Multi-species consensus network of DNA strand displacement for concentration-to-strand translation
aa r X i v : . [ q - b i o . M N ] F e b Multi-species consensus network of DNA stranddisplacement for concentration-to-strandtranslation
Toshiyuki Yamane ∗ , Eiji Nakamura † and Koji Masuda ‡ IBM Research - Tokyo, Kawasaki-shi, Kanagawa-ken 212-0032, Japan
February 23, 2021
Abstract
We propose novel chemical reaction networks to translate levels of con-centration into unique DNA strand species, which we call concentrationtranslators. Our design of the concentration translators is based on com-bination of two chemical reaction networks, consensus network and con-version network with any number of chemical species. We give geometricanalysis of the proposed CRNs from the viewpoint of nonlinear dynami-cal systems and show that the CRNs can actually operate as translator.Our concentration translators exploit DNA strand displacement (DSD)reaction, which is known for a universal reaction that can implement ar-bitrary chemical reaction networks. We demonstrate two specific typesof concentration translators (translator A and B) with different switchingbehavior and biochemical cost and compared their characteristics compu-tationally. The proposed concentration translators have an advantage ofbeing able to readout the concentration of targeted nucleic acid strandwithout any fluorescence-based techniques. These characteristics can betailored according to requirements from applications, including dynamicrange, sensitivity and implementation cost. keywords chemical reaction network, consensus network, DNA strand displacement, con-centration translator, heteroclinic orbits, nonlinear dynamical systems
Synthetic biologists have created a variety of artificial biological circuits (here-inafter simply called synthetic circuit), including logic gates, analog circuits,toggle switches, oscillators, and signal amplifiers [11][3][5][4][1]. While some of ∗ corresponding author, [email protected] † current affiliation: [email protected], Department of Mechanical and Aerospace Engineer-ing, University of California, Los Angeles, USA ‡ [email protected] Geometrical analysis of concentration trans-lator
This section describes the concentration translator with arbitrary dimensionas combination of concensus network and conversion network, from viewpointof geometrical theory of nonlinear dynamical systems. We analyze those twonetworks separately in subsection 2.1 and subsection 2.2 and then describe howthe combination of these two netwokrs can operate as an translator in subsection2.3.
The process of consensus formation has been of practical interest in some re-search areas such as distributed computing and sensor networks. For example,consensus on complete graphs was described in [7], where each node has binary(or ternary) states, for example, 1 for yes, 0 for no (and e for undecided). Afterpassing the states among the nodes, the network reaches consensus, dependingon the initial fraction of the states. Later, the consensus network using chemi-cal reaction systems of DNA strand displacement was introduced in [2]. Theirconsensus network is formally given by the following chemical reaction systemwith two main chemical species O and O as follows: O + O → XO + X → O O + X → O , where X denotes a secondary buffer chemical species. However, their modelsand analysis has been limited to this two dimensional case and the propertiesand structures of the system with multi-states remain to be investigated dueto the nonlinearity of the system. For nonlinear systems, one cannot generallyhope to find analytical solutions in an explicit way. Nonetheless, geometricalqualitative analysis can very often provide us with useful insight on the behaviorof the systems [6], and we will perform such kind of analysis for the multi-speciesconsensus networks. We start with the following rate equation of 2-speciesconsensus network given by d [ O ] dt = [ O ] [ X ] − [ O ] [ O ] , (1) d [ O ] dt = [ O ] [ X ] − [ O ] [ O ] , (2) d [ X ] dt = 2 [ O ] [ O ] − [ O ] [ X ] − [ O ] [ X ] , (3)where [ O ] , [ O ] and [ X ] describe concetrations of corresponding chemical species.We set the reaction constants to be 1 for simplicity. Though the system involvesthree variables, we can eliminate the secondary variable [ X ] and reduce them totwo dimensional system using the mass conservation law [ O ] + [ O ] + [ X ] = K as follows: d [ O ] dt = [ O ] ( K − [ O ] − O ]) , (4)3 [ O ] dt = [ O ] ( K − O ] − [ O ]) . (5)There are four fixed points of the reduced system;([ O ] , [ O ]) = (0 , , (0 , K ) , ( K, , ( K/ , K/ . (6)The eigen value analysis at these four fixed points show that the fixed points(0 , K ) and ( K,
