Chapter 8 – Theoretical Biophysics  385

b The most likely droplet size occurs at the maximum value of ΔG with respect to N,

i.e. when the first derivative of –​AN +​ BN^2/​3 with respect to N is zero. So:

–​A+​(2/​3)Nc

–​1/​3 =​ 0, thus:

Nc =​ (2B/​3A)³

The nucleation energy barrier is given by ΔGmax, i.e. ΔG when N =​ Nc or:

ΔGmax =​ B(2B/​3A)^2/​3 –​ A(2B/​3A) =​ (2B⁵/​3A)^2/​3 –​ 2B/​3

c Here A =​ 2 × 10^–​3 kBT and B =​ 4.5 × 10^–​2 kBT so:

Nc =​ ((2 × 4.5 × 10^–​2)/​(3 × 2 × 10^–​3))³ =​ 3375 molecules

ΔGmax =​ (–​2 × 10^–​3 × 3375) +​ (4.5 × 10^–​2 × 3375^2/​3) =​ 3.4 kBT

With tight-​packing, the radius R at the critical droplet radius Rc relates to the

radius of one protein R1 by:

Nc × (4/​3)πR1

3 =​ (4/​3)πRc

3 so:

Rc =​ Nc

(1/​3)R1 =​ 33751/​3 × (20/​2) =​ 150 nm

The probability of a given change in free energy ΔG is proportional to the

Boltzmann factor exp(–​ΔG/​kBT) so the ratio of number of droplets here is:

C =​ exp(–​ΔG(R=​1.25Rc))/​exp(–​ΔGmax(R=​Rc)) =​ exp(ΔGmax(R=​Rc) –​ ΔG(R=​1.25Rc))

For droplets of radius R=​1.25Rc, the volume greater than the critical radius droplets

by a factor of 1.25³ or ~1.95, so N =​ 1.95 × 3375 or ~6580 molecules, giving:

ΔG(R=​1.25Rc) =​ (–​2 × 10^–​3 × 6580) +​ (4.5 × 10^–​2 × 6580^2/​3) =​ 2.7 kBT

So:

C ≈ exp(3.4/​2,7) ≈ 3.5.

d The range for radius of 140–​160 nm is centered on the predicted critical radius of

150 nm in this case, while 180–​190 nm is ~25% larger than this radius. From the

analysis of part (c), we would therefore predict a greater number of droplets by

a factor of ~3.5 (i.e. ~3.5 × 352 or ~1200 droplets) at the higher radius since the

corresponding change in free energy is smaller for larger droplets. But here fewer

droplets were observed at this higher radius by a factor of 352/​55 or ~6. However,

the total number of these protein molecules per cell is ~7000 is only a little larger

than the number of molecules predicted to be present in single droplets with a

radius ~25% larger than the critical droplet radius. In other words, the number

of molecules present in the cell is not sufficient to sustain the growth of droplets

much beyond this, which might account for the far lower proportion of larger

droplets observed than predicted.