o-limits of points in int S* may be limit c,vcles or perhaps strange attractors. But as
we show in .... T. tn il ancl morr' than 1.X, ': I in A, one sees thar ... Let l, , l, and l"
be the lines obtained as intersection of Pr , Pz and 2, with. S, and consider the ...
Xi''fi '?t',Xrfl *J:f #hSä?,ilffi XI:K&.",tJ.?:ldLondon
Vol. 32, No. 3, June l9?9 Printed ia Bclgium
D y n a m i c aSl y s t e m us n d e rC o n s t a nO t r g a n i z a t i o nl l.l . Cooperativeand CompetitiveBehaviorof Hypercycles P. ScHusrnn,x K. SrcnruNo,rANo R. Worrr* * Institut für Theoretische chemie und strahlmchemie der (Jnitsersittit wien.
nf ,rrnstitut rür * 0,-Y,f'Jf 'J'ii; i,o ", "1!; ,ä: r, rudthorgasse4,r0s0 Received April 13, 1978; revised June 7, l97g
Qualitative analysis for a system of differential equations playing an important role in a theory of molecular self-orsanization.
l. hvrnooucrrox This note deals with the ordinary differential equation xi:
x;(hix;_,_ @)
i :
1 , . . . ,f l ,
(l.l)
where A; ) 0 are constants,indices are counted mod z and n
@ - , L: T
krxixi-,
(t.2)
More precisely,it considersthe restriction of (l.r) to the (invariant) simplex S" defined by n
xi)O
and
I
r, -
t.
(1.3)
;_1
Equation (1.1) plal's a central role in the recent theory of self-organization of biological macromoleculeswhich focuses on the notion of the catalytic hypercycle(F], [2]). We interpret x, as (relative)concentrationof the speciesi. These speciesform a cycle: the growth of speciesi is catalysed by its ,,pre_ decessor"i - I through reactionsof ]\{ichaelis-Mententyp". o acts ,,selection as pressure" by keepingthe total concentrationfixed. There existsa unique fixed point C in int S, given by the relations , krxr_, : _ hjxj-rtogether with (1.3). It is shown in [3] that if all i;are equal, then C is a sink for n : 2 and 3, no longer a sink but still asymptoticaliystabre for n - 4 and unstable for n )- 5.
357 0022_039 6I 79 10603 57_I 2$02.00$0 Copyright @ 1979 by Academic press. Inc. AII rights of reproduction in any form resened.
358
SCHI-]STER. SIGMUND.
AND
WOLFF
In Section 2 of this paper we show that (1.1) is cooperative in the sense that no species goes extinct, or more precisel-v that bd S, is a repeller. The o-limits of points in int S* may be limit c,vcles or perhaps strange attractors. But as we show in Section 3, the only attractor for a "short hvpercvcle" (n ..! 3) is the fixed point C. Actually, we prove a more general result on intcrnai equilibration which allov's us to show that if several short hyperc,vclescompete (under the constraint of constant total concentration), then all but one of them will vanish. In Section 4, u'e discuss the biological relevance of this model for the formation and seiection of hyperc.vcles and shou' how it can account for the "once for ever" decisions rvhich occured at manv steps of the evolution of selfreproductive biopoiymers.
2. \'sn For 1 (
i, j {n
and r; }
HvpnRcvcrp IS CooPERATIVTI 0 rve have
- A;.\)-r) (i: )' {-l'} tr,',-,
i:.i)
The boundar-v of S", is inväriant, and contains subfaces of fixed points. I-et F denote this set of fixeti points. LsMNIe 1.
If x e bd Sn, tlte o-limit of x is a subselof F.
