jabsim

A jax-based package for simulating ODE models of biological systems, where all variables are non-negative. This enforcement of non-negativity, which neither scipy.solve_ivp nor diffrax solvers can give you, is why you may need jabsim. This is achieved by clamping the state variables to zero whenever they are negative

jabsim is powered by the jax package for high-performance computing and parallelisation. This means that jabsim simulations can be jit-compiled and parallelised on a GPU or TPU using jax.vmap or shard mapping.

How to use jabsim

1. Make an ODE function which calculates the derivative $\frac{dx}{dt}$ from the arguments t, x, par in this order.
   • What do the arguments stand for?
     • t is the time at this point in the simulation
     • x is the state vector at this time point
     • args is a tuple of extra arguments passed on to the ODE function.
   • IMPORTANT: if you don't know jax, there are a few differences to keep in mind:
     • If you normally use numpy functions in your ODE, import jax.numpy as jnp and use it instead of np. jax with jax.numpy will get automatically installed as a dependency of jabsim when you install it.
     • Be careful using loops and if-statements. If you're an amateur programmer, just avoid doing all that. Otherwise, have a look at the JAX documentation .
2. Import jabsim and call jabsim.sim to simulate. The arguments are as follows:
   • par: a list, array or dict of model parameters as in your ODE function
   • model_ode: the ODE function you created
   • x0: the initial state vector as a 1D np.array or jnp.array
   • tf: tuple only. The ODE will be simulated for time points between tf[0] and tf[1] (inclusive).
   • savetimestep: interval between the time points at which the trajectory is saved
   • simulator: string specifyingthe simulation method to use
     • "euler": Euler simulator.
     • "rk4": Runge-Kutta 4th order simulator. Slower per ODE integration step but more accurate, hence allowing larger steps for the same accuracy.
   • ode_steps_in_savetimestep: number of ODE integration steps within one timestep
     • e.g. if savetimestep=0.5 hours and ode_step_in_savetimesteps=100, there will be 100 integration steps per 0.5 hour, so the ODE integration step size will be 0.5/100=0.05 hours.
     • high ode_steps_in_savetimestep number increases accuracy but increases runtimes
   • return_numpy: if True (by default, it is) the output will be in the np.array format, otherwise it will be jnp.array.
3. Running jabsim.sim() will return the arrays ts and xs as np.array or jnp.array, as well as a boolean value success.
   • ts: array of timepoints between tf[0] and tf[1] with savetimestep hours, seconds or whatever units you are using between each two consecutive point
   • xs: system trajectory saved as an array at the time points in ts - axis 0 for time, axis 1 for entries in the state vector (i.e. xs.shape[0]=len(ts)).
   • success: boolean value; True if no entry in xs is nan or inf, False otherwise.

Notes

• In practice, 500 steps per hour (e.g. savetimestep=0.5, ode_steps_in_savetimestep=250 or savetimestep=1.0, ode_steps_in_savetimestep=500) works well for the RK4 solver. For the Euler solver, 1e4 steps per hours is reasonably good.
• For benchmarking, you can also set simulator="scipy" to simulate your ODE with scipy.solve_ivp (but without any of the delicious jax features of the solvers above). In that case, don't use the arguments ode_steps_in_savetimestep and savetimestep. Instead, you can optionally specify:
  • solver: string describing any solver which may be used with scipy.solve_ivp. By default, we have solver="LSODA".
  • tols: dictionary of relative and absolute tolerances for the scipy solver. By default, tols={'rtol': 1e-6, 'atol': 1e-9}.
  • dt0: starting integration step size. By default, dt0=0.1.
• If you want to make use of jax parallelisation, make sure to set return_numpy=False so that the solver would operate with jnp.array objects only.

Example

Let us integrate a simple one-dimensional ODE $\frac{dx}{dt} = a x^2$. For the initial condition $x_0=1$ and $a=0.4$, this has the analytical solution $x = \frac{1}{1-0.4t}$. This means we can verify that for savetimestep=0.5, jabsim.sim() produces ts=np.array([0, 0.5, 1.0]) and xs=np.array([1.0, 1.25, 1.66666667]). All entries in xs are finite, hence success=True.

# import jabsim
import jabsim

# import jax.numpy for numpy operations
import jax.numpy as jnp

# our model ODE function returning a list of one element
def model_ode(t, x, args):
    # unpack args - get the dictiory of parameters
    par, = args
    
    # use jnp to square x 
    # (here you could just as well use x[0]**2, we just want to make a point)
    x_squared = jnp.square(x[0])
    
    # return dx/dt as a list - with one entry for a one-dimensional ODE
    return [par['a'] * x_squared]

# our dictionary of paramneters
par = {'a': 0.4}

ts, xs, success = jabsim.sim(
    model_ode=model_ode,
    args=(par,),
    x0=jnp.array([1.0]),
    tf=(0.0, 1.0),
    savetimestep=0.5,
    simulator='rk4',
    ode_steps_in_savetimestep=10,
)

# print the timne
print(ts)
print(xs)
print(success)

Citation

If you find this package useful in your work, please cite the paper below: code for its Showcase 2 served as jabsim's direct ideological precursor.

@article{Gallup2024,
	author = {Gallup, Olivia and Sechkar, Kirill and Towers, Sebastian and Steel, Harrison},
	title = {Computational Synthetic Biology Enabled through JAX: A Showcase},
	journal = {ACS Synth. Biol.},
	volume = {13},
	number = {9},
	pages = {3046},
	year = {2024},
	doi = {10.1021/acssynbio.4c00307}
}

The original JAX package should be cited as:

@software{jax2018github,
  author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Yash Katariya and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander{P}las and Skye Wanderman-{M}ilne and Qiao Zhang},
  title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
  url = {http://github.com/jax-ml/jax},
  version = {0.3.13},
  year = {2018},
}