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Recycling, reproduction, and life's origins
BioSystems, 15 (1982) 89--97
89
Elsevier/North-Holland Scientific Publishers Ltd.
RECYCLING, R E P R O D U C T I O N , AND LIFE'S ORIGINS
G.A.M. KING
Physics and Engineering Laboratory, DSIR, Lower Hutt, New Zealand (Received July 7th, 1981) (Revision received October 13th, 1981 ) Kinetic considerations make it most improbable that any reproducing system could arise spontaneously in a prebiotic soup containing a large variety of organic molecules, as commonly postulated. This batch process can be contrasted with a completely recycling network of reactions maintained by an influx of energy. So long as the network includes at least two bimolecular reactions it is likely to support pathways for chemical reproduction. However, such reproducing systems will be simple in both kinetic and structural terms. Subsequent evolution will lead to much more complex reproducing structures although the kinetic complexities, measuring the varieties of reactions between these structures and their media, will remain relatively simple.
I. Introduction Chemical activity necessarily consumes the reactants. If the activity persists longer than the time required to consume stocks of reactants, they must be regenerated through a cyclic sequence of reactions. In a closed system all stocks are finite and persistent activity then requires a completely recycling n e t w o r k of reactions maintained by an influx of energy -- radiant energy, for example. It is thought that the present day biosphere follows such a pattern of recycling in nearly all respects. The purpose of this paper is to examine some properties of completely recycling networks in case they may have some relevance to studies on the origins o f life, much as Morowitz (1977) has suggested. A most relevant property, developed in Section II, is the likely appearance o f autocatalytic subsystems within networks which contain bimolecular reactions. Section III enquires into the conditions under which these autocatalytic pathways can manifest "autocatalytic kinetics" -- an exponential increase in concentrations. Attention is thus shifted from the complete closed system to an open medium in it, isolated from
the rest except for supply by physical transport. The reactions of a recycling network must be specific in order to avoid a proliferation of products. This fact is used qualitatively in Sections II and III. In Section IV the specificity of reaction is examined quantitatively for the particular case of a single autocatalytic system. As the number of reactants in the medium increases, the number of alternatives to the autocatalytic pathway increases rapidly. Autocatalytic growth occurs, then, only if the specificities of reaction increase very greatly. From this we deduce a need for selection to achieve a growing system of even modest complexity. Section V refers to earlier work on the selection for physical structure which can confer specificity on an autocatalytic particle. The Discussion takes the various concepts which have been developed and applies them in general terms to the problem of life's origins.
II. Single medium Consider a closed set of chemical transformations in a single medium. They can be described by a completely recycling network
0303-2647/82/0000--0000/$02.75 © 1982 Elsevier/North-Holland Scientific Publishers Ltd.
90 of chemical reactions. In this Section, we are concerned neither with the way that energy is supplied to maintain the chemical activity nor with the kinetics of reaction. The interest here is in the pathways for the flow of materials and the conditions under which parts of the network may be autocatalytic pathways. These conditions are deduced in twelve progressive steps. (1) As every active substance is recycled, it is both consumed and produced. It is an intermediate in the network of reactions. (2) Confine reactions to unimolecular and bimolecular yielding one product or two. All more complex reactions will be broken down into these. (3) Because of (1) and (2), we can use the language of chain reactions (e.g. Laidler, 1965), classifying them as initiating, propagating, branching, and terminating reactions. (4) If the energy input is steady, the system moves towards a stable steady state. (Again because of (2), we are not concerned with " e x o t i c " behaviour which might be produced by ter-molecular reactions --Nicolis and Prigoglne, 1977; King, unpublished). (5) Under the steady state conditions of (4), we can use the idea of parity to get relations among parts of the recycling network. A reaction contributes to a parity sum by the increase in the number of intermediates on traversing that reaction. Thus, a branching reaction contributes ÷1 to the parity sum, a propagating reaction zero, and a terminating reaction (removing a single intermediate) c o n t r i b u t e s - 1. The parity sum for the complete recycling network is zero. (6) Take any reaction yielding two products. Parity requires that it be compensated somewhere in the network by a reaction consuming two reactants. The network thus contains at least one bimolecular reaction.
(7) Treat the two products from (6) as the intermediates from a branching reaction. As the network is completely recycling, it is possible to trace a cyclic pathway from each of the products back to the branching reaction. In a complex network, this may be done in many ways. (8) Take any selected pair of the cycles passing through the branching reaction in (7) and take a parity sum over this sub-system. Count as intermediates only those substances both produced in one reaction of the sub-system and consumed in another. There are two distinct cases: (a) The parity sum is zero. (b) The sum is +1. (9) In case (8a), the +1 in parity contributed by the branching reaction has been compensated by a terminating reaction where the cycles rejoin. Two intermediates, one from each cycle, have reacted together to produce only one intermediate common to both cycles. This case is characterised by reactions among the intermediates of the sub-system (see Fig. la). (10) In case (8b), the first intermediate comm o n to both cycles is independently produced from the last distinct intermediates of the separate cycles, so that both reactions producing it are propagating. The + 1 in parity from the branching reaction is not compensated within the sub-system although, of course, it must be compensated elsewhere in the total recycling network. This case is characterised by the absence of reactions among the intermediates of the sub-system (see Fig. lb). (11) The latter case (8b and 10) allows material to be taken up from the rest of the recycling network and incorporated in the sub-system. In the absence of other restraints, reactions of the subsystem thus cause growth in the concentrations of its intermediates. The sub-system is autocatalytic. (12) During the steady state postulated in (4), autocatalytic accumulation in the
91 (b) The network contains at least two bimolecular reactions.
