Synchronous Sequential Computations with Biomolecular Reactions

Abstract

We present a methodology for implementing synchronous sequential computation using molecular reactions. Such systems perform computations in terms of molecular concentrations, i.e., molecules per unit volume, whereas the traditional electronic systems perform computations in terms of voltages, i.e., energy per unit charge. Thus far, several researchers have already proposed molecular reactions to implement static logical and arithmetic functions such as addition, multiplication, exponentiation, square root, and logarithms. In this paper, we propose two mechanisms to implement a multi-phase clock using molecular reactions. In addition, we synthesize memory by transferring concentrations between molecular types in the alternating phases of the clock. We illustrate how our methodology can be used to construct finite impulse response (FIR) filter, an infinite impulse response (IIR) filter and a four-point, two-parallel fast Fourier transform (FFT). We also show how these molecular reactions can be translated into DNA strand displacement reactions and validate our designs through chemical kinetics simulations at the DNA reactions level. Our proposed methodology is conceptual but has potential in developing synthetic biological constructs for biochemical sensing and drug delivery.

Keywords

Appendix: \({\cal L}_1\) Controller to Synthesize a Multi-Phase Clock

Fig. 14.12

a A deadzone nonlinearity. b A block diagram of the reference signal generator for our adaptive controller to generate the mutli-phase clock. The gain $k_\mu$ defines the relative phase of the oscillations. The magnitude and frequency depend on the choice of saturation level $\nu_{\max}$ and the gain k _r . The bias b defines the value at which the oscillations occur

Let the functions ${\rm sgn}(\xi)$ and ${\rm dz}(\xi)$ denote the “sign”and the “dead–zone”functions respectively. An instance of the dead–zone function is shown in Figure 14.12a; the theory used by us is valid for some other definitions of the dead–zone function as well. If ξ is a vector, then we assume that these functions are applied to each component of the vector independently. For example

$$\rm sgn\Bigg(\Big[\xi_1~\xi_2~\xi_3\Big]^\top\Bigg) = \Bigg[{\rm sgn}\Big(\xi_1\Big)~{\rm sgn}\Big(\xi_2\Big)~{\rm sgn}\Big(\xi_3\Big)\Bigg]^\top.$$

With a slight abuse of notation, we shall interchangeably uses time–domain and frequency–domain notation to refer to signals. For example, $\xi(t)$ denotea ξ as a function of time and $\Xi(s)$ denotes its Laplace transform. We approach controller synthesis via a 2-stage process. In the first stage, we develop a feedback controller. In the second stage, we develop a method of exciting sustainable oscillations. The feedback controller in our design plays two roles: (1) it is used to ensure that the system tracks the reference commands generated by oscillation excitation block, and (2) it is used to compensate for the environmental uncertainties and disturbances to reduce their effect on the closed-loop system behavior. For this purposes we choose ${\cal L}_1$ adaptive controller, which enables fast adaptation and provides guaranteed transient performance while preserving robustness of the control system.

The ${\cal L}_1$ adaptive control theory was originally developed for the systems with fast computing capability (Hovakimyan and Cao 2010), which allow complicated mathematical calculations at relatively large speeds; however some of the ${\cal L}_1$ adaptive architectures can be suitable for implementation in chemical reactions. Namely, for the problem in this paper, we choose ${\cal L}_1$ adaptive controller with switching adaptation laws (Kharisov and Hovakimyan 2012). This architecture has adaptation laws with simple structure and does not require large values of any of the parameters or signals. Thus, the adaptive controller is designed to ensure that the system outputs x ₁, x ₂ track the given reference signals $r_1(t)$ and $r_2(t)$ with the performance specifications given by the desired system

$$\dot x_{{\rm des}_1}(t) = \frac{1}{\tau^*}\Big( r_1(t) - x_{{\rm des}_1}(t)\Big),\ \dot x_{{\rm des}_2}(t) = \frac{1}{\tau^*}\Big( r_2(t) - x_{{\rm des}_2}(t)\Big),$$

(14.16)

where $x_{{\rm des}_1}(t)$ and $x_{{\rm des}_2}(t)$ are the desired values of the states x ₁ and x ₂, respectively, and $\tau^*$ is a nominal value of the uncertain system parameter τ.

