Biomedical Engineering Reference
In-Depth Information
1. Start with A, for which the volume is
3
π
R
A
.
2. Surround B with a regular grid of side 2
R
B
, and examine which grid points lie inside B
but exclude those that lie in region X. To do this, test if they are within distance
R
A
of
nucleus A.
3. Also exclude points which lie outside atom B.
4. Asimple ratio of points then gives the contribution to themolecular volume fromatomB.
The process can be easily extended to any polyatomic system. This is a pretty simple
problem; suppose there are
N
atoms and we take a grid of
m
one-dimensional points for
each atom. This means that we will have to consider (
N
−
1)
×
m
3
points in space. If
N
=
101 and
m
=
50 the number is 12.5
×
10
6
; a large number of thankfully very simple
calculations.
Rather than take a regularly spaced cubic array of points around atomB, we can surround
atom B by an (imaginary) cube of side 2
R
B
and choose points at random inside this cube.
Here is the revised algorithm.
1. Choose a point at random and examine whether it lies inside atomA, in which case we
accept it.
2. If the point lies inside atom B and not in region X, we accept it.
3. Increment the number of points and return to 1.
A simple proportionality between the number of successful points and the total number
tried gives the required volume.
This calculation is an example of the
Monte Carlo
technique (denoted MC in this text).
MC is a generic term applied to calculations that involve the use of random numbers for
sampling; it became widely used towards the end of the Second World War by physicists
trying to study the diffusion of neutrons in fissionable material.
In fact, the MC method is usually attributed to a French eighteenth-century naturalist
named Búffon, who discovered that if a needle of length
l
were dropped at random onto a
floor consisting of wooden planks with uniform separation
d
>
l
, then the probability of
the needle crossing the join between two planks is 2
l
/π
d
. Búffon's subsequent experiments
enabled him to make an accurate estimation of π .
In the case of the molecular volume calculation, the problem scales as the number
of atoms. That is good news. We can easily identify worse problems; for example, the
trapezium rule enables us to approximate a definite integral
b
1
2
d (
f
(
a
)
f
(
x
) d
x
=
+
2
f
(
a
+
d
))
+···+
f
(
b
))
a
as the sum of trapezia shown in Figure 9.2. If we choose
n
points along the
x
axis then we
have to evaluate the function (
n
+
1) times in order to evaluate the sum.
For the corresponding two-dimensional integral
b
d
f
(
x
,
y
)
d
x
d
y
a
c
we would have tomake (
n
1)
2
function evaluations and the number of function evaluations
is seen to rise as the power of the dimension of the integral. That is bad news.
+
