Protein Modelling and Surface Folding by Limiting the Degrees of Freedom

Abstract

One aspect of tissue engineering represents modelling of the extracellular matrix of connective tissue as the fiber network arrangement of the matrix determines its tensile strength. In order to define the correct position of the e.g. collagen in a structure, an optimized tertiary structure must be characterized. Existing approaches of protein models consider random packing of rigid spheres. We propose an alternative strategy to model protein structure by focusing on the folding. Our model considers (a) segments of amino-acid peptides or beads, (b) hydrogen bond distances, and (c) the distance geometry as functional components rather than minimizing distances between the centers of atoms. We reduced the molecular volume by using concepts from low dimensional topology, such as braids and surfaces, via differential geometry. A braid group maintains the continuity of a sequence while the spatial minimization is performed, and guarantees the continuity during the process. We have applied this approach to different examples of known protein sequences using ab initio protocols of ProteoRubix Systems™. Sequence files of three different proteins types were tested and modeled by ProteoRubix™ and compared to models derived by other methods. ProteoRubix™ created near-identical models with minimal computational load. This model can be expanded to large, multi-molecular network structures.

Appendices

Appendix 1

1.1 NP

Suppose the bead involves $d$ dihedral angles. Let $\phi^{ * } = \left( {\phi_{1}^{ * } , \ldots ,\phi_{n}^{ * } } \right) \in \left[ {0,360} \right]^{d}$ be an optimal solution to the constrained optimization problem

(1.1) $\mathop {\min }\limits_{{\phi \in \left[ {0,360} \right]^{d} }} \left\{ {v(\phi :\phi \; {\text{is arotation about the bond i}})} \right\}$

Then there is a maximal number $n > 0$.

(1.2) The $p^{n}$ hard problem of an exhaustive search over the angles $0 \le \phi_{i}^{1} < \cdots < \phi_{i}^{P} \le 360$ to find an approximate optimizer $\overline{\phi }$ to $\phi^{ * }$ may be possible for a modern computer.

(1.3) There is exactly one solution to (4) in $\left| {\overline{{\phi_{i} }} - p,\overline{{\phi_{i} }} + p} \right|$ which would be $\phi^{ * }$ and can be approximated using a given constrained optimization algorithm (B 1).

The convergence ball for the constrained optimization algorithm provides a candidate for p in the proposition. Using this proposition, we can obtain an acceptable initial condition for a constrained optimization algorithm.

Appendix 2

1.1 Boundary Determination to Prevent Overlap

A path is a one-dimensional sub-manifold ${\text{M}}$ of ${\text{R}}^{ 3}$, so that, for any point ${\text{x}} \in {\text{M}}$ there is a local parameterization near x. ${\text{C}}^{\text{k}} \left( {{\text{k}} \ge 2} \right)$ denotes the curvature of the path and ${\text{D}}$ denotes the coordinates identifying the path. The output of each iteration is a set of coordinates in three dimensions, $\left( {{\text{D}} = {\text{x}}_{ 1} , {\text{x}}_{ 2} ,\ldots , {\text{x}}_{\text{n}} } \right)$ identifying a path. We denote by length bond is the polygonal arc around the path (Fig. 15). The curvature ${\text{C}}^{\text{k}}$ and the arc-length are non-regular. Let ${\text{x}} = {\text{x(t),}}$ with ${\text{a}} \le {\text{t}} \le {\text{b}}$ and consider a partition [15]:

${\text{a}} = {\text{t}}_{ 0} < {\text{t}}_{ 1} < \cdots < {\text{t}}_{\text{n}} = {\text{b}}$, of an interval (a, b).

The sequence (a, b) are the boundaries of a single coil) gives an approximation to the polygon arc C. As illustrated the length between two points (a, b), where ${\text{D}}$ are segments of arc-length given by:

$$ \lambda ( {\text{D)}} = \sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{D}}_{\text{j}} = } \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {\left\| {{\text{x}}_{\text{i}} - {\text{x}}_{{{\text{i}} - 1}} } \right\|} = \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {\left\| {{\text{x}}\left( {{\text{t}}_{\text{i}} } \right) - {\text{x}}\left( {{\text{t}}_{{{\text{i}} - 1}} } \right)} \right\|} $$

(2.1)

