## Abstract

The expression of a gene is characterised by its transcription factors and the function processing them. If the transcription factors are not affected by gene products, the regulating function is often represented as a combinational logic circuit, where the outputs (product) are determined by current input values (transcription factors) only, and are hence independent on their relative arrival times. However, the simultaneous arrival of transcription factors (TFs) in genetic circuits is a strong assumption, given that the processes of transcription and translation of a gene into a protein introduce intrinsic time delays and that there is no global synchronisation among the arrival times of different molecular species at molecular targets.

In this paper, we construct an experimentally implementable genetic circuit with two inputs and a single output, such that, in presence of small delays in input arrival, the circuit exhibits qualitatively distinct observable phenotypes. In particular, these phenotypes are long lived transients: they all converge to a single value, but so slowly, that they seem stable for an extended time period, longer than typical experiment duration. We used rule-based language to prototype our circuit, and we implemented a search for finding the parameter combinations raising the phenotypes of interest.

The behaviour of our prototype circuit has wide implications. First, it suggests that GRNs can exploit event timing to create phenotypes. Second, it opens the possibility that GRNs are using event timing to react to stimuli and memorise events, without explicit feedback in regulation. From the modelling perspective, our prototype circuit demonstrates the critical importance of analysing the transient dynamics at the promoter binding sites of the \({\mathsf {DNA}}\), before applying rapid equilibrium assumptions.

### Keywords

- Gene regulation
- Stochastic modelling
- Long lived transients
- DNA looping

Tatjana Petrov’s research was supported by SNSF Advanced Postdoc. Mobility Fellowship grant number \(P300P2\_161067\), the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and the DFG Centre of Excellence 2117 ‘Centre for the Advanced Study of Collective Behaviour’ (ID: 422037984). Claudia Igler is the recipient of a DOC Fellowship of the Austrian Academy of Sciences. Thomas A. Henzinger’s research was supported in part by the Austrian Science Fund (FWF) under grant Z211-N23 (Wittgenstein Award).

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## Notes

- 1.
Models used in this paper will count 23 and 6 distinct \({\mathsf {DNA}}\) binding configurations.

- 2.
The ratio between the binding and unbinding rate.

- 3.
Notice that these six parameter combinations are different than those used for the model with looping.

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## Acknowledgements

We are very grateful to Moritz Lang, Tiago Paixao and Jakob Ruess, for their feedback during the manuscript preparation.

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## Appendices

### Appendix 1.A Parameter Values

Table 1 lists the parameter ranges used for our case study example. We next explain the choice of each of the parameters with respect to their biological context.

**1.A.1 Stochastic Scaling Constant**

The stochastic scaling of rates and concentrations is done with a standard scaling rate for *E. coli* cell \(N= 10^9\) [26].

**1.A.2 Protein Production and Degradation**

The protein production is taken 0.5 molec.s\(^{-1}\) ([41], caption of Fig. 2) and the degradation rate is taken 0.001 s\(^{-1}\) (corresponding to the halflife of \(12\min \), consistent with [26]).

**1.A.3 RNAP Rates**

On rate, off rate and number of RNAP molecules are consistent with the orders of values reported in [5, 14, 35].

**1.A.4 Activator**

The activation mechanism is inspired by the activation of the \(\mathsf {PRM}\) promoter in the lysogenic state by protein \({\mathsf {CI}}\) in the regulation of \(\lambda \)-phage: \({\mathsf {CI}}\) competes with \({\mathsf {Cro}}\) to bind to the promoter sites, and, when bound, it recruits \({\mathsf {RNAP}}\) (increases \(\mathsf {PRM}\) activity). The mechanism with looping, explained at mechanistic detail level in [35], contains three left and three right operators, leading to 1200 different DNA binding states. We model a mechanism with two states for the activator without looping (‘bound’ or ‘not bound’) and with four binding states for the activator with looping (see Fig. 5). The on-rate, off-rate well as the number of activators is taken from [35] (page 82) When activator is bound, the recruitment of \({\mathsf {RNAP}}\) is increased by factor 10 or 50 ([34] and [27] respectivelly) (Table 3).

