System Definition
Definition
Given a genetic alphabet \(\langle \Sigma, \sim \rangle\), a Modular
Cloning System \(S\) is defined as a mathematical sequence
\[(M_l,\ V_l,\ e_l)_ {\ l\ \ge -1}\]
where:
- \(M_l \subseteq \Sigma^\star \cup \Sigma^{(c)}\) is the set of modules
of level \(l\)
- \(V_l \subseteq \Sigma^{(c)}\) is the set of vectors of level \(l\)
- \(e_l \subseteq E\) is the finite, non-empty set of asymmetric,
Type IIS restriction enzymes of level \(l\)
Definition: \(k\)-cyclicity
A Modular Cloning System \((M_l, V_l, e_l)_ {l \ge -1}\) is said to
be \(k\)-cyclic after a level \(\lambda\) if:
\[\begin{split}\begin{array}{ll}
\exists k \in N^\star, & \\
\forall l \ge \lambda, & \\
& \begin{cases}
M_{l+k} \subseteq M_l \\
V_{l+k} \subseteq V_l \\
e_{l+k} \subseteq e_l
\end{cases}
\end{array}\end{split}\]
Definition: \(\lambda\)-limit
A Modular Cloning System \((M_l, V_l, e_l)_ {l \ge -1}\) is said to
be \(\lambda\)-limited if:
\[\forall l \ge \lambda,
M_l = \emptyset,
V_l = \emptyset,
e_l = \emptyset\]
Modules
Definition
For a given level \(l\), \(M_l\) is defined as the set of modules \(m \in \Sigma^\star \cup \Sigma^{(c)}\)
for which:
\[\begin{split}\begin{array}{l}
\exists ! (S, n, k) \in e_l, \\
\exists ! (S^\prime, n^\prime, k^\prime) \in e_l, \\
\exists ! (s, s^\prime) \in S \times S^\prime, \\
\exists ! (x, y, o_5, o_3) \in (\Sigma^\star)^4, \\
\\
\quad \exists ! t \in \Sigma^\star,
\left\{ \begin{array}{lll}
\exists ! b \in \Sigma^\star,\ & m = (s \cdot x \cdot o_5 \cdot t \cdot o_3 \cdot y \cdot \widetilde{s^\prime} \cdot b)^{(c)}, & \text{ if } m \in \Sigma^{(c)}\\
\exists ! u, v \in (\Sigma^\star)^2, & m = u \cdot s \cdot x \cdot o_5 \cdot t \cdot o_3 \cdot y \cdot \widetilde{s^\prime} \cdot v, & \text{ if } m \not \in \Sigma^{(c)}
\end{array} \right.
\end{array}\end{split}\]
with:
- \(|x| = n\)
- \(|y| = n^\prime\)
- \(|o_5| = abs(k)\)
- \(|o_3| = abs(k^\prime)\)
Note
This decomposition is called the canonic module decomposition, where:
- \(t\) is the target sequence of the module \(m\)
- \(b\) is the backbone of the module \(m\) (if \(m\) is circular)
- \(u\) and \(v\) are called the prefix and suffix of the module \(m\) (if \(m\) is not circular)
- \(o_5\) and \(o_3\) are the upstream and downstream overhangs respectively.
Property
\(\forall \langle \Sigma, \sim \rangle\), \(\forall l \ge -1\),
\(\forall e_l \subset E\):
\[M_l \text{ is a rational language }\]
Demonstration
Let there be a genetic alphabet \(\langle \Sigma, \sim \rangle\)
and a Modular Cloning System \((M_l, V_l, e_l)_ {l \ge -1}\) over
it.
\(\forall l \ge -1\), the regular expression:
\[\begin{split}\begin{array}l
\bigcup_{\begin{array}l(S, n, k) \in e_l \\ (S\prime, n\prime, k\prime) \in e_l\end{array}}
\Sigma^\star \cdot S \cdot \Sigma^n \cdot \Sigma^{abs(k)} \cdot \Sigma^\star \cdot \overline{(S | S^\prime)} \cdot \Sigma^\star \cdot \Sigma^{abs(k\prime)} \cdot \Sigma^{n\prime} \cdot \widetilde{\,S\prime\,} \cdot \Sigma^\star \\
\end{array}\end{split}\]
where:
matches a sequence \(m \in \Sigma^\star \cup \Sigma^{(c)}\) if and only if
\(m \in M_l\).
\(M_l\) is regular, so given Kleene’s Theorem, \(M_l\) is rational.
Vectors
Definition
For a given level \(l\), \(V_l\) is defined as the set of vectors \(v \in \Sigma^{(c)}\)
for which:
\[\begin{split}\begin{array}{l}
\exists ! (S, n, k) \in e_l, \\
\exists ! (S^\prime, n^\prime, k^\prime) \in e_l, \\
\exists ! (s, s^\prime) \in S \times S^\prime, \\
\exists ! (x, y, o_5, o_3) \in (\Sigma^\star)^4, \\
\\
\quad \exists ! (b, p) \in (\Sigma^\star)^2,
\exists ! b \in \Sigma^\star,\ v = (o_3 \cdot b \cdot o_5 \cdot y \cdot \widetilde{s} \cdot p \cdot s\prime \cdot x)^{(c)} \\
\end{array}\end{split}\]
with:
- \(|x| = n\)
- \(|y| = n^\prime\)
- \(|o_5| = abs(k)\)
- \(|o_3| = abs(k^\prime)\)
- \(o_3 \ne o_5\)
Note
This decomposition is called the canonic vector decomposition, where:
- \(p\) is the placeholder sequence of the vector \(v\)
- \(b\) is the backbone of the vector \(v\)
- \(o_3\) and \(o_5\) are the upstream and downstream overhangs respectively.
