Design and performance of in vitro transcription rate regulatory circuits

Transcription

1 Conference on Decision an Control (CDC), Design an performance of in vitro transcription rate regulatory circuits Elisa Franco an Richar M. Murray Abstract This paper proposes a synthetic in vitro circuit that aims at regulating the rate of RNA transcription through positive feeback interactions. This esign is ual to a previously synthesize transcriptional rate regulator base on self-repression. Two DNA templates are esigne to interact through their transcripts, creating cross activating feeback loops that will equate their transcription rates at steay state. A mathematical moel is evelope for this circuit, consisting of a set of ODEs erive from the mass action laws an Michaelis Menten kinetics involving all the present chemical species. This circuit is then compare to its regulatory counterpart base on negative feeback. A global sensitivity analysis reveals the funamental features of the two esigns by evaluating their equilibrium response to changes in the most crucial parameters of the system. I. INTRODUCTION AND BACKGROUND Builing biological machinery out of known components with the same confience as one can buil a silicon chip is one of the crucial objectives of scientists who approach synthetic biology with a quantitative min. This will not only allow to expan the pool of available molecular architectures but also help to gain a better unerstaning of the characteristics, moularity an evolvability of existing complex biological networks still to be unravele [1]. It is funamental to focus on basic functional motifs that are wiely iffuse in nature, as they can be consiere elementary builing blocks of large scale systems [2]. Builing a circuit out of biological components is simplifie when operating in vitro: a higher control over the environment an over unwante reactions permits to monitor more precisely the functional response of the esigne system. Utilizing few components is also beneficial to the same purposes. Recognizing the extraorinary importance of unerstaning basic network motifs in a controllable environment, the topic of this paper is the esign an moeling of an in vitro circuit that aims at regulating the transcription rate of RNA through a positive feeback interconnection. This work follows a previously propose negative feeback-base circuit having the same objective [4]. Transcription is a funamental part of the central ogma of molecular biology an is naturally regulate in the cell: for instance it can be turne on or off by bining of transcription factors, or by seconary structure formation in the nascent RNA (see [5] an references cite therein). The ynamics of several genes Research supporte in part by the Institute for Collaborative Biotechnologies through grant DAAD19-03-D-0004 from the U.S. Army Research Office. The authors are with the Division of Engineering an Applie Sciences, California Institute of Technology, Pasaena, CA elisa,murray@cs.caltech.eu. can be couple, an it is an interesting question whether there exist mechanisms that match the transcription rates of two or more genes. The circuit escribe in this paper is compose of two ouble strane DNA (sdna) species couple through their transcripts with a mechanism of cross activation: if one of the two transcripts is in stoichiometric excess, it is esigne to increase the prouction of the other RNA species by releasing a single strane DNA (ssdna) activator molecule. Thanks to this positive feeback loop, at equilibrium the two transcription rates are equal. This feature is attractive because it is a first step towars the esign of concentrationfollowing circuits. The objective of our future work is in fact to unerstan how positive an negative feeback motifs can be use in orer to keep a molecular species of interest at a esire concentration level, or make it track a certain concentration time profile. The first in vitro transcriptional switches were esigne an synthesize by Kim [9], [8] as a possible biological implementation of neural networks. More complex cell free environments for quantitative analysis have been propose in [11], where protein signaling patterns are consiere. However, the computational power of a simple setting comprising only nucleic acis an few enzymes has been theoretically proven to be superior [7] by virtue of its simplicity. The same thermoynamics principles utilize to buil transcriptional switches are useful to construct several other systems presenting a circuit like behavior [12], [15] or even to create nanomolecular evices [3], [13]. A further motivation in focusing our attention on nucleic acis lies in their important role in the control of gene expression, which is being acknowlege an stuie with increasing interest [5]. This paper, builing on previous work that analyze a negative feeback rate regulator [4], proposes its positive feeback counterpart. The performance an features of the two ifferent esigns are compare through a global sensitivity analysis with respect to their feeback interconnection an the environment enzymatic levels. These two architectures base on transcriptional switches create a regulatory mechanism not consiere before. Following the proceure in [4], the positive feeback rate regulator was esigne an mathematically moele starting from the occurring biochemical reactions; this new system is currently being synthesize in laboratory. The employe pool of biological machinery is of interest because it can be use to construct a variety of molecular evices with ifferent functionalities, espite its simplicity an low number of components.

