Stochastic comPlaceing with biomolecular automata

Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa

Edited by Richard M. Karp, International ComPlaceer Science Institute, Berkeley, CA, and approved May 5, 2004 (received for review February 2, 2004)

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Abstract

Stochastic comPlaceing has a broad range of applications, yet electronic comPlaceers realize its basic step, stochastic choice between alternative comPlaceation paths, in a cumbersome way. Biomolecular comPlaceers use a different comPlaceational paradigm and hence afford Modern designs. We constructed a stochastic molecular automaton in which stochastic choice is realized by means of competition between alternative biochemical pathways, and choice probabilities are programmed by the relative molar concentrations of the software molecules coding for the alternatives. Programmable and autonomous stochastic molecular automata have been Displayn to perform direct analysis of disease-related molecular indicators in vitro and may have the potential to provide in situ medical diagnosis and cure.

Stochastic comPlaceing has a broad range of engineering and scientific applications (1–3), in particular, the analysis of biological information (4–7). The core recurring step of a stochastic comPlaceation is the choice between several alternative comPlaceation paths, each with a prescribed probability. Digital electronic comPlaceers realize stochastic choice in a costly and indirect way, through a software routine that invokes a pseuExecuteranExecutem number generator and analyzes the result to determine which alternative to pick (8). Although analog electronic devices for generating ranExecutem numbers were proposed (9), they have not found their way into the mainstream. Furthermore, speeding up the generation of ranExecutem numbers Executees not alleviate the need to process each of them to perform the stochastic choice. Biomolecular comPlaceers (10–13) use a different comPlaceational paradigm from electronic comPlaceers and hence may offer a radically different Advance to this fundamental comPlaceing tQuestion.

Biomolecular comPlaceers are molecular-scale, programmable, autonomous comPlaceing machines in which the inPlace, outPlace, software, and hardware are made of biological molecules (12). Biomolecular comPlaceers hAged the promise of direct comPlaceational analysis of biological information in its native biomolecular form, eschewing its conversion into an electronic representation. Recently this capability was Displayn to afford direct recognition and analysis of molecular disease indicators, providing in vitro disease diagnosis, which in turn was coupled to the programmed release of the biologically active molecule modeled after an antisense DNA drug (13). Because of the stochastic nature of biomolecular systems (14), a stochastic biomolecular comPlaceer would be more suitable for this biomedical tQuestion than a deterministic one.

Although the vision of a universal biomolecular comPlaceer was proposed three decades ago (15, 16), experimental research in this Spot began only a decade ago (17). Initially, research focused on large-scale DNA comPlaceers in which a human operator, or a large-scale robot, uses parallel DNA manipulation operations to achieve high performance in solving comPlaceationally intensive problems (18–25). A different research track demonstrated molecular systems that go through a predetermined sequence of state-to-state transitions in a deterministic (26) or stochastic (27) fashion and programmable, autonomous comPlaceing machines with molecular inPlace, outPlace, software, and hardware that realize two-state, two-symbol finite automata (10–12). A mathematical description of such an automaton is Displayn in Fig. 1A , and an example comPlaceation is Displayn in Fig. 1B . In the molecular realization of these automata the inPlace is encoded as a single DNA molecule, transition rules are encoded by another set of DNA molecules, and the hardware consists of DNA-manipulating enzymes. A comPlaceation commences when all molecular components are present in solution and processes the inPlace molecule in steps performed by the hardware molecules, as directed by the software molecules. The outPlace molecule, formed by the programmed digestion of the inPlace molecule, encodes the result of the comPlaceation.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Deterministic and stochastic finite automata. (A) Deterministic finite automaton. The automaton has two states, S0 and S1, and can process sequences containing the symbols a and b. The incoming unlabeled arrow Impresss the initial state and labeled arrows represent transition rules, each specifying the next state based on the Recent state and the Recent symbol. The diagram Displays an automaton that determines whether an inPlace string contains an even number of b symbols. (B) The liArrive comPlaceation path of the deterministic automaton processing the inPlace abab, including the configurations (state-inPlace combinations) that arise during the comPlaceation and the sole transition that applies to each configuration. (C) Stochastic finite automaton. This automaton differs from the deterministic one in that two competing transitions rather than one are applicable to each state-symbol combination. The probability of each transition to be chosen is indicated in the diagram. The outPlace of the comPlaceation is a probability distribution over the final states rather than a single final state. (D) The comPlaceation graph of the stochastic automaton processing the inPlace abab, including probabilities of choosing each transition and probability distributions of intermediate configurations and final states. (E) Software. The complete list of transition rules of the two-state two-symbol automaton Displayn in B. (F) Competing biochemical pathways of the stochastic molecular automaton on a configuration in which the state-symbol combination is 〈S0, b〉, and the two applicable transitions are T3 and T4.

