Our minds love linear but we live in a nonlinear world

5 min readNov 24, 2022

**Escherichia coli** cells. Source: https://www.cdc.gov/ecoli/general/index.html

Why are we humans oft times focused on how bacteria, and other microorganisms reproduce? We can omit viruses because biologically they are not free-living microbes — — they must inhabit a host cell from which all of their reproductive capacity is obtained, i.e. they might be thought of as parasites, they take all but nothing is given back except death to the host cell.

Most commonly reported outbreaks caused by bacteria are toxic strains of Escherichia coli and Salmonella spp.. We know that not all E. coli cause severe illness in humans and other mammals because they are part of our inborn, in situ microflora. However, E.coli are one of many bacteria that put humans and other species at risk. The STEC0157 strain is notorious. In the United States, “According to the Centers for Disease Control, about 265,000 STEC infections occur in the United States each year. The STEC O157 strain causes about 36% of these infections and non-O157 STEC strains cause the rest.” .

For a list of some of the more commonly harmful bacteria, visit this site: https://www.canada.ca/en/public-health/services/antibiotic-antimicrobial-resistance/antibiotic-resistant-illnesses-bacteria.html

Bacterial Infections can be highly contagious and rapidly transmitted within and among groups. Listeria, another bacterium, made headlines as I am writing this post.

Events such as these are commonly reported: Recent 2022 outbreaks

Outbreak Investigation of Salmonella: Seafood (October 2022): https://www.fda.gov/food/outbreaks-foodborne-illness/outbreak-investigation-salmonella-seafood-october-2022#:~:text=The%20FDA%2C%20along%20with%20CDC,Arizona%20by%20Mariscos%20Bahia%2C%20Inc

Food safety: https://www.foodsafetynews.com/tag/2022-outbreaks/

Listeria: https://www.livescience.com/listeria-outbreak-meat-cheese?utm_campaign=368B3745-DDE0-4A69-A2E8-62503D85375D ; “Contaminated meat and cheese from deli counters have sickened 16 people in six states.”

What does a simple growth curve look like?

Figure 1 demonstrates a hypothetical example showing what we would hope to see, simple, linear and manageable. Let’s create a story to fit: We begin with a culture of cells and using a well calibrated cell-sorter so that are2 cells transferred to a broth that inhibits growth. We repeat this procedure another 20 equally spaced times. After plotting the data in Figure 1 we see a simple linear growth curve.

After 21 cycles the total predicted cells are n=42, i.e. 2 cells are added each cycle (generation) and the equation can be written: Yn=2x. Obviously this is not a realistic, biologically based growth model. Organisms reproduce and their population sizes. That is fundamental to their evolution.

**Figure 1. An example of what the simple linear growth of a “bacterium” might look like in a broth inhibiting reproduction. Graph by Zack Florence.**

The Exponential Growth Model

Compare the hypothetical growth profile in Figure 1 to that in Figure 2. In 21 generations we see that the growth of one E. coli cell has divided by fission and accelerated exponentially to produce a population in 21 generations that may exceed 2 million cells!

Quoting: “ When conditions are favourable such as the right temperature and nutrients are available, some bacteria like Escherichia coli can divide every 20 minutes. This means that in just seven hours one bacterium can generate 2,097,152 bacteria. After one more hour the number of bacteria will have risen to a colossal 16,777,216. That’s why we can quickly become ill when pathogenic microbes invade our bodies.”

Figure 2 shows how the growth by a bacterium like E. coli may look when an organism divides at such a rate. The population doubles approximately every 20 minutes, i.e. 21 generations in 7 hours. Compare that with a human generation which amounts to 20–30 years each.

Exponential growth without limit: Yt = No(2^t)

where Yt is the number of cells at time t,
No is the initial population at t=0 ,
2^t is the rate of reproduction (doubling per generation) and time where 1 generation=20 minutes.

You might immediately think — — this is not feasible, no organism can reproduce so rapidly and increase to infinity: if so, you would be correct.

**Figure 2. This analysis was motivated by a paragraph within the following link:** https://microbiologysociety.org/why-microbiology-matters/what-is-microbiology/bacteria.html . Graph by Zack Florence.

There are Limits to Growth: the logistic growth model

In Figure 2 growth appears to be infinite which makes no ecological sense. There are many factors that can limit the growth of a population, whether E.coli or humans. The question then arises what must be the limit? The logistic growth model is commonly applied to determine the limit of growth over time, space and at an intrinsic rate.

Figure 3 displays the logistic growth of bacteria with an upper limit. The “S” shaped function is sigmoidal and there is an initial slower rate of increase (lag) which transitions to the exponential phase, then an inflection point where the exponential growth transitions to a slower growth rate (negative) and appears asymptotic at the limit: if the numbers in the population increase above the limit, then negative feedback from lack of energy within the system should bring the population back to its limit. Sustained growth is a function of population size and available energy. Without supplements to resources at some point the population crashes.

**Figure 3. Data from** **Bittinger, Brand and Quinanilla (2006)**. Graphic by Zack Florence.

In Summary

The fundamental difference in the previous examples is this, assuming x=time:

The predictor (time), x axis, is linear in this equation: Y=2x.
However, in the nonlinear exponential model, x is nonlinear: Y= 2^x

The evolution of population growth

Figure 4 summarizes the four phases of the evolution of a population. Again, we can extrapolate to many systems in biology, and some scientists would include the evolution of our Universe.