Synthesizing genetic sequential logic circuit with clock pulse generator
© Chuang and Lin; licensee BioMed Central Ltd. 2014
Received: 28 April 2014
Accepted: 15 May 2014
Published: 28 May 2014
Rhythmic clock widely occurs in biological systems which controls several aspects of cell physiology. For the different cell types, it is supplied with various rhythmic frequencies. How to synthesize a specific clock signal is a preliminary but a necessary step to further development of a biological computer in the future.
This paper presents a genetic sequential logic circuit with a clock pulse generator based on a synthesized genetic oscillator, which generates a consecutive clock signal whose frequency is an inverse integer multiple to that of the genetic oscillator. An analogous electronic waveform-shaping circuit is constructed by a series of genetic buffers to shape logic high/low levels of an oscillation input in a basic sinusoidal cycle and generate a pulse-width-modulated (PWM) output with various duty cycles. By controlling the threshold level of the genetic buffer, a genetic clock pulse signal with its frequency consistent to the genetic oscillator is synthesized. A synchronous genetic counter circuit based on the topology of the digital sequential logic circuit is triggered by the clock pulse to synthesize the clock signal with an inverse multiple frequency to the genetic oscillator. The function acts like a frequency divider in electronic circuits which plays a key role in the sequential logic circuit with specific operational frequency.
A cascaded genetic logic circuit generating clock pulse signals is proposed. Based on analogous implement of digital sequential logic circuits, genetic sequential logic circuits can be constructed by the proposed approach to generate various clock signals from an oscillation signal.
Synthetic biology is an emerging interdisciplinary research field, which concentrates on understanding the behaviors of biological system from system-level as well as creating an artificial genetic circuit based on the principles of systems biology, mathematics and engineering[1–4]. Analogous to an electronic circuit, the synthetic genetic circuit also includes some standard biological components to assemble the biochemical process of living organisms and achieve specific functionality. Based on a bottom-up approach, more complicated bio-computing modules can be expected to perform more complex functions via integrating a variety of biological devices, like very-large-scale integration circuits in electronics. By using mathematical models to capture the quantitative and qualitative characteristics of biological systems, the customized genetic circuits with specific functions can be designed from the system perspective[5–8]. For drug development and disease treatment, synthetic biology brings a useful and rapid direction through inserting the designed genetic circuits into the host cells to improve or modify the disease state of organisms. In addition, there are still potential applications in biofuels, biotechnology, bioremediation, and bioenergy remained to be developed.
Inspired by electronic circuits, several synthetic genetic circuits have recently been created, such as toggle switch, genetic oscillator, pulse generator, genetic counter, logic evaluator, sensor, filter, and cell-cell communicator. The former twos are based on protein-protein interaction without any external input to control their behaviors. Toggle switch applies two repressor genes repressing each other to cause bi-stable phenomenon, like as a memory device. By cascading odd number of repressor genes in the cycle chain, a genetic oscillator can be synthesized to generate a stable oscillation signal in the protein response and applied in the control of dosage of drugs, or regarded as a synchronous mechanism for cell-cell communication[10–13]. A pulse generator generates an instantaneous stimulating signal and then resets to the original state by using time difference between the input and the corresponding delayed signals[14, 15]. If the input signal is a periodic clock signal, then a clock pulse signal can be synthesized. Biosensor and filter are designed to detect the concentrations of specific molecular signal and range.
Boolean logic gate is an essential unit of a computer in digital logic circuits. To bring the insight of digital logic circuit design in electronic systems into biological systems, the more complicated bio-computing processes can be easily constructed by combining a variety of genetic logic gates. The genetic logic gates constructed are based on different genetic transcriptional reactions to express various logic behaviors[16–22]. To use genetic components such as promoter, ribosomal binding site (RBS), repressor/activator genes and reporter gene, genetic logic gates with different logical operations have been assembled, such as NOT, Buffer, AND, OR, XOR. Through synchronous cascades of these genetic logic gates based on the topology of digital logic circuits, more complicated genetic logic circuits can be synthesized, such as multiplexer, half adder, combinational logic circuits, memory, and sequential logic circuits[23–30]. A genetic sequential logic circuit works with a counters, which is composed of some basic devices such as SR latch and flip-flop, has been developed in[25, 29].
