3.3. kda.diagrams

3.3.1. Diagram Generation

The diagrams module contains code to generate partial diagrams (undirected spanning trees), directional diagrams, and flux diagrams using the diagram method developed by King and Altman [1] and Hill [2].

enumerate_partial_diagrams(G)

Quantifies the number of partial diagrams (undirected spanning trees) that can be generated from a kinetic diagram.

Parameters:

G (NetworkX.MultiDiGraph) – A kinetic diagram

Returns:

n_partials – The number of unique partial diagrams (spanning trees) that can be generated from a graph represented by the input rate matrix.

Return type:

int

Notes

This implements Kirchhoff’s matrix theroem [3] by generating the adjacency matrix from the input diagram, generating the Laplacian matrix from the adjacency matrix, then getting the cofactor matrix of the Laplacian matrix. All cofactors are equal and equal to the number of undirected spanning trees.

A more sophistocated version of this function is available in the NetworkX library [4] (see here).

generate_partial_diagrams(G, return_edges=False)

Generates all partial diagrams (undirected spanning trees) for a kinetic diagram.

Parameters:

• G (NetworkX.MultiDiGraph) – A kinetic diagram

• return_edges (bool) – Binary used for determining whether to return NetworkX diagram objects (primarily for plotting) or the edge tuples (generally for calculations).

Returns:

partials – Array of NetworkX.Graph where each graph is a unique partial diagram with no loops (return_edges=False), or a nested array of unique edges for valid partial diagrams (return_edges=True).

Return type:

ndarray of NetworkX.Graph

generate_directional_diagrams(G, return_edges=False)

Generates all directional diagrams for a kinetic diagram using depth-first-search algorithm.

Parameters:

• G (NetworkX.MultiDiGraph) – A kinetic diagram

• return_edges (bool) – Binary used for determining whether to return NetworkX graph objects (primarily for plotting) or the edge tuples (generally for calculations).

Returns:

directional_diagrams – Array of all directional diagram edges made from 3-tuples (return_edges=True) or array of all directional diagrams (return_edges=False) for G.

Return type:

ndarray or ndarray of NetworkX.DiGraph

generate_flux_diagrams(G, cycle)

Generates all flux diagrams for a specific cycle in the kinetic diagram.

Parameters:

• G (NetworkX.MultiDiGraph) – A kinetic diagram

• cycle (list of int) – List of node indices for cycle of interest, index zero. Order of node indices does not matter.

Returns:

flux_diagrams – List of flux diagrams. Diagrams contain the input cycle where remaining edges follow path pointing to cycle. Cycle nodes are labeled by attribute 'is_target'.

Return type:

list of NetworkX.MultiDiGraph

generate_all_flux_diagrams(G)

Generates all flux diagrams for a kinetic diagram.

Parameters:

G (NetworkX.MultiDiGraph) – A kinetic diagram

Returns:

all_flux_diagrams – List of lists of flux diagrams, where each list is for a different cycle in G.

Return type:

list of lists of NetworkX.MultiDiGraph

References

[1]

Edward L. King and Carl Altman. A Schematic Method of Deriving the Rate Laws for Enzyme-Catalyzed Reactions. The Journal of Physical Chemistry, 60(10):1375–1378, October 1956. doi:10.1021/j150544a010.

[2]

Terrell L. Hill. Studies in irreversible thermodynamics IV. Diagrammatic representation of steady state fluxes for unimolecular systems. Journal of Theoretical Biology, 10(3):442–459, April 1966. doi:10.1016/0022-5193(66)90137-8.

[3]

Maumita Chakraborty, Sumon Chowdhury, Joymallya Chakraborty, Ranjan Mehera, and Rajat Kumar Pal. Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review. Complex & Intelligent Systems, 5(3):265–281, October 2019. doi:10.1007/s40747-018-0079-7.

[4]

Aric A. Hagberg, Daniel A. Schult, and Pieter J. Swart. Exploring Network Structure, Dynamics, and Function using NetworkX. In Gaël Varoquaux, Travis Vaught, and Jarrod Millman, editors, Proceedings of the 7th Python in Science Conference, 11 – 15. Pasadena, CA USA, 2008.

Functions

enumerate_partial_diagrams(G)	Quantifies the number of partial diagrams (undirected spanning trees) that can be generated from a kinetic diagram.
generate_all_flux_diagrams(G)	Generates all flux diagrams for a kinetic diagram.
generate_directional_diagrams(G[, return_edges])	Generates all directional diagrams for a kinetic diagram using depth-first-search algorithm.
generate_flux_diagrams(G, cycle)	Generates all flux diagrams for a specific cycle in the kinetic diagram.
generate_partial_diagrams(G[, return_edges])	Generates all partial diagrams (undirected spanning trees) for a kinetic diagram.