Analysing and simulating energy-based models in biology using BondGraphTools

Abstract

Like all physical systems, biological systems are constrained by the laws of physics. However, mathematical models of biochemistry frequently neglect the conservation of energy, leading to unrealistic behaviour. Energy-based models that are consistent with conservation of mass, charge and energy have the potential to aid the understanding of complex interactions between biological components, and are becoming easier to develop with recent advances in experimental measurements and databases. In this paper, we motivate the use of bond graphs (a modelling tool from engineering) for energy-based modelling and introduce, BondGraphTools, a Python library for constructing and analysing bond graph models. We use examples from biochemistry to illustrate how BondGraphTools can be used to automate model construction in systems biology while maintaining consistency with the laws of physics.

Graphical abstract

Appendix A: Parameters for multisite phosphorylation

When only kinetic data are available, the energetic parameters cannot be determined uniquely due to the relative nature of chemical potential and the absence of reference values. This parameter undeterminacy has been discussed in previous papers, where sets in parameter space with equivalent kinetic behaviours were defined [15, 21, 24]. However, for this model, we will avoid the issue of parameter uncertainty for simplicity and refer the reader to the above articles for further information.

We make the following assumptions on the energetic parameters:

• We set $\mu _\text {ATP} = 50\ {\hbox {kJ}/\hbox {mol}}$, $\mu _\text {ADP} = 0\ {\hbox {kJ}/\hbox {mol}}$ and $\mu _\text {Pi} = 0\ {\hbox {kJ}/\hbox {mol}}$ since the free energy of ATP hydrolysis $\varDelta G = \mu _\text {ADP}+\mu _\text {Pi}-\mu _\text {ATP}$ typically varies between $-50$ kJ/mol and $-60$ kJ/mol in physiological contexts.

• We set $K_E = K_F = K_{S_0} = 1\ {\hbox {nM}^{-1}}$ (corresponding to a free energy of formation of zero at a standard concentration of 1 nM).

• We assume that the energetics of phosphorylation are identical for each site, i.e. $K_{S_{i+1}} = \gamma K_{S_{i}}$. Hence, in conjunction with the above assumption, $K_{S_i} = \gamma ^i$. For reasons that we justify later, we have chosen $\gamma = 3.47 \times 10^4$.

Note that implicit in the above assumptions are that ADP, Pi, $\mathrm {S}_0$, E and F are the “reference species” whose chemical potentials are used to define the potentials of the rest of the species. It is relatively straightforward to adapt the assumptions when the more standard chemical free energies of formation are used, in which the chemical potentials of each species are referenced to their constituent elements in their standard states.

The rest of the parameters can be determined from the kinetic parameters in Thomson and Gunawardena [30]. Omitting chemostats, each of the enzyme-catalysed reactions follows the generic structure

$$\begin{aligned} \mathrm {S}_u + \mathrm {X} \mathrel {\mathop {\rightleftharpoons }^{a}_{b}} \mathrm {XS}_u \mathrel {\mathop {\rightleftharpoons }^{c}_{d}} \mathrm {S}_v + \mathrm {X} \end{aligned}$$

(A.1)

where $\mathrm {S}_u$ and $\mathrm {S}_v$ are the input substrates, $\mathrm {X = E} \text { or } \mathrm {F}$ and $\mathrm {XS}_u$ is the complex. The kinetic constants of Thomson and Gunawardena are given in Table 3. Since the reaction scheme does not explicitly account for chemostats, they have been absorbed into the kinetic parameters.

We need to determine the rate constants of both reactions and the species constant of $\mathrm {XS}_u$. These can be determined using the following equations, which are derived by writing kinetic parameters in terms of the energetic parameters:

$$\begin{aligned} r_a&= a/\left( K_{S_u} K_E e^{A_\text {cs}^f/RT}\right) \end{aligned}$$

(A.2)

$$\begin{aligned} K_{\mathrm {XS}_u}&= b/r_a \end{aligned}$$

(A.3)

$$\begin{aligned} r_b&= c/K_{\mathrm {XS}_u} \end{aligned}$$

(A.4)

where $A_\text {cs}^f$ is the potential of the reactant chemostat (if present). Similarly, we can define $A_\text {cs}^r$ as the potential of the product chemostat. Thus, for kinases (E), $A_\text {cs}^f = \mu _\text {ATP}$ and $A_\text {cs}^r = \mu _\text {ADP}$. For phosphatases (F), $A_\text {cs}^f = 0$ and $A_\text {cs}^r = \mu _\text {Pi}$.

The remaining kinetic constant $d = r_b S_v A_\text {cs}^r$ is assumed to be zero in Thomson and Gunawardena, but this is impossible in a real system, which requires all reactions to be reversible. Thus, we choose the final parameter $\gamma$ to minimise the magnitude of these rate constants, or more precisely, we minimise their squared sum

$$\begin{aligned} J = \sum _{i=1}^4 \left( d_i^E\right) ^2 + \sum _{i=1}^4 \left( d_i^F\right) ^2 \end{aligned}$$

(A.5)

where $d_i^X$ is the parameter d for the ith kinase ($\mathrm {X=E}$) or phosphatase ($\mathrm {X=F}$) reaction. A value of ${\gamma = 3.47 \times 10^4}$ will minimise J. The full list of energetic parameters is given in Tables 4 and 5.

Table 3 Kinetic parameters used in the Thomson and Gunawardena model [30]

Table 4 Species parameters K for the four-site phosphorylation example

Table 5 Reaction parameters r for the four-site phosphorylation example