Proc. Natl. Acad. Set. USA
Vol. 73, No. 8, p
Mathematics an B .2639-2643, August 1976
       Biochemistry

Linking numbers and nucleosomes

(DNA double helix/twist/writhing number/chromosome structure/simian virus 40)

F. H. C. CRICK

Medical Research Council Laboratory of Molecular Biology, Hills Road, Cambridge CB2 ZQH, England

Contributed by F. H. C. Crick, May 21,1976

ABSTRACT  In considering supercoils formed by closed
double-stranded molecules of DNA certain mathematical con-
cepts, such as the linking number and the twist, are needed. The
meaning of these for a closed ribbon is explained and also that
of the writhing number of a closed curve. Some simple examples
are given, some of which may be relevant to the-structure of
chromatin.

It is not easy to think clearly about the way in which double-
stranded DNA twists into various coils and supercoils, The
subject has been greatly clarified by the mathematician F.
Brock Fuller in a paper entitled "The writhing number of a
space curve" (1). This paper is written in a clear, concise. manner
but its very compactness makes it difficult to grasp for the av-
erage molecular biologist. This note is an expansion and clari-
fication of part of his paper. See also an earlier paper by White
(2) and a further paper by Fuller (3).

The reader should recall two elementary facts about helic&
and handedness. The first is that a right-handed helix is right-
handed from whatever position one looks at it. If it is turned end
to end it stays right-handed. The second is that if a right-handed
helix is viewed in a mirror, or inverted through a center of
symmetry, it becomes left-handed, and vice versa.
The basic ideas

The essential concept we use is that of a ribbon. This ribbon can
be thought of as a pair of lines-its two edges. Mathematically
these are considered to be a minute distance apart. In reality
the ribbons we will be considering will have finite width but
we shall have the physical restriction that our ribbons cannot
interpenetrate. We shall mainly be considering closed ribbons,
which join back on themselves. It is assumed that each edge
joins only with itself-and not with the other edge as in a
Miibius strip. To underline this and to relate our ideas to the
physical structure of the double helix of DNA, whose two chains
run antiparallel rather than parallel, we put arrows, all pointing
the same way, on one edge of the ribbon (in an arbitrary chosen
direction) and label the other edge with arrows in the opposite
direction. If we have occasion to break lines and join them we
can only join lines pointing in the same direction. That is, we
assume that we cannot join, by chemical bonds, a DNA back-
bone of one polarity to one of opposite polarity.

The line running along the center of the ribbon, which we
shall call its axis, is also important. For a closed ribbon it joins
back on itself. It does not have a direction.

Now we have to grasp three distinct but related concepts.
These are: (a) the Linking Number, L; (b) the Twist, T; (c) the
Writhing Number, W. The first important thing to realize is
that the first two, L and T, are properties of a rfbbon. They
have, in general, no meaning for a single curve such as the ar8
of the ribbon. The Writhing Number, on the other hand, is the
property of a closed curve, such as the ribbon axis. Its value
depends on the exact shape of the curve in space, but not where
the curve is in space ("is invariant under rigid motions") nor
on the scale ("invariant under dilatations"). The mirror image

of any curve has a writhing number of the same magnitude but
of opposite sign. Thus, the writhing number of any curve which
is its own mirror image (such as a circle) is necessarily zero. A
curve which has a center of symmetry also has a zero writhing
number.

The essence of Fuller's definition of the writhing number is
the equation:

W-L-T

In short, although both L and T are properties of a ribbon, their
difference (where they are suitably defined) is a property only
of the ribbon's axis and not at all of the way in which the ribbon
is twisted about that axis.

The meaning of L, T, and W

We must now state more precisely what is meant by L, T, and
W.

The linking number, L, is roughly speaking the number of
times the closed line along one edge of the ribbon is linked, in
space, with the closed line along the other edge. For example,
a ribbon forming a simple (untwisted) circle has linking number
zero, since the two distinct circles formed by the edges are not
linked in space. The linking number for a closed ribbon is
necessarily an integer but as we shall see it can be positive or
negative. It is unaltered under all deformations of the ribbon
which do not tear it ("which deform it smoothly") and is
therefore a topological property. We shall not define it more
precisely here but later in this paper we give an algorithm for
calculating it.

