![]() The steady state binary representation for each parity checker B1, B2, and B3 are marked on top of the gates' trajectories. The starting time of the 3-bit counter is set by by the first time that the last bit switches state, as explained in the text. (B) We show an example trajectory for each module. Q 1, Q 2, and Q 3 are the outputs of each bit level, and Q ¯ 1, Q ¯ 2, and Q ¯ 3 are the variables to be fed back. The only difference between the 3-bit counter and the 2-bit counter is that the second carry bit module uses the connector type 2. (A) The diagram shows a modular design of the 3-bit counter following the pattern of “counter-connector-counter”. The 3-bit counter design schematics The inputs are shown in the middle panel, for ease of reference. The parameters for this working 2-bit counter are: the scaling factor ξ = 0.025, the degradation rate γ = 0.025, the equilibrium constant K = 0.011, the cooperativity index n = 1.5, the duration of external signal δ = 300, the amplitude of the external signal A = 20, the shifted hill coefficient maximum value y m a x = 3, the shifted hill coefficient minimum value y m i n = 0.002, and the initial and between signals relaxation times Δ 0 = Δ = 5000. The representation of the 2-bit counter “ B2- B1″ is used to indicate the output state “ Q 2- Q 1” for the counter. Due to the plotted long relaxation time (relative to input pulse width) between adjacent pulses, each green dot in the graph, if zoomed in, represents one input pulse as shown in Figure 5. The counter starting time is determined by the first time that the last bit switches state, as explained in the text. (B) We show an example trajectory of each module aligned with the schematic design in (A). The output of the connector module can be a diffusible small molecule which serves as the input for the second parity checker. The connector type 1 takes Q 1 and the external input signal as inputs. ![]() The signal Q 1 is the output while Q ¯ 1 is the feedback signal for the first parity checker. The 2-bit counter design (A) Schematic design of the 2-bit counter. Mathematical biosciences Synthetic biology Systems biology. This work can be viewed as a step toward obtaining circuits that are capable of finite automaton computation in analogy to digital central processing units. An optimization framework is used to determine appropriate gate parameters and to compute bounds on admissible pulse widths and relaxation (inter-pulse) times, as well as to guide the construction of novel gates. The N-bit counter is then obtained by interconnecting (using diffusible chemicals) a set of N single-bit counters and connector modules. The design starts with a single-bit counter. This allows scalability and bypasses constraints on the maximal number of circuit genes per cell due to toxicity or failures due to resource limitations. The design is based on distributed computation with specialized cell types allocated to specific tasks. An N-bit counter reads sequences of input pulses and displays the total number of pulses, modulo 2 N. on Logic Programming, 1994.A design for genetically encoded counters is proposed via repressor-based circuits. Datalog with integer periodicity constraints. van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 134–191. In Proc 3rd International Conference on Database Theory, pages 187–201, Paris, 1990. A closed form for datalog queries with integer order. Computation, Finite and Infinite Machines. Computer Aided Verification, LNCS 697, pages 333–346. Research Report LIENS-96-3, Ecole Normale Supérieure, Paris, Feb. Symbolic verification with gap-order constraints. A decompositional approach for computing least fixedpoint of datalog programs with z-counters. Technical Report 1998-02, Laboratoire Spécification et Vérification, ENS Cachan, Mar. ![]() A closed form evaluation for extending timed automata. Decidabihty of model checking for infinite-state concurrent systems. on Princinples Of Programming Languages (POPL), 1978. Automatic discovery of linear restraints among variables of a program. Computer Aided Verification, volume 1254 of LNCS, Haifa, Israel, 1997. Symbolic model checking of infinite state systems using presburger arithmetic. Symbolic verification with periodic sets. Linear operators and regular languages (ii). Wolper.An automata-theoretic approach to branching time model checking. on Automata, Languages and Programming, Warwick, LNCS 443, pages 322–335.
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