國立中興大學電機工程學系 博士學位論文. 利用頻率合成器設計基因時鐘 Design of Genetic Clock Using Frequency Synthesizer

國立中興大學電機工程學系 博士學位論文 利用頻率合成器設計基因時鐘 Design of Genetic Clock Using Frequency Synthesizer 指導教授 : 林俊良 Chun-Liang Lin 研究生 : 莊佳華 Chia-Hua Chuang 中華民國一百零三年六月

誌謝 本論文得以順利完成, 承蒙恩師林俊良教授於求學期間細心指導與諄諄教誨, 使學生在專業領域上得以更深入地鑽研與拓展, 尤其是研究方法和態度的明確指引, 不僅是一生受用不盡的寶藏, 更是學生由衷感佩的精神典範, 在此獻上誠摯的敬意與謝忱 同時亦感謝口試委員陳博現教授 黃榮興教授 吳謂勝教授 林昱成教授的指導, 對本論文提供寶貴的意見與建議, 使本論文得以更加周詳與完整 四年的博士研究生涯, 是條漫長且艱辛的學習道路, 不僅是研究上的考驗與挑戰, 更是待人處世之道的磨練, 其中的甘苦, 相信只有走過的人才能體會 每當遇到挫折時, 感謝學長與學弟們, 在學業及生活中的指導與關懷, 使研究得以更加順利進行, 所建立的深厚友誼與每一句歡笑, 都將成為我美好的回憶 另外, 亦感謝在無時無刻幫助過我的人 最後, 謹以本論文獻給我最敬愛的父母與家人, 感恩他們多年來不辭辛勞的養育 栽培與無怨無悔的付出, 讓我衣食無憂的取得博士學位 ; 以及所有關心我的師長與朋友們, 感謝你們長久以來在實質與精神上, 不斷的給我支持與鼓勵 本研究亦承蒙行政院國家科學委員會專題研究計畫 (NSC-98-222-E-5-87-MY3 和 NSC--222-E-5-5-MY3) 經費補助 i

摘要 本論文提出含有倍頻器與除頻器功能的基因頻率合成器電路, 利用已存在的基因震盪器合成具有震盪器基頻的整數倍或倒整數倍之方波訊號 目前, 合成基因震盪器已成功地建構在大腸桿菌中產生週期的震盪現象, 於此基礎上, 利用數個串接的基因緩衝器可建構類似電子電路中的波形整形電路 (waveform-shaing circuit), 重新調整輸入震盪訊號的邏輯高與低的準位, 輸出具有不同責任週期 (duty cycle) 的脈衝寬度調變 (ulse-width-modulation, PWM) 訊號 對於基因倍頻器電路設計, 可藉著數個基因邏輯互斥或閘整合基因波形整形電路所產生的不同責任週期之脈衝寬度調變訊號, 合成頻率為震盪器基頻的整數倍之方波訊號 對於基因除頻器電路設計, 利用數位循序邏輯電路的拓墣可建構同步的基因計數器電路, 以基因波形整形電路所產生的週期脈衝訊號觸發, 合成頻率為震盪器基頻的倒整數倍方波訊號 模擬結果證明本研究提出的基因波形整形電路可將震盪訊號轉換成不同責任週期的脈衝寬度調變訊號, 所設計的基因頻率合成器電路也能以震盪訊號為基礎合成具有震盪器基頻的整數倍或倒整數倍之方波訊號 關鍵字 : 合成生物 基因震盪器 方波訊號 邏輯電路 頻率合成器 ii

Abstract This dissertation resents genetic frequency synthesizer circuits as a frequency multilier or a frequency divider based on an existing genetic oscillator to synthesize a clock signal whose frequency is an integral or inverse integral multile of that of genetic oscillator. In the renowned literature, the synthetic genetic oscillator has been successfully built in Escherichia coli to generate a eriodic oscillation. On the basis of this fact, an analogous electronic waveform-shaing circuit is constructed by a series of genetic Buffers to shae logic high/low levels of an oscillation inut with a basic sinusoidal cycle and generate a ulse-width-modulated (PWM) outut with various duty cycles. For genetic frequency multilier circuit design, the clock signal with the integral multile of frequency of genetic oscillator is synthesized through some of the genetic logic XOR gates integrating a variety of PWM signals. For genetic frequency divider circuit design, a synchronous genetic counter circuit based on the toology of the digital sequential logic circuit is triggered by a series of clock ulses to synthesize the clock signal with the inverse integral multile frequency to the genetic oscillator. Simulation results demonstrate the roosed design. Keywords: synthetic biology, genetic oscillator, clock signal, logic circuit, frequency synthesizer. iii

Contents 誌謝... i 摘要... ii Abstract... iii Contents... iv List of Figures... vi List of Tables... x Chater Introduction.... Background....2 Motivation... 3.3 Literature Review... 4.4 Contributions... 4.5 Dissertation Organization... 6 Chater 2 Mathematical Model... 8 2. Genetic Logic Circuit... 8 2.2 Genetic Oscillator... Chater 3 Design of Genetic Waveform-Shaing Circuit... 3 3. Creating a Waveform-Shaing Circuit... 3 3.2 Regulation of Duty Cycle... 3 3.3 Realizing a Genetic Waveform-Shaing Circuit... 4 Chater 4 Design of Genetic Frequency Synthesizer Circuit... 8 4. Genetic Frequency Multilier Circuit... 8 4.2 Genetic Frequency Divider Circuit... 2 4.2. Genetic Fli-Flo... 2 iv

4.2.2 Genetic Counter... 22 Chater 5 Simulation Results... 3 5. Genetic Logic Gates... 3 5.2 Synthesis of Genetic Oscillator... 3 5.3 Synthesis of PWM Signals... 32 5.4 Synthesis of Clock Signals... 36 Chater 6 Conclusions and Future Works... 39 6. Conclusions... 39 6.2 Future Works... 4 Acknowledgments... 42 References... 43 List of Publication... 9 v