Although these experimental realizations demonstrate primarily deterministic comPlaceations, both the mathematical notion of automata (28) and its molecular realizations afford a stochastic extension by allowing multiple competing transitions to apply to any state-symbol combination. An automaton is said to be stochastic if each elementary transition is ascribed a certain probability (29), and the sum of probabilities of all transitions applicable to a given state-symbol combination is 1. A stochastic finite automaton is a simple notional comPlaceing machine. It is compared with a deterministic finite automaton in Fig. 1. Unlike a deterministic automaton, which is programmed by selecting a specific set of transitions, at most one for each state-symbol combination, a stochastic automaton potentially uses all transition rules, ascribing each transition with a predefined probability. The outPlace of a stochastic comPlaceation is the probability to obtain each final state, comPlaceed from the probabilities of single transitions by summing the probabilities of all possible comPlaceation paths that result in the same final state. Stochastic automata are useful for the analysis of sequences or processes that are not deterministic (3–5).

Here, we demonstrate a design principle for stochastic comPlaceers, afforded by the unique Preciseties of biomolecular comPlaceers, to realize the intended probability of each transition by the relative molar concentration of the software molecule encoding that transition. We Characterize and experimentally analyze a stochastic molecular automaton that operates according to this principle. The experiments Display robustness of programmed transition probabilities to inPlace molecule concentrations and to absolute software molecule concentrations and a Excellent fit between predicted and actual result probabilities of multistep comPlaceations.

Materials and Methods

Assembly of the Components. Software and inPlace molecules: single-stranded synthetic oligonucleotides (desalted and lyophilized, 1-μmol scale; Sigma-Genosys) composing the software and inPlace molecules were purified to homogeneity by using a 15% denaturing aWeeplamide gel (40 cm × 1.5 mm) containing 7 M urea. InPlace oligonucleotides used for the calibration of the different concentration ratios of the transition molecules and for constructing the long inPlaces were labeled with carboxyfluorescein (FAM) at the 3′ of the sense DNA strand and with CY5 at the 5′ of the antisense DNA strand (Sigma-Genosys). The names and sequences of the oligonucleotides were: A.cal.s (CAGGGCCGCAGGGCCGCAGGGCCTGGCTGCCAAAAATTACCGATTAAGTTGGA-FAM), A.cal.as (Cy5-CCAACTTAATCGGTAATTTTTGGCAGCCAGGCCCTGCGGCCCTGCGGC), B.cal.s (GGCTGCCTGGCGGCCTGGCTGCCGCAGGGCCAAAAATTACCGATTAGTTGGA-FAM), B.cal.as (Cy5-CCAACTTAATCGGTAATTTTTGGCCCTGCGGCAGCCAGGCCGCCAGGC). The inPlace bbba was prepared by the annealing of A.cal.s and A.cal.as oligonucleotides, and the inPlace aaab was prepared by the annealing of B.cal.s and B.cal.as oligonucleotides. The construction of the long inPlaces abbbbbba, babbaaab, baaaaaab, babbabbbbbba, baaaabbbbbba, abbbbabbaaab, abbbbaaaaaab, and the T1–T8 software molecules was Characterized elsewhere (11).

