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Biophysical Chemistry 127 (2007) 123 – 128 http://www.elsevier.com/locate/biophyschem Natural process – Natural selection Vivek Sharma, Arto Annila ⁎ Department of Physical Sciences, Institute of Biotechnology, and Department of Biological and Environmental Sciences, University of Helsinki, Finland Received 20 December 2006; received in revised form 16 January 2007; accepted 16 January 2007 Available online 24 January 2007Abstract Life is supported by a myriad of chemical reactions. To describe the overall process we have formulated entropy for an open systemundergoing chemical reactions. The entropy formula allows us to recognize various ways for the system to move towards more probable states.These correspond to the basic processes of life i.e. proliferation, differentiation, expansion, energy intake, adaptation and maturation. We proposethat the rate of entropy production by various mechanisms is the fitness criterion of natural selection. The quest for more probable states results inorganization of matter in functional hierarchies.© 2007 Elsevier B.V. All rights reserved.Keywords: Catalysis; Entropy; Evolution “Life is chemistry” is a cliché but it is one that deserves a re- change dS / dt = ΣvjAj / T due to reaction velocities vj and affinitiesinspection. For a long time, it has been understood that chemical Aj = Σμj is known as well . Here we derive the formula ofreactions lead to chemical equilibrium. Gibbs was first to realize entropy to describe evolution of an open system undergoingthat this stationary state, where chemical potentials μ on both numerous chemical reactions.sides of a reaction formula are equal, corresponds to themaximum entropy . Chemical reactions as well as other 1. Entropy of an open systemprocesses, e.g. diffusion, heat flow from hot to cold and ioncurrents in electric fields that evolve towards increasing entropy The relationship between a state of a system and itsare all called natural processes . Does this mean that life is a probability P was formulated by Boltzmann already somenatural process towards high-entropy states? 140 years ago . For a homogeneous system entropy is  The question has remained open despite many studies [3–8].Since high-entropy states are often associated with high X N S ¼ RlnPcR ni ½lnðgi =ni Þ þ 1 ð1Þdisorder and ordered structures are distinctive features of life, i¼1it is customarily thought that living processes work to reduceentropy rather than to increase it . However, no firm proof where identical particles in the total numbers N = Σni arehas been given and it has remained obscure what prevents us distributed among levels of (kinetic) energy Ei relative tofrom deriving characteristics of living matter from the fun- thermal energy RT e.g. given per mole. The indistinguishabledamental principles. Is it a missing concept or a misconception particles ni are often considered non-interacting or only weaklyor something else? interacting, i.e. the occupancy of each level gi is assumed to be a Nevertheless, one open question is easy to define. Even if the constant.condition for maximum entropy i.e. dS = ⇔Σμj = 0 is known, This must be the restriction why the formalism cannotthe formula of S itself has remained unknown. This state-of- account for an evolution of a homogeneous system to an openaffairs is peculiar particularly because the rate of entropy heterogeneous system via chemical reactions. It is important to notice that when various chemical reactions are running simultaneously substrates used in one reaction are away from ⁎ Corresponding author. other reactions. These dynamic bounds correspond to proba- E-mail address: email@example.com (A. Annila). bilities and should be included in the total probability of the0301-4622/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.bpc.2007.01.005
124 V. Sharma, A. Annila / Biophysical Chemistry 127 (2007) 123–128 partition of compounds made of three elemental constituents h, k and l is ::: ðH−h100 − −h320 Þ! P321t ¼ Àh321 Á : : : −h321 Þ! 3 þ 2 þ 1 !ðH−h100 − k321 l321 ðK−k100 − : : : −k320 Þ! ðL−l100 −: : : −l320 Þ! Â Â ð3Þ ðK−k100 −: : : −k321 Þ! ðL−l100 −: : : −l321 Þ! where H, K and L denote the total numbers of constituents. TheFig. 