List of Figures Figure. Process for creating a genetic frequency synthesizer.... 5 Figure 2. Constructions of a class of genetic logic gates. (a) NOT gate; (b) Buffer; (c) AND gate; (d) OR gate; (e) XOR gate; (f) NAND gate; and (g) NOR gate.... 5 Figure 3. Toology of a reressilator.... 52 Figure 4. I/O characteristics of the waveform-shaing circuit design.... 52 Figure 5. Ideal PWM signals with different duty cycles. (a) Clock ulse from low to high; (b) Less than 5% duty cycle; (c) 5% duty cycle; (d) More than 5% duty cycle; and (e) Clock ulse from high to low.... 53 Figure 6. Promoter activity functions of genetic Buffer with (a) different Hill constants; and (b) different Hill coefficients.... 54 Figure 7. Toology of the designed genetic waveform-shaing circuit.... 55 Figure 8. Ideal PWM signals for the frequency doubler design. (a) 25% duty cycle; (b) 75% duty cycle; and (c) Outut signal of logic XOR of (a) and (b).... 55 Figure 9. Ideal PWM signals for the frequency triler design. (a) 6.7% duty cycle; (b) 5% duty cycle; (c) 83.3% duty cycle; (d) Outut signal of logic XOR of (a) and (b); and (e) Outut signal of logic XOR of (c) and (d).... 56 Figure. Toology of the designed frequency doubler.... 57 Figure. Toology of the designed frequency triler... 57 Figure 2. Ideal clock signals with (a) A basal eriod; (b) Double basal eriod; and (c) Quadrule basal eriod.... 58 Figure 3. Toology of the engineered clocked JK fli-flo. (a) A rising edge-triggered one; and (b) A falling edge-triggered one.... 59 vi

Figure 4. A class of the genetic clocked JK fli-flos. (a) A rising edge-triggered one; and (b) A falling edge-triggered one.... 6 Figure 5. Toology of the genetic clocked JK fli-flo. (a) A rising edge-triggered one; and (b) A falling edge-triggered one.... 6 Figure 6. Toology of the synchronous genetic counter for the clock signal with double basal eriod. (a) A rising edge-triggered JK fli-flo; and (b) A falling edge-triggered JK fli-flo.... 6 Figure 7. Karnaugh ma of the synchronous genetic counter for the clock signal with quadrule basal eriod. (a) J ; (b) K ; (c) J 2 ; and (d) K 2... 6 Figure 8. Toology of the synchronous genetic counter for the clock signal with quadrule basal eriod. (a) Two rising edge-triggered JK fli-flos; and (b) Two falling edge-triggered JK fli-flos.... 62 Figure 9. Karnaugh ma of the synchronous genetic counter for the clock signal with eight-fold basal eriod. (a) J ; (b) K ; (c) J 2 ; (d) K 2 ; (e) J 3 ; and (f) K 3... 63 Figure 2. Toology of the synchronous genetic counter for the clock signal with eight-fold basal eriod.... 64 Figure 2. Toology of the synchronous genetic counter for the clock signals with 2 δ fold basal eriod.... 65 Figure 22. Karnaugh ma of the synchronous genetic counter for the clock signal with trile basal eriod. (a) J ; (b) K ; (c) J 2 ; and (d) K 2.... 66 Figure 23. Toology of the synchronous genetic counter for the clock signal with trile basal eriod.... 66 Figure 24. Ideal signals of the synchronous genetic counter for the clock signal with trile basal eriod. (a) Clock ulse signal from low to high; (b) Clock ulse signal from vii

high to low; (c) Outut signal Q of the st genetic JK fli-flo; (d) Outut signal Q 2 of the 2nd genetic JK fli-flo; (e) Outut signal Q 3 of the 3rd genetic JK fli-flo; and (f) Outut signal of logic OR of (d) and (e).... 67 Figure 25. Karnaugh ma of the synchronous genetic counter for the clock signal with five-fold basal eriod. (a) J ; (b) K ; (c) J 2 ; (d) K 2 ; (e) J 3 ; and (f) K 3.... 68 Figure 26. Toology of the synchronous genetic counter for the clock signal with five-fold basal eriod.... 69 Figure 27. Ideal signals of the synchronous genetic counter for the clock signal with five-fold basal eriod. (a) Clock ulse signal from low to high; (b) Clock ulse signal from high to low; (c) Outut signal Q of the st genetic JK fli-flo; (d) Outut signal Q 2 of the 2nd genetic JK fli-flo; (e) Outut signal Q 3 of the 3rd genetic JK fli-flo; (f) Outut signal Q 4 of the 4th genetic JK fli-flo; and (g) Outut signal of logic OR of (e) and (f).... 7 Figure 28. I/O resonses of a class of genetic logic gates with single inut. (a) NOT gate; and (b) Buffer.... 7 Figure 29. I/O resonses of a class of genetic logic gates with double inuts. (a) AND gate; (b) OR gate; (c) XOR gate; (d) NAND gate; and (e) NOR gate.... 72 Figure 3. Resonses of the genetic oscillator with (a) χ =.5; and (b) χ =.5.... 73 Figure 3. Relationshi between regulation coefficient χ and basal eriod of the genetic oscillator.... 74 Figure 32. A PWM signal with D = 6.7%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed PWM signal.... 75 viii

Figure 33. A PWM signal with D = 25%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed PWM signal.... 76 Figure 34. A PWM signal with D = 5%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed PWM signal.... 77 Figure 35. A PWM signal with D = 75%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed PWM signal.... 78 Figure 36. A PWM signal with D = 83.3%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed PWM signal.... 79 Figure 37. A clock ulse signal with D = %. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed clock ulse signal.... 8 Figure 38. A clock ulse signal with D = 9%. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed clock ulse signal.... 8 Figure 39. A PWM signal with D = %. (a) I/O characteristic curve in each stage; and (b) Resonse of the designed DC signal in each stage.... 82 Figure 4. Resonse of the designed frequency doubler.... 83 Figure 4. Resonse of the designed frequency triler.... 83 Figure 42. Resonse of the designed clock signals with double and quadrule basal eriods.... 84 Figure 43. Resonse of the designed clock signal with trile basal eriod.... 84 Figure 44. Resonse of the designed clock signal with five-fold basal eriod.... 85 ix

List of Tables Table. True table of the clocked JK fli-flo.... 86 Table 2. Excitations of the clocked JK fli-flo.... 86 Table 3. State table for design of the clock signal with double basal eriod.... 87 Table 4. State table for design of the clock signal with quadrule basal eriod.... 87 Table 5. State table for design of the clock signal with eight-fold basal eriod.... 88 Table 6. State table for design of the clock signal with trile basal eriod.... 88 Table 7. State table for design of the clock signal with five-fold basal eriod.... 89 Table 8. Error comarison for PWM signals with different duty cycles.... 89 x