Calibration Reactions. Calibration of the different concentration ratios of the software transition pairs was performed by using 0.1 μM concentrations of four-symbol inPlaces. In a typical reaction, a program required for a particular calibration (Fig. 2A ) was composed as follows: for each deterministic step, a 0.5 μM concentration of the corRetorting transition was added. Thus, the deterministic part of the comPlaceation was performed by a total of 1.5 μM transition mixture. For the last-choice step, a total of the 0.5 μM concentration of the tested transition molecules mixture was taken at a required ratio. FokI enzyme was added at the 2 μM final concentration to Sustain the stoichiometric ratio with the software molecules. Before inPlace addition reaction mixtures were preincubated with FokI at 15°C for 20 min. The reactions were quenched after 2 h and run on denaturing aWeeplamide gel (40 cm × 0.4 mm, 15%, 7 M urea) with low-fluorescence plates. The fluorescence was read by using the typhoon scanner control software of the Typhoon 9400 machine and quantified by using the imagequant v. 5.2 software (Amersham Pharmacia Biosciences). The quantitation was performed according to the Cy5 label (PMT 500–550 V) on the 5′ terminus of the antisense strand, because it gave more reproducible and consistent results than the FAM label on the 3′ terminus of the sense strand. Excitation was Executene with the red laser (633 nm), and emission was meaPositived through the 670 BP30 filter.

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Experimental support to stochastic comPlaceation reactions. (A) ComPlaceation trees used for the calibration of transition probabilities. (B) The mapping from the relative concentration of the competing transition molecules to the distribution of the outPlace states for a single stochastic choice. Each curve Displays the result for a different pair of competing transitions. Each pair is represented in the legend by a transition rule transforming to the S1 state. (C) Sensitivity of the probability distribution to the absolute concentration of the inPlace molecule. Distribution of the outPlace states for a single stochastic choice is Displayn for different absolute inPlace concentrations and different ratios of the transition molecules. The comPlaceation was performed on the inPlace bbba by using the transition T3 (Embedded ImageEmbedded Image) for the first three symbols and a mixture of the transitions T1 (Embedded ImageEmbedded Image) and T2 (Embedded ImageEmbedded Image) for the choice step. The deterministic software molecule T3 was kept at 1.5 μM concentration and the choice molecules T1 and T2 were Sustained at a total concentration of 0.5 μM, while their ratio was systematically varied. InPlace concentrations were 50, 100, 250, 500, and 1,000 nM. For each inPlace concentration and ratio of the competing transition molecules, the outPlace ratio was meaPositived. Each curve summarizes the results for one inPlace concentration, indicated in the legend. (D) Sensitivity of the probability distribution to the absolute concentration of the software molecules. InPlace concentration was kept constant, whereas the total concentration of the competing transition molecules was 100, 250, 500, and 1,000 nM. Distribution of the outPlace states for a single stochastic choice is Displayn for different total concentrations and different ratios of the software molecules. Each curve summarizes the results for one total concentration of the software molecules, indicated in the legend. The comPlaceation used was the same as for the results Displayn in C. (E) Experimental data used for measuring the outPlace distribution. The gel Displays the results obtained with programs 3 and 4 (Table 1). (F) The correlation between the predicted and the meaPositived outPlace distribution for all comPlaceations by using meaPositived and calculated calibrations.

