The discovery is described in **Nature 406, 287-290 (2000)**.

**Helical Close Packings of Ideal Ropes**

Sylwester Przybył , Piotr Pierański

**1. Introduction**

The phenomenon of coiling is observed in various biological and physical
systems such as the tendrils of climbing plants [1]_{, }one-dimensional
filaments of bacteria [2] or cylindrical stacks of phospholipid membranes
interacting with an amphiphilic polymers [3]. In some cases the phenomenon
occurs in conditions in which the helical structures created by coiling
become closely packed. A single, closely packed helix is one of them. In
a different context its formation was studied in Monte-Carlo simulations
by Maritan et. al. [4]. Below we present a simple analytical arguments leading
to the determination of the parameters of the optimal closely packed helix.

Take a piece of a rope of diameter *D* and try to arrange it into
a right-handed helix, see Fig.1, described parametrically by the set of
equations:

(1)

Fig. 1 The radius and pitch of a helix defined by eq. (1)

The helix is well defined if its radius *r* and pitch *P* are
specified. As easy to check experimentally, when the helix is formed on a
real rope, not all values of *r* and *P* are accessible. Being material,
the rope cannot be arranged into shapes, which violate its self-impenetrability.
If, for instance, one chooses to form a helix with *r*=2*D*, its
pitch cannot be made smaller than about 1.003 *D*. This is the value
at which the consecutive turns of the helix become closely packed. For a smaller
pitch overlaps would occur. See Fig. 2.

Fig. 2 When at a given radius the pitch of the rope helix is too small overlaps appear.

*r*=2; *P*=1.003 on the left and *P*=0.5
on the right.

A general question arises: *which are the limits for the radius and
pitch values of a helix formed with a rope of the diameter D*?

The question was posed some time ago in discussions connected with the problem of ideal knots [5]. Calculations we present here were directly stimulated by recent paper by Maritan et al. who performed Monte Carlo simulations of the rope confined within a box. As indicated by the authors, the pitch to radius ratio of the optimal helix they discovered matches very well the value of the ratio found in the a -helix discovered by nature in the evolution processes. Possible implications of their results were discussed by Stasiak and Maddocks [6].

The symbolic algebra and numerical techniques we present below are analogous to those we applied considering the problem of the close packings of two ropes twisted together [7].

**2. Ideal rope**

Considerations presented below are valid for the so-called ideal ropes,
i.e. ropes, which, from a physical point of view are completely flexible,
but at the same time perfectly hard. Assume that the axis of such a rope
is shaped into a smooth curve *C*. At any point of the curve its tangent
vector ** t** is defined. The rope is ideal, if each of its sections,
perpendicular to the tangent vector

Let the ideal rope be shaped into a closely packed helix *H* of a
radius *r*>*D*. To understand better, what we mean by the „closely
packed helix”, we may imagine that the rope is wound as tightly as possible
on a cylinder of diameter (2*r*-*D*).

on a cylinder of diameter (2*r*-*D*).

**3. Closely packed helix limited by its doubly critical self distance**

Consider the ideal rope shaped into a helix *H* whose consecutive
turns touch each other. Which, for a given *r*, should be the pitch
*P* of the helix, to keep its turns closely packed? We shall answer
the question.

Consecutive turns of the helix touch each other if the minimum of the
distance from any point *P*_{1} of the helix to the points located
in the beginning of the next turn is equal *D*. Let *P*_{2}
be the point at which this minimum distance is reached. Let *t*_{1}
and *t*_{2} be the vectors tangent of *H* at *P*_{1}
and *P*_{2}, respectively. Obviously, in such a situation,
the
vector is perpendicular both to* t*_{1} and *t*_{2}.
belongs both to the plane S_{1 }located
at *P*_{1} and perpendicular to *t*_{1 }and
to the plane S_{2} located at *P*_{2}
and perpendicular to *t*_{2}. Thus,
belongs to the line along which S_{1}
and S_{2} cross. Let *P*_{1 }located
at (*x*_{1}, *y*_{1}, *z*_{1}) be
given; let it be the point of *H* defined by x* _{1}*
= 0:

(2)

Components of the tangent vector *t*_{1} located at
*P*_{1} are equal:

(3)

Consequently, the S_{1} plane going
through *P*_{1} and perpendicular to *t*_{1}
is defined by the equation:

, (4)

Let

(5)

be the coordinates of the *P*_{2} point located in the vicinity
of the next turn, i.e. at the x values close to
2p . The point must belong to the S_{1} plane. Consequently its coordinates given
by (5) must fulfil equation (4), what gives:

(6)

The square of distance between *P*_{1} and *P*_{2}
should be equal *D*^{2} what results in equation:

(8)

which in a parametric manner describe the relation *P*_{CP1}(*r*)
between *P* and *r* which must be fulfilled by helices whose consecutive
turns are closely packed. (Notice that x becomes
here a free parameter which serves to describe the shape of the *P*_{CP1}(*r*)
relation.) The shapes of the *P*(x ) and
*r*(x ) functions are shown in Fig. 4, where
we present them in the potentially interesting interval of xÎ (p , 2p ).