1. Diagram presenting a distribution of matter among compounds in term (h321 / 3 +k321 / 2 +l321 / 1)=HKL321 =N321 is equal to thenumbers Nj in classes j of increasing Gibbs free energy along vertical axis. The numbers of particular entities. Similarly to the single constituentdistribution will change in chemical reactions, i.e. synthesis and degradation system, the factor H!K!L! for each indistinguishable class of basic(arrows) that deliver (red) or absorb (blue) energy (wavy arrows) as illustrated constituents will cancel. In general the resulting distribution offor an endothermic reaction N1 + N4 ⇄ N5 and for an exothermic reaction entities is multinomial with respect to its various basic constituents.N5 ⇄ 5N1. Next, we consider probabilities that depend on the avail- ability of substrates Nk, the Gibbs free energy difference between the substrates and products ΔGjk = Gj – ΣGk as well assystem as will be shown below in analogy to the basic external energy ΔQjk that may couple to the reactionsderivation of classical statistics  for ideal gas. From here on we work to reveal the general principles ! Nj DQjk −DGjk Nthat govern living matter. We understand that living matter is Pja ¼ j Nk exp uCj j ð4Þcomplex and our statistical description is only a starting k RTpoint that will require further elaboration to account fully for where we assume constant temperature T. Motions of substratesany specific case. Nevertheless knowledge of underlying prin- are considered statistically independent from each other, e.g. inciples is valuable. In essence we ask how numerous chemical a solution, thus denoted by the product Πk and the energyreactions will distribute matter among various chemical contributions that expressed relative to thermal energy RT arecompounds. regarded as reactants as well. The bracket is raised to the power The probability P for a distribution of elemental constituents, of the number of products Nj because the substrates and as-e.g. atoms in total numbers N = Σnj among various classes j of sociated energy differences may incorporate in any of the pos-compounds Nj, is obtained by inspecting the level diagram of sible products. We rename Pja as ΓjNj to have a concise notationGibbs free energy (Fig. 1). The first level holds stable elemental in analogy to Eq. (1).constituents in numbers n1. The associated probability is the The overall probability is obtained as the product of proba-familiar binomial P1b = N! / n1!(N − n1)!, since all n1 are identical bilities over all levels, i.e. for all compounds that participate inthe order of compounds at the level is immaterial. Second level reactions of the systemhouses metastable compounds that are composed of twoelemental constituents. These compounds are referred to as P ¼ P1a P1b P2a P1b P3a P3b: : :metastable because they keep spontaneously degrading to the !Nj C Nj 1 DQjk −DGjk ð5Þstable compounds. The associated probability is P2b = (N − n1)! / ¼j ¼j j Nk exp j¼1 Nj ! j¼1 Nj ! k RT(n2 / 2)!(N − n1 − n2)!. The factorial (n2 / 2)! gives the numberof combinations associated with the number of compoundsN2. The probabilities of subsequent levels are expressed Finally, entropy S per mole is obtained by taking logarithmaccordingly for S = RlnP and applying Stirlings approximation X ðN −n1 −: : : −nj−1 Þ! ScR Nj ½lnðCj =Nj Þ þ 1Pjb ¼ : ð2Þ ðnj =jÞ!ðN −n1 −n2 −: : : −nj Þ! j¼1 X ! DQjk −DGjk The notation by including the term nj / j for the compound ¼R Nj ln j Nk exp =Nj þ 1stoichiometry, i.e. assembly rules of an entity, opens the way j¼1 k RTfrom homogeneity to heterogeneity. In analogy to the derivation !of classical statistics for indistinguishable particles  the 1X Xfactor N! will cancel since all elemental constituents are ¼ Nj lk þ DQjk − lj þ RT T j¼1 kidentical, e.g. all carbon atoms are alike. Obviously living matter comprises many different atoms and 1Xnot only one type of constituents as outlined above. The ¼ Nj ðAj þ RT Þ: ð6Þ T j¼1partition of matter among compounds in a many-constituentsystem is obtained analogously to the single-constituent systemwhen the stoichiometry of an entity with respect to the basic where the affinity Aj = Σμk + ΔQjk − μj includes also externalconstituents is denoted. For example, the probability P321t for a energy when it couples to reaction. The Eq. (6) designates all
