4.6 Equivalence Based on Pathway Decomposition
4.6.3 Pathway Decomposition
However, the above definition will still interpret pathways like (A→i, A→i, i→B, i→B) as correctly implementingA+A→B+B. To prevent this, we again take note of the fact that this path- way can be thought of as a composition of two smaller pathways each implementing (A→i, i→B).
Clearly, if a module is merely a composition of smaller modules, it should not be considered a module in the first place. Therefore, we introduce the following notion of decomposability on pathways. For better understanding, Fig. 4.6 provides some examples of decomposable formal pathways.
Definition. A formal pathway p is decomposable if p can be partitioned into two nonempty subsequences (which need not be contiguous) that are each formal pathways. If a pathway is not decomposable, it is said to beprime.
Now, we have an important theorem that almost immediately follows, which says that any formal pathway in a system can be retrieved from “composing” prime formal pathways. However, one should also notice that this does not imply that every formal pathway has a unique decomposition into prime pathways. For example, the pathway (A→i, B →i, i→C, i→D) can be decomposed in two different ways: (A→i, i→C) and (B→i, i→D), and (A→i, i→D) and (B→i, i→C).
Theorem 4.1. Any formal pathway can be generated by interleaving one or more prime formal pathways.
Proof. Trivial.
Finally, we are ready to articulate what we mean by implementing a CRN using pathways.
Definition. A CRN is regular if every prime formal pathway implements some formal reaction.
Equivalently, a CRN is regular if every prime formal pathway is regular.
Theorem 4.2. If a CRN Cis regular, for any formal CRNC0,C ∪ C0 is regular.
Proof. This immediately follows from the fact that any pathway of length greater than one that contains a formal reaction is decomposable.
This means that an irregular CRN contains some formal pathway that cannot be interpreted as implementing any formal reaction or series of reactions. In such cases, we will say that the given CRN does not implement any formal CRN by pathway decomposition (of course, it can still be a non-modular implementation that is correct in some other sense). On the other hand, if the
A i j i j B
C k D
A
B i
j k
D
l E
m F
A i
j
B
k E
B
C D
Figure 4.6: Some examples of decomposable formal pathways. The partition of reactions is marked by different colors. Note that only in the last case, a pathway is not regular after decomposition.
given CRN C is regular, it is now obvious what formal CRN C0 that C should be interpreted as implementing. We callC0 theformal basisofC.
Definition. The set of prime formal pathways in a given CRN is called theelementary basisof the CRN. Theformal basis is the set of (initial state,final state) pairs of the nonfutile pathways in the elementary basis.
One final problem is that since the formal basis and regularity both only concern formal pathways, we might fail to catch a problem that arises with non-formal pathways. To avoid this problem, we introduce the following property.
Definition. Letpbe a pathway with a formal initial state andT its final state. Then, a pathway p0= (r1, . . . , rk) is said to be theclosing pathwayofpifp0 can occur in T andT⊕r1⊕ · · · ⊕rk
is a formal state. A CRN is confluentif every pathway with a formal initial state has a closing pathway.
This means that the given CRN is capable of cleaning up all the intermediate species that it produced. For example, the CRN{A→i, i+B →C}will not be confluent because if the system starts from the state{A}, it will immediately turn into{i}and thisimolecule will fail to be removed.
Similarly to before, we will say that non-confluent CRNs do not implement any formal CRN, but it can be a non-modular implementation that is correct by some other definition.
For a more subtle example, let us consider the CRN {A→i+B, i+B→B}, which is clearly confluent according to the definition above. In fact, there will be no problem with this implemented CRN when it is operating by itself. However, when intermediate species require some formal species in order to get removed, the implemented CRN might not work correctly when there are other reactions in the test tube as well. For instance, if the above implementation runs in an environment that also contains the reactionB→C, there is no longer a guarantee thatiwill always be removed, i.e., the CRN consisting of these three reactions will not be confluent. Thus, we often want to enforce that CRNs should be able to remove intermediate species without any help from formal species.
Definition. A closing pathway isstrongif its reactions do not consume any formal species. A CRN isstrongly confluentif every pathway with a formal initial state has a strong closing pathway.
Theorem 4.3. If a CRNCis strongly confluent, for any formal CRNC0,C ∪C0is strongly confluent.
Proof. Supposepis a pathway ofC ∪ C0 that has a formal initial state. Clearly, if we remove from p all reactions that belong to C0 and call the resulting pathway p0, then p0 has exactly the same intermediate species in its final state aspdoes. This is because the removed reactions are all formal reactions. SinceC is strongly confluent,p0 has a strong closing pathway, and so too doesp.
In a sense, strong confluence means that we can always get an immediate (formal) interpretation of a chemical state (or of the intermediate species in a state). From this point, whenever we say
confluence, we will implicitly mean strong confluence. Similarly, we will use the word ‘closing pathway’ to mean ‘strong closing pathway.’
A+B→i i+C→D
i→E confluent
A→i i+B→C not confluent
A→i i→A i+B→C confluent Figure 4.7: Some examples of confluent and non-confluent CRNs