Received: September 24, Research Article. pubs.acs.org/synthbio

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1 wdb ACSJCA JCA..65/W Unicode research.3f (R3.6.i:32 2. alpha 39) 25/7/5 :3: PROD-JCAVA rq_ /3/26 7:52:52 JCA-DEFAULT pubs.acs.org/synthbio Cheical Reaction Networks for Coputing Polynoials 2 Sayed Ahad Salehi,* Keshab K. Parhi, and Marc D. Riedel 3 Departent of Electrical and Coputer Engineering, University of Minnesota, Minneapolis, Minnesota 5555, United States ABSTRACT: Cheical reaction networks (CRNs) provide a 5 fundaental odel in the study of olecular systes. Widely used 6 as foralis for the analysis of cheical and biocheical systes, 7 CRNs have received renewed attention as a odel for olecular coputation. This paper deonstrates that, with a new encoding, 9 CRNs can copute any set of polynoial functions subject only to the liitation that these functions ust ap the unit interval to itself. These polynoials can be expressed as linear cobinations of 2 Bernstein basis polynoials with positive coefficients less than or 3 equal to. In the proposed encoding approach, each variable is represented using two olecular types: a type- and a type-. The value is the ratio of the concentration of type- olecules to 5 the su of the concentrations of type- and type- olecules. The proposed encoding naturally exploits the expansion of a 6 power-for polynoial into a Bernstein polynoial. Molecular encoders for converting any input in a standard representation to 7 the fractional representation as well as decoders for converting the coputed output fro the fractional to a standard representation are presented. The ethod is illustrated first for generic CRNs; then, an exaple is apped to DNA strand- 9 displaceent reactions. 2 KEYWORDS: olecular coputing, polynoials, DNA strand-displaceent reaction, ass-action kinetics 2 t has long been recognized that, viewed fro a atheatical 22 I standpoint, a set of cheical reactions can exhibit rich 23 dynaical behavior. On the coputational front, there has 2 been a wealth of research into efficient ethods for siulating 25 cheical reactions, ranging fro ordinary differential equations 26 (ODEs) 2 to stochastic siulation. 3 On the atheatical front, 27 entirely new branches of theory have been developed to 2 characterize cheical dynaics. The idea of coputation 29 directly with cheical reactions, as opposed to writing 3 coputer progras to analyze cheical systes, dates back 3 to the seinal work of Adlean. 5 In this context, a cheical 32 reaction network (CRN) transfors input concentrations of 33 olecular types into output concentrations and thus iple- 3 ents coputation. It should be noted that the equilibriu 35 concentrations of the output olecules are considered as the 36 coputed output of the syste. 37 The question of the coputational power of cheical 3 reactions has been considered by several authors. Magnasco 39 deonstrated that cheical reactions can copute anything that digital circuits can copute. 6 Soloveichik et al. deonstrated that cheical reactions are Turing Universal, 2 eaning that they can copute anything that a coputer 3 algorith can copute. 7 This work was applicable to a discrete, stochastic odel of cheical kinetics. The coputation is 5 probabilistic; the total probability of error of the coputation 6 can be ade arbitrarily sall (but not zero). 7 Either explicitly or iplicitly, prior work has considered two types of encodings for the input and output variables of 9 CRNs:,9. The value of each variable corresponds to the 5 concentration of a specific olecular type; we will call this 5 the direct representation The value of each variable is represented by the difference 53 between the concentrations of a pair of olecular types; we will 5 call this the dual-rail representation In this paper, we introduce a new representation that we call 56 the fractional representation. A pair of olecular types is 57 assigned to each variable, e.g., (X,X ) for a variable x. The 5 value of the variable is deterined by the ratio 59 x [ X [ X + [ X () 6 Evidently, the value is confined to the unit interval [,. The proposed encoding ethod is inspired by prior work in designing stochastic circuits. 