Theoretical analysis of the cost of antagonistic activity for aquatic bacteria in oligotrophic environments

Abstract

Many strains of bacteria produce antagonistic substances that restrain the growth of others, and potentially give them a competitive advantage. These substances are commonly released to the surrounding environment, involving metabolic costs in terms of energy and nutrients. The rate at which these molecules need to be produced to maintain a certain amount of them close to the producing cell before they are diluted into the environment has not been explored so far. To understand the potential cost of production of antagonistic substances in water environments, we used two different theoretical approaches. Using a probabilistic model, we determined the rate at which a cell needs to produce individual molecules in order to keep on average a single molecule in its vicinity at all times. For this minimum protection, a cell would need to invest 3.92 × 10(-22) kg s(-1) of organic matter, which is 9 orders of magnitude lower than the estimated expense for growth. Next, we used a continuous model, based on Fick's laws, to explore the production rate needed to sustain minimum inhibitory concentrations around a cell, which would provide much more protection from competitors. In this scenario, cells would need to invest 1.20 × 10(-11) kg s(-1), which is 2 orders of magnitude higher than the estimated expense for growth, and thus not sustainable. We hypothesize that the production of antimicrobial compounds by bacteria in aquatic environments lies between these two extremes.

Keywords: Bacterial antagonism; community ecology; diffusion; microbial ecology; microbial physiology.

Figure 1

Schematic illustration of the scenarios considered in this study. Cells are shown as red circles, and antagonistic molecules as black dots. In the lower left side, an example of a random trajectory of a molecule is represented as a black line. In the lower right side, a gradient represents concentrations of antagonistic molecules, with darker shades indicating higher concentrations. On the left side, a cell keeping on average one molecule of an antagonistic substance in its vicinity is considered. As shown, this would grant the cell an absolute minimum of protection against competing cells, and the associated production cost is considered a lower boundary for resource investment. Note that in these conditions, competing cells still could be close to the producing strain and not encounter the antagonistic molecule. These production costs were calculated using in silico simulations in this work, analyzing the random movements of individual molecules, and modeling the distribution of many such simulations (probabilistic modeling). On the right side, another extreme scenario is illustrated, where a cell would obtain full protection by maintaining certain concentration of antagonistic molecules around itself. The ideal concentration was arbitrarily chosen to be the minimum inhibitory concentration, as explained in the text. A cell in an intermediate scenario is also shown for illustration, although it is not explicitly investigated in this work.

Figure 2

Simulation to illustrate “random walk” movements of a particle away from a cell (black circles) that produced it. (A) Trajectories of three different particles produced by the same cell at different time points are shown as colored continuous lines (different colors are used to distinguish the three independent particles; red dots, circles and squares mark the start and end positions of the three independent particles), since the time they are released, until they reach a distance of 3 cell radiuses from the cell surface for the first time. Notice that they will approach the cell again with probability 0.5 at the next step, or drift further away. The number of steps taken to reach that position, is indicated for each case. Note that the length of each step is greatly exaggerated in this figure, as compared to the scales of real molecules and bacteria. (B) The trajectories are plotted around the same depiction of the cell, putting them in the same time-frame. (C) The trajectories are rotated, putting them in the same spatial frame, illustrating how these bi-dimensional trajectories can be studied in one dimension if the only quantity of interest is the distance from the cell. This same reasoning can be applied for a three-dimensional scenario. (D) Distribution of the number of steps taken to reach 3 cell radiuses in 1000 simulations with the same scales as in panel (A). The red line indicates an Inverse Gaussian Distribution fitted to the data. Note that an equally well fit can be attained using a Birnbaum-Saunders Distribution, but the Inverse Gaussian is preferred for theoretical reasons.

Figure 3

Simulation experiment showing the relation between parameter μ of the Inverse Gaussian Distribution, and the square of the number of steps n²_b needed to reach a radius n_b from the surface of the cell, measured in step-lengths. For each point, 1000 particles were allowed to wander with steps of fixed length and random direction, simulating Brownian motion, until they reached the radius indicated in the x axis, and the number of steps taken were recorded. The values correspond to parameter μ, obtained by fitting the Inverse Gaussian Distribution to the corresponding 1000 times. We notice that this relation can most likely be obtained analytically, but do not count with a mathematical proof at the moment.