The arc-length can be bounded from above and from below. The upper bound is given by:

$$ \rho_{ + } \left( {{\text{K}},{\text{D}}} \right){ = }\frac{ 1}{{\lambda \left( {\text{D}} \right)}}\sum\limits_{{\left( {{\text{K}} \circ {\text{D}}{}_{\text{j}}} \right) \cap {\text{D}} \ne \emptyset }} {\lambda \left( {{\text{K}} \circ {\text{D}}_{\text{j}} } \right)} $$

(2.2)

And the lower bound is:

$$ \rho_{ - } \left( {{\text{K}},{\text{D}}} \right) = \frac{ 1}{{\lambda \left( {\text{D}} \right)}}\sum\limits_{{\left( {{\text{K}} \circ {\text{D}}_{\text{j}} } \right) \subset {\text{D}}}} {\lambda \left( {{\text{K}} \circ {\text{D}}_{\text{j}} } \right)} $$

(2.3)

where ${{\uprho}}_{ + } ( {\text{K,D)}}$ is the ratio of the total measure of the set in the system ${\rm K}$ (is the volume minimization) so that the transformation ° (projection) of the segments and the curve ${\text{C}}$ give the lower and the upper bound$\left( {\text{a,b}} \right)$.

$$ b = \rho_{ + } = \mathop {\lim }\limits_{\lambda \left( D \right) \to \infty } \sup \rho_{ + } \left( {K,D} \right) = \mathop {\lim }\limits_{\lambda \to \infty } {\text{Sup}}\rho_{\begin{subarray}{l} + \\ \lambda \left( D \right) \ge \lambda \end{subarray} } \left( {K,D} \right) $$

(2.4)

$$ a = \rho_{ - } = \mathop {\lim }\limits_{\lambda \left( D \right) \to \infty } \inf \rho_{ - } \left( {K,D} \right) = \mathop {\lim }\limits_{\lambda \to \infty } \mathop {\inf }\limits_{\lambda \left( D \right) \ge \lambda } \rho_{ - } \left( {K,D} \right) $$

(2.5)

Hence the boundaries of ${\text{C}}$ are given in (2.4) and in (2.5).

Fig. 15

The sequence [$\left( {a,b} \right)$ is length of a single coil] gives an approximation to the polygon arc $P$; the length between two points (a, b), where $D$ is segments of arc-length

Appendix 3

The geometry structure of the protein is defined by a braid (see B 2 for a description of a chain as a collection of beads forming a braid). The jth molecule of the chain is fitted to a conveniently shaped open bead Sj (see B 3) with is 0 center located at the center of the bead and the radius ${\text{r}}_{\text{i}}$ has size such that the ith bead does not overlap with the jth bead when ${\text{i}} \ne {\text{j}}.$

The radii in Fig. 16 ${\text{r}}_{\text{i}}$ are chosen so that the intersection of the closure of any two beads $\mathop {{\text{S}}_{\text{i}} }\limits^{ - }$ and $\mathop {{\text{S}}_{\text{j}} }\limits^{ - }$ is a single point ${\text{p}}_{\text{ij}} ,$ (see B 3). The point ${\text{p}}_{\text{ij}} ,$ is the origin of a right and a left vector ${\text{v}}_{\text{iR}} , {\text{v}}_{\text{jL}}$. In this process it is important to translate (projection) and rotate these vectors. The mathematics of this construction justifies geometry of the bead construction.

Fig. 16

The radii are chosen so that the intersection of the closure of any two beads $\mathop {S_{i} }\limits^{ - }$ and $\mathop {S_{j} }\limits^{ - }$ is a single point $p_{ij} ,$. The point $p_{ij} ,$ is the origin of a right and a left vector $v_{iR} ,v_{jL}$

Appendix 4

With our model of collagen in mind, we next introduced the concept of the braid group. The braid was defined as the union of the backbones creating a string representing the amino acids. The collagen has three strands (as a group) or coils and each strand has a back bone, represented as the union of all points x (t_i − 1, t_i) that are generated:

$$ {\text{Bonds}} = \mathop \cup \limits_{{{\text{n}} = 1}}^{\text{N}} \left\{ {{\text{x}}\left( {{\text{t}}_{{{\text{i}}\, - \, 1}} , {\text{t}}_{\text{i}} } \right)} \right\} $$

(4.1)

