The General Analytical Rebridging Algorithm

The general analytical rebridging algorithm is a set of subroutines capable of solving the rebridging problem in molecular simulations. It is freely distributed under the GNU Public License.

Description

The general analytical rebridging algorithm solves for solutions that direct a linear segment of the molecule towards a fixed end. It conserves the bond lengths and bond angles. By defining the appropriate rigid units, the user can choose to conserve some torsional angles, e.g., those for non-$\sigma$ bonds, as well. It can be applied to cause a local conformational change of the internal part of a molecule. It can also be used to ``graft'' a molecule onto another molecule. The algorithm accommodates a general backbone geometry, from simple poly-ethylene to peptides and proteins. Although the algorithm focuses on solving the configuration of the backbone, side chains with free ends are allowed.

Consider a arbitrary segment of the molecular backbone shown in Fig. 1. The bonds shown in bold lines are chosen to be torsionally constrained. Given the the two bonds u₁, u₆ in the 3-d space, the algorithm finds all the solutions that re-insert in a valid way the backbone units within the dotted region. The variables to be solved are the six torsional angles $\phi$ _i, i = 1,..., 6. The number of variables is six, because the rebridging solution enforces six geometrical constraints towards the end.

Figure 1: A backbone segment selected to be rebridged. Only the backbone atoms are shown. A change of the driver angles $\phi$₀ and $\phi$₇ breaks the connectivity. The dotted area represents the region in which the positions of the backbone atoms must be restored. The thick solid lines represent pi bonds or rigid molecular fragments within which no rotation is possible.

Download sources

The general analytical rebridging algorithm is written in ANSI C (download do_rebridge_1.1.3.tar.gz, last updated 9 January 2006). An older version is still available (download do_rebridge_1.0.tar.gz, last updated 12 December 2000). The function to be called is do_rebridge. To make use of the subroutine

• Uncompress the file with gunzip do_rebridge.tar.gz
• Extract the file with tar -xvf do_rebridge.tar
• Copy all of the extracted files to your source directory
• Call the do_rebridge subroutine from your code

We ask that you cite our paper [1] in any publications that result from the use of the do_rebridge subroutine.

Preparing the data input

The rebridging algorithm asks the user to input a set of parameters that describe the backbone geometry. These parameters can be calculated from a current valid configuration.

We denote û_i, i = 1,..., 6, as the unit axis of the ith torsional bond. Let r_i be the 3-d position of the atom that ends the ith torsional bond. The idea is to construct a closed loop consisting of seven joints with u₁, u₂,.., u₇ and of seven links with vectors a₁,a₂, ..., a₇, as shown in Fig. 2.

Figure 2: The geometry of the closed, 7-revolute mechanism, consisting of 7 joints and 7 links. The joints are represents by u₁, u₂, ..., u₇. The links are represented by a₁, a₂, ..., a₇. Each link is perpendicular in three dimensions to the two adjacent joints. The unit axes of the joints and links are defined as û_i and â_i, respectively.

Each link is perpendicular to the two adjacent joint vectors. How the links are calculated is given below. For i = 1,..., 5, calculate the following parameters:

1. Find the shortest vector, a_i, that connects the lines containing the two consecutive torsional bonds û_i and û_{i + 1}. Choose the direction for a_i so that it goes from û_i to û_{i + 1}. Let a_i be the length of a_i. Define â_i as the unit axis of a_i. If the lines containing û_i and û_{i + 1} intersect with each other, a_i = 0 and a_i is a null vector. In this case, we choose â_i as the unit axis perpendicular to both û_i and û_{i + 1} and passing through the intersection. Choose the direction of â_i so that it is parallel with û_ix û_{i + 1}.
2. Calculate $\alpha$_i, the ``twist'' angle, defined by û<sub>i</sub>, â<sub>i</sub>, and û<sub>i + 1</sub>. The ``twist'' angle $\alpha$_i is zero when û_i and û_{i + 1} are parallel to each other. In the a_i = 0 case, the â_i chosen in the previous step makes $\alpha$_i always positive.
3. Find the shortest vector connecting r_i and the line containing â_i, and calculate the projection of this vector on û_i. This projection is called S_i.
4. Find the shortest vector connecting r_{i + 1} and the line containing â_i, and calculate the projection of this vector on - û_{i + 1}. This projection is called T_i.

