Goldfarb D. Biophysics DeMYSTiFied

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102 Biophysics DemystifieD

those between vibrational states, are very small relatively to the Gibbs energy

change of binding or of a conformational transition. Therefore the much smaller

energy changes are negligible and can be ignored. Practically speaking, they con-

tribute only to the degeneracy of a particular Gibbs energy level. Therefore it is

most common in biophysical statistical mechanics to write the partition function

as follows:

−

∑

∆

MAX

(5-16)

where DG

represents the Gibbs energy change in going from the reference

state to the ith state of the system.

chapter 5 statistical Mechanics 103

Quiz

Refer to the text in this chapter if necessary. Answers are in the back of the

book.

1. Which of the following statements about statistical mechanics are true?

S1. Statistical mechanics is the branch of mechanics that focuses on statistics.

S2. Statistical mechanics provides a molecular interpretation of a system.

S3. Statistical mechanics provides the mechanical tools necessary to measure

statistical properties of molecules.

S4. Statistical mechanics relates statistical averages of particles (atoms, molecules,

or residues) to the overall thermodynamic measurements of a system.

A. S1 and S2

B. s3 and s4

C. S1 and S3

s2 and s4

2. How many ways are there to distribute 8 units of energy among five molecules?

(Hint: Reread the section Finding Distributions.)

A. 1

B. 3

C. 7

3. Given the distribution one molecule with 4 units of energy and four molecules

with 1 unit of energy each, how many ways are there to arrange the molecules

in the same distribution?

a. 5

B. 10

D. 20

4. Given the distribution one molecule with 3 units of energy, one molecule with

2 units of energy, and three molecules with 1 unit of energy each, how many

ways are there to arrange the molecules in the same distribution?

a. 5

B. 10

D. 20

5. What is the probability of finding the system in each distribution of the three

distributions, a, b, and c, shown in Fig. 5-4?

4%, 12%, and 4%

20%, 60%, and 20%

C. 10%, 80%, and 10%

D. .2%, .6%, and .2%

104 Biophysics DemystifieD

6. Given a system with a certain total amount of energy and a certain total number

of molecules, the system will be most likely to distribute its energy among its

molecules according to

a. which distribution has the most number of permutations.

B. which energy has the most number of distributions.

c. which molecule has the most energy.

which distribution has the most number of molecules.

7. Suppose we have a particular distribution of energy among a set of molecules.

For convenience we will call it distribution X. How do we calculate the probabil-

ity of distribution X relative to all possible distributions of the energy for this set

of molecules?

a. number of permutations for distribution X divided by total number of distributions

B. number of distributions for distribution X divided by the total number of distributions

c. number of distributions for distribution X divided by the total number of permutations

for all possible distributions

number of permutations for distribution X divided by the total number of permutations

for all possible distributions

8. The Boltzmann distribution is

a. the most probable distribution for any significant number of molecules.

not the only distribution that occurs.

c. so much higher in probability than all other distributions, that for all practical purposes,

with any significant number of molecules it can be considered to be the only distribution

that occurs.

all of the above

9. A biophysicist wants to use a two-state approximation to model the conforma-

tional state of a solution of lipids. That is, we assume that the lipids can exist in

only two energy states. Assuming a Boltzmann distribution and using E

= 2.5 3

–20

J and E

= 3.0 3 10

–20

J, what is the fraction of lipids in each energy level at

295 K?

a. F

= 77.3% and F

= 22.7%

B. F

= 22.3% and F

= 77.7%

c. F

= 76.3% and F

= 23.7%

D. F

= 89.0% and F

= 11.0%

10. When an energy level has multiple states with the same amount of energy, then

that energy level is said to be

a. diminished.

decadent.

c. degenerate.

D. depraved.

Quar

De Broglie’s photon

sin

Electr yclosity

lec

ty closit

Relativist

Ong

105

We now turn our attention to the various forces that affect biomolecules, sta-

bilizing or destabilizing their various conformations. Wherever there is a force,

you should think of energy as well. The stronger a particular force is, the larger

is the energy difference associated with a movement with or against that force.

Therefore all of these forces affect the energy state of the biomolecules. It is

important to understand that biomolecules run the gamut from molecules that

are made up of only a few atoms, to relatively enormous macromolecules that

are made up of literally thousands or tens of thousands of atoms. As a result,

particularly in the larger molecules, it is possible that many different forces can

come into play at the same time. All of the forces we will speak about are elec-

tronic in nature (i.e., they ultimately involve the interaction of positive and/or

negative charges). These forces manifest themselves in a large variety of ways

and we classify them as different forces in order to categorize them according

to their different behavior.

chapter

Forces Affecting

Conformation in

Biological Molecules

106 Biophysics DemystifieD

CHAPTER OBJECTIVES

In this chapter, you will

Learn the forces that influence the conformation of biomolecules.

