Pressurized spherocylinders with helical reinforcement

Cesar L. Pastrana, 2022 (c)

This software was used in the manuscript "Pressure-induced shape-shifting of helical bacteria" by Pastrana C.L, Qiu L., Armon S., Gerland U., and Amir A. Soft Matter 19(12), 2023: 2224-2230.

Description

The surface is described using a discrete bead and spring model. The total energy of the system at a pressure $p$ is expressed as:

$$ E = \frac{k_s}{2}\sum_{\langle i,j\rangle} (r_{ij} - r_{ij}^0)^2 + k_b\sum_{\langle \alpha,\beta\rangle}(1 - \hat{n_\alpha}\cdot\hat{n}_\beta) - pV $$

where the first sum runs over all pairs of connected notes $i,j$ constituting an edge of the mesh and $r_{ij} = |r_i - r_j|$; and the second sum is over pair of triangles $\alpha,\beta$ sharing an edge, $\hat{n}$ are their respective normal vectors and $k_b$ is a bending stiffness. For a detailed description of the relation between the discrete $k_x$ and the continum variables (Young modulus, thickness and Poisson ratio) see Soung H.S. and Nelson D, R. (Phys. Rev. A 38,1005--1018, 1988).

In this work, the relax configuration used as input is a spherocylinder. A helical domain with the desired properties is reinforced to be nearly undeformable.

The non-linear conjugate gradient algorithm is used to find the minimal energy configuration.

Compilation

A Makefile is provided. To build, simply execute:

make

in the main folder. This will generate the compiled code shape in the bin folder.

Execution

The simulation is launched by simply executing ./shape in the bin folder. Pressurisation proceeds in a step-wise fashion from $0$ to a target pressure $p$ atm, steps of $\Delta p$ provided by the user.

Input

The program requires as input the following files:

• init_coords.dat: Array of $N\times 3$ with the $N$ vertices coordinates defining the surface in the relax configuration. The units of the input coordiantes coordinates are nm.
• mesh.dat: Array of $T\times 3$ with the $T$ triangles defining the connectivity of the triangulated surface.
• params.conf: This file contains the simulation parameters. The meaning of each parameter is indicated in commented blocks in the file. The main parameters are the target pressure $p$ (PRESSURE), the step $\Delta p$ (DP) and the reinforcement factor $K$ (K_SPR_FACTOR).

For the appropiate determination of the helical region to be reinforced, the spherocilinder should have the main body oriented along the $z$ axis. Moreover, the lowest part of the cylinder should be at the $z$ = 0. The coordinates of the hemispherical caps are located at the end of the init_coords.dat file and the number of particles per cap needs to be specified in the variable N_CAP of the file params.conf. Spherocyilinders with custom length and radius can be be generated using the accesory Python script in src/genspherocyl/genspherocyl.py.

Output

The following output files are generated after execution of shape:

• ./minim_coords.dat: Coordinate at the target pressure.
• ./press/X_press.dat: Coordinates for each step of pressurization.
• ./summary/pressures.dat: Pressures in each step.
• ./summary/output.dat: Summaary of mechanical and mesh parameters employed.
• ./helix/main_helix_vertexes.dat: Indices of the verticies defining the main helix (layer zero)
• ./helix/helix_vertexes.dat: Every index involved in the reinforced area