Proof . l,et x be on the boundary, that is, on some face of S", . Thus assume xi r --' 0, x, ': 0, t,., .> 0,..., ri+x)> 0 and r:t-r'., -- 0' We shall shorn' bv induction that ,1,-' C, .tr*r + 0,..., trl+r r * 0. ri * 0 and xrir,-r convergesmonotonicallv io some iimit zr ' trndeed' since x, 1 .-= 0, one has i; --10, and so rr ,, i lor :i;;r.;eh >- 0' Also, hv {2.1) (1)
y;
l;;l
'
- Ä , - t \ ' , { ,. t\ , ,I
0
and hence xJxr*, trt,r. If h were strictll' positive,.rii'rri., 1 rvould decreaseat -k;'ri, rvhich implies zri :0. rvhich in turn leastexponentiallvwith factor i m p l i e sh : 0 . (2) fti :,- 0 and r;..r/rr+2convergesto some limit z',*, (rvhich may be -f m). For sufficientlylargel, xi*tlxi*, is monotone.Indeed,bv (2.1)
(#) : ki+,8,*,(f;X;;- *:) If vo)
ko*rlho*r, then x*tlxin,
(2.2)
is increasing,qtherwise it is ultimately
BEHAVIOR
OF HYPERCYCLES
decreasing.In anv case the limit 2,.,, is approached monotonicallv. If z; > 0, then rr * 0 obviously implies tr; r,r.+ 0. Assume now ?i : 0, which according to (2.2) implies ti+t I co. If r,*, : Q, then clearly fi+l + 0. There remains the case that zo*, ;. 0. Supp,sc x,*, does not converge to 0, There exists, then, a 3 > 0 and a sequence /r * i, oo with ro*r(t*) ) ö for k : 1,2,... . Set T - (krr,)-t log2. Sincej,.*, ( kr*rxi*t and x,*r(to) ) ö, one has .{i ' r(l)
Let r :
::.,
for tr-T
{ /{tr.
!kr*, and choosee > 0 such that - l]. e[l - exp(r?El2)] < 2,,*,[exp(r?ö/2)
Now chooseA so large that / x; A. " r '' -' tt, - i;)u, and
D''tl< '
I ,1* for t)- t*-
T. This implies by (2.2) that for tu-
(2.3) T { r S /,
r)1_rö2) (\ f :i +.2./r ) - ( + \J,-r/ and thus xr -r(1r)
x,.
,(1u .< tr,;*z(/r) ffi5
T)
. exP(-r7ö'/2) -,,-
{ (o,*, -l- e) exp(-rT6i2) < ti+r - € which is a contradictionto (2.3). Thus .r,*, - Q. The proof that r,+2 - 0,...,xi+k_r+ 0 is analogous.Hence ,r(/) converges to some subface of the boundary with the property that whenever .t, > 0 t h e n 4 * , : 0 . O n s u c ha s u b f a c e@ ( x ) : 0 a n d h e n c ej r - 0 f o r i = . . 1 , . . . , n . Thus Lemma I is proved. Lpnnre 2.
xeF implies@(x) :
0.
Proof. Suppose tri - 0 for some 7, and fi : 0 for i : 1,...,z. Since *i+t :0 one has either Ar*rx,- O(") : 0 (and hence @(x) - 0) or else xi+r : 0' In the latter caseone repeatsthis. since some tri+rühas to be strictly positive,one gets finally @(x) : 0.
360
SCHUSTER. SIGMUND.
AND
1VOLFF
if bd .S, is a repeller,i.e. if there An ODE on S, will be calledcooperative existsan e ) 0 such that with 1(.) : {x e S,: 0 < d(x, bd S") ( e} (where d is Euclidean metric), the initial condition x(0) e int S, implies that x(t)t'(e) for all sufficientlvlarge l. (1.1)zscooperatiz:e. TirnoRnlr. Thesystem Proqf.
Let P(x) :
xruz"' ;rr,. One has P:P(s-n@)
with s . s(x) = Li:rkrr,-r. Let m; min{s(x)- nrD(x):x e S,l. \ote that. , . : , 0 . B u t m i n { s ( x ) :x € S n } } 0 , a n d h e n c eor n e m a v c h o t t s ca n i n g e n e r a lm /14 '' 0 rvith m .:. ,V < min{s(x): x e S"} and set 1,
I m ,'J'1.\\re deiine J = {x e.S,: s(x) - n@(x) > II)
and
B + ^5'"\t I is an open neighbourhood of -F. Since b). L"*ttt"* x(t) e .'l for all sulhcientlv iarge I, the sct D(x) = i?' ': {}'x(t)e -{ forall t elT.t{'
i :rr,ii I x '' i,,1 !,, irnplies
- ii?'