0) are stable, and the origin (0,0) are unstable. On the otherhand, ([ O ] , [ O ]) = ( K/ , K/
3) is a fixed point of saddle type since the eigenvalues of the Jacobian at ( K/ , K/
3) are − K/ , K/ O ] = 0 and [ O ] + 2[ O ] = K for [ O ] , [ O ] = 0 and 2[ O ] + [ O ] = K for [ O ].Summarizing all these calculations, we can draw the phase portrait as shownin Fig 2(left). The line [ O ] = [ O ] separates the phase space into two regions,and we can see the system can operate as consensus network. The remarkablefeature of the system is that the existence of the orbits connecting two fixedpoint with two different properties, i.e, saddle and stable/unstable fixed points,which is called heteroclinic orbit. The existence of heteroclinic orbits charac-terizes the overall structure of consensus network because all orbits behave likethese heteroclinic orbits.Figure 2: (left) Phase portrait of 2-species consensus network. K = 20. Thedashed lines and axes are nullclines. The dashed arrows show the vector fieldon the nullclines. (right)2 N fixed points (white: the origin, black: stable pointson the vertices, gray: saddle points on the faces) and local picture of behaviouraround the fixed point [ O ] = . . . = [ O N ] = K/ (2 N −
1) (red).This geometric analysis can be extended to the following consensus networkswith N -species. d [ O i ] dt = [ O i ]([ X ] − X j = i [ O j ]) (7)4 [ X ] dt = 2 X i,j,i = j [ O i ][ O j ] − [ X ] X i [ O i ] . (8)Similar to the two-species case, using the law of mass conservation [ O ] + . . . +[ O N ] + [ X ] = K , we have d [ O i ] dt = [ O i ]( K − [ O i ] − X j = i [ O j ]) . (9)The phase space of consensus network with N species is a hyper tetrahedronin N dimensional Euclidean space, [ O i ] ≥ , [ O ] + . . . + [ O N ] ≤ K . We havetwo choices of the nullclines [ O i ] = 0 or K − [ O i ] − P j = i [ O j ] = 0 for each[ O i ] and therefore there are 2 N fixed points in the N -species consensus net-work. The dynamics of multi-species consensus network is characterized bythe 2 N fixed points on the faces and vertices of the hyper tetrahedron andthe heteroclinic orbits connecting them. The fixed point located inner of thehyper tetrahedron is [ O ] = . . . = [ O N ] = K/ (2 N − N dimensional vector K/ (2 N − · ( N − , − , . . . , − − K/ (2 N −
1) with the eigenvector (1 , . . . ,
1) and the other eigen values are all K ( N − / (2 N − > N fixed points and the local picture of behaviour aroundthe fixed point [ O ] = . . . = [ O N ] = K/ (2 N − N − O i ]’s and lead to thelow dimensional subspace along the heteroclinic orbits.Fig.3(above) shows the fixed points and heteroclinic orbits connecting themin the 3-species consensus network. Note that a multi-species consensus net-work naturally contains many sub-consensus networks with fewer species in-cluding the trivial consensus network with only one species [ O i ] → K as shownin Fig.3(below). This is because multi-species consensus network reduces tosmaller ones if we set some of the variables equal to zero as [ O i ] = [ O j ] = . . . = [ O k ] = 0 or set some variables to be equal as [ O i ] = [ O j ] = . . . = [ O k ].The overall dynamics of the consensus network follows one of heteroclinic orbitsdepending on its initial state and is attracted to lower dimensional subspace.Then, the dynamics again follows another heteroclinic orbit of the lower dimen-sional consensus network embedded in that subspace, and finally reaches oneof the stable fixed points on the axis. In summary, the structure of the multi-species consensus network can be described by hierarchically organized networkof heteroclinic orbits. We define upconversion networks (or simply upconverters) as chemical reactionnetworks which coonvert one species O i to next one O i +1 , in a successive way.For example, upconversion network with two output species [ O ] and [ O ] isgiven by I + G → O O + G → O , I ] is a input species, and G and G are gate species. The two-speciesupconversion network is described by the following differential equations. d [ I ] dt = − [ I ][ G ] , (10) d [ G ] dt = − [ I ][ G ] , (11) d [ G ] dt = − [ O ][ G ] , (12) d [ O ] dt = [ I ][ G ] − [ O ][ G ] , (13) d [ O ] dt = [ O ] [ G ] . (14)Using the conservation law [ O ]+[ O ]+[ G ] = G (:= [ G ](0)) , [ O ]+[ O ]+[ I ] = I (:= [ I ](0)) and [ O ] + [ G ] = G (:= [ G ](0)), we can eliminate [ I ] , [ G ] and6 G ] and we have d [ O ] dt = ( I − [ O ] − [ O ])( G − [ O ] − [ O ]) − [ O ] ( G − [ O ]) , (15) d [ O ] dt = [ O ] ( G − [ O ]) . (16)Introducing a new variable P = [ O ] + [ O ], we have dPdt = ( I − P )( G − P ) . This is a closed form equation only for P , and assuming G > G , we can findeasily the final state of P, [ O ] and [ O ] as follows:(a) If I < G , then P → I, [ O ] → , [ O ] → I and [ I ] → G < I < G , then P → I , [ O ] → I − G , [ O ] → G and [ I ] → I > G , then P → G , [ O ] → G − G , [ O ] → G and [ I ] → I − G .The phase portrait of the upconversion network is shown Fig 4. In case(a) and (b), the system has a single global fixed point at the intersection of[ O ] + [ O ] = I and the edges of the rectangle. It moves along the edges while I increases from 0 toward G . On the other hand, in case (c), it stays at ( G − G , G ). The analysis described here can be extended to the higher dimensionalupconversion network. The intersection point moves along the edges of thehyper-cube as I increases from zero, and finally stays at a point on an edgewhen I > G .Figure 4: Phase portrait of upconversion network. G = 20 , G = 8.7 .3 Combining the two networks together The translator proposed in this paper (see translator A in Section 3.1) can be un-derstood as collaboration of consensus network and upconversion network. Thesimplest chemical reaction network combining two networks can be described inFig.5(left): The dynamics is given by the following rate equation involving thesix variables [ I ] , [ G ] , [ G ] , [ O ] , [ O ] and [ X ]. d [ I ] dt = − [ I ][ G ] ,d [ G ] dt = − [ I ][ G ] d [ G ] dt = − [ O ][ G ] d [ O ] dt = [ I ][ G ] − [ O ][ G ] − [ O ][ O ] + [ O ][ X ] d [ O ] dt = [ O ][ G ] − [ O ][ O ] + [ O ][ X ] d [ X ] dt = 2[ O ][ O ] − [ O ][ X ] − [ O ][ X ] , Following the same arguments in Section 2.1 and Section 2.2, we can elimi-nate [ G ] and [ X ] using the conservation laws and we have d [ I ] dt = − [ I ]([ I ] + G − I ) , (17) d [ G ] dt = − [ O ][ G ] (18) d [ O ] dt = [ I ] ([ I ] + G − I ) − [ O ] [ G ] − [ O ] [ O ]+ [ O ] ( I − [ I ] − [ O ] − [ O ]) , (19) d [ O ] dt = [ O ] [ G ] − [ O ][ O ] + [ O ] ( I − [ I ] − [ O ] − [ O ]) (20)At first, the upconversion dominates the overall dynamics because the initialpoints are zero on [ O ] − [ O ] plane, where the vector field of consensus networkvanishes. After the dynamics of upconversion network reaches its stable points,the entire dynamics switches to the consensus network. As was described inSection 2.2, if I is small and [ I ] goes to 0, the system reduces to the followingconsensus network: d [ O ] dt = [ O ] ( I − [ O ] − O ]) , (21) d [ O ] dt = [ O ] ( I − O ] − [ O ]) . (22)On the other hand, if I is large enough and [ I ] goes to I − G , the systembecomes d [ O ] dt = [ O ] ( G − [ O ] − O ]) , (23) d [ O ] dt = [ O ] ( G − O ] − [ O ]) . (24)8s is shown in Fig.5(right), the final state depends on state of the systemwhen the switching from upconversion and consensus network occurs, whichexplains how the combination of consensus network and upconversion networkwork as a translator of concentration of input chemical species I .Figure 5: (left) Chemical reaction network combining consensus network andupconversion network. (right) phase portrait of consensus network(CN) andupconversion network (UN) In this section, we demonstrate that how the chemical reaction networks de-scribed in the previous can be implemented by DSD reaction and behave underrealistic experimental setup. Specifically, we consider two types of chemical re-action network, translator A and tranlator B. Translator A is a composite ofconsensus network and upconverter described in Section 2 and translator B iscomposed of upconverters and downconverters.