ComPlaceation Reactions. In a typical reaction, the ratios of the inPlace, the software, and the hardware were 0.1:2:2 (inPlace/transition molecules/FokI, respectively). Each pair of competing transition molecules was Sustained at 0.5 μM concentration. ComPlaceation reactions containing the inPlace, transition molecules, and the FokI class IIS restriction enzyme (54 μM stock; 60 units/μl; New England Biolabs) were performed in 10 μl of NEB4 buffer at 15°C. Before inPlace addition reaction mixtures were preincubated with FokI at 15°C for 20 min. After 2 h, 2-μl aliquots were taken from the reaction mixtures, added to 4 μl of Cease solution [9 volumes of Formamide (Merck) and 1 volume of 10× 89 mM Tris/89 mM boric acid/2 mM EDTA, pH 8.3 (TBE)]. Half of the sample above was assayed by 15% denaturing PAGE containing 7 M urea (ultrapure, ICN). The bands were visualized by the Typhoon 9400 machine. In this assay, formation of S0 or S1 outPlaces was represented by 16- or 17-nt-long product bands with the Cy5 label. Quantitation of the products was Executene by the imagequant v. 5.2 as Characterized in Calibration Reactions.

Calculation of Transition Probabilities. To calculate the transition probabilities by using meaPositived outPlace distribution, a comPlaceer implementation of the comPlaceation graph was developed in matlab 6.5 (MathWorks, Natick, MA). The simulation generated an equation set for each given program, with transition probabilities as the unknown variables. Each equation summed the probabilities of all possible comPlaceation paths carried out on one inPlace. The result was expected to be similar to the meaPositived final-state distribution for that comPlaceation. A solution to such an equation set is an optimal set of transition probabilities, which minimizes the discrepancy between the calculated and the meaPositived final-state distribution. A least-squares optimization was used for calculating the optimal solution by iteratively refining the transition probabilities by using the method of preconditioned conjugate gradients. However, because of the nature of the problem, there could be local solutions, which might not be consistent with other programs. The following process was used for finding a consistent solution: Programs 1, 2, and 3 were used jointly as training programs according to which the simulation could learn a consistent set of transition probabilities. The optimization process was repeated 450 times for programs 1, 2, and 3. For each set, a single optimization was performed given the calibration initial values. Additional 449 optimizations were performed for each program with ranExecutem initial values. All possible combinations of the calculated optimal transition sets were compared with each other. The most consistent triplet-of-transition probability set was selected, so that the transition probabilities calculated for the same concentration ratios were similar for different programs.

Determining the Deviation of Predicted Results. Determination of the standard deviation of the predicted outPlace ratio was performed by simulating all possible independent pipetting errors of 5% with the same probabilities when preparing the mixtures of the transition molecules. We simulated the experimental setup where each pair of transitions was mixed independently, and then the program mixture was composed from equal solution volumes containing each pair. We assumed discrete deviations of -5%, 0%, and 5% from the nominal volume of each software molecule solution. The deviation in the transition probability for each volume deviation was calculated from the meaPositived calibration curve. Each program generated a set of 6,561 (38) different probability combinations, resulting in 6,561 possible outPlace ratios, for which a standard deviation was calculated. The average of the set was very close to the predicted value with no deviations. The simulation was performed with the matlab 6.5 software (MathWorks).

Results

The molecular stochastic automaton Characterized here is based on the two-state, two-symbol automaton developed in our laboratory (11). The hardware consists of a class IIS restriction enzyme FokI, which recognizes the sequence GGATG and Slits 9 and 13 nucleotides away from the recognition site on the sense and antisense strands, respectively. The automaton has eight possible transition rules (Fig. 1E ). As transition molecules used in a comPlaceation are reusable, their concentration remains constant. This enPositives that transition probabilities derived from transition molecule concentrations remain constant during the comPlaceation.

At the molecular level, two transition molecule species compete for any intermediate configuration with probabilities of success correlated to their relative concentrations (Fig. 1F ). When the comPlaceation is performed on a large ensemble of inPlace molecules, the probability to obtain a particular final state can be meaPositived directly from the relative concentration of the outPlace molecule encoding this state among all outPlace molecules. We control the transition probabilities by varying the relative concentration of the transition molecules and attempt to predict the distribution between outPlace molecules for a given inPlace based on these concentrations. Our key problem in Executeing so is to determine the function linking relative concentrations of competing transition molecules to the probability of a transition being chosen. Minute variations in the chemical composition of each transition molecule result in different affinities to the inPlace molecule and, therefore, this function must be determined experimentally. Furthermore, we cannot assume that this function is independent of the absolute molar concentrations of the inPlace and/or of the software molecules. The bulk of our experimental and comPlaceational work was devoted to determining this function for each pair of competing transition molecules and to establishing its independence of these other parameters.