•

Learn the relative strengths of these forces.

•

Understand the conditions under which each force becomes significant.

•

Learn to calculate the value of various forces in biomolecules.

•

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 107

Chemical Bonds

Molecules are atoms connected together. The bonds that connect atoms

together in molecules are called chemical bonds. There are, however, many

types of bonds and forces that influence the exact location of the atoms

within a molecule, and so influence the conformation (shape) of the mole-

cule. These we will discuss later in this chapter. But we need to first under-

stand chemical bonds, since chemical bonds provide the primary structure

of molecules.

A chemical bond between two atoms always involves one or more of the

outer electrons of the atoms. These outermost electrons that participate in

chemical bonding are called valence electrons. There are two main types of

chemical bond: A covalent bond occurs when one or more of the valence elec-

trons from each atom are shared between the two atoms. An ionic bond occurs

when one or more of the valence electrons are completely stripped away from

an atom and are “donated” to the neighboring atom.

Just as a reminder (you’ve probably heard this before in an introductory

chemistry or physics course) a couple of points to keep in mind: First, we think

of electrons orbiting the nucleus as kind of an electron cloud. One possible

reason for this is because, from our perspective, each electron moves so fast and

orbits the nucleus in so many different directions that even in the simplest

atom, hydrogen, the single electron orbiting the nucleus can be thought of as a

cloud of negative charge buzzing about. From a quantum mechanical point of

view, we say that the cloud represents the probability of finding the electron at

a given point in space. The denser the cloud at any given point, the higher the

probability of finding the electron at that point at any moment in time. From a

colloquial point of view, we say the electron, in its buzzing about, “on average

spends more time” where the cloud is dense and “on average spends less time”

where cloud is thin.

The other point to keep in mind is that the electrons orbit the nucleus in

layers called orbitals, with various numbers of electrons in each orbital. As we

mentioned, it is the outermost electron cloud that participates in chemical

bonding, with one or more of its electrons.

Covalent Bond

There is always an attractive force, in an atom, between the negatively charged

electron cloud and the positively charged nucleus. In the case of a covalent

bond, two atoms come close enough together (under the right conditions) for

108 Biophysics DemystifieD

one or more outer electrons from each atom to begin orbiting both nuclei of

the atoms. Electrons orbiting both nuclei are said to be shared. Covalent bonds

always involve sharing pairs of electrons (one from each atom for every pair of

electrons that is shared). Both nuclei feel an attractive force toward the shared

electrons. Like two people wrapped in a single blanket, the force between the

shared electrons and the two nuclei is the chemical bond that holds the two

atoms together.

In practice, the electrons in a covalent bond may not be equally shared by

both atoms. One of the atoms may have a stronger pull on the electrons than

the other. This can be because one atom has a more positively charged nucleus

or because its nucleus is more exposed to, or closer to, the shared electrons.

When this happens, the electron cloud of the shared electrons leans closer to

the atom with the stronger pull, giving that atom a slightly negative charge. The

other atom will have a slightly positive charge. In such a case the covalent bond

is said to be polar, meaning that the electric charge is not evenly distributed

across the bond. See Fig. 6-1.

Covalent bond Polar covalent bond

–

FIguRE 6-1•Acovalentbondresultswhenoneor

more outer electrons (from each atom) are shared by

(orbit) both atoms (left). A polar bond is a type of

covalent bond where one atom has a stronger pull on

the shared electrons (right).

Ionic Bond

In the extreme case of a polar bond, one of the atoms may have such a strong

pull on the electrons that an electron is completely stripped away from the

“weaker” atom. This “lost” electron, instead of being shared, only orbits the atom

with the strong pull. The end result is that each atom becomes an ion. The atom

that loses an electron becomes a positive ion. While the atom that gains an

electron becomes a negative ion. The force holding the two atoms together no

longer results from both nuclei being attracted to the shared electrons. Instead

it results from the electrostatic attraction between ions, one positively charged

and the other negatively charged.

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 109

Notice that it’s not so black and white between covalent and ionic bonds. At

one extreme, the bonding electrons are equally shared between the two atoms.

In the middle case, the covalent bond becomes a polar bond. Within the realm

of polar covalent bonds it is possible to have various degrees to which the elec-

trons lean more toward one atom than the other, creating various degrees of

polarity in the bond. In the extreme case one or more electrons are no longer

shared at all. You can see that there is somewhat of a continuum between com-

pletely evenly shared electrons (covalent bonds) and completely ionized atoms

(ionic bonds). In practice, however, if the electrons are shared even slightly,

then the bond is referred to as covalent (although it may be highly polar). Only

when an electron is completely transferred from one atom to another do we

refer to the bond as ionic.

