Coordination Problems in Evolution: Eigen’s Paradox

Introduction

Lately I’ve writ­ten cou­ple of posts that dis­cuss co­or­di­na­tion prob­lems. Not the ideal­ized, game-the­o­ret­i­cal stuff but rather the real, messy co­or­di­na­tion prob­lems en­coun­tered by real peo­ple in the real world. Here, I will ex­plore very differ­ent ter­ri­tory. I will look at co­or­di­na­tion prob­lems be­tween be­tween molecules, chro­mo­somes, cells and in­di­vi­d­u­als as they oc­curred and as they were solved in the course of biolog­i­cal evolu­tion.

This ar­ti­cle is based on the book “The Ma­jor Tran­si­tions in Evolu­tion” by John May­nard Smith and Eörs Sza­th­máry.

Be­fore pro­ceed­ing I would like to say few words about why I chose that par­tic­u­lar book, al­though it was pub­lished in 1995 and thus misses a lot of re­cent re­search.

First, it was writ­ten by widely rec­og­nized ex­perts in the field. That may not have been that im­por­tant if I was writ­ing about a differ­ent topic, but evolu­tion­ary biol­ogy is no­to­ri­ously tricky, sub­tle and prone to mi­s­un­der­stand­ing. Some­times it gen­er­ates crack­pot ideas, which, nonethe­less, some­times turn out to be true. A lay­man, or even a pop­u­lar sci­ence writer, is likely to get lost.

John May­nard Smith is one of the big names of evolu­tion­ary biol­ogy of 20th cen­tury. He owe to him the in­tro­duc­tion of game the­ory into evolu­tion­ary biol­ogy. He’s the au­thor of the cen­tral idea in the field, so called evolu­tion­ar­ily sta­ble strat­egy, which is, to put it shortly, an ap­pli­ca­tion of the con­cept of Nash equil­ibria to biolog­i­cal, evolv­ing sys­tems.

Eörs Sza­th­máry is less known, but he did a lot of work on the topic of ori­gin of life.

Se­cond, the book is con­cerned with the big changes in the evolu­tion­ary his­tory. It doesn’t spend much time on evolu­tion-as-usual, on how a spe­cific bone or or­gan evolved. Rather, it dis­cusses the events which sig­nifi­cantly changed the na­ture of evolu­tion it­self: How did the life be­gan? How we’ve got the first self-repli­cat­ing molecules? How did the cell origi­nated? How did the mul­ti­cel­lu­lar or­ganisms?

One would ex­pect a book on such a grand sub­ject to be a least a bit hand-wavy. Sur­pris­ingly though, it’s not. In­stead, the au­thors dive deep into the de­tails of each in­di­vi­d­ual topic, they dis­cuss chem­i­cal de­tails of the re­ac­tions in ques­tions, their yield and speed, how would they sur­vive in the com­pe­ti­tion of other re­ac­tions go­ing on nearby and so on. They dis­cuss the game-the­o­retic con­sid­er­a­tions of form­ing an eu­kary­otic cell or an in­sect so­ciety. They de­scribe the minu­tiae of in­trage­nomic con­flict and the in­ter­play be­tween the de­vel­op­ment and evolu­tion.

Third, in the in­tro­duc­tion Smith and Sza­th­máry note that many (but not all) of the tran­si­tions they are go­ing to dis­cuss are, ac­tu­ally, solu­tions to co­or­di­na­tion prob­lems. They don’t use that ex­act term, but it’s pretty clear what they mean:

One fea­ture is com­mon to many of the tran­si­tions: en­tities that were ca­pa­ble of in­de­pen­dent repli­ca­tion be­fore the tran­si­tion can repli­cate only as part of a larger whole af­ter it. … Given this com­mon fea­ture of the ma­jor tran­si­tions, there is a com­mon ques­tion we can ask of them. Why did not nat­u­ral se­lec­tion, act­ing on en­tities at the lower level (repli­cat­ing molecules, free-liv­ing prokary­otes, asex­ual pro­tists, sin­gle cells, in­di­vi­d­ual or­ganisms), dis­rupt in­te­gra­tion on the higher level (chro­mo­somes, eu­kary­otic cells, sex­ual species, mul­ti­cel­lu­lar or­ganisms, so­cieties)?

In fact, think­ing about co­or­di­na­tion prob­lems was what made them write the book in the first place:

One of the stim­uli for at­tempt­ing the work was our re­al­iza­tion that a model one of us had de­vel­oped to an­a­lyze the ori­gin of com­part­ments con­tain­ing pop­u­la­tions of molecules was for­mally and math­e­mat­i­cally similar to a model that the other had de­vel­oped to an­a­lyze the evolu­tion of co­op­er­a­tive be­havi­our in higher an­i­mals.