2 II. CIRCUIT DESCRIPTION AND MODELING A. Circuit esign The first objective of this work is that of proposing a new synthetic transcription rate regulatory network relying on positive feeback, as an alternative to the negative feeback base circuit escribe in [4]. The two esign ieas are schematically compare in Figure 1a an 1b. Transcriptional circuits are compose of nucleic acis an few enzyme species [9], [8]. The core of these circuits are DNA templates ( nucleoties long) esigne to be transcribe into RNA ( nucleoties long) in the presence of the enzyme RNA polymerase (R p ). This process can be switche on or off by isplacement of part of the enzyme bining area (the promoter). The ssdna sequences allowing completion of the promoter are calle activators (25 35 nucleoties long), which can be sequestere by ssdna (or RNA) sequences calle inhibitors (25-35 nucleoties long). In the positive feeback rate regulator, two templates T 1, T 2 are incomplete in their promoter region: activators A 1, A 2 can bin the templates completing the promoter an allowing R p to operate the transcription of RNA species R 1, R 2. Transcription is normally off ue to the presence of two ssdna inhibitor strans S 1, S 2, that sequester the activators. When transcription is initiate, the two RNAs are esigne to bin to each other, forming a ouble strane complex potentially available for further processing. By construction, either prouct in excess with respect to the other will promote the prouction of the other species by bining to its inhibitor, thereby releasing the corresponing activator. Since both transcripts have this cross-activation feature, at steay state their prouction rates shoul be equal, as emonstrate in Section III. RNase H (R h ), the other enzyme species present, allows egraation of DNA-RNA hybris, introucing a further level of ynamic aaptation. This architecture is schematically escribe in Figure 1 a). bins to S 1 with free energy of 40 kcal/mol: the secon will thus be a more favorable reaction. The stran esign metho consists of fining the esire complementarity regions an energetic constraints. This proceure can be automate by using Monte-Carlo optimization of a user efine scoring function where the free energy gain of unwante seconary structures is suitably weighte (in house software of the E. Winfree Lab at Caltech). Our esign for the strans is shown in Figure 2: following the iea propose in [4], the RNA transcripts will have mirrore sequences in orer to satisfy complementarity an cross activation. As in the negative feeback rate regulatory circuit, this esign can prouce unwante interactions among the strans. Specifically there will be a further off state of the templates ue to the bining of R i to T j. Moreover, given that each RNA species also encoes for its activator complementary sequence, there coul be a self-inhibitory action for each subsystem: this effect can be limite to R i bining to free A i if the activator toehol region is not present in the RNA sequence (therefore R i will not be able to strip off A i from the template T i ). The system behavior can be monitore by labeling the DNA strans with fluorescent yes an quenchers [10], as etaile in Figure 2. Fig. 2. Design for the positive feeback regulator, sub circuit of inex 1. The arrow tick at the en of the strans inicates the 5 to 3 irection. Starting from the 5 (left): fluorophore (cyan circle); a 1 region (orange) incluing part of the promoter region (blue box); initiation sequences (cyan); complementary toehol th S2 region (light purple); complementary a 2 region (light blue); a 1 region (orange); toehol th S2 region (light green) an at the 3 en hairpin region (brown). The hairpin is necessary to avoi spurious elongation of the RNA stran uring transcription [8]. The sequence of the transcript R 1 comprises all the regions of T 1 right after the promoter. Starting from the 3 en (left) for A 1 : quencher (black circle); toehol th a1 region (green); activator a 1 region (re) comprising part of the promoter. Fig. 1. Schematic representation of a) positive an b) negative feeback rate regulators The mechanism that allows turning on an off the templates is known as branch migration. Nucleic aci strans provie with toehol regions [14] that remain expose in boun states can be isplace by specifically esigne RNA or DNA molecules. In our case, the RNA transcripts encoe for the inhibitor toehol an can initiate the stran isplacement. For instance, we can esign A 1 so that it bins to S 1 with free energy of 35 kcal/mol, an R 1 so that it B. Mathematical moeling The chemical reactions occurring in the system are use to erive a set of orinary ifferential equations (ODEs). Throughout this erivation, the issociation constants are omitte when assume to be negligible. For enzymatic reactions, we assume that the concentration of enzymes is consierably lower than that of the DNA molecules, allowing the classical steay state assumption for Michaelis-Menten kinetics. The use of a eterministic continuous moel is well justifie in this context: the experimental setting of transcriptional circuits is such that the molecular concentrations are in the orer of 10 9 units per microliter. This case is ramatically ifferent from, for instance, certain transcription factors in cellular environments, which coul be 9 orers of magnitue less concentrate; stochastic moeling is necessary in those cases.