Chater Introduction. Background Synthetic biology is an emerging interdiscilinary research field, which focuses on understanding the behavior of biological systems from system-level as well as creating an artificial genetic circuit based on the rinciles of systems biology, mathematics and engineering [-6]. Analogous to electronic circuits, the synthetic genetic circuits also include some standard biological comonents to assemble the biochemical rocess of living organisms and achieve some secific functionality. Based on the bottom-u aroach [7], more comlicated bio-comuting modules can be exected in the near future to erform more comlex functions via integrating a variety of biological devices, like very-large-scale integration circuits in electronics. By using mathematical models to cature the quantitative and qualitative characteristics of biological systems, the synthetic genetic circuits with secific functions can be designed and analyzed from the system ersective [8-3]. For drug develoment and disease treatment, synthetic biology brings a useful and raid direction through inserting the designed genetic circuits into the host cells to imrove or modify the disease state of organisms. In addition, there are still otential alications in biofuels, biotechnology, bioremediation, and bioenergy remained to be develoed. Insired by electronic circuits, several synthetic genetic circuits have recently been created, such as toggle switch, genetic oscillator, ulse generator, genetic counter, logic evaluator, sensor, filter, and cell-cell communicator [3-26]. The former twos are based on rotein-rotein interaction without any external inut to control their behaviors. Toggle switch alies two reressor genes reressing each other to cause bi-stable

henomenon, like as a memory device [4, 5]. A genetic oscillator can be synthesized by cascading odd number of reressor genes in the cycle chain to generate a stable oscillation signal in the rotein resonse and it acts as a synchronous mechanism for harmonizing cell-cell communication [6-23]. A ulse generator generates an instantaneous stimulating signal and then resets to the original state by using time difference between the inut and the corresonding delayed signals [24, 25]. When the inut signal is a eriodic clock signal, a clock ulse signal can be synthesized. Biosensor and filter are designed to detect the concentrations of secific molecular signal and range [3]. As Boolean logic gates are the essential units of comuters in electronic circuits, genetic logic gates can be served as fundamental elements for constructing a bio-comuter in digital genetic logic circuits in the future. To bring the insight of digital logic circuit design in electronic systems into biological systems, the more comlicated bio-comuting rocesses can be easily constructed by combining a variety of genetic logic gates. The genetic logic gates constructed are based on different genetic transcritional reactions to exress various logic behaviors [27-37]. To use genetic comonents such as romoter, ribosomal binding site (RBS), reressor/activator genes and reorter gene, genetic logic gates with different logical oerations have been assembled, such as NOT, AND, OR, and XOR. Through synchronous cascades of these genetic logic gates based on the toology of digital logic circuits, more comlicated genetic logic circuits can be synthesized, such as multilexer, half adder, combinational logic circuits, memory, and sequential logic circuits [38-5]. There are many engineering aroaches roosed to identify circuit toology and estimate system arameters for design of a class of synthetic genetic logic circuits. From the engineering ersective, the urose of these aroaches is to convert a 2

synthetic design roblem into an otimal tracking roblem with system identification. A robust synthetic design aroach based on H otimization control theory is roosed to design robust synthetic genetic circuits under the stochastic molecular erturbations by regulating synthesis and degradation rates of roteins [5, 52]. Other comutational evolutionary algorithms are also used to deal with this roblem [53-56]. A genetic algorithm (GA) is alied to mimic the mechanism of natural selection to search for the otimal arameters to achieve the desired erformance for synthesizing robust genetic logic circuits [53, 54]. In addition to the arameter otimization roblem, the structural otimization roblem also needs to be considered [55-57]. In [55, 56], a real structural genetic algorithm (RSGA) is develoed to synthesize a class of genetic logic circuits with the minimal number of genes while ensuring satisfactory erformance. Designing a genetic circuit with multile functions is another issue for reducing the size of synthesized genetic circuits [58, 59]. To construct a romoter-rbs library via measuring the intensity of fluorescence of reort roteins and select alicable romoter-rbs comonents from the constructed libraries, a class of robust genetic circuits has been theoretically realized in the genetic systems by solution searching algorithm [6-63]..2 Motivation Oscillation is a rhythmic natural henomenon and widely occurs in hysical, biological, chemical, and social systems. In biological systems, it has been discovered at various levels of biological organizations, ranging from neuronal rhythm, biochemical oscillation, and circadian clock and controls several asects of cell hysiology [2]. For different cell tyes, it is sulied with various rhythmic frequencies. A widesread tye is circadian rhythm and a 2-hour rhythm has recently been found in the mouse liver 3

[64]. How to synthesize a secific clock signal is a reliminary but a necessary ste to further develoment of more comlicated biologic genetic circuits or even a biological comuter in the future..3 Literature Review Several systematic aroaches have been roosed to synthesize secific oscillation signals. A sinusoidal wave with secific amlitude, frequency and hase is considered as a reference signal and the desired oscillation is generated by tracking the signal via otimization aroaches. The GA has been used to search for the otimal solution for the transcrition and degradation rates of the existing genetic oscillators in [53, 54]. In [55], the RSGA has been develoed to construct genetic oscillator with a chea structure. A robust synthetic genetic oscillator is designed based on the H control theory under stochastic erturbations [5, 52]. For cell-cell communication, synchronized genetic oscillator designs have been roosed to synchronize a oulation of oscillation signals [65, 66]. A genetic circuit with multile functions is designed to synthesize the oscillation signal with half original frequency [59]. Another asect is to assemble the different oscillation signals to synthesize the desired oscillation frequency. Based on Fourier theory, the frequency-doubling oscillation has been realized by using the hase difference between two oscillation signals with the same frequency and an AND gate [67]. The concet for constructing a synchronous counter to synthesize the clock signal with a multile of basal frequency of genetic oscillator has been roosed [43, 48]. However, the clock ulse triggering the counter circuit doesn t be considered..4 Contributions This dissertation resents genetic frequency synthesizer circuits with frequency 4

multilier and frequency divider based on the synthesized genetic oscillator and the toology of the digital logic circuits to synthesize the clock signals whose frequency is multile and inverse multile to genetic oscillator. The design rocess can simly be exlained as illustrated in Fig.. There are three sub-circuits to be built: a genetic oscillator, a genetic waveform-shaing circuit, a frequency multilier or a frequency divider. A genetic oscillator is first constructed to generate the fundamental oscillation signal at a base frequency via engineering aroaches. A genetic waveform-shaing circuit shaes the oscillation inut into ulse-width-modulated (PWM) outut with various duty cycles. Through using a frequency multilier or frequency divider, a clock signal with secific frequency is synthesized. The main contributions of this dissertation are summarized as follows:. Analogous to an electronic waveform-shaing circuit, a genetic waveform-shaing circuit constructed by several genetic Buffers with a cascaded toology is roosed, which regulates time duration of logic high/low levels of an oscillation signal in the basic sinusoidal cycle and reshaes the oscillation signal into a PWM signal with different duty cycles by regulating different threshold levels of the Buffer. The PWM signal can be regarded as a clock ulse signal with its frequency is coherent to that of the genetic oscillator. The clock ulse signal is served as the rising or falling triggered edges of a clock signal with base frequency. 2. A genetic frequency multilier circuit is roosed to synthesize the clock signal with the multile of frequency of the genetic oscillator through using some of genetic logic XOR gates and PWM signals. The rising and falling edges of an ideal clock signal can be determined and the PWM signals with the different duty cycles are obtainable from genetic waveform-shaing circuit. Integrating these PWM signals via genetic logic XOR gates, the desired genetic clock with the integer 5

