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Duello
Virial Coefficient and Dissociation Constant Estimation for Rigid Macromolecules

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Introduction

Duello is a tool to calculate the potential of mean force (PMF) between two rigid bodies, performing a statistical mechanical average over inter-molecular orientations using subdivided icosahedrons. For each mass center separation, R, the static contribution to the partition function, $\mathcal{Z}(R) = \sum_{\mathbf{\Omega}} e^{-V(R,\mathbf{\Omega})/k_BT}$, is explicitly evaluated to obtain the potential of mean force, $w(R) = -k_BT \ln \mathcal{Z}(R)$ and the thermally averaged energy,

$$ U(R) = \frac{\sum V(R,\mathbf{\Omega}) e^{-V(R,\mathbf{\Omega})/k_BT}} {\mathcal{Z}(R)} $$

where $V(R,\mathbf{\Omega})$ is the total inter-body interaction energy and $\mathbf{\Omega}$ represents a 5D angular space (e.g. two spherical coordinates for each body plus a dihedral angle around the connection line).

The osmotic second virial coefficient, which has dimensions of volume, reports on exactly two-body interactions:

$$ \begin{align} B_2 & = -\frac{1}{16\pi^2} \int_{\mathbf{\Omega}} \int_0^{\infty} \left ( e^{-V(R,\mathbf{\Omega})/k_BT} - 1 \right ) R^2 dR d\mathbf{\Omega}\\ & = -2\pi \int_0^{\infty} \left ( e^{-w(R)/k_BT} -1 \right )R^2 dR \\ & = B_2^{hs} -2\pi \int_{\sigma}^{\infty} \left ( e^{-w(R)/k_BT} -1 \right )R^2 dR\\ \end{align} $$

where $B_2^{hs} = 2\pi\sigma^3/3$ is the hard-sphere contribution and $\sigma$ is a distance of closest approach where $w(R\lt \sigma)=\infty$ is assumed. For systems with net attractive interactions, the dissociation constant, $K_d$, can be estimated by,

$$ K_d^{-1} = 2 N_A\left (B_2^{hs} - B_2\right ) $$

Installation

Binary packages are available for Linux and MacOS through PyPI.org:

pip install duello

If you have a Rust toolchain installed, you may alternatively build and install directly from the source code:

cargo install --git https://github.com/mlund/duello

Usage

The command-line tool duello does the 6D scanning and calculates the angularly averaged potential of mean force, A(R) which is used to derive the 2nd virial coefficient and twobody dissociation constant, $K_d$. The two input structures should be in .xyz format and all particle names must be defined in the topology file under atoms. The topology also defines the particular pair-potential to use, see below. Note that currently, a coulomb potential is automatically added and should hence not be specified in the topology. The program is written in Rust and attempts to use all available CPU cores.

duello scan \
    --mol1 cppm-p18.xyz \
    --mol2 cppm-p18.xyz \
    --rmin 37 --rmax 50 --dr 0.5 \
    --top topology.yaml \
    --resolution 0.8 \
    --cutoff 1000 \
    --molarity 0.05 \
    --temperature 298.15

Preparing PDB files

The following uses pdb2xyz to create a coarse grained XYZ file and Calvados topology for Duello:

pip install pdb2xyz
pdb2xyz -i 4lzt.pdb -o 4lzt.xyz --pH 7.0 --sidechains
duello scan \
  -1 4lzt.xyz -2 4lzt.xyz \
  --rmin 24 --rmax 80 --dr 0.5 \
  --resolution 0.6 \
  --top topology.yaml \
  --molarity 0.05

If pdb2xyz give errors, you may be able to correct your PDB file with pdbfixer.

Examples

Ready run scripts examples are provided in the scripts/ directory:

Command	Description
scripts/cppm.sh	Spherical, multipolar particles using the CPPM model
scripts/calvados3.sh	Two coarse grained lysozyme molecules w. Calvados3 interactions

Interaction models

Each macromolecule is represented by a rigid constellation of beads with properties defined under atoms in the topology file. The inter-molecular energy, $V(R,\Omega)$ is calculated by summing all pairwise interactions between beads using a customizable pair potential, $u_{ij}$. If needed, different pair-potentials can be explicitly defined for specific atom pairs.

The provided examples illustrate the following schemes:

• Screened Coulomb + AshbaughHatch, for the Calvados model.
• Screened Coulomb + WeeksChandlerAndersen for the CPPM model.

Many more pair-potentials are available through the interatomic library, e.g. LennardJones, HardSphere etc.

Warning: The electrostatic term, Coulomb is always automatically added and should therefore not be specified in the topology.

Development

This is for development purposes only and details how to create and publish a binary package on pipy.org.

Create pip package using Maturin via a Docker image:

docker run --rm -v $(pwd):/io ghcr.io/pyo3/maturin publish -u __token__ -p PYPI_TOKEN

For local Maturin installs, follow the steps below.

pip install ziglang pipx
pipx install maturin # on ubuntu; then restart shell
maturin publish -u __token__ -p PYPI_TOKEN --target=x86_64-unknown-linux-gnu --zig

MacOS targets can be generated without --zig using the targets x86_64-apple-darwin and aarch64-apple-darwin.

rustup target list
rustup target add x86_64-apple-darwin