Determining Flexibility of Molecules Using Resultants of Polynomial Systems

coauthor Evangelos Coutsias

We solve systems of multivariate polynomial equations in order to understand flexibility of three dimensional objects, including molecules.   

Protein flexibility is a major research topic in computational chemistry. In general, a polypeptide backbone can be modeled as a polygonal line whose edges and angles are fixed while some of the dihedral angles can vary freely. It is well known that a segment of backbone with fixed ends will be (generically) flexible if it includes more than six free torsions. Resultant methods have been applied successfuly to this problem. In this work we focus on non-generically flexible structures (like a geodesic dome) that are rigid but become continuously movable under certain relations. The subject has a long history: Cauchy (1812), Bricard (1896), Connelly (1978).

In a previous work, we began a new approach to understanding flexibility, using not numeric but symbolic computation. We describe the geometry of the object with a set of multivariate polynomial equations, which we solve with resultants. Resultants were pioneered by Bezout, Sylvester, Dixon, and others.  The resultant appears as a factor of the determinant of a matrix containing multivariate polynomials. We describe a method to find these factors "early". Given the resultant, we described an algorithm, Solve, that examines it and determines relations for the structure to be flexible. We discovered in this way the conditions of flexibility for an arrangement of quadrilaterals in Bricard, which models molecules.  Here we significantly extend the algorithm and the molecular structures.  We consider the cylo-octane molecule.

The Brusselator and Other Polynomial Systems In Chemical Autocatalytic Reactions

This was part of the Elimination Session, http://aca2009.etsmtl.ca/sessions/elimination.html

Autocatalytic reactions are chemical reactions in which at least one of the products is also a reactant. This "feed-back loop" yields a system of nonlinear polynomial equations. A famous example is called the Brusselator. Although the classic Brusselator equations are trivial, the basic idea generalizes to more interesting equations.
  We will examine the systems of equations that result from two- and three-dimensional configurations of interacting Brusselators. We have up to eight equations in eight variables and up to twelve parameters. We find that all are solvable with Dixon resultant methods. We will describe how  Groebner Bases fail on all but the simplest cases. We will show other examples of autocatalytic reactions.

Dixon Beats Groebner: "Almost Linear" Polynomial Equations Arising in GPS Systems and in Nash Equilibria

We  apply the Dixon-EDF resultant method to several sets of multivariate polynomial equations. They arise in two applications.  The first is GPS, or global positioning systems. We show that a 3D affine transformation problem can be completely solved symbolically with Dixon-EDF. Other symbolic techniques failed. One of these systems has 6 equations in 6 variables and 12 parameters.  Another has 9 equations in 9 variables and 18 parameters. In both systems, every equation has (total) degree three.

Secondly, we use Dixon-EDF to solve several sets of equations that arise from the study of Nash equilibria. This is an important topic in economic game theory. We examine the cases of three or four  players with two pure strategies each.  The latter produces a set of 8 equations with 8 variables and 32 parameters. Then we look at a classic problem due to Nash, simplified three-man poker (with 4 equations, 4 variables, 44 parameters), and lastly at a "cube game"  (8, 8, 4). These are found in the book by Sturmfels and the papers by Datta. Apparently we are the first to provide fully symbolic solutions to these games.

All of these problems are solvable with Dixon-EDF.  Apparently all are intractable with other methods. We report on failed attempts to solve these with Maple12, using both its builtin Groebner bases command and its implementation of Faugere's fgb algorithm.

There is another common thread in these two apparently disparate subjects: all the equations are of degree one in each variable. That is, in every equation no variable is squared.  In only one of the equations is any parameter squared.  Indeed, often we find that every equation is of total degree two in the variables. These are in some sense the simplest non-linear equations; we call them "almost linear."  Yet Groebner bases methods fail repeatedly as Dixon-EDF succeeds.

Comparing acceleration techniques for the Dixon and Macaulay resultants

later published in MATCOM

The Bezout-Dixon resultant method for solving systems of polynomial equations lends itself to various heuristic acceleration techniques, previously reported  by the present author, which can be extraordinarily effective. In this paper we will discuss how well these techniques apply to the Macaulay resultant.  In brief, we find that they do work there with some difficulties, but the Dixon method is greatly superior. That they work at all is surprising and begs theoretical explanation.

Polynomials Everywhere

For about twenty years, I have worked on the borderline between mathematics and computer science. I’ve become an applied mathematician, but in a sense that didn’t exist thirty years ago.

  I think the general public does not have a very good idea of what mathematicians do, so let me start with some generalities.  Before roughly 1985, almost all mathematicians in academic positions in the United States and Europe were pure mathematicians. ...

Fermat Address to International Congress of Mathematicians

Fermat is an interactive system for mathematical experimentation.  It is a super calculator -- computer algebra system, in which  items being computed can be  integers (of arbitrary size), rational numbers, real numbers, complex numbers, modular numbers, finite field elements, multivariable polynomials, rational functions, or polynomials modulo other polynomials.  The main areas of application are multivariate rational function arithmetic and matrix algebra over rings of multivariate polynomials or rational functions. Fermat does not do simplification of transcendental functions or symbolic integration.

Comparison of Polynomial-Oriented Computer Algebra Systems

published in SIGSAM bulletin, 2000.

Thanks to Alexandra Seremina, a translation of this paper into Spanish is available at:

http://www.azoft.com/people/seremina/edu/computer-algebra-systems.html

(I do not speak Spanish and cannot verify the translation.)

See also

http://www.fordham.edu/lewis/cacomp.html

[Now nine years old, but still worth a look.]

Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields. By "exact symbolic" we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra systems ("CA systems"). Over the last dozen years several large CA systems have become widely available, such as Axiom, Derive, Macsyma, Magma, Maple, Mathematica, and Reduce. They tend to have great breadth, be produced by profit-making companies, and be relatively expensive. However, most if not all of these systems have difficulty computing with the polynomials and matrices that arise in actual research. Real problems tend to produce large polynomials and large matrices that the general CA systems cannot handle.

In the last few years several smaller CA systems focused on polynomials have been produced at universities by individual researchers or small teams. They run on Macs, PCs, and workstations. They are freeware or shareware. Several claim to be much more efficient than the large systems at exact polynomial computations. The list of these systems includes CoCoA, Fermat, MuPAD, Pari-GP, and Singular.