Goldfarb D. Biophysics DeMYSTiFied

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122 Biophysics DemystifieD

because electrostatic interaction depends only on the distance from charge to

charge, and turning a dipole clockwise or counterclockwise relative to a point

charge does not change the distance from the point charge to each end of the

dipole; only turning the dipole on an angle toward or away from the point

charge changes this distance.

However, turning a dipole clockwise or counterclockwise relative to another

dipole will change the distances from each end of one dipole to each end of the

other dipole. So dipole-dipole interactions must consider two different angles

for each dipole, making a total of four angles. See Fig. 6-7.

FIguRE 6-7•Dipole-dipoleinteractions.Twoanglesmustbeconsideredfor

each dipole: one angle in the xy plane between the x-axis and the dipole, and

the other angle in the yz plane between the y-axis and the dipole.

to keep things

simple, we orient the axes so that the distance r between the two dipoles is

along the x-axis.

The potential energy between two dipoles can be written as

ddF

−

12 11 22

(,,,)

αβαβ

(6-8)

Compare this equation with Eq. (6-5). To emphasize the form of the poten-

tial energy function, we write simply F(a

, b

, a

, b

), where F is some function

of the four angles a

, b

, a

, b

(see Fig. 6-7). The important points to remember

here are

The potential energy falls off with the

• cube of the distance r

as com-

pared with the square of the distance for a charge-dipole interaction,

and compared with 1/r for a charge-charge interaction. This demon-

strates the trend mentioned earlier that when multiple charges interact

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 123

simultaneously, the more charges there are, the higher the power on

the inverse relationship with distance, and the shorter the range of

interaction.

For a fixed orientation between dipoles (such as one may find in a

•

folded protein or molecular complex), the potential energy depends

on this orientation, and the orientation is defined by four different

angles.

Thermal Averaging in Dipole-Dipole Interactions

When the orientation between two dipoles is not constant, but is allowed

to move freely by the random thermal motion of molecules, then thermal

averaging can be done using the Boltzmann distribution to calculate the

average over all possible orientations, as was noted for the charge-dipole

interaction. For the case of dipole-dipole interactions, the average potential

energy is given by

kT r

−2

(6-9)

Notice that (as was the case with a charge-dipole interaction) thermal aver-

aging has doubled the power on the inverse distance dependence compared

with the fixed orientation case. Dipole-dipole potential energy can indeed be

significant, but only at very short distances.

Induced Dipoles

The electron cloud of a neutral, nonpolar molecule can be distorted in the pres-

ence of an electric field so that the molecule becomes polar. This is called an

induced dipole. The electric field that induces the polar quality of the molecule

typically comes from a nearby ion or dipole.

We can measure the tendency of the electron cloud of an atom or molecule

to be distorted from its normal (nonpolar) shape by an electric field. We call

this measurement polarizability, defined as the ratio of the induced dipole

moment to the electric field strength.

(6-10)

Notice that what this equation says is that a large degree of polarizability

means a given electric field induces a large dipole moment. A small degree of

polarizability means the same given electric field induces a small dipole moment.

124 Biophysics DemystifieD

(Notice also that a

is the symbol for polarizability, not to be confused with our

use of a as the angle between a dipole and a point charge.)

The standard formula for the strength of an electric field due to a point

charge is given by

(6-11)

where E is the electric field strength, q is the amount of charge,

is the dielec-

tric constant of the medium (also called permittivity), and r is the distance from

the point charge.

If we substitute Eq. (6-11) for E in Eq. (6-10) and then solve for the dipole

moment, we get

(6-12)

This is the formula for the dipole moment of an induced dipole. We can

substitute this formula into Eq. (6-5), to get the potential energy due to the

interaction of a charge with an induced dipole.

−

(6-13)

In Eq. (6-13) we’ve made an additional assumption. We know that an induced

dipole will orient itself in the direction of the inducing electric field. If we assume

that the interacting point charge is also the source of the inducing electric field,

then we can assume the dipole will orient itself in the direction of the charge.

Making this assumption leads to cos a = 1 [compare Eq. (6-13) with Eq. (6-5)].