Fig.4 Shapes the *P*(x
) and *r*(x ) functions given by (7). *D*=1.

The shape of the *P*_{CP1}(*r*) relation obtained within
the parametric plot is shown in Fig.5.

Fig.5 *P*_{CP1}(*r*) relation described
parametrically by *P*(x ) and *r*(x ) functions shown in Fig. 3.

As seen in the figure, the *P*_{CP1}(*r*) relation contains
two branches. Which of them presents the physically sensible solution we may
find out checking within which range of x the function
of the square distance

(9)

in which *P* and *r* are in the relation described by (8) displays
a minimum versus Dx . (When the minimum exists,
the turn-to-turn distance we are calculating becomes identical with the *doubly
critical self distance* introduced by Simon [8]:
, where *h* is the helix parameterised by arc-length, *h*(*x*)
and *h*(*y*) are points located on the helix, (*h*(*x*)-*h*(*y*))
is the vector which joins the points and *h’*(*x*) and *h’*(*y*)
are the tangent vectors at *h*(*x*) and *h*(*y*), respectively.)
To reach the aim, we substitute (8) into (9), expand it into a Taylor series
truncated at the (Dx )^{2} term and differentiate
it twice versus Dx . The second derivative obtained
in such a manner equals:

. (10)

Fig. 6 presents it within the interesting (p , 2p ) range.

Fig. 6 The second derivative of the square distance function.

The second derivative is positive in the interval (4.49341, 2p ). Replotting the *P*_{CP1}(*r*)
relation only within this range reveals which of its branches presents the
required solution. See Fig. 7.

Fig.7 The dependence of the pitch *P* on the radius
*r* found in closely packed helices. *D*=1.

The result we obtained stays in agreement with the intuition: a closely
packed helix, whose radius is squeezed, increases its pitch. As seen in
the figure, *r*=0.5 seems to limit the squeezing process. Is it really
the case?

**4. Closely packed helix limited by its curvature**

There exists another mechanism, which limits the set of possible (*r*,*P*)
values of the helices formed on an ideal rope. It stems from the fact that
the ideal rope of diameter *D* cannot have a local curvature larger
than 2/*D*. The following heuristic reasoning indicates the source of
the limitation. Let *h*(*x*) be a helix of curvature k parameterised by arc-length *x*. Let *h*’(*x*)
be the field of its tangent vectors. Imagine that a disk of diameter *D*,
centred on *h*(*x*) and perpendicular to *h*’(*x*) is
swept along the helix. The circular border of the moving disk determines within
the space the surface of the ideal rope. Consider the traces *d*(*x*_{1})
and *d*(*x*_{2}) of the disk in two consecutive positions
*h*(*x*_{1}) and *h*(*x*_{2}) separated
by an infinitesimal arc *dx*. Because of the non-zero curvature of the
helix along which the disk was swept disks *d*(*x*_{1})
and *d*(*x*_{2}) are not parallel to each other – they are
inclined by angle k*dx*. When k >2/*D* the disks overlap. Consequently, the
surface of the rope determined by border of the swept disk becomes non-smooth.
Fig.8 illustrates the situation.

The curvature of a helix defined by equation (1) equals:

(12)

As easy to find, equation k =2/*D* is
fulfilled if

(13)

Fig. 9 presents relation (13) together with the discussed above relation
(8). One can see that their plots cross. As a result some parts of the *P*_{CP1}(*r*)
and *P*_{CP2}(*r*) solutions become inaccessible.

Fig.9 *P*_{CP1}(*r*) and *P*_{CP2}(*r*)
solutions plotted together. The mutually inaccessible parts

of the solutions are marked with a dashed line. *D*=1.

Numerically determined coordinates of the crossing point are as follows:

(14)

Mutually accessible parts of the *P*_{CP1}(*r*) and
*P*_{CP2}(*r*) solutions define within the (*r*,*P*)
plane a border line *P*_{CP}(*r*) below which one cannot
go; helices found in this forbidden region are impossible to build with
the perfect rope. Figure 10 presents the border *P*_{CP}(*r*)
line together with the images of the closely packed helices located at a
few representative points of it .

