
A colloid is a solution that has particles ranging between 1 and 1000 nanometers in diameter, in
which the particles are evenly distributed throughout the solution, rather than settling at the bottom
of a container.
1
quantity, corresponding to an integer multiple of some basic molecular component
(monomer). Becker and Doring [3] introduced a system of kinetic equations, in
which clusters form by monomers colliding with each other and then grow via subsequent collisions between clusters and monomers. The main simplifying assumption is that interactions between clusters are ignored, which is reasonable when the
cluster density is relatively small. If un , n ≥ 2, denotes the concentration of clusters
of size n and u1 denotes the concentration of monomers then the Becker-Doring
(BD) equations take the form
dun
= Jn−1 − Jn , n ≥ 2
dt
du1
= −J1 − ∑n≥1 Jn ,
dt
(12.1.1a)
(12.1.1b)
where we have introduced the particle fluxes
Jn = an u1 un − bn+1 un+1 .
(12.1.2)
Here an and bn denote the rates of aggregation and partial fragmentation of a cluster
of size n. More precisely, equations (12.1.1a) and (12.1.1b) are a slightly modified
version of the original BD equations, whereby the total mass of the system is conserved [1, 38]. The original model took the monomer concentration u1 to be fixed
[3], which was also assumed in the model of actin polymerization (see Sect. 4.1). In
recent years the BD equations (12.1.1a) and (12.1.1b) have been applied to a wide
range of chemical and biological processes including micelle and vesicle formation
[46, 10], protein aggregation [5], viral capsid assembly [19], and the formation of
robust protein concentration gradients [41]. There have also been several mathematical studies of the existence and uniqueness of steady-state solutions and large-time
asymptotics [1, 38, 47]. See Wattis [48] for a recent mathematical review of the BD
equations.
an
bn+1
Xn+1
Xn
Fig. 12.1: Aggregation/fragmentation steps of the Becker-Doring model. Here Xn denotes a cluster
of size n.
2
12.1.1 Constant monomer concentration
Let us begin the analysis of the BD equations in the simpler case that the monomer
concentration is fixed, u1 (t) = u for all t, and the forward and backward rate coefficients are size-independent, an = a, bn = b. This model was briefly studied in Sect
4.1 within the context of actin polymerization. We will follow closely the analysis
of Wattis and King [47]. We first look for an equilibrium solution that satisfies detailed balance. That is, setting Jn = 0 for all n ≥ 1 yields the following solution for
the concentrations:
au n−1
u ≡ θ n−1 u.
(12.1.3)
ueq
n =
b
Requiring that un → 0 as n → ∞ implies that an equilibrium solution only exists for
θ < 1 or u < b/a; the corresponding mass density is
ρ0 ≡
u
∑ nun = (1 − θ )2 .
(12.1.4)
n≥1
If θ ≥ 1 then one has to consider a more general class of steady-state solutions
for which Jn = J 6= 0. There is then a constant flux through the system and detailed balance no longer holds. Iteratively solving the constant flux conditions
auun − bun+1 = J yields the steady-state solution
J(θ n − θ )
n−1
uss
=
u
θ
−
.
(12.1.5)
n
au2 (θ − 1)
If θ < 1 then un → −J/b(1 − θ ) as n → ∞. It follows that un becomes negative at
large n for J > 0, whereas for J < 0 the concentrations remain bounded away from
zero, implying that the mass density is infinite. Hence, the only steady-state solution
is the equilibrium solution when θ < 1. In the case θ > 1, there exists a steady-state
2
solution uss
n = u with J = au (1 − 1/θ ) = u(au − b). This is the least divergent
solution in the sense that all other values of J produce solutions with un → ±∞ as
n → 0. It turns out that in the large t limit, the system approaches the least divergent
solution when θ > 1 and the equilibrium solution when θ < 1. Finally, for θ = 1
one finds that
J(n − 1)
ss
un = u 1 −
.
(12.1.6)
bu
The least divergent solution is un = u with J = 0, which is a mixture of the two other
cases.
Another way to differentiate between the two cases θ > 1 and θ < 1 is terms of
the behavior of the function
∞
V=
∑ un
n=1
log(un /uθ n−1 ) − 1 .
It can be shown that
3
(12.1.7)
dV
= − ∑ (auun − bun+1 )(log(auun ) − log(bun+1 ) ≤ 0,
dt
n≥1
with equality only holding at equilibrium. For θ < 1, the equilibrium is a minimizer
of V , whereas V is not bounded below when θ > 1. Hence, only in the former case
is V a Lyapunov function that guarantees convergence to the equilibrium solution.
For θ > 1 the steady-state mass density is infinite, suggesting that the approach to
steady-state is not uniform in n. However, asymptotic approximation methods can be
used to determine the large-time behavior [47]. (In fact, in the case of monodisperse
initial data for which un (0) = 0 for all n > 1, one can write down an exact solution
for un (t) [47]. However, exact solutions for more general rate coefficients are not
known.)
Large-time kinetics for θ < 1
eq
Since the system is expected to converge to the equilibrium solution un = uθ n−1 ,
eq
we introduce the rescaling ψn (t) = un (t)/un so that ψn (t) → 1 as t → ∞. The BD
equations reduce to
1 dψn
= ψn−1 (t) − ψn (t) − θ ψn (t) + θ ψn+1 (t),
b dt
n ≥ 2.
(12.1.8)
with ψ1 (t) = 1. We also assume that at t = 0 there are no clusters of infinite size so
that ψn (0) → 0 and n → 0. In order to generate an approximate solution for ψn (t),
we suppose ψn (t)is a slowly varying function of n at large times; this assumption
will be verified below. Taking the continuum limit of equation (12.1.8) by setting
Ψn (t) = ψ(n,t) and Taylor expanding Ψ (n ± 1,t) to second order about n yields the
advection-diffusion equation
∂Ψ
b
∂ 2Ψ
∂Ψ
= (1 + θ ) 2 − b(1 − θ )
∂t
2
∂n
∂n
1
(b)
0.8
concentration un(t)
concentration un(t)
(a)
0.6
equilibrium
0.4
0.2
0
transition region
40
80
120
160
aggregate size n
1
0.8
0.6
transition region
0.4
increasing time
0.2
0
200
(12.1.9)
20
40
60
80
aggregate size n
100
Fig. 12.2: Large-time asymptotics of cluster distribution function un (t) for fixed t. (a) Case θ < 1.
(b) Case θ > 1 shown at at times t = 200, 400, 600 with a = u = 1, b = 0.9. Speed of the front is
V = 0.1 and the width of the front grows with t. [Redrawn from Wattis [48].]
4
with effective diffusivity D = b(1 + θ )/2 and drift velocity V = b(1 − θ ). The
boundary conditions are Ψ (0,t) = 1 and Ψ (n,t) = 0 for n → ∞. Since we are dealing with the large n,t regime, we expect the solution to be of the form of a traveling
wave Ψ (n,t) = f (n − V t) with f (x) → 1 as x → −∞ and f (x) → 0 as x → ∞; this
satisfies the boundary conditions. It follows that we have the asymptotic solution
n −V t
1
,
Ψ (n,t) = erfc √
2
4Dt
where erfc(x) is the complementary error function
2
erfc(x) = √
π
Z ∞
x
2
e−y dy.
Hence,
b au n
un (t) ∼
erfc
2a b
n − [b − au]t
p
2[b + au]t
!
,
t → ∞.