The architecture of translator A and corresponding master equations are shownin Fig.6a. Here we consider the case of 5 outputs species as an example, althoughthe number of outputs can be arbitrarily increased as explained later in thispaper. Reaction (1) is upconverters, and reaction (2) - (6) compose an extendedconsensus network. O i , G i , and X represent output strands, gate strands, andbuffer strand respectively. While the original consensus network by Chen [2]involves two species, the presented consensus network in this paper involves allof the output strands (here we exemplify the case of 5 output species) sharingthe single buffer strand X . k j,i is a reaction rate constant, where j indicatesthe reaction equation numbers, and i is the indexing of related strand species( k i,j is defined only for i listed in the parenthesis following each equation). The9eaction dynamics follows a set of differential equations shown below. d [ G i ] dt = − k ,i [ O i ][ G i ] , ( i = 0 , , , ,
4) (25) d [ O i ] dt = k ,i − [ O i − ][ G i − ] − k ,i [ O i ][ G i ] − k ,i [ O i ][ O i +1 ] − k ,i [ O i ][ O i +2 ](26) − k ,i [ O i ][ O i +3 ] − k ,i [ O i ][ O i +4 ] + k ,i [ O i ][ X ] , ( i = 0 , , , , , d [ X ] dt = X i =1 k ,i [ O i ][ O i +1 ] + X i =1 k ,i [ O i ][ O i +2 ] + X i =1 k ,i [ O i ][ O i +3 ] (27)+2 k , [ O ][ O ] − X i =1 k ,i [ O i ][ X ]Rate constant k for undefined i is regarded as zero. For simplicity, we assumethat all k ,i are the same and also k ,i , k ,i , k ,i and k ,i are the same. Here weredefine the rate constants of the upconverter ( k ,i ) and the rate constants insidethe consensus network ( k ,i , k ,i , k ,i , k ,i ) as k UC and k CN respectively. First,we assume all bimolecular rate constants to be 1 . × M − s − unless otherwiseindicated. This value is in a realistic range of rate constant for bimolecular DSDreaction [15]. This point is explained in more detail later. Note that we do notincorporate reverse reaction for each formal reaction, because DSD reaction canbe designed to suppress reverse reaction.We have already analyzed the mechanism of translator A in Section 2, wecan also understand its functionality from viewpoint of chemical reaction. Weconsider functions of the upconverters and the consensus network separately.First, in the case that the consensus network does not takes place (reaction (2)- (6) are absent), upconverters (reaction (1)) convert input strands into largerindexed output strands while consuming gate strands ( G i ) until input strandsor gate strands are used up. Fig.6b shows the output strand concentrationsversus initial input strand concentration provided only by the upconverters after20 hours reaction time. The concentrations of gate strands are indicated onthe plot. As shown in Fig.6b, the major strand species changes successivelycorresponding to the input concentration. This behavior is derived from thegradient of the gate strand concentration. As input strands increase from zero, G is used up at a certain input level so that the subsequent increase of inputstrand causes accumulation of O . In this manner, the major strand speciesswitches in turn. This switching behavior is essential to single out the majorstrand species by consensus network as explained below.In the next step, we consider the functions provided by both the upconvertersand the consensus network. Along with the successive production of outputstrands driven by upconverters, the consensus network (reaction (2)-(6)) leavesthe major strand species. The consensus network in our translator is extendedfrom the original consensus network of Chen [2] such that more than two speciescan make consensus. Our consensus network is composed of ten non-catalyticreactions (reaction (2)-(5) for each i ) and five catalytic reactions (reaction (6) for i = 1 , , . . . , X is a buffer signal strand which is shared by all the consensusnetwork reactions. In the consensus network reactions, all the output strandsreact each other first to generate buffer strand X by non-catalytic reactions (2)- (5). Subsequently, buffer strands are consumed by catalytic reactions (6). Thereaction rate of the catalytic reactions is in proportion to the concentration of10ach output strand species, and as a result the major output strand populationgrows faster and finally dominates. The remaining strand composition after20 hours is shown in Fig.6c. In a wide range of input strand concentration,only single output strand becomes dominant. The output strand concentrationincreases proportionately as input concentration increases, and subsequentlyoutput strand species switch at certain input concentrations where each gatestrand is used up. Finally output strand concentration saturates when the gatestrand G is used up. This switching behavior is exactly what realizes thefunction of our translator.Now we try to qualitatively understand this switching behavior. The behav-ior is governed by the concentrations of the gate strands and rate constants ofreactions. First, as already mentioned, the gate strand concentrations shouldhave gradient in order to switch the major strand corresponding to the in-put strand concentration, and