Analysis of primitive genetic interactions for the design of a genetic signal differentiator

Abstract We study the dynamic and static input–output behavior of several primitive genetic interactions and their effect on the performance of a genetic signal differentiator. In a simplified design, several requirements for the linearity and time-scales of processes like transcription, translation and competitive promoter binding were introduced. By experimentally probing simple genetic constructs in a cell-free experimental environment and fitting semi-mechanistic models to these data, we show that some of these requirements can be verified, while others are only met with reservations in certain operational regimes. Analyzing the linearized model of the resulting genetic network, we conclude that it approximates a differentiator with relative degree one. Taking also the discovered nonlinearities into account and using a describing function approach, we further determine the particular frequency and amplitude ranges where the genetic differentiator can be expected to behave as such.


Supplementary Data
A Data pre-processing In this section we denote the data obtained in the experiments discussed in Sections 4.1 and 4.2 with y ∈ R. There are mainly three issues with these data, exemplarily depicted in Fig. 12 A and B as grey crosses. i First, the measurements are corrupted with noise, i.e.
where f (t) is some deterministic process generating the noise-free data and ρ the gaussian noise. This is particularly the case for the malachite green fluorescence measurements. Second, the time points at which the measurements are obtained are not uniformly spaced due to inconsistent preparation times of the experiments. This leads to a heterogeneous distribution of the measurements along the time axis. And last, for malachite green, a substantial part of the measured signal stems from some background signal caused by unbound malachite green, leading to the need of correcting the signals by subtracting the background part. However, due to the non-uniform temporal spacing of the measurements, a correction of the background requires some kind of model or interpolation scheme of the data.
We therefore assume that the measurement noise ρ is i.i.d. and model the timeseries for each experimental condition as a gaussian process, i.e. y ∼ GP µ, k(t, t , θ) + 2 δ tt where µ ∈ R is a constant mean, k is chosen as a squared exponential kernel parametrized with θ and δ tt being the Kronecker delta. Now let y (ctrl) and y (e) be the fitted gaussian processes of a control experiment without any DNA and some other experimental condition with predicted mean µ (ctrl) , µ (e) and predicted standard deviations σ (ctrl) , σ (e) as derived in [34] and depicted in Fig. 12 as dashed blue lines (mean) and light blue shaded area (standard deviation). The background corrected signalỹ (e) is then determined byμ like depicted in Fig. 12 C.
holds. Now, 0 ≤ A:B i for both i = [1, 2] follows directly from v and with it can be seen that A:B 2 violates (25). It remains to realize that to conclude that A:B 1 is the only biologically meaningful solution. vi

D pTar promoter characterization
The time series data of the pTar characterization experiment is depicted in Fig. 13.

E Limitations of the Describing Function approach
The way the Describing Function approach has been used in Section 4.3.1, we assume that higher harmonics can be neglected in the output signal. This, however, may not the be case for every combination of parameters A, A 0 and ω of the input signal given in (22) If p rel ≈ 1, this indicates that higher harmonics can be neglected. As shown in Fig. 14, this is not always the case. For large values of A A 0 and input frequencies in the range ω ∈ [10 −2 , 10 0 ], the value of p rel drops below 0.8, suggesting that the output signal will significantly be influenced by frequency components other than the basis frequency ω. This means that the output signal will have a distorted shape. viii