Typed Modular Cloning System¶

System Definition¶

Definition

Given a genetic alphabet \(\langle \Sigma, \sim \rangle\), a Typed Modular Cloning System \(S\) is defined as a mathematical sequence

\[(M_l,\ V_l,\ \mathcal{M}_l,\ \mathcal{V}_l,\ e_l)_ {\ l\ \ge -1}\]

where:

\((M_l, V_l, e_l)_{l \ge -1}\) is a standard Modular Cloning System
\(\mathcal{M}_l \subseteq \mathcal{P}(M_l) \to \mathcal{P}(M_l)\) is the set of module types of level \(l\)
\(\mathcal{V}_l \subseteq \mathcal{P}(V_l) \to \mathcal{P}(V_l)\) is the set of vector types of level \(l\)

Types¶

Definition

\(\forall l \ge -1\), we define types using their signatures (i.e. the sets of upstream and downstream overhangs of elements using this type):

\[\begin{split}\begin{array}{ll} \forall t \in \mathcal{M}_l,& \begin{cases} Up(t) &= \bigcup_{m \in t(M_l)} \{ up(m) \} \\ Down(t) &= \bigcup_{m \in t(M_l)} \{ down(m) \} \end{cases} \\ \forall t \in \mathcal{V}_l,& \begin{cases} Up(t) &= \bigcup_{v \in t(V_l)} \{ up(v) \} \\ Down(t) &= \bigcup_{v \in t(V_l)} \{ down(v) \} \end{cases} \end{array}\end{split}\]

Corollary

\(\forall l \ge -1\),

\[\begin{split}\begin{array}{lll} \forall t \in \mathcal{M}_l,&\ t(M_l) &= \{ m \in M_l\ |\ up(m) \in Up(t),\ down(m) \in Down(t) \} \\ \forall t \in \mathcal{V}_l,&\ t(V_l) &= \{ v \in V_l\ |\ up(v) \in Up(t),\ down(v) \in Down(t) \} \end{array}\end{split}\]

Property: Structural equivalence of module types

Given a valid (resp. unambiguous) (resp. complete) assembly

\[m_1 + \dots + m_k + v \xrightarrow{e_l} A \subset (\Sigma^\star \cup \Sigma^{(c)})\]

then if there exist \(t \in \mathcal{M}_l\) such that

\[\begin{split}\begin{cases} \lvert Up(t) \rvert = \lvert Down(t) \rvert = 1 \\ m_1 \in t(M_l) \end{cases}\end{split}\]

then \(\forall m_1\prime \in t(M_l)\),

\[m_1\prime + \dots + m_k + v \xrightarrow{e_l} A \subset (\Sigma^\star \cup \Sigma^{(c)})\]

is valid (resp. unambiguous) (resp. complete).