Ex­po­nen­tial Growth, Ephemeral & Un­sus­tain­able

by Roberto

A few months ago, an idea dawned on me: I should write a se­ries of posts on pop­u­la­tion dy­nam­ics. By no means a for­mal quan­ti­ta­tive treat­ment of the sub­ject. In its stead, a sub­jec­tive in­ter­pre­ta­tion of a topic that has cap­tured my at­ten­tion for nearly half a cen­tury. Growth, no growth, col­lapse. Re­peat. It's the his­tory of mi­cro­bial cul­tures. It's also Earth's nat­ural his­tory. Wor­thy of look­ing at. Since I must start some­where, let me start with ex­po­nen­tial growth.

In last Monday's post I wrote what I'll call the pre­am­ble of the se­ries. If you've not read it yet, you might want to read it now. Stop, click on this link and I'll see you in a bit. But, if right now is not the right mo­ment, if you're feel­ing rushed, if events around you are in­ces­santly ac­cel­er­at­ing, leav­ing with way too much to do and so lit­tle time, jump a para­graph. Or. Per­haps. Those are rea­sons that should make this the per­fect mo­ment for you to... Stop, click on this link and I'll see you in a bit.

Hav­ing just fin­ished read­ing Arthur Helman's mag­nif­i­cent book The Cave and the Light: Plato Ver­sus Aris­to­tle and the Strug­gles for the Soul of West­ern Civ­i­liza­tion, it was no sur­prise that the set­ting of an ex­plorer in a cave emerged spon­ta­neously in my mind. I twisted Plato's al­le­gory to drive the point home that the pace of ex­po­nen­tial growth can eas­ily catch you by sur­prise. At first growth seems un­de­tectable, it's easy to get lost in a dream world. Then, be­fore you know it, it's too late. It's over. The con­se­quences could be dire. Your cul­tures could have reached sta­tion­ary phase. Or worse. Much worse.

Earth is said to har­bor 10³⁰ or­gan­isms, the over­whelm­ing ma­jor­ity mi­crobes. Most of these liv­ing in caves of their own in the deep sub­sur­face. Us mi­cro­bi­ol­o­gists love this num­ber even though we can­not even be­gin to phan­tom this "su­per-gar­gan­tuan" scale. As I re­lated be­fore, we now have an even larger quan­tity to be amazed by: the to­tal num­ber of or­gan­isms that have ever ex­isted. Dur­ing the four bil­lion years of life on Earth, 10⁴⁰ or­gan­isms have lived out their lives. An unimag­in­ably large num­ber.

Let's play with those two num­bers to get a bet­ter sense of ex­po­nen­tial growth. I'll lead you by the hand through a Gedanken Ex­per­i­ment (be­cause, in fact, there's no way the phys­i­cal ex­per­i­ment can be done). We'll start with a sin­gle cell of our hum­ble friend, the bac­terium E. coli, weigh­ing in at 10^-12 grams. We'll grow it un­der con­di­tions where it dou­bles every twenty min­utes. Im­por­tantly, we will pro­vide it with un­lim­ited space and re­sources such that it can grow ex­po­nen­tially for­ever. We'll now quan­ti­tate the pop­u­la­tion size af­ter 10, 100 and 132 dou­blings. That's easy: 2¹⁰, 2¹⁰⁰ and 2¹³². We're not all that fa­mil­iar with base 2 ex­po­nents so I'll con­vert them to base 10: 10³, 10³⁰, 10⁴⁰. Re­call the dou­bling time is twenty min­utes. You do the math: when grow­ing ex­po­nen­tially, it took E. coli two days to reach 10⁴⁰ cells, that's the to­tal num­ber of or­gan­isms that have ever lived! Whether you are a sea­soned mi­cro­bi­ol­o­gist (and you've known this since for­ever) or are a new­bie to bac­te­r­ial growth, this fact will never cease to amaze you. Now cal­cu­late the weight of those 10⁴⁰ E. coli. Start­ing out at a wimpy 10^-12 grams, af­ter two days of ex­po­nen­tial growth those bac­te­ria weigh 10²⁸ grams. Is that a lot or not so much? What do you know that weighs 10²⁸ grams? Right. Planet Earth! Af­ter a mere two days of ex­po­nen­tial growth, the bac­te­ria weigh as much as Earth! I call that "the awe­some power of the ex­po­nen­tial." I've taken you through this lu­di­crously ex­treme ex­am­ple of ex­po­nen­tial growth to make the case that con­tin­u­ous, un­re­stricted ex­po­nen­tial growth is both ephemeral and un­sus­tain­able. Ephemeral and un­sus­tain­able, com­mit that to mem­ory.