Hydrogen Bonds and Water

Water is an amazing molecule, and life as we know it could not exist without

water. Entire volumes have been written about the chemical and physical prop-

erties of water. What concerns us here is that water is a highly polar molecule.

The molecule is shaped somewhat like the letter V with a slight positive charge

at each end and a somewhat negative charge in the middle. See Fig. 6-8.

Water is a permanent dipole. Actually, it can be treated as two dipoles, or as

a V-shaped “tri-pole” with a negative charge in the middle and positive charges

at the ends. Sometimes we treat each “leg” of a water molecule as a dipole,

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 125

because typically only one side (at a time) of the water molecule interacts with

other charges or dipoles. However, it is possible, for example, depending on the

size and shape of the interacting molecules, for all three charges of the water

molecule to interact with several charges on another molecule. In such a case

we can also approximate a water molecule as a single dipole with a dipole

moment, d, as indicated in Fig. 6-8. This makes sense since the sum of the par-

tially positive charges on the hydrogen atoms is equal to the partially negative

charge on the oxygen atom.

The permanent dipole nature of water causes water molecules to attract one

another and to line up in or orderly arrangement. The slightly positive hydro-

gen atom of one water molecule orients in the direction of the negative oxygen

atom of a neighboring water molecule. See Fig. 6-9. The attractive force between

the positively charged hydrogen atom on one water molecule and the nega-

tively charged oxygen atom on the other molecule is called a hydrogen bond.

These bonds contain potential energy. When oriented as such, since the dipoles

tend to orient toward one another, the interaction is effectively a charge-charge

interaction. That is, because the dipoles are lined up end to end, for all intents

and purposes it is as if two point charges are interacting: the slightly positively

charged hydrogen on one water molecule with the somewhat negatively charged

oxygen on another water molecule. Due to the orientation of the dipoles, each

charge can “see” only one end of the dipole on the other molecule.

FIguRE 6-8•Awatermoleculeconsistsofanoxygenatom

with two hydrogen atoms. the oxygen atom is significantly

larger than the hydrogen atoms.

the oxygen atom has a

stronger pull on the electron cloud, resulting in a slight posi-

tive charge d

on each of the hydrogen atoms and a some-

what negative charge d

on the oxygen atom. the direction of

the dipole moment d is shown.

104.45°

–

•

126 Biophysics DemystifieD

Hydrogen Bonds in General

Water is not the only molecule capable of hydrogen bonding. Whenever a

hydrogen atom in a molecule has a slightly positive charge, it can interact with

a negative or partially negative charge elsewhere (either on another molecule

or on other parts of the same molecule).

Notice that a hydrogen bond is not an interaction with a single hydrogen

atom or even a hydrogen molecule. A single hydrogen atom is nonpolar. A

hydrogen molecule is also nonpolar, consisting of two hydrogen atoms cova-

lently bonded together in a nonpolar bond. Even if we had a single hydrogen

atom in the form of a positively charged ion, then the interaction would be

ionic. This is not what we mean by hydrogen bond.

When we speak of a hydrogen bond, it is important to remember that the

hydrogen atom is always covalently bonded to an atom other than hydrogen,

typically a much larger atom. The larger atom has a stronger pull on the elec-

tron cloud, leaving the hydrogen with a slightly or partially positive charge. It

is this polar covalently bonded hydrogen that can participate in a hydrogen

bond by being attracted to a negative or partially negative charge.

–

•

–

Hydrogen

bonds

FIguRE 6-9•Hydrogenbondinginwater.(Courtesy of Wikimedia

Commons.)

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 127

The physical property that describes the amount of pull a particular atom

has on electrons is called electonegativity. Atoms that have a strong pull on elec-

tron clouds are said to be highly electronegative. In biomolecules, the most com-

monly found highly electronegative atoms include nitrogen and oxygen. There

are many places in proteins, DNA, and other biomolecules where hydrogen is

covalently bonded to oxygen or nitrogen in a highly polar bond. This leaves the

hydrogen with a partially positive charge and able to participate in hydrogen

bonds.

