Phenomenological model

4.2 Displacillator: a de novo strand displacement oscillator

4.2.3 Inferring signal strand concentrations

4.2.3.2 Phenomenological model

To avoid the rigid stoichiometry assumptions in Section 4.2.3.1, which are hard to justify, we present here an alternative phenomenological model where reactant and product stoichiometries are fit parameters. This model, presented in Equations4.13-4.18, attempts to infer signal con- centrations in an internally consistent way. For the purposes of this section, Helper_i(t)for an autocatalytic moduleiis defined as the total-Helper concentration for that module at timet.

r_BA·(B + A)−−−−→^kBABB p_BA·B, rate =kBABB·B·A (4.13) r_CB·(C + B)−−−−→^kCBCC p_CB·C, rate =kCBCC·C·B (4.14) r_AC·(A + C)−−−−→^kACAA p_AC·A, rate =k_ACAA·A·C (4.15) φ−−−−→^kleakA A, rate =kleakA (4.16) φ−−−−→^kleakB B, rate =k_leakB (4.17) φ−−−−→^kleakC C, rate =k_leakC (4.18) Rate constantskBABB, kCBCC, kACAAare in /M /s; rate constantskleakA, kleakB, kleakCare in M /s.

ri andpidetermine effective stoichiometries for the reactants and products, respectively. Along with the initial conditions(A(0),B(0),C(0)), which are fitted for each experiment, these comprise the parameters (Θ) of the phenomenological model.

How do we interpret the parametersr_i andp_i? For a given autocatalytic module,r_iis inter- preted as the average number of reactants consumed per unit consumption of total-Helper species for that module. Similarly,p_iis interpreted as the average number of products released per unit consumption of total-Helper species for that module.

Interpreting the data. We now outline how, given a parameter setΘand the data setDcon- taining measurements of total-Helper concentrations, we infer concentrations A(t), B(t), and C(t).

We define the leak-corrected consumption of total-Helper as below:

Flow^Θ,D_BABB(t) := HelperBBs(0)−HelperBBs(t)−kleakB·t (4.19) Flow_CBCC^Θ,D (t) := Helper_CCk(0)−Helper_CCk(t)−k_leakC·t (4.20) Flow^Θ,D_ACAA(t) := HelperAAq(0)−HelperAAq(t)−kleakA·t. (4.21) This leak correction involves an implicit assumption: that gradual leaks of signal strands arise pri- marily from the Produce-Helper gradual leak mechanism. This is a good approximation because the Produce-Helper gradual leak arises from two fuel species which are both at high concentra- tion, in contrast to the React-second input gradual leak where only the React species is at high concentration. In addition, for Design 4, the React-second input gradual leaks have a rate constant of about 1 - 5 /M /s, whereas Produce-Helper gradual leaks have a relatively larger rate constant between 5 - 15 /M /s.

By taking time derivatives of equations4.19-4.21, we can infer instantaneous rates of total- Helper consumption through each autocatalytic module.

FlowRate^Θ,D_BABB(t) :=−dHelper_BBs(t)

dt −k_leakB (4.22)

FlowRate^Θ,D_CBCC(t) :=−dHelperCCk(t)

dt −kleakC (4.23)

FlowRate^Θ,D_ACAA(t) :=−dHelper_AAq(t)

dt −k_leakA. (4.24)

The interpretation ofriandpias average number of reactants consumed and products released per unit consumption of total-Helper provides a clean way of inferring concentrations of signal strands from measurements of total-Helper concentrations, given a set of model parametersΘ.

A^Θ,D(t) = A(0)−r_BA·Flow^Θ,D_BABB(t) + (p_AC−r_AC)·Flow^Θ,D_ACAA(t) +k_leakA·t (4.25) B^Θ,D(t) = B(0)−rCB·Flow^Θ,D_CBCC(t) + (pBA−rBA)·Flow^Θ,D_BABB(t) +kleakB·t (4.26) C^Θ,D(t) = C(0)−rAC·Flow^Θ,D_ACAA(t) + (pCB−rCB)·Flow^Θ,D_CBCC(t) +kleakC·t. (4.27) Model predictions.Given the reactions and rates (4.13-4.18), we may generate model predic-

tions by numerically simulating the following mass action ODEs. We define

F^Θ_BABB(t) :=kBABB·B(t)·A(t) (4.28) F^Θ_CBCC(t) :=k_CBCC·C(t)·B(t) (4.29) F^Θ_ACAA(t) :=kACAA·A(t)·C(t). (4.30)