Now I'll pose a re­lated ques­tion. When con­di­tions are pro­pi­tious for growth and there are nu­tri­ents avail­able, mi­cro­bial pop­u­la­tions grow. But, since ob­vi­ously they can­not grow for­ever, which processes limit their num­bers? Of course, that de­pends.

If nu­tri­ents re­main avail­able there might be a dy­namic equi­lib­rium be­tween cell births and deaths. It's un­likely that this bal­ance will be achieved by nu­tri­ent fluxes alone. No, the pri­mary process dri­ving death rates when nu­tri­ents are avail­able is likely pre­da­tion; think preda­tor-prey Lotka-Volterra dy­nam­ics. These days phages are very much in fash­ion, so many read­ers will pro­pose them as the main bac­te­r­ial preda­tors. My fa­vorites for top preda­tors are not phages but pro­tists, al­though I ad­mit that's just a hunch.

Fig. 1. E. coli sur­vival dur­ing pro­longed in­cu­ba­tion. Source: Roberto Kolter

But what hap­pens if there are no preda­tors around, what lim­its pop­u­la­tion num­bers then? What causes the end of ex­po­nen­tial growth? Nu­tri­ent de­ple­tion will cer­tainly re­sult in the ces­sa­tion of growth. But poi­son­ing of the en­vi­ron­ment will also ar­rest growth. Where might we ob­serve this? Here's one ob­vi­ous an­swer: in the very ar­ti­fi­cial con­di­tions of grow­ing bac­te­ria in the lab­o­ra­tory as pure cul­tures! Which brings me to the topic of bac­te­r­ial sur­vival in the ab­sence of growth. While I could take this nar­ra­tive in many di­rec­tions, I want to fo­cus on one ob­ser­va­tion we made some forty years ago. E. coli in­cu­bated in a low con­cen­tra­tion of a com­plex medium (e.g. 0.1X LB) will grow ex­po­nen­tially for a few hours and then stop at just un­der 10⁹ cells/ml. In­ter­est­ingly, this num­ber re­mains al­most con­stant for the next ten days. In con­trast, when the same bac­terium is grown in a much higher con­cen­tra­tion of the same com­plex medium (e.g. 1X LB), the pop­u­la­tion sat­u­rates at about 10¹⁰ cells/ml. But af­ter two days most of the pop­u­la­tion dies, sta­bi­liz­ing at about 10⁹ cells/ml. By us­ing more of the very same re­sources the bac­te­ria grow more. But in the process, they poi­son their en­vi­ron­ment, ul­ti­mately lead­ing to the demise of most of the pop­u­la­tion. With­out get­ting too far lost on the well-known lim­i­ta­tions of LB medium, sim­ply let those num­bers sink in. With­out a preda­tor to con­trol their num­bers, the fate of these pop­u­la­tions is de­fined by the amounts of re­sources they use and the ef­fect that their meta­bolic by-prod­ucts have on their en­vi­ron­ment.

Fig. 2. Hu­man pop­u­la­tion from 10,000 BCE to the present. Adapted from source.

Now con­sider the pop­u­la­tion dy­nam­ics of hu­mans, as plot­ted on the right. For mil­len­nia the pop­u­la­tion re­mained rea­son­ably con­stant and very low rel­a­tive to the present day. How­ever, for the last cou­ple hun­dred years, the pop­u­la­tion grew dra­mat­i­cally. Our num­bers were (more or less) 1 x 10⁹ in 1800, 2 x 10⁹ in 1928, 4 x 10⁹ in 1974 and 8 x 10⁹ in 2023. Looks like we are quickly ap­proach­ing 10¹⁰ hu­mans on Earth. Hu­mans, like our E. coli in pure cul­ture, have not had a very ef­fec­tive preda­tor for a long time. Cer­tainly not li­ons or tigers. Even mi­cro­bial pathogens, which could in prin­ci­ple dec­i­mate us, have failed to keep the pop­u­la­tion at bay. Hu­man in­tel­li­gence, as man­i­fested in as­ton­ish­ing in­no­va­tions, now keep the hu­man pop­u­la­tion rel­a­tively safe from pathogens. Wit­ness the dra­mat­i­cally dif­fer­ent preda­tor ef­fec­tive­ness of Yersinia pestis dur­ing the Me­dieval Black Death com­pared that of SARS-CoV‑2 dur­ing the COVID-19 pan­demic just a few years ago.