1.1 ${\cal L}_1$ Control System Architecture

Since ${\cal L}_1$ adaptive controller ensures closeness of the control system to the desired system 14.16, we design the oscillation excitation scheme for the system (14.16). To demonstrate feasibility of such an approach, we use simple nonlinear oscillation excitation of the following form:

$$\begin{aligned} r_1(t) &= \mu_1(t) - k_\mu \mu_2(t),\ r_2(t) = \mu_2(t) + k_\mu \mu_1(t)\\ \mu_1(t) &= \int_0^t k_r \nu_1(\tau) d\tau, \ \mu_2(t) = \int_0^t k_r \nu_2(\tau) d\tau \\ \dot \nu_1(t) & =\begin{cases} 0, &\begin{array}{l} \mbox{if}|\nu_1(t)|> \nu_{\max} ~\mbox{and}\\ \mbox{sign}\Bigg(\nu_1(t) \Big(x_1(t)-b\Big)\Bigg)> 0,\end{array}\\ x_{{\rm des}_1}(t) - b, & \;\,\mbox{otherwise},\end{cases}\\ \dot \nu_2(t) & =\begin{cases} 0, &\begin{array}{l} \mbox{if}|\nu_2(t)|> \nu_{\max} ~\mbox{and}\\ \mbox{sign}\Bigg(\nu_2(t) \Big(x_2(t)-b\Big)\Bigg)> 0,\end{array}\\ x_{{\rm des}_2}(t) - b, & \;\,\mbox{otherwise}.\end{cases}\end{aligned}$$

The block diagram of this scheme is shown in Fig. 14.11b. The ${\cal L}_1$ adaptive controller is comprised of the state predictor, switching adaptation laws, and the control law. The state predictor is given by

$$\begin{aligned} \dot{\hat{x}}_1(t) &= \frac{1}{\tau^*}\Bigg( \Big[T_{12}^{tot}\Big]^* \Omega\Big(r I_2,K_I,n\Big) + \hat{\sigma}_1(t) - \hat{x}_1(t)\Bigg),\\ \dot{\hat{x}}_2(t) &= \frac{1}{\tau^*}\Bigg(\Big[T_{21}^{tot}\Big]^*\Big(1 - \Omega\Big(r A_1,K_A,m\Big)\Bigg) + \hat{\sigma}_2(t) - \hat{x}_2(t)),\end{aligned}$$

where $\hat{x}_1(t)$ and $\hat{x}_2(t)$ are the predictions for $\Big[T_{12}A_2\Big](t)$ and $\Big[T_{21}A_1\Big](t)$ respectively; and $\hat{\sigma}_1(t) \in \mathbb{R}$, $\hat{\sigma}_2(t) \in \mathbb{R}$ are the adaptive estimates governed by the following adaptation laws:

$$\begin{aligned} \hat{\sigma}_1(t) &= -\Delta_\sigma{\rm sgn} \left[{\rm dz}_{\epsilon_\sigma}\left(\tilde x_1(t)\right)\right],\\ \hat{\sigma}_2(t) &= -\Delta_\sigma{\rm sgn} \left[{\rm dz}_{\epsilon_\sigma}\left(\tilde x_2(t)\right)\right],\end{aligned}$$

where $\tilde x_1(t) \triangleq \hat{x}_1(t) - {\Big[T_{12}A_2\Big]}(t)$, $\tilde x_2(t) \triangleq \hat{x}_2(t) - {\Big[T_{21}A_1\Big]}(t)$; ${\rm sgn}(\!\cdot\!)$ and ${\rm dz}(\!\cdot\!)$ stand for sign and dead–zone functions; $\epsilon_\sigma \in \mathbb{R}^+$ is the dead–zone interval; and $\Delta_\sigma \in \mathbb{R}^+$ is a design constant.

In ${\cal L}_1$ adaptive control theory the control signal performs compensation for the system uncertainty within the bandwidth of a lowpass filter. Notice that in our case the plant contains an input nonlinearity. This nonlinearity is invertible within admissible control input ($K_I> 0$ and $K_A>0$). Therefore, to allow compensation for the system uncertainty, we use a virtual control signals $v_1(t)$ and $v_2(t)$ and define the systems control signals using the nonlinear inversion compensation:

$$K_I(t) = \frac{\Big[r I_2\Big](t)}{\left( \frac{1}{v_1(t)} - 1 \right)^\frac{1}{n}},\ K_A(t) = \frac{\Big[r A_1\Big](t)}{\left( \frac{1}{1-v_2(t)} - 1 \right)^\frac{1}{m}}.$$

