He M., Petoukhov S. Mathematics of Bioinformatics: Theory, Methods and Applications

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DNA KNOTS AND LINKS 103

ribbon is deformed in space. Link, twist, and writhe can be either positive or

negative. Link is always an integer, whereas twist and writhe can take any real

values.

1 . Writhing. Global contortions of circular DNA are described as writhe .

The writhing number describes the supertwisting or supercoiling of the helix

in space. It is the number of turns that the duplex axis makes about the super-

helix axis. Writhe describes the supercoiling, the coiling of the DNA coil. It is

a measure of the DNA ’ s superhelicity (supercoiling) and can be positive or

negative. Writhe is a measure of the coiling, bending, or nonplanarity of the

axis of the double helix. A right - handed coil is assigned a negative number

(negative supercoiling) and a left - handed coil is assigned a positive number

(positive supercoiling). When a molecule is relaxed and contains no supercoils,

the linking number = the twist number since W = 0. The linking number of

relaxed DNA is L = N /10.5 , where N is the number of base pairs in the DNA

fragment.

2 . Twisting. Twist is the number of helical turns in the DNA: that is, the

complete revolutions that one polynucleotide strand makes about the duplex

axis in the particular conformation under consideration. Twist is normally the

number of base pairs divided by 10.4: that is, the number of bases per turn of

the helix. Twist is altered by deformation and is a local phenomenon. The total

twist is the sum of all the local twists. Twist is a measure of deformation due

to a twisting motion. Twist and writhe are interconvertible. In part because

chromosomes may be very large, segments in the middle may act as if their

ends are anchored. As a result, they may be unable to distribute excess twist

to the rest of the chromosome or to absorb twist to recover from underwind-

ing; the segments may become supercoiled, in other words. In response to

supercoiling, they will assume an amount of writhe, just as if their ends were

joined.

3 . Linking number. This is a topological property that determines the

degree of supercoiling. It deﬁ nes the number of times a strand of DNA winds

in the right - handed direction around the helix axis when the axis is constrained

to lie in a plane. It is the number of times that one DNA strand crosses about

the other when the DNA is made to lie ﬂ at on a plane. If both strands are

covalently intact, the linking number cannot change. Link is thus a topological

invariant, remaining unaltered even if the two curves are deformed in space —

as long as neither is cut. Topology theory indicates that the sum of T and W

equals the linking number: L = T + W . For example, in the circular DNA of

5400 base pairs, the linking number is 5400/10 = 540. When a molecule is

relaxed and contains no supercoils, the linking number = the twist number

since W = 0. Thus, if there is no supercoiling, W = 0 and T = L = 540. If there

is positive supercoiling, W = + 20 and T = L − W = 520. In special cases in

which the axis of the double helix remains in a plane or on the surface of a

sphere, twist equals the linking number and there is no writhe, but all other

104 STRUCTURES OF DNA AND KNOT THEORY

cases are considerably more complex. Supercoiling can even be caused by an

increase in the linking number (although this does not occur in nature).

4 . Density. The density of supercoiling is useful to deﬁ ne as a property that

distinguishes DNAs varying signiﬁ cantly in size. Superhelical density is the

number of supercoils per turn of helix. It is denoted by the Greek letter sigma

and is deﬁ ned as the number of turns that have been added or subtracted in

the supercoiled DNA compared to the relaxed state, divided by the total

number of turns in the DNA if it were relaxed (which would normally be

bp/10.5). Typically, sigma is between − 0.05 and − 0.07 (5 to 7% underwinding)

in isolated natural DNA.

5 . Link - altering enzymes. The functionality of DNA is related to its topol-

ogy, which is maintained by enzymes that are capable of altering it. Nature has

come up with particular enzymes that control the knottedness (as well as other

topological states such as twist - induced supercoiling) of DNA. The exact

ability of these enzymes to locate a knot in a circular DNA is an unresolved

question in molecular biology. Known as topoisomerases, these enzymes

change the structure by altering the DNA link of a molecule. This is achieved

by breaking one of the strands temporarily, passing the other strand through

it and then resealing the bonds. This effectively changes the linking number in

the DNA. The enzymes are of two types:

a . Type 1. These function by creating transient single - strand breaks in

DNA, altering the link by one, cutting one strand, and passing the other

strand through the break.

b . Type 2. These alter the link by two, by breaking both strands of the

double helix at the same time and passing a segment of the double helix

through the break.

