Thermodynamically-consistent, reduced models of gene regulatory networks

Abstract

Synthetic biology aims to engineer novel functionalities into biological systems. While the approach has been widely applied to single cells, the scale of synthetic circuits designed in this way is limited by factors such as resource competition and retroactivity. Synthetic biology of cell populations has the potential to overcome some of these limitations by physically isolating synthetic genes from each other. To rationally design cell populations, we require mathematical models that link between intracellular biochemistry and intercellular interactions. The interfacing of agent-based models with systems biology models is particularly important in understanding the effects of cell heterogeneity and cell-to-cell interactions. In this study, we develop a model of gene expression that is suitable for incorporation into agent-based models of cell populations. To be scalable to large cell populations, models of gene expression should be both computationally efficient and compliant with the laws of physics. We satisfy the first requirement by applying a model reduction scheme to translation, and the second requirement by formulating models using bond graphs. We show that the reduced model of translation faithfully reproduces the behaviour of the full model at steady state. The reduced model is benchmarked against the full model, and we find a sub-stantial speedup at realistic protein lengths. Using the modularity of the bond graph approach, we couple separate models of gene expression to build models of the toggle switch and repressilator. With these models, we explore the effects of resource availability and cell-to-cell heterogeneity on circuit behaviour. The modelling approaches developed in this study are a bridge towards rationally designing collective cell behaviours such as synchronisation and division of labour.

Energy-based Analysis of Biochemical Oscillators Using Bond Graphs and Linear Control Theory

“…A synthetic oscillator using DNA transcription, known as the repressilator, was introduced by Tyson et al [13] and a bond graph model of the repressilator is given by Pan et al [42]. The repressilator is a complex system and was simplified by giving the mRNAs a length of 8 : the corresponding bond graph model [42] has 53 species, of which 7 are chemostats, giving 46 states. However, as shown below, the linearised system of equation ( 9), and thus L 0 (14), can be well-approximated by a third order system: pole/zero cancelling pairs are removed using the minimal realisation function minreal function and order further reduced using the balanced order reduction balred function of the control systems library python-control [58].…”

“…However, as shown below, the linearised system of equation ( 9), and thus L 0 (14), can be well-approximated by a third order system: pole/zero cancelling pairs are removed using the minimal realisation function minreal function and order further reduced using the balanced order reduction balred function of the control systems library python-control [58]. The parameters are given in [42].…”

“…The Bode diagram is based on the frequency response L(jω) of L(s) and thus can be used for system of arbitrarily high order; the root locus provides further insights but is restricted to low-order systems or, as discussed in § 5, a reduced order system. These ideas are illustrated by the example of this section; the Glycolytic Oscillator of Sel'kov [11] is discussed in § 4 and the Repressilator of Tyson et al [59], as modelled by Pan et al [42] is discussed in § 5.…”

“…As biochemical oscillators may arise from complex biochemical systems, a modular hierarchical bond graph approach [39,[52][53][54] is used in this paper. Oscillators have been considered in the bond graph context by Stramigioli and van Dijk [55], and a bond graph model of a repressilator [13] is developed by Pan et al [42].…”

“…A key feature of biochemical oscillators is that the oscillation can only continue with a sustained supply of energy [41,42]. This key feature can be included in the oscillator model using the bond graph approach [43][44][45][46][47] introduced into Systems Biology by Oster et al [48,49] and further developed by Gawthrop and Crampin [50]; an introduction to the approach is given by Gawthrop and Pan [51].…”