Geometry of space curves and applications to polymers’ conformations investigation
A. Ivanov, A. Mishchenko, A. Tuzhilin
Moscow State University, Russia
Contents
Abstract:
The main aim of the present work is to demonstrate how to use geometrical characteristics of space polygonal lines to investigate polymers’ conformations. The geometrical characteristics suggested below, are discrete analogue of the curvature and the torsion of the space curves which are well-known in Classical Differential Geometry. Main Theorem of Curves Theory says that a space curve can be uniquely reconstructed (up to a motion) starting from its curvature and torsion functions. It turns out that complicated curves may have sufficiently simple curvature and torsion functions. The latter makes us expect to obtain a more simple representation of complicated forms of polymers, which may lead to some progress in the conformations investigations.
We put
v(s) = g’(s) – the velocity vector (its absolute value is equal to 1),
k(s) = | g’’(s) | – the curvature,
n(s) = g’’(s) /| g’’(s) | – the principal normal vector (we suppose that the curvature is not zero, such curves are referred as biregular),
(v, n) – the Frenet frame,
{v’ = k n; n’ = -k v} – Frenet Theorem.
Theorem. For any positive smooth function f(s), there exists a plane curve g(s), where s is the natural parameter, such that its curvature k(s) is equal to f(s). Any such curves differ from each other by a plane motion.
Let us give several examples of plane curves with given curvature functions.
If k(s) = 0, then g(s) is a straight line.
If k(s) = const 0, then g(s) is a circle.
Here we see a more complicated example, a helix.
Notice that simple curvature functions correspond to complicate curves often. To illustrate this we list a series of plane curves with the curvature function of the form
k(s) = a cos2(s) + 1.
And here we see an example of a curve with more complicated curvature function.
Now let us consider the case of space curves.
Let g(s) = (x(s), y(s), z(s)) be a smooth parametric plane curve, i.e. the functions x(s), y(s), and z(s) are smooth, and let s be the natural parameter, i.e. the absolute value of the velocity vector g’(s) = (x’(s), y’(s), z’(s)) is equal to 1.
We put
v(s) = g ’(s) – the velocity vector (its absolute value is equal to 1),
k(s) = | g’’(s) | – the curvature,
n(s) = g’’(s) /| g’’(s) | – the principal normal vector (we suppose that the curvature is not zero, i.e. the curve is biregular),
b(s) = [v(s), n(s)] – the binormal vector (here [ , ] stand for the cross product),
(v, n, b) – the Frenet frame,
{v’ = k n; n’ = -k v + x b; b’ = - x n} – Frenet Theorem,
x(s) = < b’(s), n(s)> – the torsion.
Theorem. For any positive smooth function f(s) and any smooth function g(s) there exists a biregular space curve g(s), where s is the natural parameter, such that its curvature k(s) is equal to f(s) and its torsion x(s) is equal to g(s). Any such curves differ from each other by a space motion.
Let us give several examples of space curves with given curvature and torsion functions.
If k(s) = 0, then g(s) – is a straight line, and the torsion x(s) is not defined.
If x(s) = 0, then g(s) – is a plane curve.
If k(s) = const 0 and x(s) = 0, then g(s) – is a circle.
If k(s) = const 0 and x(s) = const 0, then g(s) – is a space helix (i.e. a screw line).
We also include another example. In the right hand part, the current curvature function is shown in red, and the torsion function does not change (it is shown in black).
Polymers appearing in biology usually have rather complicated structure. An example of such polypeptide is shown in the next Figure.
Structure of a Nucleosome (fragments of DNA are shown in light gray).
We illustrate our main ideas on the case of proteins conformation modeling. The space from of a protein can be described by a polygonal line joining the consequent alpha-carbons (the carbon atoms which the radicals are joined with).
To describe the geometry of space polygonal lines we use the following local geometrical characteristics of them:
There are several ways to define the analogue of curvature and torsion for a polygonal line. To obtain a geometrically sensible characteristics, it is natural to demand that the resulting values possess the following property. If we consider a polygonal line inscribed into a biregular curve, then the values tend to the curvature and the torsion of the curve, as the polygonal line tends to the curve. (However, it turns out that for the polypeptides under consideration this condition is not very important, since the corresponding polygonal lines have the edges of nearly the same length.)
Let us give the most natural ways to define the concepts of curvature and torsion for a polygonal line:
the torsion is proportional to the volume of the parallelepiped spanned on to three consequent edges divided by the product of the areas of the two consequent triangles.
Below we give the results of the numerical experiments demonstrating the behavior of the curvature and torsion analogue for fragments of real proteins. The curvature is shown in red, and the torsion is shown in blue.
Notice, that up to a scalar multiplier the graphs obtained are quite similar to each other. But this effect can be explained mostly by the property of nearly the same edges lengths of the polygonal lines under consideration, see Remark above.
Recall that in the case of biregular curves, the constancy of curvature and torsion property is a characteristic property of space helixes (the screw lines). It is well known that proteins usually contain spiral fragments which are referred as a-helixes. Let s mention that the fragments corresponding to the a-helixes can be distinctly seen in our graphs. Namely, they correspond to nearly horizontal fragments of the both graphs. Besides, flat fragments of the space curves can be characterized by the torsion zeroing. Such fragments are also known in proteins description as so called b-layer. The fragments corresponding to the b-layers can be distinguished in he graphs as the ones where the torsion oscillates is a neighborhood of zero. Thus, one of possible applications of the concepts defined above is a numerical finding of a-helixes and b-layers.
Another possible application is a statistical analysis of relations between the curvature and torsion functions describing a protein molecule and the amino acids sequence corresponding to the molecule. But here we need some more complicated model which is worked out by our group now.
Also it is possible to use the obtained characteristics to model conformation changing of proteins. The idea is to construct a continuous deformation of the curvature and torsion of the initial polygonal line to the curvature and torsion of the terminal polygonal line through some appropriate class of functions, and than to construct the corresponding polygonal line for each intermediate set of values. Below we give two possible realizations of this idea.
HUMAN PRION PROTEIN,
Animation of a deformation which changes curvature and torsion of the entire polygonal line simultaneously.
HUMAN PRION PROTEIN
Animation of a deformation which changes curvature and torsion inside a small frame only.
To model long range interactions between amino acids forming a protein also, we generate a model of an elastic girder, corresponding to a protein molecule. Such girder consists of elastic edges joining not only the neighboring alpha-carbons (as in the polygonal line described above) but also the alpha-carbons separated by one or two other alpha-carbons. In other words, we add to the polygonal line the links joining each vertices separated by one or two consecutive vertices. These edges supposed to have the same elastic properties. Than we also add one or several elastic edges corresponding to long range interactions which evoke the protein tertiary structure appearance. After all, we look an equilibrium position under Hook’s low assumptions. Here we give an example of numerical experiments results describing the behavior of a helix whose end points are attracted to each other.