Many topoisomerase enzymes sense supercoiling and either generate or

dissipate it as they change DNA topology.

Energy Associated with Various Structures

The energy of a molecule changes if there is a change in pitch (i.e., the number

of bases per full turn) or bending of the double - helix ring. Even a small change

in the pitch of DNA results in a large increase in energy

1 . Minimum energy. Linear DNA assumes the B - conﬁ guration because it

is the conﬁ guration of minimum energy. Linear molecules of DNA assume a

conﬁ guration known as the b - conﬁ guration. Deviation from this relaxed state

increases the energy of the DNA molecule, although circular DNA of large

diameter increases it least.

2 . Higher energy. In the ring form, too, the DNA double helix tries to attain

the state of minimum energy. The DNA ring approximates the b - conﬁ guration

of the linear molecule while trying to attain the state of minimum energy. This

packaging of DNA deforms it physically, thereby increasing its energy. Such

CHALLENGES AND PERSPECTIVES 105

an increase in stored (potential) energy within the molecule is then available

to drive reactions such as the unwinding events that occur during DNA rep-

lication and transcription. Too much stored energy is not necessarily a good

thing, though. In nature, this problem is addressed by having DNA form super-

coils, in which the helical axis of the DNA curves itself into a coil. Supercoiling

or the formation of a superhelix structure minimizes the excess energy that

builds up when DNA molecules are deformed during the packing process.

At this point, we should mention that supercoiling is not necessarily the

only solution to the problem of normalizing the number of bases pairs per

helix in an unwound piece of DNA. You could also separate the two strands

by breaking the hydrogen bonds between complementary bases in contiguous

base pairs until the remaining DNA has the correct number of bases per turn.

In terms of energy needed, though, it requires a lot more energy to break the

hydrogen bonds than to supercoil. Nevertheless, strand separation does occur

during replication and transcription, and it turns out that it is the physics of

the underwinding that facilitates strand separation. Cruciform structures also

require some unpairing of the base pairs, and again, it is the underwinding that

maintains the required strand separation.

4.4 CHALLENGES AND PERSPECTIVES

We can gain insight into the unknotting of DNA by using principles of topol-

ogy. Topologists study the invariant properties of geometric objects, such as

knots. Tightly packed DNA in the genes must quickly unknot itself in order

for replication or transcription to occur. Principles of topology give cell biolo-

gists a quantitative, powerful, and invariant way to measure the properties of

DNA. Principles of knot theory have helped elucidate the mechanisms by

which enzymes unpack DNA. Additionally, topological methods have been

inﬂ uential in determining the left - handed winding of DNA around histones.

Measuring changes in a crossing number have also been instrumental in under-

standing the termination of DNA replication and the role of enzymes in

recombination. In the area of DNA structure, several subareas are particularly

amenable to mathematical analysis:

• A complete analysis of the packaging of DNA in chromatin. Only ﬁ rst -

order coiling into core nucleosomes is understood. By far the largest

compaction of DNA comes from higher - order folding.

• Presentation of the topological invariants that describe the structure of

DNA and its enzymatic transformations. The goal is to be able to predict

the structure of interstate or products from enzymatic mechanisms and

in turn to predict mechanisms from structure.

• An analysis of the reciprocal interaction between secondary and higher -

order structures. This includes the phenomena of bending, looping, and

phasing.

106 STRUCTURES OF DNA AND KNOT THEORY

This work has implications for both biology and mathematics. Mathematics

will be affected in both topology and geometry. Renewed interest in the study

of embedding invariants for graphs has occurred because of the enumeration

and classiﬁ cation topoisomers; the study of random knots has been used to

study macromolecules in dilute solution, and tangle calculus and Dehn surgery

theory have been used in the study of DNA enzyme mechanisms.