2,5 Such circuits operate on randoized bit streas with the values of variables represented as the fraction of s versus s in the streas. In a sense, the ain contribution of this paper is the application of this theory fro stochastic circuit design to CRNs. On the basis of the fractional representation in eq, we propose a CRN fraework for coputing univariate polynoials that ap the unit interval [, to itself. We deonstrate that a CRN exists that coputes any such polynoial. The full syste consists of an encoder, the coputation CRNs, and a decoder, as shown in Figure. The encoder converts the input olecular type, X (for [X ), into two olecular types, X and X, such that Received: Septeber 2, f 7 75 XXXX Aerican Cheical Society A DOI:.2/acssynbio.5b63

2 ACS Synthetic Biology Figure. Whole syste perforing coputation in fractional representation. [ X [ X [ X + [ X 76 The decoder converts the ratio of two olecular types, Y and 77 Y, into a single olecular type, Y, as the final output such that [ Y [ Y [ Y + [ Y 7 We describe the design of the encoder and decoder in the 79 Encoding and Decoding section. We first illustrate the coputation CRN block with a siple exaple. Consider the following CRN X + X S + X + X X + X 2S + X + X X + X S2 + X + X 2 (a),,,,,,,, 2 2, 2, 2 2, 2, Y Y 3 (b) Set the initial concentrations as [ B,.25nM [ B,.75 [ B.75nM b, [ B, + [ B, [ B,.75nM [ B,.25 [ B.25nM b, [ B, + [ B, [ B 2,.5nM [ B2,.5 [ B.5nM b2 2, [ B2, + [ B2, Although not obvious, it ay be shown that this CRN 6 coputes the function x x x + 7 (2) where x. 9 Note that any unit could have been used in this paper for the 9 olecular concentrations; nm has been used due to the 9 practical utility. 92 The CRN is coposed of two sets of reactions: the three 93 reactions in group (a) are referred as control generating 9 reactions, and the six reactions in group (b) represent the 3 2 transferring reactions. The control generating reactions generate the olecules that control the transferring reactions (siilar to the way that the control bits select outputs fro inputs with ultiplexors in electronic circuits). However, the control olecules represent analog values and transfer inputs to outputs proportionally. We note that the transferring reactions are conceptually siilar to the olecular reactions proposed in ref 3 for ipleenting Markov Chains. We provide details regarding the synthesis ethod in the Synthesizing CRNs for Coputing Polynoials section. Here, we siply note that, given a polynoial y(x), the first step is to convert it to its Bernstein polynoial equivalent g(x). For the polynoial y(x) in eq 2 3 g() x x + x x + x [( ) [2 ( ) (A discussion of the ath behind this is given in the Proof Based on the Mass-Action Kinetics section.) Note that the coefficients of the Bernstein polynoial correspond to the values of b i for i,,2. These values are used to initialize the olecular types B i, and B i, for i,,2. In fact, coputing with cheical reaction networks consists of two parts. First, choose a CRN as a eans of building the dynaical syste. Second, siulate a purposefully chosen dynaical syste to equilibriu. By introducing the B i, and B i, species, the concentrations of which are tie-invariant and fixed to what would have been rate constants, we propose changes to the first part that result in the sae dynaical syste siulated in the second part. Suppose we want to evaluate y(x) at x.5. We would initialize X X.5 nm such that x [ X.5 [ X + [ X (3) () We would set the initial concentration of the other types to zero. The control generating reactions use X and X to produce the control olecules