Reversible binding

We can initialize a system of a simple reversible binding interaction as follows:

from kappybara.system import System

system = System.from_kappa(
    mixture={"A(x[.])": 100, "B(x[.])": 100},
    rules=[
        "A(x[.]), B(x[.]) -> A(x[1]), B(x[1]) @ 1",
        "A(x[1]), B(x[1]) -> A(x[.]), B(x[.]) @ 1",
    ],
    observables={"AB": "|A(x[1]), B(x[1])|"},
)

or equivalently from a .ka-style string:

system = System.from_ka(
    """
    %init: 100 A(x[.])
    %init: 100 B(x[.])

    %obs: 'AB' |A(x[1]), B(x[1])|

    A(x[.]), B(x[.]) <-> A(x[1]), B(x[1]) @ 1, 1
    """
)

100 instances of molecules of type A and of type B, each with an empty binding domain x, are created, and we track the number of AB complexes.

We’re going to simulate this system and plot its behavior, marking certain times of interest. We’ll first simulate until time 1:

times = []
while system.time < 1:
    system.update()
times.append(system.time)

We’ll now manually instantiate 50 new A and B molecules each, start tracking the number of free A, and simulate until there are no more than 10 free A in the mixture:

system.mixture.instantiate("A(x[.]), B(x[.])", 50)

system["A"] = "|A(x[.])|"
while system["A"] > 10:
    system.update()
times.append(system.time)

Now let’s simulate some more time:

while system.time < 2:
    system.update()
times.append(2)

The default simulator provides the most features since it’s written directly in Python, but models can be offloaded to KaSim, a compiled Kappa simulator, for faster simulation. For example, we could’ve run:

system.update_via_kasim(time=1)

Finally, let’s plot the history of the quantities we tracked:

import matplotlib.pyplot as plt

system.monitor.plot(combined=True)
for time in times:
    plt.axvline(time, color="black", linestyle="dotted")
plt.show()

The system equilibrates relatively early. Then new A is added and the number of AB complexes increases, reaching a higher equilibrium concentration.