Splines & Geometry¶

This page explains how MDNA uses spline interpolation to map arbitrary 3D shapes onto DNA structures.

From Control Points to DNA¶

MDNA's construction pipeline converts a set of 3D control points into a full atomic DNA structure through spline-based frame generation.

graph LR
    CP["Control Points<br/>(n × 3)"] --> SPL["B-Spline<br/>Interpolation"]
    SPL --> ARC["Arc-Length<br/>Parameterization"]
    ARC --> FR["Frame Distribution<br/>(0.34 nm spacing)"]
    FR --> TW["Twist Application<br/>(bp_per_turn / dLk)"]
    TW --> OUT["Reference Frames<br/>(n_bp × 4 × 3)"]

Spline Interpolation¶

The SplineFrames class uses B-spline interpolation (via scipy.interpolate.splprep) to create a smooth 3D curve through the control points.

Key parameters:

Parameter	Default	Description
`degree`	3	Spline degree (cubic)
`closed`	False	Whether the spline forms a closed loop
`num_points`	1000	Internal evaluation resolution

For closed DNA (circular=True), the spline is made periodic so the curve smoothly connects back to the start.

Arc-Length Parameterization¶

After fitting the spline, MDNA reparameterizes it by arc length. This ensures that base pair origins are distributed at equal spacing along the curve, regardless of how the control points are distributed.

The arc length is computed by numerical integration of the spline derivative magnitude.

Frame Generation (Bishop Frames)¶

At each distributed point, MDNA computes an orthonormal frame (origin + 3 basis vectors) using the Bishop frame method:

Tangent (\(\hat{T}\)): First derivative of the spline (normalized)
Normal (\(\hat{N}\)): Derived from a reference "up" direction, orthogonalized against the tangent
Binormal (\(\hat{B}\)): Cross product of tangent and normal

The Bishop frame method is preferred over the Frenet-Serret frame because:

It avoids singularities at inflection points (where curvature is zero)
It produces minimal torsion — frames rotate smoothly along the curve without sudden flips
It is well-defined even for straight segments

The initial frame orientation is propagated incrementally along the curve, ensuring continuity.

Twist Application¶

After generating untwisted frames, DNA twist is applied:

Default Twist¶

\[\theta_{\text{twist}} = \frac{360°}{\text{bp\_per\_turn}} \approx 34.3° \text{ per step}\]

with bp_per_turn = 10.5 (canonical B-DNA).

Linking Number Control¶

For circular DNA, the total twist can be modified by specifying \(\Delta Lk\):

\[Lk = Tw + Wr\]

where \(Tw\) is twist and \(Wr\) is writhe. The dLk parameter adds extra twist that, after minimization, redistributes into twist and writhe according to the elastic properties of the structure.

Frame Format¶

The output frames have shape (n_bp, 4, 3):

Row	Content	Description
0	Origin	Position of the base pair center (nm)
1	\(\hat{b}_L\)	Long axis (roughly along H-bonds)
2	\(\hat{b}_D\)	Short axis (in base plane)
3	\(\hat{b}_N\)	Normal to the base plane

These frames serve as the coordinate systems into which atomic base structures are placed by the StructureGenerator.

Predefined Shapes¶

The Shapes class provides factory methods returning control point arrays:

Shape	Method	Key Parameters
Straight line	`Shapes.line(length)`	`length`
Circle	`Shapes.circle(radius)`	`radius`
Helix	`Shapes.helix(...)`	`height`, `pitch`, `radius`, `num_turns`

All return numpy arrays of shape (n, 3) suitable for passing to mdna.make(control_points=...).

Practical Tips¶

Number of control points

You need at least 4 control points for cubic spline interpolation. More points give finer control over the shape.

Scaling

When you provide both control_points and n_bp (or sequence), the spline is scaled so that exactly n_bp base pairs fit along the curve at 0.34 nm spacing.

Inferring n_bp

If you provide only control_points without sequence or n_bp, MDNA automatically determines the number of base pairs from the spline arc length.