number of frequency of the genetic oscillator can be realized. Different from the genetic frequency multilier design [67], we roose a generalized form to synthesize the clock signals while oerating based on a genetic oscillator. 3. A genetic frequency divider circuit is roosed to synthesize the clock signal with the inverse multile of frequency of the genetic oscillator. Through Karnaugh ma [68] in the digital logic theory is alied to determine the inut signals of the rising or falling edge-triggered genetic JK fli-flos in each level, a synchronous genetic counter circuit is triggered by the clock ulse signal to realize the genetic clock with its frequency is an inverse integral multile to the genetic oscillator. Different from the genetic counter circuit design [43, 48], we introduce the generalized form based on the toology of digital logic circuit to synthesize the clock signals while oerating based on a genetic oscillator. Advantage of the roosed aroach is that it is simle to construct comlex genetic logic circuits via bottom-u aroach with less comutational time. Simulation results in silico show erformance of frequency multilier and frequency divider while oerating at a genetic oscillator..5 Dissertation Organization This dissertation is organized as follows: Chater 2 introduces the dynamic model of synthetic genetic logic circuits. For the different logic exressions, the models and toologies of a class of genetic logic gates are illustrated. Besides, the different-tye genetic oscillators to generate the desired oscillation signals by otimization aroaches are stated. Chater 3 resents an analogous electronic waveform-shaing circuit with genetic Buffers based on a cascaded toology to convert oscillation inut into a variety of PWM 6

signals with different duty cycle through regulating the threshold level of Buffer. Chater 4 designs genetic frequency synthesizer circuits with frequency multilier and frequency divider to generate the clock signal with a multile and inverse multile basal frequency to genetic oscillator. Chater 5 illustrates numerical examles to demonstrate effectiveness of the roosed genetic frequency synthesizer circuit and confirm the erformance of the roosed method. Chater 6 makes a brief conclusion and rovides the suggested issues for further research. 7

Chater 2 Mathematical Model In genetic systems, genetic exression is comosed of transcrition and translation rocesses. DNA is transcribed to messenger RNA (mrna) when an enzyme RNA olymerase (RNA) binds to the corresonding romoter. The transcrition rate can be controlled by transcrition factors (TFs), which are roteins binding to the oerator or romoter and are divided reressor and activator. Protein is roduced from mrna when ribosome binds to mrna in the translation rocess. By alying mathematical models to describe the biochemical reaction of genetic systems, a synthetic genetic circuit with a secific function can be synthesized from the system's ersective. 2. Genetic Logic Circuit Consider the dynamic model of the synthetic genetic logic circuit with L genes described by a class of nonlinear Hill differential equations [44] ( ),, m = α f u γ m + α i i i i i i = βm λ, i=,, L i i i i i () where m i and i denote, resectively, concentrations of mrna and rotein, α i is the transcrition rate of mrna, β is the synthesis rate of rotein, α i, is the basal i roduction rate, γ i and λ i are, resectively, the degradation rates of mrna and rotein, fi ( ) is the romoter activity function which describes the nonlinear transcritional behavior and reflects the strength of the interaction between regulated rotein and RNA, and u is the concentration of TF which comes from other gene s roduces or inducers to regulate the transcrition rate of target genes. For a gene with an oerator site which can bind a reressor or activator TF, the 8

romoter activity functions for the genetic logic NOT and the Buffer are described, resectively, as and f f NOT Buffer ( u) ( u) = u + K n n u K = u + K n (2) (3) where f NOT and f Buffer are romoter activity functions for logic NOT and Buffer, resectively, u is concentration of a reressor or activator TF, n is the Hill coefficient which denotes the binding cooerativity between TF and the corresonding romoter, and K is the Hill constant which is roortional to the lengths or affinities of the TF binding sites inserted in the romoter region of target genes. For logic NOT gate, the inut is a reressor and the gene roduces a rotein only in absence of the reressor; otherwise, the resence of the reressor obstructs the bound of RNA and romoter. For genetic Buffer, the inut is an activator which advances the bound of RNA and romoter to roduce rotein. The frameworks for the two logic gates are illustrated in Figs. 2(a) and 2(b), resectively. For a gene with two oerator sites, which can bind two reressor TFs or activator TFs, the romoter activity functions are described in accordance with their logic functions as f ( u, u ) AND 2 n n2 u u 2 K K = u u 2 u u 2 + + + K K K K 2 n n2 n n2 2 2 (4) 9

and f f f ( u, u ) OR 2 ( u, u ) XOR 2 ( u, u ) NAND 2 f n n2 n n2 u u 2 u u 2 + + K K K K = u u 2 u u 2 + + + K K K K 2 2 n n2 n n2 2 2 n n2 u u 2 + K K = u u 2 u u 2 + + + K K K K 2 n n2 n n2 2 2 n n2 u u 2 + + K K = u u 2 u u 2 + + + K K K K 2 n n2 n n2 2 2 ( u, u ) = n n2 n n2 NOR 2 u u 2 u u 2 + + + K K K K 2 2 (5) (6) (7) (8) where f AND, f OR, f XOR, f NAND and f NOR are, resectively, romoter activity functions of logic AND, OR, XOR, NAND and NOR gates, u and u 2 are concentrations of reressor or activator TFs, K and K 2 are Hill constants for u and u 2, resectively, and n and n 2 are the corresonding Hill coefficients. The genes exressions for the logic AND, OR and XOR gates are regulated by two activator TFs. For the AND gate, the rotein is roduced only in the resence of both TFs. The gene of the OR gate is activated when any one of two TFs (or both) resents. For the XOR gate, the gene is activated in the resence of any one of two TFs; it is inhibited when both TFs are resent or absent. For the logic NAND and NOR gates, the transcritional behaviors are regulated by two reressor TFs. The gene of the NAND

gate is inhibited only in the resence of both TFs. For the NOR gate, the gene is activated when both TFs are absent. Their frameworks are illustrated, resectively, in Figs. 2(c)-2(g). Because the half-life of mrna is shorter than that of the corresonding rotein, the effects of transcrition and translation can be combined and the romoter and RBS are considered as a romoter-rbs art to regulate the gene exression. One can rewrite () as ( ),,,, = ρ f u λ + ρ i= L (9) i i i i i i with where ρ αβ α β = i i, i i i, ρ, i= γi γ () i ρ i and ρ,i are the new synthesis and basal roduction rates of rotein. The dynamic model of 2L differential equations () is reduced to the dynamic systems with L differential equations (9). 2.2 Genetic Oscillator The oscillation caability deends not only on the network toology but also on the system arameters. Currently, the simlest synthetic genetic oscillator can be synthesized from a single gene reressing itself with a delayed negative feedback loo. An extension of the simlest oscillator is called as a reressilator which consists of three genes ( laci, tetr, ci ) reressing the each other in the limit cycle shown in Fig. 3. The roduct of the first reressor gene, laci from E. coli, inhibits the transcrition of the second gene, tetr from the tetracycline-resistance transoson Tn, whose rotein roduct in turn inhibits the exression of the third reressor gene, ci from the λ hage. Finally, ci inhibits laci exression, comleting the negative feedback cycle

