Models of quorum sensing for multi-agent systems

Quorum sensing is a system of stimulus and responses correlated to population density that is used by bacteria to coordinate gene-expression. I am looking for a simple computational/mathematical model of quorum sensing that abstracts away from the details of the mechanism implementing it inside the agent, but keeps the key inter-agent properties like diffusion rate, range, and timing.

Is there a standard abstract mathematical model of quorum sensing used by biologists?

I am not interested in the particulars of a specific organism, but would like a general model I could apply to capture the 'gist' for any organism that relies on quorum sensing for part of its behavior.

Bernardini et al. (2007) provided an extension to P-systems incorporating the basics of quorum sensing, and Romero-Campero & Pérez-Jiménez (2008) have used their approach to model bioluminosity in vibrio fischeri. This approach is conceptually appealing to me, but that is because I am predominantly a computer scientists. Although P-system can be used for modeling biological systems (Ardelean & Cavaliere, 2003), they still feel fundamentally computer-science-y and are typically not published in orthodox biological venues. This makes me suspect there is a more standard approach among biologists, probably via dynamic systems and diffusion equations.

• It may not be what you are looking for, but there is a quorum sensing model available here at the EBI repository. Sep 10, 2012 at 18:04

I found this paper[1], which might be relevant; it uses is a more chemically inspired approach. Another paper[2] also might be interesting, it takes a more dynamical systems approach.

My personal instinct would be to use ordinary differential equations: generate a population of cells in random positions, assign each cell levels of any relevant molecular species, and come up with a list of ODEs that describe the rates of change of the species in a particular cell in terms of gene expression, degradation, and diffusion. You should then be able to simulate the time evolution of the system just by solving the (large) system of ODEs given your initial conditions. Something like the approach in this paper[3] (read the supplemental info), only your cells would move around during the simulation, rather than remaining still as part of a tissue. The implementation would get thorny, because you would have to accommodate the motion of the cells as you solve the ODEs. I imagine that this could be accomplished by simply treating the initial positions of the cells as part of the system's initial conditions and then defining the ODEs that govern the rate of change in each dimension (either randomly, or perhaps they move in a coordinated manner) so that this aspect of the system can evolve during the integration.

References:

Since the accepted answer is nearly a decade old, I'm adding a new answer to address a recent piece of research that presents a good framework for modeling quorum sensing over small distances.1

First, the authors consider two types of signal diffusion systems.

1. Absorbing -- the signal molecule is imported into the recipient cell, where it interacts with its receptor to induce a response. This process is irreversible (the signaling molecule is "used up" once imported). Many peptide-based quorum sensing systems found in Gram-positive bacteria are absorbing systems.2
2. Non-absorbing -- the signal molecule interacts with a receptor on the surface of a recipient cell, inducing some response cascade. The signal then dissociates and diffuses into the intercellular space, where it can interact with other recipient cells. Examples of non-absorbing systems include autoinducer networks in Vibrio species 3 and acyl-homoserine lactone quorum sensing systems in Proteobacteria.4

Next, they made three modeling predictions, and tested each prediction in synthetic communities grown in microfluidic chambers. Their model organism, B. subtilis, was transformed with signal-recipient gene pairs from both absorbing and non-absorbing systems, though the bulk of their paper is spent discussing the less well-studied absorbing modality.

1. Signal response in signal-receiving cells ($$Y$$) is inversely proportional to their distance from the signal source boundary ($$x$$) and is a function of the communication range ($$\lambda$$). For a absorbing systems, the authors define $$\lambda$$ as "...the distance from the boundary of signal producers over which the signal response reduces one order of magnitude ... on a natural log scale". Unsurprisingly, they predict absorbing systems to have much shorter signaling ranges than non-absorbing systems, assuming a high signal uptake rate among recipient cells.

$$Y(x) \propto e^{-x/\lambda}$$

1. In absorbing systems, communication range ($$\lambda$$) is a function of signal diffusion rate ($$D$$) and signal uptake rate ($$\alpha$$). Using the Stokes-Einstein equation 5 and an estimated radius of 0.5 nm for small peptides, the authors estimate that signaling molecules diffuse through water at a rate of ~ 400 μm2s-1. $$\lambda ≅ \sqrt{\frac{D}{\alpha}}$$

2. In both absorbing and non-absorbing systems, the signal concentration (and thus the signal response $$Y$$) is related to the width of the cluster of signal-producing cells ($$W$$). Simply, more cells produce more molecules. For non-absorbing systems, the signal concentration is unbounded. However, the concentration of signaling molecules in absorbing systems saturates when the cluster radius is larger than $$\lambda$$.

$$Y \propto W^2$$

The authors find experimental justification for each of their model predictions (I suspect the predictions wouldn't have been included in the paper, otherwise). For the nitty-gritty math behind their models and other assumptions / simplifications, read Section 2 from the supplementary discussion, Mathematical modeling of quorum-sensing designs. An excerpt:

While bacterial communities consist of cells, we use a continuous framework to model the community. That is, instead of modeling cells that produce and receive signal molecules explicitly, we simplify the model by assuming a constant production rate per unit volume and a decay rate per unit volume of the community.

References

1. van Gestel J, Bareia T, Tenennbaum B, Dal Co A, Guler P, Aframian N, Puyesky S, Grinberg I, D'Souza GG, Erez Z, Ackermann M, Eldar A. Short-range quorum sensing controls horizontal gene transfer at micron scale in bacterial communities. Nat Commun. 2021 Apr 19;12(1):2324.
2. Neiditch MB, Capodagli GC, Prehna G, Federle MJ. Genetic and Structural Analyses of RRNPP Intercellular Peptide Signaling of Gram-Positive Bacteria. Annu Rev Genet. 2017 Nov 27;51:311-333.
3. Papenfort K, Bassler BL. Quorum sensing signal-response systems in Gram-negative bacteria. Nat Rev Microbiol. 2016 Aug 11;14(9):576-88.
4. Schuster M, Sexton DJ, Diggle SP, Greenberg EP. Acyl-homoserine lactone quorum sensing: from evolution to application. Annu Rev Microbiol. 2013;67:43-63.
5. Miller CC. The Stokes-Einstein law for diffusion in solution. Proc Roy Soc. 1921;106:724.