3 The mass action reactions are, for i {1, 2}, j {2, 1}: Activation Inhibition Release Output formation Unwante interactions The target enzymatic reactions are: k Ti A T i + A i i k Ti A T ia i + S i S i i k Ai S A i + S i i R i + A js j k Ri A j S j R i + S j k Ri S j R i + R j k Ri R j k Ri A R i + A i i R i + T j k Ri T j T ia i T i + S ia i A is i R is j + A i R is j R ir j R ia i R it j + k ON ii k catonii R p + T ia i R p T ia i R p + T ia i + R i k ON ii + k OF F i R p + T i R p T i kcatof Fi R p + T i + R i k OF F i R h + R is j k + H Sj k H Sj R h R is j k cathsj R h + S j The enzymatic processes involving unwante complexes: R p + R it j R h + R ia i R h + R it j k + OF F ij k OF F ij R p R it j k + H Ai k H Ai R h R ia i k + H Tj k H Tj R h R it j k catof Fij k cathai k cathtj R p + R it j + R j R h + A i R h + T j Given equations (1), (2), an (3) it is straightforwar to erive a set of ODEs as follows: t [Ti] = kt ia i [T i] [A i] k Rj T i [R j] [T i] + k cathti [Rh R jt i] t [Ai] = kt ia i [T i] [A i] k Ai S i [A i] [S i] k Ri A i [R i] [A i] + k Rj A i S i [R j] [A is i] + k cathai [R h R ia i] t [Si] = ka is i [A i] [S i] k Ti A i S i [T ia i] [S i] k Rj S i [R j] [S i] + k cathsi [R h R js i] t [Ri] = kr ia j S j [R i] [A js j] k Ri R j [R i] [R j] k Ri T j [R i] [T j] k Ri S j [R i] [S j] k Ri A i [R i] [A i] + k catonii [R p T ia i] + k catof Fi [R p T i] + k catof Fji [R p R jt i] t [RiTj] = + kr it j [R i] [T j] k cathtj [R h R it j] t [RiRj] = + kr ir j [R i] [R j] (1) (2) (3) (4) Where [T i ] represents the concentration of species T i. The molecular complexes that appear on the right han sie of the above equation can be expresse as a function of the states with some stanar steps. Mass conservation immeiately yiels the ynamics of [T i A i ], [A i S i ], [R i S j ]. Assuming that bining of the enzyme is faster than transcription or egraation in equations (2) an (3), an efining the Michaelis Menten coefficients (e.g. for the on state of the template k MONii = k ONii+k catonii k +ONii ), it is possible to use mass conservation laws to obtain explicit expressions for the enzyme concentrations. Due to space limitations we only report the complete expression for the term forr p boun to [T i A i ]: [R tot p ] [R pt ia i] = (1 + P ) [T i A i ] i,j k MONii + [T i] k MOF Fi + [R it j ] (5) k MOF Fij The nonlinear set of equations (4) was numerically analyze using the MATLAB oe23s solver. The parameter values use in these simulations are reporte in Table I. These parameters are taken from the literature [8] an from experimental ata fitting on the negative feeback regulator (ata not shown here), an they are chosen so that the two sub-circuits are ientical. This is a simplifying assumption that helps to provie intuition on the performance of the circuit by just creating an imbalance in the concentration of the strans. In particular, utilizing the parameters liste in Table I, an initial conitions T tot 1 = 100nM, A tot 1 = 100nM, S1 tot = 100nM, T2 tot = 200nM, A tot 2 = 200nM an S2 tot = 200nM. The enzymatic concentrations are R tot p = 20nM an R tot h = 2nM. These initial conitions are chosen base on the amounts normally utilize in an experimental setting an are targete for reaction volumes of 70µL. The ynamics of T 1, T 2 on an of the total amount of RNA prouce are shown in Figure 3, simulate over a 6 hour time winow. Fig. 3. a) Time profile of the templates in on state. b) Time profile of the total amount of prouce RNA transcripts III. POSITIVE AND NEGATIVE FEEDBACK DESIGN A. Rate regulation COMPARISON The moels for the positive an negative feeback rate regulator [4] can be simplifie by eliminating negligible ynamics an unwante interactions. The negligible ynamics

