Chain geometry

The geometrical parameters that we set are the relative positions and orientations of the two points on the polymer we are looking at. The orientation is defined by a tangent, an arrow along the contour of the polymer, and a twist.

1. End-to-end displacement: the relative position of the 2nd end from the 1st end. Depending on the menu item selected, the program takes into account the probability of:
   
   
   • vector: the second end being displaced from the first by a given vector. The x, y, and z components of the displacement vector are entered into the three fields below.
   • distance: the two ends being separated by a given distance. (This means separated by that distance, not separated by that distance or less.) The separation distance is given in the field immediately below.
   • summed: no condition on the positions of the two ends.

   If a displacement vector is specified, then the probability density will involve a factor of 1 / length³ (the precise units can vary). If only their distance is constrained, then the program returns a probability per unit distance, or ~ 1 / length. If the program sums the distribution over all displacements then distance does not enter into the final answer at all.

   The displacement can be given in any of the following units:

   • Lp: multiples of the bending persistence length given in the 2nd field on the form.
   • nm: nanometers
   • A: Angstroms
   
   
2. Initial tangent: Specifies the tangent vector (i.e. the arrow parallel to the polymer in the 'forward' direction) at the first point we are considering. This tangent can enter into the calculation in one of two ways:
   
   
   • vector: the initial tangent is constrained to lie along the vector, whose x, y, and z components are specified in the fields below. The vector can be of any length -- the program will rescale it to be a unit vector -- and only its direction is important. (It does need to have a direction, so at least one of the fields needs to be nonzero.)
   • averaged: the probability distribution is averaged over all initial tangent vectors.
   
   
3. Final tangent: specifies the final tangent vector (the tangent at the second point on the polymer). The final tangent enters the calculation in one of two ways determined by the menu selection:
   
   
   • vector: the final tangent is constrained to lie along the given vector whose x, y, and z components are specified.
   • summed: the distribution the program returns will be summed over all final tangents.

   If the final tangent is constrained (i.e. the 'vector' option is checked for the final tangent), the program computes a distribution per unit solid angle 'steradian'. (A solid angle is the 2-dimensional equivalent of an angle; a sphere has a surface area of 4*pi in units of the radius, so tangent space occupies 4*pi steradians.) Summing the distribution over all final tangents is equivalent to ignoring the final tangents, so no factor of steradians appears in the final answer.

4. Twist: gives the relative twist of the two ends (i.e. the twist angle of the second end minus the twist angle of the first end). Twist angles are measured about the tangent vector, so if the two ends have opposite tangents and zero relative twist, then their corresponding sides face oppositely. The twist enters the final distribution in one of two ways:
   
   
   • angle: the relative twist is constrained by the value of this field. The twist is an angle given in radians -- to convert, multiply by "pi / 180" (which can be written out and calculated in the field).
   • summed: the resultant distribution is summed over all twists.

   If the relative twist is constrained, the program computes a distribution per radian of twist. (Twist is an angle, in contrast to a tangent). If the program sums over twists, then no factor of radians appears in the final answer.

   It is hard to talk about summing tangents without summing twists, so the program will automatically sum both if either the initial tangent is averaged or the final tangent is summed over.