Prior observations of phage-host systems in vitro have led to the conclusion that susceptible host cell populations must reach a critical density before phage replication can occur. Such a replication threshold density would have broad implications for the therapeutic use of phage. In this report, we demonstrate experimentally that no such replication threshold exists and explain the previous data used to support the existence of the threshold in terms of a classical model of the kinetics of colloidal particle interactions in solution. This result leads us to conclude that the frequently used measure of multiplicity of infection (MOI), computed as the ratio of the number of phage to the number of cells, is generally inappropriate for situations in which cell concentrations are less than 10 7 /ml. In its place, we propose an alternative measure, MOI actual , that takes into account the cell concentration and adsorption time. Properties of this function are elucidated that explain the demonstrated usefulness of MOI at high cell densities, as well as some unexpected consequences at low concentrations. In addition, the concept of MOI actual allows us to write simple formulas for computing practical quantities, such as the number of phage sufficient to infect 99.99% of host cells at arbitrary concentrations.It has long been observed that when bacteriophage are mixed with susceptible host bacteria, the number of phage in the culture supernatant does not increase until after an eclipse period of, generally, 30 to 40 min at 37°C has passed. This period of time is explained as the time the phage requires to inject its genome into the host, express its genes, and assemble progeny phage and release them into the environment. Additionally, when host cell densities are very low, it has been observed that there is a longer delay before phage numbers increase over the numbers of input phage. This period has been explained as the time needed for the host cells to reach a "replication threshold" (16) or "proliferation threshold" (7,8) density. This density has been reported to be approximately 10 4 cells per ml for the multiple phage-host combinations tested (16) and has been said to have broad implications for the propagation of phages in natural environments and in terms of their use as antimicrobial therapies (7,8). The mechanism of this delay in phage replication has not been widely investigated or discussed.One explanation for the apparent threshold density would be a requirement on the part of the phage for the host cell to be in a particular metabolic state and that this state is only reached when the cell density is 10 4 CFU/ml or more. Small molecules called autoinducers or quorum factors are known to be secreted into the environment by bacteria and, by their accumulation as the number of cells increases, to allow the bacteria to monitor their local population density (3). These soluble signaling molecules alter the expression of dozens of genes and thereby regulate the metabolic state when the sensing bacteria are expo...

Abstract. The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in x.

Rational and soliton solutions of the KP hierarchy in the subgrassmannian Gr 1 are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, τ , which are equivalent to bispectrality of the associated wave function, ψ, are identified. In particular, it is shown that there exists a bound on the degree of all time variables in τ if and only if ψ is a rank one bispectral wave function. The action of the bispectral involution, β, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions and particle systems, it is demonstrated that β is a linearizing map of the Calogero-Moser particle system and is essentially the map σ introduced by Airault, McKean and Moser in 1977 [2].

This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order r > 1. As a result, the bispectrality of a large family of algebras of rank r is demonstrated. An involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters, generalizing Wilson's rank one bispectral involution. Spectral geometry and the relationship to the Sato grassmannian are discussed.

Pairs of n × n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r × r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs (L, Λ) of bispectral matrix differential operators is different than those previously studied in that L acts from the left, but Λ from the right on a common r × r eigenmatrix.