**5. Physical properties of the optimal helix**

Monte Carlo simulations performed by Maritan et al. were aimed to find those shapes of the rope, for which the radius of gyration becomes minimised. Radius of gyration is a geometrical property of a curve. It is defined as the root mean square distance of a set of points (obtained by a discretisation of the curve) from its centre of mass. Does the optimal helix minimises some physical properties of the curve?

Let us imagine that equal masses, *m*=1, are distributed along an
ideal rope at equal distances *dL.* The rope is then shaped into a closely
packed helix. Which is the energy of the gravitational interaction between
the masses? Obviously, the energy depends on the Euclidean distance *dS*
between them. See Fig. 11. The distance *dS* is always smaller than
*dL* and its value depends on the parameters of the helix into which
the rope is shaped. As shown above, values of the parameters are limited
by the turn-to-turn distance (doubly critical self distance) and the curvature
of the helix.

Fig. 11 The relation between *dL* and *dS*.

Calculations we performed show, that for a given *dL*, *dS*
reaches its minimum within the optimal helix. See Fig. 12.

Fig. 12 *dS* vs. *r* for closely packed helices.

Consequently, the energy of gravitational interactions reaches its minimum
as well. One can certainly find a few other cases, in which the optimal
helix of Maritan et al. proves also to be optimal.

**6. Discussion**

We have shown that looking for closely packed helices formed with the
ideal rope one has to consider two cases: helices limited by the turn-to-turn
distance and helices limited by the local curvature. As indicated by Maritan
et al._{ }[4], the two cases can be brought into one: helices limited
by the *global curvature*, a notion introduced by O. Gonzalez and J.
Maddocks [9]. The radius of the global curvature at a given point *P*
of a space curve *C* is defined as the radius of smallest circle which
goes through the chosen point and any other two points *P*_{1},
*P*_{2} belonging to *C* and different from *P*.
Putting a limit on the global curvature of a helix one limits both the local
curvature and the closest distance between its consecutive turns. As a result
the union of accessible parts of the presented above partial solutions *P*_{CP1}(*r*)
and *P*_{CP2}(*r*) can be seen as a single solution *P*_{CP}(*r*).
The solution shown in Fig.10 answers the problem formulated as follows:
*which is the relation between the pitch P and the radius of helices whose
global curvature equals 2/D? *Asking simpler, synthetic questions helps
finding simpler, synthetic answers.

There exists another, equivalent formulation of the problem. Instead of
the global curvature one can use the notion of the *injectivity radius
*[10]. The injectivity radius of a smooth curve *K* is the maximum
radius of the disks which centred on each point of *K* and perpendicular
to its tangent remain disjoint. In terms of the injectivity radius the closely
packed helices can be seen as *the helices whose injectivity radius is
equal to the predetermined radius of the used tube*.

**REFERENCES**

. A. Goriely and M. Tabor, Phys. Rev. Lett. **80**, 1564 (1998).

2. R. E. Goldstein, A. Goriely, G. Huber and C.Wolgemuth, Phys. Rev. Lett.
**84**, 1631 (2000).

3. V. Frette et. al. Phys. Rev. Lett. 83. 2465 (1999); I. Tsafrir, M.-A. Guedeau-Boudeville, D. Kandel and J. Stavans, preprint: cond-mat/0008435.

4. Maritan A. *et al*. Nature **406**, 287 (2000).

5. Katritch V. *et al*. Nature **384**, 142 (1996); *Ideal Knots*,
eds. Stasiak A., Katritch V. and Kauffman L. H., World Scientific, Singapore,
1998.

6. Stasiak A. and Maddocks J. H., Nature **406**, 251 (2000).

7. Pierański P. in *Ideal Knots*,
eds. Stasiak A., Katritch V. and Kauffman L. H., p.20, World Scientific,
Singapore, 1998; Przybył S. and Pierański P.,
ProDialog, **6**, 87 (1998), in Polish.

8. J. O’Hara in *Ideal Knots*, eds. Stasiak A., Katritch V. and Kauffman
L. H., p.20, World Scientific, Singapore, 1998

9. Gonzalez O. and Maddocks J. H., Proc. Natl. Acad. Sci. USA **96**,
4769 (1999).

10. See e.g. E. Rawdon in *Ideal Knots*, eds. Stasiak A., Katritch
V. and Kauffman L. H., p.20, World Scientific, Singapore, 1998.

For Rob Kusner