(12.1.10)
For large times t, the cluster distribution function un (t) has three regions as illustrated in Fig. 12.2(a): (i) For n V t the system has effectively reached equilibrium;
(ii) For n = V t + O(t 1/2 ) there is a transition region; (iii) For n V t the concentration is still asymptotically small. Note that as time increases the transition region
moves to larger n and widens.
Large-time kinetics for θ ≥ 1
The case θ > 1 can be analyzed along similar lines to θ < 1. Now one rescales
the concentrations according to ψn (t) = un (t)/u since the system converges to the
steady-state solution un (t) = u. Taking the continuum limit in n with Ψ (n,t) = ψn (t)
results in the advection-diffusion equation
∂Ψ
∂Ψ
b
∂ 2Ψ
= (1 + θ ) 2 − b(θ − 1)
.
∂t
2
∂n
∂n
(12.1.11)
We thus obtain the asymptotic solution
u
un (t) ∼ erfc
2
n − [au − b]t
p
2[b + au]t
!
,
t → ∞.
(12.1.12)
The evolution of this traveling front solution is illustrated in Fig. 12.2(b). Finally, in
the special case θ = 1 the continuum limit yields a pure diffusion equation, and the
approach to the equilibrium solution un = u occurs via a stationary diffusive wave
according to
5
un (t) ∼ u erfc
n
√
2 bt
,
t → ∞.
(12.1.13)
12.1.2 Constant mass formulation
The analysis of the full BD equations (12.1.1a) and (12.1.1b) is much more involved due to the presence of nonlinearities. Formally speaking, multiplying equation (12.1.1a) by n, summing with respect to n and adding equation (12.1.1b) gives
d
∑ nun = −J1 − ∑ Jn + ∑ n(Jn−1 − Jn )
dt n≥0
n≥2
n≥1
= −J1 − ∑ Jn + 2(J1 − J2 ) + 3(J2 − J3 ) + . . . = 0.
n≥1
This implies that the total mass is conserved:
∑ nun (t) = ∑ nun (0) ≡ ρ0 .
n≥1
(12.1.14)
n≥1
One subtlety regarding the above derivation of the conservation equation is that it
has been assumed one can reverse the order of infinite summation and differentiation. It turns out that for certain choices of the n-dependent transition rates an , bn ,
commutativity breaks down, reflecting the fact that the total mass of the equilibrium
solution is less than the initial mass [1, 38], see below. The physical interpretation
of such behavior is that an equilibrium solution no longer exists, since there is an
irreversible transfer of monomers to ever larger clusters. The nucleation of these
clusters can ultimately lead to crystallization of a solid from solution, for example.
A first step in the analysis of the BD equations (12.1.1a) and (12.1.1b) is to look
for steady-state solutions Jn (t) = J for all n ≥ 0. The only physical solution is the
equilibrium solution J = 0, otherwise du1 /dt → −∞. Hence
bn+1 un+1 = an u1 un ,
n≥1
which on rearranging and iterating gives
un = Qn un ,
Qn =
an−1 an−2 . . . a1
.
bn bn−1 . . . b2
(12.1.15)
Here u1 = u and, in contrast to Sect. 12.1.1, u has to be determined self-consistently
from the mass conservation condition
ρ0 = F0 (u) ≡
6
∑ nQn un
n≥1
(12.1.16)
We will assume that for a given choice of an and bn the infinite series defining F0 (u)
has a finite radius of convergence u = R. There then exists a steady-state solution
provided that equation (12.1.16) has a solution for which u ≤ R, see below.
Approach to equilibrium can be established by constructing an appropriate Liapunov function [38]. That is, consider the function
∞
L=
∑ un [log(un /Qn ) − 1] .
(12.1.17)
n=1
Differentiating with respect to t gives
dun
dL
=∑
log(un /Qn )
dt
n≥1 dt
= (−J1 − ∑ Jn ) log(u/Q1 ) + ∑ (Jn−1 − Jn ) log(un /Qn )
n≥2
n≥2
= J1 log(u2 Q21 /Q2 c21 ) + J2 log(u3 Q2 /uu2 Q3 ) + J3 log(u4 Q3 /u3 uQ4 ) + . . .
= J1 log(u2 Q21 /Q2 c21 ) + ∑ Jn log(un+1 bn+1 /uun an ) + . . .
n≥2
2
= (a1 u − b2 u2 ) log(u2 b2 /a1 u2 ) + ∑ (an uun − bn+1 un+1 ) log(bn+1 un+1 /an uun )
n≥2
≤ 0,
(12.1.18)
Moreover, L is bounded from below. First note that each term
f (un ) ≡ un [log(un /Qn ) − 1]
is a convex function of un so that at an arbitrary value u∗n ,
f (un ) − f (u∗n ) ≥ f 0 (u∗n )[un − u∗n ],
which implies that
L≥
∑ (u∗n [log(u∗n /Qn ) − 1] + (un − u∗n ) log(u∗n /Qn )) .
n≥0
Let us choose u∗n = Qn zn so that
L
≥ ∑n≥0 (nun log z − Qn zn )
= (ρ0 − c) log z − ∑ Qn zn
n≥0
= (ρ0 − c) log z − F0 (z) > −∞.
It follows that L must approach a limit as t → ∞ such that dL/dt → 0. Since every
term on the right-hand side of the penultimate line of Eq. (12.1.18) is non-positive,
it follows that the individual terms approach zero:
Jn ≡ an uun − bn+1 un+1 → 0
7
as t → ∞,
n ≥ 0.
Therefore,
u n − Qn u n → 0
as t → ∞,
n≥1
(12.1.19)
However, we still need to determine how u behaves as t → ∞. From the conservation
equation, we have
(12.1.20)
lim ∑ nun (t) = ρ0 .
t→∞
n≥1
On the other hand, Eq. (12.1.19) tells us that
lim nun (t) =
∑ t→∞
h
i
n
nQ
lim
u(t)
.
n
∑
n≥1
n≥1
t→∞
(12.1.21)
If we can interchange the two limit operations t → ∞ and n → ∞, then we obtain the
asymptotic results
lim u(t) = u,
(12.1.22)
t→∞
with
ρ0 = F0 (u).
(12.1.23)
Ball et al [1] carried out a rigorous analysis of the BD equations, in which they
investigated the existence of steady-state solutions and mass conservation under various conditions on the rate constants an , bn and the initial statesun (0). One of their
major findings was that in certain regimes, the system exhibits a form of metastability. We distinguish three different cases (see also [38, 48]).
Case I: R = ∞.
In this case the function F0 (u) has an infinite radius of convergence and so 0 ≤
F0 (u) < ∞ for all 0 ≤ u < ∞ with F0 (u) → ∞ as u → ∞. Given any finite initial
mass ρ0 , there exists a unique equilibrium solution u1 = u for which F0 (u) = ρ0 . A
simple example is obtained by taking aggregation rates an = 1 and fragmentation
rates bn = n, which implies that fragmentation dominates in large clusters. Then
Qn = 1/n! and
un
F0 (u) = ∑ n = ueu .
n≥1 n!
This is a monotonically function of u with F0 (0) = 0 and F0 (∞) = ∞. Thus there
exists a unique solution to the equation F0 (u) = ρ0 .
Case II: 0 < R < ∞ and F0 (R) = ∞.
Now the function F0 (u) has a finite radius of convergence and as u → R− , we have
F0 (u) → ∞. For any initial mass ρ0 , the equation F0 (u) = ρ0 has a unique solution
for u ∈ [0, R). In particular, for large initial masses, there is an upper limit to the
equilibrium monomer concentration. A simple example occurs for n-independent
8
aggregation-fragmentation rates, an = a, bn = b, as in the case of linear polymerization and the formation of cylindrical micelles (see Sect. 12.3). Then Qn = (a/b)n−1
and
au n−1
u
=
.
