gpac 1.0.1

gpac stands for "General-Purpose Analog Computer". This package makes it easy to specify ordinary differential equations (ODEs) and view their solutions.

gpac Python package

Table of contents

• Overview
• API
• Installation
• Examples
  • Plotting ODEs
  • Getting trajectory data of ODEs
  • Chemical reaction networks

Overview

This is a Python package for simulating General-Purpose Analog Computers as defined and studied by Claude Shannon. It's primarily a front-end to scipy and sympy making it easier to specify systems of ODEs, numerically integrate them, and plot their solutions. It also has support for a very common model governed by polynomial ODEs, the of continuous mass-action chemical reaction networks.

This is ostensibly what pyodesys does as well, and that package is much more powerful and configurable than gpac. The purpose of gpac is primarily to be simpler to use for common cases of ODEs, at the cost of being less expressive. For example, gpac has some functions (plot and plot_crn) to do plotting in matplotlib, which is easier than manually getting the ODE data through integrate_odes and passing it along to the matplotlib plot function. This is possible if you want to have more control over how things are plotted than is possible with the gpac plotting functions; however in most cases you can configure what you need in plot and plot_crn either by passing keyword arguments (which are passed along to the matplotlib plot function), or by calling functions in matplotlib.pyplot (e.g., yscale) after calling gpac's plot or pyplot.

Note: Some of the relative links below are intended to be used only on the GitHub page for this project: https://github.com/UC-Davis-molecular-computing/gpac#readme They will not work if you are reading this document on PyPI, for example.

API

The API for the package is here: https://gpac.readthedocs.io/

Installation

Python 3.7 or above is required. There are two ways you can install the gpac package, pip or git:

A. pip: The easiest option is to install via pip by typing the following at the command line:

pip install gpac

B. git: The other option is to clone the git repo. You may need to install git first: https://git-scm.com/book/en/v2/Getting-Started-Installing-Git

1. Install the dependencies by typing the following at the command line:

   pip install numpy scipy matplotlib sympy
   

2. Clone this repo by typing the following at the command line:

   git clone https://github.com/UC-Davis-molecular-computing/gpac.git
   

3. Add the directory into which you cloned the repo (it should be the gpac subdirectory under your working directory where the git clone above was executed) to your PYTHONPATH environment variable. See https://www.geeksforgeeks.org/pythonpath-environment-variable-in-python/ for example if you don't know how to alter PYTHONPATH. After doing this you should be able to import the gpac package in your Python scripts/Jupyter notebooks with import gpac. Try testing this out in the Python interpreter:

   $ python
   Python 3.9.12 (main, Apr  4 2022, 05:22:27) [MSC v.1916 64 bit (AMD64)] :: Anaconda, Inc. on win32
   Type "help", "copyright", "credits" or "license" for more information.
   >>> import gpac
   >>>
   

Examples

See more examples in the Jupyter notebook notebook.ipynb.

Plotting ODEs

ODEs are specified by creating sympy symbols and expressions (or if you like, Python strings), represented as a Python dict odes mapping each variable---a single sympy symbol or Python string---to an expression representing its time derivative, represented as a sympy expression composed of sympy symbols (or for convenience you can also use Python strings, or if the derivative is constant, a Python int or float).

Every symbol that appears in any of the expressions must also be a key in this dict.

The initial values are specified as a Python dict initial_values mapping variables (again, sympy symbols or strings) to their initial values (floats). If you leave out a symbol as a key to initial_values, it is assumed to have initial value 0.

Finally, you can specify the times at which to solve for the ODEs as an iterable of floats t_eval. (This is optional; if not specified it uses the time values 0.0, 0.01, 0.02, 0.03, ..., 0.98, 0.99, 1.0)

Remaining parameters are optional (see below for examples of them). See API documentation for integrate_odes and plot for more details.

import sympy
import gpac
import numpy as np

a,b,c = sympy.symbols('a b c')

# ODEs specified as dict mapping each variable to expression describing its derivative.
# key representing variable can be a sympy Symbol or string.
# value representing derivative can be a sympy Expr, string, or (if constant) int or float.
odes = {               # represents ODEs:
    a: -a*b + c*a,     # d/dt a(t) = -a(t)*b(t) + c(t)*a(t)
    b: -b*c + a*b,     # d/dt b(t) = -b(t)*c(t) + a(t)*b(t)
    'c': '-c*a + b*c', # d/dt c(t) = -c(t)*a(t) + b(t)*c(t)
}
initial_values = {
    a: 10,
    b: 1,
    c: 1,
}
t_eval = np.linspace(0, 5, 200)

gpac.plot(odes, initial_values, t_eval=t_eval, figure_size=(12,3), symbols_to_plot=[a,c])

Getting trajectory data of ODEs

If you want the data itself from the ODE numerical integration (without plotting it), you can call integrate_odes (replace the call to plot above with the following code).

t_eval = np.linspace(0, 1, 5)

solution = gpac.integrate_odes(odes, initial_values, t_eval=t_eval)
print(f'times = {solution.t}')
print(f'a = {solution.y[0]}')
print(f'b = {solution.y[1]}')
print(f'c = {solution.y[2]}')

which prints

times = [0.   0.25 0.5  0.75 1.  ]
a = [10.          4.84701622  0.58753815  0.38765743  3.07392998]
b = [1.         6.84903338 9.63512628 3.03634559 0.38421121]
c = [1.         0.3039504  1.77733557 8.57599698 8.54185881]

The value solution returned by integrate_odes is the same object returned from scipy.integrate.solve_ivp.

Chemical reaction networks

There are also functions integrate_crn_odes and plot_crn, which take as input a description of a set of chemical reactions, derives their ODEs, then integrates/plots them. They both use the function crn_to_odes, which converts chemical reactions into ODEs.

Reactions are constructed using operations on Specie objects returned from the function species:

# plot solution to ODEs of this CRN that computes f(x) = x^2, using the gpac.crn module
# 2X -> 2X+Y
# Y -> nothing
x,y = gpac.species('X Y')
rxns = [
    x+x >> x+x+y,
    y >> gpac.empty,
]
initial_values = {x:5}
t_eval = np.linspace(0, 5, 100)

# plot trajectory of concentrations
gpac.plot_crn(rxns, initial_values, t_eval=t_eval, figure_size=(20,4))

See notebook.ipynb for more examples.

Although they appear similar, a Specie object (such as x and y returned from the gpac.species function above) is different from a sympy.Symbol object. The Specie object is intended to help specify reactions using the notation above with the symbols +, >>, and | (as well as the k and r functions for specifying non-unit rate constants, see example notebook). However, any of the following objects can be a key in the initial_values parameter to plot_crn and integrate_crn_odes: Specie, sympy.Symbol, or str. They will be normalized to sympy.Symbol objects in keys of the returned dict representing the ODEs.