Biomedical Engineering Reference
In-Depth Information
A final point for consideration is that we must also take account of the chemical envir-
onment of a given atom. An sp carbon atom is different from an sp
2
carbon atom and so
on. It is traditional to speak of
atom types
in MM.
Our idea is to treat the force field as a set of constants that have to be fixed by appeal to
experiment ormore rigorous calculation. InMMwe take account of nonbonded interactions,
and also the chemical sense of each atom. A valence force field that contains nonbonded
interactions is often referred to as a
Urey-Bradley force field
.
4.4.1 Bond Stretching
If we consider phenylanine (the structure of which is shown in Figure 4.2) we can identify
a variety of bond types including C(sp
2
)
O, and so
on. If we assume that Hooke's law is adequate, then each bond stretch between atom types
A and B makes a contribution to the total molecular potential energy of
−
C(sp
2
),C(sp
2
)
−
C(sp
3
),O
−
H, C
=
2
k
AB
R
AB
−
R
e,AB
2
1
U
AB
=
(4.10)
in an obvious notation. Parameter
k
AB
is the force constant,
R
AB
the instantaneous bond
length and
R
e,AB
the equilibrium bond length.
O
HO
NH
2
Phenylanine
Figure 4.2
Structure of phenylanine
Other scientists recommend the Morse potential:
D
1
]
2
U
AB
=
−
[−
α(
R
AB
−
exp
R
AB,e
)
whilst some recommend the addition of extra terms to the simple harmonic expression
k
1
R
AB
−
R
e,AB
2
k
2
R
AB
−
R
e,AB
4
U
AB
=
+
(4.11)
4.4.2 Bond Bending
Next we have to consider the bond bending vibrations. It is usual to write these as harmonic
ones, typically for the connected atoms A-B-C
2
k
ABC
θ
ABC
−
θ
e,ABC
2
1
U
ABC
=
(4.12)
