Decoupled molecules with binding polynomials of bidegree (n, 2)
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Abstract
We present a result on the number of decoupled molecules for systems binding two different types of ligands. In the case of n and 2 binding sites respectively, we show that there are \(2(n!)^{2}\) decoupled molecules to a generic binding polynomial. For molecules with more binding sites for the second ligand, we provide computational results.
Keywords
Ligand binding Binding polynomials Numerical algebraic geometryMathematics Subject Classification
82B05 92C40 14Q99 65H04 68W301 Introduction
In biology, a ligand is a substance that binds to a target molecule to serve a given purpose. A classical (Bohr et al. 1904; Hasselbalch 1917) and intensively studied (Barcroft 1913; Hill 1913) example is oxygen, which binds reversibly to hemoglobin to be transported through the bloodstream. Reversible mutual binding of different molecules is also a key feature in biological signal transduction (Changeux and Edelstein 2005; Cho et al. 1996; Gutierrez et al. 2009; Ha and Ferrell 2016) and gene regulation (Gutierrez et al. 2012).
For two different types of ligands, the binding polynomial has two variables (representing the activities of both ligands in the environment) and \((n_1+1) \cdot (m_1+1)\) coefficients \(\left( a_{i,j}\right) _{i=0 \ldots n_1;j=0,\ldots ,n_2}\). Given an arbitrary binding polynomial for two types of ligands, it is not in general possible to find a molecule without interactions between all binding sites and possessing this binding polynomial. However, molecules can be found in which the binding sites for the same type of ligand do not interact and only interactions between sites for different ligands are nontrivial (Martini et al. 2013b, c). Contrary to the case of one type of ligand, where the decoupled sites representation is unique up to permutation of the roots, there are several different decoupled molecules. It has been shown previously that in the case of n and 1 binding sites for the two ligands, respectively, there are n! decoupled molecules. The situation becomes more complicated for general systems of \(n_1\) and \(n_2\) binding sites. The main goal of this paper is to prove the following theorem.
Theorem 1.1
The decoupled molecules of a fixed binding polynomial of bidegree (n, 2) are the solutions to a system of \(3n+2\) unknowns: the \(n+2\) binding energies and the 2n interaction energies. For generic binding polynomials, the number of complex solutions to this system equals \(4(n!)^3\). These come in \(2(n!)^{2}\) classes under relabeling of the sites.
2 Background and framework
In this section, we briefly recap the algebraic framework as well as past results, and, in doing so, fix various notations. Most importantly, we introduce some shorthand notation for molecules with (n, 2) sites for Sects. 3 and 4.
2.1 Single type of ligand
The binding behavior of systems with one type of ligand is governed by the energies required to bind to each site of the target molecule and the way different binding sites interact with each other. Following the notation of Martini and Ullmann (2013), we identify target molecules with these parameters.
Definition 2.1
Definition 2.2
The following theorem is also known as the decoupled sites representation. It implies that any molecule with real binding and interaction energies can be uniquely represented by a molecule with neutral interaction energy, provided that complex binding energies are allowed. Its proof consists of a reformulation of Vieta’s formulas.
Theorem 2.3
(Martini and Ullmann 2013, Proposition 2) For any molecule \(\mathrm{N}\) there exists a decoupled target molecule \(\mathrm{M}\), unique up to relabelling of the sites, such that \(\mathrm{P}_{\mathrm{M}} = \mathrm{P}_{\mathrm{N}}\).
2.2 Multiple types of ligands
In case of \(d>1\) types of ligands, we consider each binding site to be only able to take up to one type of ligand (Martini et al. 2013c). This is sensible, as we can model a single binding site capable of binding to two types of ligands as two binding sites with interaction energies set so that the two sites can never be saturated at the same time.
For our purposes, let us assume that \(d = 2\). We write \(n_{1}\) and \(n_{2}\) for the number of sites capable of binding to the first and second ligand, respectively.
Definition 2.4

\(g_{T_1},\dots ,g_{T_{n_1}}\) and \(g_{S_1},\dots ,g_{S_{n_2}}\) are the binding energies,

\(w_P\) for \(P\subset \{T_1,\dots ,T_{n_1},S_1,\dots ,S_{n_2}\}\) with \(P=2\) are the interaction energies.
Similar to the case \(d=1\), there is a natural \(S_{n_1}\times S_{n_2}\) action that corresponds to relabelling the sites.
Definition 2.5
For \((n_1,n_2)=(n,1)\), the decoupled sites representation takes the following form:
Theorem 2.6
(Martini et al. 2013c, Corollary 2) For any molecule \(\mathrm{N}\) with (n, 1) sites there exist, up to relabelling of the sites, and counted with multiplicity, n! decoupled molecules \(\mathrm{M}\) of the same type such that \({\mathrm{P}_\mathrm{N}= \mathrm{P}_\mathrm{M}}\).
2.3 Decoupled molecules with (n, 2) sites