In biological systems, there are different rhythm frequencies depending on cell types. A 12-hour rhythm has been recently found in the mouse liver. For this reason, there are many engineered approaches proposed to synthesize the specific oscillation signals. In, the frequency-doubling oscillation can be constructed by using Fourier theory. A genetic circuit with multiple functions is designed to synthesize the oscillation signal with half original frequency. Another aspect is to use regulated protein to control the transcription and degradation rates of target gene in an existing network structure[33–35]. A robust synthetic genetic circuit is designed based on H ∞ control theory by regulating degradation rates of mRNAs and proteins in stochastic perturbational environments[33, 34]. For cell-cell communication, synchronized genetic circuit designs are proposed to synchronize a population of oscillation signals[36, 37]. To construct a promoter-RBS library from microarray data and find suitable promoter-RBS components, a robust genetic circuit has been theoretically realized in the genetic systems by a systematic approach[38, 39].
This paper proposes an artificial genetic sequential logic circuit with a function of frequency divider based on the periodic oscillation signal from a repressilator and analogous topology of the digital logic circuits in electronics. The proposed genetic sequential logic circuit is triggered by a clock pulse signal to generate a clock signal whose frequency is an inverse integer multiple to the genetic oscillator. Similar to an electronic waveform-shaping circuit, a genetic waveform-shaping circuit constructed by several genetic Buffers in series is designed, which regulates time duration of logic high/low levels of an oscillation signal in the basic sinusoidal cycle and reshapes the oscillation signal into a pulse-width-modulated (PWM) signal with different duty cycles by regulating the different threshold levels of the Buffer. The PWM signal can be regarded as a pulse signal with the frequency is coherent to that of the genetic oscillator. The clock pulse signal is served as the rising or falling triggered edges of a clock signal with base frequency. In the digital logic theory, Karnaugh map is applied to determine the input signals of the rising or falling edge-triggered genetic JK flip-flops in each level. A synchronous genetic counter circuit is triggered by the clock pulse signal to realize the genetic clock with its frequency is an inverse integer multiple to the genetic oscillator.
For our proposed genetic pulse generator design, the periodic property of genetic oscillator is considered and the clock pulse signal is generated by utilizing the existing synthetic genetic oscillator constructed by three repressor genes which repress each other in the closed loop. Different from the genetic counter circuit design[25, 29], we introduce a generalized form based on the topology of digital logic circuits for synthesizing a clock signal with an inverse multiple of clock frequency to the genetic oscillator. The major advantage of the proposed approach is that it is easy to construct complex genetic sequential logic circuits via bottom-up approach with less computational time. Simulation results in silico show performance of the synthesizing clock pulse signal, and the clock signal with double, triple, quadruple basal periods while operating at the same genetic oscillator.
Dynamic model of synthetic genetic logic circuits
By applying mathematical models to describe the biochemical reactions of genetic systems, a synthetic genetic circuit with a specific function can be synthesized from the system's perspective.
where m i and p i denote, respectively, concentrations of mRNA and protein for the gene i, λ i and γ i are, respectively, the degradation rates of mRNA and protein, α i is the transcription rate of mRNA, β i is the synthesis rate of protein, α i,0 is the basal production rate, f i (⋅) is the promoter activity function which describes the nonlinear transcriptional behavior and reflects the strength of the interaction between regulated protein and RNA polymerase (RNAp), and u is the concentration of transcription factor (TF) which is produced from other gene(s) or inducer(s) to control the transcription rate of target genes.
where f AND, f OR, f XOR, f NAND and f NOR are, respectively, promoter activity functions of logic AND, OR, XOR, NAND and NOR gates, u 1 and u 2 are concentrations of repressor or activator TFs, K 1 and K 2 are Hill constants for u 1 and u 2, respectively, and n 1 and n 2 are the corresponding Hill coefficients. For logic AND, OR and XOR gates, the transcriptional behaviors are regulated by two activator TFs with different binding sites. Two repressor TFs control the genetic expressions of logic NAND and NOR gates. Their construction frameworks are shown in Figure 1(c)-(g).
Here, ρ i and ρ 0,i are new synthesis and basal production rates of the protein. The dynamic model of 2L differential equation (1) is reduced to the dynamic system with L differential equation (9). For real-world implementation, fetching the corresponding promoter-RBS parts from the promoter-RBS library, the synthetic genetic circuit can be realized in the genetic systems.