[6]. The dynamic model of the reressilator is described by ( ) = ρ f λ () i i NOT, i j i i where i and are concentrations of roteins for ( i, j) ( laci, ci ), ( tetr, laci ) j or ( ci, tetr ). For other ring oscillator design, the oscillation behavior can be generated by a number of reressor and activator genes in which the number of reressor genes must be odd. To achieve the desired oscillations, one can let () to track a sinusoidal wave described by ( ω ϕ) y = Asin t+ + y (2) d d, where y d is the oscillation signal with the desired amlitude A, basal frequency ω, hase ϕ and y d, is the base level to ensure nonnegative rotein concentration. For more details regarding synthetic genetic oscillator design by otimization algorithms one is referred to [53-56]. 2

Chater 3 Design of Genetic Waveform-Shaing Circuit 3. Creating a Waveform-Shaing Circuit In electronics, a waveform-shaing circuit is designed to shae the inut signal to the desired form according to an inut and outut (I/O) characteristic curve. For the oscillation inut and the clock outut, the I/O characteristic curve of the desired waveform-shaing circuit is dislayed in Fig. 4. A ste function (dashed line) with a threshold level y T is used in electronics. For the inut signal with its value larger than the threshold level, it is treated as logic high. Otherwise, it is referred to logic low. However, in real biological world, an ideal ste function doesn t exist. A sigmoid function (solid line in Fig. 4) might be used instead. From the I/O characteristic curve of a sigmoid function, there are two oerational regions: saturation and transition. The inut signal in the saturation region can be cut-off and hold on the high level or the low level for aroximation. In the transition region, the gain in the oeration oint y T must be more than (normalized) because it ensures that the inut which is larger or less than the threshold level will be amlified or shrunk. By cascading the next sigmoid function, the oscillation inut signal will gradually reach the saturation region and remain in the high or low level. 3.2 Regulation of Duty Cycle According to this idea, a waveform-shaing circuit can be used to regulate the eriod of the logic high/low levels of an oscillation signal in a sinusoidal cycle and generate a PWM signal with different duty cycle defined by 3

D T T on = % (3) where D is the duty cycle, T is the basal eriod of oscillation signal (2) with 2π ω and T on being the eriod of logic high in a basal eriod. For the PWM signal with different duty cycle, the threshold is obtained by considering Ton yt = Asin ( ω t+ ϕ) + yd,, t = th ± (4) 2 with sin ϕ th = ( ), th [ T ] (5) ω ω To select the threshold level aroaching to yd, + A, a clock ulse served as a rising triggered edge is generated and shown in Fig. 5(a). When the threshold level is chosen between d, y and yd, + A, the outut PWM signal will be generated between % to 5% duty cycle shown in Fig. 5(b). Similarly, the threshold level is selected to equal the base level of oscillation signal y d,. That means the outut signal has the balanced eriod for logic high and logic low, i.e. the PWM signal with 5% duty cycle shown in Fig. 5(c). When the threshold level is chosen between yd, A and y d,, the outut PWM signal within 5% to % duty cycle will be generated and shown in Fig. 5(d). For the clock ulse regarding as a falling triggered edge shown in Fig. 5(e), one can choose the threshold level which is close to yd, A. In other words, the PWM signals with different duty cycles can be synthesized from an oscillation signal via a waveform-shaing circuit in different threshold levels. 3.3 Realizing a Genetic Waveform-Shaing Circuit In genetic logic circuits, a genetic Buffer [56] is roosed to serve as a buffer 4

between two cascade genetic circuits to enhance logic signal transfer. It s used here to aid the genetic waveform-shaing circuit design: ( ) = ρ f u, K, n λ + ρ, k =,, M (6) k k Buffer, k k k k k k, k with its steady-state solution is easily obtained as ρ f u K n ( ) ρ = k, k k, ss Buffer, k k, k, k λ + k λ (7) k where is the outut concentration of the kth Buffer, k, ss denotes its steady-state k concentration, u k, K k and n k are resectively the inut concentration, Hill constant, and Hill coefficient of the kth Buffer and ρ k, λ k and ρ,k are, resectively, synthesis, decay and basal rates of the kth Buffer. The second term in the right side of (7) is minimal level and ρk λ k is the difference between minimal level and maximal level. For a Buffer gate, the relationshi between inut concentration and romoter activity function for the different Hill constants is shown in Fig. 6(a). Figure 6(b) shows the relationshi between inut concentration and romoter activity function for the different Hill coefficients. Outut concentration of the genetic Buffer is the half maximal outut concentration when the inut concentration equals K k and thus K k refers to the threshold level y T. In each stage, the corresonding inuts and the threshold levels are given by yd, k = uk = k,< k M (8) and 5

K yt, k = = ρ + ρ (9), < k M 2λ k k, k k In the first stage, the inut signal is the oscillation signal in (2) and the threshold level is chosen according to the desired duty cycle in (4). In the next stage, the inut signal is the outut concentration of the revious Buffer and the threshold level is the half maximal outut level in the revious one. The toology of the roosed genetic waveform-shaing circuit is dislayed in Fig. 7. The oscillation signal from roduct of any gene of the reressilator activates the transcrition exression of the first gene in the genetic waveform-shaing circuit, whose roduct activates the next gene. Stage by stage, the oscillation can be reshaed to the cris clock signal or PWM signal. However, one can observe that the roblem of slow convergence to the maximal level is occurred for the larger threshold level K k in Fig. 6(a). To resolve this roblem, one can again cascade a Buffer with the design arameters of (7) in the last stage of genetic waveform-shaing circuit to comensate the outut level. The gain at the oerating oint K k is obtained as A k k, ss ρknk = = u 4λ K k u k k k= Kk (2) where A k is the gain of the kth Buffer. The gain is roortional to the Hill coefficient n k and the synthesis rate ρ k and is inversely roortional to the Hill constant K k and the decay rate λ k at the oerating oint uk Kk =. To ensure that the necessary condition of the gain at the oerating oint K k, A k should be exceeding. At first, one chooses the aroriate Hill constant for the desired synthesized PWM signal 6

according to (4) and then selects Hill coefficient n k, synthesis rate ρ k and decay rate λ k satisfying (2) to guarantee that the inut concentration will be amlified or decayed when its concentration is larger or less than the threshold level K k. From the system arameters in the revious stage, one choose the suitable system arameters in the next stage satisfying (9) and (2). follows: Design stes of the roosed genetic waveform-shaing circuit are summarized as Ste. Construct a genetic oscillation signal with basal frequency given by (2). Ste 2. Choose the Hill constant in the first stage for the PWM signal with desired duty cycle by (4). Ste 3. Choose the adequate synthesis rate, decay rate, and Hill coefficient satisfying (2) in the first stage. Ste 4. Choose the Hill constant satisfying (9) and the adequate synthesis rate, decay rate and Hill coefficient satisfying (2) in the next stage, and reeat M 2 times for the ste. Ste 5. Choose the synthesis and decay rates satisfying (7) to maintain the maximal level in the last stage. Ste 6. Generate the PWM signal by (6). 7