the switching values, that indicates the inputconcentrations on which the output strand switch from one strand species toanother strand species, are mainly determined by the concentration of eachgate. Regarding the rate constants, we focus on the relative ratio of rate con-stants because absolute values only change the timescale in which the translatorworks. Relatively higher rate constants of the upconverters ( k UC ) than thoseof consensus network ( k CN ) result in more drastic switching behavior as shownin Fig.6d, because the output strand concentrations more directly follow theconcentrations prepared by the upconverters as shown in Fig.6b. On the otherhand, with a higher-rate consensus network, more strands are converted intolarger-indexed strand species than with a lower-rate consensus network. Thisis because an amount of the larger-indexed strand species always exceeds thanthat of the lower-indexed one due to the gradient of the gate strand concentra-tion. As a result, the switching values shift to lower input concentration withhigher k CN . Therefore, the dynamic range of the translator can be adjustedby both the gate strand concentrations and rate constants of each reaction. Itshould be noted that isolation of a single output strand is not so clear in thelower input range while it’s clear in the higher input range. This is because thereaction rate is slower with the lower input due to lower reactants’ concentrationso that the time required to reach a steady state is longer than that with higherinput concentration. The mathematical analysis of transient dynamics of thetranslator is described in detail in appendix section.Biochemical implementation of translator A is shown in Fig.7, which is basedon the previous work by Soloveichik et al [12]. DNA sequences are representedby arrows which direct from 5’ to 3’. Each of DNA strands included in thereaction equations comprises two types of sequence domain: a representativedomain of each strand species represented by a lowercase letter, and toeholddomains represented by t, by which a DSD reaction can be initiated. In addi-tion to the strand species indicated in the reaction equations in Fig.6a, there areother strand species involved in the reactions, called auxiliary strands, whichare highlighted by the pink boxes in Fig.7. We assume that there is an exces-sive amount of the auxiliary strands. Thereby we can approximate all formalreactions shown in Fig.6a to be bimolecular reactions, because only bimolecularelementary reactions indicated by the dotted square lines in Fig.7 are rate-limiting steps with non-excess amounts of reactants. The gray boxes in Fig.7 indicate waste strands which do not participate in any subsequent reactionsincluding the reverse reaction of each elementary reaction. Although the reverse11eactions occur slightly, the reaction rates of the reverse reactions are so slowto be negligible.The kinetics of DSD reaction can be well-predicted by mathematical model,as shown by the work of Zhang and Winfree, in which the mathematical modelshowed good agreement with experimental results within an order of magnitude[15]. According to their work, a rate constant of a DSD reaction can be con-trolled by the number of bases and GC contents of the toeholds over 6 ordersof magnitude (1 . − . × M − s − ), under an assumption that there is nosecondary structure in the toehold domain. Therefore, 10 and 10 M − s − weused in the computational analysis is a plausible value for a rate constant of aDSD reaction. We also propose translator B which does not have consensus network but stillhave a similar network structure as shown in Fig.8a. Biochemical implementa-tion of translator B is shown in Fig. 9. Translator B is composed of upconverters(reaction (1)), the same as those of translator A, and also the downconverters(reaction (2) - (5)) which are unique to translator B. The downconverters con-vert larger-indexed output strand species into smaller-indexed output strandspecies. There are two major differences between the consensus network andthe downconverters. First, the downconverters do not involve any buffer strands,so output strands directly react each other. Second, the downconverters com-pete with the upconverters, whereas the consensus network involves competi-tions among the members of the consensus network for winning the majority.Therefore, in translator B, the ratio of reaction rates of upconverters and down-converters have an essential role for determining the switching behavior. Thereaction dynamics of translator B follow a set of differential equations as shownbelow. d [ G i ] dt = − k ,i [ O i ][ G i ] , ( i = 0 , , , ,
