Engineering Scalable Digital Circuits From Non-digital Genetic Components

Abstract

Synthetically engineered single-cellular biological systems could be designed to classify patterns of chemical signals with high specificity and invoke appropriate responses. This requires cells to produce accurate logical computation over their multiple inputs and then trigger cellular response in a binary form like the signals YES and NO. However, current engineered biological systems, as a rule, are built from components like combinatorial promoters that, although displaying ’logic like’ capabilities, fall short of supporting true binary (Boolean) computation. Consequently misclassification of inputs or errors in processing commonly occur that in turn lead to an incorrect cellular response. Here we show how that increased nonlinearity combined with noise suppression leads to genetic circuits capable of true Boolean logic operation able to support scalable logic circuit design.

Appendix

Let the circuit $D \dashv E \dashv Z$ represents a double inversion module, where the protein D is repressing the synthesis of the protein E, and the protein E is repressing the synthesis of the protein Z. According to the thermodynamic model of transcription [2], the stationary concentration of the protein E is dependent on the concentration of D as following,

$$\begin{aligned} C_{E}=\omega \;\frac{\varOmega ^{m}}{C_{D}^{m}+\varOmega ^{m}}, \end{aligned}$$

(4.12)

where the coefficients $\omega$ and $\varOmega$ are some constants, m is integer, $m=1,\; 2,\; 3,\; 4,\;\ldots$. It is assumed that $\omega >0$ and $\varOmega >0$. Here $m=1$ corresponds to a case of the protein D that is a simple repressor of E. In contrast to $m=1$, the case $m>1$ means the protein D is assembled into a complex to be the repressor of E. For examples, $m=2$ corresponds to a dimer, and $m=4$ means the complex is a tetramer, e.t.c. We can write a similar equation for a relationship between the concentrations of the proteins Z and E,

$$\begin{aligned} C_{Z}=\lambda \; \frac{\varLambda ^{i}}{C_{E}^{i}+\varLambda ^{i}}, \end{aligned}$$

(4.13)

where the coefficients $\lambda$ and $\varLambda$ are some positive constants, $\lambda >0$ and $\varLambda >0$, the index i is integer, $i=1,\; 2,\; 3,\; 4,\;\ldots$.

By substitution Eq. (4.12) into Eq. (4.13), we obtain the transfer function of the double inversion module,

$$\begin{aligned} C_{Z}= & {} \lambda \frac{\varLambda ^{i} \left( C_{D}^{m}+\varOmega ^{m} \right) ^{i}}{\omega ^{i}\varOmega ^{i m}+\varLambda ^{i}\left( C_{D}^{m}+\varOmega ^{m} \right) ^{i}} \nonumber \\= & {} \lambda \;\frac{\varLambda ^{i} \sum _{k=0}^{i}\left( {\begin{array}{c}i\\ k\end{array}}\right) C_{D}^{m k}\varOmega ^{(i-k)m}}{\omega ^{i}\varOmega ^{i m}+\varLambda ^{i} \sum _{k=0}^{i}\left( {\begin{array}{c}i\\ k\end{array}}\right) C_{D}^{m k}\varOmega ^{(i-k)m}}, \end{aligned}$$

(4.14)

where $\left( {\begin{array}{c}i\\ k\end{array}}\right)$ are the binomial coefficients.

In case $i=1$, Eq. (4.14) can be simplified,

$$\begin{aligned} C_{Z}= \frac{\lambda }{\varLambda } \; \frac{ C_{D}^{m}+\varOmega ^{m}}{C_{D}^{m} + \left( \dfrac{\omega }{\varLambda }+1\right) \varOmega ^{m} }. \end{aligned}$$

(4.15)

It is easy to find a similarity between Eq. (4.15) and the thermodynamic model of transcription with a simple activator [2], $D\rightarrow Z$,

$$\begin{aligned} C_{Z}= \nu \; \frac{ C_{D}^{n}+\varGamma ^{m}}{C_{D}^{m} + M^{m}}. \end{aligned}$$

(4.16)

Indeed, Eqs. (4.15) and (4.16) are identical when $M^{m}=(\omega /\varLambda +1)\varOmega ^{m}$, $\varGamma =\varOmega$ and $\nu =\frac{\lambda }{\varLambda }$. Therefore, the double inversion module can be replaced by the single activator module with the same order m of the complex, i.e. a dimer activator can be used instead of a dimer repressor and a single represser together. On one hand, the simplification of the genetic circuit could be an advantage. On the other hand, the double inversion module has an advantage over the activation module. The number of free parameters in the double inversion module is greater than in the activation module therefore the circuit with double inversion module could easily be tuned to parameter levels of interest. For example, we need a module with a very low saturation level in limit $C_{D}\rightarrow 0$. Then, Eq. (4.14) is transformed into the following,

$$\begin{aligned} \lim _{C_{D}\rightarrow 0} C_{Z}= \lambda \frac{\varLambda ^{i} \varOmega ^{i m}}{\omega ^{i}\varOmega ^{i m}+\varLambda ^{i}\varOmega ^{i m}} = \lambda \frac{1}{\left( \dfrac{\omega }{\varLambda }\right) ^{i}+1}. \end{aligned}$$

(4.17)

According to Eq. (4.17), the low saturation level is independent from the parameters $\varOmega$ and m. If the ratio $\omega /\varLambda >1$ then $\lim _{C_{D}\rightarrow 0} C_{Z}$ rapidly approaches to zero with growing i. Therefore the low level limit can be reduced by increasing both the ratio $\omega /\varLambda$ and i.

In contrast to the low saturation level, the high saturation level is only dependent on one parameter, $\lim _{C_{D}\rightarrow \infty }C_{Z}=\lambda$.