The mass action ODEs are dA

dt ^Θ

=−rBA·F^Θ_BABB(t) + (pAC−rAC)·F^Θ_ACAA(t) +kleakA (4.31) dB

dt ^Θ

=−r_CB·F^Θ_CBCC(t) + (p_BA−r_BA)·F^Θ_BABB(t) +k_leakB (4.32) dC

dt Θ

=−rAC·F^Θ_ACAA(t) + (pCB−rCB)·F^Θ_CBCC(t) +kleakC. (4.33) Given the initial concentrations of A, B, and C specified inΘ, we may numerically solve the mass action ODEs above to generate model predictions for A(t), B(t), and C(t). Further, the pre- dicted curves for A(t), B(t), and C(t) imply a prediction of the instantaneous rates (in nM/hr) through each autocatalytic module. There is a subtlety involved in this calculation. The instanta- neous rate (in nM/hr) of consumption of thereactantsof each autocatalytic module is calculated as specified in equations4.28-4.30. However, since we measure total-Helper concentrations, we would like to calculate the instantaneous rate of consumption oftotal-Helper species, rather than the reactants, for each autocatalytic module. Sincer_iis interpreted as the average number of reactants consumed per unit total-Helper consumption, all we need to do is to divide equations4.28-4.30by the correspondingri; this gives us the instantaneous rate of consumption of total-Helper species for each module, according to our model.

Parameter fitting. We simultaneously fit (i) the model predicted signal concentrations (A^Θ(t), B^Θ(t),C^Θ(t)) to the signal concentrations inferred from data (A^Θ,D(t),B^Θ,D(t),C^Θ,D(t)) and (ii) the model predicted instantaneous rate of total-Helper consumption for each autocatalytic module (calculated asF^Θ_BABB(t)/rBAand so on) to the same quantity inferred from data using equations 4.22-4.24.

We briefly address the fact that the two fitting criteria, (i) and (ii) above, are in different units (nM and nM/hr respectively). As a quick preliminary solution to this issue, we just add the two

Parameter Best-fit value (r_BA,r_CB,r_AC) (1.2 , 0.8, 0.8) (pBA,pCB,pAC) (2.0 , 1.4, 1.9) (kleakB,kleakC,kleakA) (2, 2, 6) in /M /s

(kBABB,kCBCC,kACAA) (1.2×10⁴,7.9×10³,6.4×10³)in /M /s

Table 4.1: Best-fit values for parameters from the phenomenological model. The leak rate constants k_leakB,k_leakC,k_leakAare provided in terms of the equivalent bimolecular Produce-Helper leak rate constants they imply, assuming Produce and Helper species are at 100 nM each.

error functions corresponding to (i) and (ii) above to obtain the objective function which is sub- sequently minimized by nonlinear least squares. Essentially, this procedure involves an implicit weighting arising from the different units. Currently, the optimal error value has a ratio of approx- imately 20 : 1 in favor of the first component. Further analysis with a more principled choice of weights may result in better fits.

Interpreting stoichiometric best fit parameters. The best fit values (Θopt) for the parameters are listed in Table4.1. The initial concentrations of (A, B, C) for each experiment, which were fitted around expected values, are omitted for brevity. We now attempt to interpret the stoichiometric best fit parameters.

Sincer_i is the average number of reactants consumed per unit of total-Helper consumption, r_i <1implies that more than one unit of total-Helper is consumed, on average, per unit consump- tion of reactants. This indicates the effectiveness of the catalytic Helper mechanism. While at first glancerBA= 1.2may be surprising, we point out that in this preliminary analysis we do not have estimates for the error bars associated with these parameters; therefore, it is hard to interpret the best fit parameters at face value.

Since pi is the average number of products released per unit of total-Helper consumption, pi < 2 implies sub-stoichiometric release of products per unit of total-Helper consumed. This could arise from incompletion effects in the produce step, possibly due to mechanisms similar to that of Figure3.13(b).

Lastly,p_i/r_i, the average number of products released per unit consumption of reactants, is

approximately (1.7, 1.6, 2.2) for the three modules.

Results from the phenomenological model. The results of the data analysis with the phe- nomenological model are provided in Figures4.7and 4.8. In particular, the phase and velocity plots (panels (e) and (h) of Figure4.8) clearly illustrate the triangular orbit which is characteristic of the Displacillator.

a b c

0 15 30 45 60

5 10 15

0 15 30 45 60

!1 0 1

Time (hours) Concentration of A, B, C

0 15 30 45 60

5 10 15

0 15 30 45 60

!1 0 1

Time (hours)

0 15 30 45 60

5 10 15

0 15 30 45 60

!1 0 1

Time (hours)

Signal concentration (nM) d[Signal]/dt (nM/hr)

Net production of A, B, C

Addition: (Ap, Br, Cj) = (0, 10, 13) nM Addition: (Ap, Br, Cj) = (11, 0, 13) nM Addition: (Ap, Br, Cj) = (11, 10, 3) nM

Signal concentration (nM) d[Signal]/dt (nM/hr) Signal concentration (nM) Signal concentration (nM)

Net production of A, B, C Net production of A, B, C Concentration of A, B, C Concentration of A, B, C

Figure 4.7: a. Concentrations of A, B, and C inferred from Helper concentration measurements based on the phenomenological model in Section4.2.3.2. b. Time derivatives of A, B, and C. The order of the peaks is as expected in experiments with all three initial conditions.