Chater 4 Design of Genetic Frequency Synthesizer Circuit This chater rooses a genetic frequency synthesizer circuit as a frequency multilier or a frequency divider based on the synthesized genetic oscillator to generate a clock signal with the multile frequency or inverse multile frequency to that of the genetic oscillator. 4. Genetic Frequency Multilier Circuit Frequency multilier is a device that generates an outut signal whose frequency is a multile to the inut signal. The genetic clock with a multile of frequency of genetic oscillator can be synthesized by using PWM signals with different duty cycles to generate rising and falling edges. To construct the genetic clock signal with N-fold frequency of genetic oscillator, the N number of threshold levels is selected as ( ) y = Asin ω t + ϕ + y, ε =,, N (2) Tε ε d, with where yt ε T 2π = + ( ε ) = (22) 4N ω tε th 2, T is the value of threshold level for the synthesis of frequency multilier and t h in (5). From (3) and (4), one must generate N number of PWM signals with the following duty cycles for N-fold basal frequency where D ε is the duty cycle for each threshold level. D ε 2ε = % (23) 2N For a frequency doubler with N = 2, one can generate two PWM signals with 25% and 75% duty cycles. Similarly, three PWM signals with 6.6%, 5% and 83.3% 8

duty cycles are synthesized for a frequency triler with N = 3, and so on. Figures 8 and 9 show the ideal PWM signals for the design of frequency doubler and frequency triler. From Figs. 8 and 9, these PWM signals are integrated to synthesize the clock signal by using logic XOR gate described by ( ) = ρ f u, u λ + ρ, m=,, N (24) m m XOR, m 2, m m m, m where m denotes the concentration of genetic logic XOR gate, ρ m, λ m and ρ,m are, resectively, its synthesis, decay and basal rates, and the inuts u,m and u 2,m are any two PWM signals or the outut of any genetic logic XOR gate. Figure 8(c) is the outut signal of logic XOR of the PWM signals with duty cycle 25% and 75%, resectively shown in Figs. 8(a) and 8(b). The toology of genetic frequency doubler circuit and the corresonding signals are shown in Fig.. The signal of Fig. 9(d) is the outut of the logic XOR of the PWM signals in Figs. 9(a) and 9(b). The clock signal with trile basal frequency of genetic oscillator in Fig. 9(e) is generated by using a logic XOR of the signals in Figs. 9(c) and 9(d). Toology of the genetic frequency triler circuit and the corresonding signals are shown in Fig.. Design stes of the roosed genetic frequency multilier circuit are summarized as follows: Ste. Construct a genetic oscillation signal with base frequency given by (2). Ste 2. Find N number of threshold levels yt, ε =,, N in (2) for the desired clock ε signal with N-fold basal frequency of the oscillation inut. Ste 3. Construct the corresonding genetic waveform-shaing circuit via (6). Ste 4. Generate N number of PWM signals with the duty cycles defined by (23). Ste 5. Generate the clock signal via N- number of XOR gates via (24). 9

4.2 Genetic Frequency Divider Circuit Frequency divider in electronics is a device that generates an outut signal whose frequency is an inverse multile to that of the inut signal. A sequential logic circuit, counter, is used to achieve this function, which is constructed by a series of fli-flos and triggered by clock ulse to generate the clock signals with multi-fold basal eriod. Figure 2 illustrates an ideal clock signal while triggering in the rising edge of clock signal with the basal eriod. 4.2. Genetic Fli-Flo In electronics, there are four kinds of fli-flos: RS fli-flo, D fli-flo, T fli-flo and JK fli-flo. In which, the JK fli-flo has two inuts and four ossible logic outut, and it is thus considered for the following design. For a rising edge-triggered JK fli-flo, its toology is dislayed in Fig. 3(a). There are two AND gates with three inuts and two NOR gates with two inuts. For a falling edge-triggered JK fli-flo, its toology is dislayed in Fig. 3(b). There are two NAND gates with three inuts and two NAND gates with two inuts. The genetic JK fli-flos based on the toology of the digital logic circuits are also divided into the rising edge-triggered one and the falling edge-triggered one shown in Fig. 4. The true table and excitation table are illustrated in Tables and 2, resectively. For the rising edge-triggered genetic JK fli-flo, one uses two genetic AND gates with two inuts to relace a logic AND gate with three inuts and its model is described by ( ) ( ) = ρ f, λ, W W AND K CLK W W = ρ f, λ, V V AND J CLK V V ( ) ( Q ) = ρ f, λ, R R AND W Q R R = ρ f, λ, S S AND V S S 2

( Q ) ( S, Q) = ρ f, λ, Q Q NOR R Q Q = ρ f λ Q Q NOR Q Q (25) where CLK is the concentration of clock ulse from low to high,, W V, R, S, Q, and denote, resectively, the rotein concentrations of the genes W, V, R, Q S, Q, and Q, ρ W, ρ V, ρ R, ρ S, ρ Q, and ρ Q are the corresonding synthesis rates, and λ W, λ V, λ R, λ S, λ Q, and λ Q are the corresonding decay rates. The rising edge-triggered genetic JK fli-flo becomes active only when the clock ulse goes from low to high. There are four genetic AND gates and two NOR gates and the toology is dislayed in Fig. 5(a). The roteins K and CLK activate the transcrition of the gene W. The roteins J and CLK activate the transcrition of the gene V. The roducts of the genes W and Q activate the transcrition of the gene R and the roducts of the genes V and Q activate the transcrition of the gene S. The roteins R and inhibit the transcrition of the gene Q and the Q roteins S and Q inhibit the transcrition of the gene Q. For the falling edge-triggered genetic JK fli-flo, the model is described by ( ) ( ) = ρ f, λ, W W NAND K CLK W W = ρ f, λ, V V NAND J CLK V V ( ) ( Q ) ( Q ) ( R, Q) = ρ f, λ, R R NAND W Q R R = ρ f, λ, S S NAND V S S = ρ f, λ, Q Q NAND S Q Q = ρ f λ Q Q NAND Q Q (26) where CLK is the concentration of clock ulse from high to low. The falling 2