4 are represente by transcription reactions that may occur when the templates are off (not boun to their activator), or when they are boun to an RNA species. The unwante interactions are those ue to the specific esign choice, but may be eliminate with a more sophisticate esign. The bining reactions between RNA species an DNA templates fall in this latter category; for the positive feeback-base circuit, the bining between the RNA prouct an its own activator is also consiere an unwante interaction. These simplifications are be helpful in orer to stuy the funamental features of the two types of feeback. Referring to the negative feeback circuit ynamics erive in [4], we can write the following simplifie equations, where the off transcription reactions an the bining of R i T j have been eliminate: t [Ti] = kt ia i [T i] [A i] + k Ri T i A i [R i] [T ia i] t [Ai] = kt ia i [T i] [A i] + k cathii [R h R ia i] t [Ri] = (6) kr ir j [R i] [R j] k Ri T i A i [R i] [T ia i] + k catonii [R p T ia i] t [RiRj] = + kr ir j [R i] [R j] For the positive feeback circuit just introuce, the equations can be simplifie as follows, by neglecting the off transcription reactions, the bining of R i T j an of R i A i : t [Ti] = kt ia i [T i] [A i] + k Ri T i A i [R i] [T ia i] t [Ai] = kt ia i [T i] [A i] k Ai S i [A i] [S i] + k Rj A i S i [R j] [A is i] t [Si] = ka is i [A i] [S i] k Ti A i S i [T ia i] [S i] k Rj S i [R j] [S i] + k cathsi [R h R js i] t [Ri] = kr ia j S j [R i] [A js j] + k catonii [R p T ia i] k Ri R j [R i] [R j] k Ri S j [R i] [S j] t [RiRj] = + kr ir j [R i] [R j]. It is possible to prove that for both esigns the steay state transcription rate of R 1 is equal to that of R 2. For the negative feeback circuit, write [Ri tot ] = [R i ] + [R i A i ] + [R i R j ]. Taking the erivative with respect to time, one can immeiately see that t [R i] = 0 = t [R ia i ], provie that the concentration of two species reaches an equilibrium. Therefore one is left with t [Rtot 1 ] = t [R 1R 2 ] = t [Rtot 2 ]. Analogously for the positive feeback circuit, where [Ri tot ] = [R i ] + [R i S j ] + [R i R j ], one shows that when all the other species in the solution have reache a steay state, t [Rtot 1 ] = t [Rtot 2 ] = t [R 1R 2 ]. Note that the notation [R i ] inicates the concentration of the ith RNA species which is not boun to other molecules. The transient ynamics of [Ri tot ] can be evaluate by explicitly writing their expression, where several terms will cancel out. For the negative feeback regulatory circuit one is left with the equation: t [Rtot i ] = k catonii [R p T i A i ] k cathii [R h R i A i ]. For the positive feeback regulator one (7) gets: t [Rtot i ] = k catonii [R p T i A i ] k caths j [R h R i S j ]. It is clear from these expressions that, if one assumes ientical bining parameters for the two sub-circuits, in orer to experimentally verify t [Rtot 1 ] = t [Rtot 2 ] one shoul be able to measure both concentrations of [T i A i ] an [R i A i ] (negative feeback circuit) or [R i S j ] (positive feeback circuit). While by using fluorescent yes one can easily estimate [T i A i ] an use it as an inicator of the transcription rate; measuring [R i S j ] may not be as straightforwar. B. Sensitivity analysis The basic architecture of the positive an negative feeback rate regulatory circuits is such that the objective of equating the transcription rates of two sub-circuits will always be fulfille, when the initial conitions allow the system to reach a steay state (excluing the srna species R 1, R 2 which is not egrae). It is of interest to unerstan what is the variability of performance ue to changes of certain critical parameters of the system. Local sensitivity analysis restricte to a first orer Taylor series approximation was insufficient for this class of nonlinear systems. In fact, the sensitivity matrix technique [6] i not yiel meaningful results, ue to the large integration times an the type of nonlinearities. The analysis is therefore explore numerically, by tuning the parameters in a certain range an observing the variation in the solution of the ODEs. Attention will be restricte to a limite number of interesting parameters: the feeback strength (self inhibition an cross activation bining rates), the concentration of R p an of R h. The feeback strength can be moulate by changing the length of the toehols. The enzymatic concentration is often a source of experimental uncertainty, as venors only provie information about the activity of the protein, efine as moles of substrate converte per unit time. Figures 4 an 5 show the simulation results for the steay state amount of templates an the transcription rate versus fol change in the parameter of interest. It is meaningful to analyze the steay state behavior of the templates since it is the most irectly measurable concentration. The prouction rate of RNA will also be evaluate, since the objective of the esign is that of