Splining

This section gives an in-depth overview of the steps taken in the splining module.

At a Glance : Smoothing Pixelated Traces

The splining.py module handles all the functions associated with smoothing the ordered pixel-wise trace from the “ordered_tracing” step, in order to produce curves which more closely follow the samples structure.

The quality of the resultant metrics and smoothed coordinates will depend on the splining method chosen, whether the ordering worked successfully, and whether the skeleton matches the underlying sample conformation.

This smooths the ordered trace by using an average of splines through the ordered coordinates (spline method) or using the mean coordinate of a rolling window (rolling_window method), helping to resolve length errors in jagged in the skeletons. It adds the contour length and end-to-end euclidean distance to the all_mol_statistics.csv and the sum and average of these respectively to the all_statistics.csv.

Some quick FYI’s:

• Constricted Traces - Using the rolling_window method with a rolling window size that is not far below the persistence length can result in over-smoothing where the spline is dragged away from regions of high curvature.

• Unrepresentative Splines - The spline method’s parameters can be quite temperamental so it’s recommended that spline_linear_smoothing, spline_circular_smoothing and spline_degree are not changed unless understood.

• Unrepresentative Splines 2 - The spline method may produce unwanted results if the spline_step_size is too long. This is because the way the splines are averaged assumes that each spline point is sampled close to that of another’s.

Processing Steps

Splining Method

1. Trace Coordinate Subsets

The first stage is to use the ordered coordinates and the spline_step_size parameter to define how many splines we want to average together. As the spline_step_size is the distance (in nm) at which to take every i’th coordinate for its spline. This means the number of splines to average is calculated below:

$$ N = max(\text{spline step size} \div \text{px to nm}, 1) $$

The coordinates used for each spline are then obtained from the initial ordered trace, i.e. the spline coordinate indexes: [1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], where spline 1 takes every 4th coordinate, starting at position 0, then spline 2 takes every 4th coordinate starting at position 1, etc.

2. Obtaining Splines

For each of these coordinate subsets, we use the Sci-Py library to find the B-Spline representation of the 2D curve using the smoothness parameters spline_linear_smoothing, spline_circular_smoothing, and the spline_degree, found in the configuration file. The spline_linear_smoothing and spline_circular_smoothing define the degrees of smoothing for circular or linear molecules obtained in the Ordered Tracing step, which also tells TopoStats whether to make the spline periodic if circular. Larger smoothing values mean more smoothing and smaller values indicate less smoothing. The spline_degree should take odd values, or even with large smoothing. Read more on the Sci-Py documentation above.

These splines are then averaged together via their index i.e. first spline coordinates are averaged, then second etc… until a single averaged B-Spline remain. However, because the splined are averaged on their index, it is assumed that each index is within close proximity, thus, the spline_step_size must be small enough for this assumption to hold true and for the splining to work as intended.

Rolling Window Method

1. Rolling Average

This method simply uses a rolling average of the ordered trace coordinates within the rolling_window_size to produce the new smoothed trace. The rolling window will move one coordinate along each time, and continues until the start of the window returns back to its initial position.

For linear smooth traces, the same as above occurs however, the initial and final coordinates are also added to the smoothed trace as to not drastically reduce the length of the smoothed trace.

Outputs

The <image>_<threshold>_ordered_traces image shows the direction of ordered coordinates.

For each grain, the following new columns are added to the grainstats.csv file:

Column Name	Description	Data Type
total_contour_length	The total length along the splined trace of all identified molecules.	float
average_end_to_end_distance	The average distance from two endpoints of the spline of all identified linear molecules.	float

 

For each molecule found by the ordering algorithm(s), the following new columns are added to the molstats.csv file:

Column Name	Description	Data Type
contour_length	The length along the splined trace of the molecule.	float
end_to_end_distance	The distance from two endpoints of the spline of the linear molecule.	float

 

Note: Most information obtained during the Splining processing can be obtained from the <image_name>.topostats file found within the processed folder and contains a multitude of molecule related objects such as:

• bounding box

• spline coordinates

• tracing statistics of contour length and end to end distance

Diagnostic Images

There are no diagnostic images produced in this step.

Possible Uses

This module would lend itself useful for accurately measuring the lengths of complex objects within samples, and obtaining an accurate representation of the underlying conformation of the sample.

We have used this module to accurately measure the length of topologically complex DNA samples such as knots, catenanes, and theata-curves (replication intermediates). Additionally, we’ve used the end-to-end distance to measure the conformational variation of linear DNA in the presence of NDP-52 to show that it may also play a role in topological regulation. See more here..