) la
r
) ) lb
O + O 2 -~ 03 O + 03 ~ 02 + 02 02 ~ O + O
)
~~
~ -_-
Bimolecular reactions are needed, not only to compensate for the branching reaction, but also to supply material to the autocatalytic sub-system, building up the molecule split in the branching. If the branching is unsymmetrical the network contains at least three bimolecular reactions, b u t if symmetrical the n e t w o r k may contain as few as two. There is a familiar example of a symmetrical autocatalyst (Chapman, 1930);
X-Y IC
Fig. 1. Sub-systems of a completely recycling network. (a) Sub-system with one branching and one terminating reaction. This is not autocatalytic. (b) Sub-system with one branching reaction and all others propagating. This is unsymmetrically autocatalytic. (c) A symmetrical autocatalytic system.
sub-system (11) is offset by other reactions of its intermediates, consuming them. Away from the steady state, the sub-system may grow or decline depending on the relative importance of the autocatalytic reactions and the other reactions of its intermediates. The foregoing analysis shows that not all completely recycling chemical networks need contain an autocatalytic sub-system, b u t the conditions leading to autocatalysis are not severe and it may appear quite readily. The conditions are: (a) A branching sub-system that can be cyclically closed without reactions among its intermediates.
(A) (B) (C)
When supplied with atomic oxygen in (A) and (B), molecular oxygen undergoes autocatalytic growth, with (B) the branching reaction. Growth is offset by dissociation of the autocatalyst in (C), restoring atomic oxygen and completing the recycling. In the upper atmosphere, where this system and its derivatives assume some importance, 02 greatly predominates over O and departures from the steady state are small. Condition (a), the absence of reactions among the intermediates of an autocatalytic s u b s y s t e m , is a particular facet of a general relationship -- that a cyclic set of reactions is a restricted set from among those which are energetically possible with the chemical species present (King, 1978). As a completely recycling network follows strict conservation of parity, this restriction can be made explicit. For simplicity, we treat just a set of bimolecular reactions, b u t the same conclusions follow for mixed unimolecular and bimolecular reactions. Because of parity, all the bimolecular reactions are exchanges, yielding two products. Because recycling is complete, each product is also a reactant. However, it may react with more than one of the other reactants. Let there be r reactants and the ith one reacts with Ji others. Then, the number of reactions is ~ Z r Ji, and they yield Zr Ji product terms. If any Ji exceeds unity, this sum
92 must exceed r, the number of distinct products. Therefore, some of the products must be "repeats" formed either in separate reactions or as pairs of identical products from single reactions. The number of such repeats is (Zr J i - r), and it gives directly the number of autocatalytic sub-systems. Thus, the reactions of a recycling system with autocatalytic sub-systems, although a restricted set, are less restricted than those of a recycling system without autocatalysis. It is convenient to distinguish the intermediates of an autocatalytic sub-system from the other intermediates o f the recycling network, calling them the " c o n s t i t u e n t s " of the autocatalytic system. Those intermediates that react directly with the constituents promoting the autocatalytic growth can then be called the autocatalytic "reagents". The restriction of reactivity around the autocatalytic sub,system therefore takes the form -constituents react only with specific reagents and n o t with one another, and the reagents also do not react with one another. In Section IV, the question of specificity will be treated quantitatively, the "allowed" and " f o r b i d d e n " reactions being just the extremes of a continuum of reaction rates.
III. Physical transport The discussion so far has assumed the ability to observe the various intermediates in a completely recycling system and hence to deduce the reactions among them. In fact, knowledge of an autocatalytic sub-system is likely to arise, not from complete knowledge of the total system, but from observing the kinetic behaviour of one or a few intermediates. When a substance shows an exponential increase in concentration, it is assumed to be catalysing its own production from a constant supply of reagents -- autocatalytic kinetics. Because the reactions in the autocatalytic set are cyclic, growth may be started with less than a full set of constituents, even one being enough. There is an induction period while the ratios of con-
stituents adjust to those appropriate for exponential growth, and they all then grow with a c o m m o n exponential law. This common behaviour allows us to deduce the fact of autocatalysis while at the same time it conceals the details of its reactions. On the other hand, exponential growth requires the presence of all the reagents in concentrations adequately exceeding their corresponding thresholds (see next Section). A shortfall in any one reagent means that the reactions causing autocatalytic decay predominate. By manipulating the reagents, one can investigate the autocatalytic reactions. If the intention is just to observe autocatalytic kinetics, it is first necessary to accumulate all the reagents in adequate concentrations. If the reagents are supplied through the recycling network, their consumption by reactions with the autocatalytic constituents must be prevented, and this requires physical separation. We move from a single medium to considering separated media related through physical transport. The extent of physical separation needed to effect accumulation is influenced by the restriction of reactivity implicit in the cyclic reaction set of an autocatalytic system. Consider a system where a constituent, C, is consumed in a branching reaction to yield constituents, X and Y. Name as the X and Y segments those parts of the cycles passing through X and Y having no c o m m o n constituents (see Fig. lb), and name the part with common constituents the C segment. If a reagent reacting in the X segment is withheld, the Y and C segments together act as a catalytic cycle, continuing to consume their reagents so they do not accumulate. Likewise, if a reagent for the Y segment is withheld, reagents for the X and C segments do not accumulate. However, if a reagent for the C segment is withheld, the constituent reacting with that reagent accumulates while all others decline. As a result of their decline, all other reagents suffer lower rates of consumption and thus accumulate. To effect the accumulation of reagents, then, it is enough to
93 separate only one of them provided it is a reagent for the c o m m o n segment of the autocatalytic system. Two points are worth noting: Firstly, accumulation can take place because reagents do n o t react one with another. Secondly, a symmetrical system with Y--- X has no X and Y segments, all reactions occurring on the c o m m o n segment (see Fig. lc). The physical exclusion of any single reagent is thus enough to effect accumulation of them all when the system is symmetrical. Materials are excluded from the reacting medium by a change of phase, say, by the ebullition of gas or precipitation of a solid from a liquid medium. Autocatalytic growth resumes when the depleted reagents are resupplied to the medium, and it takes the exponential law only if the time for resupply is short compared with the characteristic exponential time. Some mixing process thus introduces the reagents and disperses them uniformly through the medium. Growth can occur only in a mixing zone. Moreover, if the mixing is continuous, the materials excluded from the medium must be different from the reagents mixed into the medium. Therefore, some chemical changes must occur in the external environment as well as physical transport. N o w consider the case where the time of resupply is long compared with the characteristic exponential time. The interesting case, contrasting with the autocatalytic kinetics of the batch case discussed above, is when supply is steady. The autocatalytic system grows until reagents are consumed as fast as supplied. At this point, the decay of autocatalytic constituents just offsets growth and growth stops. It is likely that the limitation occurs in a single reagent more than in the others, so that its concentration fails to increase while the others still accumulate. In the course of time, this "limiting reagent" becomes the sole supply determining the concentrations of autocatalytic constituents. More generally, fluctuations in physical transport will cause fluctuations in supply. The autocatalytic behaviour will then lie
between those for the " b a t c h " and "constant s u p p l y " modes, approximating to the former when the fluctuations are extreme and to the latter when they are slight.