In the study of kinetoplast DNA, topology and the theory of interacting

particles have been brought together in a unique way. Finally, in the study of

DNA – protein interactions, theorems from differential geometry and differen-

tial topology have been recast in different frameworks to solve helical period-

icity problems. Determination of the conﬁ guration of closed circular DNA

brings together the ﬁ elds of geometry and topology and nonlinear partial dif-

ferential equations, or topology and Monte Carlo techniques. These will

involve extensive use of computational techniques, including the creation of

new codes to use nonlinear partial differential equations to solve elasticity

problems for closed circular rods.

Differential geometry is the branch of mathematics that applies the methods

of differential calculus to study the differential invariants of manifolds.

Topology is the mathematical study of shape. It deﬁ nes and quantizes proper-

ties of space that remain invariant under deformation. These two ﬁ elds have

been used extensively to characterize many of the basic physical and chemical

properties of DNA. Speciﬁ c examples of particular note follow.

The recent review of Dickerson (1989) summarizes how such geometric

concepts as tilt, roll, shear, and propeller twist have been used to describe the

secondary structure of DNA (i.e., the actual helical stacking of the bases, which

forms a linear segment of DNA). In addition, these concepts can be used to

describe the interaction of DNA with ligands such as intercalating drugs

(Wang et al., 1983 ).

From the time that closed circular DNA was discovered, it has been clear

that such DNA exhibits physical and chemical properties that differ in funda-

mental ways from those of related linear DNA. Using differential geometry

and topology, both molecular biologists and mathematicians have been able

to explain many of the properties of these molecules from two basic charac-

teristics of the linking number: ﬁ rst, that it is invariant under deformations;

and second, that it is the sum of the two geometric quantities, twist and writhe

(White, 1969 ). Among the major applications are:

• An explanation for and extent of supercoiling in a variety of closed

DNAs (Bauer, 1978 )

• An analysis of the enzymes that change the topology of a DNA chain

(Cozzarelli, 1980 ; Wasserman and Cozzarelli, 1986 )

• An estimation of the extent of winding in nucleosomes (Travers and

Klug, 1987 )

• A determination of the free energy associated with supercoiling (Depew

and Wang, 1975 )

CHALLENGES AND PERSPECTIVES 107

• Quantitative analysis of the binding of proteins and small ligands to DNA

(Wang et al., 1983 )

• A determination of the helical repeat of DNA in solution and DNA

wrapped on protein surfaces (White et al., 1988 )

• A determination of the average structure of supercoiled DNA in solution

(Boles et al., 1990 )

Topology and, in particular, knot and link theory of closed space curves

have been used extensively to elucidate additional intertwining of closed

DNA caused by catenation of two closed duplexes or knotting of a single

duplex. In particular, recent developments in polynomial invariants for links

and knots have been used to describe the structure of DNA and to charac-

terize the action of recombinases (Wasserman and Cozzarelli, 1986 ; White

et al., 1987 ).

Additional areas of mathematics recently have developed interactions with

biology. Three - dimensional topology and low - dimensional differential geom-

etry are two examples. Theorems about the global topological invariants of

curves and ribbons in three - dimensional space have been instrumental in

studying the structural conformation of closed circular DNA. These mathe-

matical ideas apply to supercoiling in closed DNA, topoisomerases, nucleo-

some winding, the free energy associated with supercoiling, and binding

between proteins and DNA. These applications were carried out by experi-

mentalists, often in collaboration with mathematicians. As collaborative work

continues and our knowledge of the role of conformational changes of biologi-

cal macromolecules grows, the biological problems to be solved become more

complicated and the mathematical questions deepen. For example, molecular

biology has renewed interest in embedding invariants for graphs (used in

studying topoisomers), the study of random knots (used to study solutions of

macromolecules), and the tangle calculus (used in the study of the DNA

enzyme mechanism).

Structural biology includes analysis of the topological and geometric struc-

ture of DNA and proteins. It also includes molecular dynamics simulation and

drug design. Basic work must be done related to the structure and folding of

crystalline and hydrated proteins. For many proteins, the structure is dictated

by the sequence, so this area is closely related to genomics. Molecules are in

continual motion in nature, but nuclear magnetic resonance spectroscopy

(NMR) and x - ray crystallography necessarily involve snapshots. Mathematical

and computational methods are essential to complement experimental struc-

tural biology by allowing the addition of motion to molecular structures.