edge-triggered genetic JK fli-flo becomes active only when the clock ulse goes from high to low. This circuit is comosed of six genetic NAND gates with its toology structure shown in Fig. 5(b). The roteins K and CLK inhibit the transcrition of the gene W. The roteins J and CLK inhibit the transcrition of the gene V. The roducts of the genes W and Q inhibit the transcrition of the gene R and the roducts of the genes V and Q inhibit the transcrition of the gene S. The roteins S and inhibit the transcrition of the gene Q and the roteins Q R and Q inhibit the transcrition of the gene Q. 4.2.2 Genetic Counter The counter circuit in electronics works on the rising or falling edge of the clock and count the number of clock ulses. For synthesizing 2 δ -fold base eriod, the δ number of fli-flos is at least needed. Based on the feature, one firstly generates the clock ulse using the roosed genetic waveform-shaing circuit, and uses the clock ulse to trigger the genetic counter based on the toology of digital logic circuit. According to the Karnaugh ma (K-ma) in the digital logic theory [68], the inut signals of each genetic JK fli-flo and the toology of genetic counter circuit can be determined. To synthesize the clock signal with double basal eriod, one can use a rising or falling edge-triggered genetic JK fli-flo. According to the stimulations given in Table 2, the digitized state resonses are summarized in Table 3. The synchronous genetic counter circuits with one rising or falling edge-triggered JK fli-flo are constructed resectively as follows 22

( ) ( ) ( ) ( Q ) ( Q ) ( S, Q ) = ρ f, λ, W W AND K CLK W W = ρ f, λ, V V AND J CLK V V = ρ f, λ, R R AND W Q R R = ρ f, λ, S S AND V S S = ρ f, λ, Q Q NOR R Q Q = ρ f λ Q Q NOR Q Q (27) or ( ) ( ) ( ) ( Q ) ( Q ) ( R, Q ) = ρ f, λ, W W NAND K CLK W W = ρ f, λ, V V NAND J CLK V V = ρ f, λ, R R NAND W Q R R = ρ f, λ, S S NAND V S S = ρ f, λ, Q Q NAND S Q Q = ρ f λ Q Q NAND Q Q (28) with inuts of genetic JK fli-flo given by = = (29) K J where CLK is the clock ulse signal from low to high for (27) and from high to low for (28). Figure 6 shows the toology of the synchronous genetic counter circuit for the clock signal with double basal eriod. Similarly, the clock signal with quadrule basal eriod is synthesized by using two rising or falling edge-triggered JK fli-flos. The digitized state resonses are illustrated in Table 4 according to the stimulations given in Table 2. The K-ma is shown in Fig. 7 and the synchronous genetic counter circuit for the rising edge-triggered JK fli-flos is constructed as 23

ρ AND ( ) = ρ AND ( ) = ρ AND ( ) = ρ = ρ AND ( Q ) NOR ( Q ) = ρ NOR ( S ) Q ρ AND ( ) V = ρ 2 V ( 2 AND ) = ρ AND ( ) = ρ AND ( ) Q2 = ρ NOR ( Q ) 2 = ρ NOR ( S ) 2 Q 2 = ρ AND (, ) = f, λ, W W K CLK W W f, λ, V V J CLK V V f, λ, R R W Q R R f, λ, S S V S S f, λ, Q Q R Q Q f, λ, Q Q Q Q = f, λ, W2 W2 K2 CLK W2 W2 f, λ, J2 CLK V2 V2 f, λ, R2 R2 W2 Q2 R2 R2 f, λ, S2 S2 V2 S2 S2 f, λ, Q2 Q2 R2 Q2 Q2 f, λ, Q2 Q2 Q2 Q2 f λ G G K2 Q2 G G (3) with inuts of each genetic JK fli-flo given by = =, K J = = K2 J2 Q (3) where CLK is the clock ulse signal from low to high and G is the clock signal with quadrule basal eriod. One also can use two falling edge-triggered JK fli-flos to assemble the genetic counter with the clock ulse signal from high to low to relace (3). The toology of the synchronous genetic counter circuit for the clock signal with quadrule basal eriod is dislayed in Fig. 8. To synthesize the clock signal with eight-fold basal eriod, three rising or falling edge-triggered JK fli-flos are used. According to the stimulations given in Table 2, the digitized state resonses are summarized in Table 5. The K-ma is dislayed in Fig. 9 and the synchronous genetic counter circuit with three rising edge-triggered JK 24

fli-flos is constructed as W = ρ ( ) W f AND K, CLK λw W, V = ρ ( ) V f AND J, CLK λv V, R = ρ ( ) R f AND W,, Q λ R R S = ρ ( ) S f AND V, λ, Q S S Q = ρ ( ) Q f NOR R, λ, Q Q Q = ρ fnor ( S, ), Q Q Q λ Q Q W = ρ ( ) 2 W f 2 AND K 2 CLK λw 2 W2 V = ρ 2 V f ( ) 2 AND J, 2 CLK λv 2 V, 2 R = ρ ( ) 2 R f 2 AND W, 2 Q λ 2 R 2 R, 2 S = ρ ( ) 2 S f 2 AND V, λ 2 S, Q2 2 S2 Q = ρ ( ) 2 Q f 2 NOR R, λ, 2 Q Q2 2 Q2 = ρ fnor ( S, ), Q2 Q2 2 Q λ 2 Q2 Q2 W = ρ ( ) 3 W f 3 AND K 3 CLK λw 3 W3 V = ρ ( ) 3 V f 3 AND J,, 3 CLK λv 3 V3 R = ρ 3 R f 3 AND ( W, ) 3 Q λ 3 R 3 R, 3 S = ρ ( ) 3 S f 3 AND V, λ 3 S, Q3 3 S3 Q = ρ ( ) 3 Q f 3 NOR R, λ, 3 Q Q3 3 Q3 = ρ fnor ( S, ), Q3 Q3 3 Q λ 3 Q3 Q3 G = ρ ( ) G f AND K,, 2 Q λ 2 G G G = ρg fand ( K, Q ) λg G,,,, 2 2 3 3 2 2 (32) with inuts of each genetic JK fli-flo given by = =, K J = = K2 J2 Q = = K3 J3 G, (33) where CLK is the clock ulse signal from low to high and G 2 is the clock signal 25