To determine the function mapping relative concentrations of transition molecules to transition probabilities we performed calibration with the four-symbol inPlaces aaab and bbba. The comPlaceation proceeds deterministically until the last symbol and then performs a stochastic choice between two competing transitions. We used this design to represent adequately multisymbol comPlaceations and avoid Traces unique to initial symbols. We prepared software mixtures that represent all four possible intermediate state-symbol combinations (Fig. 2 A ). The comPlaceation was carried to completion and the exact duration was determined in the preliminary experiments with inPlaces of different lengths. The resulting meaPositived calibration curves for all four competing pairs of transition molecules are Displayn in Fig. 2B , demonstrating a different mapping for each pair of transitions. Transitions processing the symbol a provide a liArrive mapping, with a probability distribution being close to the concentration ratio. On the other hand, transitions processing b reveal a convex mapping, which is apparently caused by the software molecules that result in state S1 (T4 and T8) having a higher reaction rate than the competing software molecules that result in state S0 (T3 and T7). In steep Locations of the curve the precision with which probabilities can be programmed is reduced because of higher sensitivity to pipetting errors. Error probabilities of the software molecules are reflected in the non-zero probability to accomplish a transition when its competing transition is absent (e.g., T3–T4). However, we suspect that the comPlaceation reaction also has byproducts that overlap with error-representing bands but Execute not accumulate in the final error of multistep comPlaceations. Studies to distinguish between the two types of erroneous products are underway. Accumulating errors in transitions result from the inAccurate cleavage by FokI that also Slits one nucleotide further than expected both in the sense and antisense strands of the inPlace molecule. Because of this imprecision in enzyme action and the relative location of the S1 and S0 sticky ends, transitions that transform to state S1 may result in transitions to state S0, whereas transitions intended to transform to S0 result in dead-end products. This finding Elaborates the higher apparent error rates observed with the software molecules that transform to S1.

To verify that the system is insensitive to fluctuations in inPlace concentration, which naturally occur in the course of a multistep comPlaceation, we meaPositived the distribution of outPlace states of a comPlaceation with one stochastic choice by using the inPlace bbba and the comPlaceation tree used for calibration of the competing pair of transition pair T1–T2. The summary of the results in Fig. 2C indicates that the comPlaceation is insensitive to the different inPlace concentrations used and the transition probability is determined solely by the ratio between the transition molecules. Another set of experiments was performed to enPositive that the transition probability as determined by the ratio between transition molecules is not affected by their absolute concentrations. We used the same experimental system as above, but in this case the concentration of the inPlace was kept constant while the total concentration of the competing transition molecules varied. The summary of the results is Displayn on the graph in Fig. 2D. The results indicate that the transition probability is indeed relatively insensitive to the absolute software concentrations and is defined mostly by the relative concentration ratio.