In order to gi\;e a sign to the linking number we must, in ef-
fect, put arrows on the two edges of the ribbon. We have al-
ready chosen to have these arrows run in opposite directions on
the two edges because of the structure of DNA. [Mathematically
this is not essential. In Fuller (1) the arrows are defined to run
in the same direction.] Then a strip which is twisted in a
right-handed manner will be given a positive linking number.
To make the sign convention quite clear we illustrate two strips,
one right-handed and one left-handed, in Fig. la and b. We can
deform these figures to give the arrowed lines shown in Fig. lc
and d, which also illustrate the convention. Fig. Ie illustrates
a (deformed) ribbon with a linking number of +2. The mirror
image of a ribbon, or the ribbon inverted through a center, has
L of the same magnitude as the original ribbon but of opposite
sign.

We must now tackle the twist 2'. The exact definition, fol-
lowing Fuller (l), is given in the &pen&r but at this stage all
the reader needs is an intuitive idea of the twist.

Note first that a simple bend (Fig. 2a) does not introduce a
twist, nor does the bend shown in Fig. 2b, in which the de-
formed ribbon lies in one plane. On the other hand the twisted
ribbon shown in Fig. 3c clearly has a twist under any definition.
The wafts of twist are chosen so that the stretch of ribbon shown
in this figure, which goes round 360" (that is, 2?r radians) is
defined to have a twist of 1. Since the twist is right-handed it

2639


2640  Mathematics and Biochemistry: Crick

Proc. Natl. Acad. Sci. USA 73 (1976)

Right handed

Left handed [L=+& )

FIG. 1.  Line drawings to show the sign convention for L. The pair
of lines in (a) can be regarded as the two edges of a ribbon on which
the arrows have been marked in opposite directions. (6) The mirror
image of (a). Drawing (c) is topologically the same as (a), while (d) is
the same as (b). If we reverse the direction of an arrow on one of the
chains of a pair we alter the sign of L. (e) A pair of lines for which L
= +2. Compare (c).

is called +l. The mirror image of a ribbon has a twist of the
same magnitude but opposite sign, so that the (left-handed)
reflected image of Fig. 2c has the twist -1.

The twist, T, of a ribbon is the integrated angle of twist (di-
vided by 2s) along its length. For a closed curve it need not be
an integer and in general it will not be. T is invariant under
rigid motions and dilatations but its value depends on the exact
shape of the ribbon in space and it is, in general, altered by ar-
bitrary deformations. It is thus not a topological property but
a metrical one. To illustrate the value of T for a slightly more
complicated but still simple case we consider the twist of a
ribbon wound flat on a cylinder and going round N times in a
right-handed helix (see Fig. 3a, where N = 2). Then the
mathematical definition of twist leads to the value

T = N sin LY

where (Y is the pitch angle of the helix.

Care must be taken over the units of twist. One can easily but
incorrectly assume that if N = 1 and cr is small then the angle
of twist is approximately equal to (Y. This is incorrect. The angle
of twist is 2~ sin a = 27r cr. However the twist T, which is the
integrated angle of twist divided by 2*, does approximate to
cy in this case, (Y being measured in radians.

The twist of a ribbon wound flat on a cylinder tends to zero
if the axis of the ribbon tends to a circle (each turn approximates
to a simple untwisted circle) while the twist of a ribbon wound
flat on a cylinder of vanishingly small radius (which gives a
ribbon like that in Fig. 2c) tends to T = N. The twist for more
complicated shapes must be calculated from the definition in
the Appendix.

From the way that T is there defined it follows that in cal-
culating it for a closed ribbon, by travelling along it, one can
do so in sections. Thus, if there are "points" A, B, and C on the
ribbon then one can calculate the value of T for the section AB,
that for BC, and that for CA. The total T will be the algebraic

T=+l

        a             b             0
FIG. 2.  (a) A ribbon which is bent but not twisted. This is also true
for (b). In the latter case the ribbon lies entirely in the plane of the
paper. (c) A ribbon which is twisted but not bent.

