Synchronous Balanced Analysis

Abstract

When modeling Chemical Reaction Networks, a commonly used mathematical formalism is that of Petri Nets, with the usual interleaving execution semantics. We aim to substitute to a Chemical Reaction Network, especially a “growth” one (i.e., for which an exponential stationary phase exists), a piecewise synchronous approximation of the dynamics: a resource-allocation-centered Petri Net with maximal-step execution semantics. In the case of unimolecular chemical reactions, we prove the correctness of our method and show that it can be used either as an approximation of the dynamics, or as a method of constraining the reaction rate constants (an alternative to flux balance analysis, using an emergent formally defined notion of “growth rate” as the objective function), or a technique of refuting models.

Appendices

Appendix A CRN Mass-action Kinetic Equations

Consider the following chemical reaction network:

Then $\mathcal {S} = \{S_1,S_2\}, \nabla ^- = \begin{bmatrix} 1&1 \\ 0&1 \end{bmatrix}, \nabla ^+ = \begin{bmatrix} 0&2 \\ 1&0 \end{bmatrix}, \mathbf {x}^{\nabla ^-} = \begin{bmatrix} x_1 x_2 \\ x_2 \end{bmatrix}$ and $\frac{d\mathbf {x}}{dt} = \begin{bmatrix} -1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} \kappa _1&0 \\ 0&\kappa _2 \end{bmatrix} \cdot \begin{bmatrix} x_1 x_2 \\ x_2 \end{bmatrix} = \begin{bmatrix} -\kappa _1 x_1 x_2 + \kappa _2 x_2\\ \kappa _1 x_1 x_2 - \kappa _2 x_2 \end{bmatrix}$

Appendix B Proof of Theorem 1

Proof

Given a resource array m, consider v a (potentially max-parallel) vector compatible at m:

$$\begin{aligned} \nabla ^- v \le m \end{aligned}$$

(13)

Then we can construct $\alpha \in \mathbb {R}_+^{|T| \times |S|}$:

$$\begin{aligned} \alpha _{ji} = \frac{\nabla ^-_{ij} \cdot v_j}{m_i} \end{aligned}$$

(14)

s.t.

$$\begin{aligned} (\alpha \star m)_j = \underset{i \in S}{\min } \left\{ \frac{\alpha _{ji}}{\nabla _{ij}^{-}} \cdot m_i\right\} =\min \left\{ \frac{\nabla ^-_{ij} \cdot v_j}{m_i} \cdot \frac{m_i}{\nabla ^-_{ij}} \mid \nabla ^-_{ij} \ne 0 \right\} = v_j \end{aligned}$$

(15)

Furthermore,

(16)

i.e. $\alpha$ is indeed a resource-allocation matrix.

If all reactions of the CRN are unimolecular, then:

$$\begin{aligned} \forall j \in T, \exists ! i_j \in S \quad \text {s.t.} \quad \nabla ^-_{i_jj} \ne 0 \implies \forall j \in T, (\alpha \star m)_j = \frac{\alpha _{ji_j}}{\nabla _{i_jj}^{-}} \cdot m_{i_j} \end{aligned}$$

(17)

hence the uniqueness of $\alpha$. $\square$

Appendix C Non-uniqueness of $\alpha$ for Bimolecular Reactions

Example 1

(Based on Fig. 1.) $m = \begin{bmatrix} 9 \\ 9 \\ 9 \end{bmatrix}$, $\nabla ^- = \begin{bmatrix} 3&0&0 \\ 2&5&0 \\ 0&3&1 \end{bmatrix}$, and $v = \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix}$, one of the 2 possible maximally parallel steps ($\{t_0 \times 2, t_1 , t_2\times 6\}$).

Then $\exists \alpha = \begin{bmatrix} \frac{6}{9}&\frac{4}{9}&0 \\ 0&\frac{5}{9}&\frac{3}{9} \\ 0&0&\frac{6}{9} \end{bmatrix}$, defined as in (14), s.t. $\alpha \star m = {\begin{bmatrix} \frac{6}{9} \cdot 9 \cdot \frac{1}{3} \wedge \frac{4}{9} \cdot 9 \cdot \frac{1}{2} \wedge \infty \\ \infty \wedge \frac{5}{9} \cdot 9 \cdot \frac{1}{5} \wedge \frac{3}{9} \cdot 9 \cdot \frac{1}{3}\\ \infty \wedge \infty \wedge \frac{6}{9} \cdot 9 \cdot \frac{1}{1} \end{bmatrix}} = \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix} = v$.

By re-allocating the excess of species A to the first reaction, we get $\alpha ' = \begin{bmatrix} 1&\frac{4}{9}&0 \\ 0&\frac{5}{9}&\frac{3}{9} \\ 0&0&\frac{6}{9} \end{bmatrix}$, a resource-allocation matrix that also verifies $\alpha ' \star m = \begin{bmatrix} 1 \cdot 9 \cdot \frac{1}{3} \wedge \frac{4}{9} \cdot 9 \cdot \frac{1}{2} \wedge \infty \\ \infty \wedge \frac{5}{9} \cdot 9 \cdot \frac{1}{5} \wedge \frac{3}{9} \cdot 9 \cdot \frac{1}{3}\\ \infty \wedge \infty \wedge \frac{6}{9} \cdot 9 \cdot \frac{1}{1} \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix} = v$ (non-uniqueness of $\alpha$ in the bimolecular case).