IV. Specificity The previous Section pointed out that the separate recycling of at least some reagents is likely to require chemical reactions in a region physically distinct from the autocatalytic medium. Physical separation thus helps to define the restriction of reactivity needed in a completely recycling network. However, in a single medium the restriction depends on specificity in the reactions themselves. Present interest is in the degree of specificity needed to support autocatalytic kinetics. For simplicity, consider a symmetrical autocatalytic system with n constituents, Ci. The treatment can easily be extended to unsymmetrical systems. Let the medium also contain m other active substances, R j , including all the reagents for the autocatalytic constituents. If ~i1 is the rate coefficient for reaction between Ci and R j , it is convenient to write: -iy = ~ij • R j
(la)
C~i = olii.
(lb)
Then rn
d C i / d t = -- Y, aij " Ci + .i=l
Oli-i
" Ci-1
"
v
(2)
wherevislfori¢ land2fori= 1, i - - l = n . The first term on the right hand side o f (2) contains n o t only a i • C i but also the possible reactions of Ci with all other materials in the medium apart from c o n s t i t u e n t s - the alternatives to the autocatalytic cycle. Write (2) as m
d In C i / d t + Z ~ q = v • ~i-~
• Ci-1/Ci
(3)
1=1
The system behaves coherently when all dln Ci/dt=
dlnC/dt=
X,
aconstant
(4)
94 Substitute (4) in (3) and multiply all (3) together. Because of the c o m m o n behaviour, all Ci cancel:
(9b) contains the concentration of the limiting reagent, Rl:
R l = Otl/fJl ~] I ~ + ~ cqj ] = 2 ~] cq
i= l
j= l
i= l
(5)
The system shows autocatalytic growth if is positive and decay if }, is negative, so there is a threshold condition when ~, is zero:
~1 I j=~1 aij/ail = 2
i= 1
where ~l is the rate coefficient for uptake of the limiting reagent and c~l is the effective rate coefficient for decay of the autocatalytic constituents, predominantly of Cl. Because all other Rj are large and therefore relatively constant, al takes the form of a first order rate coefficient:
(6) t~l= j =~l
As Y,cqj includes ~i, the condition can be written
~l Ii+
1/Sil =2 ,
i=l
(7)
where Si is the specificity of C i for reaction with Ri in a medium containing all the R's: rn
Si = oli/j~1 aij,
j :/= i.
(8)
Notice that the various a's contain the concentrations of R's (eqn. 1), so that the S's also are affected by concentration. The threshold condition can be interpreted to mean that the system will grow, displaying autocatalytic kinetics, only if the algebraic expression in (7) has a value less than 2. The "limiting reagent" case mentioned in the previous Section follows easily. As the concentrations of non-limiting reagents build up, so increase the corresponding specificities until the terms in (7) containing them all approach unity. There is left for consideration only the term in the specificity, Sh of the constituent, C h using the limiting reagent:
1 + llSl = 2
(9a)
therefore
Sl = 1
(10)
(9b)
Otlj ,
j--/=l
(11)
Equation (10) says that, under conditions of constant limited supply, the limiting reagent assumes a constant concentration. If the supply fluctuates, it can be shown that the concentration fluctuates about the value given by
(10). Each reagent has a threshold condition of the form (10). Although the various ~l are characteristics of the autocatalytic system, the al have a more complex nature, partly determined by the system and partly by the other materials in the medium. Moreover, these conditions apply when the corresponding reagents are separately limiting. If several or all of the reagents have similar importance in controlling growth, the more general relation (7) must be used, calling for somewhat higher concentrations of the reagents. This is the condition mentioned in the second paragraph of the previous section. For the present purposes, the most significant use for equation (7) is discussing the growth of autocatalytic systems as a function of their complexity. Without knowing the detailed properties of all materials in a medium, it is not possible to say whether or not a postulated system will exhibit growth. However, the degree of selection, both of rate coefficients and of concentrations, needed to produce a growing system can be assessed in general terms. It is inversely related to the probability that a system with randomly
95 assigned rate coefficients and concentrations would achieve growth. As a first step to assessing this probability, it is convenient to approximate to equation (7). Define
Then, the probability that (15) would be satisfied by chance,
(12)
is given by the incomplete gamma integral,
pq = O~/i/O~ i
Then, the term, 1/Si, in (7) can be expanded as a sum:
P=P{O<
P-
1
( y - - 1)!
yffil~p y < l ) ,
(18)
1
J(° p(y-1) . e-p dp.
(19)
m
1/Si = j~l Pij, J ¢ i.
(13)
Now expand the complete expression (7) both as an algebraic product and, in its logarithmic form, as a sum. When terms of order higher than the first are neglected, the following inequality remains: ln2= 0.693< ~
i=1
~ Pij <1,
1=1
j:/:i.
(14)
We adopt the approximation to {14),
i=1
m
Y
j=l
y=l
~ Pij =
~; py = 1,
(15)
where Y is the total n u m b e r of terms in the final sum:
Y= n ( m - 1).
(16)
Y is a measure of the system's "kinetic complexity". With p given by (12), equation (15) is a satisfactory approximation to (7). The probability that a system with randomly assigned parameters would achieve growth depends, then, on the probability density function for p. This function is not properly known, but only its general properties of unlikely large values, skew to small values, and finite mean. Fortunately, the qualitative conclusion we shall draw is not very dependent on the exact function and it is enough to take a function with the least further assumptions, the exponential: p(py)
=
e-Py.