\(g_1,\dots ,g_n,g_A,g_B\) represent the binding energies,

\(w_{1,A},\dots ,w_{n,A},w_{1,B},\dots ,w_{n,B}\) represent the nontrivial interaction energies.
3 Numerical algebraic geometry
Theorem 3.1
 (1)
\(({\underline{g}}^*,{\underline{w}}^*)\) is the only solution of \(f({\underline{g}},{\underline{w}};{\underline{a}}^*)=0\) for \(({\underline{g}},{\underline{w}})\in {\mathcal {U}}\);
 (2)
for fixed \(a'\in {\mathcal {V}}\), \(f({\underline{g}},{\underline{w}};{\underline{a}}')=0\) has only isolated solutions for \(({\underline{g}},{\underline{w}})\in {\mathcal {U}}\);
 (3)
for fixed \(a'\in {\mathcal {V}}\), the multiplicity of \(({\underline{g}}^*,{\underline{w}}^*;{\underline{a}}^*)\) as a solution of \(f({\underline{g}},{\underline{w}};{\underline{a}}^*)=0\) is the sum of the multiplicities of the solutions of \(f({\underline{g}},{\underline{w}};{\underline{a}}')=0\) for \(({\underline{g}},{\underline{w}})\in {\mathcal {U}}\).
Example 3.2
Should \(x^n+a_1' x^{n1} + \dots + a_n'=(x1)^n\), then the only solution is \({\underline{g}}'=(1,\dots ,1)\). Theorem 3.1 implies that this solution is of multiplicity n!. This will be important in the proof of Lemma 4.2.

a starting solution \((g',w';a')\in X\times Y\),

a target parameter \(a^*\in Y\),

a continuous path \(\phi :[0,1]\rightarrow Y\) with \(\phi (1)=a'\) and \(\phi (0)=a^*\),
Example 3.3
4 Generic decoupled molecules with (n, 2) sites
In this section, we show that a generic binding polynomial represents \(4\cdot (n!)^3\) decoupled molecules with (n, 2) sites. Due to the complexity of the system of polynomial equations, the proof is split in two parts. First, we study a special class of decoupled molecules and their binding polynomials. In a second step, we study their implication to the generic case.
4.1 Normalized molecules
In this subsection, we restrict ourselves to a special class of decoupled molecules and their binding polynomials. This simplifies our system of equations and allows us to show that a generic binding polynomial of such molecules represents 2n! molecules, each of multiplicity \(2(n!)^{2}\).
Definition 4.1
Lemma 4.2
Proof

any solution \((g_A,g_B;a_{0,j})\) to the two equations in the first row is of multiplicity 2 (see Example 3.2),

the solution \((g_1,\dots ,g_n;a_{i,0})=(1,\dots ,1;\left( {\begin{array}{c}n\\ i\end{array}}\right) )\) to the latter equations in the first column is of multiplicity n!,

given \(g_A=g_B=g_i=1\), the solution \((w_{i,A},w_{i,B};a_{i,2})=(1,\dots ,1;\left( {\begin{array}{c}n\\ i\end{array}}\right) )\) to the latter equations in the second column is of multiplicity n!.
Proposition 4.3
A generic normalized binding polynomial represents 2n! decoupled molecules, each of multiplicity \(2(n!)^{2}\).
Proof
4.2 A generic decoupled sites representation
Theorem 4.4
A generic binding polynomial of bidegree (n, 2) represents \(4(n!)^3\) decoupled molecules with (n, 2) sites. These come in \(2(n!)^2\) classes modulo the \(S_n\times S_2\) action that corresponds to relabelling of the sites.
Proof
Similar to the proof of Proposition 4.3, it suffices to show the claim in a euclidean open set. For that consider a generic normalized binding polynomial \({\underline{a}}^*\in \mathbb {C}^{(n+1)(2+1)1}\). Proposition 4.3 states that there are 2n! solutions \(({\underline{g}}^*,\underline{w}^*;{\underline{a}}^*)\) to \(f({\underline{g}},{\underline{w}};\underline{a}^*)=0\) of multiplicity \(2(n!)^2\), and we will now argue why each of them yields \(2(n!)^2\) solutions if we perturb \({\underline{a}}^*\) slightly, see Fig. 6.