equating such rate for the two transcripts. The initial conitions were set to T 1 = 100nM, A 1 = 100nM, T 2 = 200nM an A 2 = 200nM for the negative feeback circuit. For the positive feeback regulator: T 1 = 100nM, A 1 = 100nM, S 1 = 100nM, T 2 = 200nM, A 2 = 200nM an S 2 = 200nM. The nominal parameters utilize in the simulations are reporte in Tables I an II; such parameters have been taken from the literature [8] an from ata fits performe on analogous circuits. The tot nominal concentration of enzymes are R p = 20nM, = 2nM, base on experimental practice. This sensitivity analysis was also performe on the full moels of the two systems, giving no relevant ifference with respect to the reporte results (ata not shown). For the negative feeback circuit, Figures 4a an 4b), one can see that both high feeback gain an high R p R h tot

5 concentration will ecrease the equilibrium amount of T i A i on. For the positive feeback circuit, Figures 5a an 5b, the effect is instea that of increasing the amount of T i A i on. On the other han, the steay state rate of prouction of RNA respons in a very ifferent way to a variation of the two parameters. In Figures 4 an 4e one observes that strong self inhibition will obviously ecrease the prouction of RNA for the negative feeback circuit, while a high concentration of R p will boost it. As for the positive feeback case, in Figures 5 an 5e show that a strong cross activation will increase up to a certain saturation value the rate of prouction, while the increment is almost linear with respect to the R p amount. Finally, the concentration of R h has similar effects on the concentration of templates on an on the rate of prouction: for the negative feeback circuit, Figures 4c an 4 f, a more significant egraation of RNA- DNA hybri R i A i means more A i capable of turning the circuit on, an accoringly a higher yiel of RNA. On the other han, the opposite effect is observe in the positive feeback circuit, in Figures 5c an 5f, where egraation of R i S j puts back into the circuit the inhibitor S i. Given the above analysis, the choice of a particular esign will epen on the performance specifications. Clearly the superposition principle oes not hol since the systems are nonlinear. The two sub-circuits will reach the same prouction rate of RNA using both esigns: but feasibility constraints shoul be taken into consieration. For instance, operating at high feeback strength is better, since long toehols woul make the branch migration process faster [14]. Working with low amount of enzymes is also an avantage, as they represent the most costly component of the circuits. The negative feeback rate regulatory circuit has a clear avantage over the positive feeback one in its lower number of DNA species, which make it a simpler an cheaper circuit. As an example, suppose an overall low amount of RNA is esire. The negative feeback circuit shoul be utilize, operating at high feeback gain an low R p, as shown in Figures 6a an 6b (self inhibition bining rates ten times higher than nominal, R p = 10n). A low amount of RNA coul be obtaine also using the positive feeback regulatory circuit, at the expense of either using more R h or esigning short toehols for the cross activation process, which woul make the reactions slower. If instea the objective is a higher prouction of RNA, positive feeback loops shoul be use, with a high cross activation interconnection an low concentration of R h, Figures 6c an 6 (cross activation ten times higher, R h = 1nM). A high amount of RNA coul be obtaine also with the negative feeback loops, utilizing though more R p an short toehols (more expensive an slower reactions). IV. CONCLUSIONS AND FUTURE WORK A circuit aime at matching the transcription rate of two DNA templates has been presente in this paper. The circuit esign is base on positive feeback an is an alternative to a previously escribe circuit base on negative feeback [4]. Fig. 4. Steay state sensitivity analysis of the negative feeback rate regulator: a), b) an c) show the equilibrium concentrations of T 1 A 1 an T 2 A 2 versus the fol change of self inhibition bining rates, R p an R h amounts; ), e) an f) show the corresponing prouction rate of RNA uner the same conitions. The simulation time is of 6 hours. TABLE I POSITIVE FEEDBACK REGULATORY CIRCUIT PARAMETERS Units: [s/m] Units: [1/s] Units: [M] k Ti A i = k catonii = k MONii = k Ti A i S i = k catof Fi = k MOF Fi = k Ai S i = k catof Fij = k MOF Fij = k Ri A i S i = k cathsi =.106 k MHSi = k Ri S i = k cathti =.106 k MHTi = k Ri T j = k Ri A i = k Ri R j = TABLE II NEGATIVE FEEDBACK REGULATORY CIRCUIT PARAMETERS Units: [s/m] Units: [1/s] Units: [M] k Ti A i = k catonii = k MONii = k Ti A i R i = k cathii =.106 k MHii = k Ai R i = k Ri R j =