L=+2   k=o"     L=O
T>O        <      T=O

a

b       c

L=+2     L=O
TcO     T=O

I-I'
_..- .._,

d       e

FIG. 3.  (a) A short length of ribbon in the shape of a right-handed
helix having two complete turns. To aid visualization a cylinder has
been drawn in. The pitch angle of tbe helix is 01. The broken lines show
how the two ends of the ribbon can be joined to make it a closed rib-
bon. (b) A closed ribbon wound as a double helix, having two turns
up and two turns down. Again the cylinder is shown to help visual-
ization. (c) An untwisted ribbon. Note that by twisting(c) it can be
deformed topologically into (b). (d) A closed ribbon forming a double
helix with two left-handed turns joined along the helical axis. (e) An
untwisted ribbon. Note that twisting (e) does not immediately give
,(d) (see text). Although in (d) the helix is left-hand, L is positive for
this case.

sum of these three partial values. Notice that the definition of
T is such that it has exactly the same value, both for magnitude
and sign, if for the section AB one proceeds during the inte-
gration from A to B, or from B to A. Thus, not only can we
calculate T in sections but we can do this in either direction for
each section, as we please.

By contrast it is impossible to calculate the value of L in this
way. L is a property of the entire ribbon and cannot usefully
be calculated by travelling along it. One must view the ribbon
as a whole. There is, however, a very neat way to obtain L from
any outside view of the ribbon or of any topological deforma-
tion of the ribbon (F. B. Fuller, personal communication). The
algorithm consists of marking first each edge of the ribbon
distinctively (say, one with a red line and one with a black one),
putting the necessary arrows on both of them. One then views
the configuration (or any convenient topological deformation
of it) from any chosen point outside the configuration. If this
point is at infinity the view will be a projection but this is not
essential. One now inspects each case where the red and black
lines cross, arbitrarily choosing either the places where red is
in front of black, or black is in front of red, but not both. Each
such cross-over can be assigned a value +1 or -1, according to
the local direction of the arrows on the two lines at the cross-over
and using the conventions of Fig. 1. Then L is the algebraic sum
of all these assignments. In a loose way one can see that this will
work because each cross-over represents a place where a cut
must be made in one of the colored lines to allow it to be re-
moved to infinity (either towards one, or away from one, as the
case may be) while leaving the other colored lines in place.

The convention in Fig. 1 differs in sign from that of Fuller
(1) because we have orientated the edges of a ribbon to run in
opposite directions while Fuller orientated them to run in the
same direction. Thus, the linking numbers for closed curves
differ in sign from Fuller (1) but the linking number for ribbons
agree.

What Fuller did was first to define L and T for a ribbon, on
the lines sketched above, and then to define W = L - T. He
showed that W was a property of the axis of the ribbon rather
than of the ribbon itself. The properties of W, L, and T are
summarized in Table 1.

One special result is of interest. It has been shown (1,2) that


Proc. Natl. Acad. Sci. USA 73 (1976)   2641

Mathematics and Biochemistry: Crick

Table 1.  Properties of L, Z', and W

  L         T         W
(linking)   (twisting)   (writhing)

Is a property of:  A closed    A ribbon    A closed
            ribbon               line
It is:        Topological Metrical     Metrical
Its value is:     An integer  Any number Any number

the value of W for any closed curve lying on a sphere is zero.
Obviously this is also true if it lies on a plane.

Although in general T can have any value there is a special
case for which its values are restricted. Imagine that we are
given a particular closed curve and told that our closed ribbon
must have this curve as its axis but that we can put our ribbon
on this axis in any way we please, provided that it has no dis-
continuity anywhere (i.e., the ends are joined up smoothly).
Now because the axis of our ribbon is defined, the value of W
is fixed. No matter how we choose to put the ribbon on the curve
the value of L (which we are free to select) must necessarily be
an integer. Thus, the possible values of T will be quantized. For
example, if we are presented with a closed curve for which W
= +I.7 then T must have a value chosen from the infinite set

. . . -2.7, -1.7, -0.7, +0.3, +1.3, +2.3, . . . since L = W +
T.

Although L for a single closed ribbon must always be an in-
teger there are special cases where the effective value of L need
not be integral, Consider the case of a circular piece of DNA
which is relaxed in solution under defined conditions so that it
is the "unconstrained" state. Two sets of authors (4, 5) have
shown that in practice one does not get a single value but a
Boltzmann distribution of L, due to thermal effects. Thus, the
effective value for the unconstrained molecule is the weighted
average of these integral values of L, and this average will, in
general, not be an integer. In the same way a solution of su-
perhelical DNA molecules will have an effective nonintegral
value of L. Note that while at any moment a particular mole-
cule in solution will have definite values of L, T, and W, the
thermal motion constantly changes T and W. Only L remains
constant with time.