4) (28) d [ O i ] dt = k ,i − [ O i − ][ G i − ] + k ,i − [ O i − ][ O i ] + k ,i − [ O i − ][ O i ] (29)+ k ,i − [ O i − ][ O i ] + k ,i − [ O i − ][ O i ] − k ,i [ O i ][ G i ] − k ,i [ O i ][ O i +1 ] − k ,i [ O i ][ O i +2 ] − k ,i [ O i ][ O i +3 ] − k ,i [ O i ][ O i +4 ] , ( i = 0 , , , , , k ,i ) are identical andthat the rate constants of downconverters ( k ,i , k ,i , k ,i , k ,i ) are identical, so k ,i is represented by k UC and k ,i , k ,i , k ,i , k ,i are represented by k DC . Fig.8bshows the remaining strand composition after 20 hours of translator B operationwith k UC : 1 . × M − s − and k DC : 1 . × M − s − . Even with thesame gate strand composition, the switching values are different from that oftranslator A. The switching values of translator A are determined mainly bythe gate strand composition and partly affected by the rate constants. How-ever, the switching values of translator B are strongly dependent on the rateconstants. Fig.8c and Fig.8d shows the remaining strand compositions withdifferent rate constants. With higher k UC , the switching values shift to largerinput concentrations, while with higher k DC the switching timings shift to lowerinput concentrations. This behavior can be simply interpreted as a result fromthe competition of upconverters and downconverters.12igure 6: (a)Architecture of translator A. Input strand species is represented by O . (b)Remaining strand composition after 20 hours operations of translator Aunder the condition of k UC = 1 . × M − s − , k CN = 0 M − s − , (c) k UC = k CN = 1 . × M − s − , (d) k UC = 1 . × M − s − and k CN = 1 . × M − s − , (e) k UC = 1 . × M − s − and k CN = 1 . × M − s − . O . (b)Remaining strand composition after 20 hours operations of translatorB under the condition of k UC = k CN = 1 . × M − s − , (c) k UC = 1 . × M − s − and k CN = 1 . × M − s − , (d) k UC = 1 . × M − s − and k CN = 1 . × M − s − . 15igure 9: Biochemical implementation of translator B. Strands highlighted inpink are auxiliary strands for realizing desired functions. Strands highlightedin gray are waste strands, which no longer react with other strands. There are two points to be considered when we compare translator A and B:switching behavior and biochemical implementation cost. First, the switchingbehavior of translator B is more sensitive to the rate constants than that oftranslator A. As already mentioned, this sensitivity is a result of the competi-tion between the upconverters and the downconverters. This feature providesa tunability of a dynamic range of the input strand concentration translator.On the other hand, the sensitivity can also be interpreted as instability of thebehavior of the translator. Therefore, both concentration translators should beemployed properly according to requirements from application stand points. Itshould be noted that the switching behavior is also controlled by the gate strandconcentrations. If the maximum concentration of DNA strands in a reaction sys-tem (in other words, biochemical resources) is constant, an increase of each gatestrand concentration limits the number of output strand species to be processedwhile retaining substantial concentration. Therefore, in the present study, weset the gate strand concentration at constant when calculating the translatordynamics. Next, we discuss the biochemical implementation cost of both trans-lators. Here, the term ”biochemical implementation cost” simply means thethe number of DNA species involved in the chemical reaction networks. Table1 shows a comparison of the number of DNA strand species required to im-plement each translator circuit when the number of the output strand is N .The total biochemical cost (DNA concentration) is predominantly determinedby the concentrations of the auxiliary strands, because they should be larger16han other strand species to keep the reaction system the set of bimolecularreactions as described by the reaction equations in Fig. 6a and Fig. 8a. Notethat the absolute number of the auxiliary strand species depends on the specificbiochemical implementation, while we can still relatively compare the numberof strand species in both translators. Translator B requires a smaller number ofauxiliary strand species to be biochemically implemented, because it does notinvolve buffer strands X, which are required by translator A. However, if N islarge enough, the term of N becomes more dominant. Thus, both translatorsare comparable in terms of the cost for biochemical implementation.Table 1: Number of DNA strand species to biochemically implement each trans-lator circuit. Strand species Translator A Translator B
Output strand
N N
Buffer strand 1 0Gate strand
N N
Auxiliary strand N + 2 N N N : number of output strand In the present work, we proposed muti-species consensus networks by chemicalreaction networks and showed that they can perform as concentration-to-strandtranslators. The dynamics of the translator was understood as heteroclinicnetwork from the viewpoint of nonlinear dynamical systems. It was success-fully demonstrated that two types of translators output a unique output strandspecies corresponding to a value of the input strand concentration. Transla-tor A and B showed a slightly different behaviors which offers tunable optionsdepending on applications. Our translators map analog concentration signalto digital information, that is, set of multiple DNA strands. This functionalityprovides easy-to-use biomarkers which are potentially useful for on-site personalhealthcare systems since no costly fluolescence-based techniques are required.Such direction of research is left to futre work.
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