Protein pathways are dynamic and highly coordinated spatially and temporally, capable of performing a diverse range of complex chemistries and enzymatic reactions with precision and at high efficiency. Biotechnology aims to harvest these natural systems to construct more advanced in vitro reactions, capable of new chemistries and operating at high yield. Here, we present an efficient Multiplex Automated Genome Engineering (MAGE) strategy to simultaneously modify and co-purify large protein complexes and pathways from the model organism Escherichia coli to reconstitute functional synthetic proteomes in vitro. By application of over 110 MAGE cycles, we successfully inserted hexa-histidine sequences into 38 essential genes in vivo that encode for the entire translation machinery. Streamlined co-purification and reconstitution of the translation protein complex enabled protein synthesis in vitro. Our approach can be applied to a growing area of applications in in vitro one-pot multi-enzyme catalysis (MEC) to manipulate or enhance in vitro pathways such as natural product or carbohydrate biosynthesis.

Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$ and $s_i=r_1+\ldots +r_i$ for every $i\in\{1,2,\ldots,n\}$. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums $s_i$. Define $p(\vec{r})$ to be the number of positive sum $s_i$ among $s_1,\ldots,s_n$ and $m(\vec{r})$ to be the smallest index $i$ with $s_i=\max\limits_{0\leq k\leq n}s_k$. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let $\vec{r}_i=(r_i,\ldots,r_n,r_1,\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\vec{r}$. For $s>0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$; for $s\leq 0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$. We also give an analogous result for the class of all permutations of $\vec{r}$.

The u-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types gl(n), sp(2n) and so(2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU (p, q), SU (p) × SU (q)), (Sp(2n), U (n)) and (SO * (2n), U (n)).

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a b s t r a c tA k-container of G between u and v, C (u, v), is a set of k internally disjoint paths between u and v. A k * -container C (u, v) of G is a k-container if it contains all vertices of G.A graph G is k * -connected if there exists a k * -container between any two distinct vertices. Thus, every 1 * -connected graph is Hamiltonian connected. Moreover, every 2 * -connected graph is Hamiltonian. Zhan proved that G = L(M) is Hamiltonian connected if the edgeconnectivity of M is at least 4. In this paper, we generalize this result by proving G = L(M)is k * -connected if the edge-connectivity of M is at least max{2k, 4}. We also generalize our result into spanning fan-connectivity.