S, S, and S 2, and transferring reactions use control olecules to copute the output. The output value, y(x), is coputed as the ratio of the final concentrations of Y and Y, i.e., x [ Y [ Y + [ Y (5) The siulation results for evaluating this exaple at x.5 using a continuous ass-action kinetics odel are shown in Figure 2. As tie t, the ratio [ Y( t) [ Y( t) + [ Y( t) f2 35 approaches the correct value of y(.5) RESULTS AND DISCUSSION 37 Representation. In our ethod, the Bernstein representa- 3 tion of a polynoial is a key eleent. We briefly describe the 39 relevant atheatics. The faily of n + polynoials of the for n i ni B x in(), x( x), i,..., n i (7) are called Bernstein basis polynoials of degree n. A linear cobination of Bernstein basis polynoials of degree n, (6) 2 3 B DOI:.2/acssynbio.5b63

3 ACS Synthetic Biology Figure 2. Siulation results for the CRN ipleenting the polynoial x x x + at x.5. These were obtained fro an ODE siulation of the ass-action kinetics. n g() x b B () x in, in, 5 i () 6 is a Bernstein polynoial of degree n. The b i,n s are called 7 Bernstein coefficients. Polynoials are usually represented in power for, i.e., x a x n in, i 9 i (9) 5 We can convert such a power-for polynoial of degree n into 5 a Bernstein polynoial of degree n. The conversion fro the 52 power-for coefficients, a i,n, to the Bernstein coefficients, b i,n, is 53 a closed-for expression () i i j bin, ajn,, i n n j () j 5 () 55 For a proof of this, the reader is referred to ref. 56 Generally speaking, a power-for polynoial of degree n can 57 be converted into an equivalent Bernstein polynoial of degree 5 greater than or equal to n. The coefficients of a Bernstein 59 polynoial of degree + ( n) can be derived fro the 6 Bernstein coefficients of an equivalent Bernstein polynoial of 6 degree as b, i i i b + i, bi, + bi, i + + b, i + 62 () 63 Again, for a proof, the reader is referred to ref. 6 By encoding the values of variables as the ratio of the 65 concentrations of two olecular types, x [ X [ X + [ X 66 we can only represent nubers between and. Accordingly, 67 our ethod synthesizes functions that ap the unit interval 6 [, onto itself. The ethod can also synthesize functions that 69 ap the unit interval to the negative unit interval [,. This coputes the negative of a function that aps the unit interval to itself. As was shown in Exaple, the coefficients of the polynoials that we copute are also represented in this fractional for. Fortunately, Qian et al. proved that any polynoial that aps the unit interval onto the unit interval can be converted into a Bernstein polynoial with all coefficients in the unit interval. 5 Synthesizing CRNs for Coputing Polynoials. In this section, we present a systeatic ethodology for synthesizing CRNs that can copute polynoials. As discussed in the previous section, we assue that the target polynoial is given in Bernstein for with all coefficients in the unit interval. The ethod is coposed of two parts, designing the CRN and initializing certain types to specific values, as discussed in the following section. Designing the CRN. The CRN reactions consist of two sets of reactions that we call the control generating reactions and the transferring reactions. First, consider the control generating reactions. When our proposed CRN is coputing a polynoial of degree, each control generating reaction should have reactants. The reactions consist of all possible cobinations of olecules chosen fro X and X. These ( + ) reactions are listed in eq 2. In the first reaction of eq 2, all reactants are chosen fro olecules of X and produce olecules of S. In the second, ( ) olecules of X and one olecule of X are cobined to produce olecules of S. Siilarly, the (i + )st reaction contains i olecules of X and ( i) olecules of X. The total nuber of possible reactions, as shown in eq 2, is ( + ). X S + X X + ( ) X S + X + ( ) X 2 X + ( 2) X S + 2 X + ( 2) X 2 2 ix + ( i) X S + X + ix + ( i) X i i, X S + X (2) A degree Bernstein polynoial has ( + ) Bernstein coefficients. We consider ( + ) pairs of types (B j,, B j, ) for j,,..., to represent these coefficients. The transferring reactions produce the final output, Y or Y, fro the products of the control generating reactions, the S j s. They do so proportionally to the Bernstein coefficients. S j goes to Y if it cobines with B j, and goes to Y if it cobines with B j,. Accordingly, there are 2( + ) transferring reactions as listed in eq C DOI:.2/acssynbio.5b63