Fourth, the book strikes a good bal­ance be­tween tar­get­ing gen­eral pub­lic and tar­get­ing the ex­perts only. It re­quires you to know your high school molec­u­lar and evolu­tion­ary biol­ogy, but not much more than that. You should be vaguely fa­mil­iar with the con­cept of cit­ric cy­cle, but no­body ex­pects you to know what 1,3-biphos­pho­glyc­er­ate is. And once you know the ba­sics, the book is sur­pris­ingly ac­cessible and not hard to un­der­stand. (By the way, I see there’s a pop ver­sion of the book pub­lished by the au­thors them­selves. I haven’t read it my­self but it may be worth check­ing out.)

To sum it up, the book may be old, but it dis­cusses ex­actly the topic I am in­ter­ested in and it does so with great ex­per­tise and thought­ful­ness. I don’t think there’s a newer book that does such a good job in this area.

And af­ter all, my goal is not to sum­ma­rize the cut­ting-edge biolog­i­cal re­search but to learn a les­son about the most gen­eral pat­terns of solv­ing co­or­di­na­tion prob­lems. And those, I be­lieve, haven’t changed much in the past twenty years.

Ei­gen’s paradox

How did the first self-repli­cat­ing molecules origi­nate?

We know that with RNA and some similar molecules this pro­cess hap­pens au­to­mat­i­cally: If there are ba­sic blocks available in the en­vi­ron­ment, they will, thanks to their chem­i­cal prop­er­ties, au­to­mat­i­cally at­tach them­selves into ap­pro­pri­ate places of an ex­ist­ing sin­gle-strand RNA and form a dou­ble stranded RNA. For repli­ca­tion to pro­ceed, the two strands then have to be sep­a­rated. It have been pro­posed that this may have hap­pened in the vicinity of hy­drother­mal vents, where the molecule would ex­pe­rience both cool tem­per­a­tures, con­duc­tive to at­tach­ment of nu­cleotides to the RNA and sud­den hot tem­per­a­tures which would sep­a­rate the two strands.

Now, putting aside the speci­fics of RNA repli­ca­tion, how likely it is that such a self-repli­cat­ing molecule will sur­vive in the chaotic world?

Man­fred Ei­gen ob­serves that it de­pends on two fac­tors: On the speed of repli­ca­tion and on its fidelity. Speed al­lows molecule to be repli­cated faster than it de­cays. Fidelity en­sures that the re­sult of repli­ca­tion is, in fact, the molecule we care about and not some­thing else.

As for the fidelity fac­tor, it de­pends on the RNA size. If chance of cor­rect repli­ca­tion of a sin­gle base is, for ex­am­ple, 12, then chance of cor­rect repli­ca­tion of 2-base RNA is 14, 3-base RNA 18 and so on.

Mea­sure­ments show that the thresh­old for the RNA size is some­where around 100 bases. If the molecule is larger than that it wouldn’t be able to sus­tain it­self. It would de­volve into a mix of its bro­ken copies. (For the ex­act maths check the Wikipe­dia ar­ti­cle.)

Fol­low­ing graph shows sus­tain­able RNA size based on the mu­ta­tion rate (1 - repli­ca­tion fidelity):

eigen.png

And here we en­counter the para­dox. If we wanted longer, self-sus­tain­able RNA molecules we would need bet­ter fidelity. But bet­ter fidelity can only be achieved with the help of spe­cial­ized en­zymes. But the small­est genome able to code for such an en­zyme, and for the nec­es­sary trans­lat­ing ma­chin­ery, would re­quire a num­ber of bases greatly ex­ceed­ing 100 nu­cleotides. It’s a catch-22 situ­a­tion.

Ei­gen spec­u­lates that the small molecules would have to some­how co­op­er­ate (Lo, a co­or­di­na­tion prob­lem!) to cre­ate a sys­tem ca­pa­ble of hold­ing enough in­for­ma­tion to cre­ate more com­plex stuff. He pro­poses the “hy­per­cy­cle” model.

Hyper­cy­cle is a set of RNA molecules that cat­alyze each other’s repli­ca­tion. For ex­am­ple, if molecule A cataly­ses repli­ca­tion of molecule B, molecule B cataly­ses repli­ca­tion of molecule C and molecule C cataly­ses repli­ca­tion of molecule A, it’s a hy­per­cy­cle.