A braid is a collection of beads for which two operators $\left( { \circ , = } \right)$ can be defined. The bead in the collection can be projected using least of the squares. Let B denote this collection of beads, so ${\text{B}} = \left( {\text{braids}} \right)$, and $\left( {{\text{B,}} \circ } \right)$ is a group. We are checking the segments of the radius of bead of a single braid. The enclosed volume shrinks driven by minimization and through the homoeopathy is guaranty [2] (see B 5). We are modelling three coils, and their geometrical configuration has an equivalence class denoted by ${{\upsigma}}_{\text{i}}$ and ${{\upsigma}}_{\text{i}}^{ - 1}$. A braid is equivalent and it is called isotope if the three coils cannot pass each other or themselves without intersecting [8] Fig. 17.

Fig. 17

A braid is equivalent and it is called isotope if the three coils cannot pass each other or themselves without intersecting

${{\upsigma}}_{\text{i}} {{\upsigma}}_{{{\text{i}} + 1}} {{\upsigma}}_{\text{i}} = {{\upsigma}}_{{{\text{i}} + 1}} {{\upsigma}}_{\text{i}} {{\upsigma}}_{{{\text{i}} + 1}}$ if $1\le {\text{i}} \le {\text{n}} - 2$ [1]

Appendix 5

The distances of the projection to $P$ is given by $\left\| {b - r} \right\|$, where $v\left( {x - p} \right) = 0$ and by Pythagorean gives us that $b^{2} = c^{2} - a^{2}$, where, $a^{2} = \left\| {\frac{b}{{\left\| b \right\|_{2} }} \times \left( {Q - P} \right)} \right\|_{2}^{2}$, $c^{2} = \left\| {Q - P} \right\|_{2}^{2}$ or $\frac{{b\left( {Q - P} \right)^{2} }}{{\left\| v \right\|_{2} }}$ shown in Fig. 18a.

Fig. 18

The projection of the rotation of the rotation angles. The enclosed volume shrinks by the minimization and via homoeopathy which is guaranteed

1.1 B 1

Let $\mathop \phi \limits^{ - }$ the solution to $\phi^{ * } = \left( {\phi_{1}^{ * } , \ldots ,\phi_{n}^{ * } } \right) \in \left[ {0,360} \right]^{d}$ and $d$ dihedral angle, $\phi_{n}^{*} = \sum^{*} \to N$, $1 \le n \le k$; Let $q$, $r$ be polynomial such $\phi_{n}^{*} \left( I \right) \le q\left( {\left| I \right|} \right)$, where $I$ is the instance of the angle in our problem. Then test instance construction system for all the angles of our problem $\left( {TICA} \right)$, then $P = NP$.

Conversion We know that $\phi^{ * } = \left( {\phi_{1}^{ * } , \ldots ,\phi_{n}^{ * } } \right) \in \left[ {0,360} \right]^{d}$ is the optimal solution, where $d$ dihedral angles then $\delta = \frac{1}{2}\mathop {\min }\limits_{{\phi \in \left[ {0,360} \right]^{d} }} \left\{ {v(\phi :\phi \; is \, arotation \, \;about\; \, the\; \, bond\; \, i)} \right\}$$n$ is the maximum number of angles, $n > 0$ and $\delta > 0$. Let $\in > 0$ be given. Where $\phi^{ * }$ is continuous, there is a point $p \in \phi^{*}$, $\phi^{*} \le \frac{1}{2}\phi \left( p \right)$ where implies $\left| {\overline{{\phi_{i} }} - p,\overline{{\phi_{i} }} + p} \right| < \in$ and $v\left( p \right) \le \frac{1}{2}\phi \left( p \right)$, we have $\left| {\overline{{\phi_{i} }} - p,\overline{{\phi_{i} }} + p} \right| \le \phi^{*} + \left| {v\left( p \right)} \right| < \delta + \frac{1}{2}\phi \left( p \right) < \in$

Uniqueness using the existence and uniqueness theorem, we know that $\phi^{ * }$ is continuous, in the interval $\left| {\overline{{\phi_{i} }} - p,\overline{{\phi_{i} }} + p} \right|$ then converges.