4 ,,,,,,,,,,,, Y Y 2 (3) 2 The nuber of required reactions for the ipleentation of 22 a Bernstein polynoial of degree is equal to We also t 23 need olecular types listed in Table. 2 Initialization. We initialize the pair (B j,,b j, ) according to 25 the Bernstein coefficients b j,, i.e., we have [ Bj, bj, [ Bj, + [ Bj, 26 () 27 For siplicity, we initialize B j, and B j, such that the su 2 [B j, +[B j, is the sae arbitrary value for all js. Call the su 29 [B j,+[b j, B for all js. In fact, we first calculate the values of 22 Bernstein coefficients using eq and then initialize B j, and B j, 22 as [B j, B b j, and [B j, B[B j,. (For the exaple in 222 the introduction, we considered B nm.) 223 We initialize the corresponding olecular type in the input 22 pair (X, X ) based on the value x in at which the polynoial is 225 to be evaluated, i.e., [ X xin 226 [ X + [ X (5) All of the other interediate types, i.e., the S 227 j s as well as the 22 output types Y and Y, are initialized to zero. Proof Based on the Mass-Action Kinetics. We use an 229 ordinary differential odel of the ass-action kinetics to prove the correctness of our proposed CRN design. The control generating reactions (eq 2) produce type S 232 j whereas the transferring reactions (eq 3) consue the. 233 ACS Synthetic Biology Table. Nuber of Required Molecular Types in the Proposed CRN for a Polynoial of Degree represented olecular type 23 Therefore, the ODEs for type S j are nuber of olecular types X, X 2 S j + B i,, B i, 2 +2 Y, Y 2 total 3 +7 d[ S d[ S d[ S d[ S,,,, [ X [ B [ S [ B [ S [ X [ S ([ B + [ B ),,,, X [ [ X [ B [ S [ B [ S X [ [ X [ S([ B + [ B ) [ X [ X [ Bk, [ Sk [ Bk,[ Sk k k k [ X [ X [ S ([ B + [ B ) k k k, k, k k k At equilibriu, copute the S j s as [ S j,,,, [ X [ B [ S [ B [ S [ X [ S ([ B + [ B ) d[ Sj j [ X [ j j X [ B + [ B for all js. Accordingly, we can j j, j, (6) 237 Now, we write the ODEs for the output types Y and Y. On the basis of the transferring reactions (eq 3), we have d[ Y [ B,[ S + [ B,[ S + + [ B,[ S [ Y d[ Y [ B,[ S + [ B,[ S + + [ B,[ S [ Y (7) d[ Y At equilibriu, and d[ Y [ Y [ B [ S + [ B [ S + + [ B [ S,,, [ Y [ B [ S + [ B [ S + + [ B [ S,,, () According to the fractional encoding, the output value y is calculated as y [ Y [ Y + [ Y {[ B [ S + [ B [ S [ B [ S },,, {([ B [ S + [ B [ S [ B [ S ),,, + ([ B [ S + [ B [ S + + [ B [ S )},,, (9) With the assuption that ([B j, +[B j, ) B for all js, we have D DOI:.2/acssynbio.5b63

5 ACS Synthetic Biology y {[ B,[ S + [ B,[ S + + [ B,[ S} {([ B, + [ B,)[ S + ([ B, + [ B,)[ S + + ([ B + [ B )[ S },, [ B [ S + [ B [ S + + [ B [ S,,, [ B [ S j j, j B( [ S) B([ S + [ S + + [ S ) j j 27 (2) 2 By substituting [S i fro eq 6, j j j [ X [ X [ B j j, B y j j [ X j [ X B j B 29 (2) We know that + 25 j [ X j [ X j ([ X [ X) j due 25 to binoial theore; therefore, the denoinator can be 252 replaced by ([X +[X ). y [ B j j, B ([ X + [ X) j j j j [ X [ X [ B j j j, B j [ X [ X ([ X + [ X) b j j j j, ( x) x j 253 (22) 25 Eq 22 is exactly the expression for a Bernstein polynoial 255 representation of degree for y(x). Thus, this CRN coputes 256 y(x). Note that y is finite because [X and [X 257. Therefore, for every initial state of interest, our proposed 25 CRN coputes a stable equilibriu state. 