(14.17)

Notice that upon substituting these control signal into the system equations, we obtain

$$ \frac{d x_1}{dt}(t) = \frac{1}{\tau}\Big(\nu_1 v_1(t) - \nu_1\Big),\ \frac{d x_2}{dt}(t) = \frac{1}{\tau}\Big(\nu_1 v_2(t) - \nu_2\Big).$$

The system uncertainty due to variations of parameters of the above equation is compensated with the help of the following control law:

$$v_1(s) = k_{g_1} r_1(s) - C(s)\hat{\sigma}_1(s),\ v_2(s) = k_{g_2} r_2(s) - C(s)\hat{\sigma}_2(s),$$

(14.18)

where $k_{g_1} = 1/\nu_1^*$, $k_{g_2} = 1/\nu_2^*$, and C(s) is a stable strictly proper transfer function with unit dc gain $C(0) = 1$.

1.2 Stability and Performance Bounds for the ${\cal L}_1$ Adaptive Controller

Similar to all ${\cal L}_1$ adaptive control architectures from (Hovakimyan and Cao 2010), the analysis can be performed by defining the ${\cal L}_1$ reference system, which incorporates the low pass filter and assumes compensation of the system uncertainties only within the available bandwidth of the control channel. Then, the performance bounds can be computed as the distance between the ${\cal L}_1$ reference system and the closed–loop adaptive control system for both the system output and the control input. The ${\cal L}_1$ reference system is given by

$$\begin{aligned} \frac{\it dx_1}{\it dt} &= \frac{1}{\tau^*}\Big(\nu_1^* v_1(t) - x_1(t) + \sigma_1(t)\Big),\\ \frac{\it dx_{2}}{\it dt} &= \frac{1}{\tau^*}\Big(\nu_2^* v_2(t) - x_2(t) + \sigma_2(t)\Big),\\ v_1(s) &= C(s) (r_1(s) - \sigma_1(s)),\\ v_2(s) &= C(s) (r_2(s) - \sigma_2(s)),\end{aligned}$$

where

$$ \sigma_1(t) = \left(\frac{\tau^*}{\tau}[\nu_1-\nu_1^*\right)v_1(t),\ \sigma_2(t) = \left(\frac{\tau^*}{\tau}[\nu_2- \nu_2^*\right)v_2(t).$$

Notice that the ${\cal L}_1$ reference system involves the system uncertainty in its definition. Therefore, it can be used only for analysis purposes. This fact also implies that the stability of the ${\cal L}_1$ reference system is not guaranteed apriori. Following the same steps of the proof in Sect. 2.4 of (Hovakimyan and Cao 2010) the stability of the ${\cal L}_1$ reference system can be ensured locally by ${\cal L}_1$-norm condition similar to the Sect. 2.4. The derivations and precise equation for the ${\cal L}_1$-norm stability condition will be given elsewhere. If the ${\cal L}_1$ reference system is stable, then the performance bounds between both the system output and the control signals of the closed–loop adaptive system and the ${\cal L}_1$ reference system are given by

$$ \|x_{1}^{\rm rf} - x_{1}\|_{{\cal L}_\infty} \leq{\gamma}_{x_{1}},\ \|x_2^{\rm rf} - x_{2}\|_{{\cal L}_\infty} \leq{\gamma}_{x_{2}},\ \|v_{1} - v_{1}^{\rm rf}\|_{{\cal L}_\infty} \leq{\gamma}_{v_{1}},\ \|v_{2} - v_{2}^{\rm rf}\|_{{\cal L}_{\infty}} \leq{\gamma}_{v_{2}},$$

where $\gamma_{*}$ are computable bounds. These bounds can be obtained by combining the proofs of theorem 1 from (Kharisov and Hovakimyan 2012) and (J. Vanness et al. 2012). The proofs along with precise equations for the bounds $\gamma_{*}$ will be given elsewhere. Due to the nature of the input nonlinearity, only local results can be achieved. In other words, the system states must remain positive-valued, which can be achieved by applying $r_1(t)$ and $r_2(2)$ within the admissible set. The precise bounds on the reference commands can be derived from the ${\cal L}_1$ reference system along with 14.7 for given conservative knowledge of the uncertainty.