1.2 B 2

${\text{D}}$ is said to be covering itself if $\bigcup\limits_{\text{j}} {{\text{D}}_{\text{j}} } \supset {\text{D}}$ and each elements of at least one of ${\text{D}}$ belongs to ${\text{d}}_{\text{j}}$. The system ${\text{D}}_{\text{j}}$ is packing if ${\text{D}}_{\text{i}} \cap {\text{D}}_{\text{j}} = \emptyset$$( {\text{i}} \ne {\text{j),}}$$\bigcup\limits_{\text{j}} {{\text{D}}_{\text{j}} } \supset {\text{D}}$

If two sets ${\text{D}}_{ 1} , {\text{D}}_{ 2} ,\ldots$ have the same elements in common then each element ${\text{D}}_{ 1} , {\text{D}}_{ 2} ,\ldots$ belong to ${\text{D}} .$

1.3 B 3

Each segment can be treated as open beads, as such the coordinates belong to a set ${\text{X}}$ and for any point ${\text{p}} \subset {\text{D}}_{\text{j}}$ and ${{\updelta}} = {\text{D}}_{\text{j}}$ where the measure is positive.

So, the definition of the bead is:

$$ {\text{D}} = \left\{ {{\text{x:d(p,x)}} < {{\updelta}}} \right\} $$

1.4 B 4

Let $A$ and ${\text{B}}$ be a disjoint convex set in a convex space, then

${\text{A}} = \left\{ {{\text{x:}}\left( {{\text{x}} - {\text{D}}_{\text{i}} } \right)^{ 2} < {\text{r}}_{\text{i}} } \right\}$ and ${\text{B}} = \left\{ {{\text{x:}}\left( {{\text{x}} - {\text{D}}_{\text{j}} } \right)^{ 2} < {\text{r}}_{\text{j}} } \right\}$, the distance is given by:

${\text{dis(D}}_{\text{i}} , {\text{D}}_{\text{j}} )= {\text{r}}_{\text{i}} + {\text{r}}_{j}$. The closure of ${\text{B}}$ is given by $\mathop {\text{B}}\limits^{{\text{\_}}} = \left\{ {{\text{x:}}\left( {{\text{x}} - {\text{D}}_{\text{j}} } \right)^{ 2} \le {\text{r}}_{\text{j}} } \right\}$ then ${\text{A}} \cap \mathop {{\text{B}} = \emptyset }\limits^{{\text{\_}}}$.

$A$ is an open set by construction. A & B are the convex hull, also by construction, and then:

$\exists {\text{l(x)}} = {\text{a}}$ if

$$ \begin{aligned} {\text{x}} &\in {\text{A}}\;{\text{l(x)}} \le {\text{a}} \\ {\text{x}} &\in {\text{B}}\;{\text{l(x)}} \ge {\text{a}} \\ \end{aligned} $$

where ${\text{a}}$ is

$$ \begin{array}{*{20}c} {{\text{v}}_{\text{j}} = \left( {{\text{a}} - {\text{D}}_{\text{j}} } \right)^{ 2} } \\ {{\text{v}}_{\text{i}} = ( {\text{a}} - {\text{D}}_{\text{i}} )^{ 2} } \\ \end{array} $$

where

$$ {\text{D}}_{\text{i}} = {\text{dist}}\left( {{\text{a}} - {\text{v}}_{\text{Ri}} } \right)\;{\text{and}}\;{\text{D}}_{\text{j}} = {\text{dist}}\left( {{\text{a}} - {\text{v}}_{\text{Lj}} } \right) $$

1.5 B 5

Let ${\text{x}} \in {\text{S}}\left( {{\text{r,x}}_{ 0} } \right)$, ${\text{S}} \in \Re^{\text{n}}$ and ${\text{x}}_{ 0} \ne 0$ i.e. $p ( {\text{x)}} = {\text{x}}_{ 0} + {\text{r}}\frac{\text{x}}{{\left\| {\text{x}} \right\|}}$ (Fig. 18b) then; $r = \left\| {{\text{p}} - {\text{x}}_{ 0} } \right\| = \left\| {{\text{x}}_{ 0} - {\text{r}}\frac{\text{x}}{{\left\| {\text{x}} \right\|}} - {\text{x}}_{ 0} } \right\| = \frac{\text{r}}{{\left\| {\text{x}} \right\|}}\left\| {\text{x}} \right\| = {\text{r}}$ hence ${\text{p}} \in {\text{S}}\left( {{\text{r,x}}_{ 0} } \right)$ [2].