Overhangs
By definition, every valid level \(l\) module and vector only have a single canonic
decomposition where they have unique \(o_5\) and \(o_3\) overhangs. As such,
let the function \(up\) (resp. \(down\)) be defined as the function which:
- to a module \(m\) associates the word \(o_5\) (resp. \(o_3\)) from its
canonic module decomposition
- to a vector \(v\) associates the word \(o_3\) (resp. \(o_5\)) from its
canonic vector decomposition.
Standard Assembly
Definition: Standard MoClo Assembly
Given an assembly of level \(l\), where \(m_1, \dots, m_k \in M_l^k, v \in V_l\):
\[a:\quad m_1 + \dots + m_k \xrightarrow{\quad e_l \quad} A \subset (\Sigma^\star \cup \Sigma^{(c)})\]
and the partial order \(le\) over \(S = \{m_1, \dots, m_k\}\) defined as:
\[\begin{split}\begin{array}{l}
\forall x, y \in S^2, \\
\quad x \le y \iff \begin{cases}
x = y & \\
down(x) = up(y) & \text{ if } x \ne y\\
\exists z \in S \backslash \{x, y\}, down(x) = up(z), \ z \le y & \text{ if } x \ne y \text{ and } down(x) \ne up(y)
\end{cases}
\end{array}\end{split}\]
then a chain \(\langle S\prime, \le \rangle \subset \langle S, \le \rangle\) is
an insert if:
\[\begin{split}\begin{cases}
v \le min(S^\prime) \\
max(S^\prime) \le v
\end{cases}
\iff
\begin{cases}
down(v) = up(min(S^\prime)) \\
up(v) = down(max(S^\prime))
\end{cases}\end{split}\]
\(a\) is:
- invalid if \(\langle S, \le \rangle\) is an antichain or \(\langle S, \ge \rangle\)
has no insert.
- valid if \(\langle S, \le \rangle\) has at least one insert.
- ambiguous if \(\langle S, \le \rangle\) has more than one insert.
- unambiguous if \(\langle S, \le \rangle\) has exactly one insert.
- complete if \(\langle S, \le \rangle\) is an insert.
Corollary
If an assembly \(a\) is complete, then there exist a permutation
\(\pi\) of \([\![1, k]\!]\) such that:
\[m_{\pi(1)} \le m_{\pi(2)} \le \dots \le m_{\pi(k-1)} \le m_{\pi(k)}\]
and:
\[\begin{split}\begin{array}{lll}
up(m_{\pi(1)}) &=& down(v) \\
down(m_{\pi(k)}) &=& up(v)
\end{array}\end{split}\]
Property: Uniqueness of the cohesive ends
If an assembly
\[m_1 + \dots + m_k \xrightarrow{\quad e_l \quad} A \subset (\Sigma^\star \cup \Sigma^{(c)})\]
is unambiguous and complete, then \(\forall i \in [\![1, k]\!]\),
\[\begin{split}\left\{
\begin{array}{llll}
up(m_i) &\ne& down(m_i)& \\
up(m_i) &\ne& up(m_j), & j \in [\![1, k]\!] \backslash \{i\} \\
down(m_i) &\ne& down(m_j), & j \in [\![1, k]\!] \backslash \{i\} \\
\end{array}
\right .\end{split}\]
Demonstration
Let there be an unambiguous complete assembly
\[a:\quad m_1 + \dots + m_k \xrightarrow{\quad e_l \quad} A\]
\(up(m_i) \ne down(m_i)\)
Let’s suppose that \(\exists i \in [\![1, k]\!]\) such that
\[up(m_i) = down(m_i)\]
then \(\langle \{m_1, \dots, m_k\} \backslash \{m_i\}, \le \rangle\)
is also an insert, which cannot be since \(a\) is complete.
\(up(m_i) \ne up(m_j)\)
Let’s suppose that \(\exists (i, j) \in [\![1, k]\!]^2\) such that
\[up(m_i) = up(m_j)\]
Since the \(a\) is complete, there exists \(pi\) such that
\[m_{\pi(1)} \le m_{\pi(2)} \le \dots \le m_{\pi(k-1)} \le m_{\pi(k)}\]
and since \(a\) is unambiguous, \(\langle \{m_1, \dots, m_k\}, \le \rangle\)
is the only insert.
\(down(m_i) \ne down(m_j)\)
TODO
Property: Uniqueness of the assembled plasmid
If an assembly
\[m_1 + \dots + m_k \xrightarrow{\quad e_l \quad} A \subset (\Sigma^\star \cup \Sigma^{(c)})\]
is unambiguous, then
\[A \cap \Sigma^{(c)} = \{p\}\]
with
\[p = \left( up(v) \cdot b \cdot up(m_{\pi(1)}) \cdot t_{\pi(1)} \cdot \, \dots \, \cdot up(m_{\pi(n)}) \cdot t_{\pi(n)} \right) ^{(c)}\]
(\(n \le k\), \(n = k\) if \(a\) is complete).