Synthetic genetic sequential logic circuits
In digital logic circuits, the output of sequential logic circuits depends not only on the present inputs but also on the past inputs. For synchronous sequential circuits, a clock signal is utilized as a metronome to coordinate actions of circuits, which oscillates between high-level and low-level states. The circuits with triggered clock signals become active either in the rising edge, the falling edge, or in both of the rising and falling edges. For the sequential logic circuit triggered at the rising edge of the clock signal, it becomes active when its clock pulse goes from low to high (0 to 1), and ignores high-to-low (1 to 0) transition.
In genetic logic circuits, oscillation signal produced from a repressilator is not ideal as a clock for use in the kind of circuits relying on the change of rising or falling edge of the clock signal for state transition. Our proposed approach is to introduce the idea of a waveform-shaping circuit in electronics to genetic logic circuits, and reshape the synthesized genetic oscillation signal into a crisp clock signal or a PWM signal with different duty cycles. By regulating the size of duty cycle, the clock pulse can be generated with a rising edge or a falling edge whose frequency is coherent to the oscillation frequency. To use the clock pulse, the designed genetic counter based on the topology of an electronic sequential logic circuit is triggered to generate a clock signal with its frequency is inversely integer multiple to the genetic oscillation.
Synthetic genetic oscillator
where p i and p j are concentrations of proteins for (i, j) ≡ (lacI, cI), (tetR, lacI) or (cI, tetR). For other design, the oscillation behavior can be generated by a number of repressor and activator genes in which the number of repressor genes must be odd.
where y d is the oscillation signal with the desired amplitude A, basal frequency ω 0, phase φ and y d,0 is the base level to ensure nonnegative protein concentration. For more details regarding synthetic genetic oscillator design by optimization algorithms one is referred to.
Realizing a genetic waveform-shaping circuit
where p k is the output concentration of the k th Buffer, p k,ss denotes its steady-state concentration, u k , K k and n k are, respectively, the input concentration, Hill constant, and Hill coefficient of the k th Buffer and ρ k , γ k and ρ 0,k are, respectively, synthesis, decay and basal rates. The second term of the right-hand side of (16) is the minimal level and ρ k /γ k is the difference between the minimal and maximal levels. Output concentration of the genetic Buffer is the half maximal output concentration when the input concentration equals K k and thus K k refers to the threshold level y T .
where A k is the gain of the k th Buffer. The gain is proportional to the Hill coefficient n k and the synthesis rate ρ k and is inversely proportional to the Hill constant K k and the decay rate γ k at the operating point u k = K k . To ensure that the necessary condition of the gain at the operating point, K k should be exceeding 1. At first, one chooses the appropriate Hill constant for the desired synthesized PWM signal and then selects a suitable Hill coefficient n k , synthesis rate ρ k and decay rate γ k satisfying (19). From the system parameters in the previous stage, one proceeds to choose the appropriate system parameters in the next stage satisfying (18) and (19). From[38, 39], to realize the proposed genetic logic circuit in reality, one can find applicable promoter-RBS components from the constructed promoter-RBS library, whose I/O characteristic curves are capable of satisfying (18) and (19).
Design of genetic frequency divider circuit
Genetic JK flip-flop
The above approach is generic, by an analogous way, one is able to determine the corresponding inputs of each genetic JK flip-flop based on the engineering digital logic theory and cascade these basic flip-flips to resemble other types of genetic counters with the desired operational frequency.
To demonstrate the proposed synthetic genetic sequential logic circuit is effective to realize the function of frequency divider, the following numerical examples are illustrated to confirm the performance of the proposed method.