The parameters extracted from the above calculations are a_i, $\alpha$_i, S_i, and T_i, where i = 1,..., 5. Other parameters such as a₇ are calculated by the algorithm automatically. Finally, the vectors û₁, r₁, û₆, and r₆ that specify the end constraints must be supplied.

Function call and returned values

The arguments to be passed to do_rebridge are û₁, r₁, û₆, r₆, a_i, $\alpha$_i, S_i, and T_i, i = 1,..., 5. They are shown in do_rebridge.c as follows

void do_rebridge(double uv1[3], double uv6[3], 
		 double rv1[3], double rv6[3],
		 double a_in[6], double alpha_in[6],
		 double S_in[6], double T_in[6],
		 int *n_roots, 
		 double **uv_solve[MAXROOTS], double **rv_solve[MAXROOTS])

Here MAXROOTS is 16, the maximum number of possible solutions. The first and second arguments pass the 3-d components of the unit axes û₁ and û₆. The third and fourth arguments pass the 3-d components of r₁ and r₆. In these arguments the subscript of the arrays goes from 0 to 2. The fifth to eighth arguments pass a_i, $\alpha$_i, S_i, and T_i, where i = 1,..., 5. Note that in these arguments the subscript goes from 1 to 5. That is, a_in[1] stores a₁, S_in[5] stores S₅, and so on. The argument *n_roots returns the number of solutions found. It is always an even number between 0 and 16. The 3-dimensional array **uv_solve[MAXROOTS] returns û_i, i = 1,..., 6, for each solution. The 3-dimensional rray **rv_solve[MAXROOTS] returns r_i, i = 1,..., 6, for each solution. Both arrays have the dimensions of 16 x 8 x 3 and have the same ranges. Take uv_solve[i][j][k] as an example, the valid ranges for the i, j, and k are 0,...,*n_roots-1, 1,..., 6, and 0,..., 2, respectively. The uv_solve[i] and rv_solve[i] vectors determine the positions of the five rebridged units, as determined in the ith solution of the rebridging equations.

A minimal example of a call to do_rebridge has been graciously provided by Stepan Ruzicka (stepan.ruzicka@epfl.ch). It can be found here (download example.tar.gz, last updated 4 August 2017).

In actual Monte Carlo simulations, each solution must be weighted by a associated Jacobian so as to account for the non-uniform distribution of the solutions in the torsional angle space. The Jacobian J takes this form:

$$\begin{eqnarray} {\mathrm J} & = & \frac{ {\hat{\mathbf u}}_{6}\cdot{\hat{\mathb... ...mes{\hat{\mathbf u}}_{6}]_{i-3} \mbox{, if $i=4,\ 5$} .\nonumber \end{eqnarray}$$
Here $\hat{\mathbf e}$ is the z-axis of the reference coordinate. The determinant of the 5x5 matrix B can be evaluated by the function double mtx_det(double **a, int n) in mtxinv.c. The first argument, double **a, passes the elements of B, and the second argument n passes the size of the matrix, which is 5 in this case. The subscripts of the 2-dimensional array a go from 0 to n-1. That is, a[i-1][j-1] stores the value of${\mathbf B}_{ij}$, where i and j are 1,...,n.

Feedback

do_rebridge.c is unsupported software.

References

1) M. G. Wu and M. W. Deem, ``Analytical Rebridging Monte Carlo: Application to cis/trans Isomerization in Proline-Containing, Cyclic Peptides,'' J. Chem. Phys. 111 (1999) 6625-6632. A pdf reprint is available.