6 Fig. 5. Steay state sensitivity analysis of the positive feeback rate regulator: a), b) an c) show the equilibrium concentrations of T 1 A 1 an T 2 A 2 versus the fol change of cross activation bining rates, R p an R h amounts; ), e) an f) show the corresponing prouction rate of RNA uner the same conitions. The simulation time is of 6 hours. A mathematical moel of the positive feeback-base regulatory circuit has been erive along with the basic esign iea, showing that the theoretical properties of the circuit are as anticipate by physical intuition. A sensitivity analysis has been carrie out, comparing the performance of both versions of the rate regulatory architecture. Unergoing work is aime at experimentally verifying the two circuit properties an the traeoffs between positive an negative regulation. Future work will focus on how to esign concentration followers with transcriptional circuits, starting from the rate regulatory systems. The objective is that of unerstaning how to match the concentration of two molecules with accuracy through a specific feeback motif. This will allow us to gain more insight on how living organisms perform this feature, maintaining precise concentration levels of their molecular components. Acknowlegments The authors woul like to thank Erik Winfree, Jongmin Kim, Per-Ola Forsberg an all the members of the DNA an Natural Algorithms group at Caltech for their helpful avise uring the evelopment of this project. REFERENCES [1] Alon, U. An Introuction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC, 2006 [2] Alon, U. Network Motifs: Theory an Experimental Approaches. Nature Reviews Genetics, 2007, 8, Fig. 6. Design choices to achieve ifferent performances: a) an b) show the time profile of templates on an total prouction of RNA for the negative feeback circuit with high self inhibition an low amount of R p; c) an ) show the positive feeback circuit performance with high cross activation an low amount of R h. The prouction of RNA in b) is one orer of magnitue higher than in ). [3] Bishop, J. an Klavins, E. An Improve Autonomous DNA Nanomotor. Nano Letters, 2007, 9, [4] Franco, E., Forsberg, P.-O. an Murray, R. M. Design, moeling an synthesis of an in vitro transcription rate regulatory circuit. Proc. of the American Control Conference, June 2007, Seattle, WA. To appear. Available online: elisa/publications files/ffm ACC08.pf. [5] Isaacs, F. J., Dwyer, D. J. an Collins, J. J. RNA synthetic biology. Nature Biotechnology, 2006, 24, [6] Khalil, H. K. Nonlinear Systems. Pearson Higher Eucation, 2002 [7] Kim, J., Hopfiel J. J. an Winfree, E. Neural Network Computation by in vitro Transcriptional Circuits. Avances in Neural Information Processing Systems (NIPS), 2004, 17, [8] Kim, J., White, K. S. an Winfree, E. Construction of an In Vitro Bistable Circuit from Synthetic Transcriptional Switches. Molecular Systems Biology, 2006, 2:68 [9] Kim, J. In Vitro Synthetic Transcriptional Networks. California Institute of Technology, 2006 [10] Marras, S. A. E. an Kramer, F. R. an Tyagi, S. Efficiencies of fluorescence resonance energy transfer an contact-meiate quenching in oligonucleotie probes. Nucl. Acis Res., 30, 2002 [11] Noireaux, V., Bar-Ziv, R. an Libchaber, A. Principles of cell-free genetic circuit assembly. Proc. Of The National Acaemy Of Sciences Of The Unite States Of America, 2003, 100, [12] Seelig, G., Soloveichik, D., Zhang, D. Y. an Winfree, E. Enzyme-free nucleic aci logic circuits. Science, 2006, 314, [13] Yin, P., Choi, H. M. T., Calvert, C. R., an Pierce, N. A. Programming Biomolecular Self-Assembly Pathways. Nature, 2008, 451: [14] Yurke, B. an Mills, A. P. Using DNA to Power Nanostructures. Genetic Programming an Evolvable Machines, 2003, 4, [15] Zhang, D. Y., Turberfiel, A. J., Yurke, B., an Winfree, E. Engineering Entropy-Driven Reactions an Networks Catalyze by DNA Science, 2007, 318 (5853), 1121.