F0 (u) = ∑ n
2
b
(1
−
au/b)
n≥1
The equilibrium equation ρ0 = F0 (u) then has the unique solution
u=
2ρ0
p
.
1 + 2aρ0 /b + 1 + 4aρ0 /b
Note that u → 0 as ρ0 → 0 and u → (b/a)− as ρ0 → ∞. The radius of convergence
is R = (b/a)− .
Case III: 0 < R < ∞ and F0 (R) < ∞.
Again the function F0 (u) has a finite radius of convergence R, but there now exists
a critical mass ρc at which the monomer concentration takes its maximal value,
ρc = F0 (R). If ρ0 < ρc then there exists a unique equilibrium solution with the same
mass ρ0 . As in Case II, convergence to equilibrium is strong in the sense that
lim
t→∞
∑ n|un (t) − Qn un | = 0,
F0 (u) = ρ0 .
(12.1.24)
n≥1
This means that the two limits t → ∞ and n → ∞ are interchangeable. On the other
hand, if ρ0 > ρc then there is no equilibrium solution with the same mass as the
initial data. One finds that
lim un (t) = Qn Rn ,
t→∞
where R is the radius of convergence of the series representation of F0 . Now convergence is weak, that is, individual terms of the series (12.1.24) converge to zero, but
their sum does not2 . It follows that the two limits are not interchangeable:
∑ nun (t) = ρ, ∑ nQn Rn = F0 (R) = ρc < ρ0 .
n≥1
n≥1
The excess mass ρ0 − ρc reflects the fact that there is an irreversible transfer of
monomers to ever larger clusters (nucleation and growth), which ultimately leads to
phase separation (formation of macroscopic clusters), see Sect. 12.2. One biological
example of case III is the formation of spherical micelles, see Sect. 12.3.
2
A more familiar example of weak convergence was highlighted in [48]. Consider the dif2
fusion Requation ut = uxx on R with initial condition√u(x, 0) = e−x . One expects the integral
∞
I[u] = −∞ u(x)dx to be conserved, that is, I[u] = 2 π for all t ≥ 0. However, the solution
√
2
u(x,t) − e−x /4(t+1) / t + 1 implies that pointwise u(x,t) → 0 as t → 0. Since I[0] = 0, we see that
the limit t → ∞ does not commute with evaluating the integral, which is analogous to performing
an infinite summation.
9
12.2 Nucleation and coarsening in the BD model
We begin by briefly describing the classical theory of nucleation as illustrated in
Fig. 12.3. Suppose that one starts with a pure monomer solute and rapidly increases
the concentration of the solute beyond the critical concentration uc = R (stage I).
The solute is then said to be supersaturated and enters a nucleation stage in which
monomers are rapidly converted into small clusters, some of which survive possible
conversion back to monomers and grow into larger stable nuclei or particles (stage
II). Let nc denote the critical cluster size for nucleation. Since small clusters have
relatively large free energies, the formation of small clusters is difficult and this
generates an energy barrier ∆ Gc to the nucleation of stable large clusters. A quasisteady-state is then reached, in which stable nuclei are produced at a constant rate
J. A classical formula for the nucleation rate is
J ∼ ua(nc )e−∆ Gc /kB T ,
where u is the monomer concentration, a(nc ) is the rate at which monomers bind
to clusters of critical size nc . However, this formula is based on the assumption
that every cluster that grows beyond nc continues to grow without decaying back
to a smaller size. However the actual nucleation rate is smaller, since growth is not
irreversible, and so one must scale J by a correction factor Z, 0 < Z < 1, known as
Fig. 12.3: Illustration of the the classical mechanism for nucleation and growth of molecular aggregates in solution. (I) A rapid increase in the concentration of free monomers in solution. (II)
Once the monomer concentration S exceeds a critical value Sc , it becomes supersaturated and nucleation occurs, which significantly reduces the concentration of free monomers in solution. (III)
Following nucleation, growth and cluster coarsening occurs under the control of the diffusion of
the monomers through the solution.
10
the Zeldovich factor. Eventually, the monomer concentration becomes so depleted
that the number of stable clusters becomes approximately constant, and the system
enters stage III. In this growth and coarsening stage larger nucleated particles grow
at the expense of smaller particles, which causes the total number of particles to
decrease.
12.2.1 Nucleation
The BD equations were originally introduced as a model of nucleation in which
the monomer concentration is effectively fixed [3]. In order to understand how this
relates to the constant-mass version of these equations considered in Sect. 12.1.2,
suppose the rates an , bn are chosen so that Case III holds. If the initial monomer
concentration is just above the radius of convergence R, u1 (0) = u = R + ε, and
there are initially no clusters, then the nucleation rate for the formation of clusters
is exponentially small, and u1 (t) ≈ u for exponentially large times. Since u > R,
there does not exist an equilibrium solution, that is, nQn un diverges (albeit slowly)
as n → ∞. This suggests looking for a non-equilibrium steady-state solution for
which Jn (t) = J for all n ≥ 1, and identifying J as the nucleation rate. Such a state
is metastable, since eventually the pool of monomers will be depleted due to the
irreversible growth of arbitrarily large clusters. Rigorous results on metastable states
in the BD equations have been developed by Penrose [38]. Here we will follow a
more physical approach [37, 11].
First, note that we can use statistical mechanics to relate Qn to the free energy
change Gn for the reaction
nX1 ↔ Xn ,
where Xn denotes a cluster of size n. That is, the ratio of the forward and backward
rates of X + Xn−1 ↔ Xn is given by (see Chap. 1)
an−1 u
= e−∆ gn−1 /kB T ,
bn
where ∆ gn−1 is the change in free energy for adding one monomer to the cluster
Xn−1 . It follows that
bn = uan−1 e[Gn −Gn−1 ]/kB T
(12.2.1)
and
n
n
a j−1 u
= e−Gn /kB T ,
b
j
j=2
un−1 Qn ≡ ∏
Gn =
∑ ∆ g j−1 .
(12.2.2)
j=2
In cases where the equilibrium solution exists, we have the Boltzmann distribution
n
−Gn /kB T
ueq
.
n = u Qn = ue
Consider a steady-state solution for which Jn (t) = J > 0 for all n. Then
11
(12.2.3)
J = an uun − bn+1 un+1 ,
(12.2.4)
which on dividing by an un+1 Qn becomes
J
an
un+1 Q
=
n
un
n
u Q
n
−
un+1
.
n+1
u Qn+1
Summing this equation from n = 1 to some maximum value n = N (with Q1 = 1)
gives
N
1
uN+1
J∑
.
(12.2.5)
= 1 − N+1
n+1 Q
a
u
u
QN+1
n
n=1 n
In the metastable state of nucleation processes, the condensed (liquid) phase is thermodynamically stable, which means that clusters of sufficiently large size have a
faster growth rate then shrinkage rate, that is, uaN−1 > bN (or ∆ gN < 0) for sufficiently large N. Thus, uN−1 QN grows exponentially with N for large N. On the
other hand, the assumption of small depletion of monomers requires that uN u.
Therefore, taking the limit N → ∞ in equation (12.2.5) yields the steady-state flux
∞
J=u
1
∑ an un Qn
j=1
!−1
.
(12.2.6)
This determines the nucleation rate. Finally, returning to equation (12.2.4) and iterating from n = 1 to r gives
!
r−1
1
r
.
u r = Qr u J 1 − ∑
n
j=1 an u Qn
Substituting for J shows that
ur = ueq
r
∞
1
∑ an un Qn
j=r
!−1
.
