Analytic framework for a stochastic binary biological switch

Guilherme C. P. Innocentini, Sarah Guiziou, Jerome Bonnet, and Ovidiu Radulescu

Phys. Rev. E 94, 062413 – Published 28 December 2016

Abstract

We propose and solve analytically a stochastic model for the dynamics of a binary biological switch, defined as a DNA unit with two mutually exclusive configurations, each one triggering the expression of a different gene. Such a device has the potential to be used as a memory unit for biological computing systems designed to operate in noisy environments. We discuss a recent implementation of this switch in living cells, the recombinase addressable data (RAD) module. In order to understand the behavior of a RAD module we compute the exact time-dependent joint distribution of the two expressed genes starting in one state and evolving to another asymptotic state. We consider two operating regimes of the RAD module, a fast and a slow stochastic switching regime. The fast regime is aggregative and produces unimodal distributions, whereas the slow regime is separative and produces bimodal distributions. Both regimes can serve to prepare pure memory states when all cells are expressing the same gene. The slow regime can also separate mixed states by producing two subpopulations, each one expressing a different gene. Compared to the genetic toggle switch based on positive feedback, the RAD module ensures more rapid memory operations for the same quality of the separation between binary states. Our model provides a simplified phenomenological framework for studying RAD memory devices and our analytic solution can be further used to clarify theoretical concepts in biocomputation and for optimal design in synthetic biology.

Received 15 June 2016
Revised 18 November 2016

DOI:https://doi.org/10.1103/PhysRevE.94.062413

Physics Subject Headings (PhySH)

Genomics Proteomics

Physics of Living Systems

Authors & Affiliations

Guilherme C. P. Innocentini

Departamento de Matemática Aplicada, Universidade de São Paulo, São Paulo, Brazil

Sarah Guiziou and Jerome Bonnet

CBS, CNRS UMR 5048 - UM - INSERM U 1054, Montpellier, France

Ovidiu Radulescu

DIMNP, UMR CNRS 5235, University of Montpellier, Montpellier, France

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Vol. 94, Iss. 6 — December 2016

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Images

Figure 1
Schematic architecture and operational principle of the recombinase addressable data (RAD) module. (a) The DNA register is composed of a DNA sequence containing a promoter (driving transcription) flanked by the two recombination sites recognized by the integrase (Int). (b) In state 0, the promoter drives expression of GFP. Integrase catalyzes the SET reaction, inverting the DNA register and enables transition towards state 1, in which the promoter has been inverted and now drives transcription of RFP. Concomitant expression of integrase and excisionase (Xis) catalyze the RESET reaction by enabling transition from state 1 to state 0.
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Figure 2
Computed steady-state distributions of GFP and RFP for (a) pure $p_{1} = 1$ red state (all cells express RFP) and $ε = 0.5$ (slow switching); (b) pure $p_{0} = 1$ green state (all cells express GFP) $ε = 0.5$ (slow switching); (c) mixed $p_{0} = p_{1} = 0.5$ unimodal state (all cells express both GFP and RFP) and $ε = 10$ (fast switching); (d) mixed $p_{0} = p_{1} = 0.5$ bimodal state (half of the cells express only GFP and half only RFP) and $ε = 0.5$ (slow switching). In logarithmic variables, the double integral of the probability distribution is normalized to one. The remaining parameters are: $Δ = 40$ and $ρ = 1$ . The probabilities $p_{r}, p_{g}, p_{r g}$ indicate the fractions of cells with $ln (g) < 2, ln (r) > 2$ ; $ln (g) > 2, ln (r) < 2$ and $ln (g) > 2, ln (r) > 2$ , respectively, where the threshold 2 corresponds to the middle of the dynamical expression interval.
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Figure 3
Time-dependent distributions of a RAD unit (time is measured in units of $ρ^{- 1}$ ). In (a), (b) we illustrate setting of a pure state. The $t = 0$ state corresponds to a pure distribution with $\tilde{p_{0}} = 1$ (green) and switch flexibility $\tilde{ε} = 0.5$ . During set operation (for $t > 0$ ) the unit evolves towards a pure state $p_{1} = 1$ (red) but with different switch parameters: $ε = 10$ for (a) (fast switching) and $ε = 0.5$ for (b) (slow switching). In (c) we illustrate separation of a unimodal mixed state. An unimodal mixed state with $\tilde{p_{0}} = \tilde{p_{1}} = 0.5$ and $\tilde{ε} = 10$ (fast switching) is transformed to a bimodal mixed state by lowering the switch flexibility, $ε = 0.1$ (slow switching). Also, in (c) one can see that for fast switching the phenotype is unimodal: bacteria express both GFP and RFP. The $t = 10$ phenotype, is bimodal: $50 %$ of the cells express RFP and $50 %$ express GFP. The remaining parameters are: $N_{g} = N_{r} = 40$ and $ρ = 1$ . The rightmost columns (d)–(f) show the time dependence of the state occupancies.
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Figure 4
The RAD model compared to the toggle switch model from Ref. [17] in the symmetric situation when the mean expression is the same for the two competing genes. Parameter fitting of the RAD model enforce that the initial and final bivariate probability distributions and the transient means are the same for the two models. The negative correlation between genes indicating the separation between memory states reaches more rapidly its asymptotic value for the RAD switch. The toggle switch time series were obtained by Monte Carlo simulations. Fitting of the RAD model used analytic expressions, but the final estimates were checked by comparison to Monte Carlo simulations. The parameters of the toggle switch model are $k = 0.05, d = 0.005, b = 0.1, u = 0.2$ for $t = 0$ and $u = 0.05$ for $t > 0$ . The fitted parameters of the RAD module are $ρ = 0.005, f = h = 0.07, k_{1} = k_{2} = 0.057$ for $t = 0$ and $f = h = 0.0015, k_{1} = k_{2} = 0.053$ for $t > 0$ , suggesting that a change of the unbinding parameter in the toggle switch model can be reproduced by a change of the switching rates in the RAD switch.
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