Mathematics has made perhaps its most important contribution to cellular

and molecular biology in the area of structural biology. This area is at the

interface of three disciplines — biology, mathematics, and physics — because its

success has involved the use of sophisticated physical methods to determine

the structures of biologically important macromolecules, their assembly into

specialized particles and organelles, and even at higher levels of organization

108 STRUCTURES OF DNA AND KNOT THEORY

more recently. A wide array of methods have been employed, but we focus on

the two most powerful of these, x - ray crystallography and NMR, but with a

mention of other methods.

Mathematics plays three roles. First, computational methods lie at the heart

of these techniques because a large amount of information about local areas

or short distances is encrypted in the raw data, and it is a major computational

task to deduce a structure. Second, new mathematical methods of analysis are

continually being developed to improve ways of determining the structure.

Third, increasingly sophisticated computer graphics have been developed in

response to the need to display and interpret such structure.

In crystallography the actual process of data collection has been enhanced

by modern methods of detection (e.g., area detectors) and the use of intense

synchrotron sources so that data collection per se is rarely rate limiting. Also,

the use of modern techniques of recombinant DNA has greatly facilitated the

isolation of material for crystallization.

Until the development of two - dimensional NMR in 1978 by Richard Ernst,

the use of nuclear magnetic resonance for studying the structure of biological

macromolecules was limited by the need to represent too much information

in a limited space. With the pioneering development of the ability to represent

NMR spectra in two frequency domains, it became possible to resolve the

spectra of small proteins and oligonucleotides. A key beneﬁ t was that cross

peaks, resulting from magnetic interactions of nuclei close to one another,

could be measured. Since these cross peaks contained spatial information,

there was an immediate movement to determine the structure of these mol-

ecules at atomic resolution. The technique has been remarkably effective. The

structures of a number of proteins and oligonucleotides have been determined.

The use of NMR to determine structures has proven to be an important

complement to x - ray crystallography because the structures of many biologi-

cally important molecules [e.g., zinc ﬁ ngers by Klevit (1991) , Summers (1991) ,

and Lee et al. (1991) ] have resisted attempts at crystallization; these structures

must be studied in solution. The success of this technique has been critically

dependent on mathematics, beginning with the theoretical underpinnings by

Ernst. The determination of structures is dependent on the mathematical

technique of distance geometry that calculates all structures consistent with

the distance constraints obtained from the NMR experiment. Other methods

have included molecular dynamics and more recently the use by Altman and

Jardetzky (1989) and Altman et al. (1991) of a Kalman ﬁ lter to sample con-

formational space. There are signiﬁ cant limitations to two - dimensional NMR

for structure determinations. First, the resolution obtained from NMR is less

than that obtained from the best x - ray structures and is not sufﬁ cient to show

in detail active sites of biologically important molecules. A major mathemati-

cal challenge is to obtain such detailed structural information from structures

that are basically undetermined. One important approach is to use the struc-

ture to backcalculate the NMR data and, by iteration, improve resolution.

Better computational techniques could extend the limit.

CHALLENGES AND PERSPECTIVES 109

One cannot overestimate the importance of solving structures at atomic

resolution. It has led directly to an understanding of the replication of DNA

and its supercoiling in chromatin; the basis of protein and nucleic acid second-

ary, tertiary, and quaternary structures; how proteins act as enzymes and anti-

bodies; and how electron transfer is achieved.

The area of molecular geometry and its interface with visualization has

been underrepresented in research to date. This research, which would beneﬁ t

from the involvement of geometers and would probably contribute to new

mathematics, is a major limiting area in structural biology, especially in drug

design and protein folding. As noted above, new methods will enhance the use

of NMR for the determination of structures. Signiﬁ cant advances for solving

the phase problem mathematically are being pursued. Important advances are

being made in the ﬁ eld of computer - aided drug design.