(12.2.7)
In classical nucleation theory, the flux J is estimated using steepest descents.
Assuming that nucleation is dominated by large clusters, we can treat the cluster
size n as a continuous variable and approximate the sum in equation (12.2.6) by an
integral. Let nc denote the critical cluster size for which the term Γn = an un Qn is
minimized,
d
(an un Qn ) = 0 at n = nc .
dn
Assuming that the sum in (12.2.6) is dominated by values in a neighborhood of nc
and an is a slowly varying function of n, we can then use the approximation
12
∞
Z
∞
1
1
1
∑ an un Qn ≈ a(nc ) −∞ eG(n)/kB T dn ≈ a(nc ) eG(nc )/kB T
j=1
s
2πkB T eG(nc )/kB T
=
.
|G00 (nc )| a(nc )
Z ∞
−∞
00 (n )|(n−n )2 /k T
c
c
B
e−|G
We thus obtain the classical Zeldovich formula for the nucleation rate:
s
|G00 (nc )|
a(nc )e−G(nc )/kB T
J=u
2πkB T
dn
(12.2.8)
12.2.2 Coarsening
We now turn to the dynamics of large clusters in the large-time limit, where the
metastable state of the nucleation stage has broken down and the system has entered
the growth stage. Suppose that the rate coefficients vary with cluster size n according
to a power law of the form [38, 32]
q
an = nγ , bn = an zs + γ
(12.2.9)
n
where zs > 0, q > 0 and 0 < γ < 1. Such coefficients arise in the formation of liquid
droplets in a supersaturated vapor (for which γ = 1/3 in 3D) or the phase segregation of a quenched binary alloy. (Although the above choice of rate coefficients is
less relevant to the biological applications considered in this Chap., it provides an
analytical framework for exploring the general phenomenon of coarsening.) From
equation (12.1.15) we see that the equilibrium solution is of the form un = Qn un .
Given the rate coefficients (12.2.9) one finds that for large n [32],
C0
an−1 an−2 . . . a1
q
1−γ
−γ
Qn =
≈
n [1 + O(n )] .
exp −
bn bn−1 . . . b2
(1 − γ)zs
nα zn−1
s
It follows that the series F(u) = ∑n≥1 nQn un has the radius of convergence
R = lim
n→∞
bn+1
= zs < ∞,
an
and
ρs =
∑ nQn zns < ∞.
n≥1
Hence, the system corresponds to Case III of Sect. 12.1.2.
One of the important consequences of having power law rate coefficients is that
the evolution of large clusters in the BD model is effectively governed by the classical coarsening model of Lifshitz, Slyozov, and Wagner (LSW) [30, 45]. Here
we sketch a heuristic argument [38]; see Niethammer [32] for a rigorous deriva13
tion. First, in order to consider large times, rescale time according to τ = εt with
o < ε 1, such that
dun
1
= (Jn−1 − Jn ),
(12.2.10)
dt
ε
with the fluxes rewritten as
Jn = (nγ (u − zs ) − q)un − (bn+1 un+1 − bn un ).
(12.2.11)
Introduce a critical size n0 = ln(1/ε), say, separating small and large clusters. For
n ≥ n0 , substitute λ = εn and treat λ as a continuous variable. Rescaling the cluster
densities and fluxes, we write
un = ε 2 v(λ ,t),
Jn = ε 2 K(λ , τ).
Finally, introducing the rescaled monomer density
u − zs = ε γ U(τ),
we find that to lowest order
∂v ∂K
+
= 0,
∂τ ∂λ
K = (λ γ U(τ) − q)v.
(We have used the fact that bn+1 un+1 ≈ bn un for large n.) The final assumption is
that relatively small clusters (n < n0 ) have reached steady-state so that
un ≈ Qn zns ,
n < n0 .
It follows from mass conservation that
∞
ρ=
n0 −1
∞
n=1
n=n0
∑ nun = ∑ nun + ∑ nun
n=1
≈
n0 −1
∞
∑ nQn zns + ∑ nun
n=1
≈ ρs +
Z
n=n0
λ v(λ , τ)dλ .
We have replaced the upper limit in the first summation by ∞ since n0 1, and exploited the ε 2 scaling of un to write the second sum as an integral over λ . Combining
these results, we finally obtain the classical LSW model of coarsening:
∂v
∂
+
((λ γ U(τ) − 1)v) = 0
∂ τZ ∂ λ
λ v(λ , τ)dλ = ρ − ρs .
14
(12.2.12a)
(12.2.12b)
For simplicity, we have set q = 1. Note that multiplying equation (12.2.12a) by λ ,
integrating by λ and imposing equation (12.2.12b) shows that
U(τ) = R
R
v(λ , τ)dλ
.
λ γ v(λ , τ)dλ
(12.2.13)
The LSW model was originally formulated in terms of spherical clusters of radius
r rather than the number of monomers within the cluster[30, 45]. The two versions
can be related by noting that
4π 3
r = nν0 = λ ν0 /ε,
3
where ν0 is the volume of a single monomer (assuming close packing). Rescaling
with (4πε/3ν0 )r3 = R3 we have R3 = λ . Setting F(R, τ)dR = ν(λ , τ)dλ and γ =
1/3 then gives
∂F
∂
1
1
1
+
−
F =0
(12.2.14a)
∂ τ ∂ R R Rc (τ) R
Z
R3 F(R, τ)dR = constant.
(12.2.14b)
where a constant factor has been absorbed into τ and
R
1
F(R, τ)dR
.
= U(τ) = R
RF(R, τ)dR
Rc (τ)
(12.2.15)
LSW also showed that the pair of equations (12.2.14) has a family of self-similar
solutions (as do the more general LSW equations (12.2.12) [38, 32]). That is, if
F(R, τ), Rc (τ) is a solution then so is the re-scaled solution
k4/3 F(k1/3 R, kτ),
k−1/3 Rc (kτ)
for any positive constant k. This suggests looking for a self-similar solution of the
equations, that is, one that is invariant under this transformation. In particular,
F(R, τ) = τ −4/3 g(Rτ −1/3 ),
Rc (τ) = (τ/σ )1/3 ,
(12.2.16)
where σ is a constant and g(x) obeys an ODE obtained by substituting the similarity
solution into (12.2.14a):
dg
d 1 1 1
4g(x) + x − 3
−
g = 0,
(12.2.17)
dx
dx x x̄ x
with x̄ = σ −1/3 . The constant σ is related to g by the self-consistency equation
(12.2.15).
It turns out that there exists a family of self-similar solutions, each of which
is localized on a finite interval [0, xm ] of the similarity variable x = Rτ −1/3 . These
15
solutions are parameterized by σ , and the range of possible values of σ is determined by enforcing continuity of F(R, τ) on [0, xm ] and the normalization condition
(12.2.14b). For each of these solutions, the aggregate radius grows as τ 1/3 and the
number of clusters decreases like τ −1 . An interesting question is which self-similar
solution (if any) is selected by a given initial condition; in the original analysis of the
LSW it was assumed that the initial distribution F(R, 0) had non-compact support,
in which case a unique solution is selected. Various selection rules and asymptotic
results have been found [15, 31]
12.3 Self-assembly of phospholipids
Amphiphiles, such as the phospholipids of the plasma membrane (see Sect. 7.4.1),
can spontaneously form aggregates in aqueous solution [21, 33]. Aggregation is
driven by the fact that it is energetically favorable to form clusters, since they shield
the hydrophobic regions of individual lipids from contact with water. However, formation of clusters lowers the number of objects in the system, thus reducing the
entropy. Competition between these two factors is taken into account by the free
energy of the system. The resulting geometrical arrangement of the lipids depends
on the amphiphile concentration, the molecular volume, the length of the hydrocarbon chain, and properties of the solvent. Examples include monolayer cylindrical
Fig. 12.4: Examples of various phospholipid aggregates. The type of aggregate depends on an
effective packing parameter p = v/la, which describes the geometry of the volume occupied by an
individual molecule (eg. cone, truncated cone or cylinder). Here a is the polar head surface, v is
the tail volume, and l is the hydrophobic tail length.
