Realizing logic gates with time-delayed synthetic genetic networks

Abstract

We demonstrate the realization of fundamental logic operations, as well as a memory element, with engineered delayed synthetic gene networks. Further, we investigate the effect of time delay in different kinds of processes, on the operational range of this biological logic gate. We show that this delay can either enhance or diminish logic behavior, depending on its functional form. Lastly, we show that the desired response to inputs can be induced, even in the absence of noise, by time delay alone.

Appendix

Time delay in the degradation process is given by the following equation:

$$\begin{aligned} \frac{dx(t)}{dt} =& \frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6} \\ &{} - \gamma x(t-\tau) + D\eta(t). \end{aligned}$$

(6)

The presence of small delay changes the position and depth of the wells in the bistable potential.

The dynamics in Eq. (6) is a non-Markovian process. Using the probability density approach, the non-Markovian process can be reduced to a Markovian process, and the approximate time-delay Fokker–Plank equation is

$$ \frac{\partial P(x,t)}{dt} = -\frac{\partial[h_{\mathrm{eff}}P(x,t)]}{\partial x}+D\frac{\partial^2P(x,t)}{\partial^2x}. $$

(7)

Here the conditional average h _eff is

$$ h_{\mathrm{eff}} =\int_{-\infty}^{\infty} h(x,x_\tau)P(x_\tau, t-\tau| x,t) $$

(8)

where

$$\begin{aligned} &{x_\tau= x(t-\tau),}\\ &{h(x,x_\tau) = \frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6} - \gamma x_\tau,}\\ &{h(x) = \frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6} - \gamma x.} \end{aligned}$$

P(x _τ ,t−τ|x,t) is the zeroth order approximate Markovian transition probability density

$$\begin{aligned} &{P(x_\tau, t-\tau| x,t) = \frac{1}{\sqrt{4\pi D\tau}} }\\ &{\quad \times\exp \biggl(- \frac{(x_\tau-x-h(x)\tau)^2}{4D\tau} \biggr). } \end{aligned}$$

(9)

Substituting Eq. (9) into Eq. (8), we obtain

$$\begin{aligned} h_{\mathrm{eff}} =& (1-\gamma\tau)\frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6} \\ &{} - (1-\gamma\tau)\gamma x_\tau. \end{aligned}$$

(10)

So, the effective Langevin equation for Eq. (7) becomes

$$\begin{aligned} \frac{dx(t)}{dt} =& (1-\gamma\tau)\frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6} \end{aligned}$$

(11)

$$\begin{aligned} &{}- (1-\gamma\tau)\gamma x_\tau+ D\eta(t). \end{aligned}$$

(12)

The effective time-delay potential function of Eq. (6) is

$$\begin{aligned} U_{\mathrm{eff}} =& -(1-\gamma\tau)\int\frac{m(1+x^2+\alpha\sigma_1 x^4)}{1+x^2+\sigma_1 x^4+\sigma_1\sigma_2 x^6}\,dx \end{aligned}$$

(13)

$$\begin{aligned} &{}+ (1-\gamma\tau)\int\gamma x\,dx. \end{aligned}$$

(14)

The effect of time delay on the effective potential given by Eq. (14) is shown in Fig. 1.