(17)
(I thank Dr. J.L. Beck for deriving equation 19). Table 1, giving P as a function of Y, shows that kinetically simple systems (small Y) are much more likely to exceed by chance the threshold for growth. For example, if an autocatalytic system has only 4 reactions for uptake and the medium contains only the 4 appropriate reagents, then Y = 12. Yet the chance that it grows spontaneously is many orders of magnitude less than for a system with 2 reactions in a m e d i u m containing only its reagents (Y = 2). Conversely, while the 2reaction system could arise by chance, the 4-reaction system almost certainly requires a very large a m o u n t of selection in its rate coefficients and in the concentrations of its reagents. This qualitative conclusion clearly applies regardless of the probability density function adopted for p (eqn. 17). V. Selection
In general terms, a selected system persists while its alternatives decline to insignificance. Selection for autocatalytic growth therefore requires the continual supply of reagents which would allow a successful system to persist. Growth then leads to concentrations TABLE 1 Probability complexity
Y P
2 0.26
of autocatalytic
4 0.019
6 6 X l O -4
growth
12 9XlO -l°
as a f u n c t i o n o f k i n e t i c
20 2 X l O -t9
30 2 X l O -3a
96 of autocatalytic constituents such that, on average, growth and decay balance. The concentrations are controlled by the flux of a limiting reagent which fluctuates about its threshold concentrations, and selection is thus largely confined to the specificity of the reaction for its uptake. Although the "competition" is here between a reproducing system and less organised chemical reactions, it resembles biological competition between reproducing species, being with respect to a limiting resource. The idea of selection also applies to the initial formation of a completely recycling network of reactions, where the competition is purely chemical. The recycling network persists while its alternatives wane. Section II called a completely recycling n e t w o r k a "restricted set" of reactions, describing the o u t c o m e of the selection without reference to how it occurred. The autocatalytic systems discussed in Sections III and IV already reflect this degree of selection, because there are no reactions among the constituents and no reactions among the reagents for them. These provisions appear in the definition of kinetic complexity, Y (equation 16), which in the absence of selection would take the larger value, (n + m)2/4 (King, 1978, Section 2B). Section 3 of King (1978) has looked at the factors conferring specificity of reaction in both gaseous and liquid media. In a liquid medium specificity depends on the physical structures of the reactants themselves. The structural complexity of an autocatalytic system is related directly to the complexity of the reaction set producing it. However, some reactions in this set may be internal rearrangements of the constituents. These unimolecular reactions bear no simple relationship to the reactions for uptake from the medium, although dependent on them. Also, they are quite unrelated to the kinetic complexity of the system (eqn. 16), which counts only those bimolecular reactions between constituents and other active materials in the medium alternative to the autocatalytic pathway. For example, a bacterial cell has a
very large number of internal reactions but just a few reactions with nutrients and other materials in the medium. It is structurally very complex but its kinetic behaviour is relatively simple. The big disparity between structural and kinetic complexities is due to the "particulate" nature of this autocatalytic system, its constituents being stages in the growth of a particle. The moderate kinetic complexity means that the reactions for uptake from the medium need not be impossibly specific. Complementing this, reactions associated with the high structural complexity form a surface on the particle which exhibits high specificity for uptake. The combination of properties has been selected during a long period of biological evolution. Section 6F of King (1978) describes a process of selection which simultaneously increases the structural complexity and decreases the kinetic complexity of particulate autocatalytic systems. It depends on the conditional stability of a system containing two particulate autocatalysts which mutually exchange materials. If the exchanged materials are well supplied, the two types of particle reproduce separately but, if the exchanged materials are limiting, only a particle formed by their union can show growth. This process has been incorporated into an hypothesis for the evolution of structural complexity in biology (King, 1977a,b). VI. Discussion Conventional views on the origin of life usually disregard the possibility, or at least any significance, of recycling in the period of "chemical evolution" preceding it. They often envisage conversion of a material stock in the atmosphere into a batch of nutrients in the oceans -- the "primordial s o u p " waiting for the onset of reproduction. These hypotheses are all open to the critical objection that preceding events have not matched the greater part of the nutrients to the requirements of the initial reproducing species. Instead of being "nutrients" they are likely
97 to be "poisons". Following Section IV, the variety of active materials means large values for the kinetic complexity, Y (eqn. 16), and the probability of autocatalytic growth by even a simple system is virtually zero (see Table 1). Hypotheses which further postulate a reproducing polymer with attendant catalytic machinery for polymerization and translation imply that there are many reactions for uptake. They are committed to still larger values of Y and are correspondingly more incredible. On the other hand, a completely recycling system would have evolved chemically from a more complex system by losing those materials which otherwise disrupt the recycling. If part of the system were to reproduce chemically, following autocatalytic pathways, it would immediately receive appropriate nutrients. Further reactions would, at the same time, consume the reproducing constituents so that their materials would recycle. "Life" and "death" would appear together. Section II has shown that a completely recycling system is likely to contain an autocatalytic subsystem, provided it includes at least two bimolecular reactions. This proviso dictates a minimum kinetic complexity and the considerations in Section IV, while readily allowing the minimum, virtually exclude much greater kinetic complexity. We conclude that the first living things were kinetically simple, with only two or three reactions for uptake. The first living things were also structurally simple because they lacked the history of
selection needed to build up structural complexity (see Section V). This paper has affirmed that selection, competition, and evolution are ideas applying to chemical systems in general and that the biological usages are specialisations. Similarly, we have regarded the reproducing species of biology as particular kinds of autocatalysis, specialisation to the particulate form arising from kinetic advantage. Much more work remains to do on the development of structural complexity, and we suggest that the role of physical mixing and its variability receive careful study. Periods of autocatalytic growth occur only in media defined by mixing zones, and variability in mixing can join media which previously were isolated, and vice versa. These are important factors in the evolution of structural complexity through successive unions of reproducing particles. References Chapman, S., 1930, A theory of upper-atmospheric ozone. Mem. R. Meteorol. Soc. 3, 103--125. King, G.A.M., 1977a, Symbiosis and the evolution of prokaryotes. BioSystems 9, 35---42. King, G.A.M., 1977b, Symbiosis and the origin of life. Origins Life 8, 39--53. King, G.A.M., 1978, Autocatalysis. Chem. Soc. Rev. 7,297--316. Laidler, K.J. 1965, Chemical Kinetics (McGraw-Hill, New York). Morowitz, H.J., 1977, Perspectives on thermodynamics and the origin of life. Adv. Biol. Med. Phys. 16,151--163. Nicolis, G. and I. Prigogine, 1977, Self<~rganization in Nonequilibrium Systems (Wiley, New York).