ill
is nonemptv anci henc,: rr,e rnav Cefine i'(x) = inf Dixi forxebdS',. ..ii1l rr:rii We shor.vn 0. there is a I' rvith 7(x) ::..1. i that x(t) e Ä for t €lT', (.l1 1) T' : ll. Since,4 is open, there is a ö(r) > 0 such that d(x, y) ": ö(x) inr|iits v/fr ri 'i - 1) T' :- ll and hence I(y) ::. ?'(x) -r' cy. in particul;rl lor for I e V',Q' a : I there are finitell' manv x{1),...,x(r) e bd S,, such that ihe orrcrl icts Ki -
iy e S,,: d(y, xu)) < 3(x(j'))
cover bd S,. Put
t*
Ü.,
I]EHAVIOR
361
OF HYPERCYCLES
and
r For anvxe lthere is a 7 Define
igl,
z(x"') r.
l 7 s u c h t h a rx ( t ) e - . {f o r t e l T , ( L - + -l ) f r - 1 . l . 1 t = s u p { P ( x ) :x e 1 i .
Supposeno\\' x .I btj S" . \\-c ciairr'that there is a I ) 0 such that x(t) eJ. that x(l) -, 1 for ali 1 > 0 and lerIndeed.assurlre t, ;
inf{t ,} 0: x(t)e Br.
/, erists sinceotherwisex(t) e -4 and P(x(r)) > /'(x(0)) exp(.\[r; for ail t l> 0, which is imnossibiesince P is bounded.There is a T, ..- T s u c h t h a t x ( t ) q " . 1f o r / - t , e l T r , ( L : l ) Z , - r l l . P u t t i n s
ri + r, r (.1,4' 1)r, -- l iurl norrrrg:hrr: riLrrinr-1 thc timc [t, , tj] the orbit of x spencisa time lessthar. T. tn il anclmorr' than 1.X, ': I in A, one seesthar P & ( j ' t i ) ' : : P ( x ( r l ) )c x p ( - - r n i ' l ' , . , L I , 1 7 1 + X I , ) P(x(r')tew )2 P(x(0))eM. Similarll', l, * inf{t } r,': x(t)e Bi existsanclthere is a T" '::..7 with x(t) e -,1 for t With tL:tz+Q,-t1)72+
tre lTr, (f
i 1) 7"r,, 11.
1,
one gets P(x(ti)) >- P(x(t,))eM 2 P(x(O))ezM. P r o c e e d i nign d u c t i v e l yo n e o b t a i n sa s e q u e n c te; , k : 1 , 2 , . . .
with
P(x(ti)) )> P (x(O))ekM, which is a contradictionto P(x(t)) { P for all I ) 0. Thus x(r) has to leave
362
SCHUSTER, sIGMUND, AND woLFF
1 at some time. Note also that for any I ) 0 smaller than the first exit time one has t e lt'* , to*r] ty some A (with ti : 0), P ( x ( r ) )2 P ( x ( t ; ) ) e x p ( - , z
?)
P ( x ( O )e) x p ( - , z
7'- ,tt)
and hence P(x(r)) - P(x(0))exp(- z
T)
Let 1:min{P(x):reS"\/}. Clearlyp > O. Choosee > 0 so small that I(e)C I and that P(*) < pexp(-imlT)
forallxe/(e).
The orbit of a point x e S,\1 may possibleenter d but never ,I(e),since P ( x ( t ) )> , p e * p ( - , n l T ) Any orbit starting in 1(e) leavesI after a finite time and never returns to 1(e). Thus the theorem is proved.
3. INrrnNel EgurlrnnerroN AND Corrrpntrtrox oF SHoRTHypnncycr-ss As frameworkwe shall usethe ODE on S,: xr(Gt- Q)
4 :
(3.1)
where the G, are functionson S, and E :li-rxoGi.
3.a. Internal Equilibrationfor 2-Hypercycles Supposethat in (3.1) one has G, : h$z and G, : hzxt, where k, and k, are constants)0. Equation (3.1) then describesa systemhaving a 2-hypercycle as subsystem.For r, and x, ) 0 one has (!,
\t-n'')
A,r'
: - f -r " ' ( t*x' r
_j,t
k't
and hence ,rl
;-
kr -E
In particular,all orbits of (1.1) in intS, convergeto the fixed point C.
