3.3 Test case: engineering a strand displacement oscillator
3.3.2 Non-idealities in the DNA implementation
The simple model presented above does not include any non-idealities in the DNA implemen- tation. These non-idealities could be of several different kinds, ranging from different strand displacement reactions having very different rate constants to spurious “leak” reactions which compete with desired reaction pathways.
Broadly, there are two classes of non-idealities. The first class refers to those that are a conse- quence of imperfect molecules, e.g. errors in DNA synthesis or mis-folded complexes. The second class comprises non-idealities that are unavoidable in our CRN-to-DNA scheme even with per-
[
Produce
][A , B, C]
0 1 2 3 4 5 6 7 8 9 10 11 12 tHhoursL 50
100 150 200 250 Conc300HnML
0 1 2 3 4 5 6 tHhoursL
10 20 30 40 50 Conc60HnML
0 1 2 3 4 5 6 7 8 9 10 11 12 tHhoursL 10
20 Conc30HnML
[A , B, C]
[A , B, C]
a c
b
([A], [B], [C])
0= (30, 10, 10) nM ([A], [B], [C])
0= (60, 10, 10) nM
Figure 3.11: Modeling the DNA implementation of the oscillator at the level of individual strand displacement and toehold exchange reactions. Equations3.1to3.4, along with similar equations for the other two modules, specify the model. The fuel species are present at an initial concentra- tion of 300 nM and are not replenished. a. Concentrations of Produce molecules (dashed lines;
ProduceCApAqin red,ProduceBCjCkin blue,ProduceABrBsin orange) and signal strands as a func- tion of time starting with an initial concentration of([A],[B],[C])0= (30,10,10) nM. b. The plot in (a), zoomed in so that the oscillatory dynamics of the signal strands are visible. c. The plot in (b) with an initial concentration of([A],[B],[C])0= (60,10,10) nM, all other parameters being the same.
fect molecules, since they arise from the domain-level specification of the molecules or reaction pathways. In particular, leak reactions that arise from “blunt end” (zero-base toehold) strand displacement are good examples of this class (see Figure3.12). These leak reactions are a direct consequence of the fact that blunt end strand displacement occurs at a non-zero rate.
These non-idealities have consequences for our experiments. First, our initial conditions are imperfect because some fraction of our fuel species are “bad” — they may be mis-folded com- plexes or DNA strands with synthesis errors on them. One particular manifestation of these im- perfect initial conditions is what we measure experimentally as “initial leak” — some fraction of our fuel species release their outputs even when their inputs are not present. Therefore, our ini- tial conditions may involve a smaller concentration of “good” fuel species than we expect, some leaked signal strands or Flux molecules, and some spurious products.
Second, spurious products that form due to leak pathways may be capable of undergoing cer- tain reactions that are legitimate steps in a desired reaction pathway. Figure3.13illustrates some reactions of this kind in the case of spurious products that arose from the gradual leak pathways shown in Figure3.12. Therefore leak reactions may affect dynamical behavior in ways that are more complex than merely the unexpected release of signal strands or Flux strands.
In particular, spurious products can affect reaction stoichiometries. For example, Leaked- ReactCBrconsumes one molecule of Cj and releases no other signal strand; Leaked-ProduceBCjCk
consumes one molecule of FluxBCj and releases only one molecule of C, since Ck had already
“leaked” out (see Figure3.13). Indeed, this type of mechanism could conceivably explain “incom- pletion” effects we observe in our experiments - where triggering a particular fuel species (say ProduceBCjCk) with its input(s) (FluxBCj) produces sub-stoichiometric amounts of the output (C).
If we can account for the initial amounts of such leaked products and correspondingly de- plete the fuel concentrations (say by fitting those leaked amounts to the data), and include in our model the set of expected reactions in which spurious products may participate, we may be able to mechanistically account for our experiments without any additional assumptions that are fun- damentally different from the idealized model. The reactions we may need to include would be of the form outlined below (where L is short for “Leaked”).