89
Elsevier/North-Holland Scientific Publishers Ltd.
RECYCLING, R E P R O D U C T I O N , AND LIFE'S ORIGINS
G.A.M. KING
Physics and Engineering Laboratory, DSIR, Lower Hutt, New Zealand (Received July 7th, 1981) (Revision received October 13th, 1981 ) Kinetic considerations make it most improbable that any reproducing system could arise spontaneously in a prebiotic soup containing a large variety of organic molecules, as commonly postulated. This batch process can be contrasted with a completely recycling network of reactions maintained by an influx of energy. So long as the network includes at least two bimolecular reactions it is likely to support pathways for chemical reproduction. However, such reproducing systems will be simple in both kinetic and structural terms. Subsequent evolution will lead to much more complex reproducing structures although the kinetic complexities, measuring the varieties of reactions between these structures and their media, will remain relatively simple.
I. Introduction Chemical activity necessarily consumes the reactants. If the activity persists longer than the time required to consume stocks of reactants, they must be regenerated through a cyclic sequence of reactions. In a closed system all stocks are finite and persistent activity then requires a completely recycling n e t w o r k of reactions maintained by an influx of energy -- radiant energy, for example. It is thought that the present day biosphere follows such a pattern of recycling in nearly all respects. The purpose of this paper is to examine some properties of completely recycling networks in case they may have some relevance to studies on the origins o f life, much as Morowitz (1977) has suggested. A most relevant property, developed in Section II, is the likely appearance o f autocatalytic subsystems within networks which contain bimolecular reactions. Section III enquires into the conditions under which these autocatalytic pathways can manifest "autocatalytic kinetics" -- an exponential increase in concentrations. Attention is thus shifted from the complete closed system to an open medium in it, isolated from
the rest except for supply by physical transport. The reactions of a recycling network must be specific in order to avoid a proliferation of products. This fact is used qualitatively in Sections II and III. In Section IV the specificity of reaction is examined quantitatively for the particular case of a single autocatalytic system. As the number of reactants in the medium increases, the number of alternatives to the autocatalytic pathway increases rapidly. Autocatalytic growth occurs, then, only if the specificities of reaction increase very greatly. From this we deduce a need for selection to achieve a growing system of even modest complexity. Section V refers to earlier work on the selection for physical structure which can confer specificity on an autocatalytic particle. The Discussion takes the various concepts which have been developed and applies them in general terms to the problem of life's origins.
II. Single medium Consider a closed set of chemical transformations in a single medium. They can be described by a completely recycling network
0303-2647/82/0000--0000/$02.75 © 1982 Elsevier/North-Holland Scientific Publishers Ltd.
90 of chemical reactions. In this Section, we are concerned neither with the way that energy is supplied to maintain the chemical activity nor with the kinetics of reaction. The interest here is in the pathways for the flow of materials and the conditions under which parts of the network may be autocatalytic pathways. These conditions are deduced in twelve progressive steps. (1) As every active substance is recycled, it is both consumed and produced. It is an intermediate in the network of reactions. (2) Confine reactions to unimolecular and bimolecular yielding one product or two. All more complex reactions will be broken down into these. (3) Because of (1) and (2), we can use the language of chain reactions (e.g. Laidler, 1965), classifying them as initiating, propagating, branching, and terminating reactions. (4) If the energy input is steady, the system moves towards a stable steady state. (Again because of (2), we are not concerned with " e x o t i c " behaviour which might be produced by ter-molecular reactions --Nicolis and Prigoglne, 1977; King, unpublished). (5) Under the steady state conditions of (4), we can use the idea of parity to get relations among parts of the recycling network. A reaction contributes to a parity sum by the increase in the number of intermediates on traversing that reaction. Thus, a branching reaction contributes ÷1 to the parity sum, a propagating reaction zero, and a terminating reaction (removing a single intermediate) c o n t r i b u t e s - 1. The parity sum for the complete recycling network is zero. (6) Take any reaction yielding two products. Parity requires that it be compensated somewhere in the network by a reaction consuming two reactants. The network thus contains at least one bimolecular reaction.
(7) Treat the two products from (6) as the intermediates from a branching reaction. As the network is completely recycling, it is possible to trace a cyclic pathway from each of the products back to the branching reaction. In a complex network, this may be done in many ways. (8) Take any selected pair of the cycles passing through the branching reaction in (7) and take a parity sum over this sub-system. Count as intermediates only those substances both produced in one reaction of the sub-system and consumed in another. There are two distinct cases: (a) The parity sum is zero. (b) The sum is +1. (9) In case (8a), the +1 in parity contributed by the branching reaction has been compensated by a terminating reaction where the cycles rejoin. Two intermediates, one from each cycle, have reacted together to produce only one intermediate common to both cycles. This case is characterised by reactions among the intermediates of the sub-system (see Fig. la). (10) In case (8b), the first intermediate comm o n to both cycles is independently produced from the last distinct intermediates of the separate cycles, so that both reactions producing it are propagating. The + 1 in parity from the branching reaction is not compensated within the sub-system although, of course, it must be compensated elsewhere in the total recycling network. This case is characterised by the absence of reactions among the intermediates of the sub-system (see Fig. lb). (11) The latter case (8b and 10) allows material to be taken up from the rest of the recycling network and incorporated in the sub-system. In the absence of other restraints, reactions of the subsystem thus cause growth in the concentrations of its intermediates. The sub-system is autocatalytic. (12) During the steady state postulated in (4), autocatalytic accumulation in the
91 (b) The network contains at least two bimolecular reactions.
) la
r
) ) lb
O + O 2 -~ 03 O + 03 ~ 02 + 02 02 ~ O + O
)
~~
~ -_-
Bimolecular reactions are needed, not only to compensate for the branching reaction, but also to supply material to the autocatalytic sub-system, building up the molecule split in the branching. If the branching is unsymmetrical the network contains at least three bimolecular reactions, b u t if symmetrical the n e t w o r k may contain as few as two. There is a familiar example of a symmetrical autocatalyst (Chapman, 1930);
X-Y IC
Fig. 1. Sub-systems of a completely recycling network. (a) Sub-system with one branching and one terminating reaction. This is not autocatalytic. (b) Sub-system with one branching reaction and all others propagating. This is unsymmetrically autocatalytic. (c) A symmetrical autocatalytic system.
sub-system (11) is offset by other reactions of its intermediates, consuming them. Away from the steady state, the sub-system may grow or decline depending on the relative importance of the autocatalytic reactions and the other reactions of its intermediates. The foregoing analysis shows that not all completely recycling chemical networks need contain an autocatalytic sub-system, b u t the conditions leading to autocatalysis are not severe and it may appear quite readily. The conditions are: (a) A branching sub-system that can be cyclically closed without reactions among its intermediates.
