High-throughput screening of enzyme libraries: thiolactonases evolved by fluorescence-activated sorting of single cells in emulsion compartments

Single bacterial cells, each expressing a different library variant, were compartmentalized in aqueous droplets of water-in-oil (w/o) emulsions, thus maintaining a linkage between a plasmid-borne gene, the encoded enzyme variant, and the fluorescent product this enzyme may generate. Conversion into a double, water-in-oil-in-water (w/o/w) emulsion enabled the sorting of these compartments by FACS, as well as the isolation of living bacteria cells and their enzyme-coding genes. We demonstrate the directed evolution of new enzyme variants by screening >10(7) serum paraoxonase (PON1) mutants, to yield 100-fold improvements in thiolactonase activity. In vitro compartmentalization (IVC) of single cells, each carrying >10(4) enzyme molecules, in a volume of <10 femtoliter (fl), enabled detection and selection despite the fast, spontaneous hydrolysis of the substrate, the very low initial thiolactonase activity of PON1, and the use of difusable fluorescent products.     

Kinetics of Diffusing Polymer Encounter in Confined Cellular Microdomains

We study the mean first time that two monomers, located on the same polymer, encounter in a confined microdomain. Approximating the confined geometry by a harmonic potential well, we obtain an asymptotic expression for the mean first encounter time (MFETC) as a function of the radius ε around one monomer. By studying the end-to-end distance of the polymer in a ball using the Edwards’ formalism, we derive an other estimation of the MFETC. We validate the asymptotic formulas using Brownian simulations and derive their range of validity in terms of the polymer length. We apply the present models to compute the mean time for a gene located far away from a promoter site to be activated during looping in confined genomic territories.     

On parallel asynchronous high-order solutions of parabolic PDEs

In achieving significant speed-up on parallel machines, a major obstacle is the overhead associated with synchronizing the concurrent processes. This paper presents high-orderparallel asynchronous schemes, which are schemes that are specifically designed to minimize the associated synchronization overhead of a parallel machine in solving parabolic PDEs. They are asynchronous in the sense that each processor is allowed to advance at its own speed. Thus, these schemes are suitable for single (or multi) user shared memory or (message passing) MIMD multiprocessors. Our approach is demonstrated for the solution of the multidimensional heat equation, of which we present a spatial second-order Parametric Asynchronous Finite-Difference (PAFD) scheme. The well-known synchronous schemes are obtained as its special cases. This is a generalization and expansion of the results in [5] and [7]. The consistency, stability and convergence of this scheme are investigated in detail. Numerical tests show that although PAFD provides the desired order of accuracy, its efficiency is inadequate when performed on each grid point.In an alternative approach that uses domain decomposition, the problem domain is divided among the processors. Each processor computes its subdomain mostly independently, while the PAFD scheme provides the solutions at the subdomains' boundaries. We use high-order finite-difference implicit scheme within each subdomain and determine the values at subdomains' boundaries by the PAFD scheme. Moreover, in order to allow larger time-step, we use remote neighbors' values rather than those of the immediate neighbors. Numerical tests show that this approach provides high efficiency and in the case which uses remote neighbors' values an almost linear speedup is achieved. Schemes similar to the PAFD can be developed for other types of equations [3].This research was supported by the fund for promotion of research at the Technion.     

Existence of identities inA⊗B

The following theorem is proved: ifA andB are twoPI-algebras over a fieldF, thenA⊗ _F B is aPI-algebra. This paper was written while the author was doing his Ph.D. thesis at the Hebrew University of Jerusalem under the supervision of Professor A. S. Amitsur, to whom the author wishes to express his warm thanks.     

representations and combinatorial identities

For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the entry equals a given number . Beside the combinatorics of finite standard tableaux, the main tools here are from the Vershik-Kerov character theory of . The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.

     

Asynchronous and corrected-asynchronous finite difference solutions of PDEs on MIMD multiprocessors

A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent processes. Removing the synchronization constraint has the potential of speeding up the computation, while maintaining greater computation flexibility (e.g. differences in processors speed; differences in the data input to processors). We construct asynchronous (AS) finite difference schemes for the solution of PDEs by removing the synchronization constraint. We analyze the numerical properties of these schemes. Based on the analysis, we develop corrected-asynchronous (CA) finite difference schemes which are specifically constructed for an asynchronous processing. We present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although our discussion concentrates on the Euler scheme it should serve only as a sample, as it can be extended to other schemes and other PDEs.These schemes are implemented on the shared-memory multi-userSequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that synchronization penalty can be about 50% of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time-level, and with the problem dimension. The efficiency of the AS may reach 90% and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient but provides more accurate results for intermediate (non steady-state) values. The results show the potential of developing asynchronous finite deference schemes for steady-state as well as non steadystate problems.This research was partially supported by a grant from The Basic Research Foundation administrated by The Israel Academy of Sciences and Humanities.A reduced version of the paper was presented at the 4th SIAM Conference on Parallel Processing for Scientific Computing, Dec. 11–13, 1989, Chicago, USA.The work by this author was supported by research grant 337 of the Israeli National Council for Research and Development in the years 1990–1991.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASI-18107 while the author was in residence at the Institute for Computer Applications in Sciences and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, USA.     

The Max-Flow Min-Cut theorem for countable networks

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are “orthogonal” to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends.     

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics. Partially supported by Minerva Grant No. 8441 and by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272. Partially supported by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.     

Codimensions and trace codimensions of matrices are asymptotically equal

The codimensionsc _n and the trace codimensionst _n of thek×k matrices are asymptotically equal: lim _n→∞(t _n /c _n )=1. Sincet _n ≅q(n)·k ²ⁿ whereq(x) is a known rational function, this asymptotically givesc _n . This has applications to the codimensions of Capelli identities.     

Combinatorial sums,identities and trace identities of the 2 × 2 matrices

An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions c_n(F₂) and the trace codimensions t_n(F₂) of F₂, the 2 × 2 matrices, are asymptotically equal: $\text{c}{}_{\mathrm{n}}\text{(F}{}_2\text{) ? t}{}_{\mathrm{n}}\text{(F}{}_2\text{)}$.