A burn­ing (pun in­tended) ques­tion is: what will hap­pen to the hu­man pop­u­la­tion over the next cen­tury or two, or more? Will it sta­bi­lize? Will it so poi­son its en­vi­ron­ment lead­ing to its own demise? I've been con­tem­plat­ing pos­si­ble long-term out­comes, based not only on what de­mog­ra­phers pre­dict from their mod­els, but also con­sid­er­ing that some as­pects of pop­u­la­tion dy­nam­ics might ap­ply across vastly dif­fer­ent scales, from bac­te­ria in cul­ture to hu­mans. I know that the 10¹⁰ bac­te­ria in 1 ml and 10¹⁰ hu­mans in 1 Earth is a sheer co­in­ci­dence. Yet, the ra­tio of the size of a sin­gle bac­terium rel­a­tive to 1 ml is not too far from the ra­tio of the size of a sin­gle hu­man rel­a­tive to the bios­phere. Tan­ta­liz­ing thoughts emerge.

Do you get the sense that my fas­ci­na­tion with the ces­sa­tion of growth is evolv­ing into an ob­ses­sion? At times I've been told I talk of lit­tle else. The hu­man pop­u­la­tion dy­nam­ics and the un­cer­tainty of the fu­ture have given me lots to think about. In the next few posts, I'll share with you the path I fol­lowed the last two years in my ef­forts to come to terms with these times.

Take one more look at the growth of the hu­man pop­u­la­tion graph. Is that ex­po­nen­tial growth? It cer­tainly looks like it. Note, how­ever, that both axes of the graph are on a lin­ear scale. This gives a great vi­sual ef­fect; the pop­u­la­tion is "hit­ting a wall." But, I'll re­mind you of Elio's ad­vice: growth data should be plot­ted in semi-log pa­per. I'll go there, next week.

 

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Com­ments Ar­riv­ing

From Fer­nando Ba­quero: Grow­ing is a very equiv­o­cal word. Et­y­mo­log­i­cally, it means be­com­ing larger. This can re­fer to an in­di­vid­ual, such as a kid grow­ing into a man over time, or the ex­pan­sion of a piece of metal un­der warm con­di­tions, the fer­men­ta­tion of bread dough, or the in­fla­tion of a bal­loon. Grow­ing can also re­fer to col­lec­tive en­ti­ties, like the hu­man pop­u­la­tion of a town or bac­te­ria in a cul­ture tube. It also ap­plies to in­creas­ing knowl­edge about a spe­cific topic or the rise of a stock in a bull mar­ket. But the es­sen­tial bi­o­log­i­cal, and at large, evo­lu­tion­ary mean­ing of grow­ing is in fact di­vid­ing, branch­ing.

A tree grows by branch­ing, and the size of the tree de­pends on the num­ber of di­vi­sions and ram­i­fi­ca­tions. How­ever, the size and life of a tree are ephemeral. Leaf length and the an­gle be­tween the long axis of the leaf and the sup­port­ing stem seg­ment both de­crease with height (REF 1). In fact, every­thing is a ques­tion of the num­ber of pos­si­ble di­vi­sions in a lim­ited pe­riod of time. We can con­ceive that time is not only a di­men­sion, but also an es­sen­tial nu­tri­ent of life, which is un­sus­tain­able when the time ac­ces­si­ble to any or­gan­ism is ex­hausted (REF2).

From Roberto: Many thanks Fer­nando, for this fas­ci­nat­ing and thought-pro­vok­ing in­sight into growth, branch­ing and time. I hope our read­ers will read your ref­er­ence #2: "Evo­lu­tion and the Na­ture of Time." For the ex­plorer in the cave, that es­sen­tial nu­tri­ent, time, was ex­hausted. As for hu­man­ity, I like the rel­e­vant phrase, "time is of the essence." I at­tempt to ex­plain why I re­main op­ti­mistic about the fu­ture in the next cou­ple of posts. Will be de­lighted to hear your opin­ion.