Fig 1.
Automated flow of linear system synthesis.
Table 1.
Three primitive components, their chemical reactions, and their transfer functions.
Fig 2.
Mass-spring-damper system and its block diagrams.
(A) MSD system. (Let F = 1 N, M = 1 kg, b = 10 N s/m, and k = 20 N/m.) (B) Block diagram of the MSD model, where the triangular blocks denote gain functions with their corresponding weights, the rectangular blocks denote integrators, and the circle blocks denote mixers for summation and/or subtraction. (Let A = 2.764, B = 7.236 and C = 0.2236.) (C) Block diagram of the PI-controlled MSD model, where G is the plant shown in (B). (Assume the values of KP and KI given in Table 2.)
Fig 3.
Responses of MSD systems with and without a PI controller.
The blue, green, and red curves represent the responses in ideal, configurable biochemical implementation, and nonconfigurable biochemical implementation cases, respectively. (A) Step, impulse, and sinusoidal responses of the MSD. (B) Step responses of PI-controlled MSD. (Assume 10% rate mismatch in the MSD system.) (C) Step responses of PI-controlled MSD, where the MSD undergoes a parameter change with b = 40 N s/m and k = 60 N/m, respectively, to induce a gain change of A = 1.561, B = 38.44 and C = 0.0271.
Table 2.
Values of (KP, KI) in ideal, configurable and non-configurable implementations of the original and new MSD systems.
Fig 4.
DSD realization of catalytic reactions.
The species in green are fuels whose concentrations are fixed to 2000 nM to avoid the effect of fuel depletion. The species sp_0, sp10, and catalysis_7 represent the input u+ or u−, output y+ or y−, and catalyst x+ or x−, respectively. Species catalysis_7 reacts with the gate catalysis to begin the first stage of the reactions followed by a series of reactions involving species sp_0 and catalysis_2. At the end of the first stage, species sp16 is produced and reacts with the gate catalysis_1 to begin the second stage of the reactions. In this stage, species catalysis_7 and sp_0 are yielded to compensate for their consumption in the first stage. Finally, the output species sp10 is generated.
Table 3.
Experimental data used for measuring the rate constant of DSD catalysis.
Fig 5.
Concentration of over time in DSD realized catalytic reactions.
(A) in nM and (B)
in nM.
Fig 6.
DSD realization of the degradation reaction.
Here, deg serves as fuel with its concentration maintained at a fixed value and corresponds to catalysts z±. In the reaction, the rate constant equals the toehold binding rate 0.05 nM−1 s−1.
Fig 7.
Simulation of DSD realized degradation reactions.
(A) Concentration of over time under deg = z± = 1 nM. (B) Concentration of
over time under deg = z± = 100 nM. The three curves in each case correspond to different initial concentrations y(0)± = 10, 50, and 100 nM.
Fig 8.
DSD realization of the annihilation reaction.
Here, and
correspond to the output species y+ and y−, respectively. Two fuels (ann and ann_1) with similar structures are used to ensure that the consumption rates of y+ and y− are balanced.
Fig 9.
Simulation of DSD realized primitive components.
(A) Concentrations of y+, y−, and y = y+−y− of the integration component computing under constant inputs u+ = 2 and u− = 1 nM. (B) Concentrations of y+, y−, and y = y+−y− of the weighted summation component computing y = 2u1 + u2 under constant inputs
and
in nM.
Fig 10.
Block diagrams of naive implementation of elementary modules.
(A) Block diagram of the degree-(1, 0) module with parametric weights j1 and j2 to match the transfer function coefficients. (B) Block diagram of the degree-(2, 0) module with parametric weights e1, e2, and e3 to match the transfer function coefficients. (C) Block diagram of the degree-(2, 1) module with parametric weights f1, f2, and f3 to match the transfer function coefficients.
Fig 11.
Responses of naive implementations to a step input of amplitude 10−6 nM under different values of parameter a.
(A) Responses of under a = 0.5, 1.0, 1.5. (B) Responses of
under a = 4.0, 5.0, 6.0. (C) Responses of
under a = 0.5, 1.0, 1.5. (D) Responses of
under a = 4.0, 5.0, 6.0. (E) Responses of
under a = 0.5, 1.0, 1.5. (F) Responses of
under a = 4.0, 5.0, 6.0.
Fig 12.
Block diagrams and their transfer functions for the exact implementation of the elementary modules.
The first, second, and third rows in the table correspond to the implementations of the degree-(1, 0), degree-(2, 0), and degree-(2, 1) modules, respectively. The first, second, and third columns in the table show the block diagrams, normal transfer functions, and CRN transfer functions, respectively. A normal transfer function is obtained directly from its block diagram, whereas a CRN transfer function is obtained when the gain and summation blocks in the block diagram are realized using the gain and summation CRNs in Table 1.
Fig 13.
Responses of DSD realized naive implementations to a step input of amplitude 10−6 nM under different values of parameter a.
(A) Responses of under a = 0.5, 1.0, 1.5. (B) Responses of
under a = 4.0, 5.0, 6.0. (C) Responses of
under a = 0.5, 1.0, 1.5. (D) Responses of
under a = 4.0, 5.0, 6.0. (E) Responses of
under a = 0.5, 1.0, 1.5. (F) Responses of
under a = 4.0, 5.0, 6.0.
Fig 14.
Response comparisons on DSD realized systems under the naive and exact implementations of transfer functions G1, G2, and G3.
(A) Responses of G1 implementations. (B) Responses of G2 implementations. (C) Responses of G3 implementations. The curves in red corresponds to naive implementations (under the parameter setting a = 6 in ), while the curves in green correspond to exact implementations.
Fig 15.
Response comparisons on DSD realized systems for the naive and exact implementations of transfer function G.
(A) G implemented by a summation block adding G1, G2, and G3, assuming a = 6 for this summation block in both the naive and exact methods. (B) G implemented by superposing G1, G2, and G3, assuming that the DSD output species of G1, G2, and G3 have a common sub-domain that is recognized as the output of G.