Existing recombinase-based models

As cellular memory emerges as a defining element of higher-level synthetic biological systems, the characterisation of the requisite parts will command significant atten- tion. Hence, predictive analysis of recombinase-based genetic switches is necessary to provide engineers with reliable operational profiles. Achieving this goal is pos- sible through the inevitable progression in the efficacy of experimental procedures due to technological advancements. However, mathematical modelling approaches have the potential to provide insights that may never be physically possible in the laboratory. Wider acceptance of the merits of mathematical models in biology and increased efforts to expand collaborative experimental and computational research is therefore central to synthetic biological circuit design.

The earliest proposed model of DNA recombination in the literature pro- vides kinetic analysis of two distinct recombinases known as FLP (flippase) and Cre [Ringrose et al., 1998]. The model captures a simplistic overview of DNA deletion via a series of reversible reactions corresponding to monomeric recombinase binding to DNA attachment sites, synaptic complex formation, recombination and dissocia- tion. The universal reversibility of the reactions modelled aligns with FLP and Cre being tyrosine recombinases that exhibit bidirectional recombination and hence the model also captures the insertion of the circular DNA product back into the genetic sequence. Two pairs of reactions corresponding to two distinct aspects of synapse formation were unable to be determined experimentally, resulting in four unknown model parameters; the remaining four model parameters were established via ex- perimentation. The model was optimised to infer the four unknown parameters through fitness function minimisation. As a result, a number of dynamical proper- ties were validated including that Cre has a higher binding affinity than FLP and thus the synaptic complex is more stable for Cre which was thought to explain the 100% deletion efficiency of FLP compared to the maximum 75% excision efficiency of Cre. Furthermore, insertion was shown to be inefficient given the bidirectionality of tyrosine-mediated recombination that results in unwanted ‘re-deletion’ and, al- though it may seem intuitive that insertion efficiency would benefit from increased DNA binding affinity and rate of synapsis, such conditions are in fact detrimental to insertion efficiency since they favour re-deletion [Ringrose et al., 1998]. There- fore, this modelling investigation succeeded in highlighting both the optimal model

parameter set and the operational faults that render the mechanism unsuitable for use as a cellular memory unit.

A validated model of serine DNA recombination did not arrive until fourteen years later [Bonnet et al., 2012]. This serine model adopts a simplistic, black box approach akin to that of [Ringrose et al., 1998] however, in this case, the specific focus is DNA inversion as opposed to deletion and insertion. The model captures

in vivoDNA recombination reactions relating to the serine integrase gp35 and the RDF gp47 from the bacteriophage Bxb1 and is referred to as a rewritable RAD module. In contrast to their tyrosine cousins, serine recombinases mediate unidirec- tional recombination which makes them conducive to inducible regulation of gene expression [Bonnet et al., 2013].

The model accounts for the dynamical behaviour summarised in Fig. 3.2: integrase and RDF are both expressed in monomeric form in solution, but integrase

k-i ki ai ax yx yi yi kdi k-di k-dix kdix k-di kdi k-dix kdix Integrase monomer Integrase dimer RDF monomer Integrase-RDF tetramer kc kc Excision Integration

Empty set, representing protein degradation

DNA with AttB and AttP sites (BP state) DNA with AttL and AttR sites (LR state)

Figure 3.2: The RAD module DNA recombination reaction network taken from Bonnet et al. [2012].

alone undergoes dimerisation in solution. That is, pairs of integrase monomers bind together in solution to form dimeric complexes. Integrase dimers are then able to bind specifically to attB and attP sites on the free DNA substrate. One integrase dimer bound to each attachment site is necessary and sufficient to mediate the primary inversion event, referred to as integration since it involves integrase only. This causes the double stranded break in the DNA and the subsequent re-

