The generic geometry of steady state varieties

This repository contains files for the manuscript "The generic geometry of steady state varieties"by Elisenda Feliu, Oskar Henriksson, and Beatriz Pascual-Escudero.

File descriptions

The repository contains the following files:

• A Julia file functions.jl that contains functions for testing whether a network admits positive nondegenerate steady states when modeled with (generalized) mass action kinetics.
• Two notebooks with examples:
  • IDH.ipynb for the isocitrate dehydrogenase network in Example 4.1 of the paper.
  • 167.ipynb for the network BIOMD0000000167 from ODEbase discussed in Example 4.10 of the paper.
• A directory results that contains the following files:
  • investigated_models.csv with all consistent networks in ODEbase (as of November 2, 2023) with at least one reaction with integer stoichiometric coefficients. The networks that do not satisfy this are listed in excluded_networks.csv.
  • nondegenerate_networks.csv with all networks from investigated_models.csv that admit a positive nondegenerate steady state.
  • degenerate_networks.csv with all networks from investigated_models.csv that have a positive steady states, but all of them are degenerate.
  • generic_local_acr.csv with all networks from investigated_models.csv that satisfy the following criteria:
    • admits nondegenerate positive steady states
    • is not of full rank (after removing nonparticipating species)
    • has generic local ACR in at least one speceis.

Dependencies

The Julia code is based on Catalyst v14.4.1 and Oscar v1.1.1. For exact dependencies, see the file Manifest.toml.

Julia example

We begin by loading the functions:

include("functions.jl");

Consider the following isocitrate dehydrogenase that appears in Shinar–Feinberg's work on absolute concentration robustness, entered in catalyst format.

rn = @reaction_network begin 
    k1, X1 + X2 --> X3
    k2, X3 --> X1 + X2
    k3, X3 --> X1 + X4
    k4, X3 + X4 --> X5
    k5, X5 --> X3 + X4
    k6, X5 --> X2 + X3 
end;

The following command returns true, which means that the network admits positive steady states:

julia> is_consistent(rn)
true

The following command returns true, which means that there is a nondegenerate steady state with respect to its stoichiometric compatibility classes:

julia> has_nondegenerate_steady_state(rn, use_conservation_laws=true)
true

We check for generic local ACR with respect to the first and fourth species:

julia> generic_local_acr(rn, 1)
false

julia> generic_local_acr(rn, 4)
true

For the fourth species, we get the following polynomial as a witness for ACR:

julia> local_acr_polynomial(rn, 4)
k[4]*k[6]*x[4] - k[3]*k[5] - k[3]*k[6]

We could also do these checks on the level of the matrices that describe the associated augmented vertical system:

N = matrix(QQ, netstoichmat(rn))
B = matrix(ZZ, substoichmat(rn))
L = matrix(QQ, conservationlaws(rn))

has_nondegenerate_zero(N, B, L)
generic_local_acr(N, B, 1)
generic_local_acr(N, B, 4)
local_acr_polynomial(N, B, 4)