16
or spherical aggregates (micelles), planar bilayers, and spherical bilayers (vesicles),
see Fig. 12.4. Here we will focus on spherical and cylidrical micelles.
Let un be the concentration of a micelle of size n. A standard calculation of the
entropy in the dilute limit (see also Sect. 1.4) shows that the total free energy of the
system is
∞
∞
F = − ∑ un εn + kB T
n=2
∑ un (ln un − 1),
(12.3.1)
n=1
with the monomer density
∞
u1 = ρ0 − ∑ un .
(12.3.2)
n=2
Here ρ0 is the total density and εn = ne1 − en where en is the energy of an n-cluster,
that is, −εn is the total binding energy of the cluster. The equilibrium density of
n-clusters (n ≥ 2), assuming it exists, can be found by minimizing the free energy
with respect to un , n ≥ 2,
∂F
= 0.
∂ un
Taking into account that ∂ u1 /∂ un = −n (n ≥ 2), we find
n εn /kB T
.
ueq
n = u1 e
(12.3.3)
Comparison with equation (12.2.3) shows that the activation free energy used in the
BD model is
Gn = −εn + kB T n ln(1/u1 ).
(12.3.4)
Finally, we obtain a self-consistency condition for the monomer density in terms of
the total density ρ (see equation (12.1.16)):
∞
ρ0 = F0 (u1 ) ≡
∑ nun1 eεn /kB T .
(12.3.5)
n=1
The behavior of the system will then be determined by the n-dependence of the
energy εn .
Let us first consider a rod-like cylindrical micelle for which a the energy is typically modeled as
εn = (n − 1)αkB T,
(12.3.6)
where αkB T is the monomer-monomer bonding energy [21]. Substituting into equation (12.3.5) and using
∞
d
∞
x
∑ nxn = x dx ∑ xn = (1 − x)2 ,
n=1
x<1
n=1
we find that
ρ0 =
u1
.
(1 − u1 eα )2
17
(12.3.7)
This has the unique solution
√
1 + 2ρ0 eα − 1 + 4ρ0 eα
u1 =
2ρ0 e2α
with u1 < e−α for all 0 < ρ0 < ∞. It follows that the system is an example of Case
II within the context of the BD model (Sect. 8.A.1), since the radius of convergence
of F0 (u) is R = e−α and F0 (u) → ∞ as u → R− . Hence, cylindrical micelles form an
equilibrium, polydisperse distribution of sizes in a process known as micellization.
The average cluster size in equilibrium is then
hni ≡
∑
n≥1
ueq
n
!−1
∑ nueqn = ρ0 (1 − u1 eα )
n≥1
√
1 + 4ρ0 eα − 1
=
,
2
(12.3.8)
The behavior of spherical micelles is very different. A spherical aggregate with n
molecules will have a radius r = (3nv0 /4π)1/3 , where v0 is molecular volume. There
will be an energy 4πr2 γ associated with surface tension, where γ is the interfacial
energy per unit area. Thus, we can write the total energy of a spherical micelle for
n 1 as [21]
3
εn ∼ (n − 1)αkB T − σ n2/3 ,
(12.3.9)
2
where σ = 2γ(4πv20 /3)1/3 . In this case, equation (12.3.5) yields the self-consistency
condition
!
∞
3σ n2/3
α n−1
.
(12.3.10)
ρ0 = u1 ∑ n(u1 e ) exp −
2kB T
n=1
The radius of convergence is finite, R = e−α , but F0 (R) < ∞, corresponding to Case
III of the BD model. Hence, there is a critical micellar concentration u1 = e−α
beyond which nucleation occurs and spherical micelles grow indefinitely, resulting
in phase separation.
12.3.1 Multi-scale analysis of cylindrical micelles
Numerically solving the BD equations for cylindrical micelles, shows that if the initial monomer concentration u1 (0) e−α and un (0) = 0 for n ≥ 2, then the approach
to equilibrium (micellization ) can be divided into three stages [33], see Fig. 12.5:
(1) the monomer concentration decreases rapidly due to the formation of many small
size clusters; (2) aggregates steadily increase in size until their distribution becomes
a self-similar solution of the diffusion equation; (3) the final approach to equilibrium, which can be modeled using a Fokker-Planck equation. We shall describe a
recent multi-scale analysis of the BD equations by Neu et al [33], which captures
18
< n > /e
24
16
3
8
4
2
1
1
1
10
150 2000
105
3x106
time τ
Fig. 12.5: Log-Log plot of the mean cluster size hni/e as a function of the scaled time τ (thick
solid line), which shows three distinct stages. The black dot represents the beginning of the second
phase of growth, which asymptotes to the straight line of slope 1/2, indicating self-similar diffusive
growth. [Redrawn from Neu et al [33].]
each of these stages. (An alternative multi-scale analysis is carried out by Wattis and
King [47].)
First, combining equations (12.2.1), (12.3.4) and (12.3.6) shows that for cylindrical micelles the off and on rates bn , an−1 are related according to bn = an−1 e−α
so that
Jn = bn (eα u1 un − un+1 ) .
Setting bn = 1 and substituting into the BD equations (12.1.1a) and rearranging
yields the system of ODEs
dun
+ (eα u1 − 1)(un − un−1 ) = un+1 − 2un + un−1 ,
dt
n ≥ 2,
(12.3.11)
which is supplemented by the conservation condition ∑∞
n=1 nun = ρ0 . Suppose that
the initial condition is un (0) = ρ0 δn,1 with ρ0 e−α . We can eliminate one of the
parameters α, ρ0 by rescaling the cluster densities and time:
un = ρ0 rn ,
t
τ = eα ρ0t = ,
ε
with 0 ε < 1. Then
drn
+ (r1 − ε)(rn − rn−1 ) = ε(rn+1 − 2rn + rn−1 ),
dτ
19
n ≥ 2,
(12.3.12)
with 1 = ∑∞
n=1 nrn and initial conditions rn (0) = δn,1 . Multiplying equation (12.3.12)
by n and summing with respect to n gives
dr1
+ r1 (r1 + rc ) + ε(r1 − r2 − rc ) = 0,
dτ
(12.3.13)
where rc = ∑n≥1 rn is the total (rescaled) density of clusters and we have used ṙ1 =
− ∑n≥2 nṙn . Similarly, directly summing equation (12.3.12) with respect to n, and
combining with equation (12.3.13) yields
drc
+ r1 rc + ε(r1 − rc ) = 0.
dτ
(12.3.14)
Initial transient regime
Since r1 (0) = 1 and rn (0) = 0 for n > 1, it follows that there will be an initial regime
in which we can take the limit ε → 0 to obtain the planar system [33]
dr1
= −(r1 + rc ),
ds
with renormalized time s =
Rτ
0
drc
= −rc ,
ds
(12.3.15)
r1 (y)dy. The planar system has the solution
r1 (s) = (1 − s)e−s ,
which implies that
τ=
Z s z
e
0
1−z
rc = e−s ,
dz.
