# Is there a favoured mathematical model of bacterial growth?

One of the most regularly measured phenotypes in bacterial synthetic biology is growth. Anyone who does wet lab microbiology or synthetic biology will have performed a growth curve measurement. However, it is quite rare in synthetic biology papers for growth to be characterised using a mathematical model - even though growth curves are regularly shown in supplementary materials.

There are a plethora of growth models around - from basic logistic growth to the more biologically grounded Monod growth. I recently discovered the Richard's model which seems to be a more general form encompassing logistic, Gompertz etc. Is there a favoured model for the microbiology and/or synthetic biology community? Is there something holding the synthetic biology community back from regularly reporting growth characteristics of their cells?

• A few rather general remarks: Models are usually designed to describe specific situations using the minimal number of parameters. So the models that you listed are not necessarily mutually exclusive. One could design a more general model, but it would necessarily have more parameters and less statistical power. From statistics point of view it is also necessary to keep in mind that all these are parametric models, which can only approximate the growth curve. Mar 15, 2021 at 13:50

## 2 Answers

Some clarifications: Commonly used models for bacterial isothermal growth curves (which represent the growth curve directly) are for example:

The Monod equation expresses how the growth rate changes with the availability of substrate see a related question here.
The Monod equation can be incorporated into an exponential growth model (differential equation) like this: $$\frac{dX}{dt}=\mu X$$ with $$\mu = \mu_{max} \frac{[S]}{K_M+[S]}$$ where $$X$$ is the biomass concentration and $$[S]$$ is the substrate concentration. Or into a logistic growth model: $$\frac{dX}{dt}=\mu X \left(1 - \frac{X}{X_{max}} \right)$$ with $$\mu = \mu_{max} \frac{[S]}{K_M+[S]}$$ where $$X_{max}$$ is the carrying capacity. Other growth-substrate relations like Haldane which takes into account growth inhibition by the substrate can be used instead of Monod one.

Regarding this:

Is there something holding the synthetic biology community back from regularly reporting growth characteristics of their cells?

It depends on what branch of synthetic biology you are looking into. Model-based design engineering biology interpretation of synthetic biology goes around the DBTL cycle. In the DBTL cycle, there are always (at least) models of both growth and gene expression.

the more biologically grounded Monod growth

Why is the Monod model more biologically grounded? It is empiric and NOT based on biological considerations! I only know this equation from enzyme kinetics, not for cell-growth, where the identical Michaelis–Menten equation assumes a random collusion model. So according to this model, your cells would grow more, the more substrate (glucose?) collides against the walls!

It seems quite logical to model enzyme-kinetics or chemical reactions with random collusion models. However, the only relevant biological facts I see are the following:

1. Cells divide, which naturally suggest exponential growth.
2. Cell-cycles and circadian cycles exist, which naturally suggests a constant generation time. (within the limits that everyone knows)

So, if everyone learns about growth curves and logistic fits in their wet-lab courses, where logistic fits work perfectly fine to predict growth within the log-phase, then why would there be a need to use other - more complicated - models, while the logistic model is conceptually indicated? (Also note that the Monod model still fails to predict the lag phase and death phase.)

This leads to the final answer to the question:

Is there something holding the synthetic biology community back from regularly reporting growth characteristics of their cells?

Practical use!

1. In daily routine, the typical wet-lab scientist is merely interested in knowing the expected time until a certain optical density (OD) of the cell-culture is reached. Practically, this is done with as few measurements as possible, based on experience and a guess of a generic generation-time.
2. Exact predictions are needed in large-scale biotechnological production. In such a context all parameters must be accurately predicted and validated. Given your exact microorganisms and your production plan, small-batch trails will be performed and up-scaled. Everything must be known and planned in detail and quite often, the whole production facility is designed and built to achieve a single task! Everything, including the generated heat, O2, CO2, Glucose, (end-) concentration of the yield.. is monitored and legally binding deviation-limits must be set before-hand and met at all times. Small deviations can cause losses of millions of dollars and companies don't rely on simple models like "Monod", but stress-test every aspect and monitor and react in automated systems. There is vast knowledge regarding growth-curves and models, just you are not aware of it, as it is kept company-internal, or written in expensive bioprocess-technology books.
• Yes, that is true! Good addition! Mar 16, 2021 at 12:37
• I've been appreciating your answers, and I think you'd be a good contributor to the more interdisciplinary SynBio StackExchange we're trying to launch. Want to support the proposal? area51.stackexchange.com/proposals/125068/… Mar 30, 2021 at 11:27