The point is that the cat­a­lysts can be spe­cific. A should cat­alyze B and that’s it. No need for com­plex ma­chin­ery able to repli­cate any RNA se­quence.

But be­fore div­ing into the de­tails: Could hy­per­cy­cles even be es­tab­lished? Given the ex­is­tence of small self-repli­cat­ing RNA molecules, wouldn’t the best repli­ca­tor just crowd ev­ery­one else out and cre­ate a mono­cul­ture with no chance of form­ing a hy­per­cy­cle?

In­ter­est­ingly, no. And the rea­son is sur­pris­ing. As RNA molecules repli­cate they have a ten­dency to pair with their own coun­ter­parts. (If fact, they bind so well that the prob­lem is rather how the in­di­vi­d­ual strands get sep­a­rated af­ter the repli­ca­tion.) And the bound dou­ble-stranded RNA can­not perform its cat­alytic func­tions. It in­hibits its own cre­ation.

In other words, the pop­u­la­tion of par­tic­u­lar kind of molecule grows more slowly the more of its own kind is around. Its num­bers do not dou­ble with each gen­er­a­tion as one would naively ex­pect. Yet, other kinds of molecules are not in­hibited and can mul­ti­ply at their own pace.

It can be shown that while in the world on ex­po­nen­tially repli­cat­ing molecules the win­ner does take all, with sub-ex­po­nen­tial growth, as de­scribed above, an equil­ibrium will form con­tain­ing many differ­ent types of molecules.

Once we have the hy­per­cy­cle in place it seems to work fine. Namely, no­tice how it is self-reg­u­lat­ing in a way. If one link in the cy­cle is more effi­cient than other links it will soon run out of its cat­a­lyst and would have to wait for the rest of the cy­cle to catch up. Thus, a sin­gle com­po­nent of the hy­per­cy­cle can­not out­com­pete the rest of the cy­cle.

That be­ing said, there’s an ob­vi­ous prob­lem when we take mu­ta­tions into ac­count. If there is a mu­ta­tion that makes one RNA molecule a less effi­cient cat­a­lyst for the next step of the cy­cle it could still re­pro­duce at the same speed as the origi­nal molecule. That would mean lower con­cen­tra­tion of the well-be­haved molecule which would in turn suck the mo­men­tum out of the cy­cle. Many such free-rid­ers and the con­cen­tra­tion of co­op­er­at­ing RNAs would de­crease to the level where the hy­per­cy­cle would stop work­ing at all.

So what can be done about the par­a­site molecules? An ob­vi­ous solu­tion would be to en­close the repli­ca­tors in some kind of mem­brane. If the molecules in the com­part­ment could repli­cate only to­gether or not at all, the com­part­ment con­tain­ing par­a­sites would sim­ply “die” i.e. fail to repli­cate and be even­tu­ally out­com­peted by the “healthy” com­part­ments. (Emer­gence of mem­branes and the mechanism of the com­part­ment fis­sion is cov­ered in the book but doesn’t have much to do with the co­or­di­na­tion prob­lems, so I am go­ing to hes­i­tantly skip over it.) Alter­na­tively, the evolu­tion of the early life may have hap­pened on a sur­face of a rock, thus limit­ing ways in which molecules can in­ter­act — this is so called “pri­mor­dial pizza” model. Such an­chor­ing of molecules to a flat sur­face may have had similar effect as en­clos­ing them in­side of a mem­brane.

Sza­th­máry and Deme­ter pro­pose an al­ter­na­tive model called “stochas­tic cor­rec­tor”. The idea can be ex­em­plified as fol­lows.

Imag­ine a pop­u­la­tion of molecules con­sist­ing of “al­tru­is­tic” molecules which cat­alyze repli­ca­tion and “par­a­sites” which do not. Altru­is­tic molecules re­pro­duce less (it’s hard to be both effi­cient cat­a­lyst and effi­cient repli­ca­tor at the same time), the par­a­sitic molecules re­pro­duce more.

Let’s as­sume that the molecules are ei­ther en­closed in com­part­ments or tied to a sur­face in small patches. Each com­part­ment or patch has to be small so that the law of big num­bers doesn’t kick in and make the pro­por­tion of the molecule types in all the com­part­ments ap­prox­i­mately the same.

The com­part­ments with higher pro­por­tion of al­tru­ists are go­ing to grow faster, the com­part­ments with lower pro­por­tion of al­tru­ists are go­ing to grow slower.

