ode2tn

ode2tn is a Python package to compile arbitrary polynomial ODEs into a transcriptional network simulating the ODEs.

See this paper for details: TODO

Installation

Type pip install ode2tn at the command line.

Usage

See the notebook.ipynb for more examples of usage, including all the examples discussed in the paper.

The functions ode2tn and plot_tn are the main elements of the package. ode2tn converts a system of arbitrary polynomial ODEs into another system of ODEs representing a transcriptional network as defined in the paper above. Each variable $x$ in the original ODEs is represented by a pair of variables $x^\top,x^\bot$, whose ratio $\frac{x^\top}{x^\bot}$ follows the same dynamics in the transcriptional network as $x$ does in the original ODEs. plot_tn does this conversion and then plots the ratios by default, although it can be customized what exactly is plotted; see the documentation for gpac.plot for a description of all options.

Here is a typical way to call each function:

from math import pi
import numpy as np
import sympy as sp
import gpac as gp
from transform import plot_tn, ode2tn

x,y = sp.symbols('x y')
odes = {      # odes dict maps each symbol to an expression for its time derivative
    x: y-2,   # dx/dt = y-2
    y: -x+2,  # dy/dt = -x+2
}
inits = {     # inits maps each symbol to its initial value
    x: 2,     # x(0) = 2
    y: 1,     # y(0) = 1
}
gamma = 2     # uniform decay constant; should have gamma > max q^-; 
              #   see proof of main Theorem in paper for what q^- is
beta = 1      # constant introduced to keep values from going to infinity or 0
tn_odes, tn_inits, tn_syms = ode2tn(odes, inits, gamma=gamma, beta=beta)
gp.display_odes(tn_odes)  # displays nice rendered LaTeX in Jupyter notebook
print(f'{tn_inits=}')
print(f'{tn_syms=}')

When run in a Jupyter notebook, this will show

showing that the variables x and y have been replace by pairs x_t,x_b and y_t,y_b, whose ratios x_t/x_b and y_t/y_b will track the values of the original variable x and y over time.

If not in a Jupyter notebook, one could also inspect the transcriptional network ODEs via

for var, ode in tn_odes.items():
    print(f"{var}' = {ode}")

which would print a text-based version of the equations:

x_t' = x_b*y_t/y_b - 2*x_t + x_t/x_b
x_b' = 2*x_b**2/x_t - 2*x_b + 1
y_t' = 2*y_b - 2*y_t + y_t/y_b
y_b' = -2*y_b + 1 + x_t*y_b**2/(x_b*y_t)

The function plot_tn above does this conversion on the original odes and then plots the ratios. Running

t_eval = np.linspace(0, 6*pi, 1000)
# note below it is odes and inits, not tn_odes and tn_inits
# plot_tn calls ode2tn to convert the ODEs before plotting
plot_tn(odes, inits, gamma=gamma, beta=beta, t_eval=t_eval, show_factors=True)

in a Jupyter notebook will show this figure:

The parameter show_factors above indicates to show a second subplot with the underlying transcription factors ($x^\top, x^\bot, y^\top, y^\bot$ above). If left unspecified, it defaults to False and plots only the original values (ratios of pairs of transcription factors, $x,y$ above).

One could also hand the transcriptional network ODEs to gpac to integrate, if you want to directly access the data being plotted above. The OdeResult object returned by gpac.integrate_odes is the same returned by scipy.integrate.solve_ivp, where the return value sol has a field sol.y that has the values of the variables in the order they were inserted into tn_odes, which will be the same as the order in which the original variables x and y were inserted, with x_t coming before x_b:

t_eval = np.linspace(0, 2*pi, 5)
sol = gp.integrate_odes(tn_odes, tn_inits, t_eval)
print(f'times = {sol.t}')
print(f'x_t   = {sol.y[0]}')
print(f'x_b   = {sol.y[1]}')
print(f'y_t   = {sol.y[2]}')
print(f'y_b   = {sol.y[3]}')

which would print

times = [0.         1.57079633 3.14159265 4.71238898 6.28318531]
x_t   = [2.         1.78280757 3.67207594 2.80592514 1.71859172]
x_b   = [1.         1.78425369 1.83663725 0.93260227 0.859926  ]
y_t   = [1.         1.87324904 2.14156469 2.10338162 2.74383426]
y_b   = [1.         0.93637933 0.71348949 1.05261915 2.78279691]

Usage

Funding for this work was provided by the US Department of Energy, under award DE-SC0024467.