V. Sharma, A. Annila / Biophysical Chemistry 127 (2007) 123–128 125available jk-reaction pathways for synthesis and degradation for generator of the motive force. The system will move when thereall compounds in numbers Nj including also transition states are mechanisms for flows of matter via chemical reactions fromwhen present. These mechanisms are considered to distri- compounds to compounds. Obviously the statistical descriptionbute effectively all matter and energy to make the system. When that is based only on probabilities does not contain any specifica jk-reaction pathway is open matter will flow from substrates in knowledge about mechanisms. Thus the formalism may onlynumbers Nk to products in numbers Nj when the potential  reveal statistical consequences when reaction pathways areμk = RT ln[Nkexp(Gk / RT)] is higher than μj = RTln[Njexp(Gj / RT)] open.including external energy when it couples to the reaction. Then Naturally the flow rate dNj / dt from Nk to Nj must exactlythe system is in motion towards more probable states. Any matter balance the flow rates ΣdNk / dt from Nj to Nk not to loose anyor energy that does not involve in reactions is not considered to be matter in the process. The condition holds whena part of the system. Since the assembly rules, i.e. stoichiometries of compounds, dNj Aj X dNk ¼ rj ¼− ð8Þare inherently contained in the expression of entropy we may dt RT dt kexamine systems at various levels of details. In other words anentity can be regarded as a system itself consisting of its where rates rj depend on mechanisms, e.g. structures thatinteracting entities. For example an organism can be considered catalyze reactions. Consequently we may rewriteas a system of cells, a cell can be considered as a system ofmolecules and molecules as a system of atoms. Relevant 1interactions that characterize a particular system are contained Lc j r A2 : 2 j¼1 j j ð9Þin Gibbs free energies. Furthermore, our notation implies an ðRT Þideal dilute system and Nj should be understood as activeconcentrations when relevant. We emphasize that the study aims The quadratic form of affinities reminds us from Onsagerto explain principles that could be developed further to account reciprocal relations  that are valid near stationary non-for specific cases and their characteristics. equilibrium and equilibrium states of entropy production. The We stress the meaning of entropy as a logarithmic probability equation of motion is more general in describing a system on itsmeasure for a state of a system. The entropy formula applies for way to equilibrium or non-equilibrium stationary states.a system where constituents interact with each other, not for acollection of non-interacting compounds incapable of progres- 3. Dissipative evolutionsion. At any given moment the system comprises its con-stituents in matter and energy as well as interactions. From time The equation of motion appears similar to the Liouvilleto time the system may open to take in energy as radiation or as equation of a classical system containing kinetic and potentialmatter from its surroundings or to expel it. Typically the energy terms. However, we fail to find a transformation tosurroundings of a system comprises of other open systems. For separate affinities Aj from entities Nj to find invariants ofexample an ecosystem houses a myriad of hierarchically nested motion. According to Noethers theorem this implies a lack ofsubsystems that bathe in mutual fluxes of energy and matter. symmetry and associated conserved current. Thus we suspectThe open system description allows us to include, similar to that the equation of motion is non-integrable [13,14]. IntuitivelyΔGjk, an external energy ΔQjk that couples to endothermic jk- the non-associative algebra beyond Hilbert space results fromreactions, as well as to exclude, i.e. dissipate, energy from the time-ordered operations when various reaction pathways drawsystem arising from exothermic reactions. Indeed it is in the on same constituents affecting each others paths of motions.very nature of chemical reactions not to conserve energy within When the system is moving also potentials keep changing thatthe system. Energy may also come as high chemical potential in turn to affect the motion itself. The motion is chaotic in thematter (food) and leave the system as low chemical potential sense of Poincaré, i.e. for any non-trivial system the detailedmatter (excrement). The open system formalism is central for trajectory of motion cannot be obtained.statistics of systems undergoing chemical reactions. The obvious question is what might be the non-conserved quantity? On the basis of Eq. (9) we are not left much else but to2. Equation of motion suspect but the driving forces. Affinities are under no transformation constants of motion. We note that for matter to We will proceed to differentiate Eq. (5) or alternatively flow in chemical reactions total energy of the open system is notEq. (6), i.e. ∂(lnP) / ∂t = (1 / P)(∂P / ∂t), with respect to time to conserved. Indeed on this basis chemical reactions are classifiedobtain the equation of motion for a system undergoing chemical as endo-(ΔQjk N 0) and exothermic. A steady flow of externalreactions energy to or from the system will displace the system from the X dNj Aj equilibrium partition Peq given by law of mass-action  to aAP ¼ LP; Lc ð7Þ non-equilibrium stationary partition Pss.At j¼1 dt RT ! # X jN k −DGjk DQjk kwhere Stirlings approximation has been used. The Liouvillian ln Pss ¼ Nj ln exp þ1þ ð10Þ j¼1 Nj RT RTL for chemical reactions allows us to identify affinities Aj as the
