# A cybernetic model of bacterial chemotactic response to nutrient limitation

#### Abstract

One response bacteria have to their environment is chemotaxis, in which the cells respond to the gradient of stimuli. The strength of the chemotactic response to a nutrient depends upon the concentration of the nutrient at which the bacteria are cultivated. To create a more realistic model of chemotaxis, the cybernetic perspective is applied to the growth of the bacteria: the manipulation by the cells of their chemotactic sensitivity ($\delta$) is considered to be goal-seeking: maximization of an objective function. Hence, an objective function is sought that predicts the sensitivity dependence on the nutrient concentration in agreement with experimental measurements. Long-term, short-term and matching-law objectives are considered, using a modification of the chemotaxis model by Rivero et al. (1989). A confined-growth model is used for the long-term perspective, in which the total number of cells is maximized. Two first-order ODE's for the cell mass and substrate concentration are solved using orthogonal collocation on finite elements (OCFE), initially for constant $\delta$ (${\not=}\delta$(x)). Optimized long-term solutions are found with a GRG algorithm and $\delta$ a function of x. In the matching-law model, first- and second-order stationarity conditions are applied to locate the optimal $\delta$ profile. Solutions of the ODE's are very sensitive to the placement of the finite elements. Through analysis of the discretization error (resulting in improved element placement), the near-singular system is solved accurately. The long-term perspective yields a $\delta$-profile with sharp oscillations near the nutrient source and a transition to slow decay elsewhere, and is probably a numerical artifact. The short-term model is dismissed since it can not properly represent the response of the cells to a source of nutrient. The results of the matching-law model qualitatively resemble the experimental measurements. The principal conclusions are: (1) the long- and short-term objectives do not apply, as the computed $\delta$ profiles are not qualitatively similar to the experimental measurements, (2) the matching-law model predicts these measurements qualitatively, (3) the confined-growth model, with Monod or Moser growth kinetics, is unacceptable for highly-chemotactic cells, and (4) the placement of the elements in the OCFE method is crucial to compute accurate solution profiles.

#### Subject Area

Chemical engineering

#### Recommended Citation

Koster, Loretta Gail, "A cybernetic model of bacterial chemotactic response to nutrient limitation" (1991). Dissertations available from ProQuest. AAI9125695.
https://repository.upenn.edu/dissertations/AAI9125695

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