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the state space M is still too large, then one would llke to been very useful in analyzing such systems. This paper deals with anal e Of minimal r r a l i o n for linear systems are and apply a systematic technique to reduce M and still pregous questions for nonlinear systems. serve the input-output structure of the model. Using the ideas of controllability and observability, in the early 1960's Kalman and others carried out this program for linear systems. The similar questions for nonlin-REQUENTLY, control systems of the following form ear systems were not effectively treated until the early are used to model the behavior of physical, biological, 1970's. Based on the work of Chow [5], Hermann [9], or social systems, Haynes-Hermes [8], and Brockett [9] and working independently. Lobry [21], [22], Sussman-Jurdjevic [25], and

In paper I of this series a theory of the solubility of hydrocarbons in water was formulated in terms of the significant structure theory of liquids. In this theory, the number of water molecules that can be packed around the hydrocarbon molecule enters into the partition function of a hydrocarbon solution. This number is one measure of the size of the solvent cavity just large enough to accommodate a solute molecule. A more idealized quantity that is more easily computed is the area of the cavity surface which contains the centers of the water molecules in the first layer around the solute. It has been found that the area defined in this manner is linearly related to the logarithm of the hydrocarbon solubility in water. All cavities computed in this paper are nonspherical and depend on detailed hydrocarbon shape. The correlation suggests a semiempirical means of calculating the hydrophobic interactions between hydrocarbon moieties and of obtaining Hansch values.

Miniaturized growth systems for heterogeneous culture collections are not only attractive in reducing demands for incubation space and medium but also in making the parallel handling of large numbers of strains more practicable. We report here on the optimization of oxygen transfer rates in deep-well microtiter plates and the development of a replication system allowing the simultaneous and reproducible sampling of 96 frozen glycerol stock cultures while the remaining culture volume remains frozen. Oxygen transfer rates were derived from growth curves of Pseudomonas putida and from rates of oxygen disappearance due to the cobalt-catalyzed oxidation of sulfite. Maximum oxygen transfer rates (38 mmol liter ؊1 h ؊1 , corresponding to a mass transfer coefficient of 188 h ؊1 ) were measured during orbital shaking at 300 rpm at a shaking diameter of 5 cm and a culture volume of 0.5 ml. A shaking diameter of 2.5 cm resulted in threefold-lower values. These high oxygen transfer rates allowed P. putida to reach a cell density of approximately 9 g (dry weight) liter ؊1 during growth on a glucose mineral medium at culture volumes of up to 1 ml. The growth-and-replication system was evaluated for a culture collection consisting of aerobic strains, mainly from the genera Pseudomonas, Rhodococcus, and Alcaligenes, using mineral media and rich media. Cross-contamination and excessive evaporation during vigorous aeration were adequately prevented by the use of a sandwich cover of spongy silicone and cotton wool on top of the microtiter plates.The general practice during screening of large microbial strain collections for new enzyme activities or bioactive compounds is that each strain is handled individually. The labor and time involved in separately reviving the stock cultures on agar plates and the subsequent inoculation of tubes or Erlenmeyer flasks are major limiting factors in the progress of such screening projects. This bottleneck might be alleviated if large numbers of microbial strains could be handled in parallel rather than in sequence, preferably in a format that is compatible with current standards in the manipulation of large numbers of liquid samples, e.g., in 96-or 384-well microtiter plates.Until now, the use of microtiter plates for the growth and maintenance of microbial strains has been mainly limited to clonal libraries in Escherichia coli (9,11,20) and yeasts (2, 18). For this purpose, the apparent drawbacks of growth in microtiter plates-low aeration rates and small working volumesare generally less relevant since E. coli and yeasts can grow anaerobically. Furthermore, small amounts of biomass generally suffice as the use of high-copy-number vectors ensures high levels of the desired gene or gene product. The resulting mixed aerobic-anaerobic growth, however, can lead to undesirable variations in biomass levels from well to well, especially since oxygen diffusion rates for an individual well may be affected by its position in the microtiter plate. Such well-to-well variations are especially troublesome in...

Gas-liquid mass transfer properties of shaken 96-well microtiter plates were characterized using a recently described method. The maximum oxygen transfer capacity (OTR(max)), the specific mass transfer area (a), and the mass transfer coefficient (k(L)) in a single well were determined at different shaking intensities (different shaking frequencies and shaking diameters at constant filling volume) and different filling volumes by means of sulfite oxidation as a chemical model system. The shape (round and square cross-sections) and the size (up to 2 mL maximum filling volume) of a microtiter plate well were also considered as influencing parameters. To get an indication of the hydrodynamic behavior of the liquid phase in a well, images were taken during shaking and the liquid height derived as a characteristic parameter. The investigations revealed that the OTR(max) is predominantly dependent on the specific mass transfer area (a) for the considered conditions in round-shaped wells. The mass transfer coefficient (k(L)) in round-shaped wells remains at a nearly constant value of about 0.2 m/h for all shaking intensities, thus within the range reported in the literature for surface-aerated bioreactors. The OTR(max) in round-shaped wells is strongly influenced by the interfacial tension, determined by the surface tension of the medium used and the surface properties of the well material. Up to a specific shaking intensity the liquid surface in the wells remains horizontal and no liquid movement can be observed. This critical shaking intensity must be exceeded to overcome the surface tension and, thus, to increase the liquid height and enlarge the specific mass transfer area. This behavior is solely specific to microtiter plates and has not yet been observed for larger shaking bioreactors such as shaking flasks. In square-shaped microtiter plate wells the corners act as baffles and cause a significant increase of OTR(max), a, and k(L). An OTR(max) of up to 0.15 mol/L/h can be reached in square-shaped wells.