Defining new reactions

SimBio includes a comprehensive set of predefined reactions that relate reactants with products and a well defined kinetics. But, new reactions can be easily defined, to extend SimBio’s capabilities.

There are two main types of reactions,

• SingleReaction, which models elementary chemical reactions,

• ReactionGroup, which consists of multiple SingleReaction and/or ReactionGroup.

Single reactions

A SingleReaction converting species \(A\) and \(B\) to \(C\),

\[ n_A A + n_B B \xrightarrow{k} n_C C \]

where \(n_A\), \(n_B\), and \(n_C\) are the stoichiometric coefficients, corresponds to the following differential equations:

\[\begin{split} \begin{cases} \frac{dA}{dt} = -n_A \; R(t, A^{n_A}, B^{n_B}, k) \\ \frac{dB}{dt} = -n_B \; R(t, A^{n_A}, B^{n_B}, k) \\ \frac{dC}{dt} = +n_C \; R(t, A^{n_A}, B^{n_B}, k) \end{cases} \end{split}\]

where \(R(t, A, B, k)\) is the reaction rate.

For a reaction following mass action kinetics, the reaction rate is

\[ R(t, A, B, k) = k A B \]

For a Michaelis-Menten reaction,

\[ S + E ↔ ES → P + E \]

where an enzyme \(E\) converts a substrate \(S\) to a product \(P\), under certain assumptions can be modeled as a single-step reaction with rate:

\[ R(t, S, V_{max}, K_M) = V_{max} * S / (K_M + S) \]

To define a SingleReaction, we need to declare the species and parameters it relates, which corresponds to reactants and products, and the reaction rate function:

from simbio.components import SingleReaction, Species, Parameter


class Synthesis(SingleReaction):
    """A + B → AB"""
    A: Species
    B: Species
    AB: Species
    rate: Parameter

    def reactants(self):
        return (self.A, self.B)

    def products(self):
        return (self.AB,)

    @staticmethod
    def reaction_rate(t, A, B, rate):
        return rate * A * B

Compound reactions

Compound reactions are a convenience to englobe a series of reactions into one. They can be composed of many single or compound reactions. For instance, the MichaelisMenten reaction,

\[ S + E ↔ ES → P + E \]

which is included in simbio.reactions.enzymatic, is defined as follows:

from simbio.components import ReactionGroup, Species, Parameter
from simbio.reactions.compound import ReversibleSynthesis
from simbio.reactions.single import Dissociation


class MichaelisMenten(ReactionGroup):
    """S + E ↔ ES → P + E"""
    E: Species
    S: Species
    ES: Species
    P: Species
    forward_rate: Parameter
    reverse_rate: Parameter
    catalytic_rate: Parameter

    binding_reaction = ReversibleSynthesis(
        A=E,
        B=S,
        AB=ES,
        forward_rate=forward_rate,
        reverse_rate=reverse_rate,
    )
    dissociation_reaction = Dissociation(
        AB=ES,
        A=E,
        B=P,
        rate=catalytic_rate,
    )