(A) (B) (C)
When supplied with atomic oxygen in (A) and (B), molecular oxygen undergoes autocatalytic growth, with (B) the branching reaction. Growth is offset by dissociation of the autocatalyst in (C), restoring atomic oxygen and completing the recycling. In the upper atmosphere, where this system and its derivatives assume some importance, 02 greatly predominates over O and departures from the steady state are small. Condition (a), the absence of reactions among the intermediates of an autocatalytic s u b s y s t e m , is a particular facet of a general relationship -- that a cyclic set of reactions is a restricted set from among those which are energetically possible with the chemical species present (King, 1978). As a completely recycling network follows strict conservation of parity, this restriction can be made explicit. For simplicity, we treat just a set of bimolecular reactions, b u t the same conclusions follow for mixed unimolecular and bimolecular reactions. Because of parity, all the bimolecular reactions are exchanges, yielding two products. Because recycling is complete, each product is also a reactant. However, it may react with more than one of the other reactants. Let there be r reactants and the ith one reacts with Ji others. Then, the number of reactions is ~ Z r Ji, and they yield Zr Ji product terms. If any Ji exceeds unity, this sum
92 must exceed r, the number of distinct products. Therefore, some of the products must be "repeats" formed either in separate reactions or as pairs of identical products from single reactions. The number of such repeats is (Zr J i - r), and it gives directly the number of autocatalytic sub-systems. Thus, the reactions of a recycling system with autocatalytic sub-systems, although a restricted set, are less restricted than those of a recycling system without autocatalysis. It is convenient to distinguish the intermediates of an autocatalytic sub-system from the other intermediates o f the recycling network, calling them the " c o n s t i t u e n t s " of the autocatalytic system. Those intermediates that react directly with the constituents promoting the autocatalytic growth can then be called the autocatalytic "reagents". The restriction of reactivity around the autocatalytic sub,system therefore takes the form -constituents react only with specific reagents and n o t with one another, and the reagents also do not react with one another. In Section IV, the question of specificity will be treated quantitatively, the "allowed" and " f o r b i d d e n " reactions being just the extremes of a continuum of reaction rates.
III. Physical transport The discussion so far has assumed the ability to observe the various intermediates in a completely recycling system and hence to deduce the reactions among them. In fact, knowledge of an autocatalytic sub-system is likely to arise, not from complete knowledge of the total system, but from observing the kinetic behaviour of one or a few intermediates. When a substance shows an exponential increase in concentration, it is assumed to be catalysing its own production from a constant supply of reagents -- autocatalytic kinetics. Because the reactions in the autocatalytic set are cyclic, growth may be started with less than a full set of constituents, even one being enough. There is an induction period while the ratios of con-
stituents adjust to those appropriate for exponential growth, and they all then grow with a c o m m o n exponential law. This common behaviour allows us to deduce the fact of autocatalysis while at the same time it conceals the details of its reactions. On the other hand, exponential growth requires the presence of all the reagents in concentrations adequately exceeding their corresponding thresholds (see next Section). A shortfall in any one reagent means that the reactions causing autocatalytic decay predominate. By manipulating the reagents, one can investigate the autocatalytic reactions. If the intention is just to observe autocatalytic kinetics, it is first necessary to accumulate all the reagents in adequate concentrations. If the reagents are supplied through the recycling network, their consumption by reactions with the autocatalytic constituents must be prevented, and this requires physical separation. We move from a single medium to considering separated media related through physical transport. The extent of physical separation needed to effect accumulation is influenced by the restriction of reactivity implicit in the cyclic reaction set of an autocatalytic system. Consider a system where a constituent, C, is consumed in a branching reaction to yield constituents, X and Y. Name as the X and Y segments those parts of the cycles passing through X and Y having no c o m m o n constituents (see Fig. lb), and name the part with common constituents the C segment. If a reagent reacting in the X segment is withheld, the Y and C segments together act as a catalytic cycle, continuing to consume their reagents so they do not accumulate. Likewise, if a reagent for the Y segment is withheld, reagents for the X and C segments do not accumulate. However, if a reagent for the C segment is withheld, the constituent reacting with that reagent accumulates while all others decline. As a result of their decline, all other reagents suffer lower rates of consumption and thus accumulate. To effect the accumulation of reagents, then, it is enough to
93 separate only one of them provided it is a reagent for the c o m m o n segment of the autocatalytic system. Two points are worth noting: Firstly, accumulation can take place because reagents do n o t react one with another. Secondly, a symmetrical system with Y--- X has no X and Y segments, all reactions occurring on the c o m m o n segment (see Fig. lc). The physical exclusion of any single reagent is thus enough to effect accumulation of them all when the system is symmetrical. Materials are excluded from the reacting medium by a change of phase, say, by the ebullition of gas or precipitation of a solid from a liquid medium. Autocatalytic growth resumes when the depleted reagents are resupplied to the medium, and it takes the exponential law only if the time for resupply is short compared with the characteristic exponential time. Some mixing process thus introduces the reagents and disperses them uniformly through the medium. Growth can occur only in a mixing zone. Moreover, if the mixing is continuous, the materials excluded from the medium must be different from the reagents mixed into the medium. Therefore, some chemical changes must occur in the external environment as well as physical transport. N o w consider the case where the time of resupply is long compared with the characteristic exponential time. The interesting case, contrasting with the autocatalytic kinetics of the batch case discussed above, is when supply is steady. The autocatalytic system grows until reagents are consumed as fast as supplied. At this point, the decay of autocatalytic constituents just offsets growth and growth stops. It is likely that the limitation occurs in a single reagent more than in the others, so that its concentration fails to increase while the others still accumulate. In the course of time, this "limiting reagent" becomes the sole supply determining the concentrations of autocatalytic constituents. More generally, fluctuations in physical transport will cause fluctuations in supply. The autocatalytic behaviour will then lie
between those for the " b a t c h " and "constant s u p p l y " modes, approximating to the former when the fluctuations are extreme and to the latter when they are slight.