Hydrogen bonds in biomolecules can form between different parts of the

same molecule, for example, between different parts of a folded protein. They

also form between two different molecules, for example, when a protein binds

to DNA. And they form between biomolecules and the water that surrounds

them. As we will see, all types of hydrogen bond interactions play an important

role in the conformation and functioning of biomolecules, including the hydro-

gen bonds between water and biomolecules, and even the hydrogen bonds

among the water molecules themselves that surround a biomolecule.

Aromatic Ring Structures: Cation-Pi Interactions and Stacking

In many molecules, some atoms are covalently bonded together to form a ring.

The designation ring does not necessarily mean it is geometrically a circle. By

ring, we simply mean that if we trace from atom to atom as each atom is cova-

lently bonded to the next, eventually we get back to the first atom. The ring of

atoms can take on various shapes, for example a pentagon, or hexagon, or even

a kind of zig-zag shape like the top edge of a crown.

In biomolecules there is a common type of ring structure known as an aro-

matic ring. Aromatic ring structures have certain defining characteristics.

The atoms making up the ring lie within a plane (so the zig-zag ring is not

considered aromatic).

All of the ring atoms contribute electrons from their pi orbital to the

covalent bonds that hold the ring together.

It’s not important for this discussion to know the details of what a pi

orbital is; you only need know that an orbital is a way of mathematically

defining a portion of the overall electron cloud. And a pi orbital is just the

name of one particular orbital that has certain mathematical properties. The

covalent bonds that hold the ring atoms together in an aromatic ring consist

of electrons from both sigma orbitals and pi orbitals, but a defining

128 Biophysics DemystifieD

characteristic of aromatic rings is the fact that every atom in the ring contrib-

utes at least one pi electron.

In biomolecules, aromatic rings are almost always made of six atoms or

sometimes five if in an ionic state. Typically most or all of the ring atoms are

carbon. Sometimes one or two of the ring atoms are nitrogen, or occasionally

oxygen, while the remaining atoms are all carbon. See Fig. 6-10.

Cation-Pi Interactions

The electron clouds of aromatic structures have two unique features that affect

the way they interact with other molecules. First, you learned at the beginning

of this chapter that two atoms covalently bonded together share electrons; that

is, one or more electrons orbit both atoms in the covalent bond. Electrons from

the sigma orbitals follow this rule and are shared between any two adjacent

atoms in the ring. However, the electrons from the overlapping pi orbitals

become delocalized; that is, they form a cloud around the entire aromatic ring

and are not associated with only one or a pair of atoms on the ring. Furthermore

the pi electrons are distributed mainly above and below the plane of the ring,

with the sigma orbitals localized to the space around pairs of adjacent ring

atoms. The result is an uneven distribution of charge, with negative charge

above and below the plane of the ring and positive charge along the edges.

See Fig. 6-11.

The electronegative layers above and below the ring plane interact strongly

with cations. Furthermore, depending on the polarizability of the aromatic ring,

a cation may accentuate the uneven charge distribution. The layers of positive

and negative charge are easily modeled as two dipoles as shown in Fig. 6-11.

Interactions between cations and aromatic rings are common in proteins that

contain aromatic structures. Such cation-pi interactions are found in the binding

FIguRE 6-10•Examplearomaticmoleculesfoundinbiologicalsystems.

Pyrimidine Adenine Phenylalanine

NHH

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 129

of ligands to proteins and in the binding of proteins to one another, as well as

within protein conformations where an aromatic portion of a protein folds over

and interacts with a cation side chain on the same protein. Mathematics similar

to the dipole interactions outlined previously show that the energy of cation-pi

interactions decreases in proportion to the cube of the distance. That is, the

energy of cation-pi interactions is proportional to 1/r

Stacking

The distribution of pi electrons in aromatic rings is not static, but fluctuates in

time. These fluctuations momentarily create regions of positive and negative

charge on the plane surface of the ring. The cause may be due to the movement

of a passing ion or may be induced by other nearby fluctuating electron clouds.

When the surfaces of two aromatic rings approach each other, fluctuations in

the pi electron cloud of one ring can induce reciprocal fluctuations in the other.

This creates multiple regions of alternating partially positive and partially neg-

ative charge on the surface of the ring. The electron cloud shifts back and forth

among these regions in harmonic motion, like a spring or seesaw. See Fig. 6-12.

