Dynamic Topology: TopologyChangeByTriangularPrism - weria-pezeshkian/FreeDTS GitHub Wiki
Dynamic Topology
Requires FreeDTS version 2.2 or later.
The TriangularPrism method allows the membrane topology to evolve through fusion and fission Monte Carlo moves. These moves change the local mesh connectivity while keeping the number and positions of vertices unchanged.
Input
DynamicTopology = TriangularPrism 1 -20 MAP.xvg
The general syntax is:
DynamicTopology = TriangularPrism Period kG [MAP_file]
| Parameter | Description |
|---|---|
Period |
Number of simulation steps between topology-update attempts. 1 enables attempts at every step; 0 disables them. |
kG |
Gaussian curvature modulus, using the convention (E_G = k_G \int K,dA). |
MAP_file |
Optional topology-map file passed to the triangular-prism builder. If omitted, the builder uses its default behaviour. |
In the example above, topology moves are attempted at every simulation step, with kG = -20 and MAP.xvg supplied as the topology-map file.
How it works
The method uses a local transformation involving six vertices:
- Fusion: Two nearby, disconnected triangles are replaced by the triangulated sides of a triangular prism, creating a membrane neck.
- Fission: A suitable triangular-prism neck is removed and replaced by two triangular caps, closing the membrane on both sides.
Fusion can connect separate membrane components or create a handle within the same membrane. Fission can separate a membrane into components or remove a handle, depending on the surrounding connectivity.
At each scheduled topology update, the algorithm:
- Searches for eligible necks and, if any are found, randomly selects one for a fission attempt.
- Searches the updated mesh for eligible fusion configurations and, if any are found, randomly selects one for a fusion attempt.
- Evaluates each proposed move and either accepts it or restores the previous configuration.
Thus, each scheduled update can attempt up to one fission move and one fusion move. An attempt does not guarantee a topology change.
Geometric eligibility
Fusion candidates must be sufficiently close under periodic boundary conditions. The two triangles must not share vertices, and their vertices must not already be directly connected across the pair. The triangular-prism builder then checks for an admissible connecting topology.
Fission candidates must have the required prism connectivity, and the proposed triangular caps must satisfy the face-angle checks.
Energy and acceptance
Each move is evaluated using a Metropolis–Hastings acceptance test. The energy change includes the affected local membrane and interaction energies, together with any enabled volume, total-area and global-curvature coupling terms.
The Gaussian curvature contribution is:
| Move | Gaussian energy change |
|---|---|
| Fusion | (\Delta E_G = -4\pi k_G) |
| Fission | (\Delta E_G = +4\pi k_G) |
With this sign convention, a negative kG favours fission and opposes fusion through the Gaussian term alone. Acceptance also depends on the other energy contributions and on the relative numbers of available forward and reverse proposals.
Rejected moves restore the previous mesh connectivity and affected local state.
Thermodynamic consistency and fusion multiplicity
The acceptance rule accounts for both the energy change and the number of available forward and reverse moves. This Metropolis–Hastings correction is needed because the numbers of possible fusion and fission configurations can change after a topology update.
A pair of nearby triangles may admit multiple valid fusion configurations, corresponding to different ways of connecting their six vertices into a triangulated prism. Each admissible configuration generated by the triangular-prism builder is counted as a separate proposal. Therefore, the number of fusion proposals is the total number of valid configurations across all eligible triangle pairs, rather than simply the number of pairs.
For unbiased equilibrium sampling, the acceptance probability is
$$ P_{\mathrm{acc}} = \min\left[ 1,, \frac{N_{\mathrm{forward}}}{N_{\mathrm{reverse}}} \exp(-\beta\Delta E) \right], $$
where $N_{\mathrm{forward}}$ is the number of available proposals of the attempted move type before the move, and $N_{\mathrm{reverse}}$ is the number of reverse proposals after the trial move. Here, $\Delta E$ includes the Gaussian curvature contribution and all other applicable energy changes.
For fusion, this ratio is $N_{\mathrm{fusion}}^{\mathrm{before}}/N_{\mathrm{fission}}^{\mathrm{after}}$; for fission, it is $N_{\mathrm{fission}}^{\mathrm{before}}/N_{\mathrm{fusion}}^{\mathrm{after}}$. This correction accounts for proposal multiplicity and supports detailed balance, provided each move has a valid reverse proposal.
Example topology-map file
An example is available here: FusionMAP.xvg.
If you save it as FusionMAP.xvg, use:
DynamicTopology = TriangularPrism 1 -20 FusionMAP.xvg