(12.3.16)
It follows that τ → ∞ as s → 1− with r1 (1) = 0 and rc (1) = 1/e. These are the limiting values of r1 , rc at the end of the initial stage. Finally, setting ε = 0 in equation
(12.3.12) shos that
d
(rn es ) = rn−1 es ,
ds
which can be solved recursively to yield
n−1
s
sn −s
rn (s) =
−
e , n ≥ 2.
(12.3.17)
(n − 1)! n!
As τ → ∞, rn → (n − 1)e−1 /n!, and numerically speaking there are negligible
numbers of clusters with more than five monomers. The average cluster
√ size is
hni = 1/rc = e, which is much smaller than the equilibrium value hni ∼ ρ0 eα 1
(take ρeα 1 in equation (12.3.8)). Hence, there must be additional growth on time
scales larger tha t = O(ε).
20
Intermediate transient regime
At the end of the initial transient, we have r1 = O(ε) and rn = O(1) for n ≥ 2. This
suggests rescaling r1 = εR1 and using the original time t = ετ in equation (12.3.12)
[33]:
drn
= −(R1 − 1)(rn − rn−1 ) + (rn+1 − 2rn + rn−1 ),
dt
n ≥ 3, (12.3.18)
and
dr2
= −(R1 − 1)(r2 − εR2 ) + (r3 − 2r2 + εR1 ).
dt
(12.3.19)
Moreover, equations (12.3.13) and (12.3.14) become
dR1
2
+ R1 + R1 ,
(R1 − 1)rc − r2 + ε
dt
and
drc
+ (R − 1 − 1)rc + εR1 = 0,
dt
with rc = εR1 + ∑n≥2 rn ≈ ∑n≥2 rn . In the limit ε → 0, we have R1 − 1 = r2 /rc and
we obtain the closed system of equations
drn
r2 (rn − rn−1 )
=−
+ (rn+1 − 2rn + rn−1 ),
dt
rc
n ≥ 2,
(12.3.20)
with r1 ≡ 0 and initial conditions rn (0) = (n − 1)e−1 /n!. At large times the difference rn − rn−1 becomes small, suggesting that one can approximate rn by a continuum limit. Therefore, set [33]
rn (t) ∼ δ a r(x, T ),
x = δ k, T = δ bt.
(12.3.21)
Here the scale factor δ → 0 is introduced in order to ensure that we are working in
the large-k and large-t regiome when x, T = O(1). The index a can be found from
the conservation condition ∑k≥2 krk = 1:
1 = δ a−2
∑ (nδ )r(nδ , T )δ ∼
n≥2
provided a = 2. Similarly, rc ∼ δ
yields
δb
R∞
0
Z ∞
xr(x, T )dx,
0
r(x, T )dx ≡ δ Rc . Rescaling equation (12.3.20)
∂r
δ 2 r(2δ , T )[r(x, T ) − r(x − δ , T )]
= −
+ r(x + δ , T ) − 2r(x, T ) + r(x − δ , T ).
∂T
δ Rc
21
If we choose b = 2, then we cab divide through by δ 2 and take the limit δ → 0 to
obtain the PDE
∂ r(x, T )
r(0, T ) ∂ r(x, T ) ∂ 2 r(x, T )
.
=−
+
∂T
Rc (T )
∂x
∂ x2
(12.3.22)
Numerically speaking, one finds that in the intermediate transient regime the distribution of cluster sizes is given by a unimodal function that approaches zero as x → 0
but the peak of the distribution is arbitrarily close to zero at small times. This suggests imposing the boundary condition r(0, T ) = 0 and considering the following
solution of the resulting diffusion equation
!
2
2
∂ e−x /4T
x
√
r(x, T ) = −
(12.3.23)
= √ 3/2 e−x /4T.
∂x
2 πT
πT
The normalization is chosen so that the mass conservation condition is satisfied. In
terms of the original variables, we have
2
n
(12.3.24)
rn (t) ∼ √ 3/2 e−n /4t ,
2 πt
√
and the mean cluster size is hni ∼ πt. This is consistent with the straight-line of
slope 1/2 in the log-log plot of Fig. 12.3.
Late transient regime
The large-time limit of equation (12.3.24) does not match the equilibrium distribution
n (n−1)α
ueq
.
(12.3.25)
n = u1 e
α
Since we are working
√ in the regime ρ0 e = 1/ε 1, we have from equation (12.3.8)
that u1 ∼ ρ0 ε[1 − ε] so that
√
√ n
rneq = ρ0−1 ueq
∼ εe−n ε.
n ∼ ε 1− ε
√
√
Also hni ∼ 1/ ε, whereas at the end of stage II we have hni ∼ πt. This suggests
that the final stage occurs at times t = O(ε −1/2 ). Therefore,√try the same scaling
(12.3.21) as the intermediate regime with a = b = 2 and δ = ε:
√
rk (t) = εr(x,t), x = εk, T = εt.
Repeating the continuum analysis of the intermediate stage leads to the PDE [33]
1 ∂ r(x, T ) ∂ 2 r(x, T )
∂ r(x, T )
=−
+
,
∂T
Rc (T ) ∂ x
∂ x2
22
(12.3.26)
R
with Rc (T ) = 0∞ r(x, T )dx and boundary condition r(0, T ) = 1. Finally, it can be
shown that the solution to this PDE matches the solution of the intermediate stage
diffusion equation in the limit T → 0+ and matches the equilibrium solution in the
limit T → ∞, see Neu et al [33] for details.
12.4 Protein aggregation and amyloid fibrils
Prion diseases are fatal neurodegenerative diseases that rose to prominence during
the BSE epidemic in England in the 1980s, where there was thought to be a possible
transmission pathway from infected animals to humans. Unlike classical infective
agents such as bacteria or viruses, prion diseases involve an agent consisting solely
of a misfolded protein or prion [39, 25, 18, 20]. Pathologically folded prion protein (PrP*) corrupts normally folded PrP via a fixed conformational change, which
usually leads to the formation of protein aggregates, and this process propagates
across brain cells at a speed of around 0.1-1mm per day. The initial cause could be
an external infective agent or, more commonly, the spontaneous formation of PrP*
aggregates. Recent molecular, cellular, and animal studies suggest that the intercellular propagation of protein misfolding also occurs for a variety of aggregate-prone
proteins that are linked to non-infective neurodegenerative diseases [42, 43, 40, 16].
These include amyloid-β and tau (Alzheimer’s disease), α-synuclein (Parkinson’s
diseases), and huntingtin (Huntington’s disease).
The conformational change of a normally folded protein tends to occur via direct contact with a misfolded protein aggregate. This process is commonly called
aggregate seeding or seeded polymerization [20]. If the resulting protein aggregate is small then it remains soluble, whereas larger aggregates precipitate out of
solution under physiological conditions. Two distinct morphological types of aggregate can be identified: amorphous aggregates and amyloid fibrils. The former
have an irregular granular appearance when viewed under an electron microscope,
whereas the latter consist of highly ordered and repetitive structures in which all
the polypeptides adopt a common fold. Amyloid fibrillogenesis consists of multiple
stages, involving nucleation, polymerization, elongation and aggregate seeding. It
used to be thought that mature amyloid fibrils were ultimately responsible for the
toxic effects of protein aggregates. However, there is a widening consensus that it
is the non-fibrillar assemblies in early-stages of amyloid formation that are particularly toxic. The monomers of proteins such as tau and α-synuclein are found within
the cytosol of cells. This means that the spread of such protein aggregates requires
the internalization of the corresponding fibrils in order to seed the formation of
new aggregates. The newly formed aggregates must then be released back into the
extracellular space. The precise mechanisms of internalization and externalization
are currently unknown. Such processes are not required for prions and amyloid-β
within cells, since they are both exposed to the extracellular space. Hence, the latter
proteins can diffuse intracellularly and then infect a neighboring cell directly.