Some examples

In spite of all obvious differences between the way T and L are
calculated it is not easy to obtain an easy familiarity with them
unless one works through a few examples. We will consider two
types of example: those which clarify the concepts and those
which illustrate the pitfalls.

For clarification the reader might reasonably ask for an ex-
ample of a ribbon for which L = 0 but T is not zero and the
converse example where T = 0 and L is not zero. These are
fairly easy to display.

Consider a regular left-handed dou&s helix made of a ribbon
wound flat on a cylinder and having any desired number of
turns (see Fig. 3b). The double helix is made into a closed ribbon
by joining the two ends at the top and the two ends at the bot-
tom, as shown in Fig. 3b. Then it is easy to show that for such
a structure [which, incidentally, is illustrated in Fuller's paper
(Fig. 1 of ref. I)] the value of L is zero. The proof is easy. Con-
sider a closed ribbon wound on a cylinder in the simple way
shown in Fig. 3c. Then by inspection we see that .L. = 0 since
the two edges of the ribbon are not linked. Now imagine that
the bottom of the cylinder is held still but that the top of the
cylinder is rotated so that the whole structure is twisted. Clearly
we can generate the double helix of Fig. 3b without tearing the

ribbon so it, too, must have L = 0. Equally clearly T is not equal
to zero since the contribution to the twist is negative as we go
up one helix and also negative as we come down the other, while
it can be made zero for the two loops at top and bottom by
constructing them using twistless bends. In fact (again ne-
glecting end effects) it is N sin (Y where N is the sum of the turns
up plus the turns down and (Y is the angle of inclination of the
helix (1).

It is not quite so easy to grasp a ribbon for which T = 0 and
L # 0. In principle one can do this for any closed ribbon whose
axis has an integral (non-zero) value of W by simply laying the
ribbon on the axis in such a way that L = W. Then T = 0.
Probably the best example is one of the simplest: choose a helical
ribbon like that in Fig. 3a having Q such that N sin LY is an in-
teger. Adjust the ribbon so that it is not laid flat on the gener-
ating cylinder but is given a counter-twist to make T zero. Then
connect the two ends using twistless bends.

However, a better way to grasp the distinct nature of the L
and T is to consider once again a ribbon wound flat on the
generating cylinder as a regular right-handed helix with exactly
N turns and inclination cr, as in Fig. 3a. How do L and T change
as cr is varied and N is kept constant? The answer is that L al-
ways equals N, no matter what the value of (Y (provided it is
between 0 and x/2) but that the twist, T, is equal to N sin ~1 (and
is negative if the helix is left-handed ). Notice also that the value
of L jumps from N to -N as a changes from positive to nega-
tive; when (;Y = 0 the ribbon degenerates and L is not defined.
Such discontinuities in L only occur when a change cannot be
brought about by a smooth topological deformation from one
curve to the other, as in the cases above (3).

There is one other example that may be helpful. Can we
define a closed ribbon for which L = 0 and for which the twist
is everywhere zero? To do this in the most general way draw
any closed line (which does not intersect itself) on a plane.
Imagine a ribbon whose axis lies on this line and whose breadth
is everywhere perpendicular to the plane. Notice that the ribbon
is bent, possibly in a very elaborate way, but its twist is zero
everywhere. This example makes quite transparent the dif-
ference between bending and twisting.