**1.A.5 Repressor**

The repression mechanism is inspired by the well-studied transcriptional regulation, there is a word missing after transcription of the \({\mathsf {lac}}\) operon, the repressor \({\mathsf {LacI}}\). We take the binding and unbinding rates for the repressor from ([39], Fig. 4).

**1.A.6 Looping Rates**

The stability of the looped state is incorporated in the model by scaling down the unlooping rate. We choose the scaling factors of 100 and 1000 based on the computation of the ratio of dissociation rates for the models with and without looping ([40], Table 1; parameter *a* in [39]). The mechnism proposed in, eg. [39] suggests that the looping increases the binding rate (due to increased local concentration of TFs), while leaving the unbinding rate unchanged. As the scaled on-rates may exceed theoretical limit for diffusion-limited reactions, in our model, we incorporate the same effect by leaving the binding rate identical, and scaling down the unlooping rate.

### Appendix 1.B Kappa Models

### Appendix 1.C Supplementary Theory and Proofs

**1.C.1** **Deterministic Limit**

In the continuous, deterministic model of a chemical reaction network, the state \(\mathbf {z}(t) = (z_1,\ldots ,z_n)(t)\in {\mathbb R}^n\) is represented by listing the concentrations of each species. The dynamics is given by a set of differential equations in form

where \(k_j\) is a deterministic rate constant, computed from the stochastic one and the volume \(N\) according to \(k_j := c_jN^{|{\mathbf {a}}_{j}|-1}\) (\(|\mathbf {x}|\) denotes the 1-norm of the vector \(\mathbf {x}\)). The deterministic model is a limit of the stochastic model when all species in a reaction network are highly abundant [19]: by scaling the species multiplicities with the volume: \(Z_i(t) = X_i(t)/N\), adjusting the propensities accordingly, in the limit of infinite volume \(N\rightarrow \infty \), the scaled process \({\mathbf Z}(t)\) follows an ordinary differential Eq. (4).

**1.C.2** **Expected Output in the Transient**

The CME implies that the expectation of the marginal distribution of \(\{X_t\}\) satisfies the equations

To check (5), observe a transition from \(\mathbf {x}\) to \(\mathbf {x}+\varvec{\nu }_j\). The term \(\lambda _{j}(\mathbf {x})\mathsf {P}(\mathbf {X}_t=\mathbf {x})\) appears exactly once when summing up for the state \(\hat{\mathbf {x}} = \mathbf {x}\) as the outflow probability, and exactly once when summing up for the state \(\hat{\mathbf {x}} = \mathbf {x}+\varvec{\nu }_j\), as the inflow probability. This gives the term \((\mathbf {x}+\varvec{\nu }_j)-\mathbf {x}=\varvec{\nu }_j\cdot \lambda _{j}(\mathbf {x})p^{(t)}(\mathbf {x})\). It is worth noting that, upon scaling the rate constants, the equations for \(\mathsf {E}(\mathbf {X}_t)\) are equivalent to (4) only if all rate functions are linear, that is, when all reactions are unimolecular.

**1.C.3** **Proof for Lemma** 1

We first notice that the process \(\mathbf {X}(t)|_{\mathsf {D}}\) is indeed Markovian, because all states of \(\mathbf {X}(t)\) projected to the same state in \(\mathbf {X}(t)|_{\mathsf {D}}\) are behaviourally indistinguishable (bisimulation equivalent), due to rates between lumped states not depending on protein count. From (5), it follows that

where \(\langle x_{\mathsf {D}1j}(t)\rangle \) denotes the expected value of being in one of the active promoter configurations. The latter equals (2), since in every reachable state \(\mathbf {x}\in (D_{\texttt {0}}\uplus D_{\texttt {1}})\times \{0,1,\ldots \}\), exactly one \({\mathsf {DNA}}\) configuration takes value 1.

### Appendix 1.D Supporting Figures

See Fig. 13.

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Guet, C., Henzinger, T.A., Igler, C., Petrov, T., Sezgin, A. (2019). Transient Memory in Gene Regulation. In: Bortolussi, L., Sanguinetti, G. (eds) Computational Methods in Systems Biology. CMSB 2019. Lecture Notes in Computer Science(), vol 11773. Springer, Cham. https://doi.org/10.1007/978-3-030-31304-3_9

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