259 Note that, in general, all the rate constants in our CRNs are 26 assued to be equal to each other. More precisely, on the basis 26 of the proof, there are three categories of reactions with respect 262 to the rate constants: the control generating reactions, the 263 transferring reactions, and the last two annihilation reactions of 26 the transferring reactions. All reactions in each of these 265 categories are required to have the sae rate constant. 266 Encoding and Decoding. Our proposed CRNs perfor 267 coputations on the fractional representation in eq. In this 26 section, we present cheical reactions that convert between 269 this representation and a direct representation, where the 27 value of each variable is represented directly the concentration 27 of a olecular type. 272 Encoding. Let a olecular type X denote the direct 273 representation of the input value x and (X,X ) denote the 27 olecular pair for its fractional representation. Assue that the 275 total concentration of X and X is nm. Then, we have [ X [ X [ X [ X [ X + [ X [ X [ X [ X + [ X nm (23) Because the concentration values for X and X are the sae and subsequent stages do not consue the, type X can be directly used as type X in the fractional representation. For generating X, we ust ipleent subtraction, which is a little tricky. We designed the following reactions (eq 2) for this task. T is initialized to nm, and B is an interediate olecular type with an initial value of zero. T X + T B + X X X + B X (2) For these reactions, the ODEs are d[ X d[ B [ T [ B[ X [ X [ X [ B[ X and at equilibriu, we have d[ X [ X [ T [ B[ X d[ B [ X [ B[ X By substituting [B[X fro eq 27 to 26, we have [ X [ T [ X (25) (26) (27) (2) Eq 2 is valid when [T [X. Because [X cannot be negative, for [T [X, [X. Thus, the equilibriu ODE solution for these reactions is [ T [ X if [ T [ X [ X if [ T [ X (29) If T is initialized to nm, the reactions in 2 copute [X [X. Thus, the reactions in 2 encode the input concentration of X as a pair of concentrations (X,X ) in a fractional representation. Here, in fact, X can substitute for X, as discussed above. Note that the concentration of X is initialized to zero at the outset. Decoding. For the output of our olecular coputing syste, we convert the fractional representation back to a direct representation. If the fractional output is represented by the pair of olecules (Y,Y ) and the direct output by Y, we have [ Y [ Y [ Y + [ Y (3) In other words, we need to copute the suation of [Y and [Y and then the ratio of [Y over this suation. For this coputation, we use the reactions proposed in ref 6. We will show that the reactions in 3 copute [Y [Y +[Y E DOI:.2/acssynbio.5b63

6 32 ACS Synthetic Biology and the reactions in 32 copute the final output [ Y [ Y [ Y [ Y [ Y + [ Y Y Y + Y Y Y + Y. 33 Y (3) Y Y + Y 3 Y + Y Y (32) 35 According to the ODEs of the reactions in 3, we have d[ Y [ Y + [ Y [ Y 36 and at equilibriu d[ Y [ Y [ Y + [ Y 37 (33) 3 Siilarly, for the reactions in 32, we have d[ Y [ Y [ Y[ Y [ B,.6nM [ B,.2nM [ B,.3nM [ B,.5nM [ B 2,.5nM [ B 2,.3nM [ B 3,.2nM [ B 3,.6nM (37) We ap our design to DNA strand-displaceent reactions and evaluate it for different input values between and. The values of y coputed by these CRNs are plotted against x and shown with the target polynoial y(x) in Figure 3. Table 2 tabulates the coputed values of y(x) and the corresponding errors f3t and the equilibriu value of [Y is d[ Y [ Y [ Y [ Y 32 [ Y [ Y + [ Y (3) 32 Therefore, the set of reactions in 3 and 32 ipleent the 322 decoding of the output. 323 METHODS AND MATERIALS DNA Ipleentation. The proposed CRN for coputing 32 polynoials is general in the sense that it can be ipleented by any cheical or biocheical syste with ass-action kinetics. As a practical ediu, we choose DNA strand- displaceent reactions. Indeed, Soleveichik et al. deonstrated that DNA strand-displaceent reactions can eulate the 33 kinetics of any CRN.7 They presented a software tool that 33 aps cheical CRNs to DNA reactions. 