Monte Carlo Minimization¶

MDNA uses Monte Carlo (MC) simulations to relax DNA structures into physically consistent configurations. The minimization engine is provided by PMCpy github, a vendored subpackage located at mdna/PMCpy/.

Why Minimization?¶

When DNA is built from spline control points, the resulting structure captures the desired global shape but may contain local strain — overlapping atoms, unrealistic backbone geometries, or violated steric constraints. MC minimization resolves these issues by sampling thermal fluctuations around the reference configuration while respecting an elastic energy model.

The Rigid Base-Pair Model¶

PMCpy treats each base-pair step as a rigid body connected to its neighbors through a harmonic elastic potential parameterized by the six rigid base-pair step parameters (shift, slide, rise, tilt, roll, twist). The stiffness matrices are sequence-dependent and derived from atomistic MD simulations.

graph LR
    A["BP i"] -- "elastic<br/>potential" --> B["BP i+1"]
    B -- "elastic<br/>potential" --> C["BP i+2"]
    C -- "..." --> D["BP n"]
    style A fill:#4051b5,color:#fff
    style B fill:#4051b5,color:#fff
    style C fill:#4051b5,color:#fff
    style D fill:#4051b5,color:#fff

Each MC move proposes a small random perturbation to a base-pair step, computes the energy change, and accepts or rejects the move using the Metropolis criterion at the specified temperature.

Equilibration Modes¶

The minimize() method supports three equilibration strategies:

Full Equilibration (default)¶

dna.minimize()

Runs the full PMCpy equilibration protocol. This iteratively adjusts both the step parameters and global configuration until convergence is reached. The equilibration monitors the energy trace and terminates when fluctuations stabilize.

Simple Equilibration¶

dna.minimize(simple=True)

A lighter-weight equilibration that performs a fixed set of MC sweeps without the adaptive convergence check. Faster but may not fully relax highly strained configurations.

Writhe Equilibration¶

dna.minimize(simple=True, equilibrate_writhe=True)

Special mode for circular DNA that additionally equilibrates the writhe (\(Wr\)) while maintaining the target linking number (\(Lk\)). This requires the simple equilibration path and uses the PyLk Cython extensions for efficient writhe computation.

Writhe equilibration requires compiled PyLk

See Building PyLk Cython Extensions for compilation instructions.

Key Parameters¶

Parameter	Default	Description
`frame`	`-1`	Which stored frame to minimize (index into the frames array)
`exvol_rad`	`2.0`	Excluded-volume radius in nm — prevents base-pair overlap
`temperature`	`300`	Temperature in Kelvin for the Metropolis criterion
`simple`	`False`	Use simple (fixed-sweep) equilibration
`equilibrate_writhe`	`False`	Equilibrate writhe for circular DNA
`endpoints_fixed`	`True`	Keep the first and last base pairs fixed
`fixed`	`[]`	List of base-pair indices to keep fixed during minimization
`dump_every`	`20`	Save a snapshot every n MC sweeps
`plot`	`False`	Plot the energy trace during equilibration

Accessing the MC Trajectory¶

After minimization you can retrieve the full MC trajectory as an MDTraj Trajectory object:

dna.minimize()
traj = dna.get_MC_traj()
traj.save_pdb('mc_trajectory.pdb')

Each frame in the trajectory contains:

Argon atoms at the base-pair center positions
Helium (dummy) atoms offset along the major-groove vector

This encoding lets you visualize the MC sampling path and inspect how the structure relaxes over the course of the simulation.

Architecture¶

flowchart TD
    N["Nucleic"] -->|".minimize()"| M["Minimizer"]
    M -->|"copies frames"| MC["PMCpy Run engine"]
    MC -->|"MC sweeps"| OUT["positions + triads"]
    OUT -->|"update"| N
    MC -->|"snapshots"| TRAJ["get_MC_traj() → MDTraj"]
    style N fill:#4051b5,color:#fff
    style M fill:#7c4dff,color:#fff
    style MC fill:#00897b,color:#fff
    style OUT fill:#00897b,color:#fff
    style TRAJ fill:#ff6d00,color:#fff