ellermeyer - Mathematical Models of Microbial Growth Sean...

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Mathematical Models of Microbial Growth Sean Ellermeyer Modeling in Mathematical Biology AP4484/MAP4484) (MAP4484/MAP4484) University of Florida
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The Basic Model for Microbial Growth (Monod and others, 1940s 50s) ypotheses Hypotheses Microbial growth rate is determined by the ncentration of a single growth iting substrate concentration of a single growth limiting substrate (nutrient) in the growth medium. he growth rate adjusts itself instantaneously in The growth rate adjusts itself instantaneously in response to changes in the substrate concentration.
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s t concentration of substrate at time t x t concentration of microorganism at time t x t alled e pecific rowth te f e olony x t is called the specific growth rate of the colony. Monod hypothesized that x t x t s t  here ontinuous nd creasing nction at atisfies where is a continuous and increasing function that satisfies 0 0 and lim s s m m is called the maximal specific growth rate.
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Specifically, Monod hypothesized that has the form s m s K h s where m is the maximal specific growth rate and K h is called the half-saturation constant (because it is the value of s for which s m /2 ).
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Onemorehypothes isis: The rate of decrease of substrate is instantaneously proportional to the rate of increase of microorganism. Thus s x t Ys t where mass of organism formed Y mass of substrate consumed is called the yield constant . y ombining ll f ur odelling ypotheses By combining all of our modelling hypotheses, we obtain:
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The Monod Model for Microbial Growth in Batch Culture s t s t 1 Y m K h s t x t , s 0 s 0 0 m s t x t K h s t x t , x 0 x 0 0 This solutions of this model are in agreement with hat ould e xpected ased n e odelling what would be expected based on the modelling hypotheses: The substrate decreases. The microbe increases (as an increasingly slower rate as substrate becomes depleted) and lim t s t 0 , t
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ellermeyer - Mathematical Models of Microbial Growth Sean...

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