a React-second input leak
ReactCBCj
fB mB sB hBr
Br
FluxBCj sB
mB hCj
fC mC sC hCk
Ck +
sC +
mC s*C
sB mB f*B m*B s*B m*C
f*C
fB hBr
fB sC mC fB
s*C
sB mB f*B m*B s*B m*C
f*C
hCj
1
2
Produce-Helper leak b
HelperCCk hCk
fC fC
+ 3
ProduceBCjCk
s*B
fC mC
sC
f*C hCj
hCj* hCk sC mC fC f*C hCk*
4
+
s*B hCj hCj*
hCk hCk*
fC f*C fC f*C sC mC fC
Leaked-ReactCBr
Leaked-ProduceBCjCk
c React-Produce leak
sC mC fB
s*C
sB mB f*B m*B s*B m*C
f*C
hCj
5
ProduceBCjCk
s*B
fC mC
sC
f*C hCj
hCj* hCk sC mC fC f*C hCk*
+
ReactCBCj
sC mC fB
s*C
sB mB f*B m*Bs*B m*C
f*C s*B
fC mC
sC
f*C hCj
hCj* hCk f*C hCk*
+ hCj fC mC sC Cj Leaked-ReactCBCjProduceBCjCk
Figure 3.12: Illustrative examples of spurious “leak” pathways that arise due to blunt-end (zero base toehold) strand displacement. These pathways are illustrated in the case of the autocatalytic module C + B→2C but can occur with the other modules as well. Locations of invasion are indicated by numbered dashed arrows. a. The second input (here,Br) can invade at locations 1 (the junction) and 2 (the end of the helix) in the React species. Once strand displacement finishes, the Flux molecule may be released and a spurious species can be formed. b. A similar reaction can happen between the Helper species and the Produce species, releasing the second output of the Produce molecule (here,Ck) and resulting in a spurious species. c. Spontaneous fraying due to thermal fluctuations at the end of the helix in the React molecule may enable the Produce molecule to invade at at location 5. Strand displacement can then result in the release of the first output of the produce gate (here,Cj) and the formation of a spurious species. Notice that all of these spurious species shown here are capable of participating in some reactions that are also a legitimate part of desired reaction pathways (see Figure3.13).
a Leaked-ReactCBr consumes Cj
fC mC sC hCj
Cj Leaked-ProduceBCjCk
+
+
Leaked-ReactCBr mC sC
s*C
sB mB f*B m*B s*B m*C
f*C
fB hBr
fB
Leaked-ProduceBCjCk consumes FluxBCj, releases Cj b
+
+
s*B hCj hCj*
hCk hCk*
fC f*C fC f*C sC mC fC fC mC sC
hCj
Cj
WasteCjBr
sC mC
s*C
sB mB f*B m*B s*B m*C
f*C fC
hCj fB
hBr
BackCB sC mC fB
FluxBCj
sB mB
hCj
s*B mB sB hCj
hCj* hCk hCk*
fC f*C fC f*C
WasteBCjCk
c Leaked-ReactCBCjProduceBCjCk releases Ck
sC mC fB
s*C
sB mB f*B m*Bs*B m*C
f*C s*B
fC mC
sC
f*C hCj
hCj* hCk f*C hCk*
Leaked-ReactCBCjProduceBCjCk
+ hCk fC mC sC Ck + fC hCk fC HelperCCk
sC mC fB
s*C
sB mB f*B m*Bs*B m*C
f*C s*B
fC f*C hCj
hCj* hCk f*C hCk*
Leaked-ReactCBCjProduceBCjHelperCCk
Figure 3.13: Spurious products that are formed due to leak pathways may undergo reactions that are legitimate steps in desired reaction pathways. Therefore leak reactions may affect dynamical behavior in ways that are more complex than merely the unexpected release of signal strands or Flux strands.
L-ReactCBr+C−−→WasteCjBr+ BackCB (3.5) L-ProduceBCjCk+FluxBCj−−→WasteBCjCk+ C (3.6) L-ReactCBCjProduceBCjCk+ HelperCCk−−→C + L-ReactCBCjProduceBCjCkHelperCCk (3.7) We have explored different augmented versions of the model described in Section3.3.1that include some of the non-idealities discussed in this section. We have omitted that exploration in the interest of brevity. However, a rigorous analysis of sensitivity to parameters was not per- formed. Based on our modeling, we learned that the DNA implementation of our oscillator can tolerate significant variation in individual strand displacement rate constants (factor of 3-10, and possibly more, depending on how ideal other parameters, such as gradual leak rates, are) and bi- molecular rate constants for gradual leak pathways (of the kind described in Figure3.12) as high as 50 - 80 /M /s.