When the aromatic rings are close enough together, the fluctuations in each

ring induce and influence fluctuations in the other so that it is possible for the

charge movements to come in sync with each other.

Stacking interactions occur when two aromatic rings are stacked on top of

each other, like plates, and the reciprocal alternating fluctuations in the electron

clouds create an attractive force between adjacent rings. Where one electron

FIguRE 6-11•Inaromaticrings,theelectroncloudisdistributed

so that a layer of partially negative charge exists above and below

the plane of the ring, with partially positive charge along the ring

edge.

the electronegative layers above and below the ring plane

can interact strongly with cations (positive ions).

–

130 Biophysics DemystifieD

cloud is thin and the ring surface exhibits a partially positive charge, at the same

time in the same place on the adjacent ring, the electron cloud is thick and

exhibits a partially negative charge. The partially positive and partially negative

charges attract one another. A moment later the positive regions fluctuate to

negative, and vice-versa, so the stacked rings continue to attract each other. This

is because the rate, or frequency, of the fluctuations is the same in adjacent

rings, since fluctuations in each ring induce and accentuate the opposite fluc-

tuations in the other ring.

Aromatic stacking interactions are common in nucleic acid structures such

as DNA. They account for the majority of the force that holds DNA into its

helical shape.

Dispersion Forces

Aromatic rings are not the only molecular structures that exhibit fluctuations

in their charge distribution. In fact all electron clouds fluctuate rapidly, and in

many molecules this leads to momentary dipole moments. When two such

molecules are near each other, the fluctuating dipole moments can lead to

attractive or repulsive forces. In order for these forces to be significant (to last

more than the momentary fluctuation), the fluctuations of the two molecules

need to be in sync with one another. That is, the rate or frequency of fluctua-

tions needs to be the same in both molecules. This is possible if the structures

are somewhat similar and if their polarizability allows fluctuations in one struc-

ture to induce fluctuations in the other. This is exactly the case, as mentioned,

with stacking interactions. The charge distribution fluctuations in one aromatic

structure induce synchronous fluctuations in the adjacent aromatic structure.

Forces that arise from in-sync fluctuating charge distributions are called dis-

persion forces. There are various approaches to modeling fluctuating dipole

moments and the mathematics can get rather complicated. In general they are

–

FIguRE 6-12•Electroncloudfluctuationsinaromaticringscausealter-

nating regions of partially positive and partially negative charge. When

two aromatic rings are stacked like plates, the opposite alternating fluc-

tuation in adjacent rings creates an attractive force between the rings.

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 131

treated somewhat similarly to freely rotating dipoles, with an added complica-

tion that electron cloud fluctuations are much faster than molecular rotation.

The energy associated with dispersion forces varies as the 6th power of the

reciprocal of the distance, 1/r

, so it is a relatively short-distance interaction.

Steric Interactions

Steric interactions or steric repulsions are simply terms that describe atoms or mol-

ecules bumping into each other. It’s another way of saying that two things can’t

be in the same place at the same time. Physically what’s happening is that the

electron cloud of one atom begins to penetrate the electron cloud of another. This

leads to a very sharp rise in free energy, making it very unlikely that the penetra-

tion will continue any further. The energy of steric repulsions varies as 1/r

The potential energy, or free-energy function, of steric interactions combined

with dispersion forces looks typically like the one shown in Fig. 6-13. As the

molecular structures approach each other, at a very close distance, dispersion

forces create an attraction and the potential energy drops. Then as the molecules

move even closer together, the electron clouds begin to penetrate each other and

the potential energy rises sharply. This is the steric interaction. The potential energy

FIguRE 6-13•Potentialenergyasafunctionofatomicdis-

tance. As two molecules approach each other, dispersion

forces create an attraction. At a distance of r

, the potential

energy reaches a minimum. At this distance the dispersion

forces are at their strongest. At distances closer than r

, steric

interactions begin to take over; the electron clouds begin to

penetrate one another and the potential energy rises sharply,

making it increasingly difficult for the molecules to get any

closer together.

1.5

V (potential)

r (distance)

2 2.5