126 V. Sharma, A. Annila / Biophysical Chemistry 127 (2007) 123–128 Thus a passage from an initial state P(0) to a final state P(t) synthesis. The steady-state non-equilibrium partition of reac-is given by the dissipation. Evolution is inherently coupled with tants is obtained from Eq. (11) at the conditionstructural changes in states caused by absorption or emission of Xenergy. This conceives the idea of time . Energy within the dS ¼ 0⇔lj ¼ lk þ DQjk kopen system is not conserved in transitions towards thestationary state. The motion of a macroscopic system via DQjk −DGjkreactions means irreversible destructions of embedded micro- ⇔Nj ¼ j Nk exp : ð12Þ k RTscopic systems either by repeated mergers to larger and largersubsystems or by break downs to smaller and smallersubsystems. This is of course exactly what chemistry is. The stationary-state condition applies when entropy does not change any more, i.e. fluxes of energy to and from the4. Stationary state open system are equal. The form of the law of mass action shows that the external energy will raise the stationary At a steady state the net flow of energy in and out of the open population of high-j compounds from the ground statesystem vanishes. Then the system has become macroscopically equilibrium. These energy-powered metastable states arereversible as there is no net dissipation of energy. The stationary referred to as dissipative structures . In other words thestate may though exhibit random fluctuations or cyclic motions external energy supports high-entropy states containingas structures sporadically disintegrate and reintegrate. This can complex compounds. The open system will respond changesbe understood so that the system, often perceived as one closed in flux by evolving towards a new stationary state dictated bysystem, is in fact composed of several open systems that bathe the new flux. We emphasize that the flow of external energy toin mutual sporadic or periodic fluxes form each other. The the open system will increase entropy. In the following westationary state with random fluctuations is usually pictured as will reveal characteristics of the high-entropy non-equilibriumthermodynamic equilibrium and the one with periodic oscilla- states. Most importantly we will show that the high-entropytions as a heat engine. states will contain matter in functional structures, i.e. also The knowledge of S allows us to obtain the time derivate, i.e. order. This is obviously in contrast with the common butthe rate of entropy production due to various reactions unfounded belief that the highest entropy state equates with the state of least order. !dS X dS dNj 1 X dNj X ¼ ¼ lk þ DQjk − lj 5. Evolution to diversitydt j¼1 dNj dt T j¼1 dt k 1X ¼ mj Aj z0 ð11Þ Next we will examine Eqs. (6) and (11) analytically to T j¼1 identify various ways of entropy production. The reasoning is also illustrated by simulations (Fig. 2). The well-known last form  contains in the affinity Entropy will increase when increasing numbers of com-Aj = Σμk + ΔQjk − μj also the external energy that couples in pounds emerge from reactions as long as the system has not reached the stationary-state equilibrium given by Eq. (12). Thus already at the molecular level there is a spontaneous drive for proliferation when using the vocabulary of biology. Entropy will also increase when various kinds of compounds emerge from syntheses, i.e. the sum over j extends to new classes. Thus there is a spontaneous drive for differentiation and motion towards molecular diversity. (At this point we postpone the explanation why new classes of compounds may appear later during the evolution.) Entropy will also increase when the system acquires more compounds to its processes from surroundings. This conclusion is obvious as the numbers Nj will become larger. Thus there is a spontaneous drive for ex- pansion. Entropy will increase when more and more jk-reaction pathways open to involve more and more matter to processes.Fig. 2. Simulation of entropy given by