Riemann-Symmetric-Space-Based Models in Screening for Gene Transfer Polymers

: Today, gene transfer using polymers as transfer vectors is hardly studied. Some polymers have an excellent gene-carrying ability, but their cytotoxic and biocompatibility properties are not suitable for use. Thus, increased insight into the drug space of such structures is needed in the screening for suitable molecules. This study aimed to introduce a mathematical model of polymers suitable for genes transfer. In this regard, Riemann surfaces were used. The concerned polymers were taken from secondary published experimental data. The results show that symmetric Reimann spaces are suitable for further drug screening. The branch point values of Riemann surfaces are especially increased for the polymers suitable in gene transfer.


Introduction
The lack of gene-delivery vectors [1] is the main limiting factor in the field of gene personalized therapy [2]. Viral vectors have a low efficiency. Synthetic gene-delivery agents do not possess the required efficacy [3]. In recent years, a variety of capable polymers have been designed, specifically for gene delivery. Understanding the polymer gene-delivery mechanisms will help in the design of polymer-based gene-delivery systems, thus becoming essential tools for human gene therapy. The design criteria for the construction of useful delivery vectors are continuously evolving. Reverse drug design, fragment-based drug design, and virtual screening all failed in identifying a class candidate. Most of these methods explore the same drug space of a template compound. Thus, a mathematical model, expressed as a function, will enlarge the drug ability space. Such a model will go beyond the respective class of compounds. A way of generating such a model is to explore Cartesian coordinates, close contacts, dihedral angle values, and the internal coordinates of a certain molecule. Cartesian coordinates can provide a high starting point for projection in a vast range of mathematical spaces, compared with internal coordinates and close contacts. One can distinguish linear spaces and topological spaces. Linear spaces lack 3D dimensionality due to their algebraic nature. Linear operations performed in a linear space lead to straight lines. Their dimensionality is defined as the maximal number of independent vectors.
Topological spaces that are analytic in nature lead to continuous functions. A topological space is hard to define. An algebraic approach is used in most cases.
Polynomial equations and their geometric properties are used worldwide in algebraic geometry. A major characteristic that makes polynomial equations a valuable tool in topology is their definition from a basic arithmetic operation-addition and multiplication. This operation, when used, retrieves smooth and Riemannian manifolds.

Item
Monomers with Proven Gene Transfer Capabilities                      branching points were compared against the known gene transfer properties. The methodology is exemplified for compounds 8 and 21.    Table 2 lists the initial derivate equations generated using the Cartesian coordinates of each molecule.

Results
The equations in Table 2 were transformed using the following formula: where a = free term; b = coefficient; z = Lambert W function. The equations for each polymer are shown in Table 3. The Riemann surfaces obtained for compounds 8 and 21 that showed promising experimental results are shown in Figure 2.
The branching points computed for each structure are shown in Figure 3.
The QSAR models, developed as a single hypothesis for compounds 8 and 21, and the merge hypothesis are shown in Figure 4.

Discussion
Symmetric spaces are pseudo-Riemann manifolds [10]. A connected Riemann manifold is symmetric (space) if its curvature tensor is invariant. Broadly, a Riemann manifold is symmetric if each point exists as an isometry. Furthermore, every symmetric space is exhaustive.
Riemann symmetric spaces are considered in physics, mathematics, and chemistry. They have a central role in the homology theory. Examples of Riemann spaces include Euclidean spaces, hyperbolic spaces, and projective spaces. Riemann spaces are classified into the Euclidean type, the Compact type, and the Non-Compact type [11].
If the complex argument of a function can be mapped from a single point in the domain of multiple points, then the branching point of an analytical function is a point in the complex plane [12]. When z (branching point) = 0, under the power function f (z) = za, where "a" is a complex non-integer ("a" ∈ C, with a Z). Writing z = eiθ and taking θ in the interval 0 to 2π results in: f (e oi ) = e 0 = 1; f (e 2πi ) = e 2πia (2) so that the values of the function f (z); z = (0;2π) are different [12][13][14][15]. In Riemann surfaces, the aspect of a branch point is defined for a holomorphic function when ƒ: X → Y, from a compact connected Riemann surface X to the compact Riemann surface Y (usually the Riemann sphere). If ƒ is not constant, ƒ will be a covering map onto its image at all but a finite number of points. The points of X, where the function ƒ fails, are the bifurcations points of ƒ, and the branch point is an image of a ramification point under ƒ.
For any point P ∈ X and Q = ƒ(P) ∈ Y, there are the holo-morphic local coordinates z for X near P and w for Y near Q, in terms of which the function ƒ(z) is given by ω = zk, for some integer k. This integer is called the ramification index of P. Usually, the ramification index equals one; if the branching index is not equal to one, then P is, by definition, a ramification point, and Q is a branch point.
If Y is just the Riemann sphere, and Q is in the finite part of Y, then there is no demand to select particular coordinates. The ramification index (Equation (3)) can be determined explicitly from Cauchy's integral formula. Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is (P-any point in the space with local holomorphic coordinates) This integral is the number of times that ƒ(y) winds over the point Q. As above, P is a ramification point, and Q is a branch point if eP > 1.
A Riemann surface is a surface-like composition that encloses the complex plane with infinitely many "sheets." These sheets can have very intricate structures and interconnections. Riemann surfaces are one way of representing multiple-valued functions; another way is represented by the branch cuts.
The plot in Figure 1 shows the Riemann surfaces as the solutions of the equation: with d = 2, 3, 4, and 5, where w(z) is the Lambert W-function. The Riemann surface S of the function field K is the set of non-trivial discrete evaluations on K. Here, the set S corresponds to the ideals of the ring A of integers of K over z. Riemann surfaces provide a geometric visualization of the function elements and their analytical continuations.
Schwarz proved, at the end of the nineteenth century, that the automorphism group of a unified Riemann surface of genus g ≥ 2 is finite; then, Hurwitz showed that the group order is at most 84 (g − 1), where "g" is the genus.
In light of the computational results, polymers #4, 8, 11, 16 and 21, are the best candidates for feasible gene transfer. Experimentally, only polymers 8 and 21 showed good and acceptable results. A threshold regarding the branching point and its correlation with bioactivity was observed ( Figure 2). A branching point of~1 has an excellent correlation with gene transfer capability, cytotoxicity, and biocompatibility.
The QSAR single hypothesis models for compounds 8 and 21 revealed the importance of topology in performing bioactivity. The merge hypothesis presents a shared future for both compounds 8 and 21. Hydrogen atom accepting A-groups and donor D-groups are critical in the chemical space of compounds [16]. Furthermore, the pharmacophores demonstrated a relatively diverse set of functional groups for each hypothesis, findings that suggest a lack of specificity in describing a common pharmacophore.