340 8.2 Molecular Simulation Methods
Answers
a
The mean molecular weight of an amino acid is ~137 Da, which contains a mean
of ~19 atoms (see Chapter 2); thus, the number of atoms n in protein is
n =
×
×
(
) (
) =
19
28
103
0 137
3900
/
.
atoms
PME method scales as ~O(n loge n), and classical MD scales as ~O(n2) if no trun
cation used, so a difference of a factor n
n n
n n
log
/
log
/
e
e
2
(
) = (
) , so computation
time is
5
3900 3900
0 011
15
days
log
days
min
e
×
(
) =
=
/
.
If truncation is used instead, this scales as ~O(n), so a difference of a factor
(n/n2) = 1/n, so computation time is
5
3900
2
days
min
×
=
b
The best saving in computational time for QM MD using HF approximations
scales computation time as ~O(n2.7), so a difference of a factor (n2.7/n2) = n0.7, so
total simulation time achievable for a computational time of 5 days is
5
39000
0 015
15
0 7
ns/
ns
ps
.
.
=
=
For hybrid QM/MM, the simulation time is limited by QM. The total number of
atoms simulated by the QM MD is 0.2 × 3900 = 780 atoms. The simulation time
achievable for the same computational time as for part (a) is thus
5
39000
780
1 2
2
2 7
ns/
/
ns
.
.
(
) =
which is >1 ns. Therefore, it should be possible to observe this conformational
change.
c
The molarity is defined as the number of moles present in 1 L of the solution. The
mass of 1 L of water is 1 kg, but 1 mole of water has a mass of 18 Da, or 18 g; thus,
the molarity water is
1
18
10
55 6
3
/
.
×
(
) =
−
M.
The number of water atoms (nw) present is equal to the volume of water in liters
multiplied by the molarity of water multiplied by Avogadro’s number. The volume
of water (Vw) present is equal to the volume of the confining cuboid minus the
volume of the protein cylinder.
The square based of the confining cuboid must have a minimum edge
length w of
w
2.4 nm
(2
2.0 nm)
6.4 nm
=
+
×
=
And the minimum length l of the cuboid is