for any \({\underline{a}}'\in \pi _{{\underline{a}}}(U)\), \(U\cap X \times \{a'\})\) has only isolated solutions of \(f({\underline{g}}, {\underline{w}};{\underline{a}}')=0\),

the sum of the multiplicities of those isolated solutions is \(4(n!)^3\).
5 Further experimental results
In this section, we provide some experimental results for (n, 2) and beyond. For simplicity, we will use randomly chosen \(\underline{a} \in \mathbb {C}^{(n+1)(2+1)1}\). Moreover, we will also fix a choice of \(g_{S_1},\dots ,g_{S_{n_1}},g_{T_1},\dots ,g_{T_{n_2}}\) to factor out the natural \(S_{n_1}\times S_{n_2}\) action on the roots of System (1), see Sect. 5.1.

bertini (https://bertini.nd.edu/): A solver for polynomial equations using numerical algebraic geometry. It has builtin features for parallel pathtracking, which proved to be particularly useful for big examples.

gfan (http://home.math.au.dk/jensen/software/gfan/gfan.html): A software package for computing Gröbner fans and tropical varieties. It features a new algorithm for computing mixed volumes using tropical homotopy methods (Jensen 2016).

Singular (Decker et al. 2016): A computer algebra system for polynomial computations, with special emphasis on commutative and noncommutative algebra, algebraic geometry, and singularity theory.
5.1 Explicit solutions for (3, 2)
5.2 Mixed volumes
The Newton polytope of a polynomial is the convex hull of all exponent vectors of all monomials with nonzero coefficients. Given a polynomial system \(f_1,\dots ,f_N\) in N variables and with only finitely many solutions, the mixed volume of their Newton polytopes is an upper bound on the number of solutions that is attained provided the nonzero coefficients are generic. This is known as the Bernstein–Khovanskii–Kushnirenko Theorem (Bernstein 1975), or BKK Theorem in short.
Figure 8 shows a table with the mixed volume for various \((n_1,n_2)\) computed using gfan. We see that the number for \((n_1,1)\) and \((n_1,2)\) corresponds with the theoretical results. Sadly, there is no apparent pattern for \((n_1,n_2)\) with \(n_2>2\).
5.3 Counting solutions using Gröbner bases
Given a zerodimensional polynomial ideal \(I\unlhd \mathbb {C}[x]\), the dimension of \(\mathbb {C}[x]/I\) as a \(\mathbb {C}\)vector space equals the number of solutions counted with multiplicity, though in our specific case we only have solutions with multiplicity 1. The vector space dimension can be easily read off any Gröbner basis, but computing the Gröbner basis itself is a highly challenging task (Greuel and Pfister 2008, Section 1.8.5). In Fig. 8, red numbers mark all cases for which Gröbner bases were computable in Singular. The respective vector space dimensions (computed using the Singular command vdim) all coincided with the mixed volume. This shows that their coefficients are generic in the context of the BKK Theorem, though for all cases but (3, 3) this was already proven.
5.4 Explicit solutions for (5, 2) and (4, 3)
For the cases (5, 2) and (4, 3), highlighted blue in Fig. 8, we also tried to compute explicit roots using bertini. However, numerical instabilities arose in both cases during the computation, so that the roots computed are most likely incomplete.
For (5, 2) we obtained 28,737 roots, 63 short or \(99.8\%\) of the proven 28,800 roots. For (4, 3) we obtained 156,966 roots, 5466 short or \(97\%\) of the conjectured 162,432 roots.
6 Open questions
We close with three open questions.
Question 6.1
What is the number of solutions for \((n_{1},n_{2})\)?
For binding polynomials of bidegree (n, 1) and (n, 2), the number of decoupled molecules is given by relatively simple expressions. Assuming that the mixed volumes of the Newton polytopes equals the number of solutions, Table 8 indicates a more complicated pattern in the number of decoupled molecules for (n, 3). The smallest interesting example is the case (4, 3) for which we conjecture that the number equals 162432.
Question 6.2
How many solutions with real, positive values for \(g_{i}\) and \(w_{i,j}\) exist?
For univariate binding polynomials, the existence of complex roots suggests that the system does strongly interact and cannot be represented by a real decoupled system. In particular it is an indicator for “cooperativity” (Martini et al. 2016). It is neither clear how this concept can be translated to decoupled molecules for two types of ligands nor which characteristic different decoupled molecules share. To develop an understanding, it would be helpful to determine the number of real, positive solutions for small examples.
Question 6.3
Find an algorithm to compute the minimal interaction energy that a molecule with prescribed binding polynomial has.
Notes
Acknowledgements
Open access funding provided by Max Planck Society. The authors would like to thank Bernd Sturmfels for suggesting this collaboration and for his comments on a previous version of this paper, Corey Harris for his useful remarks and Jon Hauenstein for his advice regarding the bertini computations.
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