with eight-fold basal eriod. Similarly, one can use three falling edge-triggered JK fli-flos to assemble the genetic counter with the clock ulse signal from high to low to relace to (32). The toology of the synchronous genetic counter circuit for the clock signal with eight-fold basal eriod is shown in Fig. 2. Form the above results, to synthesize the clock signal with 2 δ -fold basal eriod in which δ is a ositive integer, a synchronous genetic counter circuit with δ number of rising edge-triggered genetic JK fli-flos is constructed as ( ) ( ) ( ) ( Q ) ( Q ) ( S Q ) = ρ f, λ, W W AND K CLK W W = ρ f, λ, V V AND J CLK V V = ρ f, λ, R R AND W Q R R = ρ f, λ, S S AND V S S = ρ f, λ, Q Q NOR R Q Q = ρ f, λ, Q Q NOR Q Q ρ AND ( ) V = ρ δ V AND ( ) = ρ AND ( ) = ρ AND ( ) Qδ = ρ NOR ( Q ) δ = ρ NOR ( S Q ) δ δ = ρ AND ( ) = f, λ, Wδ Wδ Kδ CLK Wδ Wδ f, λ, δ Jδ CLK Vδ Vδ f, λ, Rδ Rδ Wδ Qδ Rδ Rδ f, λ, Sδ Sδ Vδ Sδ Sδ f, λ, Qδ Qδ Rδ Qδ Qδ f, λ, Qδ Qδ Qδ Qδ f, λ, G G K2 Q2 G G ( ) = ρ f, λ, Gδ Gδ AND Kδ Qδ Gδ Gδ (34) with the inuts of each genetic JK fli-flo given by 26

= =, J K = = J2 K2 Q = = J3 K3 G = = Jδ Kδ Gδ 2,, (35) where CLK is the clock ulse signal from low to high, and Q, Q 2, and Q δ are, resectively, the clock signals with double, quadrule and 2 δ -fold basal eriods. Figure 2 shows the toology of the synchronous genetic counter circuit for the clock signals with 2 δ -fold basal eriod. To synthesize the clock signal with trile basal eriod, the digitized state resonses are illustrated in Table 6 according to the stimulations given in Table 2. The K-ma is shown in Fig. 22 and the synchronous genetic counter circuit with two rising edge-triggered genetic JK fli-flos and a falling edge-triggered genetic JK fli-flo is constructed as ρ AND ( ) = ρ AND ( ) = ρ AND ( ) = ρ = ρ AND ( Q ) NOR ( Q ) = ρ NOR ( S ) Q ρ AND ( ) V = ρ ( 2 AND ) = ρ AND ( ) = ρ AND ( Q ) 2 = ρ NOR ( Q ) 2 = ρ NOR ( S Q ) = f, λ, W W K CLK W W f, λ, V V J CLK V V f, λ, R R W Q R R f, λ, S S V S S f, λ, Q Q R Q Q f, λ, Q Q Q Q = f, λ, W2 W2 K2 CLK W2 W2 f, λ, V2 J2 CLK V2 V2 f, λ, R2 R2 W2 Q2 R2 R2 f, λ, S2 S2 V2 S2 S2 f, λ, Q2 Q2 R2 Q2 Q2 f, λ, Q2 Q2 2 2 Q2 Q2 27

( ) ( ) ( ) ( Q ) 3 ( Q ) 3 ( R ) 3 Q3 (, ) λ = ρ f, λ, W3 W3 NAND K3 CLK2 W3 W3 = ρ f, λ, V3 V3 NAND J3 CLK2 V3 V3 = ρ f, λ, R3 R3 NAND W3 Q3 R3 R3 = ρ f, λ, S3 S3 NAND V3 S3 S3 = ρ f, λ, Q3 Q3 NAND S3 Q3 Q3 = ρ f, λ, Q3 Q NAND 3 Q3 Q3 = ρ f G G OR Q2 Q3 G G (36) with the inuts of each genetic JK fli-flo given by = =, K K2 =, =, J Q J 2 2 Q =, = J3 Q2 K3 Q2 (37) where CLK is the clock ulse signal from low to high, CLK 2 is the clock ulse signal from high to low, and G is the clock signal with trile basal eriod. The toology of the synchronous genetic counter circuit for the clock signal with trile basal eriod is dislayed in Fig. 23 and the corresonding ideal signals are shown in Fig. 24. To synthesize the clock signal with five-fold basal eriod, the digitized state resonses are summarized in Table 7 according to the stimulations given in Table 2. The K-ma is dislayed in Fig. 25 and the synchronous genetic counter circuit with three rising edge-triggered genetic JK fli-flos and a falling edge-triggered genetic JK fli-flo is constructed as ( ) ( ) ( ) ( Q ) ( Q ) = ρ f, λ, W W AND K CLK W W = ρ f, λ, V V AND J CLK V V = ρ f, λ, R R AND W Q R R = ρ f, λ, S S AND V S S = ρ f, λ, Q Q NOR R Q Q 28

= ρ fnor ( S, ) Q λ, Q Q Q Q W = ρ ( ) 2 W f 2 AND K 2 CLK λ W 2 W2 V = ρ ( ) 2 V f 2 AND J,, 2 CLK λ V 2 V2 R = ρ ( ) 2 R f 2 AND W,, 2 Q λ 2 R 2 R2 S = ρ ( ) 2 S f 2 AND V, λ, 2 Q S 2 2 S2 Q = ρ ( ) 2 Q f 2 NOR R, λ, 2 Q Q 2 2 Q2 = ρ fnor ( S, ), Q2 Q2 2 Q λ 2 Q2 Q2 W = ρ 3 W f ( ) 3 AND K 3 CLK λ W 3 W3 V = ρ ( ) 3 V f 3 AND J, 3 CLK λ V 3 V, 3 R = ρ ( ) 3 R f 3 AND W,, 3 Q λ 3 R 3 R3 S = ρ ( ) 3 S f 3 AND V, λ, 3 Q S 3 3 S3 Q = ρ ( ) 3 Q f 3 NOR R, λ, 3 Q Q 3 3 Q3 = ρ fnor ( S, ), Q3 Q3 3 Q λ 3 Q3 Q3 = ρ fnand ( ) λ V = ρ 4 V f 4 NAND ( J ) 4 CLK λ 2 V 4 V4 R = ρ ( ) 4 R f 4 NAND W, 4 Q λ 4 R 4 R, 4 S = ρ ( ) 4 S f 4 NAND V, λ, 4 S Q4 4 S4 Q = ρ ( ) 4 Q f 4 NAND S, λ, 4 Q Q 4 4 Q4 = ρ fnand ( R, ), Q4 Q4 4 Q λ 4 Q4 Q4 G = ρ ( ) G f AND Q,, Q λ 2 G G G = ρg for ( Q, Q ) λg G,,,,,,, W4 W4 K4 CLK2 W4 W4,, 2 2 3 4 2 2 (38) with the inuts of each genetic JK fli-flo given by =, =, J Q K 3 = = J2 K2 Q =, =, J3 G K3 =, = J4 Q3 K4 Q3, (39) where CLK is the clock ulse signal from low to high, CLK 2 is the clock ulse signal 29