Eq. (6) vs. time. A model system evolvesfrom homogeneity of basic constituents to heterogeneity of compounds by non- Entropy will increase when more external energy couples tocatalyzed endothermic polymerization reactions i.e. N1 + Nj-1 ⇄ Nj and by increasing number of reactions and when energy is used moredegrading exothermic reactions Nj ⇄ jN1. The syntheses couple external energy and more efficiently to power various reactions. This isto system whereas the degradations expel energy from the system. For consistent with the notion that energy is also a reactant. Thuscomparisons S vs. t is shown when 20% more matter in terms of N1 is available there is a spontaneous drive for greater intake of energy and(blue), when reactions happen 20% faster (green) and when 20% more energycouples to the transitions (red) than in the original system (black). The time matter to the system. The set of processes to maximize theseries is produced by a for-loop where at every time increment synthesis and entropy production we refer to as adaptation. The energydegradation operations are performed according to Eq. (8). consumption will level off at the state of maturity , i.e. the
V. Sharma, A. Annila / Biophysical Chemistry 127 (2007) 123–128 127steady-state, when no more matter or energy can be recruited production is very important, since the products themselvesto the system or when no new or faster reaction pathways may act as substrates for subsequent reactions. Entropy willappear. increase when simple compounds, i.e. low-j entities, react to Importantly the metastable entities may only participate to form more complex compounds, i.e. high-j entities as long asreactions that they will reach within their lifetimes. Thus the the equilibrium is not yet reached. The emerging complexity issystem is limited in its evolution by the range of interactions in functional. The process is biased for products that facilitate thespace and time. Further increase of S may take place at a higher motion towards more probable states. Therefore, we proposehierarchical level where the system itself becomes a constituent that the rate of entropy production is the fitness criterion ofof a larger system. A system is composed of interacting entities natural selection .that themselves are systems. The Eq. (6) describes by nature the The association of natural selection with the rate of entropynested organization at various scales because the system production may at first appear only a conceptual connection anddescription contains stoichiometries, i.e. assembly rules, as perhaps only valid in a simple system. Mechanisms of entropysubstrates and energy. Therefore knowledge of constituents is production are many in contemporary biota, thus also targets foravailable to change the level of inspection. Obviously entities natural selection are many. The spontaneous rise of diversity iswill organize to a higher hierarchical structure only when the inherently biased for functional structures i.e. catalyticresulting system will produce more entropy than independent mechanisms that produce entropy rapidly. Even slight andentity systems. All the aforementioned entropy producing gradual improvements in reaction rates will with time giveprocesses are strikingly familiar to us as the processes of life yet significant contributions to S. Catalysis is ubiquitous. Allthey were exposed only by a trivial inspection of Eqs. (6) and biological structures can be regarded as catalysts or parts of(11). Therefore we arrive to the conclusion that life is a natural them to generate flow of matter towards high entropy states.process. Indeed rapid growth and expansion are characteristics of living processes. When new mechanisms appear a gradual evolution6. The selection criterion may be punctuated by rapid growth phase (Fig. 3). The knowledge of S allows us for the first time to inspect the When several systems access a common but limited pool of stability of a heterogeneous non-equilibrium stationary state bymatter and energy, the associated rates of entropy production, Lyapunov criterion . The variation with respect to δNj revealsthe terms (dS / dNj)(dNj / dt) of Eq. (11), are important. Those that − δ2S N 0 and d(− δ2S) / dt b 0, i.e. the steady state is stable.systems that produce entropy more rapidly will involve in their However when products are autocatalytic, i.e. dNj / dt ∝ Nj, thenprocesses more matter and energy than those that are slower in d(− δ2S) / dt N 0 and oscillations are expected. Such a system willtheir entropy production (Fig. 3). The competition