Then some ex­ter­nal event, say a wave wash­ing the molecules from the rock, mixes the molecules and cre­ates a new ar­range­ments of com­part­ments or patches.

It can be shown that in such a setup there will arise a sta­ble ra­tio of al­tru­ists and par­a­sites. The par­a­sites won’t crowd out the al­tru­ists. To vi­su­al­ize the mechanism, imag­ine that ev­ery com­part­ment con­tains only two RNA strands. The com­part­ment con­tain­ing two al­tru­ists will grow a lot. The com­part­ment con­tain­ing one al­tru­ist and one par­a­site will grow slower. Com­part­ment with two par­a­sites won’t grow at all. After many iter­a­tions of mix­ing the molecules and re­peat­ing the pro­cess we’ll ar­rive at a sta­ble equil­ibrium of al­tru­ists and par­a­sites.

Now think of com­part­ments with three RNA strands. In that case it’s more prob­a­ble that any par­tic­u­lar com­part­ment will con­tain at least one par­a­site (7/​8 as op­posed to 34). The equil­ibrium will there­fore con­tain more par­a­sites than be­fore. As we pro­ceed to larger and larger com­part­ments the rel­a­tive ad­van­tage of par­a­sites will grow un­til it reaches the point where the en­tire cy­cle will die off. It is there­fore of essence that the com­part­ment size re­mains small.

To sum it up, the stochas­tic cor­rec­tor can work even with­out a hy­per­cy­cle (there can be only one gen­er­al­ist cat­a­lyst molecule) but re­quires com­part­men­tal­iza­tion. Hyper­cy­cle, on the other hand, doesn’t re­quire com­part­ments but is vuln­er­a­ble to par­a­sites. One can imag­ine a his­tory where repli­ca­tion started with hy­per­cy­cles and then, af­ter mem­branes were formed, con­tinued as a stochas­tic cor­rec­tor.

The mod­els above are, ob­vi­ously, just a spec­u­la­tion. We don’t have any rem­nants of those early stages of life and so guess­ing is the best we can do. Yet, some generic pat­terns, more ex­am­ples of which we are go­ing to en­counter later, are be­gin­ning to emerge.

Chromosomes

At some point in the evolu­tion the stand-alone genes stopped com­pet­ing for them­selves and started co­op­er­at­ing by get­ting linked into chro­mo­somes.

Why would that be? Why link one’s fate to other’s rather than just keep­ing the sta­tus quo?

Ap­par­ently, link­ing into chro­mo­somes comes with dis­ad­van­tages for in­di­vi­d­ual genes. Copy­ing of the long linked RNA strands is slower. The con­ser­va­tive es­ti­mate of repli­ca­tion rate slow-down is 50%. That would make the linked gene severely dis­ad­van­taged in the com­pe­ti­tion with its stand-alone cousin. There has to be some­thing that coun­ter­bal­ances that hand­i­cap.

The au­thors sug­gest that the main rea­son for this is what hap­pens dur­ing the cell di­vi­sion.

If the daugh­ter cell needs to con­tain all the genes to sur­vive, it’s cru­cial for any gene to end up in a cell that does con­tain at least one copy of each gene. Other­wise, no mat­ter how suc­cess­ful they are in­di­vi­d­u­ally, how many copies of them­selves are pre­sent, they will end up in a dead end, trapped in­side a non-func­tional cell.

This is es­pe­cially true if the mechanism of the di­vi­sion is rather un­so­phis­ti­cated and prob­a­bil­is­tic like, say, sim­ple fold­ing of the mem­brane and even­tual ran­dom split­ting of the con­tent of the cell.

Fur­ther­more, the more genes there are the more likely it is that one of them will be miss­ing in the daugh­ter cell. It may even hap­pen that both daugh­ter cells will miss a gene and die. This, I guess, places a hard up­per limit on the num­ber of stand-alone genes in the cell. With all genes linked in a sin­gle chro­mo­some, on the other hand, it is much eas­ier to suc­ceed. The worst, though im­prob­a­ble, thing that can hap­pen is that one of the daugh­ter cells will end up with no genes at all.

To be continued

In the fol­low­ing parts of this ar­ti­cle I would like to cover top­ics such as emer­gence of the eu­kary­otic cell, ori­gin of mutli-celu­lar life, of sex and of an­i­mal so­cieties. In the end I am go­ing to spec­u­late about the very high-level, generic co­op­er­a­tion pat­terns and whether they have any sem­blance to co­or­di­na­tion pat­terns that we en­counter in hu­man so­ciety.