IV. Specificity The previous Section pointed out that the separate recycling of at least some reagents is likely to require chemical reactions in a region physically distinct from the autocatalytic medium. Physical separation thus helps to define the restriction of reactivity needed in a completely recycling network. However, in a single medium the restriction depends on specificity in the reactions themselves. Present interest is in the degree of specificity needed to support autocatalytic kinetics. For simplicity, consider a symmetrical autocatalytic system with n constituents, Ci. The treatment can easily be extended to unsymmetrical systems. Let the medium also contain m other active substances, R j , including all the reagents for the autocatalytic constituents. If ~i1 is the rate coefficient for reaction between Ci and R j , it is convenient to write: -iy = ~ij • R j
(la)
C~i = olii.
(lb)
Then rn
d C i / d t = -- Y, aij " Ci + .i=l
Oli-i
" Ci-1
"
v
(2)
wherevislfori¢ land2fori= 1, i - - l = n . The first term on the right hand side o f (2) contains n o t only a i • C i but also the possible reactions of Ci with all other materials in the medium apart from c o n s t i t u e n t s - the alternatives to the autocatalytic cycle. Write (2) as m
d In C i / d t + Z ~ q = v • ~i-~
• Ci-1/Ci
(3)
1=1
The system behaves coherently when all dln Ci/dt=
dlnC/dt=
X,
aconstant
(4)
94 Substitute (4) in (3) and multiply all (3) together. Because of the c o m m o n behaviour, all Ci cancel:
(9b) contains the concentration of the limiting reagent, Rl:
R l = Otl/fJl ~] I ~ + ~ cqj ] = 2 ~] cq
i= l
j= l
i= l
(5)
The system shows autocatalytic growth if is positive and decay if }, is negative, so there is a threshold condition when ~, is zero:
~1 I j=~1 aij/ail = 2
i= 1
where ~l is the rate coefficient for uptake of the limiting reagent and c~l is the effective rate coefficient for decay of the autocatalytic constituents, predominantly of Cl. Because all other Rj are large and therefore relatively constant, al takes the form of a first order rate coefficient:
(6) t~l= j =~l
As Y,cqj includes ~i, the condition can be written
~l Ii+
1/Sil =2 ,
i=l
(7)
where Si is the specificity of C i for reaction with Ri in a medium containing all the R's: rn
Si = oli/j~1 aij,
j :/= i.
(8)
Notice that the various a's contain the concentrations of R's (eqn. 1), so that the S's also are affected by concentration. The threshold condition can be interpreted to mean that the system will grow, displaying autocatalytic kinetics, only if the algebraic expression in (7) has a value less than 2. The "limiting reagent" case mentioned in the previous Section follows easily. As the concentrations of non-limiting reagents build up, so increase the corresponding specificities until the terms in (7) containing them all approach unity. There is left for consideration only the term in the specificity, Sh of the constituent, C h using the limiting reagent:
1 + llSl = 2
(9a)
therefore
Sl = 1
(10)
(9b)
Otlj ,
j--/=l
(11)
Equation (10) says that, under conditions of constant limited supply, the limiting reagent assumes a constant concentration. If the supply fluctuates, it can be shown that the concentration fluctuates about the value given by
(10). Each reagent has a threshold condition of the form (10). Although the various ~l are characteristics of the autocatalytic system, the al have a more complex nature, partly determined by the system and partly by the other materials in the medium. Moreover, these conditions apply when the corresponding reagents are separately limiting. If several or all of the reagents have similar importance in controlling growth, the more general relation (7) must be used, calling for somewhat higher concentrations of the reagents. This is the condition mentioned in the second paragraph of the previous section. For the present purposes, the most significant use for equation (7) is discussing the growth of autocatalytic systems as a function of their complexity. Without knowing the detailed properties of all materials in a medium, it is not possible to say whether or not a postulated system will exhibit growth. However, the degree of selection, both of rate coefficients and of concentrations, needed to produce a growing system can be assessed in general terms. It is inversely related to the probability that a system with randomly
95 assigned rate coefficients and concentrations would achieve growth. As a first step to assessing this probability, it is convenient to approximate to equation (7). Define
Then, the probability that (15) would be satisfied by chance,
(12)
is given by the incomplete gamma integral,
pq = O~/i/O~ i
Then, the term, 1/Si, in (7) can be expanded as a sum:
P=P{O<
P-
1
( y - - 1)!
yffil~p y < l ) ,
(18)
1
J(° p(y-1) . e-p dp.
(19)
m
1/Si = j~l Pij, J ¢ i.
(13)
Now expand the complete expression (7) both as an algebraic product and, in its logarithmic form, as a sum. When terms of order higher than the first are neglected, the following inequality remains: ln2= 0.693< ~
i=1
~ Pij <1,
1=1
j:/:i.
(14)
We adopt the approximation to {14),
i=1
m
Y
j=l
y=l
~ Pij =
~; py = 1,
(15)
where Y is the total n u m b e r of terms in the final sum:
Y= n ( m - 1).
(16)
Y is a measure of the system's "kinetic complexity". With p given by (12), equation (15) is a satisfactory approximation to (7). The probability that a system with randomly assigned parameters would achieve growth depends, then, on the probability density function for p. This function is not properly known, but only its general properties of unlikely large values, skew to small values, and finite mean. Fortunately, the qualitative conclusion we shall draw is not very dependent on the exact function and it is enough to take a function with the least further assumptions, the exponential: p(py)
=
e-Py.