23
December 18, 2014
11:31
IJMPB
S0217979215300029
Kinetic theory of protein filament growth
(a)
Fibrils
Int. J. Mod. Phys. B 2015.29. Downloaded from www.worldscientific.com
by Thomas Michaels on 01/06/15. For personal use only.
Fibril mass
concentration
Growth phase
Filament elongation
and multiplication
through secondary
mechanisms
Amyloid fibrils
Plateau phase
Reaction runs out of
monomers
and
reaches steady-state
Soluble
Fibrils
monomers
Monomers + fibrils
Time
Lag phase
(b)
Soluble
Soluble
monomers
form
Fibril
Fibril
Primary
Primary
nucleation
nucleation
Fig. 1.
koff
Elongation//
Elongation
Dissociation
k̄+
Fragmentation
Fragmentation //
Dissociation
Association
Association
MonomerMonomerFragmentation
dependent
dependent
secondary
secondary
nucleation
nucleation
M
d
s
n
(a) Schematic representation of a typical kinetic curve for filamentous growth. In analogy
Fig. to
12.6:
Schematic
diagram showing
the typical
time
course
thethe
kinetics
of amyloid
the (a)
process
of crystallization,
the assembly
reaction
may
beginfor
with
formation
and sub-fibril growth.
Illustration
the basic
microscopic
thatphase
underly
the growth
of filaments.
sequent (b)
growth
of nucleiof
during
the lag
phase. The events
initial lag
is followed
by rapid
growth
endingfrom
in the
plateau phase.
It is important
[Adapted
Michaels
and Knowles
[29].] to note that in general the lag time does not correspond to the waiting time to form nuclei. (b) Schematic representation of the basic microscopic
events that contribute to the growth of linear protein filaments. Part (b) is adapted from Ref. 26.
In this section we focus on the nucleation and growth of amyloid fibrils. An il2. Characteristics
of Filamentous
Growth in vitro is shown in Fig. 12.6(a).
lustration
of the typical time
course of fibrillization
and growth
The2.1.
fibrilNucleation
mass concentration
is a sigmoidal function of time that is characterized by
an initial
lag phase
a rapid
growth phase
and ending
a plateau phase
The time
course followed
of many by
protein
fibrillization
reactions
in vitroindemonstrates
a
dueseries
to depletion
of monomers.
Early
models
of biofilament
growthmulti-step
were based
of characteristic
signatures,
which
directly
reflect the complex,
na-on
the ture
nucleation-elongation
mechanism
introduced
by Oosawa and
of the aggregation process.
In particular,
the time-varying
masscollaborators
concentration to
describe
the assembly
of cytoskeletal
proteins shape
such ascharacterized
actin and tubulin
35],lagsee
of aggregates
typically
has a sigmoidal-like
by an [36,
initial
by israpid
filament
growth
beforea the
reaction
reaches
a plateau,
as
alsophase
Sect.followed
4.1. That
filaments
have
to reach
critical
length
(nucleation)
before
in Fig.
1(a). molecules is energetically favorable and rapid elongation
the shown
addition
of further
The observation
of these characteristic
kinetics has
led to the
formucan occur.
However, detailed
studies of thesigmoidal
polymerization
of sickle
hemoglobin
3–7
lation
of
nucleated
polymerization
models
of
biofilament
growth.
These
models
showed that the simple nucleation-elongation model couldn’t account for observed
are based
on the concept
of homogeneous
initially
developed
by classical
sigmoidal
profiles,
which exhibited
more nucleation
pronounced
lag phases,
a steeper
initial
growth phase, and a stronger dependence
on monomer concentration [12, 13]. This
1530002-3
motivated extending the classical model to include secondary pathways that depend on the existing aggregate population in addition to the concentration of soluble precursors; the resulting autocatalytic reactions accelerate growth via positive
24
page 3
feedback. We will consider a mathematical formulation of amyloid polymerization
developed by Knowles and collaborators [22, 7, 28, 29], which is an extends the
original Oosawa model in order to take into account the various processes shown in
Fig. 12.6(b).
Let u j (t) be the concentration of filaments composed of j monomers and let a(t)
denote the monomer concentration. Assuming a critical size nc for nucleation, we
set u j (t) ≡ 0 for j < nc and take the total mass concentration of aggregates to be
[22, 7, 28, 29]
M(t) = ∑ ju j (t).
j≥nc
The reaction kinetics based on the law of mass action is given by the following
system of ODEs:
du j
= 2k+ a(t)u j−1 (t) + 2koff u j+1 (t) − 2(k+ a(t) + koff )u j (t)
dt
∞
− ( j − 1)k− u j+1 (t) + 2k−
+ k̄+
∑
k+l= j
nc
∑
ui (t)
i= j+1
uk (t)ul (t) − 2k̄+ u j (t)
∑ ui (t)
i≥nc
+ kn a(t) δ j,nc + k2 a(t)n2 M(t)δ j,n2 .
(12.4.1)
For simplicity, we take size-independent rate constants kn , k+ , koff , k− , k̄+ and k2 for
primary nucleation, elongation, monomer dissociation, fragmentation, end-to-end
association and secondary nucleation, respectively. The terms in the first line of
equation (12.4.1) are analogous to those appearing in the BD model of aggregation/fragmentation; the prefactor 2 accounts for the fact that polymerization and depolymerization can occur at either end of a filament. (They also reduce to the simple
model of actin polymerization considered in Sect 4.1 when the monomer concentration is held fixed.) The second line represents the effects of fragmentation: the first
term involves breakage of a filament of length j at one of its ( j − 1) bonds, whereas
the second term describes the formation of a filament of length j due to fragmentation of a longer filament. Similarly, end-to-end association (annealing) is described
by the third line. Finally, the two terms on the fourth line represent nucleation processes. For example, the term kn a(t)nc δ j,nc describes the primary nucleation step
due to the coalescence of nc monomers. It is assumed hat the nucleation step is slow
so nuclei are in equilibrium with monomers, and that the concentrations of small
aggregates (1 < j < nc ) are negligible. The other term k2 a(t)n2 M(t)δ j,n2 represents
secondary nucleation, in which n2 monomers form a nucleus by temporarily interacting with the surface of an existing polymer. Note that the system of equations
(12.4.1) is supplemented by the mass conservation condition
a(t) + M(t) = Atot
so that
25
da(t)
d
=−
dt
dt
∑
ju j (t).
(12.4.2)
j≥nc
12.4.1 Moment equations
It turns out that one can obtain a closed system of equation for the first two moments
M(t) and P(t) = ∑ j≥nc u j (t) (assuming one can reverse differentiation and infinite
summation) [7, 28, 29]:
dP(t)
= k− [M(t) − (2nc − 1)P(t)] − k̄+ P(t)2 + kn a(t)nc + k2 a(t)n2 M(t),
dt
(12.4.3a)
dM(t)
= 2[k+ a(t) − koff − k− nc (nc − 1)/2]P(t) + nc kn a(t)nc + n2 k2 m(t)n2 M(t).
dt
(12.4.3b)
(Note that the higher order moments do not form a finite closed system of equations.) The various terms have an intuitive explanation. For example, the number of
filaments P(t) can increase if a bond breaks at a location greater than (nc − 1) from
either end of a filament. The average number of bonds per filament is M(t)/P(t)
and of those 2nc − 1 do not generate an extra filament on breaking. This accounts
for the [M(t)−(2nc −1)P(t)] factor in (12.4.3a). Similarly, the factor nc (nc −1)/2 in
(12.4.3b) accounts for the fact that j monomers are created if a fibril fractures at an
end closer than the critical nucleus size nc , and therefore the total rate of monomers
c −1
liberated through this mechanism from the ends of fibrils is 2k− ∑nj=1
jP.