Q' 2)
BEHAVIOR OF HYPERCYCLES
363
3.b. Int ernal Equilibration for 3- Hy percycles Supposenow that in (3.1) one has G, - krx, , Gz : h"x, and Gz : hsxz. This meanswe have a 3-hypercycleas subsystem.Note that x, + x2 + r: ( I (:1 in casen :3). We shallprove ffr ---l-, x2
hs R2
x2 --_i-, x3
hr R3
- "r3 xr
h2 , Rr
(J.Jl
(In particular,ali orbits of (l.l) in int S, convergeto C). Let / be the line through the origin (in (r, , lcz, tr3)-space) given by hrx" : krx, : Arx, , and let pt , pz and pr be the planesthrough / and the xr-, xr- and r3-axesrespectively.Note that (2.1) is valid for i : 1 and j :2. Hencep, divides R3 in such a way that in the half spacecontainingthe unit vector e, , the ratio (xr/xr) is decreasing,and in the other one it is increasing. Let l, , l, and l" be the lines obtained as intersectionof Pr , Pz and 2, with S, and consider the following Fig. l: let P, be an arbitrary point between e, and C.
Frcune I Let P, (resp. Pr) be the intersectionof Pre, (resp. Prer) with /, (resp. /r). Let Q" (resp. pr) be the intersection of Pre, (resp. Prel) with /, (resp. /r). We claim that the intersectionQrof Q"erand Qre" lies on /r.
-
364
scHUSTER,srcr\rlrND, AND woLFF
Indeed, this statement is easily seen by a projective change of cr:ordinates rvhich sends the four points el , €r, €3 and c into the points ei . el, el and c' r v i t h c o o r d i n a t e s( f - . c o , 0 ) ,( 0 , l ) , ( 0 , 0 ) a n d ( 1 , j ) ( s e e F i r . 2 ) .
- + e,
Frcune i
Let H clenote the hexagon PfrQ2Q\esP2 lving in S, and a. the pvramici r.r'ith base lI and sumrnit O ithc origin of (.", , rr,:r")-space), Consider oni: of tiie fäces of r, ()PrP. tir: Since P, , P, ancl e, are corinear. ol)rl:1, "ranrrrlr. Iies on a plane ihr.r,rqh the .r.,-a:ris'o'hicrr is tirercfore of the f'orrn 11fx, =,-.g6n..1. on the other hand, since oPrl', is not on the sa're,*ide as e, i,l'ith respeci to the plane /' . tnc ratii-r(xrr'rr) must increase.'llius any orbit through ap1p2 must enter the pvramid" The other faces of ,ztarc dealt q'ith similarlr:. Letting l)r varv from e, to c and repeating the constructinn, on" obtains a nested family of pvramids having the line i as their intersection. f'hus all orbits converge to$'ards l, i.e., (3.3) is valid.
3.c. Competition. of Short llyperct'cles C o n s i d e r n o u ' a s y s t e m c o n s i s t i n qo f 4 , 2 - h 1 ' p e r c v c l e sa n d . ' i 1l - i r r l , , - r , , . , i . . . W e m a y d e s c r i b et h i s b v a n c q u a t i o n o f t h e r y p c ( 3 . 1 ) . n a m c l ,
lor i:1,...,
\-;
( 3 . 4)
- o1i rl, : *{o)18{')rioi +2 (il &
: rj')(Aj')rl') - ot)
i';'
: ,,,,1uj,,rj,, -,,)
V,
REHAVIOR
365
OF HYPERCYCLES
on the simplex Srr,rnr. Bi- (3.a) and (3.b) ali orbits in int.Sr.y_3r, converge to the invariant subset S* clefined by the relations
- Al')rl') Äl')n,l')
; I
Al').-l')-. A.:i).r|.-' Ä(t)'0)
| l r . ' . r
\l - Y
i ,. Ä - ;,...,,v . M.,
' -'lI)-simplex. As coordinateson S*, rve useJi (i :
u'hich is an (,\ '*'here
1,...,l\ +M),
' 't'!t)g' t
t''
v.ith L(,)Ltil
.
8i
_-!'t
"..,:
n1- n?