for matter easily over-deplete N k in the production of N j. Indeedand energy is incessant because the high-entropy states are autocatalytic, i.e. rj(Nj) in Eq. (8), chemical reactions aremetastable, i.e. entities j N 1 are bound by lifetimes and degrade known to fluctuate about the equilibrium . On the othercontinuously to more basic constituents. The living systems are hand hypercycles  and systems with catalytic networks, e.g.continually and rapidly regenerating themselves to attain and cells and ecosystems are mostly stable. Obviously the openmaintain high entropy states. Under these conditions the rate of systems may also suffer time to time from perturbations when there are changes in external or internal flows. 7. Discussion The formula of entropy for an open system is mathematically simple. However, the description is profoundly new in comparison with the traditional view of a closed weakly or non-interacting system. Entropy of Eq. (6) does not only enumerate particles but also includes energies, i.e. Gibbs free energy differences and external energy, required for transitions. The formula of S describes the inter-exchange of energy and matter. In endothermic reactions energy is bound to compounds and in exothermic reactions released from the system. IncomingFig. 3. Competition for matter obtained from simulation of Eq. (6) for high-energy radiation may couple e.g. to photosynthesis orpolymerization reactions N1+Nj-1 ⇄ Nj and for degradation Nj ⇄ jN1. The totalamount of matter in compounds NjN1 is shown at the right hand scale. A model incoming high-chemical potential matter may be consumed insystem initially evolved to a non-equilibrium steady state until time tc when catabolic reactions. Subsequently the system may dissipate low-another system emerged and began to draw matter from the common pool of energy, i.e. thermal energy, or discard low-chemical potentialbasic stable constituents N1 at a 1.5 times faster rate dNj / dt. Consequently, the matter.amount of matter in compounds of the original NjN1 system began to decline It should be mentioned that the well-known partition of ideal(blue) and the amount in compounds of the invader system started to rise(green). After a redistribution of matter the overall entropy S (black) at the left gas particles ni that is customarily deduced from Eq. (1) is alsohand scale of the final dual-system reached a higher level than that of the original obtained analogously to Eq. (12) when retaining only kineticsystem. For simplicity no catalytic activity was included in the simulation. energy term Ei. Obviously gaseous particles collide with each
128 V. Sharma, A. Annila / Biophysical Chemistry 127 (2007) 123–128other when the system evolves towards equilibrium, whereas The open system statistics is consistent with thermodynam-chemical reactions proceed via specific reaction pathways. ics yielding known formula of entropy production and chemical When interactions are included in the system description, equilibrium at the stationary state. We expect the naturalentities Nj acquire properties also. To be specific, entities obtain statistics to give understanding to questions such as why lifetheir characteristics through mutual interactions and those that emerged, why handedness of proteins and nucleic acids isdo not differ from each other in interactions are identical. This ubiquitous, why our genomes are fragmented and swollen withview is consistent with modern physics and modern biology. non-coding segments. We hope to inspire by this work manyInteractions can be considered to maintain or “power” more applied and fundamental studies of open systems.metastable entities, i.e. living systems. Obviously when twoclasses of entities draw from common resources, the one that Acknowledgmentcan generate a higher rate of entropy production has anadvantage over the other. We expect that the rate of entropy We thank Christian Code, Salla Jaakkola, Martti Louhivuoriproduction, even though it is a simple concept, to account for and Peter Würtz for valuable comments and discussions.many complex phenomena attributed to natural selection.Mechanisms for entropy production of present-day biota are Referencesmany and thus obscuring the common underlying principle. Therate of entropy production is a more general criterion than what  J.W. Gibbs, The Scientific Papers of J. Willard Gibbs (OxBow Press,is customarily attributed to natural selection. The rate of entropy Woodbridge, CT, 1993–1994).  D. Kondepudi, I. 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