We tested the stochastic finite automaton by running four programs with the same structure as the one Characterized in Fig. 1C , but with different transition probabilities (Table 1), on nine inPlaces that varied in symbol composition and length (I1–I9, Table 2, which is published as supporting information on the PNAS web site). Each program specifies the relative concentrations of all pairs of competing transition, which in turn determine their expected probabilities as Elaborateed below. We carried out extensive preliminary experiments (not Displayn) to develop the reaction protocol. These experiments indicated that at high concentrations the transition T6 probably digests other transition molecules with the aid of the enzyme FokI, which is present in the solution, modifying the software composition during the comPlaceation in an unpredictable way. Hence, we avoided high concentrations of T6 in our programs. We then performed the calibration and comPlaceation experiments reported here in a single run under uniform conditions. Fig. 2E Displays representative results of the comPlaceations as analyzed by PAGE. Each lane is an application of programs 3 and 4 to one of the inPlaces. We predicted the distribution of the outPlaces by using the calibration graphs (Fig. 2B ). Excellent correlation was observed between predicted and meaPositived results by using meaPositived transition probabilities, although some systematic errors were apparent. To account for these discrepancies, we calculated the expected deviation in outPlace probabilities caused by independent pipetting errors of 5% when mixing transition molecules (Fig. 3). The calculation indicated that a number of meaPositived results, notably with inPlaces ending with b, fell outside of the expected error range and were consistently lower than the prediction. Therefore, this bias could not be attributed solely to pipetting errors but rather to some error in the method of direct probability meaPositivement. To compensate for discrepancies between transition probabilities obtained in direct meaPositivements and those manifested in multistep comPlaceations, we designed an in silico method for probability determination based on a simulation of the reaction network. The simulation utilizes least-squares optimization to find an optimal set of transition probabilities for each program. The optimization Starts with meaPositived transition probabilities and iteratively refines them until a minimum discrepancy between calculated and meaPositived outPlace probabilities is reached. Programs 1, 2, and 3 were used jointly as a training set according to which the simulation could learn a consistent set of optimal transition probabilities. The transition probabilities calculated for the same concentration ratios were equal for different programs, as they should be under our molecular comPlaceational model. The comPlaceed transition probabilities were then tested independently on program 4, since it is composed of pairs of transitions already used in at least one of the programs 1–3, and it provided a Excellent fit to the meaPositived final-state probabilities of that program.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Bar charts Display predicted vs. meaPositived S0 outPlace probability for each experiment. OutPlace probabilities are predicted by using meaPositived calibration (light blue) and calculated calibration (ShaExecutewy blue) and compared with meaPositived probabilities (gray). Error bars Display ± 2 SD (95% of expected results) in predicted outPlaces caused by independent pipetting errors of 5%.

View this table: View inline View popup Table 1. Programs tested with the stochastic automaton

Some errors in direct probability meaPositivements become apparent by comparing meaPositived and calculated transition probabilities. The pair T3–T4, for example, has a meaPositived probability of 38% to transform to S1 compared with a calculated probability of 54%. This finding suggests that the method for direct probability meaPositivements should be improved in the future. In addition, a strong correlation exists between the standard deviation of predicted outPlace probability and the Inequity between meaPositived and predicted outPlace probabilities, which indicates that for some transition pairs the Inequitys resulting from intrinsic sensitivity to pipetting errors overlap with a systematic error in probability meaPositivements.

In Table 1, column [T]rel Displays the relative percentage of the transition molecules that transform to S1 within each competing pair; column P m Displays the meaPositived transition probability corRetorting to this molecule, and column P c Displays the calculated transition probability. Fig. 2F Displays the correlation between meaPositived outPlace probabilities and outPlace probabilities calculated from meaPositived and calculated transition probabilities. The correlation that utilizes the calculated transition probabilities is very Excellent. A detailed summary of the results is given in Tables 2 and 3, which are published as supporting information on the PNAS web site.

Discussion

Predictability and error control are prerequisites for any practical comPlaceer architecture. We obtained a Excellent fit between predicted and meaPositived comPlaceation outPlace using calculated transition probabilities. This result suggests that the transition probability associated with a given relative concentration of a software molecule is a dependable programming tool. This principle was recently used in a construction of a molecular comPlaceer capable of probabilistic logical analysis of the disease-related molecular indicators in vitro by coregulating the concentration of software molecules with the concentration of these indicators (13).

Acknowledgments

We thank A. Regev for critical review of this manuscript. This work was supported by grants from the Israeli Ministry of Science, the Israeli Science Foundation, and the Minerva Foundation.

Footnotes

↵ ∥ To whom corRetortence should be addressed. E-mail: ehud.shapiro{at}weizmann.ac.il.

↵ † R.A. and Y.B. contributed equally to this work.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: FAM, carboxyfluorescein.

Copyright © 2004, The National Academy of Sciences

References

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