Ionic bonds are most common in crystalline solids. But in fluid solutions,

such as inside cells, ions typically float about somewhat freely. Each ion

experiences random attractive and repulsive forces from neighboring ions

that are also floating about. This is due to the thermal (kinetic) energy of

the molecules.

There are cases where ions in solution bond to specific ionized regions of

biomolecules. One example is the case of sodium (Na

) ions binding to the

negative phosphate groups (PO

) on the backbone of the DNA double helix.

The sodium ions help to stabilize the double-helical structure by reducing the

repulsive force between the negatively charged phosphate groups along the

DNA backbone. This stabilization effect can be demonstrated experimentally

by measuring the amount of energy it takes to unwind the DNA helix at vari-

ous salt (sodium) concentrations. As the sodium concentration is increased, it

takes more energy to unwind the DNA helix. The fact that more energy is

needed to disrupt the DNA helix indicates that the helix is more stable at

higher sodium concentrations. This implies that somehow sodium stabilizes the

helical structure.

We will explore molecular conformations and their stability in detail in the

next chapter. The case of sodium ions and the DNA backbone was brought here

only as an example of ionic behavior in solution. Ions play an important role in

many biological processes and affect the stability of molecular conformations.

However, the binding of ions in solution is more aptly treated as a case of ligand

binding than as ionic bonds in crystalline solids. Therefore, we will not concern

ourselves with ionic bonds as such when discussing biological systems. Rather,

going forward, when we refer to chemical bonds, it should be understood that

we mean covalent bonds.

110 Biophysics DemystifieD

Movement in Chemical Bonds and Molecular Conformation

Covalent bonds typically have various amounts of flexibility. This flexibility,

depending on the particular bond, can allow for movements such as free rota-

tion (like a wheel on an axle), and springlike movements such as bending,

compression, extension, and twisting. See Fig. 6-2.

(a)

(d)

Stretching Tw isting

(e)

Rotation Bending Compression

(b) (c)

FIguRE 6-2•Typesofmovementscommonlyfoundin

chemical bonds. (a) Free rotation. (b) Bending. (c) compres-

sion. (d)

extension or stretching. (e) twisting (limited rota-

tion). motions (b) through (e) are all springlike motions:

they occur with a restoring force, a force that somewhat

opposes their motion and tends to restore the atoms back

to their original position.

The springlike motions (bending, compression, extension, and twisting)

occur with a restoring force. That is, just like a spring, the chemical bond exerts

a force that somewhat opposes their motion and tends to restore the atoms

back to their original position. When a force causes one of these motions in a

molecule, the balance between the force causing the motion and the restoration

force can also result in vibration of the atoms back and forth. This can give rise

to various vibrational states in the molecule, where particular covalent bonds

vibrate at specific frequencies.

All of these motions, either alone or together, allow for a large number

of conformations even in relatively simple biomolecules. For example,

imagine a simple molecule consisting of four atoms connected together,

one after the other. Now let’s say these particular covalent bonds allow

free rotation and bending. Figure 6-3 shows some possible conformations

for this molecule. To help with our discussion, the atoms in Fig. 6-3

are labeled 1, 2, 3, and 4, and the different conformations are labeled

A, B, C, and D.

chapter 6 Forces AFFecting conFormAtion in BioLogicAL moLecULes 111

If there is a strong attraction between the two outside atoms (atoms 1 and 4),

then conformation C might be the most favorable: The strong attraction draws

atoms 1 and 4 as close together as they can get without breaking any of the

covalent bonds.

If the attraction between atoms 1 and 4 is strong but the bonds are some-

what rigid, allowing only a limited amount of bending, then a conformation

similar to B would be more likely: Atoms 1 and 4 are drawn together, but only

as far as the partially flexible bonds will allow. If on the other hand there is a

repulsive force between atoms 1 and 4, then we might see a conformation such

as A or D, where, as a result of the repulsive force, atoms 1 and 4 prefer to be

as far away from each other as possible.

Notice we haven’t said anything yet about what forces might account for

such an attraction or repulsion between the different atoms of the molecule.

That’s what the rest of this chapter is about. The important point here is to get

(a)

(b)

(c)

(d)

FIguRE 6-3•Somepossiblecon-

formations of an imaginary mole-

cule made of four atoms connected

together in a line.

the atoms are

labeled 1, 2, 3, and 4. the different

conformations are labeled A, B,

and D.