(17)
(I thank Dr. J.L. Beck for deriving equation 19). Table 1, giving P as a function of Y, shows that kinetically simple systems (small Y) are much more likely to exceed by chance the threshold for growth. For example, if an autocatalytic system has only 4 reactions for uptake and the medium contains only the 4 appropriate reagents, then Y = 12. Yet the chance that it grows spontaneously is many orders of magnitude less than for a system with 2 reactions in a m e d i u m containing only its reagents (Y = 2). Conversely, while the 2reaction system could arise by chance, the 4-reaction system almost certainly requires a very large a m o u n t of selection in its rate coefficients and in the concentrations of its reagents. This qualitative conclusion clearly applies regardless of the probability density function adopted for p (eqn. 17). V. Selection
In general terms, a selected system persists while its alternatives decline to insignificance. Selection for autocatalytic growth therefore requires the continual supply of reagents which would allow a successful system to persist. Growth then leads to concentrations TABLE 1 Probability complexity
Y P
2 0.26
of autocatalytic
4 0.019
6 6 X l O -4
growth
12 9XlO -l°
as a f u n c t i o n o f k i n e t i c
20 2 X l O -t9
30 2 X l O -3a
96 of autocatalytic constituents such that, on average, growth and decay balance. The concentrations are controlled by the flux of a limiting reagent which fluctuates about its threshold concentrations, and selection is thus largely confined to the specificity of the reaction for its uptake. Although the "competition" is here between a reproducing system and less organised chemical reactions, it resembles biological competition between reproducing species, being with respect to a limiting resource. The idea of selection also applies to the initial formation of a completely recycling network of reactions, where the competition is purely chemical. The recycling network persists while its alternatives wane. Section II called a completely recycling n e t w o r k a "restricted set" of reactions, describing the o u t c o m e of the selection without reference to how it occurred. The autocatalytic systems discussed in Sections III and IV already reflect this degree of selection, because there are no reactions among the constituents and no reactions among the reagents for them. These provisions appear in the definition of kinetic complexity, Y (equation 16), which in the absence of selection would take the larger value, (n + m)2/4 (King, 1978, Section 2B). Section 3 of King (1978) has looked at the factors conferring specificity of reaction in both gaseous and liquid media. In a liquid medium specificity depends on the physical structures of the reactants themselves. The structural complexity of an autocatalytic system is related directly to the complexity of the reaction set producing it. However, some reactions in this set may be internal rearrangements of the constituents. These unimolecular reactions bear no simple relationship to the reactions for uptake from the medium, although dependent on them. Also, they are quite unrelated to the kinetic complexity of the system (eqn. 16), which counts only those bimolecular reactions between constituents and other active materials in the medium alternative to the autocatalytic pathway. For example, a bacterial cell has a
very large number of internal reactions but just a few reactions with nutrients and other materials in the medium. It is structurally very complex but its kinetic behaviour is relatively simple. The big disparity between structural and kinetic complexities is due to the "particulate" nature of this autocatalytic system, its constituents being stages in the growth of a particle. The moderate kinetic complexity means that the reactions for uptake from the medium need not be impossibly specific. Complementing this, reactions associated with the high structural complexity form a surface on the particle which exhibits high specificity for uptake. The combination of properties has been selected during a long period of biological evolution. Section 6F of King (1978) describes a process of selection which simultaneously increases the structural complexity and decreases the kinetic complexity of particulate autocatalytic systems. It depends on the conditional stability of a system containing two particulate autocatalysts which mutually exchange materials. If the exchanged materials are well supplied, the two types of particle reproduce separately but, if the exchanged materials are limiting, only a particle formed by their union can show growth. This process has been incorporated into an hypothesis for the evolution of structural complexity in biology (King, 1977a,b). VI. Discussion Conventional views on the origin of life usually disregard the possibility, or at least any significance, of recycling in the period of "chemical evolution" preceding it. They often envisage conversion of a material stock in the atmosphere into a batch of nutrients in the oceans -- the "primordial s o u p " waiting for the onset of reproduction. These hypotheses are all open to the critical objection that preceding events have not matched the greater part of the nutrients to the requirements of the initial reproducing species. Instead of being "nutrients" they are likely
97 to be "poisons". Following Section IV, the variety of active materials means large values for the kinetic complexity, Y (eqn. 16), and the probability of autocatalytic growth by even a simple system is virtually zero (see Table 1). Hypotheses which further postulate a reproducing polymer with attendant catalytic machinery for polymerization and translation imply that there are many reactions for uptake. They are committed to still larger values of Y and are correspondingly more incredible. On the other hand, a completely recycling system would have evolved chemically from a more complex system by losing those materials which otherwise disrupt the recycling. If part of the system were to reproduce chemically, following autocatalytic pathways, it would immediately receive appropriate nutrients. Further reactions would, at the same time, consume the reproducing constituents so that their materials would recycle. "Life" and "death" would appear together. Section II has shown that a completely recycling system is likely to contain an autocatalytic subsystem, provided it includes at least two bimolecular reactions. This proviso dictates a minimum kinetic complexity and the considerations in Section IV, while readily allowing the minimum, virtually exclude much greater kinetic complexity. We conclude that the first living things were kinetically simple, with only two or three reactions for uptake. The first living things were also structurally simple because they lacked the history of
selection needed to build up structural complexity (see Section V). This paper has affirmed that selection, competition, and evolution are ideas applying to chemical systems in general and that the biological usages are specialisations. Similarly, we have regarded the reproducing species of biology as particular kinds of autocatalysis, specialisation to the particulate form arising from kinetic advantage. Much more work remains to do on the development of structural complexity, and we suggest that the role of physical mixing and its variability receive careful study. Periods of autocatalytic growth occur only in media defined by mixing zones, and variability in mixing can join media which previously were isolated, and vice versa. These are important factors in the evolution of structural complexity through successive unions of reproducing particles. References Chapman, S., 1930, A theory of upper-atmospheric ozone. Mem. R. Meteorol. Soc. 3, 103--125. King, G.A.M., 1977a, Symbiosis and the evolution of prokaryotes. BioSystems 9, 35---42. King, G.A.M., 1977b, Symbiosis and the origin of life. Origins Life 8, 39--53. King, G.A.M., 1978, Autocatalysis. Chem. Soc. Rev. 7,297--316. Laidler, K.J. 1965, Chemical Kinetics (McGraw-Hill, New York). Morowitz, H.J., 1977, Perspectives on thermodynamics and the origin of life. Adv. Biol. Med. Phys. 16,151--163. Nicolis, G. and I. Prigogine, 1977, Self<~rganization in Nonequilibrium Systems (Wiley, New York).