,-
t. fttr
t
1,...,\'
and
h\'t Pat bli\bIi\ "1 ',-3
_
btitbtit "2 ''t
Llt)bttt "3 "2
Note that r:n Sx, -yi'; 0 and l)=1r;: 1 . T h e r e l a t i o nJ i : I means that there existsjust the ith hypercvcle.in internal equilibrrium.The restriction of (3.4) to the invariant subsimplexS* becomes li
:
Yikli yi -' O) (J.) )
rvith t -.. I...., ,\ , ,,JI
.tilrl
_ \r , l , f ,. r,:
'l
! t - ' i
: ,
i
r i 5 1 1 r : s t i g i 1 i 3 ; iev a s y ' t c a n a i i . ' s t . ., ; r , . , i.,th, r rhrr. lru ,:r1..,. . '
: i i l i l c l / r r rä f t i h { ' : . i l r :
,'
.'
,rir, i ::
.j-:it:,::iii.--f
i. ,, ..11r: ,trr-ti;11::
'.t.r:l
arc:h3',*t-iir-es,,i',i
,i' :r'::r I ir:i::,i ,
' , . , ' r i - . . \ . . , . , ,r.' ,. : I
ilrirs
,.
:
; ,l
."::t:i,irj'r':i11t:i:,i:.rrlrir;iü,t:i::j.,.
'
l ' i t r : s i r i r : ; i n i , l . . . , : t . i i t r r i ; i i r . l , l i i , -:
,:rrCtiOn the r t ; l l7 ; : i
i, ,r1 ille the iV(j measure
ir';perc\,cle
s u n i v e s a n d r e a c h c si n : c r n a i e q u i i i b l i l i r , : \\'e do ltot knolv h{iil io orolc a \:(,rfcsi.rundingexclusiL;nliirrcipie for the corrlpetition of longer h'"'perc,vcles, nor ho\r' to describc rhrir attractors. Finaliy, iet us note that for the comrretition of two 2-rrvpcrcy.cles(i.e. (3.4) with rV -. 2, II : 0) one obtains by a simpie computation that
- Ä11)r(t) Ä!r)x1tr htzlrtzt- Alzr;ar
-
lvlt5L.
This showsthat the planethrough the line joining the two attractorsAr(hl\ x\r) Alt'"!t', slz)-. *tzt .- 0) and Ar(*\t,:..rf) : 0, h\ztrtzt- h\z)x;ztlare invariant.
-
P-@--
366
SCHUSTER. SIGNTUND. AND
WOLFF
It means that the internal equilibration of the two hypercycles occurs in a well-balanced way. It would be interesting to know whether a corresponding fact holds for longer hypercycles.
4. DrscussroN: Tur.FonlrarroN
AND SELECTIoNop HvprRCycLES
Any attempt to explain the evolution of the genetic code has to deal u'ith a kind of "existence and uniqueness" problem: (A)
it has to show horv such an extremely improbable machinery could
emerge; (B) it must account for the very strange fact that there is only one such code for the multitude of living cells on earth. The biochemical theory of hypercycles as developed in [2] is a step towards the solution of this double task. We want to shou'here hou'the simple mathematical model (1.1) reflects this. Problem (A) relates to the notion of self-reproductive automata. As v. Neumann showed in [4], the "complexity" of such an automaton has to be above a certain threshold. This level can be estimated in the biochemical context: as shown in !] and 12] it exceeds the capacity of the primitive biopolymers likely to be found in the "primordial soup," so that they have to cooperate in order to fulfill their task. The hypercycle is a biochemical device allowing macromolecular information carriers of comparatively low grade to pool their information. The theorem in Section 2 means that this form of cooperation is stable in the sense that small perturbations cannot "kill off" members of the hypercycle. The proof in Section 2 proceeds in a way which sheds some light on a possible course of hypercycle formation. We have to assume that the species are formed by mutations, i.e. by random fluctuations introducing from time to time small positive concentrations r, . We start with a svstem which is not yet complete, i.e. where xi :0 for some /'t. By Lemma 1, the x(t) approaches the fixecl point set and asymptotically seems to be inert. All but the concentrations of the "end species" (where x, ) 0 but x;..1 : 0) are extremely small. Consider some mutation which introduces one of the previously non existing sfecies without yet completing the hypercycie: the system still remains on the boundary and after some (possibly drastic) changes in concentration approachcs again some seemingly inert state. But when finally the svstem is completed by a mutation creating the last missing member, the state x(f) enters int S* and the long term behaviour changes its character. The attractor norv lics in the interior of the concentration simplex and apart {rom the lou. dimensional cases no longer consists of fixed points. From a seemingly dead quasiequilibrium emerges a pulsating form of dynamical cooperation.