332 We illustrate with the following target function x + x x + x 333 (35) 33 The CRN includes reactions for the encoder, coputation, 335 and decoder parts. The Bernstein polynoial for y(x) is 2 g() x x + x x + x x + x [( ) 3 5 [3 ( ) 2 3 [3 2 ( ) (36) 337 Fro the Bernstein coefficients, we initialize the types (B i,,b i, ) 33 for i,,2,3 as Figure 3. Values of y(x) coputed by a DNA ipleentation of proposed CRN. Blue line: target y(x). Red stars: coputed by DNA reactions. For the DNA ipleentation, we used the paraeters based 36 on the exaples in ref 7. The axiu strand displaceent 37 rate constant is q ax 6 M s, and the initial concentration 3 of auxiliary coplexes is set to C ax 5 M. If the 39 concentration of auxiliary species, C ax, is uch larger than the 35 axiu concentration of other species (i.e., in proposed 35 CRNs, C ax nm), then, as described in ref 7, we can 352 assue that over the siulation tie the auxiliary concen- 353 trations reain effectively constant. Therefore, DNA reactions 35 correctly eulate the CRN independent of the auxiliary 355 concentrations. Note that, for this assuption, the siulation 356 tie and reaction rates should not be very large values Although these requireents have been et in our siulations, 35 errors exist. 359 As we describe later, the error stes fro the fact that each 36 olecular reaction is ipleented by a sequence of DNA 36 F DOI:.2/acssynbio.5b63

7 f ACS Synthetic Biology Table 2. Accuracy of a DNA Strand Displaceent Ipleentation of a CRN Coputing x + x x + x 3 Using the Proposed Method 362 strand displaceent reactions; the concentrations of auxiliary 363 olecules, C ax, is bounded. In fact, if C ax, the DNA 36 siulation results converge to ODE siulation results. Further 365 details concerning the analysis of errors when ipleenting 366 CRNs with DNA strand displaceent reactions, as well as a 367 proof of convergence of a DNA ipleentation to the target 36 CRN, can be found in the Supporting Inforation of refs and. 37 Using the ethod presented in ref 7, each cheical reaction 37 with reactants and nonzero products can be eulated by DNA strand displaceent reactions. For exaple, biolec- 373 ular reactions are apped to three DNA strand displaceent 37 reactions. To illustrate this, we present a sequence of DNA 375 strand displaceent reactions that are used to siulate a 376 biolecular reaction with three products. 377 As described in refs 7 and, three DNA reactions, RR3 37 shown in Figure, ipleent the olecular reaction k i A + B A + B + C. Uniolecular reactions without prod- 379 uct, e.g., Y, can be ipleented by a single DNA strand f5 3 displaceent reaction. The DNA reaction shown in Figure 5 x in coputed y(x) error (%) eulates the reaction A k i. The toehold of strand A binds to Figure 5. DNA strand displaceent reaction that eulates reaction A ki. its copleentary part of gate olecule G and produces 32 double strand W and single strand W 2. Because W and W 2 33 cannot bind together, the reaction is unidirectional. 3 Table 3 suarizes the nuber of cheical and DNA strand displaceent reactions for each group in our proposed 36 ethod for coputing the polynoial of degree. 37 Table 3. Nuber of Cheical and DNA Strand- Displaceent Reactions for Each Group of the Proposed CRN for Coputation of a Bernstein Polynoial of Degree group of reactions type of cheical reaction nuber of cheical reactions nuber of DNA reactions control reactions with + ( + ) ( + ) generating reactants transferring biolecular (2 + 2) 3 uniolecular 2 2 without product total CONCLUSIONS We have introduced a new encoding for coputation with 39 CRNs: the value corresponding to each variable consists of the 39 ratio of the concentration of a olecular type to the su of two 39 types. On the basis of this fractional representation, we 392 proposed a ethod for coputing arbitrary polynoials that 393 ap the unit interval [, to itself or to [,. This is a rich 39 class of functions. 395 Coputation of polynoials with cheical kinetics has been 396 attepted before by Buisan et al. 6 Copared to our ethod, 397 their ethod requires fewer olecular types and fewer 39 reactions ( olecular types and 3 olecular reactions for t3 k i Figure. DNA strand displaceent reactions that eulate reaction A + B A + B + C. G DOI:.2/acssynbio.5b63