We now consider a few surprising examples. As we have just
emphasized a regular helix with, say, two left-handed turns and
which has its ends joined together outside the helix as shown in
Fig. 3a has L = -2. But what is the value of L if the ends are
joined together by passing the ribbon, untwisted, along the axis
of the helix, as shown in Fig. 3d? (To avoid end effects the joins
at top and bottom should be constructed using twistless bends
and not exactly as illustrated here.) The main features of this
new structure seem so similar to that of the old one that one feels
at first that L cannot be too different. Bolder spirits often guess
that L = 0. However, the surprising result is that L = +2 instead
of -2. Joining the ends of the ribbon up the center has changed
the sign of L. This can easily be proved by the algorithm de-
scribed earlier and indeed is a good example to try one's hand
on, but a neater proof is as follows Consider once again a ribbon
for which L is obviously zero but this time imagine one arm of
the ribbon on the surface of a cylinder and the other up the axis,
as shown in Fig. 3e. Now, as before, imagine the bottom of the
cylinder to be stationary and the top to be rotated so that we
generate a structure with two left-handed turns of flat ribbon
on the outside and a twisted branch of the ribbon up the axis
also having two left-handed turns. We now calculate the
writhing number. The outside section of the ribbon contributes
-2 sin a to the twist while the contribution of the central section
is -2. Thus, T = -2( I + sin cu). Since L = 0 we have W = +2( 1
+ sin (Y) for this structure. However the axis of the ribbon fol-


2642  Mathematics and Biochemistry: Crick

FIG. 4.  (a) A ribbon with three left-handed helical turns, the
ends uncrossed. (b) A similar ribbon having a little less than three
turns but with crossed ends.

lows exactly the same configuration as the structure we are
interested in, illustrated in Fig. 3d. Thus, for this structure, too,
W = +2( + sin a). However, for Fig. 3d the value of T is clearly
-2 sin cr since the central part of the ribbon is untwisted. Thus,
since L = W + T we obtain:

L = +2(1 + Sin a) -2 sin * = +2

This calculation illustrates in a neat way one of the uses of the
writhing number, W. It is sometimes easier to calculate L and
T for a structure having the same axis as the one in which we
are interested. Then if for our structure of interest, the calcu-
lation of T is easy, we can immediately obtain L for it, or vice
versa.

The reader may be disturbed by the neglect of "end effects."
This is usually justified though in real cases the whole ribbon
must be completely specified. It may come as a surprise that
any piece of ribbon can always have its ends joined together by
an extension of the ribbon such that the extension has zero twist.
This is done by using configurations which bend but are not
twisted, such as those shown in Fig. 2a and b. Naturally in
considering end effects one must be satisfied that this can be
done without causing difficulties with L but in the cases quoted
above this is not a problem.

As a second example, consider again a ribbon wound flat on
a cylinder for a number of left-handed turns, not in this case
necessarily an integer. However, instead of leading off the
ribbon in the obvious way, as shown in Fig. 4a, we make a
twistless bend at each end of the helix and lead off the ribbon
in the opposite directions, as shown in Fig. 4b. Again the result
is surprising. If the number of left-handed turns is just under
3, as for the structure shown in Fig. 4b, then L is not near -3
nor, as one might perhaps guess, near -2 but is close to -1.
Again this can easily be checked by the cautious use of the al-
gorithm given earlier. In fact, the value of L for structures of
this type is plotted against N in Fig. 5. It will be seen that at each
integer value of N there is a discontinuity, as might be expected
since, for example, a structure with N slightly less than 3 cannot
be smoothly deformed into one with N slightly greater than 3.
To convert one to the other the ribbon must pass through itself,
as one can easily verify with a wire model.

The reader may be puzzled by the fact that L, as displayed
in Fig. 5, is not always an integer. What does it mean to say that
when N = -2$, L = -I&? What is implied is that if a very
large number of structures are joined end to end then the total
value of L divided by the number of such structures is equal to
-l%, again neglecting end effects due to making the ribbon
into a continuous one. If the axes of the ribbon as it emerges on
the two sides of each substructure are parallel (as in our exam-
ple) then the value of L per substructure will only be an integer
if the stretches of ribbon at these two points are untwisted rel-
ative to each other. For the case quoted above, if N = -2lh, it
will be found to be twisted by just half a turn. Thus, in using the
algorithm to calculate L one should view a whole series of such
substructures from one viewpoint (see Appendfr) and find the
average value of L per substructure. Strictly, L has only

Proc. Natl. Acad. Scf. USA 73 (1976)

FIG. 5.  The value of the Linking Number, L, is plotted against
the number of turns, N, for structures like that shown in Fig. 4b. The
marked point corresponds roughly to the values chosen for Fig. 4b
itself. Both L and N are shown negative here because the helix has
been taken as left-handed. For the definition of L, see the Appen-
dix.

meaning for the whole of a closed curve and attempts to
factorize ft fnto sections are only justfffed in special cases.
The structure of chromatin

