Arrow

“... even if the previous millisecond is closer to us than the birth of the universe, it is equally out of reach.” ― Jean-Christophe Valtat, Luminous Chaos

Concept

This library implements a generalized version of the Gillespie Algorithm, a stochastic approach to numerically solving discrete systems. Each iteration, the algorithm will calculate the propensities for each reaction given a rate and the counts of the reactants present in the current state of the system, then selects one reaction to occur and the interval of time between the previous reaction and the current reaction. Iterating this produces a trajectory (or history) of the state vector over the course of the simulation.

Installation

Add the following to your requirements.txt, or run pip install stochastic-arrow to install it from PyPI:

stochastic-arrow

NOTE: If upgrading from a version older than 1.0.0, check if the arrow datetime package is installed. If so, uninstall arrow before upgrading stochastic-arrow, then reinstall arrow.

> pip show arrow
> pip uninstall arrow
> pip install stochastic-arrow
> pip install arrow

Usage

The stochastic_arrow library presents a single class as an interface, StochasticSystem, which operates on a set of reactions (encoded as a numpy matrix of stoichiometrix coefficients) and associated reaction rates:

from stochastic_arrow import StochasticSystem
import numpy as np

# Each row is a reaction and each column is a molecular species (or other
# entity). The first reaction here means that the first and second elements
# combine to create the third, while the fourth is unaffected.
stoichiometric_matrix = np.array([
    [1, 1, -1, 0],
    [-2, 0, 0, 1],
    [-1, -1, 1, 0]], np.int64)

# Once we have a matrix of reactions, we can
# construct the system.
system = StochasticSystem(stoichiometric_matrix)

Now that the system has been instantiated, we can invoke it with any initial state vector and set of reaction rates and then run it for a given time interval:

# This gives the initial state of the system (counts of each molecular species,
# for instance).
import numpy as np
state = np.array([1000, 1000, 0, 0])

# We also specify how long we want the simulation to run. Here we set it to one
# second.
duration = 1

# Each reaction has an associated rate for how probable that reaction is.
rates = np.array([3.0, 1.0, 1.0])

Once we have an initial state and rates, we can run the simulation for the given duration. evolve returns a dictionary with five keys:

• steps - the number of steps the simulation took
• time - at what time point each event took place
• events - the events that occurred
• occurrences - the number of times each event occurred (derived directly from events)
• outcome - the final state of the system

result = system.evolve(duration, state, rates)

If you are interested in the history of states for plotting or otherwise, these can be derived from the list of events and the stoichiometric matrix, along with the inital state. reenact_events will do this for you:

from stochastic_arrow import reenact_events

history = reenact_events(stoichiometric_matrix, result['events'], state)

Testing

stochastic_arrow uses pytest. To test it:

> make clean compile
> pytest

NOTE: make compile without an explicit clean might not fully build the extension.

There are more command line features in test_arrow:

> python -m stochastic_arrow.test.test_arrow --complexation

> python -m stochastic_arrow.test.test_arrow --plot

> python -m stochastic_arrow.test.test_arrow --obsidian

> python -m stochastic_arrow.test.test_arrow --memory

> python -m stochastic_arrow.test.test_arrow --time

More examples:

> python -m stochastic_arrow.test.test_hang

> pytest -m stochastic_arrow/test/test_arrow.py

> pytest -k flagella

Changelog

Version 1.0.0

• Rename module to stochastic_arrow to avoid name conflict (Issue #51). All users must update their import statements to use stochastic_arrow instead of arrow.

Version 0.5.2

• Update to Cython 0.29.34. (Cython 3.0.0 is now in beta.)

Version 0.5.1

• Update to Cython 3.0.0a11 for compatibility with Python 3.11. Add arrow.pxd to work around a Cython 3.0.0 bug.
• Stop using deprecated numpy.distutils to avoid warnings and prepare for its removal in Python 3.12.
• Make test_arrow.py --plot compatible with Python 3.
• Fix PytestReturnNotNoneWarning warnings from pytest 7.2.0.

Version 0.5.0

• Add the arrow_hang unit test which catches a nasty edge-case (Issue #48), fix the bug, and make the code more robust to some other potential bugs.

Version 0.4.4

• Can pickle StochasticSystem instances.

Version 0.3.0

• Introduced backwards-incompatible API change for supplying rates at evolve() time rather than __init__() for StochasticSystem.