ligation of opposing ends of the intermediate genetic fragment that gives rise to the composite attL and attR sites. RDF monomers only bind to integrase dimers already bound to DNA, forming a synaptic tetramer. The binding of two RDF monomers to each integrase:DNA synaptic complex is necessary and sufficient to mediate the secondary inversion event, referred to as excision since both integrase and RDF are involved. RDF is also able to bind integrase:DNA complexes in the BP state, thus inhibiting integration; there is no RDF binding to integrase in solution [Bonnet et al., 2012]. The BP and LR DNA states are tagged with GFP and RFP respectively to provide clear readout of the recombination efficiency of the system. The model describesin vivo DNA recombination and, as such, the expression and degradation of recombinase proteins represent a key dynamical element;in vitro studies such as [Ringrose et al., 1998] are void of environmental pressures and hence concentrations of recombinases are synthesised experimentally. In all, the model is comprised of nine distinct variables and eight parameters representing the integration-excision DNA inversion mechanism.

The model was used to identify the operational properties of the RAD module required for delivering digital information storage. The key feature of the system is the efficiency of switching between DNA states. This is ascertained by estab- lishing the concentration every molecular entity in the DNA state of interest and computing the evolution of the summed total over time. The sum of the relevant concentrations is referred to as the total register of the system and is calculatedin silicoby summing each ODE corresponding to molecular entities in the same DNA state. Experimentally, the total register of the system is measured directly as the intensity of GFP or RFP output, with increased fluorescence signifying increased recombination efficiency. Presuming that the module initially adopts the BP state, the ‘set’ operation constitutes an efficient ‘on’ switch transitioning the system to the LR state via induced integrase and an absence of excisionase; the ‘reset’ operation therefore constitutes an efficient ‘off’ switch that reproduces the initial BP state via induced integrase and RDF.

The ability of the system to demonstrate robust ‘hold’ states, whereby the most recent set or reset operation is maintained in the absence of inducer, is an- other important criterion with regards to temporal control of the module. Switching efficiency is dependent on the concentrations of integrase and RDF in the system however, these quantities are difficult to determine numerically. As a result, the relative ratios of recombinase expression and degradation rates were examined with respect to the percentage switching efficiency they produced. A vast array of ratios were tested to provide expected dynamical responses with which to direct exper-

imentation. That is, no experimental data was used to inform the selection of individual parameter values or to provide specific outputs for model validation pur- poses, and thus the model was suitably non-dimensionalised to offer operational conditions to be emulated experimentally. All model parameters corresponding to protein:DNA interactions were set equal to 1 with the basal and induced ratios of recombinase expression and degradation rates set at 0.1 and 10 respectively. RAD module operations were initially investigated in separation; set operations were iso- lated with approximately 95% switching efficiency and a robust hold state in the absence of RDF, which is intuitive given that the set function encompasses the in- tegration reaction that is solely mediated by integrase. Isolated investigation of the reset operation revealed reversibility in vivo despite experimental evidence of approximately 100% switching efficiency regarding Bxb1 integrase and excisionase

in vitro [Ghosh et al., 2006]. This highlights the influence of noisy cellular condi- tions on recombinase expression and degradation capable of causing re-integration of the reset inverted genetic sequence that would be observed as reversibility and, ultimately, system failure.

A full set-reset cycle was also attempted however, it was observed that the majority of set and reset functions that exhibited high efficiency in isolation were unable to give rise to sufficiently robust full cycles required of a digital storage module. For example, the high concentrations of RDF required for efficient reset operations is sufficient to corrupt efficient set operations since the presence of RDF is inhibitory in the integration reaction. It was concluded that the model is capable of identifying the appropriate ranges of recombinase expression and degradation rates required of a reliable set-hold-reset-hold operative cycle [Bonnet et al., 2012]. That said, assembling the appropriate genetic constructs to realise this predicted functionality proved to be particularly challenging due to difficulties associated with inducing the expression of recombinase proteins and the timing of such induction events withinE. coli. Establishing a functional RAD module was eventually achieved through an ad hoc approach involving ∼400 trials, reiterating the aforementioned experimental limitations as well as the ramifications of noise and stochastic biological processes. Given the uncertainty surrounding particular aspects of the system, such as reversibility of the excision reaction and the search for the conditions necessary for optimal RAD module operation, it is clear that modelling DNA recombination warrants further consideration.

3.2

Formulating a mechanistic model of

in vitro

RAD