The above example may perhaps be relevant to the structure
of chromatin. The number of nucleosomes (or beads, or
v-bodies) on the simian virus 40 mini-chromosome was first
estimated by Griffith (6). From measurements on gels, using
the closed, circular, supercoiled DNA of simian virus 40 or
polyoma virus, several workers (7-9) have estimated that the
number of "supercoils" is about -1 or -1'/4 per nucleosome.
This is done by using a "relaxing enzyme" and counting the
number of different supercoiled species produced by spreading
them out on an agarose or similar gel. The bands seen are clearly
discontinuous, reflecting the integral nature of L. What is being
measured is, in our terminology, the difference between the
mean L for the mini-chromosome and the mean L for the
completely relaxed form of DNA (in that medium) without
supercoils. This estimate also fits with earlier, less direct,
methods (10-12). Note that at any temperature there is, for
unnicked, closed, circular DNA molecules, a Gaussian distri-
bution of L about its mean value (4,5).

The above example (Fig. 5) shows that a small modification
to a structure with, in the loose sense, about three "supercoils"
can produce a value of L, per nucleosome, not far from the
observed value of -I. Whether this is the true explanation of
the rather low observed value of L remains to be seen. If a
model were required for which L was near zero a rather neat
solution is to have the DNA folded as in Fig. 4b but with N a
little less than 2 rather than a little less than 3.

What emerges clearly from the above examples is that,
without some experience, it is not always obvious how to esti-
mate the value of L for a structure and that one can easily make
considerable errors. Thus, L should always be calculated
carefully. It is fortunately easy to obtain an approximate value
for a structure by constructing it from a piece of flexible ribbon
of, say, some dressmaking material. One then simply pulls the
ribbon "straight" and counts the number of twists. This will
show if a gross error has been made and can also be used to as-
tonish one's colleagues. tiowever, for more exact work a careful
estimate should be made from a model of the structure itself.
A "kink" in DNA, for example although it mainly produces a
bending, will often impart a small amount of twist and this may
have to be allowed for. Moreover it is not obvious that the DNA
in the nucleosome will be exactly in the normal B configuration
unless other evidence suggests this.

In this paper we have been considering only configurations
but mechanical properties not at all. An approach to these more
difficult problems is sketched in the latter part of Fuller's paper
0).


Mathematics and Biochemistry: Crick

            APPENDIX
The definition of twist

For an exact definition see ref. 1, where it is called the total
twfst number.

At any point on the axis of the ribbon we define a vector, X,
pointing along the axis and tangential to it at that point. We now
define another vector, U, perpendicular to X and passing
through the two edges of the ribbon at that point (the ribbon
is strictly considered to have an extremely small width). As our
point moves along the axis of the ribbon the vector U rotates
about the vector X. Then the twist is the integrated angle of this
rotation in radians, divided by 27r, integrated right around the
closed ribbon. It need not be an integer. If the rotation is
right-handed the twist is positive.
The calculation of L for Fig. 5

The special viewpoint required to calculate L correctly for Fig.
5 is defined as the view from infinity, perpendicular to a par-
ticular plane. Let there be M identical substructures, of the type
shown in Fig. 4b, joined end-t-end without any deformation.
Consider the axis of the ribbon itself where it enters the string
of M substructures and also the axis of the ribbon where it leaves
the string. Then the required plane is the plane containing these
two short straight lines.

Calculate, using the cross-over algorithm, the total contri-
bution to L from the entire string of M substructures. Then the
average L per substructure is defined as the limit L/M as M
tends to infinity. For this example the above viewpoint always

Proc. Natl. Acad. Scf. USA 7s (1976)   2643

allows one to construct a joining ribbon (of the general type
shown by broken lines in Fig. 3a) which contributes nothing
additional to the algorithm for L.

I wish especially to thank Prof. Fuller, for many useful points made
in correspondence and for allowing me to quote unpublished work,
and Dr. Graeme Mitchison of this laboratory for helpful explanations
and discussion, and in particular for the neat proof shown in Fig. 3b
and c. I also thank Dr. Aaron Klug and Prof. J. Vinograd for valuable
comments.

1.

2.
3.

4.

5.

6.
7.

8.


9.
10.
11.


12.

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