In order to illustrate the analysis of these equations, we shall consider the simpler
case of no end-to-end association (k̄+ = 0) and no secondary nucleation (k2 = 0). We
will also exploit the fact that the dominant term in equation (12.4.3b) is monomer
association 2k+ a(t)P(t). We then obtain the pair of equations
dP(t)
= k− [M(t) − (2nc − 1)P(t)] + kn [Atot − M(t)]nc ,
dt
dM(t)
= 2k+ [Atot − M(t)]P(t) + nc kn [Atot − M(t)]nc .
dt
(12.4.4a)
(12.4.4b)
Suppose that initially there are only monomers so that a(0) = Atot and P(0) =
M(0) = 0. Assuming that a(t) ≈ Atot still holds for t > 0, we obtain the linear equations
dP0 (t)
= k− [M0 (t) − (2nc − 1)P0 (t)] + kn Antotc ,
dt
dM0 (t)
= 2k+ Atot P0 (t) + nc kn Antotc .
dt
Differentiating equation (12.4.5a) with respect to t shows that
26
(12.4.5a)
(12.4.5b)
d 2 P0
dP0
= 2k− k+ Atot P0 (t) − k− (2nc − 1)
+ nc k− kn Antotc .
dt 2
dt
Assuming k+ Atot k− , we can drop the term in dP0 (t)/dt to give
P0 (t) = C+ eκt −C− e−κt −
nc k− kn Antotc
,
κ2
κ=
It follows that
p
2k+ k− Atot .
2k+ Atot C+ eκt +C− e−κt − 2 .
κ
The initial condition M(0) = 0 implies that C+ +C− = 2. Moreover, since the timeindependent contribution to P0 (t) is negligible when k+ Atot k− , we can set C+ ≈
C− = C so that P(0) = 0. Finally, comparing Ṗ0 (t) and k− M0 (t) shows that
M0 (t) =
4Ck+ k− Atot
= kn Antotc ,
κ
that is
kn Antotc
.
2κ
We thus see that the mass concentration evolves as
C=
M0 (t) =
kn Antotc κt
e + e−κt − 2 .
2k−
(12.4.6)
For small times t κ −1 , we have slow quadratic growth, M0 (t) ∼ t 2 , whereas at
large times the growth is exponential, M0 (t) ∼ eκt . Hence, the linear approximation
a(t) = Atot does not account for saturation at large times.
Better agreement at large times can be obtained by keeping the saturating term
k+ [Atot − M(t)] in equation (12.4.4b). The resulting nonlinear ODE has a unique
fixed point at M∗ = Atot , P∗ = M∗ /(2nc − 1). An approximate time-dependent solution can be obtained by converting equation (12.4.5) to an integral equation and
solving the latter iteratively, starting from the linear solution (P0 (t), M0 (t)). Since
primary nucleation is only going to play a major role at small times, and is already
taken into account by (P0 (t), M0 (t)), we set kn = 0 in (12.4.5) and then formally
integrate to give
!
R
0
c −1)k− (t−t ) M(t 0 )dt 0
k− 0t e−(2n
h
i
>
R
t
[P(t), M(t)] =
≡ A [P(t), M(t)]> . (12.4.7)
0
0
Atot 1 − e−2k+ 0 P(t )dt
The n-th order iterative solution is then defined according to
[Pn (t), Mn (t)]> = A n [P0 (t), M0 (t)]> .
(12.4.8)
For example, substituting for P0 (t) shows that the first order iterated solution for
M(t) is
27
Fractional fibril mass concentration
Kinetic theory of protein filament growth
1.0
M
M0
M2
0.8
0.6
fixed-point
operator
0.4
M0
0.2
early-time
solution
0.0
0
20
A
M1
repeated application
of fixed-point operator
M2
...
M
self-consistent
solutions
40
60
Time ! h
exact
solution
80
100
Fig. 2. Schematic
representation
of the
analysis
of with
biofilament
Fig. 12.7:
Schematic representation
of theself-consistent
iterative solution of biofilament
growth
primary nu- growth and comcleation, elongation,
monomer
fragmentation.
The numerical
solution
of equation
parison of the numerical
solution
ofdissociation
Eq. (4) and
(solid
line) with
the firstand
second-order analytical
(12.4.1) (solid line) is compared with the first- and second-order analytical solutions (dashed line,
solutions, Eqs.long-dashed
(17) andline).
(23)
(dashed line, long-dashed line). The dotted-dashed line is the early
The dotted-dashed line is the early time linearized solution. Parameter values
4
−1
−1
time linearizedaresolution.
Calculation
as in
1.s−1 and nc = 2. [Adapted
k+ = 5 M s , kn = 5 × 10−5parameters
M −1 s−1 , Atot = 1µM,
k− Table
= 2 × 10−8
from Michaels and Knowles [29].]
!∞
i
where Ei(x) = − −x e−s /sds is hthe −2kexponential
integral
function. Substituting
κt
−κt −2)/κ
+ (C+ e +C− e
M1 (t)
=
A
1
−
e
,
(12.4.9)
tot
27
Eq. (22) into Eq. (13) yields
where we have neglected a term in the primary exponential that grows linearly with
" ∞M$0 (t) at small times
" and the solution
%
t. Clearly, we recover the solution
M1 (t) sat2p−2
p
q
#
#
M2at(t)large times. Indeed, even the (−C
(pκt)
+ ) is a good
urates
first iterate
to the
pκt approximation
=
1
−
exp
e
−
exact
solution
[7],
see
Fig.
12.7.
2
mtot
p p!
q!
p=1
q=0
12.5 Exercises
+
p
C−
p2 p!
"
e−pκt −
2p−2
#
q
(−pκt)
q!
%&%
.
(23)
q=0
Problem 12.1. Oosawa model of nucleation-elongation. Consider a simplified
of the
moment
equations
(12.4.4) in which
fragmentation is solutions,
ignored and Eqs. (17) and
The accuracyversion
of the
firstand
second-order
self-consistent
monomer association is assumed to dominate. As shown by Oosawa [36], one ob(23), is shown
through comparison with the numerical integration of the
tainsin
theFig.
pair of2equations
moment equations. The improved
accuracy
of the second-order self-consistent result
da
dP
= kn a(t)nc ,
= −2k+ a(t)P(t).
becomes clear when considering
the case
of no breakage. In this limit, Eq. (23)
dt
dt
matches Oosawa’s exact solution, Eq. (21), up to O(t4 ).
28
6. Connection to Perturbative Renormalization Group
In the previous section, we have shown that self-consistent techniques can be used
as an approximation method to address the nonlinear nature of the moment equations underlying filamentous growth phenomena. As a background for discussing
this approximation scheme, we have solved the linearized moment equations, obtained by fixing the concentration of monomers at the initial value. The resulting
Suppose that a(0) = Atot and P(0) = 0. Obtain the following solution for the mass
concentration M(t) = Atot − a(t):
q
2
M(t) = Atot [1 − sech(β −1/2 λt)β ], λ = 2kn k+ Antotc , β = .
nc
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