Synthetic genetic oscillator
Synthesis of PWM signals
Synthesis of clock signals
The goal of synthetic biology is to design genetic circuits with specific functions by using the approaches of mathematics and engineering. Several genetic logic gates have recently been developed and experimentally realized[16–22]. In[16–18], simplified schemes of various genetic logic gates have been constructed by using basic biological components such as activator/repressor genes, reporter gene, promoter and RBS. For a gene with multiple binding sites, the transcription rate of the gene can be regulated by the same number of regulated proteins. The modified protease can control the degradation rate of protein. For logic NOT gate, the gene with a repressor regulated protein can generate the I/O characteristic curves with an inverse sigmoid function. A complementary form of NOT gate is “Buffer” which is regulated by an activator protein to reshape the I/O characteristic curves for enhancing sharpness of a biologic response with a sigmoidal function. For logic AND gate with two binding sites, the genetic expression can be regulated by two activator proteins. In, the activator proteins HrpR and HrpS control the promoter of the corresponding reporter gene to realize logic AND gate. By cascading a NOT gate in the outputs of AND/OR gates, the NAND/NOR gates can be synthesized[20–22]. In, a reporter gene with different promoters (lac, BAD and lux) and different RBS parts (rbs30, rbs33, rbsH, etc) can determine the values of Hill constant and coefficient and achieve the different NOT gates with different threshold levels. Similarly, one also can choose different promoter and RBS components with repressor or activator proteins to construct the corresponding system parameters for other logic gates. Through synchronous cascades of these genetic logic gates based on the topology of electronic circuits, the toggle switch and oscillator have been characterized in real world by a class of Hill differential equations, and realized in Escherichia coli[9, 10]. The previous papers have exhibited the possibility of realization of a class of fundamental biological devices in real world which form a basis for implementing more complicated combinational or sequential biological circuits. To realize the proposed genetic waveform-shaping circuit and genetic counter using a systematic approach, we first can identify the different promoter-RBS parts to achieve the basic logic gates, and then assemble these logic gates with the topologies in Figures 4,7,8,9, and10.
Reporter protein or green fluorescent protein (GFP) has been used in real-world experiments to reflect genetic expression. By using a flow cytometer, the intensity of fluorescence of GFP is measurable. The system parameters in the dynamic model of genetic circuits can be identified from these measurement data using system identification methods. In[38, 39], a class of robust genetic circuits has been constructed by selecting the applicable promoter-RBS component from a promoter-RBS library. Therefore, to realize our proposed genetic logic circuits including the genetic waveform-shaping circuit and genetic counter, one first establishes the measurement device which exhibits fluorescence concentrations of a series of a repressor or activator gene with different promoter-RBS components and TFs via fluorescence measurement and then rebuild a promoter-RBS library with information of system parameters in terms of our mathematical model describing the behaviors of genetic logic gates by using the system identification methods[30, 35]. To select the adequate promoter-RBS component generating the I/O characteristic curves of Buffers and satisfying the designed conditions (18) and (19) from the rebuilt promoter-RBS library via an optimization algorithm (such as), the clock pulse signals can be synthesized based on the proposed cascaded genetic logic circuit. Similarly, one can also choose the suitable promoter-RBS components to realize the various genetic logic gates and assemble these logic gates based the proposed topology. The designed genetic counters can be triggered by the synthesized clock pulse signals to generate the clock signals with multi-basal periods.
In sequential logic circuits, the triggered signals could be rising or falling edge of a clock signal. One can use clock pulse signals to replace the rising or falling edge to trigger the sequential logic circuits such as our proposed genetic counters. The proposed genetic counters with a function of frequency divider based on the topology of electronic circuits can be used to generate clock signals with multi-basal period. In other words, one can utilize the proposed approach to generate clock signals whose periods is an integer multiple of 24-hour from a cell with circadian rhythm.
This paper has proposed a synthetic genetic sequential logic circuit as a frequency divider. The synthesized clock frequency is inversely multiple to that of the genetic oscillator which generates the fundamental sine wave. Through controlling threshold level of the genetic Buffer, the proposed waveform-shaping circuit regulates time duration of logic high/low levels in a basic sinusoidal cycle for an oscillation input and generates ideal pulse signals with the coherent frequency to the genetic oscillator. Regarding the generated pulse signal as rising/falling edge of the clock signal with base frequency, a genetic synchronous counter circuit based on the topology of digital sequential logic circuit is triggered by a pulse wave form to synthesize a clock signal with the inverse multiple frequency to the genetic oscillator. Experimental results in silico show the synthesizing genetic clock with double, triple, and quadruple basal periods while operating based on a genetic oscillator. By extending the proposed design principle, a class of multi-basal period clock signals can be generated in a straightforward manner.
This research was sponsored by National Science Council, Taiwan, ROC under the Grant NSC-101-2221-E-005-015-MY3.
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