Washout rate

The ICR flow rate was five to eight times faster than the CSTR. The overall conversion of sugars in the ICR at a 12 hour retention time was 60%, At this retention time, the ICR was eight times faster than CSTR, but in the CSTR an overall conversion rate of 89% was obtained. At the washout rate for the chemostat, the ICR resulted in a 38% conversion of total sugars. Also, the organic acid production rate in the ICR was about four times that of the CSTR. At a higher retention time of 28 hours, the conversion of glucose in the ICR and CSTR are about the same, but the conversion of xylose reached 75% in the ICR and 86% in the CSTR. 

II. Chemostat In a chemostat, the medium is delivered at a constant rate, to keep constant growth rate. The nutrient medium is supplied to the culture vessel at a constant rate by a peristaltic pump used to control the washout rate. The rate of media flow is often set at approximately 20% of culture volume per day. Air is pumped into the algal culture vessel through an air compressor controlled by a flow meter and carried in two flasks of sterile water. This bubbling air has three effects (i) it supplies C02 and 02 to the culture, (ii) allows circulation and agitation of the cultures and (iii)... 

S° the constant concentration input of the limiting nutrient (S° > 0) D the constant washout rate (D > 0) ... 

S° the constant concentration input of the limiting nutrient (S° > 0) D the constant washout rate (D>0) m1 the maximal specific growth rate of P. subcapitata (up > 0) m2 the maximal specific growth rate of C. vulgaris (m2> 0) ... 

Several F-labeled fatty acid derivatives have been successfully prepared and evaluated as potential FAO assessing tracers [19-24]. Methyl-branched-chain (w- F-fluorofatty acids, such as 3-methyl-(3-MFHA) and 5-methyl-17-[ F]fluoro-heptadecanoic acid (5-MFHA), have been reported [23]. In a comparative study, it was found that rw-[ F]fluoropalmitic acid (FPA) exhibits the highest myocardial uptake, followed by 5-MFHA and 3-MFHA. FPA possesses the fastest myocardial washout rate, and 3-MFHA the slowest. In lipid analysis studies, 5-MFHA... 

In ocular drug delivery, the high rate of tear turnover, and the blinking action of the eyelids lead to short precorneal residence times for applied eye drops. Typically, the washout rate reduces the concentration of the drug in a tear film to one-... 

The appropriate initial conditions are S(0) > 0 and x(0) > 0. The number of parameters in the system is excessive, so some scaling is in order. First of all, note that and D (the input concentration and the washout rate) are under the control of the experimenter. The term has units of concentration and D has units of reciprocal time. Equations (2.1) and (2.2) may be rewritten as... 

Another question that can be raised concerns the validity of our assumption that all of the removal is accounted for by the washout term. If, for example, a competitor s mortality rate is a significant fraction of the washout rate D, then the assumption is not valid. In this case, the removal rate for that competitor should be the sum of D and the mortality rate. Another possibility is that a filter on the output might slow the washout of an organism but not the nutrient. This could result in that organism s removal rate being less than D. A natural question, then, is whether a species-specific removal rate changes the outcome. 

Figure 9.1. Operating diagram for two-species competition with a varying communication rate E different from the washout rate D. In region VI, Q = [foj in region V, Q = [ , 2]-, in region II, = l o. iU in regions I and IV, fi = f o, i, 2l in region III, Q, = q, , 2, . (From [STa], Copyright 1989, Journal of Mathematical Biology. Reproduced by permission.)...

The S is the input concentration of nutrient (to the leftmost vessel), and D is the washout rate. These two parameters are under the control of the experimenter. The terms 7 and y, are the yield coefficients. For convenience, one can scale substrate concentrations S, by S , time by /D (making m, nondimensional and D = 1), and microorganism concentrations by and to obtain the less cluttered system... 

Mathematically and experimentally there is no reason to connect the vessels linearly, to restrict the source to the left-hand vessel, or to keep the washout rates D equal so long as the volume of the fluid in each vessel is kept constant (see [S7]). We next describe a class of gradostat models which is sufficiently general to include all cases of biological interest and yet remain mathematically tractable. 

The focus in this chapter will be on the possibility of coexistence of two competitors competing for a single nutrient in a chemostat with an oscillatory washout rate. Therefore, an exhaustive study of sufficient conditions for competitive exclusion to hold, as was carried out in Chapter 1, will not be made here. The reference [BHW2J may be consulted for results of this type. 

The main change in this system compared to (5.1) of Chapter 1 is that the constant washout rate D of (5.1) is replaced by a positive, continuous, periodic function >(/) ... 

When (3.3) holds for the /th competitor, that competitor can survive in the chemostat in the absence of competition and with its concentration oscillating in response to the periodically varying washout rate. This is the content of the next result. 

Very little information exists on the robustness of the parameter region for which coexistence occurs. Studying the case where the nutrient concentration, rather than the washout rate, is varied periodically, Hsu [Hsu2J obtains very interesting information about the parameter region corresponding to coexistence. Perturbation methods are used in [SI] to explore this region in the case studied by Hsu. See also [SFA] for other numerical work in both cases. 

Finally, the washout rate D in the chemostat and the population death rate d are assumed to be constant, independent of /. Therefore, the removal rate of the organism is given hy D — D + d. 

The variables and the units are those which have been used since Chapter 1 S(t) is the nutrient concentration at time t, Xx(t) is the concentration of plasmid-bearing organisms at time t, and X2(0 is the concentration of plasmid-free organisms at time t S is the input concentration of the nutrient, and D is the washout rate of the chemostat. These are the operating parameters. The mj term is the maximal growth rate of x, and a, is the Michaelis-Menten (or half-saturation) constant of x,. These are assumed to be known (measurable) properties of the organism that characterize its growth and reproduction. A plasmid is lost in reproduction with probability q, and y is the yield constant. 

The principal open questions that arise in the treatment of the chemo-stat with periodic washout rate, discussed in Chapter 7, are analogous to those mentioned in connection with Chapter 6. Namely, can sufficient conditions be given for the uniqueness of the positive periodic solution (fixed point of the Poincare map) that represents the coexistence of the two populations What can be said of the case of more than two competitors ... 

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