Modeling bacterial chemotaxis
Bacteria migrate in response to chemical gradients using flagella that rotate through solution to swim. Often the way this is done is by sensing the chemical concentration at one time, and comparing it with the concentration at a later time. If they are attracted to the chemical (say it is food), they will have a mechanism to swim towards it. See for example this wikipedia article Links to an external site. on chemotaxis.
When bacteria swim, there are two kinds of behavior. One is motion in, more or less, a straight line, but the other is that they occasionally "tumble" that reorient the bacteria in a direction. In order to swim towards food, it appears to follow the following rule: if the concentration is increasing, it will tumble less often, if it is decreasing, it'll tumble more often. This way if it is going in the wrong direction it can quickly try a new one, but if it seems to be going the right way, there's no need to change directions.
The code hw5/chem.py lets you play around with this problem.
prob_tumble gives the probability of tumbling given the concentration difference delta_c that was sensed. concentration_linear and concentration_well are two concentration profiles that you can try out.
- 1. Run the code with both these concentration profiles and describe the results. There is already code to histogram position. This is important in understanding the bacterial density in steady state for the case of a concentration well. compare_to_histogram compares this to that of a Gaussian. Set "graphics = 0" in order to see this output in a reasonable time!
- 2. Are the results the same as for a Brownian particle in a quadratic potential? What are the differences? Should you expect this to be the same or different?
- 3. Can you come up with a model for the bacterial density that works for different concentration profiles? (This is hard!)
- 4. The above strategy is employed by bacteria and only uses one sensor. This is because a bacterium is small and measurements of concentration are noisy. So the ability to detect the direction of a concentration gradient is limited by the small size and by noise. Give situations where receptors and cell size are sensitive enough to detect gradients with enough certainty to be useful.
- 5 Now try to understand this problem physically. Consider a cell in a static chemical gradient. If a cell has infinite time and many receptors, will it be able to detect concentration gradients? '''Hint: look at this paper Links to an external site. by Berg and Purcell.
- 6 If the cell makes a mistake and detects a gradient in the wrong direction, this will slow down its search for food. If it waits long enough, it can obtain more accurate answers but this delays how rapidly it can search. On the other hand, the tumbling mechanism, although imperfect is reasonably good at finding food, however the random tumbling consumes more energy than if the bacterium were to move initially in the correct direction. So we have these trade-offs:
- noise in the gradient measurement versus time needed to take a measurement.
- tumbling strategy using one receptor versus gradient measurement using many receptors.
- A bacterium does not want to spend a long time finding food. Cells that find it fast will be at a selective advantage, on the other hand energy is expended in a strategy that requires too much movement.
Try to develop a theory of the optimal strategy for single cell chemotaxis. Your answer should depend on the above trade-offs, which will be dependent on cell size, typical gradients in concentration, and possibly energy expenditure. Is there are minimum size where the mechanism of tumbling by itself is always superior to tumbling plus measurements of gradients?
This is obviously hard and a lot of research on cell migration already exists and more than 20,000 articles are published on this subject per year. Therefore doing a literature search should answer many questions about chemotaxis, however you will find that many questions are still unanswered. Here Links to an external site. is a review of models of bacterial chemotaxis that might be useful. By analytic estimates, computer simulation, and researching the existing literature, you have the opportunity to try to answer questions in this exciting field. These are other papers that are of interest:
https://arxiv.org/pdf/1512.02542.pdf Links to an external site.
https://www.annualreviews.org/doi/pdf/10.1146/annurev.bi.44.070175.002013 Links to an external site.