Related to the structure of crystalline and hydrated proteins is the question

of how proteins fold. For many proteins the folded structure and organelle

formation (e.g., ribosomes) are dictated by the sequence. Reduction of the

folding code has resisted intense efforts, but very recently important new

approaches have been developed that have revealed signiﬁ cant new informa-

tion. For example, two laboratories have shown that relatively short polypep-

tides can have signiﬁ cant secondary structure. This ﬁ nding is important because

it validates a piecemeal approach to protein folding, where secondary struc-

ture can be considered apart from tertiary structure. The second is the mini-

malist approach of DeGrado et al. (1989) , in which model structures with

predicted motifs are synthesized by chemical means. Experimental advances

such as these, together with the explosive expansion of the available data and

the development of more powerful decoding methods, means that members

of families of protein folding codes will soon be readily identiﬁ able. Once

again this area requires mathematical innovation.

Three - dimensional structures as determined by x - ray crystallography and

NMR are static since these techniques derive a single average structure. In

nature, molecules are in continual motion; it is this motion that allows them to

function (a static molecule is as functional as a static automobile). Mathematical

and computational methods have been able to complement experimental struc-

tural biology by adding motion to the molecular structure. These techniques

have been able to bring molecules to life in a most realistic manner, reproducing

experimental data of a wide range of structural, energetic, and kinetic proper-

ties. Systems studied have extended from pure liquid water, through small

solutes in water, to entire proteins and segments of DNA in solution.

The methods used for these calculations provide a glimpse of how simula-

tion can be used generally in biology. Starting with a three - dimensional struc-

ture, a mathematical formulation for the forces between atoms gives the total

force on each atom. These net forces are then used in Newton ’ s second law of

motion to give the accelerations, which are then integrated to give a numerical

trajectory. The trajectory provides a complete description of the system, giving

the position and velocity of every atom as a function of time. It is remarkable

110 STRUCTURES OF DNA AND KNOT THEORY

that simple forces and classical mechanics seem to give such a faithful picture

of molecular motion.

In summary, many doubts and suspicions exist in understanding of the

genetic language. How was life information accumulated and evolved in the

DNA sequence? How can we understand the possible function of the large

amount of nongenic DNA in the genome and extract life information from

the DNA sequence under the background of strong noises? What principle

governs the functional networks in a genome? How can we predict the molec-

ular structure from its sequence information?

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5 Protein Structures, Geometry,

and Topology

Proteins play crucial roles in almost every biological process. They are respon-

sible in one form or another for a variety of physiological functions. They

function as catalysts, they transport and store other molecules (e.g., oxygen),

they provide mechanical support and immune protection, they generate

movement, they transmit nerve impulses, and they control growth and differ-

entiation. Proteins are linear polymers built of monomer units called amino

acids . The construction of a vast array of macromolecules from a limited

number of monomer building blocks is a recurring theme in biochemistry.

Does protein function depend on the linear sequence of amino acids? The

function of a protein is directly dependent on its three - dimensional structure.

Remarkably, proteins fold up spontaneously into three - dimensional structures

that are determined by the sequence of amino acids in the protein polymer.

Thus, proteins are the embodiment of the transition from the one - dimensional

world of sequences to the three - dimensional world of molecules capable of

diverse activities.

In this chapter we introduce protein primary structures, secondary struc-

tures, tertiary structure, and quaternary structure by geometric means. We also

discuss the classiﬁ cation of proteins, physical forces in proteins, protein motion

(folding and unfolding), and basic methods for secondary structure and ter-

tiary structure prediction.

5.1 INTRODUCTION

A protein comprises a sequence of amino acids, which are the building blocks

of proteins. There are 20 amino acids that commonly appear in proteins.

Recently, a twenty - ﬁ rst naturally occurring amino acid was found (Atkins and

Gesteland, 2002 ). Each amino acid is represented by one or more sequences

of three RNA nucleotides, known as a codon or triplet . The combination of

four possible nucleotides in groups of three yields 64 codons. Amino acids are

coded by more than one codon. An organelle performs the translation of

Mathematics of Bioinformatics: Theory, Practice, and Applications, By Matthew He and

Sergey Petoukhov

112