Design and analysis of DNA strand displacement devices using probabilistic model checking

by Matthew R. Lakin, David Parker, Luca Cardelli, Marta Kwiatkowska, and Andrew Phillips

InterfaceVolume 9(72):1470-1485

July 7, 2012

©2012 by The Royal Society

Toehold-mediated DNA branch migration and strand displacement.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Initial transducer gate code, with additional definition for signal strands.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

(a) Initial species and (b) expected final species for the transducer gate.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Deadlock states for the two faulty transducers in series.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Corrected transducer gate code, with an additional ‘new a’ declaration that prevents crosstalk between different gates.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Plot showing the probability for each possible outcome of the faulty transducer pair, after T seconds.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Plot showing expected percentage of leftover reactive gates in the final state of the system against the number of parallel buggy transducers—that is, the parameter N in the system

S(N,x0)| T(N,x0,x1)| T(N,x1,x2).

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Plot showing the expected time to terminate for chains of corrected transducer gates; that is, we vary the parameter k in the system S(1,x0)| T2(1,x0,x1)| . . .| T2(1,x{ k − 1} ,xk).

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Catalyst gate code, presenting two different gate implementations: one that carries out garbage collection reactions and one that does not.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

(a) Initial species and (b) expected final species for the catalyst gate C(1,x,y,z).

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

(a) Initial species and (b) expected final species for the alternative catalyst gate C_NoGC(1,x,y,z).

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Plot of expected time to completion for N parallel copies of catalyst gates with (solid line with filled circles) and without (solid line with filled triangle) garbage collection.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

DNA strand displacement (DSD) code for a catalyst gate, which extends the C_NoGC gate from figure 9 by using the constant keyword from the DSD language to abstract away from population

changes due to accumulation of waste and depletion of fuel.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Surface plot which shows the probability of reaching a consensus of X, for various initial populations of X and Y. (Online version in colour.).

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Probability of reaching a consensus of X, plotted against (X0 − Y0 )/N, which is the difference between the initial populations of X and Y, relative to the total initial population N = X0 + Y0 .

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society

Elementary reduction rules of the DSD programming language.

Matthew R. Lakin et al. J. R. Soc. Interface 2012;9:1470-1485

©2012 by The Royal Society