8 a coplete polynoial of degree ). However, unlike our approach, their CRNs are dependent on reaction rates. In fact, 2 for each coefficient of the desired polynoial, they need a 3 distinct reaction rate. This is unrealistic. Note that our approach only requires a single rate. 5 Soloveichik et al., 7 as well as earlier work, 6,9,2 attepted to 6 achieve Turing universality with cheical reactions. Although it 7 is possible to copute polynoials with their CRNs, they did not provide a systeatic fraework for doing so. 9 The fractional representation that we propose is a non- standard representation. However, we note that it is siilar to encodings found in nature. Many biological systes have 2 species with two distinct states. For exaple, it is coon for 3 an enzye to have active and inactive states. The ratio of the concentrations of the two states is a eaningful value. This is 5 quite analogous to our representation. 6 Clearly, the priary interest of this work is theoretical. CRNs 7 are a fundaental odel of coputation, abstract yet conforing to the physical behavior of cheical systes. 9 Delineating the range of behaviors of such systes has 2 intellectual erit. These results ay also have practical 2 applications. 22 Control theory has played a rearkable role in atheatical 23 biology, providing a fraework for odeling and designing the 2 dynaic behavior of systes such as biological oscillators Polynoials play a central role in control and oscillation. In 26 fact, the transfer function of a control syste, which is the ratio 27 of its output to its input in the Laplace doain, is the ratio of Az () a+ az++ anz two polynoials, i.e., H() z n. 2 Non- 2 Bz () b+ bz++ bz 29 linear feedback in oscillators can be ipleented by 3 polynoials. 25,26 3 Practitioners in synthetic biology are striving to create 32 ebedded controllers, viruses and bacteria that are 33 engineered to perfor useful olecular coputation in situ 3 where needed, for instance, for drug delivery and biocheical 35 sensing. Such ebedded controllers ay be called upon to 36 perfor coputation such as filtering or signal processing. 37 Coputing polynoial functions is at the core of any of 3 these coputational tasks. 39 In future work, we will attept to optiize the CRNs that we propose for coputing polynoials, reducing the nuber of olecular types as well as the nuber of reactions. We will 2 also attept to generalize the ethod to copute a wider class 3 of operations. ACS Synthetic Biology AUTHOR INFORMATION 5 Corresponding Author 6 *E-ail: saleh22@un.edu. 7 Notes The authors declare no copeting financial interest. 9 ACKNOWLEDGMENTS 5 The authors gratefully acknowledge nuerous constructive 5 coents of the reviewers. This research is supported by the 52 National Science Foundation Grant CCF REFERENCES 5 () Horn, F., and Jackson, R. (972) General Mass Action Kinetics. 55 Arch. Ration. Mech. Anal. 7, (2) E rdi, P., and To th, J. (99) Matheatical Models of Cheical 57 Reactions: Theory and Applications of Deterinistic and Stochastic 5 Models, Manchester University Press. (3) Gillespie, D. (977) Exact Stochastic Siulation of Coupled 59 6 () Strogatz, S. (99) Nonlinear Dynaics and Chaos with (5) Adlean, L. (99) Molecular Coputation of Solutions to 6 65 (6) Magnasco, M. O. (997) Cheical Kinetics is Turing Universal (7) Soloveichik, D., Cook, M., Winfree, E., and Bruck, J. (2) () Chen, H., Doty, D., and Soloveichik, D. 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