BEIIAVIOROF HYPERCYCLES
367
Probiem (B) has to do rvith the competition of hvpcrcvcles.\\rhile we cannot 1)ro'r an exclusion principle in full generalitr', u'e mav account for the..once for ever" decisions if u.e make the natural irssunrptionthat fluctuations leading to efficient mutations are t.erv small and do not occur frequentlv. 'I'hus consider thc competition of the hl.irercvcles11, (l < / 0 ibr I :. l '... n, r.iti .iil ,rther c.ncentrltions cxccpt those ,f one or several end-speciesxltt are0. Since rj', - 0 anri rlr -, 0, .rja is ilecreasi'g and Il]r .jslr) remains bounded au'a'from 0. A minor modification shorvs thai the theorem in section 2 is still valid for the rjr) and thus that @ remains bouncled awav from 0. Hence ;rjr) converges to 0, i.e. all orbits from ,o enter G. The same holds for all orbits starting from some suitable open neighb.rhood I/ of E in the concentration simpiex. If the time interval which precedes the mutation completirg rl, is large enough, the concentrations of those species which are .rot have ".,al.p".ies become so small that the fluctuation sends the system into some state in tr/. Hence the system will converge to some attractor of the purc hypercycre /1, .
368
SCHUSTER.SIGMUND. AND WOLFF
If the time interval up to the next fluctuation is sufficiently large, the state will not leave G under such a small perturbation and hence will still converge to an attractor of Hr. This is valid even if further hypercycles /1, are completed: the concentrations of all their species will vanish. This does not mean, of course, that evolution ends with the first hvpercycle. But it shorvs that the only possible concurrents of H, are those hypercycies having some species i in common with 11r. Such a hypercvcle 11 rvill supersede II, ifr i is a better .,,: hr*, catalysator for its ,F/-successorthan for its f1r-successor. (Indeed, kt1r.'1 implies (*l!tl*r*t) + 0 etc.). Hence mutations introducing nerv specics mav yield "improved" hypercvcles and extinguish their ancestors. The inheritance of members of the previous hyperc,vcle is a mathematical paraphrase of the "once for ever" decisions in the formation of the celiular mechanism, where v'e have "linear descendencv" of prebiotic organisms instead of the familiar, many-branched "descendency tree" of Darwinian evolution. This fact is amplv validated biochemically by the universality of the genetic code, the uniqueness of chiralities etc.
AcxNowtroclrrNts The authors thank Prof. Dr. Manfred Eigen for many stimulating discussions. Generous supply with computer time by the "Interuniversitäres Rechenzentrum, Wien" and financial support by the Austrian "Fonds zur Förderung der Wissenschaftlichen Forschung" as well as the "Bundesministerium für Wissenschaft und Forschung" is gratefully acknowledged.
RrrunrNcns 1. M. ErcrN, Selforganization of Matter and the Evolution of Biological Macromolecules, Naturwissenschaften 58 (197 1) 465*526. 2, M. Etcrw eNo P. ScuusrEn, The Hypercycle, a principle of natural selforganization. Part A: Emergence of the hypercycle, NaturwissensehaJten64 (1977) 541-,s65; Part B: The abstract hypercycle, 65 (1978), 7-41;Part C: The realistic hyperc-vcle, 65 (1978),
341-369. 3. P. Scnusrun, K. StcltuNo, aNo R, Wolrr, Dynamical systems under constant organization. I: Topological analysis of a family of non-linear differential equations, Bzl/. Math. Biophys. 40 (1978), 143-769. 4. I. v. NruneNN, The general and logical theory of automata (1951), in "Collected Works," Vol. 5, Pergamon, Long Island City, N.Y., 1963.
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